http://wiki.awf.forst.uni-goettingen.de/wiki/api.php?action=feedcontributions&user=Fehrmann&feedformat=atomAWF-Wiki - User contributions [en]2024-03-29T10:00:06ZUser contributionsMediaWiki 1.19.3http://wiki.awf.forst.uni-goettingen.de/wiki/index.php/Stratified_samplingStratified sampling2024-02-10T10:07:49Z<p>Fehrmann: /* Estimator of the total */</p>
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<div>{{Languages}}{{Ficontent}}<br />
Stratified sampling is actually not a new [[lectuenotes:Sampling design and plot design|sampling design]] of its own, but a procedural method to subdivide a [[population]] into separate and more homogeneous sub-populations called [[strata]] (Kleinn 2007<ref name="kleinn2007">Kleinn, C. 2007. Lecture Notes for the Teaching Module Forest Inventory. Department of Forest Inventory and Remote Sensing. Faculty of Forest Science and Forest Ecology, Georg-August-Universität Göttingen. 164 S.</ref>). In some situations it is useful from a statistical point of view, or required for practical and organizational reasons, to subdivide the population in different strata.<br />
The major characteristic is that independent sampling studies are carried out in each stratum where all strata are considered as sub-populations of which the parameters need to be estimated. If [[Simple random sampling]] is applied, we call that '''stratified random sampling'''. <br />
The only difference between random sampling and stratified random sampling is that the last consists of various sampling studies and the only thing we have to consider is how to combine the estimations that come from the single strata in order to produce estimations for the total population.<br />
<br />
Stratified sampling is efficient especially in those cases where the [[variability]] inside the strata is low and the differences of means between the strata is large (Akca 2001<ref name="akca">Akca, A. 2001. Waldinventur. J.D. Sauerländer's Verlag. Frankfurt am Main, 193 S.</ref>). In this case we can achive a higher [[accuracy and precision|precision]] with the same [[sample size]]. Beside statistical issues there are further arguments for stratification.<br />
<br />
We can distinguish two general approaches for stratification, the so called pre-stratification in which strata are formed '''''before''''' the sampling study starts, and the post-stratification, where we generate strata in course of the sampling or even afterwards based on the data. In the first case, that is described in this artice, the strata must be defined and - in case of geographical strata - delineated to define the [[Population|sampling frame]].<br />
<br />
The precondition for a meaningful partitioning of a population in non-overlapping strata is the availability of prior information that can be used as stratification criteria (de Vries 1986<ref>de Vries, P.G., 1986. Sampling Theory for Forest Inventory. A Teach-Yourself Course. Springer. 399 p.</ref>). In forest inventories this information might be available in form of forest management or GIS-data or can be derived from remote sensing data like aerial photos. Most efficient from a statistical point of view, is the stratification of a population proportional to the target value of the inventory. As this target value is typically not known before the inventory, forest variables that are correlated to this value are used as stratification criteria. In large managed forest areas age classes or forest types might be for example good stratification criteria if the estimation of [[volume per ha]] is targeted. <br />
<br />
<br />
===Arguments for stratification===<br />
Sometimes it is useful to subdivide the [[population]] of interest in a number of sub-populations and carry out an independent sampling in each of these strata. There are statistical as well as practical considerations that make this technique very favorable and interesting for large area forest inventories. Not without reason almost all [[national forest invetories]] are based on stratification.<br />
<br />
;Statistical justifications:<br />
*The spatial distribution of [[Sample point|sample point]]s inside the population is more evenly, if these points are selected in single strata,<br />
*It is possible to make an individual optimization of [[:Category:Sampling design|sampling]] and [[:category:plot design|plot design]] for each stratum,<br />
*One usually increases the [[accuracy and precision|precision]] of the estimations for the total population,<br />
*Separate estimations for each of the strata are produced in a pre-planned manner,<br />
*It is guaranteed that there are actually sufficient observations in each one of the strata,<br />
*It is possible to produce estimations with defined precision level for sub-populations.<br />
<br />
;Practical justification:<br />
*The possibility to optimize the [[Sampling design and plot design|inventory design]] separately for each stratum is very efficient and helps to minimize costs,<br />
*To facilitate inventory work (particularly field work): independent [[field campaign|field campaigns]] can be carried out in each stratum,<br />
*It allows a better specialization of field crews (e.g. botanists).<br />
<br />
===Stratification criteria===<br />
<br />
For the partitioning of a population we can imagine very different stratification criteria. If the reason for stratification is not an improvement of the [[accuracy and precision|accuracy]] of estimations, the stratification variables must not necessarily be [[correlation|correlated]] to the target value. Under special conditions it might be useful to stratify even if there are no statistical justifications. For example might a political boundary dividing a forest area be a good reason for stratification, even if the forest is very homogeneous, if afterwards estimations for both parts should be derived seperately. Other examples for meaningfull criteria are:<br />
<br />
;Geographical startification:<br />
*Eco zones,<br />
*[[Forest types]],<br />
*Site and soil types,<br />
*Topographical conditions,<br />
*Political boundaries or properties,<br />
*...<br />
<br />
Further one can imagine to use the expected inventory costs (regarding time consumption) as criterion. These costs are typically correlated to the above mentioned geographical conditions. It might be for example reasonable to stratify a forest area by slope classes, if time consumption for field work differs significantly between flat terrain and steep slopes.<br />
<br />
;Subject matter stratification<br />
*Species,<br />
*Species groups (e.g. commercial / non-commercial),<br />
*Tree sociological classes,<br />
*Age classes in plantation forests,<br />
*...<br />
<br />
==Statistics==<br />
<br />
The only new concept that needs to be introduced in stratified random sampling is how to combine the estimates derived for different strata. <br />
The [[Estimator|estimator]]s for stratified random sampling are based on simple considerations about [[linear combination]]s (Kleinn 2007<ref name="kleinn2007">Kleinn, C. 2007. Lecture Notes for the Teaching Module Forest Inventory. Department of Forest Inventory and Remote Sensing. Faculty of Forest Science and Forest Ecology, Georg-August-Universität Göttingen. 164 S.</ref>). <br />
When we have two '''''independent''''' random variables <math>Y_1\,</math> and <math>Y_2\,</math> and if we are interested in the sum of the two <math>Y_1+Y_2\,</math>, then<br />
<br />
:<math>E(Y_1+Y_2)=E(Y_1)+E(Y_2)\,</math> and<br />
<br />
:<math>var(Y_1+Y_2)=var(Y_1)+var(Y_2)\,</math><br />
<br />
{{info<br />
|message=Simple:<br />
|text=The [[expected value]] for the sum of both is equal to the sum of both single expected values. It is intuitively clear that we can sum up the totals to derive an overall total. It is different if we consider the variables to be means! In this case we have to weight the single means according to the size of the strata.<br />
}}<br />
<br />
If we consider <math>Y_1\,</math> and <math>Y_2\,</math> as estimations from the two strata 1 and 2, then we can apply the principles derived from these considerations for stratified sampling.<br />
<br />
However, to calculate the overall [[Mean|mean]] from two estimations, we have to weight the single means in order to account for possibly different sizes of the sub-populations <math>N_1\,</math> and <math>N_2\,</math>. If both strata are of the '''same size''', we can calculate the mean by:<br />
<br />
:<math>\frac 12 (Y_1+Y_2)=\frac 12 Y_1+\frac 12Y_2=c_1Y_1+c_2Y_2\,</math><br />
<br />
The factor <math>c_i\,</math> can be interpreted as a weight for the single estimations from stratum 1 and 2. Because both are of equal size in this example <math>c_1=c_2\,</math> holds.<br />
A more typical case would be that we deal with strata of unequal sizes.<br />
<br />
{{info<br />
|message=Example:<br />
|text=Weighting of single partial results (or estimations) is important, if they stem from different sized sub-populations and we want to calculate a mean. A simple example: You should calculate the mean body weight of 50 students in a classroom. You have a mean value derived for the 15 ladies (55 Kg) and a mean body weight for the 35 men (73 Kg). If you would calculate an unweighted mean (64 Kg) it would be wrong. Correct is 15/50*55+35/50*73=67,6 Kg! The weights 15/50 and 35/50 are an expression of the share of the respective group on the total population. This weight is equal to the selection probability for simple random sampling.<br />
}}<br />
<br />
<br />
The weights must be proportional to the size of the sub-populations. In case of forest inventories the population is typically the total forest area (we are selecting [[sample point]]s from this continuum), so that the individual stratum areas can be used to derive weighting factors. The sum of all weights must be 1, so that:<br />
<br />
:<math>\sum c_i=1\,</math><br />
The expected value E for '''different sized''' strata is then:<br />
<br />
:<math>E(c_1Y_1+c_2Y_2)=E(c_1Y_1)+E(c_2Y_2)=c_1E(Y_1)+c_2E(Y_2)\,</math> , where <math>c_1 \not= c_2\,</math> or<br />
<br />
:<math>E(\sum c_iY_i)=\sum c_iE(Y_i)\,</math><br />
<br />
and the variance:<br />
<br />
:<math>var(c_1Y_1+c_2Y_2)=var(c_1Y_1)+var(c_2Y_2)=c_1^2var(Y_1)+c_2^2var(Y_2)\,</math> or<br />
<br />
:<math>var(\sum c_iY_i)=\sum c_i^2var(Y_i)\,</math><br />
<br />
{{info<br />
|message=Note:<br />
|text=If we expand (or like in this case relate) the variance with a constant factor, this factor must be squared, because variance is a squared measure!<br />
}}<br />
<br />
===Notation===<br />
<br />
{|<br />
|-<br />
! !! <br />
|-<br />
| <math>L\,</math> || Number of strata <math>h=1, ... , L \,</math> ||<br />
|-<br />
| <math>N\,</math> || Total population size ||<br />
|-<br />
| <math>N_h\,</math> || Size of stratum <math>h (N=\sum N_h)\,</math> ||<br />
|-<br />
| <math>\bar y\,</math> || Estimated population mean ||<br />
|-<br />
| <math>\bar y_h\,</math> || Estimated mean of stratum <math>h\,</math> ||<br />
|-<br />
| <math>n\,</math> || Total sample size ||<br />
|-<br />
| <math>n_h\,</math> || Sample size in stratum <math>h\,</math>||<br />
|-<br />
| <math>S^2_h\,</math> || Sample variance in stratum <math>h\,</math> ||<br />
|-<br />
| <math>\tau\,</math> || Total ||<br />
|-<br />
| <math>\tau_h\,</math> || Total in stratum <math>h\,</math> ||<br />
|-<br />
| <math>\hat \tau_h\,</math> || Estimated total in stratum <math>h\,</math> ||<br />
|-<br />
| <math>c_h\,</math> || Relative share of stratum <math>h\,</math> or weight of stratum ||<br />
|-<br />
| <math>\hat {var} (\bar y)\,</math> || Estimated error variance ||<br />
|-<br />
| <math>\hat {var} (\hat \tau)\,</math> || Estimated error variance of the total ||<br />
|-<br />
| <math>k_h\,</math> || Estimated costs for the assessment in stratum <math>h\,</math> ||<br />
|}<br />
<br />
===Estimator for the mean===<br />
<br />
The estimator for the mean for stratified random sampling is derived based on the considerations above and analog to the estimator of [[Simple random sampling|simple random sampling]] as<br />
<br />
:<math>\bar y = \sum_{h=1}^L \frac{N_h}{N} \bar y_h = \frac {1}{N} \sum_{h=1}^L N_h \bar y_h\,</math><br />
<br />
===Estimator for the error variance===<br />
<br />
The estimator for the error variance (selection without replacement) is:<br />
<br />
:<math>\hat {var} (\bar y) = \sum_{h=1}^L \left\lbrace \left( \frac {N_h}{N} \right)^2 \hat {var} (\bar y_h) \right\rbrace = \frac{1}{N^2} \sum_{h=1}^L N^2_h \frac {N_h-n_h}{N_h} \frac {S^2_h}{n_h}</math><br />
<br />
In this case <math>N_h-n_h/N_h\,</math> is a [[finit population correction]] that is necessary if the strata are small and/or the sample size is large. It is typically applied if the relation between sample size and population is larger than 0.05 (Akca 2001<ref name="akca">Akca, A. 2001. Waldinventur. J.D. Sauerländer's Verlag. Frankfurt am Main, 193 S.</ref>).<br />
<br />
{{info<br />
|message=Note:<br />
|text=A finite population correction is important in case that we apply a selection without replacement and the population size is significantly reduced by drawing the samples. As consequence the selection probabilities are changing with every sample we draw (because the remaining population decreases) what is corrected by this factor.<br />
}}<br />
<br />
<br />
This estimator looks complex, but can be read as the weighted linear combination of simple random sampling estimators applied to the strata. The weighting factor is in this case<br />
<br />
:<math>c^2=\frac {N_h^2}{N^2}\,</math><br />
<br />
Without finite population correction the error variance is:<br />
<br />
:<math>\hat {var} (\bar y) = \frac{1}{N^2} \sum_{h=1}^L N^2_h \frac {S^2_h}{n_h}</math><br />
<br />
===Estimator of the total===<br />
<br />
:<math>\hat\tau = N\bar y = \sum_{h=1}^L \hat \tau_h = \sum_{h=1}^L N_h \bar y_h\,</math><br />
<br />
The estimated error variance of the estimated population total follows then with:<br />
<br />
:<math>\hat{var}(\hat {\tau}) = \hat{var}(N \bar y) = N^2 \hat{var}(\bar y)</math><br />
<br />
To calculate the [[confidence interval]] for the estimation we need information about the standard error and the sample size. In stratified random sampling one faces the difficulty with the number of [[dergees of freedom]]. While it is a direct function of sample size (<math>DF=n-1\,</math>) in simple random sampling, we deal with <math>h\,</math> different sample sizes that are combined. If the variances among these strata differ significantly it is not possible to simply join the degrees of freedom from different strata. This statistical problem is known as '''''Behrens-Fischer problem''''' or '''''Welch-Satterthwaite problem'''''. These statisticians proposed an approximation formula that can be used to calculate the "effective number of degrees of freedom" so that t-statistics can be approximately correct applied.<br />
<br />
:<math>DF_e=\frac {\left(\sum_{h=1}^L g_h S_h^2 \right)^2}{\sum_{h=1}^L \frac {g_h^2 S_h^4}{n_h-1}}\,</math><br />
<br />
==Sample size==<br />
<br />
To determine the necessary [[sample size|sample size]] that is always dependent on the error probability, the allowed error and the variability inside the population, we have to consider that the variance inside the different strata is different. These different variances must be weighted to calculate the necessary sample size.<br />
<br />
{{info<br />
|message=Note:<br />
|text=The "necessary" sample size is the estimated number of samples we need to derive an estimation with a defined [[confidence interval]]. This interval is determined by the predetermined error probability <math>\alpha\,</math>, the corresponding t-value from the t-distribution and the allowed error we define for the inventory. Typically this error is 10% for standard forest inventories.<br />
}}<br />
<br />
Compare the following formula with the formula provided for [[Sample size|simple random sampling]]:<br />
<br />
:<math>n = \frac {t^2 \sum \frac {N^2_h S^2_h}{c_h}}{N^2 A^2}\,</math><br />
<br />
where <math>c_h = n_h/n\,</math>, or the share of samples that is in stratum <math>h\,</math>.<br />
<br />
{{info<br />
|message=Note:<br />
|text=To calculate the overall sample size it is obviously necessary to know the share of samples in the different strata?! That sounds strange, because we like to calculate the sample size here. If we consider that the sample size in each stratum is influencing the expected error, it is clear that we need to have this information. Thats why we have to define the allocation scheme before we start.<br />
}}<br />
<br />
In all sample size calculations one has to know ''before'' how to assign the total number of samples to the different strata: the set of <math>c_h\,</math> must be predetermined!<br />
<br />
In small populations (or large samples) we have to consider the finite population correction and the above formula becomes:<br />
<br />
:<math>n=\frac {\sum_{h=1}^L \frac {N_h^2 S_h^2}{c_h}}{\frac {N^2 A^2}{t^2}+ \sum_{h=1}^L N_h S_h^2}\,</math><br />
<br />
==Allocation of sample size to the strata==<br />
Three factors are relevant in respect to the decision of how to allocate the samples to the strata:<br />
<br />
*Stratum sizes (The bigger the stratum the more samples)<br />
*Variability inside the strata (The more variability the more samples)<br />
*Inventory costs that might vary between strata (The more costly the fewer samples).<br />
<br />
In case that all strata are of same size (equal area) and the variability is also equal, we can use<br />
<br />
:<math>n_h = \frac {n}{L}\,</math><br />
<br />
so a '''uniform allocation''' of samples to the strata. As mentioned above this situation is not realistic and further stratification would not be superior in this case.<br />
<br />
If the number of samples per stratum should be determined proportional to the stratum size (e.g. area) we have a '''proportional allocation''' with:<br />
<br />
:<math>n_h = n \frac {N_h}{N}\,</math><br />
<br />
In this case one would ignore possibly different variances inside the strata. If we like to consider the variances, we need some prior information about the conditions inside the strata that is sometimes available from case studies or forest management data. In this case one can apply the '''Neyman allocation''':<br />
<br />
:<math>n_h = n \frac {N_h S^2_h}{\sum_{h=1}^L N_h S^2_h}\,</math><br />
<br />
If one has to consider inventory costs that might vary significantly between strata, this information can be included as additional factor. In this case one is able to calculate the '''optimal allocation with cost-minimization''': <br />
<br />
:<math>n_h = n \frac {\frac {N_h S^2_h}{\sqrt {k_h}}}{\sum_{h=1}^L \frac{N_h S^2_h}{\sqrt {k_h}}}\,</math><br />
<br />
{{info<br />
|message=Note:<br />
|text=It is obvious that we need <math>n</math> to calculate the allocation. At the same time we need <math>n_h</math> (the result of this calculation) to determine the sample size. This dilemma can be solved by an iterative process, where we predetermine relative shares (<math>c_h</math>) to derive <math>n</math> in a first step.<br />
}}<br />
<br />
==Summarizing==<br />
<br />
Depending on the allocation scheme that should be used one needs the following information to implement stratified random sampling:<br />
*number of strata,<br />
*size of strata or relative share on the total population,<br />
*estimations for the variance inside the strata,<br />
*eventually information about the expected inventory costs in different strata.<br />
<br />
Further one has to predefine the target precision (A) (that is the allowed error expressed as 1/2 of the confidence interval width), the error probability <math>\alpha</math> that is typically 0.05 and the corresponding t-value from the t-distribution (in case of sample size > 30 this value is approximately 2).<br />
<br />
Based on these Informations one may <br />
*choose an appropriate allocation scheme,<br />
*derive the weights <math>c_h\,</math> for the strata,<br />
*calculate the sample size,<br />
*and allocate them to the strata.<br />
<br />
{{Exercise<br />
|message=Stratified sampling examples<br />
|alttext=Example<br />
|text=Example for Stratified sampling<br />
}}<br />
<br />
==Comments==<br />
<br />
Stratification is a powerful procedure to reduce the error variance if the above mentioned preconditions (homogeneous strata with significant differences of strata means) is fulfilled. In this case one takes a maximum of variability out of the sample.<br />
Some a priori information is necessary as the subdivision of the population must be defined before the samples are taken. If these information is missing one may apply techniques like [[Double_sampling#Double_sampling_for_stratification_.28DSS.29|double sampling for stratification]] that employs stratification without requiring a priori delineation of strata (the strata sizes are estimated in the course of a two-phase sampling process).<br />
In this chapter stratified random sampling was introduced. However stratification can also be applied for other sampling techniques. It is important to note that one may apply whatever sampling technique that is appropriate inside the different strata. That means for example that one can optimize the [[Sampling design and plot design|Inventory design]] as well as the [[Sampling design and plot design|Plot design]] according to the conditions inside the strata.<br />
<br />
{{info<br />
|message=Example:<br />
|text=A forest area can be subdivided in different stands of different species or age classes. It is now possible to apply different plot designs (e.g. different sizes of sample plots) in the respective strata. While we perhaps need larger plots for the old stands, smaller plots might be sufficient in younger and more dense stands.<br />
}}<br />
<br />
==References==<br />
<references/><br />
<br />
{{SEO<br />
|keywords=stratified random sampling,strata,population,sampling technique,sub-population<br />
|descrip=Stratified sampling is a method to subdivide a population into separate and more homogeneous sub-populations called strata.<br />
}}<br />
<br />
[[Category:Sampling design]]<br />
[[Category:Article of the month]]</div>Fehrmannhttp://wiki.awf.forst.uni-goettingen.de/wiki/index.php/Cluster_samplingCluster sampling2024-01-29T12:15:22Z<p>Fehrmann: /* Notation */</p>
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<div>{{Ficontent}}<br />
Cluster Sampling (CS) is here presented as a variation of [[:Category:sampling design|sampling design]] as it is done in most textbooks as well. However, in strict terms it is not a sampling design but just a variation of [[Lectuenotes:Sampling design and plot design|response design]]: The major point in cluster sampling is that for each [[random selection]] not only one single sampling element is selected but a set (cluster) of sampling elements; thus, a cluster consists of a group of observation units, which together form a sampling unit (Kleinn 2007<ref name="kleinn2007">Kleinn, C. 2007. Lecture Notes for the Teaching Module Forest Inventory. Department of Forest Inventory and Remote Sensing. Faculty of Forest Science and Forest Ecology, Georg-August-Universität Göttingen. 164 S.</ref>). The selection itself can be done according to any [[:category:sampling design|sampling design]] ([[Simple random sampling|simple random]], [[Systematic sampling|systematic]], [[Lecturenotes:stratification|stratified]] etc.). The spatial configuration of cluster plots (in terms of number of sub-plots and their distance) is a matter of optimization based on the [[Intracluster Correlation Coefficient]] (ICC) and practical considerations on costs.<br />
<br />
Cluster sampling can be applied to any type of sampling element. If, on a production belt of screws one screw is selected randomly and the next 5 are also taken, then this set of 6 screws forms one observation unit consisting of 6 screws, because only one randomization (selection of the first screw) had been done to select this observation unit of 6. In fact, most basic [[:Category:Plot design|plot designs]] as used in [[forest inventory]] can be viewed as cluster plots, where the cluster consists of a number of individual trees (this holds for [[fixed area plots]], for [[Bitterlich sampling|Bitterlich]] plots etc.).<br />
In large area forest inventory, it is common that not single compact plots are laid out at each sample point but clusters of sub-plots. There, sub-plots are laid out in various geometric shapes and distances between them.<br />
<br />
[[file:clusterplots.png|center|500px]]<br />
<br />
The above figure shows clusters with different geometric spatial arrangements of sub-plots. Each dot depicts one sub-plot. It is important to understand that the entire set of sub-plots is the observation unit or cluster (better: the cluster-plot). [[Sample size]] is determined by the number of clusters and not by the number of sub-plots. <br />
<br />
{{info<br />
|message=Info<br />
|text=This does often cause confusion but it is easy: we may call the entire cluster a cluster-plot; where the plot does not come in one compact piece (as with a circular fixed area plot) but is sub-divided into various distinct pieces. A cluster-plot can thus be viewed as a “funny shaped” plot.<br />
}}<br />
<br />
It is a good terminology practice to always refer to “plot” if we talk about independently selected observation units. Therefore, the entire cluster is the plot (also called cluster-plot), and not the sub-plot. By erroneously referring to sub-plots as plots, one may cause confusion that may also lead to severe confusion about estimation as well. <br />
<br />
Coming back to an example, in the following figure the basic principle of cluster sampling is illustrated based on a population of 48 elements that is grouped into ''N''=24 clusters of size ''m''=2. In cluster sampling, this population of clusters constitutes the [[Lecturenotes:Population|sampling frame]] from which sample elements are drawn. With each random selection a set of two individual elements is drawn which, however, does produce but '''one observation''' for estimation. That means the total or mean of the two elements (that is: an “aggregated” value) and not the two individual values is then further processed for estimation.<br />
<br />
[[file:clustersampling2.png|center]]<br />
<br />
It has been mentioned that cluster plots, consisting of ''m'' sub-plots are standard in large area forest inventory, such as many [[National Forest Inventory|National Forest Inventories]].<br />
<br />
[[File:clusterdesign1.png|right|thumb|300px|Example of systematically arranged cluster plots (each with four rectangular sub-plots).]]<br />
[[File:clusterdesign2.png|right|thumb|300px|clusterdesign with 3 circular sub-plots along a river.]]<br />
[[File:clusterdesign3.png|right|thumb|300px|Cluster sampling design in a forest fire area in East Kalimantan (Schindele 1989)<ref>Schindele, W. 1989. Field Manual for Reconnaissance Inventory on Burned Areas, Kalimantan Timur. FR-Report No.2.</ref>]]<br />
<br />
==Notation==<br />
As with [[Lecturenotes:stratification|stratified sampling]], some notation needs to be introduced when developing the estimators for cluster sampling: <br />
<br />
:{|<br />
| <math>N\,</math> || Total population size = number of clusters in the population ||<br />
|-<br />
| <math>M\,</math> || Number of sub-plots in the population, <math>M= \sum_{i=1}^N m_i\,</math> ||<br />
|-<br />
| <math>\bar M\,</math> || Mean cluster size in the population, <math>\bar M=\frac{M}{N}</math> ||<br />
|-<br />
| <math>n\,</math> || Sample size = number of clusters in sample ||<br />
|-<br />
| <math>m_i\,</math> || Number of sub-plots in cluster ''i'', i.e. cluster size ||<br />
|-<br />
| <math>\bar m\,</math> || Mean cluster size in the sample||<br />
|-<br />
| <math>y_i\,</math> || Observation in cluster ''i'', i.e. total over all sub-plots of cluster ''i'' ||<br />
|-<br />
| <math>\bar y_i</math> || Mean per sub-plot in cluster plot ''i'' = <math>\bar y_i = \frac {y_i}{m_i}</math> ||<br />
|-<br />
| <math>\bar y_c\,</math> || Estimated mean per cluster in the sample = <math>\bar y_c = \frac{1}{n} \sum_{i=1}^n y_i</math>||<br />
|-<br />
| <math>\bar y\,</math> || Estimated mean per sub-plot in the sample = <math>\bar y = \frac{1}{n} \sum_{i=1}^n \bar y_i</math>||<br />
|}<br />
<br />
==Cluster sampling estimators==<br />
Clusters may have the same size <math>m</math> in terms of number of subplots for all clusters or variable size <math>m_i</math>. Obviously, this needs to be taken into account in estimation. <br />
In this section, we present the estimator for the situation of cluster-plots of equal size under [[Simple random sampling|random sampling]]. When the population consists of clusters of unequal size, then the [[ratio estimator]] will likely yield more precise results, because the ratio estimator uses the cluster size as ancillary variable.<br />
<br />
When all clusters have the same size, then <math>m_i=\bar m</math> is constant. The estimated mean per sub-plot can then be calculated from the estimated mean per cluster as:<br />
<br />
:<math>\bar y = \frac{\bar y_c}{\bar m}\,</math><br />
<br />
and the estimated [[error variance]] of the mean per sub-plot is:<br />
<br />
:<math>\hat {var}_{cl} (\bar y) = \frac {1}{\bar m^2} \frac{N-n}{N} \frac{S_{y_i}^2}{n} \,</math><br />
<br />
This is reminiscent of the error variance for simple random sampling, and only the first term is new:<math>S_{y_i}^2</math> is the variance per cluster-plot; when we are interested in the variance per sub-plot, then the per-cluster variance needs to be divided by <math>\bar m^2 </math>, similar to the estimated mean per sub-plot with the difference that <math>\bar m</math> is squared.<br />
<br />
The estimated variance per cluster can be derived by calculating the variance over the per-cluster-observations:<br />
<br />
:<math>S_{y_i}^2=\frac{\sum_{i=1}^n (y_i-\bar y_c)^2}{n-1}</math><br />
<br />
The estimated total results as usual from multiplying the mean with the total; in cluster sampling, one may take the cluster mean and the number of clusters, or the mean per sub-plot and the number of sub-plots: <br />
<br />
:<math>\hat \tau = N*\bar y_c = M * \bar y \,</math><br />
<br />
where the error variance is<br />
<br />
:<math>\hat {var} (\hat \tau) = N^2 \hat {var} (\bar y_c) = M^2 \hat {var} (\bar y) \,</math><br />
<br />
The estimated error variance of the mean per sub-plot can alternatively be calculated from:<br />
<br />
:<math>\hat {var}_{cl} (\bar y) = \frac {N-n}{N} \frac {1}{\bar m^2} \frac {\bar m^2 \sum_{i=1}^n (\bar y_i - \bar y)^2}{n-1} = \frac {N-n}{N} \frac {1}{n} \frac {\sum_{i=1}^n (\bar y_i - \bar y)^2}{n-1} \,</math><br />
<br />
where <math>S_{y_i}^2</math> from the above error variance formula was replaced with <math>S_{y_i}^2 = \bar m * S_{\bar y_i}^2</math>. It becomes obvious that this is exacly the [[simple random sampling]] estimator with the sub-plots as observation units and the variance of the sub-plots<br />
<br />
:<math>S_{\bar y_i}^2 = \frac {\sum_{i=1}^n (\bar y_i - \bar y)^2}{n-1} \,</math><br />
<br />
as estimated population variance. But remember that the sample size ''n'' is still the number of clusters, despite the fact that it is possible to use the sub-plot observations for estimation as well.<br />
<br />
==Cautions in cluster sampling==<br />
<br />
<br />
Unfortunately, an often committed error in cluster sampling is to treat the sub-plots as if they would be the independently selected sampling units. As a consequence the sample size appears to be much bigger than it actually is. And, because sample size ''n'' comes in the denominator of the error variance and the [[standard error]] estimator, this artificial inflation of sample size leads to a smaller value of the error variance. While, of course, smaller error variance is always welcome, in this case the smaller value comes from an erroneous calculation!<br />
<br />
<br />
When interest is in producing point and interval estimates for the target variables, then an analysis of individual sub-plot observations is not help- nor useful. However, if interest is in assessing aspects of the spatial distribution of the variables of interest, then the sub-plot information may be of great interest because the cluster plot contains some information about the spatial relationships at defined distances.<br />
<br />
{{Exercise<br />
|message=Cluster sampling examples<br />
|alttext=Example<br />
|text=Examples for Cluster sampling<br />
}}<br />
<br />
==Comparison to SRS==<br />
If we compare cluster sampling to [[simple random sampling]] of the same number of sub-plots, then simple random sampling is expected to yield more [[accuracy and precision|precise]] results, because a much higher number of samples is being selected, covering more different situations of the [[population]]. While the cluster-plot, being a larger plot, will cover more variability within the plot, thus decreasing the variability between the plots; this smaller variance does usually not compensate the much smaller [[sample size]].<br />
{{info<br />
|message=In simple words<br />
|text=As in cluster sampling the observation from one cluster plot is derived as mean over all sub-plots, it can be assumed that more of the given variance can be captured '''inside''' a plot (or inside each single observation). The spatially dispersed sub-plots cover a wider range of conditions and by calculating the mean there is a compensating effect. Remember: The criterion for the quality of estimates is the [[standard error]] calculated based on the variance '''between''' the cluster plots (or observations). If more variance can be captured within the plots, the variance between the plots will decrease and the standard error will be lower (Means: an increase in [[Accuracy and precision|precision]]).On the other hand, combining sub-plots to clusters leads to smaller [[sample size]], as each cluster gives only one observation. Both effects can compensate each other.<br />
}}<br />
<br />
<br />
However, the general statement of the preceding paragraph must be looked at in more detail: to what extent cluster sampling is statistically less [[relative efficiency|efficient]] than simple random sampling depends exclusively on the [[Spatial autocorrelation|covariance-structure]] of the [[Population|population]]. This is dealt with in more detail in [[spatial autocorrelation]], [[Intracluster Correlation Coefficient]] and precision.<br />
Cluster sampling is widely used in large area forest inventories, such as [[National Forest Inventory|National Forest Inventories]]. The major reason does not lie in the statistical performance but in practical aspects: once a field team has reached a field plot location (which in some cases may take many hours or even some days), then one wishes to collect as much information as possible at this particular sample point. Therefore, it appears straightforward to establish a large plot and not a small one. Further on, large plots can be laid out in different manners: either as a compact individual plot – which, on the average, does not capture a lot of variability - or subdivided into sub-plots that are at a certain distance from each other – which tends to capture a lot more of variability. And, in addition, this spatial variability can also be analyzed.<br />
<br />
==Spatial autocorrelation and precision in cluster sampling==<br />
[[Spatial autocorrelation]] is a concept which is very relevant for cluster plot design optimization. Spatial correlation can be described with the covariance function of observations at a distance ''d'' apart from each other and says, essentially, that observations at smaller distances tend to be more similar (higher correlated) than observations at larger distances. As mentioned in the sections on plot design, what we strive to have on one sample plot is as much variability as possible. We do not wish to have highly correlated observations because that would mean that by knowing one value we can fairly well predict the other values; so, no need to make costly additional observations once one observation is made.<br />
<br />
(1)The larger the distance between two observations (e.g. sub-plots) the lesser correlated the observations tend to be, and (2) we wish to have uncorrelated observations. These two statements are important for cluster design planning: when we assume a constant distance between sub-plots of, say, 100 m for all cluster plot designs in the above figure, then we can compare the expected statistical performance of these geometric shapes of which all have 8 sub-plots (except the triangle with 9). The most spatially compact shape is the cross; there, the average distance between sub-plots is smallest and we expect that to cause the highest intra-cluster correlation and, therefore, the lowest precision. The simple line will produce the most precise results because the largest possible distances occur in that cluster plot shape. Of course, these statements need always to be seen in the direct context of the particular covariance function: if the distance between sub-plots is such that even two neighboring sub-plots are uncorrelated – then there is no difference in terms of statistical precision between the different geometric cluster plot shapes.<br />
{{Info<br />
|message=Obs!<br />
|text=Of course, not only statistical criteria apply if cluster-plots are designed; as always it is also cost and time considerations. And in that sense the square cluster has some advantages over the statistically more efficient line, because in the square cluster, after having finished the observations on the sub-plots, the field teams return to the starting point and need not to have a relatively long and unproductive (in the sense of sample observations) way back to the starting point.<br />
}}<br />
<br />
The similarity of observations within a cluster can be quantified by means of the [[Intracluster Correlation Coefficient]] (ICC), sometimes also referred to as intraclass correlation coefficient.<br />
<br />
=References=<br />
<references/><br />
<br />
<br />
<br />
[[Category:Sampling design]]<br />
[[Category:plot design]]<br />
[[Category:Article of the month]]</div>Fehrmannhttp://wiki.awf.forst.uni-goettingen.de/wiki/index.php/Line_samplingLine sampling2023-09-26T14:08:15Z<p>Fehrmann: /* Using sample lines as sample selection tools */</p>
<hr />
<div>{{Ficontent}}<br />
Line sampling uses one-dimensional lines as [[plot design|observation units]], just as we may use for many purposes fixed area [[fixed area plots|sample plots]] (two-dimensional observation units) or points (dimensionless observation units). Line sampling does not refer to the so-called transects which usually mean elongated narrow strips, that is: two-dimensional plots. <br />
Depending on the dimensionality of the observation units, different types of observations can be made on them. There are three major applications of line sampling <ref name="kleinn2007">Kleinn, C. 2007. Lecture Notes for the Teaching Module Forest Inventory. Department of Forest Inventory and Remote Sensing. Faculty of Forest Science and Forest Ecology, Georg-August-Universität Göttingen. 164 S.</ref>:<br />
<br />
*It may be observed which portion of a sample line comes to lie in forest. These portions can be used to estimate the [[Forest Definition|forest cover]] percent. This type of sampling is called '''line intercept sampling'''.<br />
<br />
*It may be observed how many times a sample line intersects with a line feature in the landscape such as forest edges, roads or creeks. The number of intersections can be used to estimate the total length of these line features. This type of sampling is called '''line intersect sampling''' (LIS).<br />
<br />
*It may be observed whether a population element is intersected or not by a sample line. By this, we may use the line samples to make a [[Random selection|random selection]] of population elements in the absence of an a priori [[population|sampling frame]].<br />
<br />
A good reference for line sampling is deVries (1986<ref name="deVries">de Vries PG. 1986. Sampling Theory for Forest Inventory. Springer-Verlag Berlin. 399p.</ref>) or earlier related studies by deVries.<br />
<br />
{{info<br />
|message=To remember!<br />
|text=It is very easy to mix up the two different line sampling techniques. It is important to remember that Line '''Intercept''' Sampling is completely different from Line '''Intersect''' Sampling. While we just look at relative shares of line segments (here the observation units) inside a target class in Line Intercept Sampling, the second technique is applicable to estimate the length of linear features just by counting the intersections with the sample line. <br />
}}<br />
<br />
==Line intercept sampling==<br />
<br />
Line intercept sampling is used to estimate the cover of defined classes in the landscape, for example forest or forest types. Line samples are randomly placed over the area of interest and on each sample line it is observed which proportion comes to lie in the target area class. The observation per sample line can take on values between 0 and 1. From these observations, the [[mean]], [[variance]] and [[error variance]] can be estimated along the known [[estimator]]s.<br />
Line intercept sampling is typically applied in aerial photographs when rapid area estimations are desired. However it is also applied in the field: in large area forest inventory [[cluster sampling]], one may use, for example, the connecting lines between [[sub-plots]] to do line intercept sampling. Figure 1 illustrates this approach by overlaying a systematic sample of equally oriented lines of n = 24 and a random sample of randomly located and randomly oriented lines of n = 3.<br />
<br />
[[Image:SkriptFig_95.jpg|thumb|500px|'''Figure 1.''' Application of line intercept sampling on an aerial photograph.]]<br />
<br />
Let <math>l_{i(in)}</math> be the length of the sample line ''i'' which is inside the target condition class such as forest, <math>l_{i(total)}</math> be the total length of the sample line ''i''; then, the observation <math>y_i</math> which is made on line ''i'' is <br />
<br />
:<math>y_i = \frac {l_{i(in)}} {l_{i(total)}}</math><br />
<br />
The estimated proportion <math>\hat p</math> in the region off interest is then estimated from ''n'' randomly placed sample lines from <br />
<br />
:<math>\hat p=\frac {\sum_{i=1}^n y_i}{n}</math> <br />
<br />
and the variance in the [[population]] is estimated from <br />
<br />
:<math>s^2=\frac {\sum_{i=1}^n (y_i - \hat p)^2}{n-1}</math> <br />
<br />
The error variance is <math>s_p^2 = \frac {s^2}{n}</math><br />
<br />
These estimators hold for [[simple random sampling]] and sample lines of the same length. It may be that sample lines have different length; for example if the lines cross the entire region of interest and are randomly placed. Then, the [[Ratio estimator|ratio estimator]] may be applied to increase the [[Accuracy and precision|precision]] of estimation, using the length of the individual sample line as [[Ancillary variable|ancillary variable]] (see also [[sampling with unequal selection probabilities]]).<br />
<br />
==Line intersect sampling==<br />
<br />
The terms line intersect sampling and line intercept sampling should not be confused: line intersect sampling bases on counting intersections of the sample lines with line features in the landscape. It is obvious that there will be more intersections with line samples, the longer the line feature in the landscape is.<br />
By simply counting the intersections (and knowing the total length of the sample lines), one can then produce an estimation of the total length of the line feature. This is frequently used for the estimation of the length of the forest edge or for the length of the road network in a region of interest in which that sample line is located. <br />
Line intersect sampling goes back to an experiment from the 18th century:<br />
<br />
===Buffon’s needle problem (1777)===<br />
<br />
The French noble man [[Wikipedia:Georges-Louis_Leclerc,_Comte_de_Buffon|Comte de Buffon]] researched into the basic questions: If a needle of length <math>l_i</math> is thrown randomly on an area completely covered by equidistant parallel lines with the distance ''W'' (and the line length <math>l_i</math> being shorter than ''W''), what is the probability that the needle intersects with one of the parallels (This example is also known as [[Wikipedia:Buffon's_needle|Buffon's needle problem]]).<br />
<br />
[[Image:SkriptFig_96.jpg|700px|thumb|'''Figure 2.''' Illustration of Buffon´s needle problem: the probability is to be estimated that a needle intersects with one of the parallel lines.]]<br />
<br />
This is illustrated in Figure 2. The solution follows the standard procedure when determining probabilities: we must try and define the size of the total population of needles and then identify that part of the population of needles that intersect with one of the parallels. The ratio of these two is the searched probability.<br />
<br />
The motivation of Mr. Buffon was obviously to investigate the characteristics of gambling.<br />
<br />
This problem can be solved by geometric probabilities and is illustrated in Figure 3 on the left, where we concentrate on only one of the parallels. After this, the position of a particular needle ee′ relative to the parallels is defined<br />
<br />
*by the center point ''M'' of the needle and its distance m to the nearest parallel, and<br />
<br />
*by the orientation, that is the angle φ between the orientation of the needle and the orientation of the parallels.<br />
<br />
About the line shown in Figure 3, a rectangle is centered with the width ''W'' and the length ''L''; remember ''W'' is the distance between the parallels and ''L'' is the total length of the parallel. Within this rectangle, only specific combinations of the above mentioned factors do lead to an intersection of the needle with the centered line. That means: as a function of these two factors, one can determine whether the needle intersects or not. The population of needles is defined by all points of the area of interest and for each point by all possible orientations; where the area of interest is formed by the rectangle ''WL''. If the distance of the needle from the parallel is larger than <math>l_i/2</math>, an intersection is not possible at all so that we can subdivide the area of interest from the outset in two domains: Needle positions nearer than <math>l_i</math> to the parallel (here, the intersection depends on the angle φ) and those farther away than li for which the intersection is impossible.<br />
<br />
Because of symmetry it is sufficient to investigate what happens at one side of the perpendicular line, and it is sufficient to look at angles φ from 0 to 90°, or in radians from 0 to <math>\frac {\pi}{2}</math>. Thus, it is imperative that<br />
<br />
:<math>0 \le m \le \frac {l_i}{2}</math> and <math>0 \le \varphi \le \frac {\pi}{2}</math><br />
<br />
where ''m'' and φ are stochastically independent variables and equally distributed in the correspondent intervals. We can now identify those combinations of ''m'' and φ for which an intersection takes place. To do so, let’s introduce the distance ''x'' between the midpoint ''M'' of the needle and the intersection point ''S'' (Figure 3, left), for which it is imperative that <br />
<br />
[[File:Buffon's needle corrected.PNG|thumb|300px|right|The red and blue needles are both centered at x. The red one falls within the grey area, contained by an angle of 2θ on each side, so it crosses the vertical line; the blue one does not. The proportion of the circle that is grey is what we integrate as the center ''x'' goes from 0 to 1]]<br />
<br />
:<math>x \le \frac {l_i}{2}</math><br />
<br />
The variable ''x'' is a function of m and φ and can, thus, be replaced by <br />
<br />
:<math>\frac {m}{sin \varphi}</math> <br />
<br />
what finally results in <br />
<br />
:<math>m \le \frac{l_i}{2}sin(\varphi)</math> <br />
<br />
As a result, for all needle positions / combinations below the line <br />
<br />
:<math>m = \frac{l_i}{2}sin(\varphi)</math> <br />
<br />
an intersection takes place (Figure 3 right). That graph can also be read as follows: imagine a fixed value for the distance <math>m_x</math>, represented by a parallel to the abscissa passing through the value <math>m_x</math> at the ''y''-axis; for all values of φ left to the intersection with the function, there is no intersection with the parallels, but there is intersection for all angles φ right to the intersection with the function.<br />
<br />
[[Image:SkriptFig_97.jpg|center|thumb|1000px|'''Figure 3.''' Intersection probability as a function of the orientation of the needle and the distance of the needle to the nearest line (from deVries 1986<ref name="deVries" />)]]<br />
<br />
Finally, the probability of intersection is the ratio of the dotted area and the total area. That total area is the rectangle with side length ''W/2'' and <math>\frac {\pi}{2}</math>; and the area of intersection is <br />
<br />
:<math>A_{intersect} = \int_{\varphi=0}^{\frac {\pi}{2}}\, \frac {l_i}{2} sin(\varphi)\, d\varphi = \frac {l_i}{s}[-cos \varphi]_0^{\frac {\pi}{2}} = \frac {l_i}{2}</math><br />
<br />
Thus, the probability of intersection of the needle with one of the parallels is given by the simple expression<br />
<br />
:<math>\pi_{i1} = \cfrac {\cfrac{1}{2}l_i}{\cfrac {1}{2}\pi * \cfrac {1}{2}W} = \frac {2l_i}{\pi W}</math><br />
<br />
Observe that this term holds only for an experimental arrangement like illustrated in Figure 2. If we are interested in placing sample lines on an area of interest in order to estimate certain population characteristics like in Figure 3 left, we need to take the probability into account that ''M'' is within the rectangle ''WL''. This is simply given by <math>\pi_{i2} = </math>''WL/A'' and the final probability of intersection is<br />
<br />
:<math>\pi_i = \pi_{i1} * \pi_{i2} = \frac {2l_iL}{\pi A}</math><br />
<br />
As expected, the probability of intersection depends on the density of sample lines, given by line length per area ''L/A'', and the length of the needle <math>l_i</math>. Note that the term <math>\pi_i</math> is used here to indicate so-called [[inclusion probability|inclusion probabilities]] (see [[Horvitz-Thompson estimator]]), which must be distinguished from the number <math>\pi</math>.<br />
<br />
By re-arranging the latter formula we see that the total line length ''L'' can be determined by the number of intersections, resulting from <math>\pi_i, l_i</math> and ''A'' by <br />
<br />
:<math>L = \frac {\pi_i \pi A}{2l_i}</math> <br />
<br />
where <math>\pi</math> needs to be identified experimentally. One may also imagine a simple experiment to empirically determine the value of <math>\pi</math> by simply counting the number of intersections: <br />
<br />
:<math>\pi = \frac {2Ll_i}{\pi_i A}</math><br />
<br />
<br />
===Applications of line intersect sampling===<br />
<br />
Line intersect sampling is applied in forest inventory practice in particular for estimating the [[Estimating the length of the forest edge|forest edge]] length and for estimating the length of forest roads. Matérn (1964<ref> Matérn B. 1964. A method of estimating the total length of roads my means of a line survey. Studia Forestalia Fennica Suecica 18:68-70.</ref>) presented the line intersect sampling approach for the latter purpose and estimated total line length as line length per unit area with <br />
<br />
:<math>\hat \tau = \frac {\pi Am}{2L}</math> <br />
<br />
where m is the number of intersections of sample lines and the target line feature This formula results directly from combining the above derived probability of intersection <math>\pi_i</math> with the so-called [[Horvitz-Thompson estimator]], which is given by <br />
<br />
:<math>\hat \tau = \sum \frac {y_i}{\pi_i}</math>. When <math>\pi_i</math> <br />
<br />
is replaced with the above given expression, a general estimator for estimating characteristics of line features with line intersect sampling is <br />
<br />
:<math>\hat \tau = \frac {\pi A}{2L} \sum \frac {y_i}{l_i}</math> <br />
<br />
This estimator immediately results in the above given estimator from Matérn if we replace <math>y_i</math> with <math>l_i</math>, which is the length of the needle or object intersected, respectively.<br />
<br />
===Using sample lines as sample selection tools===<br />
Sample lines may also be used as a tool for sample selection in the absence of a [[population|sampling frame]]. Imagine an aerial photograph with many isolated objects on it like shrubs, or pieces of woody debris on the forest floor. One could then number the shrubs and randomly select some (this would be building a sampling frame first and then do the random selection). However, we may also throw a sample line of defined length randomly onto the area and select the shrub that is being intersected by the line. If the objects (shrubs) have different sizes, then this is sampling with unequal selection probabilities where the selection probability has to do with the length of the sample line and the size and shape of the objects. Given the same area, an object will have a higher selection probability if it has a narrow elongated shape.<br />
<br />
[[Image:SkriptFig_98.jpg|thumb|1000px|'''Figure 4.''' If line intersect sampling is used as a tool to select sample objects, a “needle” is to be defined on each object. Only intersection of the sample line with this needle leads to the selection of that object.]]<br />
<br />
Following the basic features of line sampling and in order to make that approach operational, we define a “needle” on each element, a line that characterizes that object. Only if the sample line intersects with this needle, the object is going to be selected. Figure 98 illustrates that for the example of woody debris on the ground; again: not the intersection of the object with the sample line makes it to be selected, but the intersection with the needle defined on the object.<br />
<br />
The length of the needle defines the [[inclusion probability]] of that particular object. If we measure the value of the target variable on each selected object, then we have the standard estimation situation of [[sampling with unequal selection probabilities]]. This technique is applied as standard element of sampling designs for down woody debris.<br />
<br />
Following this, if we are interested in estimating the total volume of coarse woody debris on a given area of interest, one may apply the general estimator given in the preceding chapter. The only thing that has to be done is to determine the volume of the single logs intersected, what, for example, can be done with Huber’s formula: <br />
<br />
<math>y_i = \frac {\pi}{4}d_i^2 l_i</math><br />
<br />
where <math>d_i</math> is the diameter at the point of intersection. Thus, the final estimator is <br />
<br />
:<math>\hat \tau = \frac {\pi^2 A}{8L} \sum d_i^2</math> <br />
<br />
and only one diameter measurement per object is necessary.<br />
<br />
An excellent and complete discussion of line intersect sampling with more fields of application can be found in the text book of Gregoire and Valentine (2008<ref>Gregoire TG and HT Valentine. 2008. Sampling Strategies for Natural Resources and the Environment. Chapman and Hall / CRC. 474p.</ref>) in chapter 9.<br />
<br />
{{info<br />
|message=simple<br />
|text= In this example line elements are used to select target objects in an area of interest. The target variable is not measured with or along the lines, but might be any other variable of interest measured on the selected objects. Following the above explanation it is possible to determine the [[inclusion probability]] of the object that have to be known for unbiased [[sampling with unequal selection probabilities]]. <br />
}}<br />
<br />
==References==<br />
<br />
<references/><br />
<br />
<br />
<br />
[[Category: Plot design]]</div>Fehrmannhttp://wiki.awf.forst.uni-goettingen.de/wiki/index.php/DendrometerDendrometer2023-07-13T07:34:57Z<p>Fehrmann: /* General description */</p>
<hr />
<div>{{Ficontent}} {{video|link=https://youtu.be/wCbrwpTAYxI}}<br />
==Kramer's Dendrometer==<br />
[[File:dendrometer.jpg|thumb|left|Dendrometer]]<br />
[[File:dendrometer2.png|thumb|300px]]<br />
[[File:dendrometer3.png|thumb|300px|Measurement of [[tree height]] and partitioning of volume]]<br />
[[file:dendrometer01_wzp1.jpg|thumb|300px|BAF1]]<br />
<br />
==General description==<br />
The Dendrometer (according to [http://de.wikipedia.org/wiki/Horst_Kramer Kramer]) is a simple measurement device that is suitable for the estimation of [[basal area]] per hectare based on the principle of [[Bitterlich sampling]], stand volume and rough estimates of tree height and volume partitioning of single trees (Kramer 1990<ref name="kramer">Kramer, H. 1990. Nutzungsplanung in der Forsteinrichtung. 2., überarbeitete und erweiterte Auflage. J.D. Sauerländer’s Verlag, Frankfurt/Main.</ref>, Akca 2001<ref name="akca">Akca, A. 2001. Waldinventur. J.D. Sauerländer's Verlag. Frankfurt am Main, 193 S.</ref>). The Dendrometer II is manufactured and distributed by the <br />
[http://www.uni-goettingen.de/en/67094.html Department of Forest Inventory and Remote Sensing] at the university of Göttingen based on details developed by Prof. Kramer in the early 1980s. Kramer's idea was to develop a simple and affordable alternative to the [[Relascope|Mirror Relascope]] developed based on Walter Bitterlich's invention of the "angle count method".<br />
<br />
===Height measurement===<br />
It is possible to roughly estimate [[tree height]] based on the [[The geometric principle|geometric principle]] with the Dendrometer. Therefore hold the device vertically in front of the eye (opening on top) and aim at the tree to be measured. Adjust the distance of the Dendrometer from the eye or your distance from the tree until the points ''k'' and ''h'' correspond to the tree tip resp. to the bottom of the trunk. The relation of the section ''hi'' to line ''hk'' is 1:10. the height of point ''i'' corresponds to 1/10 of the tree height. In order to calculate the tree’s height measure the distance from ''i'' on the tree to the tree base and multiply by 10. It should be considered that the measurement of tree heights is just a "side function" of this device, while the main purpose is in estimation [[basal area]].<br />
<br />
===Estimation of basal area per hectare===<br />
The estimation of [[basal area]] is based on the principle of [[Bitterlich sampling|relascope sampling]] invented by [http://de.wikipedia.org/wiki/Walter_Bitterlich Walter Bitterlich] (Bitterlich 1952<ref name="bitterlich">BITTERLICH, W., 1952. Die Winkelzählprobe. Forstwissenschaftliches Centralblatt, 215-225.</ref>). <br />
Aim at the dbh of each tree in a 360° sweep (from your standpoint). The basal area is a function of the angle and the number of stems which appear wider than the respective angle opening. Trees with width smaller than ''fg'' are not included in the sample (counted).<br />
If you hold the dendrometer vertically at a distance of 50 cm before the eye ''r'' (distance to the eye is fixed with ''rs'' is 50 cm), then the plate with the width 1 creates an angle for which every counted tree corresponds to 1 m² basal area per hectare ([[Bitterlich_sampling#Choice_of_basal_area_factor|basal area factor]] = 1).<br />
<br />
When a width of 2 or 4 is used, the resulting basal area (number of trees counted) must be multiplied by the corresponding factor (2 or 4) in order to obtain estimates of basal area per hectare.<br />
With relascope sampling on slopes it is necessary to multiply the estimated basal area per hectare with the coefficient k (k=1/cos α) for the respective maximal inclination of slope from the table on the Dendrometer.<br />
<br />
===Determining value of single trees by estimating volume sections===<br />
Set up the dendrometer in the same way as for height measurement. Through the markings ''b'', ''c'', ''d'' the stem is divided to 4 sections of equal volume. <br />
<br />
===Estimating stem volume with "form height"===<br />
For the 4 main tree species in northern Germany (Ei, Bu, Fi, Ki) and a given or measured mean stand height (''hm'', which should be approximately the height of the mean basal area tree) the "form-height factor" (FH-Tarif) can be read off the table on the dendrometer. Calculate stand volume per hectare in m³ including bark with the formula <math>V=G*FH \,</math>.<br />
<br />
==Instruction manual==<br />
Thanks to the input and efforts of many international researchers visiting our department, a lot of translations are available for the short instruction manual. It can be downloaded in:<br />
<br />
*[[:File:Gebrauchsanleitung Dendrometer Englisch.pdf|English]]<br />
*[[:File:Gebrauchsanleitung Dendrometer Deutsch.pdf|German]]<br />
*[[:File:Gebrauchsanleitung Dendrometer Afrikaans.pdf|Afrikaans]]<br />
*[[:File:Gebrauchsanleitung Dendrometer Albanisch.pdf|Albanian]]<br />
*[[:File:Gebrauchsanleitung Dendrometer Arabisch.pdf|Arabic]]<br />
*[[:File:Gebrauchsanleitung Dendrometer Chinesisch.pdf|Chinese]]<br />
*[[:File:Gebrauchsanleitung Dendrometer Dänisch.pdf|Danish]]<br />
*[[:File:Gebrauchsanleitung Dendrometer Französisch.pdf|French]]<br />
*[[:File:Gebrauchsanleitung Dendrometer Griechisch.pdf|Greek]]<br />
*[[:File:Gebrauchsanleitung Dendrometer Indonesisch.pdf|Bahasa Indonesia]]<br />
*[[:File:Gebrauchsanleitung Dendrometer Italienisch.pdf|Italian]]<br />
*[[:File:Gebrauchsanleitung Dendrometer Japanisch.pdf|Japanese]]<br />
*[[:File:Gebrauchsanleitung Dendrometer Koreanisch.pdf|Korean]]<br />
*[[:File:Gebrauchsanleitung Dendrometer Rumänisch.pdf|Romanian]]<br />
*[[:File:Gebrauchsanleitung Dendrometer Russisch.pdf|Russian]]<br />
*[[:File:Gebrauchsanleitung Dendrometer Slowakisch.pdf|Slovakian]]<br />
*[[:File:Gebrauchsanleitung Dendrometer Spanisch.pdf|Spanish]]<br />
*[[:File:Gebrauchsanleitung Dendrometer Türkisch.pdf|Turkish]]<br />
*[[:File:Gebrauchsanleitung Dendrometer Ungarisch.pdf|Hungarian]]<br />
<br />
==References==<br />
<references/><br />
<br />
<br />
<br />
<br />
<br />
[[Category:Measurement devices]]</div>Fehrmannhttp://wiki.awf.forst.uni-goettingen.de/wiki/index.php/Bitterlich_samplingBitterlich sampling2022-12-12T07:57:50Z<p>Fehrmann: </p>
<hr />
<div>{{Ficontent}} {{video|link=https://youtu.be/wCbrwpTAYxI}}<br />
__TOC__<br />
[[File:4.4.1-fig59.png|right|thumb|300px|'''Figure 1''' Illustration of inclusion proportional to size (basal area) in Bitterlich sampling (Kleinn 2007<ref name="kleinn2007">Kleinn, C. 2007. Lecture Notes for the Teaching Module Forest Inventory. Department of Forest Inventory and Remote Sensing. Faculty of Forest Science and Forest Ecology, Georg-August-Universität Göttingen. 164 S.</ref>).]]<br />
Angle count sampling was developed by Walter Bitterlich (1947/48), an Austrian forester (Bitterlich 1947 <ref name="Bitterlich1947"> Bitterlich, W. 1947. Die Winkelzählmessung. Allgemeine Forst- und Holzwirtschaftliche Zeitung 58:94ff </ref>). It is some times also referred to point sampling, horizontal point sampling, variable plot sampling, angle count technique, prism cruising, angle gauge sampling, and simply Bitterlich sampling. <br />
<br />
The idea to employ [[Fixed area plots#Nested sub-plots|nested sub-plots]] was introduced because we wished to have a balanced number of [[Tree Definition|trees]] in all dimension classes; that is, we wanted to assign a higher probability of selection to the larger trees of which there are usually less in a [[Forest Definition|stand]] (see also [[sampling with unequal selection probabilities]]).<br />
<br />
For nested sub-plots, we defined a number of fixed plot areas in which the different dimension classes are observed. One may now further pursue that idea and develop a plot design in which the [[inclusion probability]] is strictly proportional to dimension. That is each tree has its particular and own [[Fixed_area_plots#The_plot_expansion_factor|plot size]] (and therefore also its particular [[inclusion probability#inclusion zone|inclusion zone]]). This is exactly what Bitterlich sampling does: from a selected [[sample point]], the neighboring trees are selected strictly proportional to their [[basal area]].<br />
<br />
While this sounds complicated, the technique itself is very simple; the only device one needs is one that produced a defined opening angle. That can be a [[dendrometer]], a [[relascope]], a [[wedge prism]] or simply your own thumb. While standing on the sample point and aiming over e.g. the thumb at the stretched out arm to the [[Diameter at breast height|''dbh''s]] of the surrounding trees; you sweep around 360° and ''count'' all trees that appear larger than your thumb. It is obvious then, that larger trees have a larger probability of being taken as sample tree. From this ''counting alone'' you can produce then an estimate of basal area per hectare. <br />
<br />
The only additional information that you need is the “calibration factor” of your measurement device, as, obviously, the number of trees counted depends on the opening angle which is produced by the instrument (in this case by your arm length and thumb width).<br />
<br />
This calibration factor is also called ''basal area factor''. Figure 1 illustrates that approach. Around each tree the individual inclusion zone is drawn. Sample points that fall into this inclusion zone will lead to an inclusion of this particular tree as sample tree. We may also imagine that these specific circular inclusion zones are concentrically placed around the sample point; then we may view them as a set of [[Fixed_area_plots#Nested_sub-plots|nested sample sub-plots]], one for each sample tree.<br />
<br><br />
<br />
==The principle of Bitterlich sampling==<br />
<br />
[[File:4.4.2-fig60.png|right|thumb|300px|'''Figure 2''' The principle of Bitterlich sampling. For each tree, there are the three depicted situations (after Kramer & Akca 1995<ref name="kramer_akca1995">Kramer H. and A. Akca. 1995. Leitfaden zur Waldmesslehre. 3rd edition. J.D. Sauerländers Verlag, Frankfurt. 266p.</ref>).]]<br />
[[File:4.4.2-fig61.png|right|thumb|300px|'''Figure 3''' Illustration of calculation of the radius of the virtual circular sub-plot for a tree with diameter <math>d_i</math> (Kleinn 2007<ref name="kleinn2007">Kleinn, C. 2007. Lecture Notes for the Teaching Module Forest Inventory. Department of Forest Inventory and Remote Sensing. Faculty of Forest Science and Forest Ecology, Georg-August-Universität Göttingen. 164 S.</ref>).]]<br />
<br />
[[File:dendrometer.jpg|thumb|A [[dendrometer]] is a simple to use instrument that can be used for Bitterlich sampling.]]<br />
[[File:Wedge prism1.jpg|thumb|Using a [[Wedge prism]] for Bitterlich sampling.]]<br />
In a first step of developing this technique we need to calculate the size of these tree‑specific virtual sample plots. For each tree it holds the statement that it is only counted (or included) if it appears at 1.3 m height larger than the opening angle of the measurement device. For each tree there are exactly three different situations as depicted in Figure 2: (1) either the tree is larger than the opening angle; then the tree is inside the virtual nested sub-plot and is counted; or (2) the tree is smaller than the opening angle (that is: completely covered by the thumb); that is a non-tally tree which is outside the virtual plot; or (3) the tree is exactly covered by the opening angle; then, the tree is exactly on the perimeter line of the virtual circular plot and the distance from the sample point to the center of that tree is (for practical purposes sufficiently good approximation to) the radius of the virtual circular plot. We need to find this radius, because the virtual plot will allow us to calculate the tree-specific expansion factor that we need to expand the per-plot observation to the per-hectare observation.<br />
<br />
The opening angle can be produced by means of a stick of length c and a horizontal panel of width 1 cm at the end (analogous to arm and thumb, for example). For the ''virtual circle plot'', for which this angle covers exactly a tree of ''dbh'' <math>d_i</math> (Figure 2) as a borderline tree, we obtain (approximately) plot radius <math>r</math> and plot area <math>F_i</math> along: <br />
<br />
:<math>\frac{d_i}{r_i}=\frac{1}{c}</math><br />
<br />
and then<br />
<br />
:<math>r=cd_i\,</math><br />
<br />
:<math>F_i={\pi}r^2={\pi}c^2{d_i}^2\,</math><br />
<br />
This is then the virtual plot in which all trees with exactly the ''dbh'' <math>d_i</math> are tallied.<br />
<br />
This corresponds exactly to the procedure with nested circular plots where also only a defined size class was observed in each nested sub-plot. If we count <math>n_i</math> trees with exactly the diameter <math>d_i</math> (or in [[diameter class]] <math>d_i</math> in case that we work with diameter classes) at a sample point, then these trees have the basal area<br />
<br />
:<math>{n_i}{g_i}=n_i\frac{\pi}{4}{d_i}^2\,</math><br />
<br />
Now, let’s expand this per-plot observation to a per-hectare figure (referring to trees with ''dbh'' <math>d_i</math> only!): the expansion factor is <math>10000/F_i</math>, as usual, where <math>F_i</math> is the area of the virtual plot. The basal area per hectare for the trees of diameter <math>d_i</math> is then<br />
<br />
:<math>G_{iHa}=\frac{10000}{F_i}n_i\frac{\pi}{4}{d_i}^2=\frac{10000}{\pi{c}^2{d_i}^2}n_i\frac{\pi}{4}{d_i}^2=\frac{2500}{c^2}n_i\,</math> <br />
<br />
Subsequently, this is done for all ''dbh''-”classes” <math>d_i</math> and summed up to produce the estimate for the total basal area per hectare comprising all <math>d</math> classes:<br />
<br />
:<math>G_{iHa}=\sum_{i}G_i=\frac{2500}{c^2}\sum_{i}n_i=\frac{2500}{c^2}N\,</math> <br />
<br />
where <math>G_{Ha}</math> = basal area per hectare of all trees. The only variable in that equation is<br />
<br />
:<math>N=\sum_{}n_i\,</math><br />
<br />
the number of observed/counted trees on the plot (over all diameter classes). This number of counted trees is multiplied with the factor <br />
<br />
:<math>k=\frac{2500}{c^2}\,</math><br />
<br />
which converts eventually the mere count into a basal-area-per-hectare value. This factor is called '''basal area factor''' and commonly denoted with <math>k</math>. It is a factor that depends exclusively on the opening angle which is used.<br />
<br />
By simply counting the <math>N</math> trees that are wider than our opening angle we can therefore produce an observation of basal area per hectare from a sample point: <br />
<br />
:<math>G_{Ha}=k*N\,</math><br />
<br />
It must be observed, however, that only basal area per hectare can be observed as simple as that. For all other variable that may be of interest (such as mean diameter or number of stems), in addition the ''dbhs'' of all trees need to be measured. That is explained further below. <br />
<br />
Basal area factors can be different. Table 1 gives the lengths <math>c</math> of this stick for basal area factors of 1,2,3 and 4 when there is a panel of 1 cm at the end of a that stick. Following the above formula, the length of that stick is calculated by <br />
<br />
:<math>c=\frac{50}{\sqrt{k}}\,</math><br />
<br />
<blockquote><br />
{|<br />
|'''Table 1.''' Length <math>c</math> of the stick with 1cm for different values of the basal area factor <math>k</math><br />
{| class="wikitable"<br />
|-<br />
! Basal area factor (<math>m^2/ha</math>)<br />
! width="100" |c-value<br />
|- <br />
|align="center"|<math>k=1m^2\,</math> <br />
|align="right"|<math>50\,</math> <br />
|-<br />
|align="center"|<math>k=2m^2\,</math><br />
|align="right"|<math>35.4\,</math> <br />
|-<br />
|align="center"|<math>k=3m^2\,</math><br />
|align="right"|<math>28.9\,</math> <br />
|-<br />
|align="center"|<math>k=4m^2\,</math><br />
|align="right"|<math>25.0\,</math> <br />
|}<br />
|}<br />
</blockquote><br />
<br />
<br />
===Exact solution===<br />
<br />
[[File:4.4.2-fig62.png|right|thumb|300px|'''Figure 4''' Critical angle principle (Kleinn 2007<ref name="kleinn2007">Kleinn, C. 2007. Lecture Notes for the Teaching Module Forest Inventory. Department of Forest Inventory and Remote Sensing. Faculty of Forest Science and Forest Ecology, Georg-August-Universität Göttingen. 164 S.</ref>).]]<br />
The principle presented above is, of course, only approximately correct, because the distance between the points at the tree where the viewing beams touch the tree is not exactly the tree diameter. For a geometrically correct calculation we would need to take that into account. The true distance between these two points is always shorter than <math>d_i</math>. Figure 4 illustrates that and is gives the basis for the following calculations.<br />
<br />
From Figure 4 we have<br />
<br />
:<math>\mbox{sin }\frac{\alpha}{2}=\frac{\frac{d_i}{2}}{r_i}\,</math><br />
<br />
and<br />
<br />
:<math>r_i=\frac{d_i}{2*\mbox{sin }\frac{\alpha}{2}}.\,</math> <br />
<br />
The area of the virtual circular plot for the tree with diameter <math>d_i</math> is<br />
<br />
:<math>A=\pi*r^2=\pi\frac{{d_i}^2}{4\left(\mbox{sin }\frac{\alpha}{2}\right)^2}\,</math><br />
<br />
and the basal area per hectare results then from <br />
<br />
:<math>G_i=10000*\frac{n_i*\frac{\pi}{4}*{d_i}^2}{\pi* \frac{{d_i}^2}{4*\left(\mbox{sin }\frac{\alpha}{2}\right)^2}}=10000*n_i*\left(\mbox{sin }\frac{\alpha}{2}\right)^2.\,</math><br />
<br />
Basal area per hectare (of all trees) is then<br />
<br />
:<math>G_{ha}=1000*\left(\sin\frac{\alpha}{2}\right)^2*\sum_{i=1}^k n_i,\,</math><br />
<br />
where,<br />
<br />
:<math>10000*\left(\sin\frac{\alpha}{2}\right)^2=k.\,</math><br />
<br />
Therefore, <br />
<br />
:<math>\sin\frac{\alpha}{2}=\frac{\sqrt{k}}{100}.\,</math><br />
<br />
This is the mathematically correct derivation. However, the differences to the approximate solution as given above are so small that they can be accepted. The following gives the results of the simplified approximate calculation of the length <math>c</math> and the correct value <math>c_c</math>:<br />
<br />
:{|<br />
|-<br />
|<math>k=1m^2\,</math><br />
|<math>c_c=49.997\,</math><br />
|instead of<br />
|<math>c=50\,</math><br />
|-<br />
|<math>k=2m^2\,</math><br />
|<math>c_c=35.352\,</math><br />
|instead of<br />
|<math>c=35.355\,</math><br />
|-<br />
|<math>k=4m^2\,</math><br />
|<math>c_c=24.995\,</math><br />
|instead of<br />
|<math>c=25\,</math><br />
|}<br />
<br />
Anyway, for all the calculations, we need to make the assumption that the tree’s cross section is a perfect circle; which it is not; so that there are more sources of geometrical inaccuracies than the one described here.<br />
<br />
==Choice of basal area factor==<br />
<br />
The choice of the ''basal area factor'' (<math>k</math>) in Bitterlich sampling is equivalent to the choice of plot size for fixed radius plot sampling. The criteria to be considered in the choice of basal area factor are visibility in the stand and average number of sample trees desired at a sample point. A decrease in the basal area factors increases the average number of sample trees tallied in. But that means also that trees at a farther distance to the sample point will be tally-trees. This will work smoothly only when visibility is accordingly<br />
<br />
In high density stands higher ''BAF'' values such as <math>4 m^2/ha</math> are used. In low density stands with good visibility, low ''BAF'' values such as <math>1 m^2/ha</math> is normally used. In temperate and boreal regions basal area factors of 1 or 2 are common; in dense tropical rain forest values of <math>9 m^2/ha</math> may be appropriate. <br />
<br />
{{info<br />
|message=Note:<br />
|text=The choice of a basal area factor refers to the entire sampling study (or in case of [[stratified sampling]] to the whole stratum). From a statistical point of view it is in principle not allowed to choose a different factor at each sampling location. Choosing a different plot size or counting factor depending on local density, would implicitly define multiple populations. This would mean the [[inclusion probability]] of a single tree would not be fixed anymore, but depending on the the local density around a sampling location. Such approaches are a matter of adaptive designs that might be notorious difficult in regard to unbiased estimators! <br />
}}<br />
<br />
Of course, the target precision of a sampling study is always also an issue. With smaller values of <math>k</math> there will be more trees per sample plot (which comes together with somewhat higher cost) and with the same sample size, one will produce a higher precision than for larger values of <math>k</math>.<br />
<br />
==Bitterlich sampling on sloped terrain==<br />
<br />
As in sampling with [[fixed area plots]], all area measures in angle count sampling are assumed to be in the map plane. When applying Bitterlich sampling in sloped terrain we need to take the slope into account; either by using an instrument that does the correction automatically, or by introducing the correction factor into the analysis if the slope is not automatically corrected (which is the case, for example, if using a stick with a panel, or the thumb); this [[slope correction]] is then to be applied from slopes of 10% and more. <br />
The correction factor is<br />
<br />
:<math>\frac{1}{\cos\alpha}\,</math><br />
<br />
so that the basal area per hectare observation from one Bitterlich sample point is <br />
<br />
:<math>G_i=k*N_i\frac{1}{\cos\alpha_i}\,</math><br />
<br />
where: <br />
<br />
:{|<br />
|-<br />
|<math>G_i=\,</math><br />
|corrected basal area (<math>m^2/ha</math>) at the <math>i^{th}</math> sampling point<br />
|-<br />
|<math>N_i=\,</math><br />
|number of trees tallied at the sampling point and<br />
|-<br />
|<math>\alpha_i=\,</math><br />
|angle of slope at the <math>i^{th}</math> sampling point.<br />
|}<br />
<br />
==Inclusion probabilities and estimators for Bitterlich sampling==<br />
<br />
For the inclusion zone approach where for each tree an inclusion zone is defined, see the corresponding article [[ Approaches to populations of sample plots|Infinite population approach]]. If used,the [[inclusion probability]] is then proportional to the size of this inclusion zone – which actually defines the probability that the correspondent tree is included in a sample.<br />
<br />
We saw, for example, that angle count sampling ([[Bitterlich sampling]]) selects the trees with a probability proportional to their [[basal area]] and we emphasized that this fact makes [[Bitterlich sampling]] so efficient for basal area estimation. In contrast, point to tree [[Distance based plots|distance sampling]], or k-tree sampling, has inclusion zones that do not depend on any individual tree characteristic but only on the spatial arrangement of the neighboring trees; therefore, point-to tree distance sampling is not particularly precise for any tree characteristic.<br />
<br />
In [[Bitterlich sampling]], the selection probability of a particular tree ''i'' results from the inclusion zone ''F<sub>i</sub>'' and the size of the reference area, for example the hectare<br />
<br />
:<math>\pi_i = \frac {F_i}{10000}</math><br />
<br />
with the Horvitz-Thompson estimator, we have the total <br />
<br />
:<math>\hat \tau = \sum_{i=1}^m \frac {y_i}{\pi_i}</math><br />
<br />
for any tree attribute <math>y_i</math>. Applied to estimating basal area <math>y_i = g_i = \frac {\pi}{4} d_i^2</math> and its per hectare estimation, we have <br />
<br />
:<math>\hat \tau = \sum_{i=1}^m \frac {y_i}{\pi_i} = \sum_{i=1}^m \cfrac {\cfrac {\pi}{4} d_i^2}{\cfrac {F_i}{10000}}</math><br />
<br />
and with <math> F_i = \pi r_i^2 = \pi c^2 \, d_i^2</math>, we have the same as [[Bitterlich sampling]]<br />
<br />
:<math>\hat \tau = \sum_{i=1}^m \frac {y_i}{\pi_i} = \sum_{i=1}^m \cfrac {\cfrac {\pi}{4} d_i^2}{\cfrac {\pi c^2 \, d_i^2}{10000}} = \frac {2500 \pi}{\pi c^2} \sum_{i=1}^m \frac {d_i^2}{d_i^2} = \frac {2500}{c^2} m</math><br />
<br />
which is the estimated basal area per hectare from one sample point where '''m''' trees were tallied. The factor 2500/c² is the ''basal area factor''.<br />
<br />
==Estimation of number of stems==<br />
<br />
Angle count sampling allows a very fast estimation of basal area per hectare by simply counting trees. This procedure alone, however, does '''not''' allow making estimations for other variables. <br />
If we are interested in estimating number of stems per hectare, for example, we need to measure the diameters of all counted trees in order to be able to explicitly calculate their virtual circle plots and to make a correct expansion of their per-plot values to the corresponding per-hectare values. <br />
<br />
Each diameter has its own expansion factor, namely<br />
<br />
:<math>EF_i=\frac{10000}{F_i},\,</math><br />
<br />
where <math>F_i</math> is the area of the corresponding ''“virtual plot”''. <br />
For example for <math>d=30cm</math> (and <math>k=2</math>, that is, <math>c=35.52</math>), we get the following expansion factor:<br />
<br />
:<math>EF_i=\frac{10000}{F_i}=\frac{10000}{\pi{c^2}{d_i}^2}=\frac{10000}{\pi{35.35}^2{0.3}^2}=28.29\,</math><br />
<br />
That tree with diameter <math>d_i=30cm</math>, counted from the particular sample point, represents 28.29 trees per hectare with that diameter class; and, because the basal area factor is <math>k=2 m^2/ha</math>, this tree does also represent <math>2m^2</math> of basal area per hectare. <br />
<br />
The calculation of <math>EF_i</math> needs to be done for all diameters found on a sample point. Each diameter represents a different number of trees per hectare. At the end, to determine the total number of trees per hectare, these values need to be added.<br />
<br />
In some textbooks there is another (equivalent – but somewhat more complicated and less intuitive) derivation of number of stems per hectare: one determines how many trees per hectare are represented by one counted tree: <br />
<br />
For <math>k=2</math> one single tree represents <math>2m^2</math> of basal area per hectare. The basal area of one tree of, for example, <math>d=30cm</math> is<br />
<br />
:<math>g={\pi}r^2=0.0707m^2\,</math><br />
<br />
It follows that one counted tree “stands for”<br />
<br />
:<math>\frac{2.0}{0.0707}=28.29</math> trees/ha<br />
<br />
in this diameter class 30cm. Obviously, the result is the same as for the more consistent expansion factor approach.<br />
<br />
If we are interested in calculating the mean diameter from a Bitterlich sample, we can not simply take the mean diameter of the sample trees (as we do it for fixed area plot sampling) but we first need to derive the diameter distribution by means of the diameter-wise expansion and then calculate the mean diameter from that distribution.<br />
<br />
===Example===<br />
<br />
Calculation of number of stems per hectare from one Bitterlich sample point.<br />
<br />
In Bitterlich sampling, the estimation of number of stems can not be done with a simple extrapolation of the counted number of trees, but we need to take into account the different selection probabilities, that is, the different sizes of the virtual nested sample plots (which is equivalent to the different size of the selection areas).<br />
<br />
Each sampled tree “represents” a different number of trees in its ''dbh'' class ‑ depending on its ''dbh''. A small tree, if sampled, represents a much larger number of small trees than a large tree if sampled. The calculation is illustrated in Table 1 for one Bitterlich sample with <math>baf=2</math>. 10 trees were observed with their respective ''dbh'' measured. ''dbh'' and ''baf'' allow calculating the diameter-specific virtual nested circular plot area from<br />
<br />
:<math>F_i={\pi}c^2d_i^2\,</math><br />
<br />
and, therefore, also the ''dbh''-specific expansion factor. Each tree represents a certain number of trees per hectare and the sum of all those figures is the estimated number of trees per hectare from this one Bitterlich sample. If ''n'' sample plots were taken, then for each one, this calculation needs to be done. The standard error of the estimation of number of stems per hectare results then from the results of these ''n'' estimations per plot.<br />
<br />
<blockquote><br />
{| <br />
|'''Table 1.''' Illustration of estimation of number of trees per hectare from one point-relascope sample with <math>baf=2</math>.<br />
{| class="wikitable"<br />
! dbh of sample tree<br />
! Area of virtual circular<br>nested sub-plot<br />
! Expansion factor from plot<br>to per-hectare values<br />
! “represented” number of<br>stems per hectare<br />
|-<br />
|align="right"|10<br />
|align="right"|39.27<br />
|align="right"|254.65<br />
|align="right"|254.65<br />
|-<br />
|align="right"|12<br />
|align="right"|56.55<br />
|align="right"|176.84<br />
|align="right"|176.84<br />
|-<br />
|align="right"|15<br />
|align="right"|88.36<br />
|align="right"|113.18<br />
|align="right"|113.18<br />
|-<br />
|align="right"|16<br />
|align="right"|100.53<br />
|align="right"|99.47<br />
|align="right"|99.47<br />
|-<br />
|align="right"|18<br />
|align="right"|127.23<br />
|align="right"|78.60<br />
|align="right"|78.60<br />
|-<br />
|align="right"|21<br />
|align="right"|173.18<br />
|align="right"|57.74<br />
|align="right"|57.74<br />
|-<br />
|align="right"|23<br />
|align="right"|207.74<br />
|align="right"|48.14<br />
|align="right"|48.14<br />
|-<br />
|align="right"|36<br />
|align="right"|508.94<br />
|align="right"|19.65<br />
|align="right"|19.65<br />
|-<br />
|align="right"|42<br />
|align="right"|692.72<br />
|align="right"|14.44<br />
|align="right"|14.44<br />
|-<br />
|align="right"|61<br />
|align="right"|1461.23<br />
|align="right"|6.84<br />
|align="right"|6.84<br />
|-<br />
|'''Total'''<br />
|<br />
|<br />
|align="right"|'''869.54''' <br />
|}<br />
|}<br />
</blockquote><br />
<br />
In this example, the expansion factor is equal to the represented number of trees per hectare because the calculation has been done for each individual tree and not for dbh-classes in which there are more than one tree.<br />
<br />
==See also==<br />
[https://www.youtube.com/watch?v=dMk9jNF-kHo Video:Walter Bitterlich, An invention goes around the world]<br />
<br />
<br />
[https://www.youtube.com/watch?v=c1NyzVuV-QU&feature=youtu.be Video:Walter Bitterlich, Eine Erfindung geht um die Welt]<br />
<br />
<br />
==References==<br />
<references/><br />
<br />
{{SEO<br />
|keywords=point sampling,bitterlich sampling,angle count sampling,horizontal point sampling,variable plot sampling,angle count technique,prism cruising,angle gauge sampling,basal area factor,stem number estimation,plot design,forest inventory,<br />
|descrip=Angle count sampling was developed by Walter Bitterlich, an Austrian forester.<br />
}}<br />
<br />
<br />
[[Category:Plot design]]<br />
[[Category:article of the month]]</div>Fehrmannhttp://wiki.awf.forst.uni-goettingen.de/wiki/index.php/Ratio_estimatorRatio estimator2021-04-14T13:04:44Z<p>Fehrmann: /* Efficiency */</p>
<hr />
<div>{{Ficontent}}<br />
[[File:5.6.1-fig93.png|right|thumb|300px|'''Figure 1''' Example of a population of 30 unequally sized strip plots; here, the ratio estimator may be applied for estimation using plot size as co-variable (DeVries 1986<ref>de Vries, P.G., 1986. Sampling Theory for Forest Inventory. A Teach-Yourself Course. Springer. 399 p.</ref>).]]<br />
<br />
There are situations in forest inventory sampling in which the value of the target variable is known (or suspected) to be highly correlated to an other variable (called co-variable or ancillary variable). We know from the discussion on [[Plot_design#Spatial_autocorrelation|spatial autocorrelation]] and the optimization of [[Cluster sampling|cluster plot design]] that correlation between two variables means essentially that the one variable does yet contain a certain amount of information about the other variable. The higher the correlation the better can the value of the second variable be predicted when the value of the first is known (and vice versa). Therefore, if such a co-variable is there on the plot, it would make sense to also observe it and utilize the correlation to the target variable to eventually improve the precision of estimating the target variable. This is exactly the situation where the ratio estimator is applied.<br />
<br />
A typical and basic example is that of sample plots of unequal size, as depicted, for example in Figure 1 where strips of different length constitute the [[population]] of samples. Another example would be clusters of unequal size: while [[cluster sampling]] only deals with the case of clusters of equal size, the ratio estimator allows to take into account the differently sized [[sample plot]]s. Here, obviously, the size of the sample plots is the co-variable that needs to be determined for each sample plot. That a high positive correlation is to be expected between most forestry-relevant attributes and plot size should be obvious: the larger the plot the more basal area, number of stems, volume etc. is expected to be present.<br />
<br />
In fact, the ratio estimator does not introduce a new sampling technique: we use here the example of [[simple random sampling]] (as [[:Category:Sampling design|sampling design]]). The ratio estimator introduces two new aspects in respect to:<br />
<br />
# [[plot design]]: observation of an ancillary variable on each plot in addition to the observation of the target variable.<br />
# [[estimation design]]: the ratio estimator integrates the ancillary variable into the estimator. <br />
<br />
{{info<br />
|message=Side-comment:<br />
|text=For those who have some knowledge or interest (or both) on the design of experiments and the linear statistical models used there (analysis of variance) it might be interesting to note that also in the field of experimental design there is a technique in which a co-variable is observed on each experimental plot in order to allow a more precise estimation of the effects of treatments: this technique is called ''analysis of co-variance''.<br />
}}<br />
<br />
<br />
The ratio estimator is called ratio estimator because, what we actually estimate from the sample plot is a ratio: for example, number of stems per hectare. The ancillary variable (“area” in this case) appears always in the denominator of the ratio. We denote the variable of interest as usual with <math>y_i</math> and the ancillary variable as <math>x_i</math>; some more on notation is given in [[double sampling]]. <br />
<br />
{{info<br />
|message=in simple words:<br />
|text=The ratio estimator involves the ratio of two variates. Imagine there are two characteristics (say x and y) we can observe for each sampled element and we have in addition information about the total of x, then the total of y can be estimated by multiplying the known total of x by a ratio of both variables. This ratio can be estimated from the means of both variables that are observed for the sampled elements. <br />
}}<br />
<br />
===Notation===<br />
<br />
<br />
:{| <br />
|<math>R\,</math>||Parametric ratio, that is the true ratio present in the population;<br />
|- <br />
|<math>\mu_y\,</math>||Parametric mean of the target variable <math>Y</math>;<br />
|- <br />
|<math>\mu_x\,</math>||Parametric mean of the ancillary variable <math>X</math>;<br />
|- <br />
|<math>\tau_x\,</math>||Parametric total of ancillary variable <math>X</math>;<br />
|- <br />
|<math>\rho\,</math>||Parametric Pearson coefficient of correlation between <math>Y</math> and <math>X</math>;<br />
|-<br />
|<math>\sigma_y\,</math>||Parametric standard deviation of the target variable <math>Y</math>;<br />
|-<br />
|<math>\sigma_x\,</math>||Parametric standard deviation of ancillary variable <math>X</math>;<br />
|-<br />
|<math>n\,</math>||Sample size;<br />
|-<br />
|<math>r\,</math>||Estimated ratio;<br />
<br />
|-<br />
|<math>y_i\,</math>||Value of the target variable observed on the <math>i^{th}</math> sampled element (plot);<br />
|-<br />
|<math>x_i\,</math>||Ancillary variable, e.g. area, for <math>i^{th}</math> plot;<br />
|-<br />
|<math>\bar y\,</math>||Estimated mean of target variable;<br />
|-<br />
|<math>\bar x\,</math>||Estimated mean of ancillary variable;<br />
|-<br />
|<math>\hat{\tau}_y\,</math>||Estimated total of target variable;<br />
|-<br />
|<math>\hat{\rho}\,</math>||Estimated coefficient of correlation of <math>Y</math> and <math>X</math>;<br />
|-<br />
|<math>{s_y}^2\,</math>||Estimated variance of target variable;<br />
|-<br />
|<math>{s_x}^2\,</math>||Estimated variance of ancillary variable.<br />
|}<br />
<br />
<br />
With the notation for the ratio estimator it should, above all, be observed, that the letters <math>r</math> and <math>R</math> are used here for ratio – and not for the [[correlation coefficient]]. That causes some times confusions. We use the Greek letter <math>\rho</math> (rho) for the parametric correlation coefficient and the usual notation for the estimation using the “hat” sign <math>\hat{\rho}</math>.<br />
<br />
===Estimators===<br />
<br />
In the first place, what we estimate with the ration estimator (as the name yet suggests) is the ratio <math>R</math> which is target variable per ancillary variable, for example: number of trees per hectare. From this, we can then also estimate the total of the target variable.<br />
<br />
The parametric ratio is <br />
<br />
:<math>R=\frac{\mu_y}{\mu_x}\,</math><br />
<br />
where this is estimated from the sample as <br />
<br />
:<math>r=\frac{\bar{y}}{\bar{x}}=\frac{\sum_{i=1}^n y_i}{\sum_{i=1}^n x_i}.\,</math><br />
<br />
It is very important to note here that we calculate a '''ratio of means''' and NOT a mean of ratios! The ratio is calculated as the mean of the target variable divided by the mean of the ancillary variable; which is equivalent to calculating the sum of the target variable divided by the sum of the ancillary variable. It would be erroneous to calculate for each sample a ratio between target variable and ancillary variable and compute the mean over all samples! For that situation (the mean of ratios) there are other estimators which are more complex!<br />
<br />
The estimated variance of the estimated ratio <math>r</math> is <br />
<br />
:<math>\hat{var}(r)=\frac{N-n}{N}\frac{1}{n}\frac{1}{{\mu_x}^2}\frac{\sum_{i=1}^n\left(y_i-rx_i\right)^2}{n-1}.\,</math> <br />
<br />
An the estimated total derives from <br />
<br />
:<math>{\tau}_y=r{\tau}_x\,</math><br />
<br />
with an estimated error variance of the total of <br />
<br />
:<math>\hat{var}(\hat{\tau}_y)={\tau_x}^2\hat{var}(r),\,</math><br />
<br />
as usual.<br />
<br />
Given the estimated ratio <math>r</math> the estimated mean derives from<br />
<br />
:<math>\bar{y}_r=r\mu_x\,</math><br />
<br />
[[File:5.4.2-fig85.png|right|thumb|300px|'''Figure 2''' The left graph gives the population of interest. The right graph depicts the adaptive cluster sampling process, where red squares are the initially selected random samples (from Thompson 1992<ref name="thompson1992">Thompson SK. 1992. Sampling. John Wiley & Sons. 343 p.</ref>).]]<br />
<br />
We see here: for the estimation of the mean we need to know the true value of the ancillary variable. In cases such as the one in Figure 2 this is easily determined, because the total of the ancillary variable is simply the total area. However, in other situations where, for example leaf area is taken as ancillary variable to estimation leaf biomass, this is extremely difficult. However, there are corresponding [[:Category:Sampling design|sampling techniques]] that start with a first sampling phase '''estimating''' the ancillary variable before in a second phase the target variable is estimated; these techniques are called [[double sampling]] or [[two phase sampling]] with the ratio estimator.<br />
<br />
The estimated mean carries an estimated variance of the mean of <br />
<br />
:<math>\hat{var}(\bar{y}_r)={\mu_x}^2\hat{var}(r)=\frac{N-n}{N}\frac{1}{n}\left\{{s_y}^2+r^2{s_x}^2-2r\hat{\rho}{s_x}{s_y}\right\}\,</math><br />
<br />
This latter estimator of the error variance of the estimated mean illustrates some basic characteristics of the ratio estimator very clearly:<br />
<br />
If all values of the ancillary variable are the same (for example, when using plot area as ancillary variable in fixed area plot sampling), then <math>{s_x}^2</math> is zero and the estimator is identical to the error variance of the mean estimator in [[simple random sampling]] where the only source of [[variability]] that enters is the estimated variance of the target variable <math>{s_y}^2</math>. Also, we see in the above term that we need to make <math>r\hat{\rho}{s_x}{s_y}</math> large so that a large term is subtracted from the variance terms in parenthesis; the only way to influence here is an intelligent choice of the ancillary variable. The ancillary variable should highly positively correlate with the target variable (there is not much we can do about the target variable except for changing the plot type and size; that will change the variance of the population – but still, because of the expected high correlation with the ancillary variable it will also affect the estimation of <math>r</math> and <math>{s_x}^2</math> and, in addition, the error variance of <math>{s_y}^2</math> the target variable appears with positive and negative sign in the above estimator. At the end, it is difficult to predict how a change in <math>{s_y}^2</math> affects the performance of the ratio estimator.<br />
<br />
==Efficiency==<br />
<br />
In the preceding chapter, we saw in fairly general terms that a high positive correlation between target and ancillary variable will make the variance smaller than the simple random sampling estimator (which would be there, if correlation would be zero). However, by re-arranging the above term, we can more specifically formulate in which situation the ratio estimator is superior to the simple random sampling estimator. As always, the relative efficiency is meaningful only when we deal with parametric values. Based on estimated values, the relative efficiency can be misleading.<br />
<br />
For the ratio estimator we can formulate from the above estimator of the error variance of the estimated mean <br />
<br />
:<math>\rho\ge\frac{R\sigma_x}{2\sigma_y}=\frac{cv(x)}{2cv(y)}\,</math><br />
<br />
where <math>cv()</math> denotes the coefficient of variation either for the target variable <math>Y</math> or the ancillary variable <math>X</math>. <br />
<br />
As an interpretation of this inequality: the ratio estimator is superior to the simple random estimator if the correlation between the two variables exceeds the above expressions.<br />
<br />
:<math>\frac{R\sigma_x}{2\sigma_y}\,</math><br />
<br />
or<br />
<br />
:<math>\frac{cv(x)}{2cv(y)}\,</math><br />
<br />
There may be situations where a variable had been observed that might be used as an ancillary variable, but we are not sure whether to use the ratio estimator or not. Then, one could estimate the [[relative efficiency]]; although the conclusion might be misleading, there are cases with such high positive correlations that application of the ratio estimator is definitively indicated.<br />
<br />
It is maybe interesting to note that - when having an ancillary variable measured and available: there is more then just one option for the estimation design, the ratio estimator and the simple random sampling estimator; and both are unbiased (or, in the case of the ratio estimator, approximately unbiased). Then, of course, one will choose the estimator with the smaller error variance, and a criterion for this is given above with the correlation between target and ancillary variable.<br />
<br />
{{Exercise<br />
|message=Ratio estimator sampling examples<br />
|alttext=Example<br />
|text=Examples for Ratio estimator<br />
}}<br />
<br />
==Characteristics of ratio estimator==<br />
<br />
When the ratio estimator is used, what we implicitly do is assuming a simple linear regression through the origin between target and ancillary variable: <math>\bar{y}=\bar{r}x</math>, where the ratio, <math>r</math>, is the [[slope coefficient]]. The implicit assumption is that the relationship between <math>Y</math> and <math>X</math> goes really through the origin. In the above example with the plot area as ancillary variable, this is certainly true; if plot area is zero, then also the observation of number of stems or basal area is zero. However, if the relationship between <math>Y</math> and <math>X</math> is such that it does not generically pass through the origin, then, there will be an estimator bias.<br />
<br />
When the intercept term of ratio estimator is 0, then, the estimators presented here will be unbiased. In contrary, if the intercept term is not equal to 0, e.g. when ancillary variable is zero but target variable is not, then, the estimator here will be biased for small sample size but approximately unbiased for large sample size. <br />
<br />
==Regression estimator== <br />
<br />
If, as addressed in the chapter above, the regression between <math>Y</math> and <math>X</math> does not go through the origin, then, one may use the so-called regression estimator which explicitly establishes a simple [[linear regression]] between target and ancillary variable. As we assume that, as usual in simple linear regression, the joint mean value <math>\bar{x},\bar{y}</math> lies on the regression line, we may estimate the mean of the target variable from the sample as<br />
<br />
:<math>\bar{y}_L=\bar{y}+b\left(\mu_x-\bar{x}\right)</math><br />
<br />
where <math>b</math> is the estimated regression coefficient and <math>\mu_x</math> is the population mean of ancillary variable which must be known.<br />
<br />
The estimated variance of the estimated mean is <br />
<br />
:<math>\hat{var}(\bar{y}_L)=\frac{N-n}{N}\frac{1}{n}\frac{1}{n-2}\left\{\sum_{i=1}^n(y_i-\bar{y})^2-b^2\sum_{i=1}^n(x_i-\bar{x})^2\right\}\,</math><br />
<br />
where the term in curled parenthesis times the factor <math>1/(n-2)</math> is the mean square error of regression.<br />
<br />
As with the ratio estimator, we assume here a simple linear regression model between the two variables. If this model holds (that is, if the relationship can realistically be modeled like that, then the regression estimator is unbiased. If this is not so, then we introduce an estimator bias of unknown order of magnitude. However, it is said that this bias is particularly relevant for small sample sizes and not so for larger sample sizes.<br />
<br />
==References==<br />
<references/><br />
<br />
{{SEO<br />
|keywords=ratio estimator,ancillary variable,double sampling<br />
|descrip=The ratio estimator is an estimation design that makesuse of an ancillary variable that is correlated to the target variable<br />
}}<br />
<br />
[[Category:Sampling design]]</div>Fehrmannhttp://wiki.awf.forst.uni-goettingen.de/wiki/index.php/Ratio_estimatorRatio estimator2021-04-14T13:03:45Z<p>Fehrmann: /* Efficiency */</p>
<hr />
<div>{{Ficontent}}<br />
[[File:5.6.1-fig93.png|right|thumb|300px|'''Figure 1''' Example of a population of 30 unequally sized strip plots; here, the ratio estimator may be applied for estimation using plot size as co-variable (DeVries 1986<ref>de Vries, P.G., 1986. Sampling Theory for Forest Inventory. A Teach-Yourself Course. Springer. 399 p.</ref>).]]<br />
<br />
There are situations in forest inventory sampling in which the value of the target variable is known (or suspected) to be highly correlated to an other variable (called co-variable or ancillary variable). We know from the discussion on [[Plot_design#Spatial_autocorrelation|spatial autocorrelation]] and the optimization of [[Cluster sampling|cluster plot design]] that correlation between two variables means essentially that the one variable does yet contain a certain amount of information about the other variable. The higher the correlation the better can the value of the second variable be predicted when the value of the first is known (and vice versa). Therefore, if such a co-variable is there on the plot, it would make sense to also observe it and utilize the correlation to the target variable to eventually improve the precision of estimating the target variable. This is exactly the situation where the ratio estimator is applied.<br />
<br />
A typical and basic example is that of sample plots of unequal size, as depicted, for example in Figure 1 where strips of different length constitute the [[population]] of samples. Another example would be clusters of unequal size: while [[cluster sampling]] only deals with the case of clusters of equal size, the ratio estimator allows to take into account the differently sized [[sample plot]]s. Here, obviously, the size of the sample plots is the co-variable that needs to be determined for each sample plot. That a high positive correlation is to be expected between most forestry-relevant attributes and plot size should be obvious: the larger the plot the more basal area, number of stems, volume etc. is expected to be present.<br />
<br />
In fact, the ratio estimator does not introduce a new sampling technique: we use here the example of [[simple random sampling]] (as [[:Category:Sampling design|sampling design]]). The ratio estimator introduces two new aspects in respect to:<br />
<br />
# [[plot design]]: observation of an ancillary variable on each plot in addition to the observation of the target variable.<br />
# [[estimation design]]: the ratio estimator integrates the ancillary variable into the estimator. <br />
<br />
{{info<br />
|message=Side-comment:<br />
|text=For those who have some knowledge or interest (or both) on the design of experiments and the linear statistical models used there (analysis of variance) it might be interesting to note that also in the field of experimental design there is a technique in which a co-variable is observed on each experimental plot in order to allow a more precise estimation of the effects of treatments: this technique is called ''analysis of co-variance''.<br />
}}<br />
<br />
<br />
The ratio estimator is called ratio estimator because, what we actually estimate from the sample plot is a ratio: for example, number of stems per hectare. The ancillary variable (“area” in this case) appears always in the denominator of the ratio. We denote the variable of interest as usual with <math>y_i</math> and the ancillary variable as <math>x_i</math>; some more on notation is given in [[double sampling]]. <br />
<br />
{{info<br />
|message=in simple words:<br />
|text=The ratio estimator involves the ratio of two variates. Imagine there are two characteristics (say x and y) we can observe for each sampled element and we have in addition information about the total of x, then the total of y can be estimated by multiplying the known total of x by a ratio of both variables. This ratio can be estimated from the means of both variables that are observed for the sampled elements. <br />
}}<br />
<br />
===Notation===<br />
<br />
<br />
:{| <br />
|<math>R\,</math>||Parametric ratio, that is the true ratio present in the population;<br />
|- <br />
|<math>\mu_y\,</math>||Parametric mean of the target variable <math>Y</math>;<br />
|- <br />
|<math>\mu_x\,</math>||Parametric mean of the ancillary variable <math>X</math>;<br />
|- <br />
|<math>\tau_x\,</math>||Parametric total of ancillary variable <math>X</math>;<br />
|- <br />
|<math>\rho\,</math>||Parametric Pearson coefficient of correlation between <math>Y</math> and <math>X</math>;<br />
|-<br />
|<math>\sigma_y\,</math>||Parametric standard deviation of the target variable <math>Y</math>;<br />
|-<br />
|<math>\sigma_x\,</math>||Parametric standard deviation of ancillary variable <math>X</math>;<br />
|-<br />
|<math>n\,</math>||Sample size;<br />
|-<br />
|<math>r\,</math>||Estimated ratio;<br />
<br />
|-<br />
|<math>y_i\,</math>||Value of the target variable observed on the <math>i^{th}</math> sampled element (plot);<br />
|-<br />
|<math>x_i\,</math>||Ancillary variable, e.g. area, for <math>i^{th}</math> plot;<br />
|-<br />
|<math>\bar y\,</math>||Estimated mean of target variable;<br />
|-<br />
|<math>\bar x\,</math>||Estimated mean of ancillary variable;<br />
|-<br />
|<math>\hat{\tau}_y\,</math>||Estimated total of target variable;<br />
|-<br />
|<math>\hat{\rho}\,</math>||Estimated coefficient of correlation of <math>Y</math> and <math>X</math>;<br />
|-<br />
|<math>{s_y}^2\,</math>||Estimated variance of target variable;<br />
|-<br />
|<math>{s_x}^2\,</math>||Estimated variance of ancillary variable.<br />
|}<br />
<br />
<br />
With the notation for the ratio estimator it should, above all, be observed, that the letters <math>r</math> and <math>R</math> are used here for ratio – and not for the [[correlation coefficient]]. That causes some times confusions. We use the Greek letter <math>\rho</math> (rho) for the parametric correlation coefficient and the usual notation for the estimation using the “hat” sign <math>\hat{\rho}</math>.<br />
<br />
===Estimators===<br />
<br />
In the first place, what we estimate with the ration estimator (as the name yet suggests) is the ratio <math>R</math> which is target variable per ancillary variable, for example: number of trees per hectare. From this, we can then also estimate the total of the target variable.<br />
<br />
The parametric ratio is <br />
<br />
:<math>R=\frac{\mu_y}{\mu_x}\,</math><br />
<br />
where this is estimated from the sample as <br />
<br />
:<math>r=\frac{\bar{y}}{\bar{x}}=\frac{\sum_{i=1}^n y_i}{\sum_{i=1}^n x_i}.\,</math><br />
<br />
It is very important to note here that we calculate a '''ratio of means''' and NOT a mean of ratios! The ratio is calculated as the mean of the target variable divided by the mean of the ancillary variable; which is equivalent to calculating the sum of the target variable divided by the sum of the ancillary variable. It would be erroneous to calculate for each sample a ratio between target variable and ancillary variable and compute the mean over all samples! For that situation (the mean of ratios) there are other estimators which are more complex!<br />
<br />
The estimated variance of the estimated ratio <math>r</math> is <br />
<br />
:<math>\hat{var}(r)=\frac{N-n}{N}\frac{1}{n}\frac{1}{{\mu_x}^2}\frac{\sum_{i=1}^n\left(y_i-rx_i\right)^2}{n-1}.\,</math> <br />
<br />
An the estimated total derives from <br />
<br />
:<math>{\tau}_y=r{\tau}_x\,</math><br />
<br />
with an estimated error variance of the total of <br />
<br />
:<math>\hat{var}(\hat{\tau}_y)={\tau_x}^2\hat{var}(r),\,</math><br />
<br />
as usual.<br />
<br />
Given the estimated ratio <math>r</math> the estimated mean derives from<br />
<br />
:<math>\bar{y}_r=r\mu_x\,</math><br />
<br />
[[File:5.4.2-fig85.png|right|thumb|300px|'''Figure 2''' The left graph gives the population of interest. The right graph depicts the adaptive cluster sampling process, where red squares are the initially selected random samples (from Thompson 1992<ref name="thompson1992">Thompson SK. 1992. Sampling. John Wiley & Sons. 343 p.</ref>).]]<br />
<br />
We see here: for the estimation of the mean we need to know the true value of the ancillary variable. In cases such as the one in Figure 2 this is easily determined, because the total of the ancillary variable is simply the total area. However, in other situations where, for example leaf area is taken as ancillary variable to estimation leaf biomass, this is extremely difficult. However, there are corresponding [[:Category:Sampling design|sampling techniques]] that start with a first sampling phase '''estimating''' the ancillary variable before in a second phase the target variable is estimated; these techniques are called [[double sampling]] or [[two phase sampling]] with the ratio estimator.<br />
<br />
The estimated mean carries an estimated variance of the mean of <br />
<br />
:<math>\hat{var}(\bar{y}_r)={\mu_x}^2\hat{var}(r)=\frac{N-n}{N}\frac{1}{n}\left\{{s_y}^2+r^2{s_x}^2-2r\hat{\rho}{s_x}{s_y}\right\}\,</math><br />
<br />
This latter estimator of the error variance of the estimated mean illustrates some basic characteristics of the ratio estimator very clearly:<br />
<br />
If all values of the ancillary variable are the same (for example, when using plot area as ancillary variable in fixed area plot sampling), then <math>{s_x}^2</math> is zero and the estimator is identical to the error variance of the mean estimator in [[simple random sampling]] where the only source of [[variability]] that enters is the estimated variance of the target variable <math>{s_y}^2</math>. Also, we see in the above term that we need to make <math>r\hat{\rho}{s_x}{s_y}</math> large so that a large term is subtracted from the variance terms in parenthesis; the only way to influence here is an intelligent choice of the ancillary variable. The ancillary variable should highly positively correlate with the target variable (there is not much we can do about the target variable except for changing the plot type and size; that will change the variance of the population – but still, because of the expected high correlation with the ancillary variable it will also affect the estimation of <math>r</math> and <math>{s_x}^2</math> and, in addition, the error variance of <math>{s_y}^2</math> the target variable appears with positive and negative sign in the above estimator. At the end, it is difficult to predict how a change in <math>{s_y}^2</math> affects the performance of the ratio estimator.<br />
<br />
==Efficiency==<br />
<br />
In the preceding chapter, we saw in fairly general terms that a high positive correlation between target and ancillary variable will make the variance smaller than the simple random sampling estimator (which would be there, if correlation would be zero). However, by re-arranging the above term, we can more specifically formulate in which situation the ratio estimator is superior to the simple random sampling estimator. As always, the relative efficiency is meaningful only when we deal with parametric values. Based on estimated values, the relative efficiency can be misleading.<br />
<br />
For the ratio estimator we can formulate from the above estimator of the error variance of the estimated mean <br />
<br />
:<math>\roh\ge\frac{R\sigma_x}{2\sigma_y}=\frac{cv(x)}{2cv(y)}\,</math><br />
<br />
where <math>cv()</math> denotes the coefficient of variation either for the target variable <math>Y</math> or the ancillary variable <math>X</math>. <br />
<br />
As an interpretation of this inequality: the ratio estimator is superior to the simple random estimator if the correlation between the two variables exceeds the above expressions.<br />
<br />
:<math>\frac{R\sigma_x}{2\sigma_y}\,</math><br />
<br />
or<br />
<br />
:<math>\frac{cv(x)}{2cv(y)}\,</math><br />
<br />
There may be situations where a variable had been observed that might be used as an ancillary variable, but we are not sure whether to use the ratio estimator or not. Then, one could estimate the [[relative efficiency]]; although the conclusion might be misleading, there are cases with such high positive correlations that application of the ratio estimator is definitively indicated.<br />
<br />
It is maybe interesting to note that - when having an ancillary variable measured and available: there is more then just one option for the estimation design, the ratio estimator and the simple random sampling estimator; and both are unbiased (or, in the case of the ratio estimator, approximately unbiased). Then, of course, one will choose the estimator with the smaller error variance, and a criterion for this is given above with the correlation between target and ancillary variable.<br />
<br />
{{Exercise<br />
|message=Ratio estimator sampling examples<br />
|alttext=Example<br />
|text=Examples for Ratio estimator<br />
}}<br />
<br />
==Characteristics of ratio estimator==<br />
<br />
When the ratio estimator is used, what we implicitly do is assuming a simple linear regression through the origin between target and ancillary variable: <math>\bar{y}=\bar{r}x</math>, where the ratio, <math>r</math>, is the [[slope coefficient]]. The implicit assumption is that the relationship between <math>Y</math> and <math>X</math> goes really through the origin. In the above example with the plot area as ancillary variable, this is certainly true; if plot area is zero, then also the observation of number of stems or basal area is zero. However, if the relationship between <math>Y</math> and <math>X</math> is such that it does not generically pass through the origin, then, there will be an estimator bias.<br />
<br />
When the intercept term of ratio estimator is 0, then, the estimators presented here will be unbiased. In contrary, if the intercept term is not equal to 0, e.g. when ancillary variable is zero but target variable is not, then, the estimator here will be biased for small sample size but approximately unbiased for large sample size. <br />
<br />
==Regression estimator== <br />
<br />
If, as addressed in the chapter above, the regression between <math>Y</math> and <math>X</math> does not go through the origin, then, one may use the so-called regression estimator which explicitly establishes a simple [[linear regression]] between target and ancillary variable. As we assume that, as usual in simple linear regression, the joint mean value <math>\bar{x},\bar{y}</math> lies on the regression line, we may estimate the mean of the target variable from the sample as<br />
<br />
:<math>\bar{y}_L=\bar{y}+b\left(\mu_x-\bar{x}\right)</math><br />
<br />
where <math>b</math> is the estimated regression coefficient and <math>\mu_x</math> is the population mean of ancillary variable which must be known.<br />
<br />
The estimated variance of the estimated mean is <br />
<br />
:<math>\hat{var}(\bar{y}_L)=\frac{N-n}{N}\frac{1}{n}\frac{1}{n-2}\left\{\sum_{i=1}^n(y_i-\bar{y})^2-b^2\sum_{i=1}^n(x_i-\bar{x})^2\right\}\,</math><br />
<br />
where the term in curled parenthesis times the factor <math>1/(n-2)</math> is the mean square error of regression.<br />
<br />
As with the ratio estimator, we assume here a simple linear regression model between the two variables. If this model holds (that is, if the relationship can realistically be modeled like that, then the regression estimator is unbiased. If this is not so, then we introduce an estimator bias of unknown order of magnitude. However, it is said that this bias is particularly relevant for small sample sizes and not so for larger sample sizes.<br />
<br />
==References==<br />
<references/><br />
<br />
{{SEO<br />
|keywords=ratio estimator,ancillary variable,double sampling<br />
|descrip=The ratio estimator is an estimation design that makesuse of an ancillary variable that is correlated to the target variable<br />
}}<br />
<br />
[[Category:Sampling design]]</div>Fehrmannhttp://wiki.awf.forst.uni-goettingen.de/wiki/index.php/Ratio_estimatorRatio estimator2021-04-08T13:29:05Z<p>Fehrmann: /* Estimators */</p>
<hr />
<div>{{Ficontent}}<br />
[[File:5.6.1-fig93.png|right|thumb|300px|'''Figure 1''' Example of a population of 30 unequally sized strip plots; here, the ratio estimator may be applied for estimation using plot size as co-variable (DeVries 1986<ref>de Vries, P.G., 1986. Sampling Theory for Forest Inventory. A Teach-Yourself Course. Springer. 399 p.</ref>).]]<br />
<br />
There are situations in forest inventory sampling in which the value of the target variable is known (or suspected) to be highly correlated to an other variable (called co-variable or ancillary variable). We know from the discussion on [[Plot_design#Spatial_autocorrelation|spatial autocorrelation]] and the optimization of [[Cluster sampling|cluster plot design]] that correlation between two variables means essentially that the one variable does yet contain a certain amount of information about the other variable. The higher the correlation the better can the value of the second variable be predicted when the value of the first is known (and vice versa). Therefore, if such a co-variable is there on the plot, it would make sense to also observe it and utilize the correlation to the target variable to eventually improve the precision of estimating the target variable. This is exactly the situation where the ratio estimator is applied.<br />
<br />
A typical and basic example is that of sample plots of unequal size, as depicted, for example in Figure 1 where strips of different length constitute the [[population]] of samples. Another example would be clusters of unequal size: while [[cluster sampling]] only deals with the case of clusters of equal size, the ratio estimator allows to take into account the differently sized [[sample plot]]s. Here, obviously, the size of the sample plots is the co-variable that needs to be determined for each sample plot. That a high positive correlation is to be expected between most forestry-relevant attributes and plot size should be obvious: the larger the plot the more basal area, number of stems, volume etc. is expected to be present.<br />
<br />
In fact, the ratio estimator does not introduce a new sampling technique: we use here the example of [[simple random sampling]] (as [[:Category:Sampling design|sampling design]]). The ratio estimator introduces two new aspects in respect to:<br />
<br />
# [[plot design]]: observation of an ancillary variable on each plot in addition to the observation of the target variable.<br />
# [[estimation design]]: the ratio estimator integrates the ancillary variable into the estimator. <br />
<br />
{{info<br />
|message=Side-comment:<br />
|text=For those who have some knowledge or interest (or both) on the design of experiments and the linear statistical models used there (analysis of variance) it might be interesting to note that also in the field of experimental design there is a technique in which a co-variable is observed on each experimental plot in order to allow a more precise estimation of the effects of treatments: this technique is called ''analysis of co-variance''.<br />
}}<br />
<br />
<br />
The ratio estimator is called ratio estimator because, what we actually estimate from the sample plot is a ratio: for example, number of stems per hectare. The ancillary variable (“area” in this case) appears always in the denominator of the ratio. We denote the variable of interest as usual with <math>y_i</math> and the ancillary variable as <math>x_i</math>; some more on notation is given in [[double sampling]]. <br />
<br />
{{info<br />
|message=in simple words:<br />
|text=The ratio estimator involves the ratio of two variates. Imagine there are two characteristics (say x and y) we can observe for each sampled element and we have in addition information about the total of x, then the total of y can be estimated by multiplying the known total of x by a ratio of both variables. This ratio can be estimated from the means of both variables that are observed for the sampled elements. <br />
}}<br />
<br />
===Notation===<br />
<br />
<br />
:{| <br />
|<math>R\,</math>||Parametric ratio, that is the true ratio present in the population;<br />
|- <br />
|<math>\mu_y\,</math>||Parametric mean of the target variable <math>Y</math>;<br />
|- <br />
|<math>\mu_x\,</math>||Parametric mean of the ancillary variable <math>X</math>;<br />
|- <br />
|<math>\tau_x\,</math>||Parametric total of ancillary variable <math>X</math>;<br />
|- <br />
|<math>\rho\,</math>||Parametric Pearson coefficient of correlation between <math>Y</math> and <math>X</math>;<br />
|-<br />
|<math>\sigma_y\,</math>||Parametric standard deviation of the target variable <math>Y</math>;<br />
|-<br />
|<math>\sigma_x\,</math>||Parametric standard deviation of ancillary variable <math>X</math>;<br />
|-<br />
|<math>n\,</math>||Sample size;<br />
|-<br />
|<math>r\,</math>||Estimated ratio;<br />
<br />
|-<br />
|<math>y_i\,</math>||Value of the target variable observed on the <math>i^{th}</math> sampled element (plot);<br />
|-<br />
|<math>x_i\,</math>||Ancillary variable, e.g. area, for <math>i^{th}</math> plot;<br />
|-<br />
|<math>\bar y\,</math>||Estimated mean of target variable;<br />
|-<br />
|<math>\bar x\,</math>||Estimated mean of ancillary variable;<br />
|-<br />
|<math>\hat{\tau}_y\,</math>||Estimated total of target variable;<br />
|-<br />
|<math>\hat{\rho}\,</math>||Estimated coefficient of correlation of <math>Y</math> and <math>X</math>;<br />
|-<br />
|<math>{s_y}^2\,</math>||Estimated variance of target variable;<br />
|-<br />
|<math>{s_x}^2\,</math>||Estimated variance of ancillary variable.<br />
|}<br />
<br />
<br />
With the notation for the ratio estimator it should, above all, be observed, that the letters <math>r</math> and <math>R</math> are used here for ratio – and not for the [[correlation coefficient]]. That causes some times confusions. We use the Greek letter <math>\rho</math> (rho) for the parametric correlation coefficient and the usual notation for the estimation using the “hat” sign <math>\hat{\rho}</math>.<br />
<br />
===Estimators===<br />
<br />
In the first place, what we estimate with the ration estimator (as the name yet suggests) is the ratio <math>R</math> which is target variable per ancillary variable, for example: number of trees per hectare. From this, we can then also estimate the total of the target variable.<br />
<br />
The parametric ratio is <br />
<br />
:<math>R=\frac{\mu_y}{\mu_x}\,</math><br />
<br />
where this is estimated from the sample as <br />
<br />
:<math>r=\frac{\bar{y}}{\bar{x}}=\frac{\sum_{i=1}^n y_i}{\sum_{i=1}^n x_i}.\,</math><br />
<br />
It is very important to note here that we calculate a '''ratio of means''' and NOT a mean of ratios! The ratio is calculated as the mean of the target variable divided by the mean of the ancillary variable; which is equivalent to calculating the sum of the target variable divided by the sum of the ancillary variable. It would be erroneous to calculate for each sample a ratio between target variable and ancillary variable and compute the mean over all samples! For that situation (the mean of ratios) there are other estimators which are more complex!<br />
<br />
The estimated variance of the estimated ratio <math>r</math> is <br />
<br />
:<math>\hat{var}(r)=\frac{N-n}{N}\frac{1}{n}\frac{1}{{\mu_x}^2}\frac{\sum_{i=1}^n\left(y_i-rx_i\right)^2}{n-1}.\,</math> <br />
<br />
An the estimated total derives from <br />
<br />
:<math>{\tau}_y=r{\tau}_x\,</math><br />
<br />
with an estimated error variance of the total of <br />
<br />
:<math>\hat{var}(\hat{\tau}_y)={\tau_x}^2\hat{var}(r),\,</math><br />
<br />
as usual.<br />
<br />
Given the estimated ratio <math>r</math> the estimated mean derives from<br />
<br />
:<math>\bar{y}_r=r\mu_x\,</math><br />
<br />
[[File:5.4.2-fig85.png|right|thumb|300px|'''Figure 2''' The left graph gives the population of interest. The right graph depicts the adaptive cluster sampling process, where red squares are the initially selected random samples (from Thompson 1992<ref name="thompson1992">Thompson SK. 1992. Sampling. John Wiley & Sons. 343 p.</ref>).]]<br />
<br />
We see here: for the estimation of the mean we need to know the true value of the ancillary variable. In cases such as the one in Figure 2 this is easily determined, because the total of the ancillary variable is simply the total area. However, in other situations where, for example leaf area is taken as ancillary variable to estimation leaf biomass, this is extremely difficult. However, there are corresponding [[:Category:Sampling design|sampling techniques]] that start with a first sampling phase '''estimating''' the ancillary variable before in a second phase the target variable is estimated; these techniques are called [[double sampling]] or [[two phase sampling]] with the ratio estimator.<br />
<br />
The estimated mean carries an estimated variance of the mean of <br />
<br />
:<math>\hat{var}(\bar{y}_r)={\mu_x}^2\hat{var}(r)=\frac{N-n}{N}\frac{1}{n}\left\{{s_y}^2+r^2{s_x}^2-2r\hat{\rho}{s_x}{s_y}\right\}\,</math><br />
<br />
This latter estimator of the error variance of the estimated mean illustrates some basic characteristics of the ratio estimator very clearly:<br />
<br />
If all values of the ancillary variable are the same (for example, when using plot area as ancillary variable in fixed area plot sampling), then <math>{s_x}^2</math> is zero and the estimator is identical to the error variance of the mean estimator in [[simple random sampling]] where the only source of [[variability]] that enters is the estimated variance of the target variable <math>{s_y}^2</math>. Also, we see in the above term that we need to make <math>r\hat{\rho}{s_x}{s_y}</math> large so that a large term is subtracted from the variance terms in parenthesis; the only way to influence here is an intelligent choice of the ancillary variable. The ancillary variable should highly positively correlate with the target variable (there is not much we can do about the target variable except for changing the plot type and size; that will change the variance of the population – but still, because of the expected high correlation with the ancillary variable it will also affect the estimation of <math>r</math> and <math>{s_x}^2</math> and, in addition, the error variance of <math>{s_y}^2</math> the target variable appears with positive and negative sign in the above estimator. At the end, it is difficult to predict how a change in <math>{s_y}^2</math> affects the performance of the ratio estimator.<br />
<br />
==Efficiency==<br />
<br />
In the preceding chapter, we saw in fairly general terms that a high positive correlation between target and ancillary variable will make the variance smaller than the simple random sampling estimator (which would be there, if correlation would be zero). However, by re-arranging the above term, we can more specifically formulate in which situation the ratio estimator is superior to the simple random sampling estimator. As always, the relative efficiency is meaningful only when we deal with parametric values. Based on estimated values, the relative efficiency can be misleading.<br />
<br />
For the ratio estimator we can formulate from the above estimator of the error variance of the estimated mean <br />
<br />
:<math>\sigma\ge\frac{R\sigma_x}{2\sigma_y}=\frac{cv(x)}{2cv(y)}\,</math><br />
<br />
where <math>cv()</math> denotes the coefficient of variation either for the target variable <math>Y</math> or the ancillary variable <math>X</math>. <br />
<br />
As an interpretation of this inequality: the ratio estimator is superior to the simple random estimator if the correlation between the two variables exceeds the above expressions.<br />
<br />
:<math>\frac{R\sigma_x}{2\sigma_y}\,</math><br />
<br />
or<br />
<br />
:<math>\frac{cv(x)}{2cv(y)}\,</math><br />
<br />
There may be situations where a variable had been observed that might be used as an ancillary variable, but we are not sure whether to use the ratio estimator or not. Then, one could estimate the [[relative efficiency]]; although the conclusion might be misleading, there are cases with such high positive correlations that application of the ratio estimator is definitively indicated.<br />
<br />
It is maybe interesting to note that - when having an ancillary variable measured and available: there is more then just one option for the estimation design, the ratio estimator and the simple random sampling estimator; and both are unbiased (or, in the case of the ratio estimator, approximately unbiased). Then, of course, one will choose the estimator with the smaller error variance, and a criterion for this is given above with the correlation between target and ancillary variable.<br />
<br />
{{Exercise<br />
|message=Ratio estimator sampling examples<br />
|alttext=Example<br />
|text=Examples for Ratio estimator<br />
}}<br />
<br />
==Characteristics of ratio estimator==<br />
<br />
When the ratio estimator is used, what we implicitly do is assuming a simple linear regression through the origin between target and ancillary variable: <math>\bar{y}=\bar{r}x</math>, where the ratio, <math>r</math>, is the [[slope coefficient]]. The implicit assumption is that the relationship between <math>Y</math> and <math>X</math> goes really through the origin. In the above example with the plot area as ancillary variable, this is certainly true; if plot area is zero, then also the observation of number of stems or basal area is zero. However, if the relationship between <math>Y</math> and <math>X</math> is such that it does not generically pass through the origin, then, there will be an estimator bias.<br />
<br />
When the intercept term of ratio estimator is 0, then, the estimators presented here will be unbiased. In contrary, if the intercept term is not equal to 0, e.g. when ancillary variable is zero but target variable is not, then, the estimator here will be biased for small sample size but approximately unbiased for large sample size. <br />
<br />
==Regression estimator== <br />
<br />
If, as addressed in the chapter above, the regression between <math>Y</math> and <math>X</math> does not go through the origin, then, one may use the so-called regression estimator which explicitly establishes a simple [[linear regression]] between target and ancillary variable. As we assume that, as usual in simple linear regression, the joint mean value <math>\bar{x},\bar{y}</math> lies on the regression line, we may estimate the mean of the target variable from the sample as<br />
<br />
:<math>\bar{y}_L=\bar{y}+b\left(\mu_x-\bar{x}\right)</math><br />
<br />
where <math>b</math> is the estimated regression coefficient and <math>\mu_x</math> is the population mean of ancillary variable which must be known.<br />
<br />
The estimated variance of the estimated mean is <br />
<br />
:<math>\hat{var}(\bar{y}_L)=\frac{N-n}{N}\frac{1}{n}\frac{1}{n-2}\left\{\sum_{i=1}^n(y_i-\bar{y})^2-b^2\sum_{i=1}^n(x_i-\bar{x})^2\right\}\,</math><br />
<br />
where the term in curled parenthesis times the factor <math>1/(n-2)</math> is the mean square error of regression.<br />
<br />
As with the ratio estimator, we assume here a simple linear regression model between the two variables. If this model holds (that is, if the relationship can realistically be modeled like that, then the regression estimator is unbiased. If this is not so, then we introduce an estimator bias of unknown order of magnitude. However, it is said that this bias is particularly relevant for small sample sizes and not so for larger sample sizes.<br />
<br />
==References==<br />
<references/><br />
<br />
{{SEO<br />
|keywords=ratio estimator,ancillary variable,double sampling<br />
|descrip=The ratio estimator is an estimation design that makesuse of an ancillary variable that is correlated to the target variable<br />
}}<br />
<br />
[[Category:Sampling design]]</div>Fehrmannhttp://wiki.awf.forst.uni-goettingen.de/wiki/index.php/Ratio_estimatorRatio estimator2021-04-08T13:20:24Z<p>Fehrmann: /* Estimators */</p>
<hr />
<div>{{Ficontent}}<br />
[[File:5.6.1-fig93.png|right|thumb|300px|'''Figure 1''' Example of a population of 30 unequally sized strip plots; here, the ratio estimator may be applied for estimation using plot size as co-variable (DeVries 1986<ref>de Vries, P.G., 1986. Sampling Theory for Forest Inventory. A Teach-Yourself Course. Springer. 399 p.</ref>).]]<br />
<br />
There are situations in forest inventory sampling in which the value of the target variable is known (or suspected) to be highly correlated to an other variable (called co-variable or ancillary variable). We know from the discussion on [[Plot_design#Spatial_autocorrelation|spatial autocorrelation]] and the optimization of [[Cluster sampling|cluster plot design]] that correlation between two variables means essentially that the one variable does yet contain a certain amount of information about the other variable. The higher the correlation the better can the value of the second variable be predicted when the value of the first is known (and vice versa). Therefore, if such a co-variable is there on the plot, it would make sense to also observe it and utilize the correlation to the target variable to eventually improve the precision of estimating the target variable. This is exactly the situation where the ratio estimator is applied.<br />
<br />
A typical and basic example is that of sample plots of unequal size, as depicted, for example in Figure 1 where strips of different length constitute the [[population]] of samples. Another example would be clusters of unequal size: while [[cluster sampling]] only deals with the case of clusters of equal size, the ratio estimator allows to take into account the differently sized [[sample plot]]s. Here, obviously, the size of the sample plots is the co-variable that needs to be determined for each sample plot. That a high positive correlation is to be expected between most forestry-relevant attributes and plot size should be obvious: the larger the plot the more basal area, number of stems, volume etc. is expected to be present.<br />
<br />
In fact, the ratio estimator does not introduce a new sampling technique: we use here the example of [[simple random sampling]] (as [[:Category:Sampling design|sampling design]]). The ratio estimator introduces two new aspects in respect to:<br />
<br />
# [[plot design]]: observation of an ancillary variable on each plot in addition to the observation of the target variable.<br />
# [[estimation design]]: the ratio estimator integrates the ancillary variable into the estimator. <br />
<br />
{{info<br />
|message=Side-comment:<br />
|text=For those who have some knowledge or interest (or both) on the design of experiments and the linear statistical models used there (analysis of variance) it might be interesting to note that also in the field of experimental design there is a technique in which a co-variable is observed on each experimental plot in order to allow a more precise estimation of the effects of treatments: this technique is called ''analysis of co-variance''.<br />
}}<br />
<br />
<br />
The ratio estimator is called ratio estimator because, what we actually estimate from the sample plot is a ratio: for example, number of stems per hectare. The ancillary variable (“area” in this case) appears always in the denominator of the ratio. We denote the variable of interest as usual with <math>y_i</math> and the ancillary variable as <math>x_i</math>; some more on notation is given in [[double sampling]]. <br />
<br />
{{info<br />
|message=in simple words:<br />
|text=The ratio estimator involves the ratio of two variates. Imagine there are two characteristics (say x and y) we can observe for each sampled element and we have in addition information about the total of x, then the total of y can be estimated by multiplying the known total of x by a ratio of both variables. This ratio can be estimated from the means of both variables that are observed for the sampled elements. <br />
}}<br />
<br />
===Notation===<br />
<br />
<br />
:{| <br />
|<math>R\,</math>||Parametric ratio, that is the true ratio present in the population;<br />
|- <br />
|<math>\mu_y\,</math>||Parametric mean of the target variable <math>Y</math>;<br />
|- <br />
|<math>\mu_x\,</math>||Parametric mean of the ancillary variable <math>X</math>;<br />
|- <br />
|<math>\tau_x\,</math>||Parametric total of ancillary variable <math>X</math>;<br />
|- <br />
|<math>\rho\,</math>||Parametric Pearson coefficient of correlation between <math>Y</math> and <math>X</math>;<br />
|-<br />
|<math>\sigma_y\,</math>||Parametric standard deviation of the target variable <math>Y</math>;<br />
|-<br />
|<math>\sigma_x\,</math>||Parametric standard deviation of ancillary variable <math>X</math>;<br />
|-<br />
|<math>n\,</math>||Sample size;<br />
|-<br />
|<math>r\,</math>||Estimated ratio;<br />
<br />
|-<br />
|<math>y_i\,</math>||Value of the target variable observed on the <math>i^{th}</math> sampled element (plot);<br />
|-<br />
|<math>x_i\,</math>||Ancillary variable, e.g. area, for <math>i^{th}</math> plot;<br />
|-<br />
|<math>\bar y\,</math>||Estimated mean of target variable;<br />
|-<br />
|<math>\bar x\,</math>||Estimated mean of ancillary variable;<br />
|-<br />
|<math>\hat{\tau}_y\,</math>||Estimated total of target variable;<br />
|-<br />
|<math>\hat{\rho}\,</math>||Estimated coefficient of correlation of <math>Y</math> and <math>X</math>;<br />
|-<br />
|<math>{s_y}^2\,</math>||Estimated variance of target variable;<br />
|-<br />
|<math>{s_x}^2\,</math>||Estimated variance of ancillary variable.<br />
|}<br />
<br />
<br />
With the notation for the ratio estimator it should, above all, be observed, that the letters <math>r</math> and <math>R</math> are used here for ratio – and not for the [[correlation coefficient]]. That causes some times confusions. We use the Greek letter <math>\rho</math> (rho) for the parametric correlation coefficient and the usual notation for the estimation using the “hat” sign <math>\hat{\rho}</math>.<br />
<br />
===Estimators===<br />
<br />
In the first place, what we estimate with the ration estimator (as the name yet suggests) is the ratio <math>R</math> which is target variable per ancillary variable, for example: number of trees per hectare. From this, we can then also estimate the total of the target variable.<br />
<br />
The parametric ratio is <br />
<br />
:<math>R=\frac{\mu_y}{\mu_x}\,</math><br />
<br />
where this is estimated from the sample as <br />
<br />
:<math>r=\frac{\bar{y}}{\bar{x}}=\frac{\sum_{i=1}^n y_i}{\sum_{i=1}^n x_i}.\,</math><br />
<br />
It is very important to note here that we calculate a '''ratio of means''' and NOT a mean of ratios! The ratio is calculated as the mean of the target variable divided by the mean of the ancillary variable; which is equivalent to calculating the sum of the target variable divided by the sum of the ancillary variable. It would be erroneous to calculate for each sample a ratio between target variable and ancillary variable and compute the mean over all samples! For that situation (the mean of ratios) there are other estimators which are more complex!<br />
<br />
The estimated variance of the estimated ratio <math>r</math> is <br />
<br />
:<math>\hat{var}(r)=\frac{N-n}{N}\frac{1}{n}\frac{1}{{\mu_x}^2}\frac{\sum_{i=1}^n\left(y_i-rx_i\right)^2}{n-1}.\,</math> <br />
<br />
An the estimated total derives from <br />
<br />
:<math>{\tau}_y=r{\tau}_x\,</math><br />
<br />
with an estimated error variance of the total of <br />
<br />
:<math>\hat{var}(\hat{\tau}_y)={\tau_x}^2\hat{var}(r),\,</math><br />
<br />
as usual.<br />
<br />
Given the estimated ratio <math>r</math> the estimated mean derives from<br />
<br />
:<math>\bar{y}=r\mu\,</math><br />
<br />
[[File:5.4.2-fig85.png|right|thumb|300px|'''Figure 2''' The left graph gives the population of interest. The right graph depicts the adaptive cluster sampling process, where red squares are the initially selected random samples (from Thompson 1992<ref name="thompson1992">Thompson SK. 1992. Sampling. John Wiley & Sons. 343 p.</ref>).]]<br />
<br />
We see here: for the estimation of the mean we need to know the true value of the ancillary variable. In cases such as the one in Figure 2 this is easily determined, because the total of the ancillary variable is simply the total area. However, in other situations where, for example leaf area is taken as ancillary variable to estimation leaf biomass, this is extremely difficult. However, there are corresponding [[:Category:Sampling design|sampling techniques]] that start with a first sampling phase '''estimating''' the ancillary variable before in a second phase the target variable is estimated; these techniques are called [[double sampling]] or [[two phase sampling]] with the ratio estimator.<br />
<br />
The estimated mean carries an estimated variance of the mean of <br />
<br />
:<math>\hat{var}(\bar{y}_r)={\mu_x}^2\hat{var}(r)=\frac{N-n}{N}\frac{1}{n}\left\{{s_y}^2+r^2{s_x}^2-2r\hat{\rho}{s_x}{s_y}\right\}\,</math><br />
<br />
This latter estimator of the error variance of the estimated mean illustrates some basic characteristics of the ratio estimator very clearly:<br />
<br />
If all values of the ancillary variable are the same (for example, when using plot area as ancillary variable in fixed area plot sampling), then <math>{s_x}^2</math> is zero and the estimator is identical to the error variance of the mean estimator in [[simple random sampling]] where the only source of [[variability]] that enters is the estimated variance of the target variable <math>{s_y}^2</math>. Also, we see in the above term that we need to make <math>r\hat{\rho}{s_x}{s_y}</math> large so that a large term is subtracted from the variance terms in parenthesis; the only way to influence here is an intelligent choice of the ancillary variable. The ancillary variable should highly positively correlate with the target variable (there is not much we can do about the target variable except for changing the plot type and size; that will change the variance of the population – but still, because of the expected high correlation with the ancillary variable it will also affect the estimation of <math>r</math> and <math>{s_x}^2</math> and, in addition, the error variance of <math>{s_y}^2</math> the target variable appears with positive and negative sign in the above estimator. At the end, it is difficult to predict how a change in <math>{s_y}^2</math> affects the performance of the ratio estimator.<br />
<br />
==Efficiency==<br />
<br />
In the preceding chapter, we saw in fairly general terms that a high positive correlation between target and ancillary variable will make the variance smaller than the simple random sampling estimator (which would be there, if correlation would be zero). However, by re-arranging the above term, we can more specifically formulate in which situation the ratio estimator is superior to the simple random sampling estimator. As always, the relative efficiency is meaningful only when we deal with parametric values. Based on estimated values, the relative efficiency can be misleading.<br />
<br />
For the ratio estimator we can formulate from the above estimator of the error variance of the estimated mean <br />
<br />
:<math>\sigma\ge\frac{R\sigma_x}{2\sigma_y}=\frac{cv(x)}{2cv(y)}\,</math><br />
<br />
where <math>cv()</math> denotes the coefficient of variation either for the target variable <math>Y</math> or the ancillary variable <math>X</math>. <br />
<br />
As an interpretation of this inequality: the ratio estimator is superior to the simple random estimator if the correlation between the two variables exceeds the above expressions.<br />
<br />
:<math>\frac{R\sigma_x}{2\sigma_y}\,</math><br />
<br />
or<br />
<br />
:<math>\frac{cv(x)}{2cv(y)}\,</math><br />
<br />
There may be situations where a variable had been observed that might be used as an ancillary variable, but we are not sure whether to use the ratio estimator or not. Then, one could estimate the [[relative efficiency]]; although the conclusion might be misleading, there are cases with such high positive correlations that application of the ratio estimator is definitively indicated.<br />
<br />
It is maybe interesting to note that - when having an ancillary variable measured and available: there is more then just one option for the estimation design, the ratio estimator and the simple random sampling estimator; and both are unbiased (or, in the case of the ratio estimator, approximately unbiased). Then, of course, one will choose the estimator with the smaller error variance, and a criterion for this is given above with the correlation between target and ancillary variable.<br />
<br />
{{Exercise<br />
|message=Ratio estimator sampling examples<br />
|alttext=Example<br />
|text=Examples for Ratio estimator<br />
}}<br />
<br />
==Characteristics of ratio estimator==<br />
<br />
When the ratio estimator is used, what we implicitly do is assuming a simple linear regression through the origin between target and ancillary variable: <math>\bar{y}=\bar{r}x</math>, where the ratio, <math>r</math>, is the [[slope coefficient]]. The implicit assumption is that the relationship between <math>Y</math> and <math>X</math> goes really through the origin. In the above example with the plot area as ancillary variable, this is certainly true; if plot area is zero, then also the observation of number of stems or basal area is zero. However, if the relationship between <math>Y</math> and <math>X</math> is such that it does not generically pass through the origin, then, there will be an estimator bias.<br />
<br />
When the intercept term of ratio estimator is 0, then, the estimators presented here will be unbiased. In contrary, if the intercept term is not equal to 0, e.g. when ancillary variable is zero but target variable is not, then, the estimator here will be biased for small sample size but approximately unbiased for large sample size. <br />
<br />
==Regression estimator== <br />
<br />
If, as addressed in the chapter above, the regression between <math>Y</math> and <math>X</math> does not go through the origin, then, one may use the so-called regression estimator which explicitly establishes a simple [[linear regression]] between target and ancillary variable. As we assume that, as usual in simple linear regression, the joint mean value <math>\bar{x},\bar{y}</math> lies on the regression line, we may estimate the mean of the target variable from the sample as<br />
<br />
:<math>\bar{y}_L=\bar{y}+b\left(\mu_x-\bar{x}\right)</math><br />
<br />
where <math>b</math> is the estimated regression coefficient and <math>\mu_x</math> is the population mean of ancillary variable which must be known.<br />
<br />
The estimated variance of the estimated mean is <br />
<br />
:<math>\hat{var}(\bar{y}_L)=\frac{N-n}{N}\frac{1}{n}\frac{1}{n-2}\left\{\sum_{i=1}^n(y_i-\bar{y})^2-b^2\sum_{i=1}^n(x_i-\bar{x})^2\right\}\,</math><br />
<br />
where the term in curled parenthesis times the factor <math>1/(n-2)</math> is the mean square error of regression.<br />
<br />
As with the ratio estimator, we assume here a simple linear regression model between the two variables. If this model holds (that is, if the relationship can realistically be modeled like that, then the regression estimator is unbiased. If this is not so, then we introduce an estimator bias of unknown order of magnitude. However, it is said that this bias is particularly relevant for small sample sizes and not so for larger sample sizes.<br />
<br />
==References==<br />
<references/><br />
<br />
{{SEO<br />
|keywords=ratio estimator,ancillary variable,double sampling<br />
|descrip=The ratio estimator is an estimation design that makesuse of an ancillary variable that is correlated to the target variable<br />
}}<br />
<br />
[[Category:Sampling design]]</div>Fehrmannhttp://wiki.awf.forst.uni-goettingen.de/wiki/index.php/Ratio_estimatorRatio estimator2021-04-08T13:19:41Z<p>Fehrmann: /* Estimators */</p>
<hr />
<div>{{Ficontent}}<br />
[[File:5.6.1-fig93.png|right|thumb|300px|'''Figure 1''' Example of a population of 30 unequally sized strip plots; here, the ratio estimator may be applied for estimation using plot size as co-variable (DeVries 1986<ref>de Vries, P.G., 1986. Sampling Theory for Forest Inventory. A Teach-Yourself Course. Springer. 399 p.</ref>).]]<br />
<br />
There are situations in forest inventory sampling in which the value of the target variable is known (or suspected) to be highly correlated to an other variable (called co-variable or ancillary variable). We know from the discussion on [[Plot_design#Spatial_autocorrelation|spatial autocorrelation]] and the optimization of [[Cluster sampling|cluster plot design]] that correlation between two variables means essentially that the one variable does yet contain a certain amount of information about the other variable. The higher the correlation the better can the value of the second variable be predicted when the value of the first is known (and vice versa). Therefore, if such a co-variable is there on the plot, it would make sense to also observe it and utilize the correlation to the target variable to eventually improve the precision of estimating the target variable. This is exactly the situation where the ratio estimator is applied.<br />
<br />
A typical and basic example is that of sample plots of unequal size, as depicted, for example in Figure 1 where strips of different length constitute the [[population]] of samples. Another example would be clusters of unequal size: while [[cluster sampling]] only deals with the case of clusters of equal size, the ratio estimator allows to take into account the differently sized [[sample plot]]s. Here, obviously, the size of the sample plots is the co-variable that needs to be determined for each sample plot. That a high positive correlation is to be expected between most forestry-relevant attributes and plot size should be obvious: the larger the plot the more basal area, number of stems, volume etc. is expected to be present.<br />
<br />
In fact, the ratio estimator does not introduce a new sampling technique: we use here the example of [[simple random sampling]] (as [[:Category:Sampling design|sampling design]]). The ratio estimator introduces two new aspects in respect to:<br />
<br />
# [[plot design]]: observation of an ancillary variable on each plot in addition to the observation of the target variable.<br />
# [[estimation design]]: the ratio estimator integrates the ancillary variable into the estimator. <br />
<br />
{{info<br />
|message=Side-comment:<br />
|text=For those who have some knowledge or interest (or both) on the design of experiments and the linear statistical models used there (analysis of variance) it might be interesting to note that also in the field of experimental design there is a technique in which a co-variable is observed on each experimental plot in order to allow a more precise estimation of the effects of treatments: this technique is called ''analysis of co-variance''.<br />
}}<br />
<br />
<br />
The ratio estimator is called ratio estimator because, what we actually estimate from the sample plot is a ratio: for example, number of stems per hectare. The ancillary variable (“area” in this case) appears always in the denominator of the ratio. We denote the variable of interest as usual with <math>y_i</math> and the ancillary variable as <math>x_i</math>; some more on notation is given in [[double sampling]]. <br />
<br />
{{info<br />
|message=in simple words:<br />
|text=The ratio estimator involves the ratio of two variates. Imagine there are two characteristics (say x and y) we can observe for each sampled element and we have in addition information about the total of x, then the total of y can be estimated by multiplying the known total of x by a ratio of both variables. This ratio can be estimated from the means of both variables that are observed for the sampled elements. <br />
}}<br />
<br />
===Notation===<br />
<br />
<br />
:{| <br />
|<math>R\,</math>||Parametric ratio, that is the true ratio present in the population;<br />
|- <br />
|<math>\mu_y\,</math>||Parametric mean of the target variable <math>Y</math>;<br />
|- <br />
|<math>\mu_x\,</math>||Parametric mean of the ancillary variable <math>X</math>;<br />
|- <br />
|<math>\tau_x\,</math>||Parametric total of ancillary variable <math>X</math>;<br />
|- <br />
|<math>\rho\,</math>||Parametric Pearson coefficient of correlation between <math>Y</math> and <math>X</math>;<br />
|-<br />
|<math>\sigma_y\,</math>||Parametric standard deviation of the target variable <math>Y</math>;<br />
|-<br />
|<math>\sigma_x\,</math>||Parametric standard deviation of ancillary variable <math>X</math>;<br />
|-<br />
|<math>n\,</math>||Sample size;<br />
|-<br />
|<math>r\,</math>||Estimated ratio;<br />
<br />
|-<br />
|<math>y_i\,</math>||Value of the target variable observed on the <math>i^{th}</math> sampled element (plot);<br />
|-<br />
|<math>x_i\,</math>||Ancillary variable, e.g. area, for <math>i^{th}</math> plot;<br />
|-<br />
|<math>\bar y\,</math>||Estimated mean of target variable;<br />
|-<br />
|<math>\bar x\,</math>||Estimated mean of ancillary variable;<br />
|-<br />
|<math>\hat{\tau}_y\,</math>||Estimated total of target variable;<br />
|-<br />
|<math>\hat{\rho}\,</math>||Estimated coefficient of correlation of <math>Y</math> and <math>X</math>;<br />
|-<br />
|<math>{s_y}^2\,</math>||Estimated variance of target variable;<br />
|-<br />
|<math>{s_x}^2\,</math>||Estimated variance of ancillary variable.<br />
|}<br />
<br />
<br />
With the notation for the ratio estimator it should, above all, be observed, that the letters <math>r</math> and <math>R</math> are used here for ratio – and not for the [[correlation coefficient]]. That causes some times confusions. We use the Greek letter <math>\rho</math> (rho) for the parametric correlation coefficient and the usual notation for the estimation using the “hat” sign <math>\hat{\rho}</math>.<br />
<br />
===Estimators===<br />
<br />
In the first place, what we estimate with the ration estimator (as the name yet suggests) is the ratio <math>R</math> which is target variable per ancillary variable, for example: number of trees per hectare. From this, we can then also estimate the total of the target variable.<br />
<br />
The parametric ratio is <br />
<br />
:<math>R=\frac{\mu_y}{\mu_x}\,</math><br />
<br />
where this is estimated from the sample as <br />
<br />
:<math>r=\frac{\bar{y}}{\bar{x}}=\frac{\sum_{i=1}^n y_i}{\sum_{i=1}^n x_i}.\,</math><br />
<br />
It is very important to note here that we calculate a '''ratio of means''' and NOT a mean of ratios! The ratio is calculated as the mean of the target variable divided by the mean of the ancillary variable; which is equivalent to calculating the sum of the target variable divided by the sum of the ancillary variable. It would be erroneous to calculate for each sample a ratio between target variable and ancillary variable and compute the mean over all samples! For that situation (the mean of ratios) there are other estimators which are more complex!<br />
<br />
The estimated variance of the estimated ratio <math>r</math> is <br />
<br />
:<math>\hat{var}(r)=\frac{N-n}{N}\frac{1}{n}\frac{1}{{\mu_x}^2}\frac{\sum_{i=1}^n\left(y_i-rx_i\right)^2}{n-1}.\,</math> <br />
<br />
An the estimated total derives from <br />
<br />
:<math>{\tau}_y=r{\tau}_x\,</math><br />
<br />
with an estimated error variance of the total of <br />
<br />
:<math>\hat{var}(r\hat{\tau}_y)={\tau_y}^2\hat{var}(r),\,</math><br />
<br />
as usual.<br />
<br />
Given the estimated ratio <math>r</math> the estimated mean derives from<br />
<br />
:<math>\bar{y}=r\mu\,</math><br />
<br />
[[File:5.4.2-fig85.png|right|thumb|300px|'''Figure 2''' The left graph gives the population of interest. The right graph depicts the adaptive cluster sampling process, where red squares are the initially selected random samples (from Thompson 1992<ref name="thompson1992">Thompson SK. 1992. Sampling. John Wiley & Sons. 343 p.</ref>).]]<br />
<br />
We see here: for the estimation of the mean we need to know the true value of the ancillary variable. In cases such as the one in Figure 2 this is easily determined, because the total of the ancillary variable is simply the total area. However, in other situations where, for example leaf area is taken as ancillary variable to estimation leaf biomass, this is extremely difficult. However, there are corresponding [[:Category:Sampling design|sampling techniques]] that start with a first sampling phase '''estimating''' the ancillary variable before in a second phase the target variable is estimated; these techniques are called [[double sampling]] or [[two phase sampling]] with the ratio estimator.<br />
<br />
The estimated mean carries an estimated variance of the mean of <br />
<br />
:<math>\hat{var}(\bar{y}_r)={\mu_x}^2\hat{var}(r)=\frac{N-n}{N}\frac{1}{n}\left\{{s_y}^2+r^2{s_x}^2-2r\hat{\rho}{s_x}{s_y}\right\}\,</math><br />
<br />
This latter estimator of the error variance of the estimated mean illustrates some basic characteristics of the ratio estimator very clearly:<br />
<br />
If all values of the ancillary variable are the same (for example, when using plot area as ancillary variable in fixed area plot sampling), then <math>{s_x}^2</math> is zero and the estimator is identical to the error variance of the mean estimator in [[simple random sampling]] where the only source of [[variability]] that enters is the estimated variance of the target variable <math>{s_y}^2</math>. Also, we see in the above term that we need to make <math>r\hat{\rho}{s_x}{s_y}</math> large so that a large term is subtracted from the variance terms in parenthesis; the only way to influence here is an intelligent choice of the ancillary variable. The ancillary variable should highly positively correlate with the target variable (there is not much we can do about the target variable except for changing the plot type and size; that will change the variance of the population – but still, because of the expected high correlation with the ancillary variable it will also affect the estimation of <math>r</math> and <math>{s_x}^2</math> and, in addition, the error variance of <math>{s_y}^2</math> the target variable appears with positive and negative sign in the above estimator. At the end, it is difficult to predict how a change in <math>{s_y}^2</math> affects the performance of the ratio estimator.<br />
<br />
==Efficiency==<br />
<br />
In the preceding chapter, we saw in fairly general terms that a high positive correlation between target and ancillary variable will make the variance smaller than the simple random sampling estimator (which would be there, if correlation would be zero). However, by re-arranging the above term, we can more specifically formulate in which situation the ratio estimator is superior to the simple random sampling estimator. As always, the relative efficiency is meaningful only when we deal with parametric values. Based on estimated values, the relative efficiency can be misleading.<br />
<br />
For the ratio estimator we can formulate from the above estimator of the error variance of the estimated mean <br />
<br />
:<math>\sigma\ge\frac{R\sigma_x}{2\sigma_y}=\frac{cv(x)}{2cv(y)}\,</math><br />
<br />
where <math>cv()</math> denotes the coefficient of variation either for the target variable <math>Y</math> or the ancillary variable <math>X</math>. <br />
<br />
As an interpretation of this inequality: the ratio estimator is superior to the simple random estimator if the correlation between the two variables exceeds the above expressions.<br />
<br />
:<math>\frac{R\sigma_x}{2\sigma_y}\,</math><br />
<br />
or<br />
<br />
:<math>\frac{cv(x)}{2cv(y)}\,</math><br />
<br />
There may be situations where a variable had been observed that might be used as an ancillary variable, but we are not sure whether to use the ratio estimator or not. Then, one could estimate the [[relative efficiency]]; although the conclusion might be misleading, there are cases with such high positive correlations that application of the ratio estimator is definitively indicated.<br />
<br />
It is maybe interesting to note that - when having an ancillary variable measured and available: there is more then just one option for the estimation design, the ratio estimator and the simple random sampling estimator; and both are unbiased (or, in the case of the ratio estimator, approximately unbiased). Then, of course, one will choose the estimator with the smaller error variance, and a criterion for this is given above with the correlation between target and ancillary variable.<br />
<br />
{{Exercise<br />
|message=Ratio estimator sampling examples<br />
|alttext=Example<br />
|text=Examples for Ratio estimator<br />
}}<br />
<br />
==Characteristics of ratio estimator==<br />
<br />
When the ratio estimator is used, what we implicitly do is assuming a simple linear regression through the origin between target and ancillary variable: <math>\bar{y}=\bar{r}x</math>, where the ratio, <math>r</math>, is the [[slope coefficient]]. The implicit assumption is that the relationship between <math>Y</math> and <math>X</math> goes really through the origin. In the above example with the plot area as ancillary variable, this is certainly true; if plot area is zero, then also the observation of number of stems or basal area is zero. However, if the relationship between <math>Y</math> and <math>X</math> is such that it does not generically pass through the origin, then, there will be an estimator bias.<br />
<br />
When the intercept term of ratio estimator is 0, then, the estimators presented here will be unbiased. In contrary, if the intercept term is not equal to 0, e.g. when ancillary variable is zero but target variable is not, then, the estimator here will be biased for small sample size but approximately unbiased for large sample size. <br />
<br />
==Regression estimator== <br />
<br />
If, as addressed in the chapter above, the regression between <math>Y</math> and <math>X</math> does not go through the origin, then, one may use the so-called regression estimator which explicitly establishes a simple [[linear regression]] between target and ancillary variable. As we assume that, as usual in simple linear regression, the joint mean value <math>\bar{x},\bar{y}</math> lies on the regression line, we may estimate the mean of the target variable from the sample as<br />
<br />
:<math>\bar{y}_L=\bar{y}+b\left(\mu_x-\bar{x}\right)</math><br />
<br />
where <math>b</math> is the estimated regression coefficient and <math>\mu_x</math> is the population mean of ancillary variable which must be known.<br />
<br />
The estimated variance of the estimated mean is <br />
<br />
:<math>\hat{var}(\bar{y}_L)=\frac{N-n}{N}\frac{1}{n}\frac{1}{n-2}\left\{\sum_{i=1}^n(y_i-\bar{y})^2-b^2\sum_{i=1}^n(x_i-\bar{x})^2\right\}\,</math><br />
<br />
where the term in curled parenthesis times the factor <math>1/(n-2)</math> is the mean square error of regression.<br />
<br />
As with the ratio estimator, we assume here a simple linear regression model between the two variables. If this model holds (that is, if the relationship can realistically be modeled like that, then the regression estimator is unbiased. If this is not so, then we introduce an estimator bias of unknown order of magnitude. However, it is said that this bias is particularly relevant for small sample sizes and not so for larger sample sizes.<br />
<br />
==References==<br />
<references/><br />
<br />
{{SEO<br />
|keywords=ratio estimator,ancillary variable,double sampling<br />
|descrip=The ratio estimator is an estimation design that makesuse of an ancillary variable that is correlated to the target variable<br />
}}<br />
<br />
[[Category:Sampling design]]</div>Fehrmannhttp://wiki.awf.forst.uni-goettingen.de/wiki/index.php/Ratio_estimatorRatio estimator2021-04-08T12:45:48Z<p>Fehrmann: /* Estimators */</p>
<hr />
<div>{{Ficontent}}<br />
[[File:5.6.1-fig93.png|right|thumb|300px|'''Figure 1''' Example of a population of 30 unequally sized strip plots; here, the ratio estimator may be applied for estimation using plot size as co-variable (DeVries 1986<ref>de Vries, P.G., 1986. Sampling Theory for Forest Inventory. A Teach-Yourself Course. Springer. 399 p.</ref>).]]<br />
<br />
There are situations in forest inventory sampling in which the value of the target variable is known (or suspected) to be highly correlated to an other variable (called co-variable or ancillary variable). We know from the discussion on [[Plot_design#Spatial_autocorrelation|spatial autocorrelation]] and the optimization of [[Cluster sampling|cluster plot design]] that correlation between two variables means essentially that the one variable does yet contain a certain amount of information about the other variable. The higher the correlation the better can the value of the second variable be predicted when the value of the first is known (and vice versa). Therefore, if such a co-variable is there on the plot, it would make sense to also observe it and utilize the correlation to the target variable to eventually improve the precision of estimating the target variable. This is exactly the situation where the ratio estimator is applied.<br />
<br />
A typical and basic example is that of sample plots of unequal size, as depicted, for example in Figure 1 where strips of different length constitute the [[population]] of samples. Another example would be clusters of unequal size: while [[cluster sampling]] only deals with the case of clusters of equal size, the ratio estimator allows to take into account the differently sized [[sample plot]]s. Here, obviously, the size of the sample plots is the co-variable that needs to be determined for each sample plot. That a high positive correlation is to be expected between most forestry-relevant attributes and plot size should be obvious: the larger the plot the more basal area, number of stems, volume etc. is expected to be present.<br />
<br />
In fact, the ratio estimator does not introduce a new sampling technique: we use here the example of [[simple random sampling]] (as [[:Category:Sampling design|sampling design]]). The ratio estimator introduces two new aspects in respect to:<br />
<br />
# [[plot design]]: observation of an ancillary variable on each plot in addition to the observation of the target variable.<br />
# [[estimation design]]: the ratio estimator integrates the ancillary variable into the estimator. <br />
<br />
{{info<br />
|message=Side-comment:<br />
|text=For those who have some knowledge or interest (or both) on the design of experiments and the linear statistical models used there (analysis of variance) it might be interesting to note that also in the field of experimental design there is a technique in which a co-variable is observed on each experimental plot in order to allow a more precise estimation of the effects of treatments: this technique is called ''analysis of co-variance''.<br />
}}<br />
<br />
<br />
The ratio estimator is called ratio estimator because, what we actually estimate from the sample plot is a ratio: for example, number of stems per hectare. The ancillary variable (“area” in this case) appears always in the denominator of the ratio. We denote the variable of interest as usual with <math>y_i</math> and the ancillary variable as <math>x_i</math>; some more on notation is given in [[double sampling]]. <br />
<br />
{{info<br />
|message=in simple words:<br />
|text=The ratio estimator involves the ratio of two variates. Imagine there are two characteristics (say x and y) we can observe for each sampled element and we have in addition information about the total of x, then the total of y can be estimated by multiplying the known total of x by a ratio of both variables. This ratio can be estimated from the means of both variables that are observed for the sampled elements. <br />
}}<br />
<br />
===Notation===<br />
<br />
<br />
:{| <br />
|<math>R\,</math>||Parametric ratio, that is the true ratio present in the population;<br />
|- <br />
|<math>\mu_y\,</math>||Parametric mean of the target variable <math>Y</math>;<br />
|- <br />
|<math>\mu_x\,</math>||Parametric mean of the ancillary variable <math>X</math>;<br />
|- <br />
|<math>\tau_x\,</math>||Parametric total of ancillary variable <math>X</math>;<br />
|- <br />
|<math>\rho\,</math>||Parametric Pearson coefficient of correlation between <math>Y</math> and <math>X</math>;<br />
|-<br />
|<math>\sigma_y\,</math>||Parametric standard deviation of the target variable <math>Y</math>;<br />
|-<br />
|<math>\sigma_x\,</math>||Parametric standard deviation of ancillary variable <math>X</math>;<br />
|-<br />
|<math>n\,</math>||Sample size;<br />
|-<br />
|<math>r\,</math>||Estimated ratio;<br />
<br />
|-<br />
|<math>y_i\,</math>||Value of the target variable observed on the <math>i^{th}</math> sampled element (plot);<br />
|-<br />
|<math>x_i\,</math>||Ancillary variable, e.g. area, for <math>i^{th}</math> plot;<br />
|-<br />
|<math>\bar y\,</math>||Estimated mean of target variable;<br />
|-<br />
|<math>\bar x\,</math>||Estimated mean of ancillary variable;<br />
|-<br />
|<math>\hat{\tau}_y\,</math>||Estimated total of target variable;<br />
|-<br />
|<math>\hat{\rho}\,</math>||Estimated coefficient of correlation of <math>Y</math> and <math>X</math>;<br />
|-<br />
|<math>{s_y}^2\,</math>||Estimated variance of target variable;<br />
|-<br />
|<math>{s_x}^2\,</math>||Estimated variance of ancillary variable.<br />
|}<br />
<br />
<br />
With the notation for the ratio estimator it should, above all, be observed, that the letters <math>r</math> and <math>R</math> are used here for ratio – and not for the [[correlation coefficient]]. That causes some times confusions. We use the Greek letter <math>\rho</math> (rho) for the parametric correlation coefficient and the usual notation for the estimation using the “hat” sign <math>\hat{\rho}</math>.<br />
<br />
===Estimators===<br />
<br />
In the first place, what we estimate with the ration estimator (as the name yet suggests) is the ratio <math>R</math> which is target variable per ancillary variable, for example: number of trees per hectare. From this, we can then also estimate the total of the target variable.<br />
<br />
The parametric ratio is <br />
<br />
:<math>R=\frac{\mu_y}{\mu_x}\,</math><br />
<br />
where this is estimated from the sample as <br />
<br />
:<math>r=\frac{\bar{y}}{\bar{x}}=\frac{\sum_{i=1}^n y_i}{\sum_{i=1}^n x_i}.\,</math><br />
<br />
It is very important to note here that we calculate a '''ratio of means''' and NOT a mean of ratios! The ratio is calculated as the mean of the target variable divided by the mean of the ancillary variable; which is equivalent to calculating the sum of the target variable divided by the sum of the ancillary variable. It would be erroneous to calculate for each sample a ratio between target variable and ancillary variable and compute the mean over all samples! For that situation (the mean of ratios) there are other estimators which are more complex!<br />
<br />
The estimated variance of the estimated ratio <math>r</math> is <br />
<br />
:<math>\hat{var}(r)=\frac{N-n}{N}\frac{1}{n}\frac{1}{{\mu_x}^2}\frac{\sum_{i=1}^n\left(y_i-rx_i\right)^2}{n-1}.\,</math> <br />
<br />
An the estimated total derives from <br />
<br />
:<math>{\tau}_y=r{\tau}_x\,</math><br />
<br />
with an estimated error variance of the total of <br />
<br />
:<math>\hat{\tau}_y={\tau_y}^2\hat{var}(r),\,</math><br />
<br />
as usual.<br />
<br />
Given the estimated ratio <math>r</math> the estimated mean derives from<br />
<br />
:<math>\bar{y}=r\mu\,</math><br />
<br />
[[File:5.4.2-fig85.png|right|thumb|300px|'''Figure 2''' The left graph gives the population of interest. The right graph depicts the adaptive cluster sampling process, where red squares are the initially selected random samples (from Thompson 1992<ref name="thompson1992">Thompson SK. 1992. Sampling. John Wiley & Sons. 343 p.</ref>).]]<br />
<br />
We see here: for the estimation of the mean we need to know the true value of the ancillary variable. In cases such as the one in Figure 2 this is easily determined, because the total of the ancillary variable is simply the total area. However, in other situations where, for example leaf area is taken as ancillary variable to estimation leaf biomass, this is extremely difficult. However, there are corresponding [[:Category:Sampling design|sampling techniques]] that start with a first sampling phase '''estimating''' the ancillary variable before in a second phase the target variable is estimated; these techniques are called [[double sampling]] or [[two phase sampling]] with the ratio estimator.<br />
<br />
The estimated mean carries an estimated variance of the mean of <br />
<br />
:<math>\hat{var}(\bar{y}_r)={\mu_x}^2\hat{var}(r)=\frac{N-n}{N}\frac{1}{n}\left\{{s_y}^2+r^2{s_x}^2-2r\hat{\rho}{s_x}{s_y}\right\}\,</math><br />
<br />
This latter estimator of the error variance of the estimated mean illustrates some basic characteristics of the ratio estimator very clearly:<br />
<br />
If all values of the ancillary variable are the same (for example, when using plot area as ancillary variable in fixed area plot sampling), then <math>{s_x}^2</math> is zero and the estimator is identical to the error variance of the mean estimator in [[simple random sampling]] where the only source of [[variability]] that enters is the estimated variance of the target variable <math>{s_y}^2</math>. Also, we see in the above term that we need to make <math>r\hat{\rho}{s_x}{s_y}</math> large so that a large term is subtracted from the variance terms in parenthesis; the only way to influence here is an intelligent choice of the ancillary variable. The ancillary variable should highly positively correlate with the target variable (there is not much we can do about the target variable except for changing the plot type and size; that will change the variance of the population – but still, because of the expected high correlation with the ancillary variable it will also affect the estimation of <math>r</math> and <math>{s_x}^2</math> and, in addition, the error variance of <math>{s_y}^2</math> the target variable appears with positive and negative sign in the above estimator. At the end, it is difficult to predict how a change in <math>{s_y}^2</math> affects the performance of the ratio estimator.<br />
<br />
==Efficiency==<br />
<br />
In the preceding chapter, we saw in fairly general terms that a high positive correlation between target and ancillary variable will make the variance smaller than the simple random sampling estimator (which would be there, if correlation would be zero). However, by re-arranging the above term, we can more specifically formulate in which situation the ratio estimator is superior to the simple random sampling estimator. As always, the relative efficiency is meaningful only when we deal with parametric values. Based on estimated values, the relative efficiency can be misleading.<br />
<br />
For the ratio estimator we can formulate from the above estimator of the error variance of the estimated mean <br />
<br />
:<math>\sigma\ge\frac{R\sigma_x}{2\sigma_y}=\frac{cv(x)}{2cv(y)}\,</math><br />
<br />
where <math>cv()</math> denotes the coefficient of variation either for the target variable <math>Y</math> or the ancillary variable <math>X</math>. <br />
<br />
As an interpretation of this inequality: the ratio estimator is superior to the simple random estimator if the correlation between the two variables exceeds the above expressions.<br />
<br />
:<math>\frac{R\sigma_x}{2\sigma_y}\,</math><br />
<br />
or<br />
<br />
:<math>\frac{cv(x)}{2cv(y)}\,</math><br />
<br />
There may be situations where a variable had been observed that might be used as an ancillary variable, but we are not sure whether to use the ratio estimator or not. Then, one could estimate the [[relative efficiency]]; although the conclusion might be misleading, there are cases with such high positive correlations that application of the ratio estimator is definitively indicated.<br />
<br />
It is maybe interesting to note that - when having an ancillary variable measured and available: there is more then just one option for the estimation design, the ratio estimator and the simple random sampling estimator; and both are unbiased (or, in the case of the ratio estimator, approximately unbiased). Then, of course, one will choose the estimator with the smaller error variance, and a criterion for this is given above with the correlation between target and ancillary variable.<br />
<br />
{{Exercise<br />
|message=Ratio estimator sampling examples<br />
|alttext=Example<br />
|text=Examples for Ratio estimator<br />
}}<br />
<br />
==Characteristics of ratio estimator==<br />
<br />
When the ratio estimator is used, what we implicitly do is assuming a simple linear regression through the origin between target and ancillary variable: <math>\bar{y}=\bar{r}x</math>, where the ratio, <math>r</math>, is the [[slope coefficient]]. The implicit assumption is that the relationship between <math>Y</math> and <math>X</math> goes really through the origin. In the above example with the plot area as ancillary variable, this is certainly true; if plot area is zero, then also the observation of number of stems or basal area is zero. However, if the relationship between <math>Y</math> and <math>X</math> is such that it does not generically pass through the origin, then, there will be an estimator bias.<br />
<br />
When the intercept term of ratio estimator is 0, then, the estimators presented here will be unbiased. In contrary, if the intercept term is not equal to 0, e.g. when ancillary variable is zero but target variable is not, then, the estimator here will be biased for small sample size but approximately unbiased for large sample size. <br />
<br />
==Regression estimator== <br />
<br />
If, as addressed in the chapter above, the regression between <math>Y</math> and <math>X</math> does not go through the origin, then, one may use the so-called regression estimator which explicitly establishes a simple [[linear regression]] between target and ancillary variable. As we assume that, as usual in simple linear regression, the joint mean value <math>\bar{x},\bar{y}</math> lies on the regression line, we may estimate the mean of the target variable from the sample as<br />
<br />
:<math>\bar{y}_L=\bar{y}+b\left(\mu_x-\bar{x}\right)</math><br />
<br />
where <math>b</math> is the estimated regression coefficient and <math>\mu_x</math> is the population mean of ancillary variable which must be known.<br />
<br />
The estimated variance of the estimated mean is <br />
<br />
:<math>\hat{var}(\bar{y}_L)=\frac{N-n}{N}\frac{1}{n}\frac{1}{n-2}\left\{\sum_{i=1}^n(y_i-\bar{y})^2-b^2\sum_{i=1}^n(x_i-\bar{x})^2\right\}\,</math><br />
<br />
where the term in curled parenthesis times the factor <math>1/(n-2)</math> is the mean square error of regression.<br />
<br />
As with the ratio estimator, we assume here a simple linear regression model between the two variables. If this model holds (that is, if the relationship can realistically be modeled like that, then the regression estimator is unbiased. If this is not so, then we introduce an estimator bias of unknown order of magnitude. However, it is said that this bias is particularly relevant for small sample sizes and not so for larger sample sizes.<br />
<br />
==References==<br />
<references/><br />
<br />
{{SEO<br />
|keywords=ratio estimator,ancillary variable,double sampling<br />
|descrip=The ratio estimator is an estimation design that makesuse of an ancillary variable that is correlated to the target variable<br />
}}<br />
<br />
[[Category:Sampling design]]</div>Fehrmannhttp://wiki.awf.forst.uni-goettingen.de/wiki/index.php/Bitterlich_samplingBitterlich sampling2021-03-12T08:41:35Z<p>Fehrmann: </p>
<hr />
<div>{{Ficontent}} {{video|link=https://youtu.be/wCbrwpTAYxI}}<br />
__TOC__<br />
[[File:4.4.1-fig59.png|right|thumb|300px|'''Figure 1''' Illustration of inclusion proportional to size (basal area) in Bitterlich sampling (Kleinn 2007<ref name="kleinn2007">Kleinn, C. 2007. Lecture Notes for the Teaching Module Forest Inventory. Department of Forest Inventory and Remote Sensing. Faculty of Forest Science and Forest Ecology, Georg-August-Universität Göttingen. 164 S.</ref>).]]<br />
Angle count sampling was developed by Walter Bitterlich (1948), an Austrian forester. It is some times also referred to point sampling, horizontal point sampling, variable plot sampling, angle count technique, prism cruising, angle gauge sampling, and simply Bitterlich sampling. <br />
<br />
The idea to employ [[Fixed area plots#Nested sub-plots|nested sub-plots]] was introduced because we wished to have a balanced number of [[Tree Definition|trees]] in all dimension classes; that is, we wanted to assign a higher probability of selection to the larger trees of which there are usually less in a [[Forest Definition|stand]] (see also [[sampling with unequal selection probabilities]]).<br />
<br />
For nested sub-plots, we defined a number of fixed plot areas in which the different dimension classes are observed. One may now further pursue that idea and develop a plot design in which the [[inclusion probability]] is strictly proportional to dimension. That is each tree has its particular and own [[Fixed_area_plots#The_plot_expansion_factor|plot size]] (and therefore also its particular [[inclusion probability#inclusion zone|inclusion zone]]). This is exactly what Bitterlich sampling does: from a selected [[sample point]], the neighboring trees are selected strictly proportional to their [[basal area]].<br />
<br />
While this sounds complicated, the technique itself is very simple; the only device one needs is one that produced a defined opening angle. That can be a [[dendrometer]], a [[relascope]], a [[wedge prism]] or simply your own thumb. While standing on the sample point and aiming over e.g. the thumb at the stretched out arm to the [[Diameter at breast height|''dbh''s]] of the surrounding trees; you sweep around 360° and ''count'' all trees that appear larger than your thumb. It is obvious then, that larger trees have a larger probability of being taken as sample tree. From this ''counting alone'' you can produce then an estimate of basal area per hectare. <br />
<br />
The only additional information that you need is the “calibration factor” of your measurement device, as, obviously, the number of trees counted depends on the opening angle which is produced by the instrument (in this case by your arm length and thumb width).<br />
<br />
This calibration factor is also called ''basal area factor''. Figure 1 illustrates that approach. Around each tree the individual inclusion zone is drawn. Sample points that fall into this inclusion zone will lead to an inclusion of this particular tree as sample tree. We may also imagine that these specific circular inclusion zones are concentrically placed around the sample point; then we may view them as a set of [[Fixed_area_plots#Nested_sub-plots|nested sample sub-plots]], one for each sample tree.<br />
<br><br />
<br />
==The principle of Bitterlich sampling==<br />
<br />
[[File:4.4.2-fig60.png|right|thumb|300px|'''Figure 2''' The principle of Bitterlich sampling. For each tree, there are the three depicted situations (after Kramer & Akca 1995<ref name="kramer_akca1995">Kramer H. and A. Akca. 1995. Leitfaden zur Waldmesslehre. 3rd edition. J.D. Sauerländers Verlag, Frankfurt. 266p.</ref>).]]<br />
[[File:4.4.2-fig61.png|right|thumb|300px|'''Figure 3''' Illustration of calculation of the radius of the virtual circular sub-plot for a tree with diameter <math>d_i</math> (Kleinn 2007<ref name="kleinn2007">Kleinn, C. 2007. Lecture Notes for the Teaching Module Forest Inventory. Department of Forest Inventory and Remote Sensing. Faculty of Forest Science and Forest Ecology, Georg-August-Universität Göttingen. 164 S.</ref>).]]<br />
<br />
[[File:dendrometer.jpg|thumb|A [[dendrometer]] is a simple to use instrument that can be used for Bitterlich sampling.]]<br />
[[File:Wedge prism1.jpg|thumb|Using a [[Wedge prism]] for Bitterlich sampling.]]<br />
In a first step of developing this technique we need to calculate the size of these tree‑specific virtual sample plots. For each tree it holds the statement that it is only counted (or included) if it appears at 1.3 m height larger than the opening angle of the measurement device. For each tree there are exactly three different situations as depicted in Figure 2: (1) either the tree is larger than the opening angle; then the tree is inside the virtual nested sub-plot and is counted; or (2) the tree is smaller than the opening angle (that is: completely covered by the thumb); that is a non-tally tree which is outside the virtual plot; or (3) the tree is exactly covered by the opening angle; then, the tree is exactly on the perimeter line of the virtual circular plot and the distance from the sample point to the center of that tree is (for practical purposes sufficiently good approximation to) the radius of the virtual circular plot. We need to find this radius, because the virtual plot will allow us to calculate the tree-specific expansion factor that we need to expand the per-plot observation to the per-hectare observation.<br />
<br />
The opening angle can be produced by means of a stick of length c and a horizontal panel of width 1 cm at the end (analogous to arm and thumb, for example). For the ''virtual circle plot'', for which this angle covers exactly a tree of ''dbh'' <math>d_i</math> (Figure 2) as a borderline tree, we obtain (approximately) plot radius <math>r</math> and plot area <math>F_i</math> along: <br />
<br />
:<math>\frac{d_i}{r_i}=\frac{1}{c}</math><br />
<br />
and then<br />
<br />
:<math>r=cd_i\,</math><br />
<br />
:<math>F_i={\pi}r^2={\pi}c^2{d_i}^2\,</math><br />
<br />
This is then the virtual plot in which all trees with exactly the ''dbh'' <math>d_i</math> are tallied.<br />
<br />
This corresponds exactly to the procedure with nested circular plots where also only a defined size class was observed in each nested sub-plot. If we count <math>n_i</math> trees with exactly the diameter <math>d_i</math> (or in [[diameter class]] <math>d_i</math> in case that we work with diameter classes) at a sample point, then these trees have the basal area<br />
<br />
:<math>{n_i}{g_i}=n_i\frac{\pi}{4}{d_i}^2\,</math><br />
<br />
Now, let’s expand this per-plot observation to a per-hectare figure (referring to trees with ''dbh'' <math>d_i</math> only!): the expansion factor is <math>10000/F_i</math>, as usual, where <math>F_i</math> is the area of the virtual plot. The basal area per hectare for the trees of diameter <math>d_i</math> is then<br />
<br />
:<math>G_{iHa}=\frac{10000}{F_i}n_i\frac{\pi}{4}{d_i}^2=\frac{10000}{\pi{c}^2{d_i}^2}n_i\frac{\pi}{4}{d_i}^2=\frac{2500}{c^2}n_i\,</math> <br />
<br />
Subsequently, this is done for all ''dbh''-”classes” <math>d_i</math> and summed up to produce the estimate for the total basal area per hectare comprising all <math>d</math> classes:<br />
<br />
:<math>G_{iHa}=\sum_{i}G_i=\frac{2500}{c^2}\sum_{i}n_i=\frac{2500}{c^2}N\,</math> <br />
<br />
where <math>G_{Ha}</math> = basal area per hectare of all trees. The only variable in that equation is<br />
<br />
:<math>N=\sum_{}n_i\,</math><br />
<br />
the number of observed/counted trees on the plot (over all diameter classes). This number of counted trees is multiplied with the factor <br />
<br />
:<math>k=\frac{2500}{c^2}\,</math><br />
<br />
which converts eventually the mere count into a basal-area-per-hectare value. This factor is called '''basal area factor''' and commonly denoted with <math>k</math>. It is a factor that depends exclusively on the opening angle which is used.<br />
<br />
By simply counting the <math>N</math> trees that are wider than our opening angle we can therefore produce an observation of basal area per hectare from a sample point: <br />
<br />
:<math>G_{Ha}=k*N\,</math><br />
<br />
It must be observed, however, that only basal area per hectare can be observed as simple as that. For all other variable that may be of interest (such as mean diameter or number of stems), in addition the ''dbhs'' of all trees need to be measured. That is explained further below. <br />
<br />
Basal area factors can be different. Table 1 gives the lengths <math>c</math> of this stick for basal area factors of 1,2,3 and 4 when there is a panel of 1 cm at the end of a that stick. Following the above formula, the length of that stick is calculated by <br />
<br />
:<math>c=\frac{50}{\sqrt{k}}\,</math><br />
<br />
<blockquote><br />
{|<br />
|'''Table 1.''' Length <math>c</math> of the stick with 1cm for different values of the basal area factor <math>k</math><br />
{| class="wikitable"<br />
|-<br />
! Basal area factor (<math>m^2/ha</math>)<br />
! width="100" |c-value<br />
|- <br />
|align="center"|<math>k=1m^2\,</math> <br />
|align="right"|<math>50\,</math> <br />
|-<br />
|align="center"|<math>k=2m^2\,</math><br />
|align="right"|<math>35.4\,</math> <br />
|-<br />
|align="center"|<math>k=3m^2\,</math><br />
|align="right"|<math>28.9\,</math> <br />
|-<br />
|align="center"|<math>k=4m^2\,</math><br />
|align="right"|<math>25.0\,</math> <br />
|}<br />
|}<br />
</blockquote><br />
<br />
<br />
===Exact solution===<br />
<br />
[[File:4.4.2-fig62.png|right|thumb|300px|'''Figure 4''' Critical angle principle (Kleinn 2007<ref name="kleinn2007">Kleinn, C. 2007. Lecture Notes for the Teaching Module Forest Inventory. Department of Forest Inventory and Remote Sensing. Faculty of Forest Science and Forest Ecology, Georg-August-Universität Göttingen. 164 S.</ref>).]]<br />
The principle presented above is, of course, only approximately correct, because the distance between the points at the tree where the viewing beams touch the tree is not exactly the tree diameter. For a geometrically correct calculation we would need to take that into account. The true distance between these two points is always shorter than <math>d_i</math>. Figure 4 illustrates that and is gives the basis for the following calculations.<br />
<br />
From Figure 4 we have<br />
<br />
:<math>\mbox{sin }\frac{\alpha}{2}=\frac{\frac{d_i}{2}}{r_i}\,</math><br />
<br />
and<br />
<br />
:<math>r_i=\frac{d_i}{2*\mbox{sin }\frac{\alpha}{2}}.\,</math> <br />
<br />
The area of the virtual circular plot for the tree with diameter <math>d_i</math> is<br />
<br />
:<math>A=\pi*r^2=\pi\frac{{d_i}^2}{4\left(\mbox{sin }\frac{\alpha}{2}\right)^2}\,</math><br />
<br />
and the basal area per hectare results then from <br />
<br />
:<math>G_i=10000*\frac{n_i*\frac{\pi}{4}*{d_i}^2}{\pi* \frac{{d_i}^2}{4*\left(\mbox{sin }\frac{\alpha}{2}\right)^2}}=10000*n_i*\left(\mbox{sin }\frac{\alpha}{2}\right)^2.\,</math><br />
<br />
Basal area per hectare (of all trees) is then<br />
<br />
:<math>G_{ha}=1000*\left(\sin\frac{\alpha}{2}\right)^2*\sum_{i=1}^k n_i,\,</math><br />
<br />
where,<br />
<br />
:<math>10000*\left(\sin\frac{\alpha}{2}\right)^2=k.\,</math><br />
<br />
Therefore, <br />
<br />
:<math>\sin\frac{\alpha}{2}=\frac{\sqrt{k}}{100}.\,</math><br />
<br />
This is the mathematically correct derivation. However, the differences to the approximate solution as given above are so small that they can be accepted. The following gives the results of the simplified approximate calculation of the length <math>c</math> and the correct value <math>c_c</math>:<br />
<br />
:{|<br />
|-<br />
|<math>k=1m^2\,</math><br />
|<math>c_c=49.997\,</math><br />
|instead of<br />
|<math>c=50\,</math><br />
|-<br />
|<math>k=2m^2\,</math><br />
|<math>c_c=35.352\,</math><br />
|instead of<br />
|<math>c=35.355\,</math><br />
|-<br />
|<math>k=4m^2\,</math><br />
|<math>c_c=24.995\,</math><br />
|instead of<br />
|<math>c=25\,</math><br />
|}<br />
<br />
Anyway, for all the calculations, we need to make the assumption that the tree’s cross section is a perfect circle; which it is not; so that there are more sources of geometrical inaccuracies than the one described here.<br />
<br />
==Choice of basal area factor==<br />
<br />
The choice of the ''basal area factor'' (<math>k</math>) in Bitterlich sampling is equivalent to the choice of plot size for fixed radius plot sampling. The criteria to be considered in the choice of basal area factor are visibility in the stand and average number of sample trees desired at a sample point. A decrease in the basal area factors increases the average number of sample trees tallied in. But that means also that trees at a farther distance to the sample point will be tally-trees. This will work smoothly only when visibility is accordingly<br />
<br />
In high density stands higher ''BAF'' values such as <math>4 m^2/ha</math> are used. In low density stands with good visibility, low ''BAF'' values such as <math>1 m^2/ha</math> is normally used. In temperate and boreal regions basal area factors of 1 or 2 are common; in dense tropical rain forest values of <math>9 m^2/ha</math> may be appropriate. <br />
<br />
{{info<br />
|message=Note:<br />
|text=The choice of a basal area factor refers to the entire sampling study (or in case of [[stratified sampling]] to the whole stratum). From a statistical point of view it is in principle not allowed to choose a different factor at each sampling location. Choosing a different plot size or counting factor depending on local density, would implicitly define multiple populations. This would mean the [[inclusion probability]] of a single tree would not be fixed anymore, but depending on the the local density around a sampling location. Such approaches are a matter of adaptive designs that might be notorious difficult in regard to unbiased estimators! <br />
}}<br />
<br />
Of course, the target precision of a sampling study is always also an issue. With smaller values of <math>k</math> there will be more trees per sample plot (which comes together with somewhat higher cost) and with the same sample size, one will produce a higher precision than for larger values of <math>k</math>.<br />
<br />
==Bitterlich sampling on sloped terrain==<br />
<br />
As in sampling with [[fixed area plots]], all area measures in angle count sampling are assumed to be in the map plane. When applying Bitterlich sampling in sloped terrain we need to take the slope into account; either by using an instrument that does the correction automatically, or by introducing the correction factor into the analysis if the slope is not automatically corrected (which is the case, for example, if using a stick with a panel, or the thumb); this [[slope correction]] is then to be applied from slopes of 10% and more. <br />
The correction factor is<br />
<br />
:<math>\frac{1}{\cos\alpha}\,</math><br />
<br />
so that the basal area per hectare observation from one Bitterlich sample point is <br />
<br />
:<math>G_i=k*N_i\frac{1}{\cos\alpha_i}\,</math><br />
<br />
where: <br />
<br />
:{|<br />
|-<br />
|<math>G_i=\,</math><br />
|corrected basal area (<math>m^2/ha</math>) at the <math>i^{th}</math> sampling point<br />
|-<br />
|<math>N_i=\,</math><br />
|number of trees tallied at the sampling point and<br />
|-<br />
|<math>\alpha_i=\,</math><br />
|angle of slope at the <math>i^{th}</math> sampling point.<br />
|}<br />
<br />
==Inclusion probabilities and estimators for Bitterlich sampling==<br />
<br />
For the inclusion zone approach where for each tree an inclusion zone is defined, see the corresponding article [[ Approaches to populations of sample plots|Infinite population approach]]. If used,the [[inclusion probability]] is then proportional to the size of this inclusion zone – which actually defines the probability that the correspondent tree is included in a sample.<br />
<br />
We saw, for example, that angle count sampling ([[Bitterlich sampling]]) selects the trees with a probability proportional to their [[basal area]] and we emphasized that this fact makes [[Bitterlich sampling]] so efficient for basal area estimation. In contrast, point to tree [[Distance based plots|distance sampling]], or k-tree sampling, has inclusion zones that do not depend on any individual tree characteristic but only on the spatial arrangement of the neighboring trees; therefore, point-to tree distance sampling is not particularly precise for any tree characteristic.<br />
<br />
In [[Bitterlich sampling]], the selection probability of a particular tree ''i'' results from the inclusion zone ''F<sub>i</sub>'' and the size of the reference area, for example the hectare<br />
<br />
:<math>\pi_i = \frac {F_i}{10000}</math><br />
<br />
with the Horvitz-Thompson estimator, we have the total <br />
<br />
:<math>\hat \tau = \sum_{i=1}^m \frac {y_i}{\pi_i}</math><br />
<br />
for any tree attribute <math>y_i</math>. Applied to estimating basal area <math>y_i = g_i = \frac {\pi}{4} d_i^2</math> and its per hectare estimation, we have <br />
<br />
:<math>\hat \tau = \sum_{i=1}^m \frac {y_i}{\pi_i} = \sum_{i=1}^m \cfrac {\cfrac {\pi}{4} d_i^2}{\cfrac {F_i}{10000}}</math><br />
<br />
and with <math> F_i = \pi r_i^2 = \pi c^2 \, d_i^2</math>, we have the same as [[Bitterlich sampling]]<br />
<br />
:<math>\hat \tau = \sum_{i=1}^m \frac {y_i}{\pi_i} = \sum_{i=1}^m \cfrac {\cfrac {\pi}{4} d_i^2}{\cfrac {\pi c^2 \, d_i^2}{10000}} = \frac {2500 \pi}{\pi c^2} \sum_{i=1}^m \frac {d_i^2}{d_i^2} = \frac {2500}{c^2} m</math><br />
<br />
which is the estimated basal area per hectare from one sample point where '''m''' trees were tallied. The factor 2500/c² is the ''basal area factor''.<br />
<br />
==Estimation of number of stems==<br />
<br />
Angle count sampling allows a very fast estimation of basal area per hectare by simply counting trees. This procedure alone, however, does '''not''' allow making estimations for other variables. <br />
If we are interested in estimating number of stems per hectare, for example, we need to measure the diameters of all counted trees in order to be able to explicitly calculate their virtual circle plots and to make a correct expansion of their per-plot values to the corresponding per-hectare values. <br />
<br />
Each diameter has its own expansion factor, namely<br />
<br />
:<math>EF_i=\frac{10000}{F_i},\,</math><br />
<br />
where <math>F_i</math> is the area of the corresponding ''“virtual plot”''. <br />
For example for <math>d=30cm</math> (and <math>k=2</math>, that is, <math>c=35.52</math>), we get the following expansion factor:<br />
<br />
:<math>EF_i=\frac{10000}{F_i}=\frac{10000}{\pi{c^2}{d_i}^2}=\frac{10000}{\pi{35.35}^2{0.3}^2}=28.29\,</math><br />
<br />
That tree with diameter <math>d_i=30cm</math>, counted from the particular sample point, represents 28.29 trees per hectare with that diameter class; and, because the basal area factor is <math>k=2 m^2/ha</math>, this tree does also represent <math>2m^2</math> of basal area per hectare. <br />
<br />
The calculation of <math>EF_i</math> needs to be done for all diameters found on a sample point. Each diameter represents a different number of trees per hectare. At the end, to determine the total number of trees per hectare, these values need to be added.<br />
<br />
In some textbooks there is another (equivalent – but somewhat more complicated and less intuitive) derivation of number of stems per hectare: one determines how many trees per hectare are represented by one counted tree: <br />
<br />
For <math>k=2</math> one single tree represents <math>2m^2</math> of basal area per hectare. The basal area of one tree of, for example, <math>d=30cm</math> is<br />
<br />
:<math>g={\pi}r^2=0.0707m^2\,</math><br />
<br />
It follows that one counted tree “stands for”<br />
<br />
:<math>\frac{2.0}{0.0707}=28.29</math> trees/ha<br />
<br />
in this diameter class 30cm. Obviously, the result is the same as for the more consistent expansion factor approach.<br />
<br />
If we are interested in calculating the mean diameter from a Bitterlich sample, we can not simply take the mean diameter of the sample trees (as we do it for fixed area plot sampling) but we first need to derive the diameter distribution by means of the diameter-wise expansion and then calculate the mean diameter from that distribution.<br />
<br />
===Example===<br />
<br />
Calculation of number of stems per hectare from one Bitterlich sample point.<br />
<br />
In Bitterlich sampling, the estimation of number of stems can not be done with a simple extrapolation of the counted number of trees, but we need to take into account the different selection probabilities, that is, the different sizes of the virtual nested sample plots (which is equivalent to the different size of the selection areas).<br />
<br />
Each sampled tree “represents” a different number of trees in its ''dbh'' class ‑ depending on its ''dbh''. A small tree, if sampled, represents a much larger number of small trees than a large tree if sampled. The calculation is illustrated in Table 1 for one Bitterlich sample with <math>baf=2</math>. 10 trees were observed with their respective ''dbh'' measured. ''dbh'' and ''baf'' allow calculating the diameter-specific virtual nested circular plot area from<br />
<br />
:<math>F_i={\pi}c^2d_i^2\,</math><br />
<br />
and, therefore, also the ''dbh''-specific expansion factor. Each tree represents a certain number of trees per hectare and the sum of all those figures is the estimated number of trees per hectare from this one Bitterlich sample. If ''n'' sample plots were taken, then for each one, this calculation needs to be done. The standard error of the estimation of number of stems per hectare results then from the results of these ''n'' estimations per plot.<br />
<br />
<blockquote><br />
{| <br />
|'''Table 1.''' Illustration of estimation of number of trees per hectare from one point-relascope sample with <math>baf=2</math>.<br />
{| class="wikitable"<br />
! dbh of sample tree<br />
! Area of virtual circular<br>nested sub-plot<br />
! Expansion factor from plot<br>to per-hectare values<br />
! “represented” number of<br>stems per hectare<br />
|-<br />
|align="right"|10<br />
|align="right"|39.27<br />
|align="right"|254.65<br />
|align="right"|254.65<br />
|-<br />
|align="right"|12<br />
|align="right"|56.55<br />
|align="right"|176.84<br />
|align="right"|176.84<br />
|-<br />
|align="right"|15<br />
|align="right"|88.36<br />
|align="right"|113.18<br />
|align="right"|113.18<br />
|-<br />
|align="right"|16<br />
|align="right"|100.53<br />
|align="right"|99.47<br />
|align="right"|99.47<br />
|-<br />
|align="right"|18<br />
|align="right"|127.23<br />
|align="right"|78.60<br />
|align="right"|78.60<br />
|-<br />
|align="right"|21<br />
|align="right"|173.18<br />
|align="right"|57.74<br />
|align="right"|57.74<br />
|-<br />
|align="right"|23<br />
|align="right"|207.74<br />
|align="right"|48.14<br />
|align="right"|48.14<br />
|-<br />
|align="right"|36<br />
|align="right"|508.94<br />
|align="right"|19.65<br />
|align="right"|19.65<br />
|-<br />
|align="right"|42<br />
|align="right"|692.72<br />
|align="right"|14.44<br />
|align="right"|14.44<br />
|-<br />
|align="right"|61<br />
|align="right"|1461.23<br />
|align="right"|6.84<br />
|align="right"|6.84<br />
|-<br />
|'''Total'''<br />
|<br />
|<br />
|align="right"|'''869.54''' <br />
|}<br />
|}<br />
</blockquote><br />
<br />
In this example, the expansion factor is equal to the represented number of trees per hectare because the calculation has been done for each individual tree and not for dbh-classes in which there are more than one tree.<br />
<br />
==See also==<br />
[https://www.youtube.com/watch?v=dMk9jNF-kHo Video:Walter Bitterlich, An invention goes around the world]<br />
<br />
<br />
[https://www.youtube.com/watch?v=c1NyzVuV-QU&feature=youtu.be Video:Walter Bitterlich, Eine Erfindung geht um die Welt]<br />
<br />
<br />
==References==<br />
<references/><br />
<br />
{{SEO<br />
|keywords=point sampling,bitterlich sampling,angle count sampling,horizontal point sampling,variable plot sampling,angle count technique,prism cruising,angle gauge sampling,basal area factor,stem number estimation,plot design,forest inventory,<br />
|descrip=Angle count sampling was developed by Walter Bitterlich, an Austrian forester.<br />
}}<br />
<br />
<br />
[[Category:Plot design]]<br />
[[Category:article of the month]]</div>Fehrmannhttp://wiki.awf.forst.uni-goettingen.de/wiki/index.php/CaliperCaliper2021-03-12T08:40:26Z<p>Fehrmann: </p>
<hr />
<div>{{Ficontent}} {{video|link=https://youtu.be/17NPkVXteiI}}<br />
==General description==<br />
The caliper is one of the most efficient tools to measure [[Tree diameter|stem diameters]] directly wherever one has direct access to the [[Tree Definition|stem]]. An ordinary caliper, constructed of metal or wood, consists of a graduated beam with two arms perpendicular to it. One arm is used at the origin of the scale and the other arm slides (Figure 1). When the beam is pressed against the tree and the arms are closed, the tree diameter can be read on the scale. <br />
<br />
There are some calipers with a set of fixed arms also. The graduations are calibrated so that, when the fork is placed on the tree, the points of tangency indicate the tree diameter. Nowadays, aluminum calipers have replaced the wooden and steel calipers. As wooden calipers have tendency to wearing and tearing whereas steel ones are too heavy to carry in forest. <br />
<br />
[[File:caliper.jpg|thumb|right|200px|Typical example of a caliper (60 cm).]]<br />
<br />
[[Electronic calipers]] had been developed that measure diameter electronically; the diameter is then either read from a display or directly stored in a memory chip or the measurement is via wireless directly transferred to a mobile computer that the field crew carries along. There are various successful examples of using electronic calipers; but in most [[Forest inventory|inventories]] the benefit in terms of rationalization of procedures is relatively modest compared to the high cost of electronic calipers.<br />
<br />
==Handling==<br />
<br />
# press the beam of the caliper firmly against the tree stem perpendicular to the axis of it to minimize maladjustment of the moveable arm<br />
# read the diameter directly from the scale<br />
<br />
{{info<br />
|message=Note:<br />
|text= It is important that all types of calipers be held perpendicular to the axis of the tree stem at the point of measurement ([[Diameter at breast height|breast height]], usually). If the stem cross sections are irregular, some times two caliper readings are recommended at right angles; then, the mean value of the two readings is taken as diameter.<br />
}}<br />
<br />
<br />
<br />
<gallery widths=300px heights=300px><br />
File:using a caliper.jpg|How to use a caliper.<br />
File:caliper-01.jpg<br />
</gallery><br />
<br />
==Applications==<br />
<br />
* [[Diameter at breast height]]<br />
<br />
<br />
==Related articles==<br />
* [[Bitterlich sampling]]<br />
* [[Diameter tape]]<br />
<br />
<br />
==References==<br />
<br />
<references/><br />
<br />
{{SEO<br />
|keywords=caliper,tree diameter,diameter at breast height,forest inventory,tree measurement,standing tree sampling<br />
|descrip=The caliper is the most common device to measure tree diameters at breast height.<br />
}}<br />
{{Studienbeiträge}}<br />
[[Category:Measurement devices]]</div>Fehrmannhttp://wiki.awf.forst.uni-goettingen.de/wiki/index.php/Caliper_vs._diameter_tapeCaliper vs. diameter tape2021-03-12T08:39:54Z<p>Fehrmann: </p>
<hr />
<div>{{Ficontent}} {{video|link=https://youtu.be/17NPkVXteiI}}<br />
==Characteristics and advantages and disadvantages of calipers and tapes.==<br />
<br />
* Tapes are easier to carry than calipers in many circumstances, in particular when understory is dense. <br />
<br />
* Measuring diameter with calipers is faster than with tape.<br />
<br />
* The caliper has an upper bound of tree diameters that can measured. Bigger trees can not be measured. With the tape, there is the possibility to measure the diameter of any big tree as one may determine the perimeter by putting various tape together.<br />
<br />
* For felled logs, calipers can more easily be used whereas it is often difficult to pass under the logs with tapes.<br />
<br />
* While there is, for geometrical reasons, a systematic overestimation when using the tape, we have more variability in caliper measurements, because for irregularly shaped trees the reading depends on the direction from which the caliper is used<br />
<br />
* On permanent observation plots, where all trees are measured in regular intervals, it is important to secure consistency between the repeated measurements. Consistency is created by using a diameter tape. When using a caliper, it is important to defined and record from which direction each tree had been measured so that the following measurements can be done from exactly the same direction (e.g. the open end of the caliper’s beam always shows to the plot center)<ref name="kleinn2007">Kleinn, C. 2007. Lecture Notes for the Teaching Module Forest Inventory. Department of Forest Inventory and Remote Sensing. Faculty of Forest Science and Forest Ecology, Georg-August-Universität Göttingen. 164 S.</ref>.<br />
<br />
==References==<br />
<references/><br />
[[Category:Tree diameter]]<br />
[[Category:Measurement devices]]</div>Fehrmannhttp://wiki.awf.forst.uni-goettingen.de/wiki/index.php/Diameter_tapeDiameter tape2021-03-12T08:39:22Z<p>Fehrmann: </p>
<hr />
<div>{{Ficontent}} {{video|link=https://youtu.be/17NPkVXteiI}}<br />
Tree diameter can also be determined from the measurement of [[circumference]] <math>C</math>. Then, assuming a perfectly circular shape of the [[Tree Definition|stems]] cross section at the height of measurement, the diameter <math>d</math> is simply calculated as <br />
<br />
:<math>d=\frac{C}{\pi}\,</math><br />
<br />
There are also diameter tapes from which the tree diameter can be directly read. Some diameter tapes are graduated on both sides, on one side with linear centimeter scale for reading circumference, whereas the other side reads the diameter scale. It is important, not to confuse the readings! <br />
<br />
[[File:diameter tape.png|thumb|right|300px|Example for a diameter tape.]]<br />
<br />
[[File:using diameter tape.jpg|thumb|right|300px|How to use a diameter tape.]]<br />
<br />
[[File:using diameter tape reading.jpg|thumb|right|300px|Direct reading of diameter (here 35.9 cm).]]<br />
<br />
Circular shape is assumed when inferring from circumference to diameter. However, trees have never a perfect circular cross section; there are always irregularities. If we then assume that the stem has a circular cross section with the shortest perimeter for a given cross-sectional area, then we come to a systematic overestimation of the cross-sectional area and also of the diameter which is derived from this cross-sectional area.<br />
<br />
Table 1 illustrates that with a simple and theoretic example: we imagine trees that have exactly <math>1m^2</math> of [[basal area]] but different elliptic shapes of their cross-sections. We are now interested what happens when we measure the perimeter and then infer to basal area of dbh assuming a circular shape. For the circle, perimeter <math>p_c=2{\pi}r</math>, and area <math>f_c={\pi}r</math>. The shape of an ellipse is defined by the two radii <math>a</math> and <math>b</math>. Then, the area of an ellipse is calculated by <math>f_e={\pi}ab</math>; there is no formula to calculate the perimeter of an ellipse <math>p_e</math> but it can only be approximated by <br />
<br />
:<math>p_e \approx \pi\left(a*b\right)\left(1+\frac{3t}{10+\left(\sqrt{4-3t}\right)}\right)\,</math> <br />
<br />
where <br />
<br />
:<math>t=\left(\frac{a-b}{a+b}\right)^2\,</math> <br />
<br />
Different ellipses are defined in Table 1 by different values of <math>a</math> and <math>b</math>, all having the same basal area of <math>1m^2</math>.<br />
<br />
<blockquote> <br />
{|<br />
|'''Table 1''' Illustration of the effect of assuming a perfect circular cross section when determining basal area and dbh by tape-measurements (Kleinn 2007<ref name="kleinn2007">Kleinn, C. 2007. Lecture Notes for the Teaching Module Forest Inventory. Department of Forest Inventory and Remote Sensing. Faculty of Forest Science and Forest Ecology, Georg-August-Universität Göttingen. 164 S.</ref>).<br />
{| class="wikitable"<br />
! colspan="8" | Area 1m²<br />
|-<br />
| width="50" align="center" |'''r''' <br />
| width="50" align="center" |'''b''' <br />
| width="50" align="center" |'''a''' <br />
| width="50" align="center" |'''t''' <br />
| width="50" align="center" |'''Area''' <br />
| '''Perimeter Ellipse''' || '''''r'' from Perimeter''' || '''Area from Perimeter'''<br />
|-<br />
| align="right" | 0.5642 <br />
| || || <br />
| align="right" | 1.0000 <br />
| align="right" |3.5449 <br />
| align="right" |0.5642 <br />
| align="right" |1.0000<br />
|-<br />
|<br />
| align="right" | 0.50<br />
| align="right" | 0.6366<br />
| align="right" | 0.0144<br />
| align="right" | 1.0000<br />
| align="right" | 3.5837<br />
| align="right" | 0.5704<br />
| align="right" | 1.0220<br />
|-<br />
|<br />
| align="right" | 0.40<br />
| align="right" | 0.7958<br />
| align="right" | 0.1095<br />
| align="right" | 1.0000<br />
| align="right" | 3.8602<br />
| align="right" | 0.6144<br />
| align="right" | 1.1858<br />
|-<br />
|<br />
| align="right" | 0.30<br />
| align="right" | 1.0610<br />
| align="right" | 0.3127<br />
| align="right" | 1.0000<br />
| align="right" | 4.6171<br />
| align="right" | 0.7348<br />
| align="right" | 1.6964<br />
|-<br />
|<br />
| align="right" | 0.20<br />
| align="right" | 1.5915<br />
| align="right" | 0.6033<br />
| align="right" | 1.0000<br />
| align="right" | 6.5157<br />
| align="right" | 1.0370<br />
| align="right" | 3.3784<br />
|-<br />
|<br />
| align="right" | 0.10<br />
| align="right" | 3.1831<br />
| align="right" | 0.8819<br />
| align="right" | 1.0000<br />
| align="right" | 12.7584<br />
| align="right" | 2.0306<br />
| align="right" | 12.9535<br />
|}<br />
|}<br />
</blockquote><br />
<br />
<br />
A circle of <math>1m^2</math> has a radius of <math>0.5642m</math> and in this case the [[perimeter measurement]] leads exactly to the correct diameter value. However, if we have an elliptical cross-section with the two radii <math>a=0.6366</math> and <math>b=0.5m</math> and measure the perimeter, then the resulting diameter is slightly larger (<math>0.5704m</math>) and also the basal area is overestimated by about <math>2%</math>. For extreme elliptical shapes, this overestimation is considerable. For smaller deviations from the circular shape the overestimation is probably in an order of magnitude that does not have major practical relevance in [[forest inventory]]. But you should know that, from a theoretical viewpoint, this overestimation is always there. <br />
<br />
In many countries, the diameter tape is the most common [[:Category:Measurement devices|device]] used in measuring [[Tree diameter|tree diameters]], because it is small, handy and inexpensive. One should take care that the tape is correctly and horizontally positioned at the point of measurement; that it is not twisted and is set firmly around the tree trunk. The particular position of zero mark in different tapes is not standardized; therefore specific care should be taken to determine the zero mark on tape before the field measurement.<br />
<br />
{{info<br />
|message=remember!<br />
|text= When applying a diameter tape we implicitly presume a circular shape of the cross sectional area! This is a model assumption and the final error occurring in case of an elliptic cross sectional area is a model error. In general the final results are only as good as the model we imply or in other words, are dependent on how good the model is able to describe the actual target variable diameter!<br />
}} <br />
<br />
<br />
==References==<br />
<references/><br />
<br />
{{SEO<br />
|keywords=diameter tape,tree diameter,diameter at breast height,forest inventory,tree measurement,standing tree sampling<br />
|descrip=The diameter tape determines the tree diameter by measuring its circumference.<br />
}}<br />
<br />
[[Category:Measurement devices]]</div>Fehrmannhttp://wiki.awf.forst.uni-goettingen.de/wiki/index.php/DendrometerDendrometer2021-03-12T08:38:16Z<p>Fehrmann: </p>
<hr />
<div>{{Ficontent}} {{video|link=https://youtu.be/wCbrwpTAYxI}}<br />
==Kramer's Dendrometer==<br />
[[File:dendrometer.jpg|thumb|left|Dendrometer]]<br />
[[File:dendrometer2.png|thumb|300px]]<br />
[[File:dendrometer3.png|thumb|300px|Measurement of [[tree height]] and partitioning of volume]]<br />
[[file:dendrometer01_wzp1.jpg|thumb|300px|BAF1]]<br />
<br />
==General description==<br />
The Dendrometer (according to [http://de.wikipedia.org/wiki/Horst_Kramer Kramer]) is a simple measurement device that is suitable for the estimation of [[basal area]] per hectare based on the principle of [[Bitterlich sampling]], stand volume and rough estimates of tree height and volume partitioning of single trees (Kramer 1990<ref name="kramer">Kramer, H. 1990. Nutzungsplanung in der Forsteinrichtung. 2., überarbeitete und erweiterte Auflage. J.D. Sauerländer’s Verlag, Frankfurt/Main.</ref>, Akca 2001<ref name="akca">Akca, A. 2001. Waldinventur. J.D. Sauerländer's Verlag. Frankfurt am Main, 193 S.</ref>). The Dendrometer II is manufactured and distributed by the <br />
[http://www.uni-goettingen.de/en/67094.html Department of Forest Inventory and Remote Sensing] at the university of Göttingen.<br />
<br />
===Height measurement===<br />
It is possible to roughly estimate [[tree height]] based on the [[The geometric principle|geometric principle]] with the Dendrometer. Therefore hold the device vertically in front of the eye (opening on top) and aim at the tree to be measured. Adjust the distance of the Dendrometer from the eye or your distance from the tree until the points ''k'' and ''h'' correspond to the tree tip resp. to the bottom of the trunk. The relation of the section ''hi'' to line ''hk'' is 1:10. the height of point ''i'' corresponds to 1/10 of the tree height. In order to calculate the tree’s height measure the distance from ''i'' on the tree to the tree base and multiply by 10. It should be considered that the measurement of tree heights is just a "side function" of this device, while the main purpose is in estimation [[basal area]].<br />
<br />
===Estimation of basal area per hectare===<br />
The estimation of [[basal area]] is based on the principle of [[Bitterlich sampling|relascope sampling]] invented by [http://de.wikipedia.org/wiki/Walter_Bitterlich Walter Bitterlich] (Bitterlich 1952<ref name="bitterlich">BITTERLICH, W., 1952. Die Winkelzählprobe. Forstwissenschaftliches Centralblatt, 215-225.</ref>). <br />
Aim at the dbh of each tree in a 360° sweep (from your standpoint). The basal area is a function of the angle and the number of stems which appear wider than the respective angle opening. Trees with width smaller than ''fg'' are not included in the sample (counted).<br />
If you hold the dendrometer vertically at a distance of 50 cm before the eye ''r'' (distance to the eye is fixed with ''rs'' is 50 cm), then the plate with the width 1 creates an angle for which every counted tree corresponds to 1 m² basal area per hectare ([[Bitterlich_sampling#Choice_of_basal_area_factor|basal area factor]] = 1).<br />
<br />
When a width of 2 or 4 is used, the resulting basal area (number of trees counted) must be multiplied by the corresponding factor (2 or 4) in order to obtain estimates of basal area per hectare.<br />
With relascope sampling on slopes it is necessary to multiply the estimated basal area per hectare with the coefficient k (k=1/cos α) for the respective maximal inclination of slope from the table on the Dendrometer.<br />
<br />
===Determining value of single trees by estimating volume sections===<br />
Set up the dendrometer in the same way as for height measurement. Through the markings ''b'', ''c'', ''d'' the stem is divided to 4 sections of equal volume. <br />
<br />
===Estimating stem volume with "form height"===<br />
For the 4 main tree species in northern Germany (Ei, Bu, Fi, Ki) and a given or measured mean stand height (''hm'', which should be approximately the height of the mean basal area tree) the "form-height factor" (FH-Tarif) can be read off the table on the dendrometer. Calculate stand volume per hectare in m³ including bark with the formula <math>V=G*FH \,</math>.<br />
<br />
==Instruction manual==<br />
Thanks to the input and efforts of many international researchers visiting our department, a lot of translations are available for the short instruction manual. It can be downloaded in:<br />
<br />
*[[:File:Gebrauchsanleitung Dendrometer Englisch.pdf|English]]<br />
*[[:File:Gebrauchsanleitung Dendrometer Deutsch.pdf|German]]<br />
*[[:File:Gebrauchsanleitung Dendrometer Afrikaans.pdf|Afrikaans]]<br />
*[[:File:Gebrauchsanleitung Dendrometer Albanisch.pdf|Albanian]]<br />
*[[:File:Gebrauchsanleitung Dendrometer Arabisch.pdf|Arabic]]<br />
*[[:File:Gebrauchsanleitung Dendrometer Chinesisch.pdf|Chinese]]<br />
*[[:File:Gebrauchsanleitung Dendrometer Dänisch.pdf|Danish]]<br />
*[[:File:Gebrauchsanleitung Dendrometer Französisch.pdf|French]]<br />
*[[:File:Gebrauchsanleitung Dendrometer Griechisch.pdf|Greek]]<br />
*[[:File:Gebrauchsanleitung Dendrometer Indonesisch.pdf|Bahasa Indonesia]]<br />
*[[:File:Gebrauchsanleitung Dendrometer Italienisch.pdf|Italian]]<br />
*[[:File:Gebrauchsanleitung Dendrometer Japanisch.pdf|Japanese]]<br />
*[[:File:Gebrauchsanleitung Dendrometer Koreanisch.pdf|Korean]]<br />
*[[:File:Gebrauchsanleitung Dendrometer Rumänisch.pdf|Romanian]]<br />
*[[:File:Gebrauchsanleitung Dendrometer Russisch.pdf|Russian]]<br />
*[[:File:Gebrauchsanleitung Dendrometer Slowakisch.pdf|Slovakian]]<br />
*[[:File:Gebrauchsanleitung Dendrometer Spanisch.pdf|Spanish]]<br />
*[[:File:Gebrauchsanleitung Dendrometer Türkisch.pdf|Turkish]]<br />
*[[:File:Gebrauchsanleitung Dendrometer Ungarisch.pdf|Hungarian]]<br />
<br />
==References==<br />
<references/><br />
<br />
<br />
<br />
<br />
<br />
[[Category:Measurement devices]]</div>Fehrmannhttp://wiki.awf.forst.uni-goettingen.de/wiki/index.php/Slope_correctionSlope correction2021-03-12T08:36:15Z<p>Fehrmann: </p>
<hr />
<div>{{Ficontent}} {{video|link=https://youtu.be/InPERYNxQ0E}}<br />
[[File:4.2.6-fig52.png|right|thumb|300px|'''Figure 7''' A diagram showing an area of reference and projected area into the map plane on a sloping terrain (Kleinn et al 2002<ref name="kleinn2002">Kleinn C., B. Traub and C. Hoffmann 2002. A note on the slope correction and the estimation of the length of line features. Canadian Journal of Forest Research 32(4):751-756.</ref>).]]<br />
[[File:4.2.6-fig53.png|right|thumb|300px|'''Figure 8''' Distribution of inclination of forest plots of the second Swiss National Forest Inventory (class 0, 0-9.99%, class 1, 10-19.99%, etc.)(Kleinn et al 2002<ref name="kleinn2002">Kleinn C., B. Traub and C. Hoffmann 2002. A note on the slope correction and the estimation of the length of line features. Canadian Journal of Forest Research 32(4):751-756.</ref>).]]<br />
[[File:4.2.6.1-fig54.png|right|thumb|300px|'''Figure 9''' Mean correction factor for line features as a function of terrain inclination <math>\alpha</math> (bold line), assuming that the lines have a uniform distribution of angular deviation from the gradient vector. The dashed line gives the standard correction factor cos(<math>\alpha</math>)(Kleinn et al 2002<ref name="kleinn2002">Kleinn C., B. Traub and C. Hoffmann 2002. A note on the slope correction and the estimation of the length of line features. Canadian Journal of Forest Research 32(4):751-756.</ref>).]]<br />
[[File:4.2.6.2-fig55.png|right|thumb|300px|'''Figure 10''' Volume over elevation estimated from the second Swiss National Forest Inventory (Kleinn et al 2002<ref name="kleinn2002">Kleinn C., B. Traub and C. Hoffmann 2002. A note on the slope correction and the estimation of the length of line features. Canadian Journal of Forest Research 32(4):751-756.</ref>).]]<br />
==Slope correction==<br />
Results from forest inventories are area-related; examples are [[volume|volume/ha]], [[basal area|basal area/ha]], [[density|number of stems/ha]]. The area to which these measures refer is the [[map area]]; that is the horizontal projection of the terrain into the [[map plane]]. If we lay out a circular [[sample plot]] with a defined radius on the slope, the projected area in the map plane will be different. This is obviously not desired but if we wish to make an extrapolation of the per-plot observation to the per hectare observation, then '''all''' related areas must refer to the same reference system, and that is the map plane. <br />
<br />
As a consequence, if we have a defined [[plot area]] that is to be used for all sample plots, this area is defined in the map plane. If plots are to be laid out in sloped terrain, the plot area on the slope must be larger such that the target plot area results when the plot is horizontally projected. This is illustrated for circular fixed area plots in Figure 7. <br />
<br />
If we define (Figure 7) the [[target plot area]] in the map plane, then this plot becomes an ellipse when projected in the oblique plane, and this ellipse has a larger area than the original plot in the map plane. If electronic range finders that can also measure [[Measuring slope|slope angle]] are used for distance measurement, then this ellipse can easily be laid out in the field where the two radii of the ellipse are automatically calculated. If <math>r</math> is the radius of the original plot in the map plane and <math>\alpha</math> the slope angle, then the two radii of the ellipse are <math>r</math> and<br />
<br />
:<math>\frac{r}{\mbox{cos }\alpha}\,</math><br />
<br />
If we do not have the possibility of making [[automatic correction]], another way of slope correction must be applied. The idea is to find the radius of the circle that has an area equal to the one of the ellipse. By that, we guarantee that the plot has the target area when projected into the map plane; by this procedure, we accept that the plot as projected into the map plane has an ellipse shape. <br />
<br />
Let´s assume that the circular sample plot has the defined area (in the plane)<br />
<br />
:<math>F_p={\pi}r^2\,</math><br />
<br />
If projected to a slope with slope angle <math>\alpha</math>, an ellipse results with radii <math>r</math> and<br />
<br />
:<math>\frac{r}{\mbox{cos }\alpha}\,</math><br />
<br />
which has the area (on the slope),<br />
<br />
:<math>F_{slope}=\pi r\frac{r}{\mbox{cos }\alpha}=\frac{\pi r^2}{\mbox{cos }\alpha}\,</math><br />
<br />
This is larger than the one in the map plane. We wish now to find the radius <math>r_{slope}</math> of the circle that has that area and we find it as<br />
<br />
:<math>r_{slope}=\frac{r}{\sqrt{\mbox{cos }\alpha}}\,</math><br />
<br />
That means: in sloped terrain, the original plot radius needs to be multiplied by the slope correction factor<br />
<br />
:<math>\frac{1}{\sqrt{\mbox{cos }\alpha}}\,</math><br />
<br />
By that, a larger circle is generated which has an area equal to the area to the ellipse that would result if we projected the original plot size into the map plane. The correction factor is obviously always larger than 1.<br />
<br />
{{info<br />
|message=Info<br />
|text=The angle <math>\alpha\,</math> refers to degree! If a slope angle is given in % it has to be converted to degree before (for e.g. 30% slope the angle in degree is calculated as arctan(30/100)=16.7 degree)!<br />
}}<br />
<br />
<br />
For application in forest inventories, slope correction factors are usually pre-calculated and tabulated. Alternatively, it is programmed in the field computers. <br />
<br />
The slope correction is commonly applied for slopes larger than 10%. For smaller slopes the deviation is so small that it is accepted without correction. <br />
<br />
Slope correction is in particular relevant, when inventorying forests in hilly and mountainous regions. Figure 8 illustrates that for the National Forest Inventory of Switzerland. There, only about 10% of the sample plots are in terrain with slopes less than 10%, and a considerable percentage (around 5%) are even on slopes with more than 100% inclination!<br />
<br />
===Slope correction for line features===<br />
<br />
In some inventories, also linear features are measured (or like in [[line sampling]] are used as [[Sampling design and plot design|observation unit]]), like length of roads; length of water (creeks); and length of [[Forest Boundary|forest boundary]]. In principle, we can proceed like with all other variables: observing what we find on the plot and use the expansion factor to extrapolate to the whole of the inventory region (or to a per-hectare-estimate). However, in sloped terrain, the correct correction procedure must obviously be different from the described general slope correction, because the actual projected length of a line feature at the slope is a function of the actual slope of the line itself which is not the overall slope of the terrain!<br />
<br />
The projection of an individual line segment must be based on its actual inclination. Application of any gross correction factor to all sampled line segments would not be correct. However, if no information on the orientation of the individual line segments is available; one may want to employ a mean correction factor, if we assume that the lines are uniformly distributed over all possible directions.<br />
<br />
The correction line in Figure 9 was derived numerically, it is approximated by <br />
<br />
:<math>\mbox{cos}^2\frac{\alpha}{2.1}\,</math><br />
<br />
It assumes uniform angular distribution of the orientation of line segments relative to the slope gradient. It gives good results for the range of slopes that matter for forest inventories, where the deviation from the numerically derived values is less than 1% for inclinations <br />
<br />
:<math>0\le\alpha\le60°\,</math><br />
<br />
===Effects of slope correction===<br />
<br />
Concern is some time expressed that the usual slope correction ‑ that, while maintaining the target plot area, changes the shape of the plot from the correct ellipse to a larger circle ‑ introduces a systematic error in particular in cases where the target variable has a gradient along the slope gradient; then, more “''average''” elements will be observed (in segment III and IV of Figure 7) in the size-adjusted circle in the field, and lesser observations are made along the gradient (that is, in segments I and II). <br />
In fact, there are variables that change with altitude, i.e. have a gradient along the slope. Figure 10 illustrates that for volume per hectare as a result from the Swiss National Forest Inventory (here, it is interesting to note the difference between coniferous and broadleaved species). Although the gradient is significant, however, it is too small to be relevant in the context of the slope correction. For volume/ha, at least, we do not expect the slope correction to cause problems.<br />
<br />
===Example: Slope correction with fixed area plots===<br />
{{info<br />
|message=Info<br />
|text=The angle <math>\alpha\,</math> refers to degree! If a slope angle is given in % it has to be converted to degree before (for e.g. 30% slope the angle in degree is calculated as arctan(0.3)=16.7 degree)!<br />
}}<br />
<br />
Suppose, we measure a 1000 m² fixed area plot on a slope of 30% (=16.7 degree) and we count 10 trees in that plot. Without correcting for slope, we calculate an expansion factor of 10 and the estimation is 100 trees per hectare.<br />
<br />
However, the 1000 m² plot on the slope (''r''=17.84''m'') becomes an ellipse when projected to the map plane with the two radii <math>r_1=17.84m</math> and <math>r_2=17.088m=r_1*\mbox{cos }\alpha</math> and an area of<br />
<br />
:<math>F_E={\pi}r_1r_2=957.8m^2=1000m^2*\mbox{cos }\alpha\,</math> <br />
<br />
Where, the factor of 10 is valid for ''all'' plots on slopes less than 10%, the correct expansion factor for the 1000 m² circle at 30% has to be larger, as the circle becomes a smaller ellipse when projected to the map plane. The calculation is as follows: <br />
<br />
:<math>\frac{10000m^2}{1000m^2}*\mbox{cos }\alpha=\frac{10000}{957.8}=10.44\,</math><br />
<br />
and the correct estimation for the number of trees per hectare is:<br />
<br />
:<math>104.4=100*\frac{1}{\mbox{cos }\alpha}\,</math><br />
<br />
{{info<br />
|message=Observe!<br />
|text=Please observe that we are not correcting the radius of a sample plot (like explained above) but assume that a plot of 1000m² was installed on the slope. Therefore we here correct the result. Don't confuse the two different approaches (and respective formulas)!<br />
}}<br />
<br />
<br />
<br />
==References==<br />
<references/><br />
<br />
{{SEO<br />
|keywords=fixed area plots,plot design,inclusion zone concept,plot expansion factor,cluster plots,nested sub-plots,slope correction,sample plots,forest inventory,<br />
|descrip=Fixed area sample plots can have various different shapes including circular, square, rectangular and strip.<br />
}}<br />
<br />
<br />
[[Category:Plot design]]</div>Fehrmannhttp://wiki.awf.forst.uni-goettingen.de/wiki/index.php/3D-_Terrestrial_laser_scanning3D- Terrestrial laser scanning2021-03-12T08:35:20Z<p>Fehrmann: </p>
<hr />
<div>{{Ficontent}} {{video|link=https://youtu.be/s-lvK3uwkqw}}<br />
<br />
Three dimensional terrestrial laser scanning (TLS) is a ground based technique to measure the position and dimension of objects in the three dimensional space. Thereby, a laser beam is emitted from a laser light source and used to scans the surface of surrounding objects in a raster-wise manner. The instrument used to direct those laser beams into varying directions is named ‘laser scanner’. In the context of the monitoring of forest resources terrestrial, or ‘ground-based’, laser scanning should be mentioned along with airborne laser scanning (ALS). While there are strong technical similarities between these two approaches the use of airborne laser scanners, which are usually mounted on small aircrafts or helicopters, will be described in another article. Here we focus on terrestrial laser scanning and hence only some general differences between both different approaches will be pointed out.<br />
<br />
<div align="center"><br />
[[File:airborne.png|300px]]<br />
[[File:terrestrial laser scanning.png|500px]]<br />
</div><br />
<br />
Modern 3D- laser scanners are available from a number of manufactures, e.g. Faro, Leica, Riegel, Topcon, Trimble or Zoller & Fröhlich and others. <br />
<br />
<div align="center"><br />
<gallery caption="Exemplary images are taken from the Websites of the companies and are in alphabetical order." heights="200px" perrow="6" align="center"><br />
File:scanner_1.png<br />
File:scanner_2.png<br />
File:scanner_3.png<br />
File:scanner_4.png<br />
File:scanner_5.png<br />
File:scanner_6.png<br />
</gallery><br />
</div><br />
<br />
==Impuls-based systems==<br />
When using the impuls-based technique the distance between the scanner and the object is determined based on the time a puls of light needed to travel the way to the object and back. This time is double the time needed to travel one way, hence the formula:<br />
<br />
:<math>Distance = 0.5*t*v\,</math><br />
<br />
with ''t'' being the total time between puls emission and puls detection measured at the instrument and ''v'' being the speed of light (~300,000 km per second), gives the distance to the object.<br />
<br />
==Phase-difference systems (or ‘continuous wave’ systems)==<br />
In the last years phase-difference laser scanning systems gained in importance as the scanning rate was increased more and more. Today up to 1 000 000 measurements per second are possible. In contrast to the impuls-based method a continuous beam is emitted from the system. The [[phase-difference]] (amplitude) of a number of combined sinus-shaped waves, so called ‘modulated waves’, between the emitted and received laser beam is measured as it depends on the distance to the object.<br />
<br />
In both cases the laser beams, either pulsed or continuous, are directed in various orientation by a system of mirrors combined with an instrument rotation. The horizontal field of view is 360 degrees while almost 310 degrees in vertical direction are possible. State of the art laser scanners offer an [[accuracy and precision|accuracy]] of about 0.1 mm. The data created by the scanners in the form of a long list containing either cartesian (XYZ)- or [[polar coordinates]] (vertical and horizontal [[angle]]) along with an additional information on the intensity (reflectance) of the received signal. These data can be visualized in 2D-image but actually describes a 3D- cloud of detected points in space, a so called ‘point cloud’.<br />
<br />
[[File:waldin2d.png|A 2D- image of a scanned forest scene. In real space the left and right image border are directly next to each other.|center]]<br />
<br />
[[File:waldin3d.png|3D- point cloud of a real forest stand. Only the object surfaces were scanned and hence all objects are ‘empty’. As points are dimensionless a certain pointsize is chosen for visualization of the data.|center]]<br />
<br />
For [[forest monitoring]] purposes the use of terrestrial 3D-laser scanners is in an early stage. While airborne laser scanners can easily be used for large scale monitoring as they are able to cover large regions, terrestrial scanners are to be carried around on the forest floor and have limited visibility throughout the [[understory]] reducing their extensive operation. The general difference in the perspectives (airborne vs. terrestrial) causes certain advantages and disadvantages of the two approaches. While the bird-perspective is more suitable for large scale forest monitoring there is a much more detailed view to the forest when scanned from the ground with high resolution terrestrial laser scanners when compared to rather large footprint of their airborne pendants. As a consequence of the two perspectives each approach is rather ‘blind’ for certain parts of the forest. In case of terrestrial scans it might be difficult to capture the uppermost [[canopy]] due to obstruction effects, while airborne laser scanners face problems in detecting the lower parts of the [[Tree Definition|trees]].<br />
<br />
Focusing on ground based laser scanning it is important that a single scan is always limited in his field of view as laser beams are reflected from the first object hit. There is no information available on what is behind this object from only one scan position. To overcome this limitation creating a real 3D replication of the environment via the combination of multiple scan perspectives on the same scene is possible. Therefore artificial objects, so called [[targets]], are placed in the area of interest to serve as fix points, which allow merging two different scans with each other (by georeferencing). Different types of targets are available (e.g. spheres, DIN-A4 papers) and precise 3D- models of the forest can be created by puzzling the data from all perspectives together. <br />
<br />
A variety of [[forest parameters]] can be obtained from terrestrial laser scanning, e.g. tree position, [[tree height|height]], [[diameter at breast height]], [[canopy base height]], [[Crown attributes|canopy dimensions]], [[Minimum crown cover|canopy cover]], [[plant area index]], [[competition indices]] as well as tree lean, sweep and [[Stem shape|taper]] and others more. <br />
<br />
Standardized procedures for data processing are rare as suitable software is not yet commercially available. Hence, forest monitoring with terrestrial laser scanners is rather at the stage of intensive research than a matter of routine in forestry. There are some exceptions and promising approaches around the globe, e.g. the company ‘treemetrics’ (http://www.treemetrics.com/) offering commercial application for forest owners based on 3D-lasescanning.<br />
<br />
==Example of a 3D point cloud==<br />
<br />
Please find an example of a 3D point cloud of a research plot in a german beech forest with this video: [http://www.youtube.com/v/JIPRxBzHgcA?version=3]. The different color (green to red) shows different levels of intensity of reflectance. <br />
<br />
[[File:tls.jpg|Screenshot of a 3D laser scan, see the video animation!|center|400px|link=http://www.youtube.com/v/JIPRxBzHgcA?version=3]]<br />
<br />
<br />
[[Category:Forest mensuration]]<br />
[[Category:Remote sensing]]</div>Fehrmannhttp://wiki.awf.forst.uni-goettingen.de/wiki/index.php/Stem_shapeStem shape2021-03-12T08:34:14Z<p>Fehrmann: </p>
<hr />
<div>{{Ficontent}} {{video|link=https://youtu.be/MFAK-iwcJjY}}<br />
[[File:2.6.7.2-fig27.png|right|285px|thumb|'''Figure 1''' Example of a taper curve characterizing the shape of a stem.]]<br />
<br />
The stem shape is characterized by the degree of decreasing [[:Category:Tree diameter|diameter]] with [[:Category:Tree height|height]]. [[Tree Definition|Trees]] may have very different stem shapes where smaller differences can visually usually not be recognized. Depending on the stem shape, the tree [[Stem volume|volume]] may be very different for one and the same [[Diameter at breast height|dbh]] and the amount of boards and other products that the sawmill can cut from one particular stem does also depend much on the stem shape.<br />
<br />
The stem shape can graphically be depicted by the so-called [[Stem volume#The taper curve|taper curve]] which gives the radius of the stem at each height, as shown in Figure 1. This requires measurements of diameters at many heights. Therefore, some simpler approaches were developed like the form factor.<br />
<br />
==Form factor==<br />
<br />
Stem shape characterization is mainly of interest because, in addition to diameter and length, the stem shape co-determines the stem volume which is one of the most important attributes for timber production oriented forest management. <br />
The form factor is a single number that intends to summarize important stem shape information such that this information is useful for volume calculation. <br />
<br />
The idea is very simple: one measures a reference diameter (dbh, for example) and tree height; from these two variables the volume of a cylinder can be calculated that is being imagined over the dbh with the tree height as length. Of course, this cylinder has a volume which is much higher than the tree volume. Then, in order to come to the tree volume, we need to introduce a reduction factor, which reduces the cylinder volume (as defined by dbh and height) to the true stem volume. This reduction factor is called form factor <math>f</math>:<br />
<br />
:<math>f=\frac{\mbox{actual stem volume}}{\mbox{volume of a cylinder over the reference diameter}}\,</math><br />
<br />
[[File:2.6.7.2-fig28.png|right|285px|thumb|'''Figure 2''' Absolute and relative form factor measurement.]]<br />
<br />
Depending on the reference diameter which is being used to determine the form factor we distinguish the absolute form factor (<math>{f}_{1.3}</math>) and the relative form factor (<math>{\lambda}_{0.9}</math> or <math>{f}_{0.1}</math>), as shown in Figure 2.<br />
<br />
*'''Absolute form factor''': dbh is taken as reference diameter, measured at the absolute height of 1.3m. The absolute form factor is usually written as (<math>f_{1.3}</math>).<br />
*'''Relative form factor''': the diameter at 10% of tree height is taken as a reference to calculate the cylinder volume. The relative form factor is usually written as <math>f_{0.1}</math>, in older text books also as <math>\lambda_{0.9}</math>. <br />
<br />
Once the form factor for a species is known (from earlier studies for example), the stem volume can easily be calculated from the simple formula<br />
<br />
:<math>V=ghf\,</math><br />
<br />
where <br />
<br />
:<math>g=\mbox{basal area},\,</math><br />
<br />
:<math>h=height\,</math><br />
<br />
and<br />
<br />
:<math>f=\mbox{form factor}.\,</math><br />
<br />
Actually the form factor can be seen as the simplest volume model, predicting tree volume as a function of dbh, height and form factor. <br />
Form factors vary between species, but also with age, site, [[Tree sociological position|competition]], [[Crown attributes|crown size]], etc. They are commonly in the order of magnitude of 0.45-0.55.<br />
<br />
Form factors can also be determined for the commercial stem length (instead for the entire tree). This form factor is then, of course, applicable only for [[Stem volume|volume calculation]] of that stem section. Only the absolute form factor is being used for commercial height. <br />
<br />
Form factors for the commercial stem length are usually in the order of magnitude of <math>f_{comm}=0.7</math> and can perfectly be used to rapidly calculate the volume of the stem section.<br />
<br />
{{SEO<br />
|keywords=stem shape,form factor,taper curve,tree growth,forest inventory,tree measurement,volume calculation,single tree variables <br />
|descrip=The stem shape is characterized by the degree of decreasing diameter with height.<br />
}}<br />
<br />
[[Category:Single tree variables]]</div>Fehrmannhttp://wiki.awf.forst.uni-goettingen.de/wiki/index.php/Fixed_area_plotsFixed area plots2021-03-12T08:31:16Z<p>Fehrmann: </p>
<hr />
<div>{{Ficontent}} {{video|link=https://youtu.be/IA-PflXW9_k}}<br />
[[File:4.2.2-fig46.png|right|thumb|300px|'''Figure 1''' With circular sample plots all trees are taken as sample trees that are within a defined distance (radius) from the sample point, which constitutes the plot center (Kleinn 2007<ref name="kleinn2007">Kleinn, C. 2007. Lecture Notes for the Teaching Module Forest Inventory. Department of Forest Inventory and Remote Sensing. Faculty of Forest Science and Forest Ecology, Georg-August-Universität Göttingen. 164 S.</ref>).]]<br />
[[File:4.2.2-fig47.png|right|thumb|300px|'''Figure 2''' Illustration of the inclusion zone approach: For fixed area circular plots, the inclusion zones are identical for all trees. Those trees are taken as sample trees in whose inclusion zones the sample point comes to lie (Kleinn 2007<ref name="kleinn2007">Kleinn, C. 2007. Lecture Notes for the Teaching Module Forest Inventory. Department of Forest Inventory and Remote Sensing. Faculty of Forest Science and Forest Ecology, Georg-August-Universität Göttingen. 164 S.</ref>).]]<br />
[[File:4.2.2-fig48.png|right|thumb|300px|'''Figure 3''' Typical diameter distribution in a natural forest (Kleinn 2007<ref name="kleinn2007">Kleinn, C. 2007. Lecture Notes for the Teaching Module Forest Inventory. Department of Forest Inventory and Remote Sensing. Faculty of Forest Science and Forest Ecology, Georg-August-Universität Göttingen. 164 S.</ref>).]]<br />
[[File:4.2.2-fig49.png|right|thumb|300px|'''Figure 4''' Nested sub-plots showing 3 circular plots having different sizes, radii, but sharing same plot center (Kleinn 2007<ref name="kleinn2007">Kleinn, C. 2007. Lecture Notes for the Teaching Module Forest Inventory. Department of Forest Inventory and Remote Sensing. Faculty of Forest Science and Forest Ecology, Georg-August-Universität Göttingen. 164 S.</ref>).]]<br />
[[File:4.2.2-fig50.png|right|thumb|300px|'''Figure 5''' Comparison of the inclusion zone approach for nested circular sub-plots (B) and for fixed circular plots (A) (Kleinn 2007<ref name="kleinn2007">Kleinn, C. 2007. Lecture Notes for the Teaching Module Forest Inventory. Department of Forest Inventory and Remote Sensing. Faculty of Forest Science and Forest Ecology, Georg-August-Universität Göttingen. 164 S.</ref>).]]<br />
[[File:4.2.2-fig51.png|right|thumb|300px|'''Figure 6''' Different combination of shapes for nested sub-plots (after Prodan et al. 1997<ref name="prodan1997">Prodan M., R. Peters, F. Cox and P. Real 1997. Mensura forestal. Serie investigación y educación en desarrollo sostenible. IICA/GTZ. 561p.</ref>.]]<br />
Fixed area sample [[Sampling design and plot design|plots]] can have various different shapes including circular, square, rectangular and strip. As to which shape is chosen depends on practical and statistical issues. In terms of practical criteria it is mainly the cost which is of interest and which translates into criteria such as terrain conditions and expected frequency of border [[Tree Definition|trees]]. Border trees cost time because it needs to be carefully checked whether they are in or not and that implies usually additional [[Distance to tree|distance]] measurements which are not required for the other trees. <br />
<br />
From a practical point of view a circular plots is most rapidly installed wherever the visibility in the stand is good. In a tropical forest with dense understory, it is much easier to establish a strip plot where the field crew walks along the central line and measures all trees up to a defined distance to the right and to the left. A circular plot, however, has the lowest expected number of border trees, because, for a given area, the circle is the geometric shape that has the shortest perimeter. <br />
<br />
Coming back to the considerations on [[Plot design#Spatial autocorrelation|spatial autocorrelation]], a long rectangular plot (strip plot) covers more different site conditions and is likely to capture more variability per plot. Thus, we may expect higher precision when using strip plots if we compare it with circular plots with the same plot area. <br />
<br />
In what refers to the plot size, for a given [[Sampling intensity vs. sample size|sampling intensity]] it is statistically more [[accuracy and precision|precise]] to establish many small plots than few large plots. However, with many small plots, again the cost will be much higher; the decision on [[Sample size]] is again a compromise between practical (cost) criteria and statistical (precision). If there is some information on number of stems per hectare, one can calculate, as a rule of thumb, the plot area such that there are on average about 15-20 trees in a sample plot. Typical plot areas are 200 m², 500 m², 1000 m² - but anything is possible. In general sample plots (or any other [[:category:plot design|plot design]] could be installed as permanent or temporary plot. Especially in context of the [[estimation on changes]] this could have a huge influence on [[Accuracy and precision|precision]].<br />
<br />
All trees that have the center of their base inside the plot area are sample trees and are being measured. If a tree is a border tree and it is not obvious whether it is in or out, a measurement must help with that decision. If the tree is exactly on the borderline, then one may count one tree and the next such tree not; but actually, when measuring distances with high accuracy, the case that a tree is exactly on the borderline should be very rare. It is only the center point of the tree base that defines whether a tree is in or out. If center point is in but the tree leans completely outside the plot, this is still an in-tree. If the center is not in the plot, but the entire tree leans into the plot, this tree is not observed.<br />
<br />
If a defined sample plot is overlapping the [[forest boundary]] or the boundary of a [[stratified sampling|stratum]], [[edge correction]] needs to be applied in order to derive [[bias|unbiased estimates]].<br />
<br />
==The inclusion zone concept==<br />
<br />
We introduced circular fixed area plots such that a [[sample point]] is selected and then all trees within a fixed distance (the plot radius r) are taken as sample trees (see Figure 1). While this plot perspective is intuitively clear and understandable, it does not allow to say anything about the per tree [[inclusion probability]]. This however, may be an important component of statistical information.<br />
<br />
We may change the perspective from the plot perspective to the tree perspective. In that approach, we build a so-called inclusion zone around each tree. The inclusion zone is a measure for the inclusion probability of that particular tree: if a sample point falls into that inclusion zone, then this particular tree is included in a sample. While this sounds maybe complicated, it is extremely helpful in understanding [[forest inventory]] sampling and also in deriving [[bias|unbiased]] [[estimator]]s for any [[:Category:Plot design|plot design]]. <br />
<br />
In the case of fixed area circular plots, the inclusion zone that is centered about each tree has the size and shape of the plot, that is circular. All trees, therefore, have the same selection probability. We now imagine the inclusion zones around all trees and select a sample point. Then, all those trees are selected as sample trees for which the sample point falls into their inclusion zone (Figure 2). Of course, at the end, exactly the same trees are selected as with the first approach (of laying out the sample plot), but in this second approach we take the selection probability explicitly into account.<br />
<br />
The inclusion zone concept can be applied also to the other plot designs that are being presented in the rest of this chapter.<br />
<br />
==The plot expansion factor==<br />
<br />
In forest inventory, we are usually interested in estimates of the total and in estimates per hectare. The immediate per-plot results are not of major interest. In order to get the estimates per hectare one needs to expand the per-plot observation. This is done with a plot expansion factor. It is important to consider that area related variables refer to the horizontal map plane and a [[slope correction]] is necessary in sloped terrain. For fixed area plots, this expansion factor from the per-plot to the per-hectare estimate is calculated from <br />
<br />
:<math>EF=\frac{10000}{\mbox{PlotSize }(m^2)}\,</math><br />
<br />
For example, with a circular fixed area plot of 200 m² or 0.02 ha, the expansion factor is 50. The per-plot observations need to be expanded (multiplied) by <math>EF=50</math> to produce the per-hectare estimates. This expansion factor is the inverse of the [[inclusion probability]] resulting from a certain plot design.<br />
<br />
The estimation procedure is illustrated with a simple example: assuming in a 5.0 ha forest stand, 5 plots with 500 m² each are randomly selected. The target variable is the number of stems and we are interested in the estimates per ha and in the estimated total, including the corresponding [[standard errors]]. After finishing the inventory, stems per plot observations are 20, 25, 30, 35 and 40 respectively for the 5 plots. It is important to note that each of the plots gives us exactly '''one''' independent estimation for the target variable.<br />
<br />
The expansion factor is <math>EF=20</math>. Therefore, the per-hectare values expanded from the per-plot values are then 400, 500, 600, 700 and 800, respectively. From these, mean number of stems per ha and its variance can be calculated. <br />
<br />
Alternatively, and may be easier, we may calculate all estimations first on a per-plot basis and apply the expansion factor later. For this example, the results per plot are:<br />
<br />
:{|<br />
|-<br />
|Mean number of stems per plot<br />
|<math>\bar{y}=30\,</math><br />
|-<br />
|with variance per plot<br />
|<math>s^2=62.5\,</math><br />
|-<br />
|and standard error<br />
|<math>s_y=\frac{sw}{\sqrt{n}}=\frac{7.91}{\sqrt{5}}=3.53\,</math><br />
|}<br />
<br />
From the standard error, the confidence interval can be calculated where we use for convenience<br />
<br />
:<math>t=2:\mbox{ }ts_y\approx2s_y=2*3.53=7.06\,</math><br />
<br />
and therefore:<br />
<br />
:<math>CI=\left[\left(\bar{y}-ts_{\bar{y}}\right)\le\mu\le\left(\bar{y}+ts_{\bar{y}}\right)\right]=</math><br />
<br />
:<math>\left[\left(30-7.06\right)\le\mu\le\left(30+7.06\right)\right]=\left[22.94\le\mu\le37.06\right]\,</math><br />
<br />
The results at ''per-hectare'' are simply derived by multiplying the per-plot results with the expansion factor, thus for example, <br />
<br />
:{|<br />
|-<br />
|mean number of stems per hectare is<br />
|<math>{\bar{y}}_{ha}=30*20=600\,</math><br />
|-<br />
|and standard error is<br />
|<math>s_{\bar{y}ha}=3.53*20=70.6\mbox{ stems/ha}\,</math>.<br />
|-<br />
|}<br />
<br />
Other estimations follow suit. However, when applying the expansion factor to the variance, it must be observed that the variance comes as squared figure so that the expansion needs to be done with <math>EF^2</math>! That is, in our example with an estimated per-plot variance of<br />
<br />
:<math>s_{plot}^2=62.5\,</math><br />
<br />
the estimated per-hectare variance is <br />
<br />
:<math>s_{ha}^2=s_{plot}^2*EF^2=62.5*20^2=25000\,</math><br />
<br />
==Cluster plots==<br />
<br />
So far, we looked at fixed area plots as compact and coherent geometrical shapes like circle and rectangle. However, a sample plot can also come “in pieces”. The most typical examples are the so-called cluster plots (a more detailed description is in the article [[Cluster sampling|cluster sampling]]) which are usually employed in large area forest inventories. Each cluster plot consists of several subunits, called [[Sub-plot|sub-plots]], which are located at a certain defined distance apart from each other.<br />
<br />
However, we wish to stress yet here that also for cluster plots it holds what was said about randomization and sample size: each randomization step yields one independently selected sample and each one of these such independently selected samples produces exactly one independent observation of the target variable. That means that the entire cluster plot produces one observation of, for example, [[basal area]], and not each sub-plot! This is a frequently committed confusion in forest inventory sampling.<br />
<br />
==Nested sub-plots==<br />
<br />
Figure 3 shows a typical diameter distribution in a [[Forest Definition|natural forest]], where there is a very high number of small trees and fewer large trees; such a distribution is also called negatively J-shaped. If we lay out a single fixed area plot with a defined radius, our sample plot mirrors what this diameter distribution shows: we will include a high number of small trees and only very few larger trees. Usually, however, we are at least as much interested in the larger trees and do not like to have a very high number of small trees on the plot. <br />
<br />
The solution to this is simple: we use nested sub-plots with different sizes as illustrated in Figure 4 for different [[size classes]] of trees each; usually, those size classes are [[Tree diameter|diameter]] classes. Then we can define the plot size for different diameter classes such that in each class (nested sub-plot) there is more or less the target number of sample trees: plots with larger radii are used for larger sample trees and smaller plots for smaller trees. <br />
<br />
Nested sub-plots segregate trees by diameter classes and the corresponding plot sizes are different; therefore, analysis must also be made separately for each [[plot size]] as there are different plot expansion factors depending on plot size. <br />
<br />
The inclusion zone concept can immediately be applied also to nested circular sub-plots as illustrated in Figure 5. There (right hand side), the size of the inclusion zone depend on the diameter of the tree: smaller trees have a smaller inclusion zone and therefore also a smaller selection probability. <br />
<br />
Nested sub-plots are not restricted to circular plots but can be applied to any plot shape. Figure 6 gives an illustration of various options. When strip plots are applied, the regeneration plot is frequently taken as a small circular plot of some few square meters. That means that also different plot shapes can be used for the differently sized plots.<br />
<br />
<br />
==References==<br />
<references/><br />
<br />
{{SEO<br />
|keywords=fixed area plots,plot design,inclusion zone concept,plot expansion factor,cluster plots,nested sub-plots,slope correction,sample plots,forest inventory,<br />
|descrip=Fixed area sample plots can have various different shapes including circular, square, rectangular and strip.<br />
}}<br />
<br />
<br />
[[Category:Plot design]]</div>Fehrmannhttp://wiki.awf.forst.uni-goettingen.de/wiki/index.php/VertexVertex2021-03-12T08:30:08Z<p>Fehrmann: </p>
<hr />
<div>{{Ficontent}} {{video|link=https://youtu.be/OrGh-4Giyq0}}<br />
[[File:vertex_transponder.jpg|thumb|300px|Figure 1: Vertex III (right), transponder (middle) and 360° adapter (left)]]<br />
[[File:transponder.jpg|thumb|300px|Figure 2: Transponder fixed at a tree)]]<br />
[[File:vertex.jpg|thumb|300px|Figure 3: Vertex III]]<br />
[[file:vertex_outdoor.jpg|thumb|300px|Figure 4: Sighting the target with Vertex III]]<br />
==General descripition==<br />
The Vertex clinometer is a device to measure distances based on ultrasonic sound and angles. It enables to directly measure [[tree height]] based on [[The trigonometric principle|the trigonometric principle]]. For the measurement of distances a transponder is needed that is fixed at the tree and is an active component that submits a signal back to the Vertex. To determine the [[The trigonometric principle|horizontal distance]] to the tree the transponder has to be fixed at a certain height (default 1,3 meter) that is defined in the settings menu. <br />
<br />
[[file:vertexheight.png|center]]<br />
<br />
The vertex clinometer allows to measure heights from variable distances to the target. The measurement results can be stored at the display to note them later down.<br />
<br />
==Handling==<br />
To switch the instrument on, press the red on-button for a while. Ensure that the internal and external temperatures are the same - if not, wait until the internal temperature has adapted. The menue pages can be selected by pressing the 'DME' and the 'IR' buttons. To switch the instrument off, press DME and IR buttons simultaneously for a while. Place the transponder directly in front of the Vertex and press 'DME' to switch the transponder on(two beeps=on, 4 beeps=out).<br />
<br />
===Distance measurement===<br />
#Switch the transponder on (when it is working, a quietly pulsative peep is hearable),<br />
#Place the transporter at the target (Figure 2),<br />
#Select the program for distance measurement,<br />
#Locate the place to measure it's distance,<br />
#Sight through the instrument pressing the 'DME button' (Figure 3) (Looking through the device is only necessary if you really whish to measure the distance exaclty to the point where the transponder is fixed),<br />
#The cross (Figure 4) will disappear and starts blinking when the measurement is done,<br />
#The horizontal distance appears at the display (if a slope angle was measured before, the display shows the horizontal distance that is calculated automatically based on the last angle measurement)<br />
<br />
===Distances within a sample plot===<br />
#Place the transponder at the 360° adapter and locate both vertically at the center point (Figure 9),<br />
#Go to the tree and locate the instrument at the tree axis to measure the distance to the [[sample plot|plot center]] in order to check whether the tree is included or not.<br />
<br />
<gallery widths=300px heights=300px><br />
file:vertex01-messung-transponder.jpg|Figure 5: Distance measurement to transponder<br />
file:vertex05-aufbau-transponder-plotmitte.jpg|Figure 9: Installation of transponder at a plot center<br />
</gallery><br />
<br />
===Height measurement===<br />
#Place the transporter at the tree in the fixed height selected in the setup menu, usually 1,3 meter (Figure 2),<br />
#Locate a place with a good view to the transponder and tree crown,<br />
#Select the program for height measurements,<br />
#Determine the horizontal distance by measuring to the transponder (Figure 5),<br />
#If the distance is measured the red cross will start blinking, indicating that you should measure the angle to the tree top. A measurement to the tree bottom is not necessary with the Vertex, because of the fixed height of the transponder),<br />
#Measure angle to tree crown (Figure 7),<br />
#Read the resulting height (Figure 8: tree height = 24,5m)<br />
<br />
<gallery widths=300px heights=300px><br />
file:vertex03-messung-baumkrone.jpg|Figure 7: tree crown measurement<br />
file:vertex04-anzeige-hoehenmessung.jpg|Figure 8: Display results<br />
</gallery><br />
<br />
<br />
{| class="wikitable"<br />
!Advantages<br />
!Disadvantages<br />
|-<br />
|slope correction implemented<br />
|height values from leaning trees are underestimated<br />
|-<br />
|}<br />
<br />
{{info<br />
|message=Note:<br />
|text= Ultrasonic signals can be affected by interfering sounds of insects, wind, rain drops or of purling water (experiences from project work in Indonesia, Africa and Germany), so that the height values are biased or distance measurement is not possible at all. The internal temperature of the Vertex needs to be the same as the external temperature - if not the values are biased. Be sure that trees measured are done in vertical axes. Be sure to calibrate the instrument regularly, preferably every day.<br />
}}<br />
<br />
==Applications==<br />
* [[The trigonometric principle | Height measurement]]<br />
* [[Distance to tree]]<br />
* [[Measuring slope]]<br />
<br />
==Related articles==<br />
* [[The trigonometric principle]]<br />
* [[Bitterlich sampling]]<br />
<br />
==Manuals==<br />
[[:File:Vertex III V 1.5 english.pdf|Vertex III Manual English]]<br />
<br />
[[:File:Vertex IV english.pdf|Vertex IV Manual English]]<br />
<br />
[[:File:Vertex IV german.pdf|Vertex IV Manual German]]<br />
<br />
=Web Links=<br />
*[http://www.haglofcg.com/ Haglöf Sweden]<br />
{{Studienbeiträge}}<br />
[[Category:Clinometer]]</div>Fehrmannhttp://wiki.awf.forst.uni-goettingen.de/wiki/index.php/SuuntoSuunto2021-03-12T08:29:14Z<p>Fehrmann: </p>
<hr />
<div>{{Ficontent}} {{video|link=https://youtu.be/EahhlAazK1Y}}<br />
==General description==<br />
The Suunto ([figure 1]) is a device that is used to measure [[Measuring slope|slope]] and [[tree height]]s within fixed distances (15m, 20m) to the tree using [[the trigonometric principle]]. Therefore the inclination angles to the tree top and bottom are measured from fixed distances. Tree height can then be read from the scales for the respective distance. Some Suunto devices have an optical rangefinder to measure these fixed distances (15m, 20m) optically and a sighting viewer to look on the height scales. The scales implemented are the same as the scales of Silva [[Clinomaster]], but there is additionally a scale for slope measurements in %.<br />
<br />
<gallery widths=300px heights=300px><br />
file: suunto-01_portrait.jpg|Figure 1a: Front<br />
file:suunto-01_portrait_back.jpg|Figure 1b: Back<br />
</gallery><br />
<br />
==Handling==<br />
# Select desired distance (15m, 20m) to the tree,<br />
# Sight the tree bottom with both eyes open (figure 2) and remember the value,<br />
# Sight the tree crown (figure 3) and remember the value,<br />
# Tree height is calculated as :<math> {h_t} = {h_c} - {h_b} </math> (see also [[the trigonometric principle]]).<br />
<br />
<br />
<gallery widths=300px heights=300px><br />
file: suunto02_measure-bottom.jpg|Figure 2: Live view trunk bottom measurement<br />
file: suunto03_measure-top.jpg|Figure 3: Live view crown measurement <br />
</gallery><br />
<br />
<br />
To measure [[tree height]]s, aim the Suunto with both eyes open so that the hairline is superimposed. When the hairline is at the right level (e.g. bottom of the trunk) and the clinometer has stopped oscillating, remember the measurement value and measure the tree crown as you did before.<br />
The difference between the crown height and the bottom value is the tree height:<br />
Htotal = hcrown – hbottom<br />
{| class="wikitable" <br />
! Advantages<br />
! Disadvantages<br />
|-<br />
|slope correction implemented<br />
|measurement result needs to be remembered<br />
|-<br />
|optical distance measurement<br />
|fixed distances to the tree (could be difficult in closed forest stands)<br />
|-<br />
|independence of power sources (no batteries needed)<br />
|in dark forest stands optical measurement is difficult, no digital storage of measurement results available.<br />
|}<br />
<br />
==Manual==<br />
[[:File:Manual PM5 10.08_d.pdf|Suunto hypsometer multilangual]]<br />
<br />
==Applications==<br />
* [[The trigonometric principle | Height measurement]]<br />
* [[:Category:Upper stem diameter|Stem diameter in different heights]]<br />
<br />
==Related articles==<br />
* [[The trigonometric principle]]<br />
* [[Bitterlich sampling]]<br />
{{Studienbeiträge}}<br />
[[Category: Clinometer]]</div>Fehrmannhttp://wiki.awf.forst.uni-goettingen.de/wiki/index.php/Blume-LeissBlume-Leiss2021-03-12T08:28:40Z<p>Fehrmann: </p>
<hr />
<div>{{Ficontent}} {{video|link=https://youtu.be/EahhlAazK1Y}}<br />
==General description==<br />
The Blume Leiss is developed to measure slope and [[tree height]] by [[the trigonometric principle]]. The device measures the elevation angle between the operator and measured points. Tree heights can directly be read from the device depending on fixed distances of 15m, 20m, 30m and 40m. For mountainous areas [[Fixed_area_plots#Slope_correction|slope correction]] factors can directly be read from the device depending on the slope.<br />
Figure 1 shows the front side of the device including:<br />
* example for the viewing while measuring distance with the vertical base measure,<br />
* the scales of height measurement for the fixed distances to the tree (15, 20, 30 and 40m) and<br />
* scale for slope measurement in °.<br />
<br />
<gallery widths=300px heights=300px><br />
file: blume-leiss01_portrait-01_messung.jpg|Figure 1: Blume Leiss front (result of an measurement)<br />
file: blume-leiss02_portrait-02.jpg|Figure 2: Blume Leiss back<br />
</gallery><br />
<br />
In this example the type of the device (BL6) has two buttons two separately lockable needles (figure 1), so that the height between the operator and the stem foot / tree crown can directly be read. The difference between the measurements will be tree height <math>h_t</math>:<br />
<math><br />
h_t=h_c-h_b<br />
</math><br />
<br />
If the slope is larger than 5 degrees, the tree heights need to be corrected by the formula $CF=\cos\alpha_i$. This correction factor (CF) has to be multiplied with the measured tree height (since we measured a slope distance and not the horizontal distance to the tree). The correction factor on the backside of the instrument assumes that the distance is measured optical (the inbuild prism) and is $CF=(\cos\alpha_i)^2$. This is because we have an inclined view on the levelling bord.<br />
<br />
The optical rangefinder (figure 2) allows the user to measure the distance to the tree by mirroring a picture of the levelling board. Figure 2 shows the backside of the device where information is given about:<br />
*Height caluation<br />
**b-a=h (slope > 5° - viewing downwards)<br />
**a+b=H (slope < 5° - viewing horizontal)<br />
**a-b=h (slope > 5° - viewing upwards)<br />
*Correction factors<br />
**correction factor depending on slope in °<br />
**slope in ° depending on slope in %<br />
<br />
==Handling==<br />
===Distance measurement===<br />
# Determine an optimal distance to the tree by checking the visibility of the tree bottom and tree top within the forest stand,<br />
# Place the vertical base measure at the tree,<br />
# Find the correct fixed distance using the optical range finder by focusing the distance leveling board (figure 3b) and move closer or farther to the tree, until the top of the mirrored picture of the base is coincident with the corresponding mark on the original picture - in this example, the mid mark of the leveling corresponds to a distance of 15m (alternatively a tape measure can be used to find the correct fixed distance).<br />
<br />
<gallery widths=300px heights=300px><br />
file: blume-leiss03_distance-leveling-board.jpg|Figure 3a: Vertical base fixed at the tree<br />
file: blume-leiss03_distance-measurement.jpg|Figure 3b: live view through range finder. The white strips of the mirrored and original picture of the base must coincide to find the respective fixed distance.<br />
</gallery><br />
<br />
===Height measurement===<br />
# Focus the bottom of the tree (figure 4a) and lock the pendulum,<br />
# Focus the top of the tree (figure 4b) and lock the second pendulum,<br />
# The front side of the device (figure 1) shows two height values - the height can directly be derived by the formulas described above, depending on the slope<br />
# Correct the derived height if necessary<br />
<br />
===Slope measurement===<br />
# Sight at the zero mark of the levelling board and lock the needle<br />
# The slope in ° can directly be read from the scale (figure 1)<br />
<br />
<br />
<gallery widths=300px heights=300px><br />
file: blume-leiss04_mess-baumfuss_markiert.jpg|Figure 4a: live view measuring tree bottom<br />
file: blume-leiss05_mess-krone_markiert.jpg|Figure 4b: live view measuring tree crown<br />
</gallery><br />
<br />
<br />
{{info<br />
|message=Note:<br />
|text= When measuring with Blume Leiss, you need to observe the pendulum check mark to be sure, that the pendulum has stopped oscillating before locking the button. All scales are visible at once - be careful of taking the necessary scale.<br />
}}<br />
<br />
{| class="wikitable"<br />
!Advantages<br />
!Disadvantages<br />
<br />
|-<br />
|slope correction implemented<br />
|dependance on fixed scales<br />
|-<br />
|optical distance measurement<br />
|fixed distances to the tree (could be difficult in closed forest stands)<br />
|-<br />
|independence of power sources (no batteries needed)<br />
|in dark forest stand optical measurement is difficult<br />
|-<br />
|<br />
|no digital storage of measurement results available.<br />
|}<br />
<br />
==Applications==<br />
* [[The trigonometric principle | Height measurement]]<br />
<br />
<br />
==Related articles==<br />
* [[The trigonometric principle]]<br />
* [[Bitterlich sampling]]<br />
{{Studienbeiträge}}<br />
[[Category:Clinometer]]</div>Fehrmannhttp://wiki.awf.forst.uni-goettingen.de/wiki/index.php/Diameter_at_breast_heightDiameter at breast height2021-03-12T08:27:33Z<p>Fehrmann: </p>
<hr />
<div>{{Ficontent}} {{video|link=https://youtu.be/17NPkVXteiI}}<br />
The standard position for [[:Category:Tree diameter|diameter measurements]] at standing trees is at breast height, defined at the height of 1.30m in most countries. But there are many countries where diameter at breast height is measured at different heights. Therefore, it is always good to find out where ''dbh'' is measured before engaging in data analysis. In the United States, for example, dbh is taken at 4.5 feet above ground which is about 1.37m, and in Korea at 1.20m. The reason why breast height developed to be a standard measure has probably to do with the ease and convenience of measurement.<br />
<br />
==Rules==<br />
<br />
[[File:2.2.2-fig9.png|right|thumb|285px|'''Figure 1''' Illustration of an instruction for dbh measurements. This example is taken from the field manual of the second national forest inventory in Germany (BMELV 2007<ref name="BMELV2007">BMELV. 2007. Survey Instructions for the 2nd National Forest Inventory (2001-2002). Reprint February 2007 2nd corrected translation February 2006, of the 2nd corrected and revised reprint, May 2001. 106</ref>).]]<br />
<br />
However, irregularities and anomalies of [[Stem shape|tree stems]] do some times prevent the measurement of a diameter at the defined height. For these cases, rules need to be defined how to proceed. This is usually done with graphs describing these specific situations and how to proceed then with the dbh measurements. It is important that all field teams who carry out the measurements follow the same definitions and instructions. In the following, some typical situations are listed and how the dbh measurements are then usually taken: <br />
<br />
*Tree on slope: If a tree is in a sloped terrain, the dbh is measured at the standard height above the forest floor on the uphill side of the tree.<br />
<br />
*While measuring the breast height from the ground level, we have to pay attention that the level from which is measured is in fact the ground and not the surface of a heap of woody debris.<br />
<br />
*If the tree is leaning, breast height is measured parallel to the lean on the top side of the tree, whereas diameter is measured perpendicular to the longitudinal axis of the trunk.<br />
<br />
*In case of buttressed tree, if the height of the buttress is more than 1m, then the breast height is measured from the point where buttress ends. In this case, reduction models are some times be applied with which the would-be diameter at breast height is estimated from the measurement of a diameter at another height.<br />
<br />
*When there is an abnormality at breast height such as crooks, swelling or knots, the dbh is measured above or below the abnormality. Some times, the rule is defined that diameter measurements are then done at equal distances above and below breast height and dbh then estimated by simply taking the mean of the two measurements.<br />
<br />
*If a tree bifurcates above breast height, then one measures dbh as usual. If the bifurcation is below breast height, each stem is measured and counted separately.<br />
<br />
==Instruments for measuring diameter at breast height==<br />
<br />
The most commonly used instruments for measuring diameters at breast height are [[Caliper|calipers]] and [[Diameter tape|diameter tapes]]. Other instruments are interesting in principle but relatively little used; examples are the [[Biltmore stick]] and [[Bitterlich´s sector fork]]. Though some of these instruments may be used to measure diameters at other defined points of the tree, there are other instruments that are more efficient for this purpose such as [[Optical caliper|optical calipers]].<br />
<br />
==References==<br />
<br />
<references/><br />
<br />
{{SEO<br />
|keywords=tree diameter,diameter at breast height,forest definitions,forest inventory,tree measurement,standing tree sampling<br />
|descrip=The diameter at breast height is the standard position for diameter measurement at standing tress.<br />
}}<br />
<br />
[[Category:Tree diameter]]</div>Fehrmannhttp://wiki.awf.forst.uni-goettingen.de/wiki/index.php/Diameter_at_breast_heightDiameter at breast height2021-03-12T08:26:46Z<p>Fehrmann: </p>
<hr />
<div>{{Ficontent}} {{{video|https://youtu.be/17NPkVXteiI}}}<br />
The standard position for [[:Category:Tree diameter|diameter measurements]] at standing trees is at breast height, defined at the height of 1.30m in most countries. But there are many countries where diameter at breast height is measured at different heights. Therefore, it is always good to find out where ''dbh'' is measured before engaging in data analysis. In the United States, for example, dbh is taken at 4.5 feet above ground which is about 1.37m, and in Korea at 1.20m. The reason why breast height developed to be a standard measure has probably to do with the ease and convenience of measurement.<br />
<br />
==Rules==<br />
<br />
[[File:2.2.2-fig9.png|right|thumb|285px|'''Figure 1''' Illustration of an instruction for dbh measurements. This example is taken from the field manual of the second national forest inventory in Germany (BMELV 2007<ref name="BMELV2007">BMELV. 2007. Survey Instructions for the 2nd National Forest Inventory (2001-2002). Reprint February 2007 2nd corrected translation February 2006, of the 2nd corrected and revised reprint, May 2001. 106</ref>).]]<br />
<br />
However, irregularities and anomalies of [[Stem shape|tree stems]] do some times prevent the measurement of a diameter at the defined height. For these cases, rules need to be defined how to proceed. This is usually done with graphs describing these specific situations and how to proceed then with the dbh measurements. It is important that all field teams who carry out the measurements follow the same definitions and instructions. In the following, some typical situations are listed and how the dbh measurements are then usually taken: <br />
<br />
*Tree on slope: If a tree is in a sloped terrain, the dbh is measured at the standard height above the forest floor on the uphill side of the tree.<br />
<br />
*While measuring the breast height from the ground level, we have to pay attention that the level from which is measured is in fact the ground and not the surface of a heap of woody debris.<br />
<br />
*If the tree is leaning, breast height is measured parallel to the lean on the top side of the tree, whereas diameter is measured perpendicular to the longitudinal axis of the trunk.<br />
<br />
*In case of buttressed tree, if the height of the buttress is more than 1m, then the breast height is measured from the point where buttress ends. In this case, reduction models are some times be applied with which the would-be diameter at breast height is estimated from the measurement of a diameter at another height.<br />
<br />
*When there is an abnormality at breast height such as crooks, swelling or knots, the dbh is measured above or below the abnormality. Some times, the rule is defined that diameter measurements are then done at equal distances above and below breast height and dbh then estimated by simply taking the mean of the two measurements.<br />
<br />
*If a tree bifurcates above breast height, then one measures dbh as usual. If the bifurcation is below breast height, each stem is measured and counted separately.<br />
<br />
==Instruments for measuring diameter at breast height==<br />
<br />
The most commonly used instruments for measuring diameters at breast height are [[Caliper|calipers]] and [[Diameter tape|diameter tapes]]. Other instruments are interesting in principle but relatively little used; examples are the [[Biltmore stick]] and [[Bitterlich´s sector fork]]. Though some of these instruments may be used to measure diameters at other defined points of the tree, there are other instruments that are more efficient for this purpose such as [[Optical caliper|optical calipers]].<br />
<br />
==References==<br />
<br />
<references/><br />
<br />
{{SEO<br />
|keywords=tree diameter,diameter at breast height,forest definitions,forest inventory,tree measurement,standing tree sampling<br />
|descrip=The diameter at breast height is the standard position for diameter measurement at standing tress.<br />
}}<br />
<br />
[[Category:Tree diameter]]</div>Fehrmannhttp://wiki.awf.forst.uni-goettingen.de/wiki/index.php/Diameter_at_breast_heightDiameter at breast height2021-03-12T08:26:22Z<p>Fehrmann: </p>
<hr />
<div>{{Ficontent}} {{video|https://youtu.be/17NPkVXteiI}}<br />
The standard position for [[:Category:Tree diameter|diameter measurements]] at standing trees is at breast height, defined at the height of 1.30m in most countries. But there are many countries where diameter at breast height is measured at different heights. Therefore, it is always good to find out where ''dbh'' is measured before engaging in data analysis. In the United States, for example, dbh is taken at 4.5 feet above ground which is about 1.37m, and in Korea at 1.20m. The reason why breast height developed to be a standard measure has probably to do with the ease and convenience of measurement.<br />
<br />
==Rules==<br />
<br />
[[File:2.2.2-fig9.png|right|thumb|285px|'''Figure 1''' Illustration of an instruction for dbh measurements. This example is taken from the field manual of the second national forest inventory in Germany (BMELV 2007<ref name="BMELV2007">BMELV. 2007. Survey Instructions for the 2nd National Forest Inventory (2001-2002). Reprint February 2007 2nd corrected translation February 2006, of the 2nd corrected and revised reprint, May 2001. 106</ref>).]]<br />
<br />
However, irregularities and anomalies of [[Stem shape|tree stems]] do some times prevent the measurement of a diameter at the defined height. For these cases, rules need to be defined how to proceed. This is usually done with graphs describing these specific situations and how to proceed then with the dbh measurements. It is important that all field teams who carry out the measurements follow the same definitions and instructions. In the following, some typical situations are listed and how the dbh measurements are then usually taken: <br />
<br />
*Tree on slope: If a tree is in a sloped terrain, the dbh is measured at the standard height above the forest floor on the uphill side of the tree.<br />
<br />
*While measuring the breast height from the ground level, we have to pay attention that the level from which is measured is in fact the ground and not the surface of a heap of woody debris.<br />
<br />
*If the tree is leaning, breast height is measured parallel to the lean on the top side of the tree, whereas diameter is measured perpendicular to the longitudinal axis of the trunk.<br />
<br />
*In case of buttressed tree, if the height of the buttress is more than 1m, then the breast height is measured from the point where buttress ends. In this case, reduction models are some times be applied with which the would-be diameter at breast height is estimated from the measurement of a diameter at another height.<br />
<br />
*When there is an abnormality at breast height such as crooks, swelling or knots, the dbh is measured above or below the abnormality. Some times, the rule is defined that diameter measurements are then done at equal distances above and below breast height and dbh then estimated by simply taking the mean of the two measurements.<br />
<br />
*If a tree bifurcates above breast height, then one measures dbh as usual. If the bifurcation is below breast height, each stem is measured and counted separately.<br />
<br />
==Instruments for measuring diameter at breast height==<br />
<br />
The most commonly used instruments for measuring diameters at breast height are [[Caliper|calipers]] and [[Diameter tape|diameter tapes]]. Other instruments are interesting in principle but relatively little used; examples are the [[Biltmore stick]] and [[Bitterlich´s sector fork]]. Though some of these instruments may be used to measure diameters at other defined points of the tree, there are other instruments that are more efficient for this purpose such as [[Optical caliper|optical calipers]].<br />
<br />
==References==<br />
<br />
<references/><br />
<br />
{{SEO<br />
|keywords=tree diameter,diameter at breast height,forest definitions,forest inventory,tree measurement,standing tree sampling<br />
|descrip=The diameter at breast height is the standard position for diameter measurement at standing tress.<br />
}}<br />
<br />
[[Category:Tree diameter]]</div>Fehrmannhttp://wiki.awf.forst.uni-goettingen.de/wiki/index.php/Measurement_with_the_relascopeMeasurement with the relascope2021-03-12T08:07:22Z<p>Fehrmann: </p>
<hr />
<div>{{Ficontent}} {{video|link=https://youtu.be/W88BzlC715o}}<br />
[[File:2.2.7-fig15.png|left|thumb|300px|'''Figure 1''' Relascope units in the mirror relascope (Kleinn 2007<ref name="kleinn2007">Kleinn, C. 2007. Lecture Notes for the Teaching Module Forest Inventory. Department of Forest Inventory and Remote Sensing. Faculty of Forest Science and Forest Ecology, Georg-August-Universität Göttingen. 164 S.</ref>).]]<br />
<br />
The [[relascope]] (the original German name is: Spiegelrelaskop) is a multifunctional hand-held measurement device developed by the Austrian forester [http://de.wikipedia.org/wiki/Walter_Bitterlich Walter Bitterlich]. It can be used for the so-called [[Bitterlich sampling|Bitterlich (or angle-count) sampling]], for the measurement of [[:Category:Upper stem diameters|upper stem diameters]] but also for the measurement of [[Measuring slope|slope angle]], [[tree height]] and other variables. A micro-scale in the device does automatically correct for slope angle. <br />
<br />
On that micro-scale there are measurement units (the so-called relascope units) which are being used for the [[Why measuring upper diameters?|measurement of upper diameters]]. Once it is determined, for a particular measurement, to how many centimeters one relascope unit corresponds, we can determine upper diameters by simply counting relascope units (Figure 1). The height, at which the measurement is taken, needs to be determined independently.<br />
<br />
Usually one determines the relascope units by aiming at the known [[Diameter at breast height|dbh]] or any other lower diamater that can be measured directly. If, for example, a tree has a dbh of 60 cm and we count 6 relascope units when aiming with the relascope at dbh, then we know that each relascope units corresponds to 10 cm. If then for example at height 10 m we count 4.5 relascope units, we can calculate the respective diameter of 45 cm from the known relation. <br />
<br />
Another possibility is to use the fixed ratio between diameter and distance of 1:200 that every small relascope unit represents. That means, if for example the horizontal distance to the tree is 10 m, one relascope unit would correspond to an object size of 5 cm. The advantage of using this fixed distance is that the relascope can be used to measure the height of the diameter measurement at the same time.<br />
<br />
The relascope is complex in design but relatively simple to use. It is quite expensive. Visibility is a problem when light is bad. Of course, counting the relascope units and determine its fractions is difficult, above all for higher angles where the scale is very narrow. <br />
<br />
==References==<br />
<br />
<references/><br />
<br />
{{SEO|keywords=upper stem diameter,relascope,tree diameter,forest inventory,tree measurement,single tree variables <br />
sampling|descrip=The relascope (German: Spiegelrelaskop) from Bitterlich can also be used for measuring upper stem diameters.}}<br />
<br />
[[Category:Upper stem diameter]]</div>Fehrmannhttp://wiki.awf.forst.uni-goettingen.de/wiki/index.php/Template:VideoTemplate:Video2021-03-12T08:04:29Z<p>Fehrmann: </p>
<hr />
<div><noinclude>This is a standard info box linking to video tutorials.<br />
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image=<br />
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link=<br />
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;[[image:video.png|20px|video.png]] <u>'''[{{{link}}} Available as video tutorial!]'''</u></div>Fehrmannhttp://wiki.awf.forst.uni-goettingen.de/wiki/index.php/The_trigonometric_principleThe trigonometric principle2021-03-12T08:03:36Z<p>Fehrmann: </p>
<hr />
<div>{{Ficontent}} {{video|link=https://youtu.be/XqupMoiafMY}}<br />
<br />
The so-called trigonometric principle of [[tree height|height measurement]] bases on the measurement of angles as illustrated in the following figure. Height values per inclination angle are then calculated by<br />
<br />
:<math>h=e*\tan \alpha \,</math><br />
<br />
if the inclination is measured in degrees or<br />
<br />
:<math>h=e*\frac{p}{100} \,</math> <br />
<br />
if inclination is measured in percent; <math>e</math> is the [[Distance to tree|horizontal distance]]. These basic formula allow calculating what is given as <math>h1</math> and <math>h2</math> in the figure.<br />
<br />
[[file:tree_height_horizontal.png|550px|center]]<br />
<br />
To calculate total [[tree height]], these two height values need to be added. However, in order to make this calculation more general and also applicable to all applications in [[Measuring slope|sloped terrain]], we say, that we subtract the two resulting height measurements, where it must be observed that in the case of the above figure the angle <math>{\alpha}2</math> is a negative angle producing a negative height value. <br />
<br />
Therefore, as a general rule to calculate total tree height, we calculate the difference between upper measurement and lower measurement as<br />
<br />
:<math>h=e*(\tan \alpha_1 - \tan \alpha_2) \,</math><br />
<br />
while always taking into account that angle measurements below the horizontal will produce negative height values. This general formula does also work for height measurements in sloped terrain where it does not make a difference whether we measure from above or from below the tree.<br />
<br />
{{info<br />
|message=Note:<br />
|text=If you just remember to calculate <math>h_1-h_2</math> (while observing the sign of <math>h_2</math>!!) you are on the save side, no matter whether the observer stands above or below the stem base in sloped terrain!<br />
}}<br />
<br />
There are various [[:Category:Measurement devices|devices]] for measuring heights along the trigonometric principle, such as [[Haga]], [[Relascope|Spiegelrelaskop]] (relascope), the [[Suunto]] and [[Blume-Leiss]] height meters. Further most of the more modern instruments, like [[Vertex]] or other devices work based on the described principle. The scales of the instruments allow reading tree height directly whenever the indicated [[horizontal distance]] from the tree is kept; these distances are usually 15m or 20m (sometimes also 30m and 40m). Only for these fixed distances the tree heights can then be read directly from the instrument. Those modern instruments that include a rangefinder (laser or ultrasonic sound) a fixed distance is not necessary.<br />
<br />
For all other distances we need to modify the value read from the instrument proportionally. That is, if we read a tree height <math>h</math> for the distance of 20m, but in fact we are 40 m away the true height measurement is <math>2h</math>. This is important as we do not always make it to find a place in the stand at exactly the prescribed distance to the [[Tree Definition|tree]]. In fact, the optimal distance to the tree for height measurements along the trigonometric principle is at about one to one and a half tree height. If we go closer, it will be more and more difficult to aim at the top of the tree (in particular for deciduous species), if we go farther away, the angle measurements become less and less accurate.<br />
<br />
==Measuring tree heights in sloped terrain==<br />
[[file:tree_height_slope.png|550px|center]]<br />
<br />
{{info<br />
|message=Note:<br />
|text=If one uses a [[:Category:measurement devices|measurement device]] which does not enable to directly measure [[horizontal distance]]s (or e.g. rely on fixed distances to the tree determined by optical principles) the measured distance has to be corrected (or the resulting tree height). To avoid corrections in sloped terrain a practical solution is to move parallel to the slope and to find a position from which one can directly measure the horizontal distance.<br />
}}<br />
<br />
==Measuring height without given distance==<br />
There may be situations in which the measurement of a distance is not possible. Then, a variation of the trigonometric principle can be applied: a pole of known length is put perpendicularly next to the tree and the viewer takes an additional angle measurement <math>{\alpha}3</math> to the top of this pole.<br />
<br />
The tree height <math>h</math> is derived as usual from<br />
<br />
:<math>h=e*(\tan \alpha_1 - \tan \alpha_2) \,</math><br />
<br />
and the length <math>l</math> of the pole derives from the same formula:<br />
<br />
:<math>h=e*(\tan \alpha_3 - \tan \alpha_2) \,</math><br />
<br />
[[file:tree_height_pole.png|550px|center]]<br />
<br />
As the length <math>l</math> of the pole is known, we can determine the horizontal distance <math>e</math> from<br />
<br />
:<math>e=\frac{l}{(\tan \alpha_3 - \tan \alpha_2)} \,</math><br />
<br />
and inserting this into the tree height formula, what eventually yields:<br />
<br />
:<math>h=\frac{(\tan \alpha_1 - \tan \alpha_2)}{(\tan \alpha_3 - \tan \alpha_2)} \,</math><br />
<br />
With this formula we calculate tree height along the trigonometric principle without having to measure the distance to the tree.<br />
<br />
<br />
<br />
<br />
<br />
{{SEO|keywords=trigonometric principle,tree height,forest inventory,tree measurement,single tree variables <br />
sampling|descrip=The trigonometric principle of tree height measurement bases on the measurement of angles.}}<br />
<br />
<br />
[[Category:Tree height]]</div>Fehrmannhttp://wiki.awf.forst.uni-goettingen.de/wiki/index.php/The_trigonometric_principleThe trigonometric principle2021-03-12T08:01:49Z<p>Fehrmann: </p>
<hr />
<div>{{Ficontent}}<br />
{{video<br />
|link=https://youtu.be/XqupMoiafMY<br />
}}<br />
<br />
The so-called trigonometric principle of [[tree height|height measurement]] bases on the measurement of angles as illustrated in the following figure. Height values per inclination angle are then calculated by<br />
<br />
:<math>h=e*\tan \alpha \,</math><br />
<br />
if the inclination is measured in degrees or<br />
<br />
:<math>h=e*\frac{p}{100} \,</math> <br />
<br />
if inclination is measured in percent; <math>e</math> is the [[Distance to tree|horizontal distance]]. These basic formula allow calculating what is given as <math>h1</math> and <math>h2</math> in the figure.<br />
<br />
[[file:tree_height_horizontal.png|550px|center]]<br />
<br />
To calculate total [[tree height]], these two height values need to be added. However, in order to make this calculation more general and also applicable to all applications in [[Measuring slope|sloped terrain]], we say, that we subtract the two resulting height measurements, where it must be observed that in the case of the above figure the angle <math>{\alpha}2</math> is a negative angle producing a negative height value. <br />
<br />
Therefore, as a general rule to calculate total tree height, we calculate the difference between upper measurement and lower measurement as<br />
<br />
:<math>h=e*(\tan \alpha_1 - \tan \alpha_2) \,</math><br />
<br />
while always taking into account that angle measurements below the horizontal will produce negative height values. This general formula does also work for height measurements in sloped terrain where it does not make a difference whether we measure from above or from below the tree.<br />
<br />
{{info<br />
|message=Note:<br />
|text=If you just remember to calculate <math>h_1-h_2</math> (while observing the sign of <math>h_2</math>!!) you are on the save side, no matter whether the observer stands above or below the stem base in sloped terrain!<br />
}}<br />
<br />
There are various [[:Category:Measurement devices|devices]] for measuring heights along the trigonometric principle, such as [[Haga]], [[Relascope|Spiegelrelaskop]] (relascope), the [[Suunto]] and [[Blume-Leiss]] height meters. Further most of the more modern instruments, like [[Vertex]] or other devices work based on the described principle. The scales of the instruments allow reading tree height directly whenever the indicated [[horizontal distance]] from the tree is kept; these distances are usually 15m or 20m (sometimes also 30m and 40m). Only for these fixed distances the tree heights can then be read directly from the instrument. Those modern instruments that include a rangefinder (laser or ultrasonic sound) a fixed distance is not necessary.<br />
<br />
For all other distances we need to modify the value read from the instrument proportionally. That is, if we read a tree height <math>h</math> for the distance of 20m, but in fact we are 40 m away the true height measurement is <math>2h</math>. This is important as we do not always make it to find a place in the stand at exactly the prescribed distance to the [[Tree Definition|tree]]. In fact, the optimal distance to the tree for height measurements along the trigonometric principle is at about one to one and a half tree height. If we go closer, it will be more and more difficult to aim at the top of the tree (in particular for deciduous species), if we go farther away, the angle measurements become less and less accurate.<br />
<br />
==Measuring tree heights in sloped terrain==<br />
[[file:tree_height_slope.png|550px|center]]<br />
<br />
{{info<br />
|message=Note:<br />
|text=If one uses a [[:Category:measurement devices|measurement device]] which does not enable to directly measure [[horizontal distance]]s (or e.g. rely on fixed distances to the tree determined by optical principles) the measured distance has to be corrected (or the resulting tree height). To avoid corrections in sloped terrain a practical solution is to move parallel to the slope and to find a position from which one can directly measure the horizontal distance.<br />
}}<br />
<br />
==Measuring height without given distance==<br />
There may be situations in which the measurement of a distance is not possible. Then, a variation of the trigonometric principle can be applied: a pole of known length is put perpendicularly next to the tree and the viewer takes an additional angle measurement <math>{\alpha}3</math> to the top of this pole.<br />
<br />
The tree height <math>h</math> is derived as usual from<br />
<br />
:<math>h=e*(\tan \alpha_1 - \tan \alpha_2) \,</math><br />
<br />
and the length <math>l</math> of the pole derives from the same formula:<br />
<br />
:<math>h=e*(\tan \alpha_3 - \tan \alpha_2) \,</math><br />
<br />
[[file:tree_height_pole.png|550px|center]]<br />
<br />
As the length <math>l</math> of the pole is known, we can determine the horizontal distance <math>e</math> from<br />
<br />
:<math>e=\frac{l}{(\tan \alpha_3 - \tan \alpha_2)} \,</math><br />
<br />
and inserting this into the tree height formula, what eventually yields:<br />
<br />
:<math>h=\frac{(\tan \alpha_1 - \tan \alpha_2)}{(\tan \alpha_3 - \tan \alpha_2)} \,</math><br />
<br />
With this formula we calculate tree height along the trigonometric principle without having to measure the distance to the tree.<br />
<br />
<br />
<br />
<br />
<br />
{{SEO|keywords=trigonometric principle,tree height,forest inventory,tree measurement,single tree variables <br />
sampling|descrip=The trigonometric principle of tree height measurement bases on the measurement of angles.}}<br />
<br />
<br />
[[Category:Tree height]]</div>Fehrmannhttp://wiki.awf.forst.uni-goettingen.de/wiki/index.php/Template:VideoTemplate:Video2021-03-12T07:56:51Z<p>Fehrmann: </p>
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<div>icon for video template</div>Fehrmannhttp://wiki.awf.forst.uni-goettingen.de/wiki/index.php/Template:VideoTemplate:Video2021-03-12T07:50:55Z<p>Fehrmann: </p>
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:''{{{text}}}''</div>Fehrmannhttp://wiki.awf.forst.uni-goettingen.de/wiki/index.php/The_trigonometric_principleThe trigonometric principle2021-03-12T07:39:17Z<p>Fehrmann: /* See also */</p>
<hr />
<div>{{Ficontent}}<br />
The so-called trigonometric principle of [[tree height|height measurement]] bases on the measurement of angles as illustrated in the following figure. Height values per inclination angle are then calculated by<br />
<br />
:<math>h=e*\tan \alpha \,</math><br />
<br />
if the inclination is measured in degrees or<br />
<br />
:<math>h=e*\frac{p}{100} \,</math> <br />
<br />
if inclination is measured in percent; <math>e</math> is the [[Distance to tree|horizontal distance]]. These basic formula allow calculating what is given as <math>h1</math> and <math>h2</math> in the figure.<br />
<br />
[[file:tree_height_horizontal.png|550px|center]]<br />
<br />
To calculate total [[tree height]], these two height values need to be added. However, in order to make this calculation more general and also applicable to all applications in [[Measuring slope|sloped terrain]], we say, that we subtract the two resulting height measurements, where it must be observed that in the case of the above figure the angle <math>{\alpha}2</math> is a negative angle producing a negative height value. <br />
<br />
Therefore, as a general rule to calculate total tree height, we calculate the difference between upper measurement and lower measurement as<br />
<br />
:<math>h=e*(\tan \alpha_1 - \tan \alpha_2) \,</math><br />
<br />
while always taking into account that angle measurements below the horizontal will produce negative height values. This general formula does also work for height measurements in sloped terrain where it does not make a difference whether we measure from above or from below the tree.<br />
<br />
{{info<br />
|message=Note:<br />
|text=If you just remember to calculate <math>h_1-h_2</math> (while observing the sign of <math>h_2</math>!!) you are on the save side, no matter whether the observer stands above or below the stem base in sloped terrain!<br />
}}<br />
<br />
There are various [[:Category:Measurement devices|devices]] for measuring heights along the trigonometric principle, such as [[Haga]], [[Relascope|Spiegelrelaskop]] (relascope), the [[Suunto]] and [[Blume-Leiss]] height meters. Further most of the more modern instruments, like [[Vertex]] or other devices work based on the described principle. The scales of the instruments allow reading tree height directly whenever the indicated [[horizontal distance]] from the tree is kept; these distances are usually 15m or 20m (sometimes also 30m and 40m). Only for these fixed distances the tree heights can then be read directly from the instrument. Those modern instruments that include a rangefinder (laser or ultrasonic sound) a fixed distance is not necessary.<br />
<br />
For all other distances we need to modify the value read from the instrument proportionally. That is, if we read a tree height <math>h</math> for the distance of 20m, but in fact we are 40 m away the true height measurement is <math>2h</math>. This is important as we do not always make it to find a place in the stand at exactly the prescribed distance to the [[Tree Definition|tree]]. In fact, the optimal distance to the tree for height measurements along the trigonometric principle is at about one to one and a half tree height. If we go closer, it will be more and more difficult to aim at the top of the tree (in particular for deciduous species), if we go farther away, the angle measurements become less and less accurate.<br />
<br />
==Measuring tree heights in sloped terrain==<br />
[[file:tree_height_slope.png|550px|center]]<br />
<br />
{{info<br />
|message=Note:<br />
|text=If one uses a [[:Category:measurement devices|measurement device]] which does not enable to directly measure [[horizontal distance]]s (or e.g. rely on fixed distances to the tree determined by optical principles) the measured distance has to be corrected (or the resulting tree height). To avoid corrections in sloped terrain a practical solution is to move parallel to the slope and to find a position from which one can directly measure the horizontal distance.<br />
}}<br />
<br />
==Measuring height without given distance==<br />
There may be situations in which the measurement of a distance is not possible. Then, a variation of the trigonometric principle can be applied: a pole of known length is put perpendicularly next to the tree and the viewer takes an additional angle measurement <math>{\alpha}3</math> to the top of this pole.<br />
<br />
The tree height <math>h</math> is derived as usual from<br />
<br />
:<math>h=e*(\tan \alpha_1 - \tan \alpha_2) \,</math><br />
<br />
and the length <math>l</math> of the pole derives from the same formula:<br />
<br />
:<math>h=e*(\tan \alpha_3 - \tan \alpha_2) \,</math><br />
<br />
[[file:tree_height_pole.png|550px|center]]<br />
<br />
As the length <math>l</math> of the pole is known, we can determine the horizontal distance <math>e</math> from<br />
<br />
:<math>e=\frac{l}{(\tan \alpha_3 - \tan \alpha_2)} \,</math><br />
<br />
and inserting this into the tree height formula, what eventually yields:<br />
<br />
:<math>h=\frac{(\tan \alpha_1 - \tan \alpha_2)}{(\tan \alpha_3 - \tan \alpha_2)} \,</math><br />
<br />
With this formula we calculate tree height along the trigonometric principle without having to measure the distance to the tree.<br />
<br />
==Video tutorial==<br />
[https://youtu.be/XqupMoiafMY Video tutorial on basics of tree height measurement]<br />
<br />
<br />
<br />
<br />
<br />
{{SEO|keywords=trigonometric principle,tree height,forest inventory,tree measurement,single tree variables <br />
sampling|descrip=The trigonometric principle of tree height measurement bases on the measurement of angles.}}<br />
<br />
<br />
[[Category:Tree height]]</div>Fehrmannhttp://wiki.awf.forst.uni-goettingen.de/wiki/index.php/The_trigonometric_principleThe trigonometric principle2021-03-12T07:38:53Z<p>Fehrmann: /* See also */</p>
<hr />
<div>{{Ficontent}}<br />
The so-called trigonometric principle of [[tree height|height measurement]] bases on the measurement of angles as illustrated in the following figure. Height values per inclination angle are then calculated by<br />
<br />
:<math>h=e*\tan \alpha \,</math><br />
<br />
if the inclination is measured in degrees or<br />
<br />
:<math>h=e*\frac{p}{100} \,</math> <br />
<br />
if inclination is measured in percent; <math>e</math> is the [[Distance to tree|horizontal distance]]. These basic formula allow calculating what is given as <math>h1</math> and <math>h2</math> in the figure.<br />
<br />
[[file:tree_height_horizontal.png|550px|center]]<br />
<br />
To calculate total [[tree height]], these two height values need to be added. However, in order to make this calculation more general and also applicable to all applications in [[Measuring slope|sloped terrain]], we say, that we subtract the two resulting height measurements, where it must be observed that in the case of the above figure the angle <math>{\alpha}2</math> is a negative angle producing a negative height value. <br />
<br />
Therefore, as a general rule to calculate total tree height, we calculate the difference between upper measurement and lower measurement as<br />
<br />
:<math>h=e*(\tan \alpha_1 - \tan \alpha_2) \,</math><br />
<br />
while always taking into account that angle measurements below the horizontal will produce negative height values. This general formula does also work for height measurements in sloped terrain where it does not make a difference whether we measure from above or from below the tree.<br />
<br />
{{info<br />
|message=Note:<br />
|text=If you just remember to calculate <math>h_1-h_2</math> (while observing the sign of <math>h_2</math>!!) you are on the save side, no matter whether the observer stands above or below the stem base in sloped terrain!<br />
}}<br />
<br />
There are various [[:Category:Measurement devices|devices]] for measuring heights along the trigonometric principle, such as [[Haga]], [[Relascope|Spiegelrelaskop]] (relascope), the [[Suunto]] and [[Blume-Leiss]] height meters. Further most of the more modern instruments, like [[Vertex]] or other devices work based on the described principle. The scales of the instruments allow reading tree height directly whenever the indicated [[horizontal distance]] from the tree is kept; these distances are usually 15m or 20m (sometimes also 30m and 40m). Only for these fixed distances the tree heights can then be read directly from the instrument. Those modern instruments that include a rangefinder (laser or ultrasonic sound) a fixed distance is not necessary.<br />
<br />
For all other distances we need to modify the value read from the instrument proportionally. That is, if we read a tree height <math>h</math> for the distance of 20m, but in fact we are 40 m away the true height measurement is <math>2h</math>. This is important as we do not always make it to find a place in the stand at exactly the prescribed distance to the [[Tree Definition|tree]]. In fact, the optimal distance to the tree for height measurements along the trigonometric principle is at about one to one and a half tree height. If we go closer, it will be more and more difficult to aim at the top of the tree (in particular for deciduous species), if we go farther away, the angle measurements become less and less accurate.<br />
<br />
==Measuring tree heights in sloped terrain==<br />
[[file:tree_height_slope.png|550px|center]]<br />
<br />
{{info<br />
|message=Note:<br />
|text=If one uses a [[:Category:measurement devices|measurement device]] which does not enable to directly measure [[horizontal distance]]s (or e.g. rely on fixed distances to the tree determined by optical principles) the measured distance has to be corrected (or the resulting tree height). To avoid corrections in sloped terrain a practical solution is to move parallel to the slope and to find a position from which one can directly measure the horizontal distance.<br />
}}<br />
<br />
==Measuring height without given distance==<br />
There may be situations in which the measurement of a distance is not possible. Then, a variation of the trigonometric principle can be applied: a pole of known length is put perpendicularly next to the tree and the viewer takes an additional angle measurement <math>{\alpha}3</math> to the top of this pole.<br />
<br />
The tree height <math>h</math> is derived as usual from<br />
<br />
:<math>h=e*(\tan \alpha_1 - \tan \alpha_2) \,</math><br />
<br />
and the length <math>l</math> of the pole derives from the same formula:<br />
<br />
:<math>h=e*(\tan \alpha_3 - \tan \alpha_2) \,</math><br />
<br />
[[file:tree_height_pole.png|550px|center]]<br />
<br />
As the length <math>l</math> of the pole is known, we can determine the horizontal distance <math>e</math> from<br />
<br />
:<math>e=\frac{l}{(\tan \alpha_3 - \tan \alpha_2)} \,</math><br />
<br />
and inserting this into the tree height formula, what eventually yields:<br />
<br />
:<math>h=\frac{(\tan \alpha_1 - \tan \alpha_2)}{(\tan \alpha_3 - \tan \alpha_2)} \,</math><br />
<br />
With this formula we calculate tree height along the trigonometric principle without having to measure the distance to the tree.<br />
<br />
==See also==<br />
[https://youtu.be/XqupMoiafMY Video tutorial on basics of tree height measurement]<br />
<br />
<br />
<br />
<br />
<br />
{{SEO|keywords=trigonometric principle,tree height,forest inventory,tree measurement,single tree variables <br />
sampling|descrip=The trigonometric principle of tree height measurement bases on the measurement of angles.}}<br />
<br />
<br />
[[Category:Tree height]]</div>Fehrmannhttp://wiki.awf.forst.uni-goettingen.de/wiki/index.php/The_trigonometric_principleThe trigonometric principle2021-03-12T07:36:35Z<p>Fehrmann: /* See also */</p>
<hr />
<div>{{Ficontent}}<br />
The so-called trigonometric principle of [[tree height|height measurement]] bases on the measurement of angles as illustrated in the following figure. Height values per inclination angle are then calculated by<br />
<br />
:<math>h=e*\tan \alpha \,</math><br />
<br />
if the inclination is measured in degrees or<br />
<br />
:<math>h=e*\frac{p}{100} \,</math> <br />
<br />
if inclination is measured in percent; <math>e</math> is the [[Distance to tree|horizontal distance]]. These basic formula allow calculating what is given as <math>h1</math> and <math>h2</math> in the figure.<br />
<br />
[[file:tree_height_horizontal.png|550px|center]]<br />
<br />
To calculate total [[tree height]], these two height values need to be added. However, in order to make this calculation more general and also applicable to all applications in [[Measuring slope|sloped terrain]], we say, that we subtract the two resulting height measurements, where it must be observed that in the case of the above figure the angle <math>{\alpha}2</math> is a negative angle producing a negative height value. <br />
<br />
Therefore, as a general rule to calculate total tree height, we calculate the difference between upper measurement and lower measurement as<br />
<br />
:<math>h=e*(\tan \alpha_1 - \tan \alpha_2) \,</math><br />
<br />
while always taking into account that angle measurements below the horizontal will produce negative height values. This general formula does also work for height measurements in sloped terrain where it does not make a difference whether we measure from above or from below the tree.<br />
<br />
{{info<br />
|message=Note:<br />
|text=If you just remember to calculate <math>h_1-h_2</math> (while observing the sign of <math>h_2</math>!!) you are on the save side, no matter whether the observer stands above or below the stem base in sloped terrain!<br />
}}<br />
<br />
There are various [[:Category:Measurement devices|devices]] for measuring heights along the trigonometric principle, such as [[Haga]], [[Relascope|Spiegelrelaskop]] (relascope), the [[Suunto]] and [[Blume-Leiss]] height meters. Further most of the more modern instruments, like [[Vertex]] or other devices work based on the described principle. The scales of the instruments allow reading tree height directly whenever the indicated [[horizontal distance]] from the tree is kept; these distances are usually 15m or 20m (sometimes also 30m and 40m). Only for these fixed distances the tree heights can then be read directly from the instrument. Those modern instruments that include a rangefinder (laser or ultrasonic sound) a fixed distance is not necessary.<br />
<br />
For all other distances we need to modify the value read from the instrument proportionally. That is, if we read a tree height <math>h</math> for the distance of 20m, but in fact we are 40 m away the true height measurement is <math>2h</math>. This is important as we do not always make it to find a place in the stand at exactly the prescribed distance to the [[Tree Definition|tree]]. In fact, the optimal distance to the tree for height measurements along the trigonometric principle is at about one to one and a half tree height. If we go closer, it will be more and more difficult to aim at the top of the tree (in particular for deciduous species), if we go farther away, the angle measurements become less and less accurate.<br />
<br />
==Measuring tree heights in sloped terrain==<br />
[[file:tree_height_slope.png|550px|center]]<br />
<br />
{{info<br />
|message=Note:<br />
|text=If one uses a [[:Category:measurement devices|measurement device]] which does not enable to directly measure [[horizontal distance]]s (or e.g. rely on fixed distances to the tree determined by optical principles) the measured distance has to be corrected (or the resulting tree height). To avoid corrections in sloped terrain a practical solution is to move parallel to the slope and to find a position from which one can directly measure the horizontal distance.<br />
}}<br />
<br />
==Measuring height without given distance==<br />
There may be situations in which the measurement of a distance is not possible. Then, a variation of the trigonometric principle can be applied: a pole of known length is put perpendicularly next to the tree and the viewer takes an additional angle measurement <math>{\alpha}3</math> to the top of this pole.<br />
<br />
The tree height <math>h</math> is derived as usual from<br />
<br />
:<math>h=e*(\tan \alpha_1 - \tan \alpha_2) \,</math><br />
<br />
and the length <math>l</math> of the pole derives from the same formula:<br />
<br />
:<math>h=e*(\tan \alpha_3 - \tan \alpha_2) \,</math><br />
<br />
[[file:tree_height_pole.png|550px|center]]<br />
<br />
As the length <math>l</math> of the pole is known, we can determine the horizontal distance <math>e</math> from<br />
<br />
:<math>e=\frac{l}{(\tan \alpha_3 - \tan \alpha_2)} \,</math><br />
<br />
and inserting this into the tree height formula, what eventually yields:<br />
<br />
:<math>h=\frac{(\tan \alpha_1 - \tan \alpha_2)}{(\tan \alpha_3 - \tan \alpha_2)} \,</math><br />
<br />
With this formula we calculate tree height along the trigonometric principle without having to measure the distance to the tree.<br />
<br />
==See also==<br />
[https://youtu.be/XqupMoiafMY Video tutorial on basics of tree height measurement]<br />
<br />
{{SEO|keywords=trigonometric principle,tree height,forest inventory,tree measurement,single tree variables <br />
sampling|descrip=The trigonometric principle of tree height measurement bases on the measurement of angles.}}<br />
<br />
<br />
[[Category:Tree height]]</div>Fehrmannhttp://wiki.awf.forst.uni-goettingen.de/wiki/index.php/The_trigonometric_principleThe trigonometric principle2021-03-12T07:35:48Z<p>Fehrmann: </p>
<hr />
<div>{{Ficontent}}<br />
The so-called trigonometric principle of [[tree height|height measurement]] bases on the measurement of angles as illustrated in the following figure. Height values per inclination angle are then calculated by<br />
<br />
:<math>h=e*\tan \alpha \,</math><br />
<br />
if the inclination is measured in degrees or<br />
<br />
:<math>h=e*\frac{p}{100} \,</math> <br />
<br />
if inclination is measured in percent; <math>e</math> is the [[Distance to tree|horizontal distance]]. These basic formula allow calculating what is given as <math>h1</math> and <math>h2</math> in the figure.<br />
<br />
[[file:tree_height_horizontal.png|550px|center]]<br />
<br />
To calculate total [[tree height]], these two height values need to be added. However, in order to make this calculation more general and also applicable to all applications in [[Measuring slope|sloped terrain]], we say, that we subtract the two resulting height measurements, where it must be observed that in the case of the above figure the angle <math>{\alpha}2</math> is a negative angle producing a negative height value. <br />
<br />
Therefore, as a general rule to calculate total tree height, we calculate the difference between upper measurement and lower measurement as<br />
<br />
:<math>h=e*(\tan \alpha_1 - \tan \alpha_2) \,</math><br />
<br />
while always taking into account that angle measurements below the horizontal will produce negative height values. This general formula does also work for height measurements in sloped terrain where it does not make a difference whether we measure from above or from below the tree.<br />
<br />
{{info<br />
|message=Note:<br />
|text=If you just remember to calculate <math>h_1-h_2</math> (while observing the sign of <math>h_2</math>!!) you are on the save side, no matter whether the observer stands above or below the stem base in sloped terrain!<br />
}}<br />
<br />
There are various [[:Category:Measurement devices|devices]] for measuring heights along the trigonometric principle, such as [[Haga]], [[Relascope|Spiegelrelaskop]] (relascope), the [[Suunto]] and [[Blume-Leiss]] height meters. Further most of the more modern instruments, like [[Vertex]] or other devices work based on the described principle. The scales of the instruments allow reading tree height directly whenever the indicated [[horizontal distance]] from the tree is kept; these distances are usually 15m or 20m (sometimes also 30m and 40m). Only for these fixed distances the tree heights can then be read directly from the instrument. Those modern instruments that include a rangefinder (laser or ultrasonic sound) a fixed distance is not necessary.<br />
<br />
For all other distances we need to modify the value read from the instrument proportionally. That is, if we read a tree height <math>h</math> for the distance of 20m, but in fact we are 40 m away the true height measurement is <math>2h</math>. This is important as we do not always make it to find a place in the stand at exactly the prescribed distance to the [[Tree Definition|tree]]. In fact, the optimal distance to the tree for height measurements along the trigonometric principle is at about one to one and a half tree height. If we go closer, it will be more and more difficult to aim at the top of the tree (in particular for deciduous species), if we go farther away, the angle measurements become less and less accurate.<br />
<br />
==Measuring tree heights in sloped terrain==<br />
[[file:tree_height_slope.png|550px|center]]<br />
<br />
{{info<br />
|message=Note:<br />
|text=If one uses a [[:Category:measurement devices|measurement device]] which does not enable to directly measure [[horizontal distance]]s (or e.g. rely on fixed distances to the tree determined by optical principles) the measured distance has to be corrected (or the resulting tree height). To avoid corrections in sloped terrain a practical solution is to move parallel to the slope and to find a position from which one can directly measure the horizontal distance.<br />
}}<br />
<br />
==Measuring height without given distance==<br />
There may be situations in which the measurement of a distance is not possible. Then, a variation of the trigonometric principle can be applied: a pole of known length is put perpendicularly next to the tree and the viewer takes an additional angle measurement <math>{\alpha}3</math> to the top of this pole.<br />
<br />
The tree height <math>h</math> is derived as usual from<br />
<br />
:<math>h=e*(\tan \alpha_1 - \tan \alpha_2) \,</math><br />
<br />
and the length <math>l</math> of the pole derives from the same formula:<br />
<br />
:<math>h=e*(\tan \alpha_3 - \tan \alpha_2) \,</math><br />
<br />
[[file:tree_height_pole.png|550px|center]]<br />
<br />
As the length <math>l</math> of the pole is known, we can determine the horizontal distance <math>e</math> from<br />
<br />
:<math>e=\frac{l}{(\tan \alpha_3 - \tan \alpha_2)} \,</math><br />
<br />
and inserting this into the tree height formula, what eventually yields:<br />
<br />
:<math>h=\frac{(\tan \alpha_1 - \tan \alpha_2)}{(\tan \alpha_3 - \tan \alpha_2)} \,</math><br />
<br />
With this formula we calculate tree height along the trigonometric principle without having to measure the distance to the tree.<br />
<br />
==See also==<br />
[https://https://youtu.be/XqupMoiafMY Video tutorial on basics of tree height measurement]<br />
<br />
{{SEO|keywords=trigonometric principle,tree height,forest inventory,tree measurement,single tree variables <br />
sampling|descrip=The trigonometric principle of tree height measurement bases on the measurement of angles.}}<br />
<br />
<br />
[[Category:Tree height]]</div>Fehrmannhttp://wiki.awf.forst.uni-goettingen.de/wiki/index.php/Stratified_samplingStratified sampling2021-03-10T14:27:51Z<p>Fehrmann: /* Notation */</p>
<hr />
<div>{{Languages}}{{Ficontent}}<br />
Stratified sampling is actually not a new [[lectuenotes:Sampling design and plot design|sampling design]] of its own, but a procedural method to subdivide a [[population]] into separate and more homogeneous sub-populations called [[strata]] (Kleinn 2007<ref name="kleinn2007">Kleinn, C. 2007. Lecture Notes for the Teaching Module Forest Inventory. Department of Forest Inventory and Remote Sensing. Faculty of Forest Science and Forest Ecology, Georg-August-Universität Göttingen. 164 S.</ref>). In some situations it is useful from a statistical point of view, or required for practical and organizational reasons, to subdivide the population in different strata.<br />
The major characteristic is that independent sampling studies are carried out in each stratum where all strata are considered as sub-populations of which the parameters need to be estimated. If [[Simple random sampling]] is applied, we call that '''stratified random sampling'''. <br />
The only difference between random sampling and stratified random sampling is that the last consists of various sampling studies and the only thing we have to consider is how to combine the estimations that come from the single strata in order to produce estimations for the total population.<br />
<br />
Stratified sampling is efficient especially in those cases where the [[variability]] inside the strata is low and the differences of means between the strata is large (Akca 2001<ref name="akca">Akca, A. 2001. Waldinventur. J.D. Sauerländer's Verlag. Frankfurt am Main, 193 S.</ref>). In this case we can achive a higher [[accuracy and precision|precision]] with the same [[sample size]]. Beside statistical issues there are further arguments for stratification.<br />
<br />
We can distinguish two general approaches for stratification, the so called pre-stratification in which strata are formed '''''before''''' the sampling study starts, and the post-stratification, where we generate strata in course of the sampling or even afterwards based on the data. In the first case, that is described in this artice, the strata must be defined and - in case of geographical strata - delineated to define the [[Population|sampling frame]].<br />
<br />
The precondition for a meaningful partitioning of a population in non-overlapping strata is the availability of prior information that can be used as stratification criteria (de Vries 1986<ref>de Vries, P.G., 1986. Sampling Theory for Forest Inventory. A Teach-Yourself Course. Springer. 399 p.</ref>). In forest inventories this information might be available in form of forest management or GIS-data or can be derived from remote sensing data like aerial photos. Most efficient from a statistical point of view, is the stratification of a population proportional to the target value of the inventory. As this target value is typically not known before the inventory, forest variables that are correlated to this value are used as stratification criteria. In large managed forest areas age classes or forest types might be for example good stratification criteria if the estimation of [[volume per ha]] is targeted. <br />
<br />
<br />
===Arguments for stratification===<br />
Sometimes it is useful to subdivide the [[population]] of interest in a number of sub-populations and carry out an independent sampling in each of these strata. There are statistical as well as practical considerations that make this technique very favorable and interesting for large area forest inventories. Not without reason almost all [[national forest invetories]] are based on stratification.<br />
<br />
;Statistical justifications:<br />
*The spatial distribution of [[Sample point|sample point]]s inside the population is more evenly, if these points are selected in single strata,<br />
*It is possible to make an individual optimization of [[:Category:Sampling design|sampling]] and [[:category:plot design|plot design]] for each stratum,<br />
*One usually increases the [[accuracy and precision|precision]] of the estimations for the total population,<br />
*Separate estimations for each of the strata are produced in a pre-planned manner,<br />
*It is guaranteed that there are actually sufficient observations in each one of the strata,<br />
*It is possible to produce estimations with defined precision level for sub-populations.<br />
<br />
;Practical justification:<br />
*The possibility to optimize the [[Sampling design and plot design|inventory design]] separately for each stratum is very efficient and helps to minimize costs,<br />
*To facilitate inventory work (particularly field work): independent [[field campaign|field campaigns]] can be carried out in each stratum,<br />
*It allows a better specialization of field crews (e.g. botanists).<br />
<br />
===Stratification criteria===<br />
<br />
For the partitioning of a population we can imagine very different stratification criteria. If the reason for stratification is not an improvement of the [[accuracy and precision|accuracy]] of estimations, the stratification variables must not necessarily be [[correlation|correlated]] to the target value. Under special conditions it might be useful to stratify even if there are no statistical justifications. For example might a political boundary dividing a forest area be a good reason for stratification, even if the forest is very homogeneous, if afterwards estimations for both parts should be derived seperately. Other examples for meaningfull criteria are:<br />
<br />
;Geographical startification:<br />
*Eco zones,<br />
*[[Forest types]],<br />
*Site and soil types,<br />
*Topographical conditions,<br />
*Political boundaries or properties,<br />
*...<br />
<br />
Further one can imagine to use the expected inventory costs (regarding time consumption) as criterion. These costs are typically correlated to the above mentioned geographical conditions. It might be for example reasonable to stratify a forest area by slope classes, if time consumption for field work differs significantly between flat terrain and steep slopes.<br />
<br />
;Subject matter stratification<br />
*Species,<br />
*Species groups (e.g. commercial / non-commercial),<br />
*Tree sociological classes,<br />
*Age classes in plantation forests,<br />
*...<br />
<br />
==Statistics==<br />
<br />
The only new concept that needs to be introduced in stratified random sampling is how to combine the estimates derived for different strata. <br />
The [[Estimator|estimator]]s for stratified random sampling are based on simple considerations about [[linear combination]]s (Kleinn 2007<ref name="kleinn2007">Kleinn, C. 2007. Lecture Notes for the Teaching Module Forest Inventory. Department of Forest Inventory and Remote Sensing. Faculty of Forest Science and Forest Ecology, Georg-August-Universität Göttingen. 164 S.</ref>). <br />
When we have two '''''independent''''' random variables <math>Y_1\,</math> and <math>Y_2\,</math> and if we are interested in the sum of the two <math>Y_1+Y_2\,</math>, then<br />
<br />
:<math>E(Y_1+Y_2)=E(Y_1)+E(Y_2)\,</math> and<br />
<br />
:<math>var(Y_1+Y_2)=var(Y_1)+var(Y_2)\,</math><br />
<br />
{{info<br />
|message=Simple:<br />
|text=The [[expected value]] for the sum of both is equal to the sum of both single expected values. It is intuitively clear that we can sum up the totals to derive an overall total. It is different if we consider the variables to be means! In this case we have to weight the single means according to the size of the strata.<br />
}}<br />
<br />
If we consider <math>Y_1\,</math> and <math>Y_2\,</math> as estimations from the two strata 1 and 2, then we can apply the principles derived from these considerations for stratified sampling.<br />
<br />
However, to calculate the overall [[Mean|mean]] from two estimations, we have to weight the single means in order to account for possibly different sizes of the sub-populations <math>N_1\,</math> and <math>N_2\,</math>. If both strata are of the '''same size''', we can calculate the mean by:<br />
<br />
:<math>\frac 12 (Y_1+Y_2)=\frac 12 Y_1+\frac 12Y_2=c_1Y_1+c_2Y_2\,</math><br />
<br />
The factor <math>c_i\,</math> can be interpreted as a weight for the single estimations from stratum 1 and 2. Because both are of equal size in this example <math>c_1=c_2\,</math> holds.<br />
A more typical case would be that we deal with strata of unequal sizes.<br />
<br />
{{info<br />
|message=Example:<br />
|text=Weighting of single partial results (or estimations) is important, if they stem from different sized sub-populations and we want to calculate a mean. A simple example: You should calculate the mean body weight of 50 students in a classroom. You have a mean value derived for the 15 ladies (55 Kg) and a mean body weight for the 35 men (73 Kg). If you would calculate an unweighted mean (64 Kg) it would be wrong. Correct is 15/50*55+35/50*73=67,6 Kg! The weights 15/50 and 35/50 are an expression of the share of the respective group on the total population. This weight is equal to the selection probability for simple random sampling.<br />
}}<br />
<br />
<br />
The weights must be proportional to the size of the sub-populations. In case of forest inventories the population is typically the total forest area (we are selecting [[sample point]]s from this continuum), so that the individual stratum areas can be used to derive weighting factors. The sum of all weights must be 1, so that:<br />
<br />
:<math>\sum c_i=1\,</math><br />
The expected value E for '''different sized''' strata is then:<br />
<br />
:<math>E(c_1Y_1+c_2Y_2)=E(c_1Y_1)+E(c_2Y_2)=c_1E(Y_1)+c_2E(Y_2)\,</math> , where <math>c_1 \not= c_2\,</math> or<br />
<br />
:<math>E(\sum c_iY_i)=\sum c_iE(Y_i)\,</math><br />
<br />
and the variance:<br />
<br />
:<math>var(c_1Y_1+c_2Y_2)=var(c_1Y_1)+var(c_2Y_2)=c_1^2var(Y_1)+c_2^2var(Y_2)\,</math> or<br />
<br />
:<math>var(\sum c_iY_i)=\sum c_i^2var(Y_i)\,</math><br />
<br />
{{info<br />
|message=Note:<br />
|text=If we expand (or like in this case relate) the variance with a constant factor, this factor must be squared, because variance is a squared measure!<br />
}}<br />
<br />
===Notation===<br />
<br />
{|<br />
|-<br />
! !! <br />
|-<br />
| <math>L\,</math> || Number of strata <math>h=1, ... , L \,</math> ||<br />
|-<br />
| <math>N\,</math> || Total population size ||<br />
|-<br />
| <math>N_h\,</math> || Size of stratum <math>h (N=\sum N_h)\,</math> ||<br />
|-<br />
| <math>\bar y\,</math> || Estimated population mean ||<br />
|-<br />
| <math>\bar y_h\,</math> || Estimated mean of stratum <math>h\,</math> ||<br />
|-<br />
| <math>n\,</math> || Total sample size ||<br />
|-<br />
| <math>n_h\,</math> || Sample size in stratum <math>h\,</math>||<br />
|-<br />
| <math>S^2_h\,</math> || Sample variance in stratum <math>h\,</math> ||<br />
|-<br />
| <math>\tau\,</math> || Total ||<br />
|-<br />
| <math>\tau_h\,</math> || Total in stratum <math>h\,</math> ||<br />
|-<br />
| <math>\hat \tau_h\,</math> || Estimated total in stratum <math>h\,</math> ||<br />
|-<br />
| <math>c_h\,</math> || Relative share of stratum <math>h\,</math> or weight of stratum ||<br />
|-<br />
| <math>\hat {var} (\bar y)\,</math> || Estimated error variance ||<br />
|-<br />
| <math>\hat {var} (\hat \tau)\,</math> || Estimated error variance of the total ||<br />
|-<br />
| <math>k_h\,</math> || Estimated costs for the assessment in stratum <math>h\,</math> ||<br />
|}<br />
<br />
===Estimator for the mean===<br />
<br />
The estimator for the mean for stratified random sampling is derived based on the considerations above and analog to the estimator of [[Simple random sampling|simple random sampling]] as<br />
<br />
:<math>\bar y = \sum_{h=1}^L \frac{N_h}{N} \bar y_h = \frac {1}{N} \sum_{h=1}^L N_h \bar y_h\,</math><br />
<br />
===Estimator for the error variance===<br />
<br />
The estimator for the error variance (selection without replacement) is:<br />
<br />
:<math>\hat {var} (\bar y) = \sum_{h=1}^L \left\lbrace \left( \frac {N_h}{N} \right)^2 \hat {var} (\bar y_h) \right\rbrace = \frac{1}{N^2} \sum_{h=1}^L N^2_h \frac {N_h-n_h}{N_h} \frac {S^2_h}{n_h}</math><br />
<br />
In this case <math>N_h-n_h/N_h\,</math> is a [[finit population correction]] that is necessary if the strata are small and/or the sample size is large. It is typically applied if the relation between sample size and population is larger than 0.05 (Akca 2001<ref name="akca">Akca, A. 2001. Waldinventur. J.D. Sauerländer's Verlag. Frankfurt am Main, 193 S.</ref>).<br />
<br />
{{info<br />
|message=Note:<br />
|text=A finite population correction is important in case that we apply a selection without replacement and the population size is significantly reduced by drawing the samples. As consequence the selection probabilities are changing with every sample we draw (because the remaining population decreases) what is corrected by this factor.<br />
}}<br />
<br />
<br />
This estimator looks complex, but can be read as the weighted linear combination of simple random sampling estimators applied to the strata. The weighting factor is in this case<br />
<br />
:<math>c^2=\frac {N_h^2}{N^2}\,</math><br />
<br />
Without finite population correction the error variance is:<br />
<br />
:<math>\hat {var} (\bar y) = \frac{1}{N^2} \sum_{h=1}^L N^2_h \frac {S^2_h}{n_h}</math><br />
<br />
===Estimator of the total===<br />
<br />
:<math>\hat\tau = N\bar y = \sum_{h=1}^L \frac {N_h}{N} \hat \tau_h = \sum_{h=1}^L N_h \bar y_h\,</math><br />
<br />
The estimated error variance of the estimated population total follows then with:<br />
<br />
:<math>\hat{var}(\hat {\tau}) = \hat{var}(N \bar y) = N^2 \hat{var}(\bar y)</math><br />
<br />
To calculate the [[confidence interval]] for the estimation we need information about the standard error and the sample size. In stratified random sampling one faces the difficulty with the number of [[dergees of freedom]]. While it is a direct function of sample size (<math>DF=n-1\,</math>) in simple random sampling, we deal with <math>h\,</math> different sample sizes that are combined. If the variances among these strata differ significantly it is not possible to simply join the degrees of freedom from different strata. This statistical problem is known as '''''Behrens-Fischer problem''''' or '''''Welch-Satterthwaite problem'''''. These statisticians proposed an approximation formula that can be used to calculate the "effective number of degrees of freedom" so that t-statistics can be approximately correct applied.<br />
<br />
:<math>DF_e=\frac {\left(\sum_{h=1}^L g_h S_h^2 \right)^2}{\sum_{h=1}^L \frac {g_h^2 S_h^4}{n_h-1}}\,</math><br />
<br />
==Sample size==<br />
<br />
To determine the necessary [[sample size|sample size]] that is always dependent on the error probability, the allowed error and the variability inside the population, we have to consider that the variance inside the different strata is different. These different variances must be weighted to calculate the necessary sample size.<br />
<br />
{{info<br />
|message=Note:<br />
|text=The "necessary" sample size is the estimated number of samples we need to derive an estimation with a defined [[confidence interval]]. This interval is determined by the predetermined error probability <math>\alpha\,</math>, the corresponding t-value from the t-distribution and the allowed error we define for the inventory. Typically this error is 10% for standard forest inventories.<br />
}}<br />
<br />
Compare the following formula with the formula provided for [[Sample size|simple random sampling]]:<br />
<br />
:<math>n = \frac {t^2 \sum \frac {N^2_h S^2_h}{c_h}}{N^2 A^2}\,</math><br />
<br />
where <math>c_h = n_h/n\,</math>, or the share of samples that is in stratum <math>h\,</math>.<br />
<br />
{{info<br />
|message=Note:<br />
|text=To calculate the overall sample size it is obviously necessary to know the share of samples in the different strata?! That sounds strange, because we like to calculate the sample size here. If we consider that the sample size in each stratum is influencing the expected error, it is clear that we need to have this information. Thats why we have to define the allocation scheme before we start.<br />
}}<br />
<br />
In all sample size calculations one has to know ''before'' how to assign the total number of samples to the different strata: the set of <math>c_h\,</math> must be predetermined!<br />
<br />
In small populations (or large samples) we have to consider the finite population correction and the above formula becomes:<br />
<br />
:<math>n=\frac {\sum_{h=1}^L \frac {N_h^2 S_h^2}{c_h}}{\frac {N^2 A^2}{t^2}+ \sum_{h=1}^L N_h S_h^2}\,</math><br />
<br />
==Allocation of sample size to the strata==<br />
Three factors are relevant in respect to the decision of how to allocate the samples to the strata:<br />
<br />
*Stratum sizes (The bigger the stratum the more samples)<br />
*Variability inside the strata (The more variability the more samples)<br />
*Inventory costs that might vary between strata (The more costly the fewer samples).<br />
<br />
In case that all strata are of same size (equal area) and the variability is also equal, we can use<br />
<br />
:<math>n_h = \frac {n}{L}\,</math><br />
<br />
so a '''uniform allocation''' of samples to the strata. As mentioned above this situation is not realistic and further stratification would not be superior in this case.<br />
<br />
If the number of samples per stratum should be determined proportional to the stratum size (e.g. area) we have a '''proportional allocation''' with:<br />
<br />
:<math>n_h = n \frac {N_h}{N}\,</math><br />
<br />
In this case one would ignore possibly different variances inside the strata. If we like to consider the variances, we need some prior information about the conditions inside the strata that is sometimes available from case studies or forest management data. In this case one can apply the '''Neyman allocation''':<br />
<br />
:<math>n_h = n \frac {N_h S^2_h}{\sum_{h=1}^L N_h S^2_h}\,</math><br />
<br />
If one has to consider inventory costs that might vary significantly between strata, this information can be included as additional factor. In this case one is able to calculate the '''optimal allocation with cost-minimization''': <br />
<br />
:<math>n_h = n \frac {\frac {N_h S^2_h}{\sqrt {k_h}}}{\sum_{h=1}^L \frac{N_h S^2_h}{\sqrt {k_h}}}\,</math><br />
<br />
{{info<br />
|message=Note:<br />
|text=It is obvious that we need <math>n</math> to calculate the allocation. At the same time we need <math>n_h</math> (the result of this calculation) to determine the sample size. This dilemma can be solved by an iterative process, where we predetermine relative shares (<math>c_h</math>) to derive <math>n</math> in a first step.<br />
}}<br />
<br />
==Summarizing==<br />
<br />
Depending on the allocation scheme that should be used one needs the following information to implement stratified random sampling:<br />
*number of strata,<br />
*size of strata or relative share on the total population,<br />
*estimations for the variance inside the strata,<br />
*eventually information about the expected inventory costs in different strata.<br />
<br />
Further one has to predefine the target precision (A) (that is the allowed error expressed as 1/2 of the confidence interval width), the error probability <math>\alpha</math> that is typically 0.05 and the corresponding t-value from the t-distribution (in case of sample size > 30 this value is approximately 2).<br />
<br />
Based on these Informations one may <br />
*choose an appropriate allocation scheme,<br />
*derive the weights <math>c_h\,</math> for the strata,<br />
*calculate the sample size,<br />
*and allocate them to the strata.<br />
<br />
{{Exercise<br />
|message=Stratified sampling examples<br />
|alttext=Example<br />
|text=Example for Stratified sampling<br />
}}<br />
<br />
==Comments==<br />
<br />
Stratification is a powerful procedure to reduce the error variance if the above mentioned preconditions (homogeneous strata with significant differences of strata means) is fulfilled. In this case one takes a maximum of variability out of the sample.<br />
Some a priori information is necessary as the subdivision of the population must be defined before the samples are taken. If these information is missing one may apply techniques like [[Double_sampling#Double_sampling_for_stratification_.28DSS.29|double sampling for stratification]] that employs stratification without requiring a priori delineation of strata (the strata sizes are estimated in the course of a two-phase sampling process).<br />
In this chapter stratified random sampling was introduced. However stratification can also be applied for other sampling techniques. It is important to note that one may apply whatever sampling technique that is appropriate inside the different strata. That means for example that one can optimize the [[Sampling design and plot design|Inventory design]] as well as the [[Sampling design and plot design|Plot design]] according to the conditions inside the strata.<br />
<br />
{{info<br />
|message=Example:<br />
|text=A forest area can be subdivided in different stands of different species or age classes. It is now possible to apply different plot designs (e.g. different sizes of sample plots) in the respective strata. While we perhaps need larger plots for the old stands, smaller plots might be sufficient in younger and more dense stands.<br />
}}<br />
<br />
==References==<br />
<references/><br />
<br />
{{SEO<br />
|keywords=stratified random sampling,strata,population,sampling technique,sub-population<br />
|descrip=Stratified sampling is a method to subdivide a population into separate and more homogeneous sub-populations called strata.<br />
}}<br />
<br />
[[Category:Sampling design]]<br />
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<br />
[[File:studienbeiträge.jpg|left|60px|link=AWF-Wiki:About]]<br />
Some early contributions to {{SITENAME}} were only possible because of [[AWF-Wiki:About|financial support from tuition fees]] of students allocated by the Georg-August-Universität Göttingen, Germany in order to enhance the quality of teaching.<br />
<br />
Did you know that we implemented an extension that allows you to compile articles in a collection, to structure and download them as PDF?<br />
You can find the function in the left toolbar ("Create a book"). Please feel free to create your own lecture notes! [[Help:books|read more]]<br />
|}<br />
<br />
<!--#######################New Page Box bottom left##############--><br />
{| <br />
|+align="middle" style="background:#7387AD; color:white"|<br />
'''Start a new page!'''<br />
| <br />
<!--<inputbox><br />
type=create<br />
width=35<br />
</inputbox>--><br />
<center><br />
Input a title for your new page and create an article! <br />
You can learn how to create articles [[Help:editing|here]]. Before you start please check whether the topic already exists by searching for keywords!<br />
</center><br />
|}<br />
</div><br />
<br />
<!--####################Box bottom right###################--><br />
<div style="float:right; width:48%; margin-bottom:1em;"><br />
{| <br />
|+align="middle" style="background:#5C739F; color:white"|<br />
'''How to participate'''<br />
|{{picture of the week}}<br />
To contribute to {{SITENAME}} (actively write or edit articles) you need to have an account. You don't need an account to read our content and everybody is invited to do so and make use of our resources! It is not possible for external users to create an account on their own. If you like to contribute to this project, please [[Help:Account|ask for an account]]!<br />
<br />
*[[Help:Account|Account]]<br />
*[[{{SITENAME}}:About|About {{SITENAME}}]]<br />
*[[{{SITENAME}}:Current events|Current events]]<br />
{{Statistic}}<br />
|}<br />
</div><br />
<!--###################Box Tagcloud#####################--><br />
{{tagcloud}}<!--Tagcloud is transcluded from template:tagcloud--><br />
</div><br />
__NOEDITSECTION__</div>Fehrmannhttp://wiki.awf.forst.uni-goettingen.de/wiki/index.php/AWF-Wiki:Community_PortalAWF-Wiki:Community Portal2020-11-09T15:25:31Z<p>Fehrmann: </p>
<hr />
<div>__NOTITLE__<br />
<div style="float:left; width:91%;max-width:1340px"><br />
{{wide image-noborder|AWF-Wiki portal banner4.jpg|1340px}}<br />
<!--####################Box top middle###################--><br />
<div style="margin-bottom:1em;"><br />
{|<br />
|+align="middle" style="background:#163776; color: white"|<br />
'''About {{SITENAME}}'''<br />
|<br />
[[image:Logo_GAUG.gif|300px|left|link=http://www.uni-goettingen.de/de/1.html]]<br />
<br />
{{SITENAME}} is a platform for sharing information, knowledge and expertise in the fields of forest mensuration, forest inventory and remote sensing. It was initiated by the [http://www.uni-goettingen.de/en/67094.html Chair of Forest Inventory and Remote Sensing] at the Georg-August-Universität Göttingen, Germany to extend academic offers to students and interested scientists. At the beginning it was planned to implement [[:category:Lecture notes|lecture notes]] in the mentioned scientific fields and to cultivate an appropriate structure of this Wiki for additional content in other categories.<br />
|}<br />
</div><br />
<br />
<!--####################Box bottom left###################--><br />
<div style="float:left; width:48%; margin-bottom:1em; "><br />
{| <br />
|+align="middle" style="background:#5C739F; color:white"|<br />
'''Categories in {{SITENAME}}'''<br />
|<br />
Content in {{SITENAME}} is organized in different main categories. You can browse to the different main topics in the given categories below, use the search function in the tool bar to search for special articles or keywords or look up the [[Forest Inventory Glossary|glossary]].<br />
<br />
*'''[[:Category:Forest inventory|Forest Inventory lecture notes]]'''<br />
*'''[[:Category:Forest mensuration|Forest Mensuration lecture notes]]'''<br />
*'''[[:Category:Remote sensing and image processing with open source software|Remote sensing and image processing with open source software]]'''<br />
*'''[[:Category:projects|Projects & Workshops]]'''<br />
<br />
===News:===<br />
* '''<span style="color:#ff0000"> NEW! Video Tutorials on YouTube:</span> we have launched a [https://www.youtube.com/channel/UCLbaMjH2JxfBogxvmmAzmVg AWF-YouTube channel] to share video tutorials as supplementary material to our lecture "Monitoring of Forest Resources"'''<br />
* '''For self-teaching purposes in times of the current Corona crisis, please make use of our [[:Category:Forest inventory|Forest Inventory lecture notes]] and other resources!'''<br />
* '''A [[Forest Inventory Glossary|glossary]] for forest inventory and statistical sampling.'''<br />
<br />
<br />
[[File:studienbeiträge.jpg|left|60px|link=AWF-Wiki:About]]<br />
Some contributions to {{SITENAME}} are only possible because of [[AWF-Wiki:About|financial support from tuition fees]] of students allocated by the Georg-August-Universität Göttingen, Germany in order to enhance the quality of teaching.<br />
<br />
Did you know that we implemented an extension that allows you to compile articles in a collection, to structure and download them as PDF?<br />
You can find the function in the left toolbar ("Create a book"). Please feel free to create your own lecture notes! [[Help:books|read more]]<br />
|}<br />
<br />
<!--#######################New Page Box bottom left##############--><br />
{| <br />
|+align="middle" style="background:#7387AD; color:white"|<br />
'''Start a new page!'''<br />
| <br />
<!--<inputbox><br />
type=create<br />
width=35<br />
</inputbox>--><br />
<center><br />
Input a title for your new page and create an article! <br />
You can learn how to create articles [[Help:editing|here]]. Before you start please check whether the topic already exists by searching for keywords!<br />
</center><br />
|}<br />
</div><br />
<br />
<!--####################Box bottom right###################--><br />
<div style="float:right; width:48%; margin-bottom:1em;"><br />
{| <br />
|+align="middle" style="background:#5C739F; color:white"|<br />
'''How to participate'''<br />
|{{picture of the week}}<br />
To contribute to {{SITENAME}} (actively write or edit articles) you need to have an account. You don't need an account to read our content and everybody is invited to do so and make use of our resources! It is not possible for external users to create an account on their own. If you like to contribute to this project, please [[Help:Account|ask for an account]]!<br />
<br />
*[[Help:Account|Account]]<br />
*[[{{SITENAME}}:About|About {{SITENAME}}]]<br />
*[[{{SITENAME}}:Current events|Current events]]<br />
{{Statistic}}<br />
|}<br />
</div><br />
<!--###################Box Tagcloud#####################--><br />
{{tagcloud}}<!--Tagcloud is transcluded from template:tagcloud--><br />
</div><br />
__NOEDITSECTION__</div>Fehrmannhttp://wiki.awf.forst.uni-goettingen.de/wiki/index.php/AWF-Wiki:Community_PortalAWF-Wiki:Community Portal2020-11-09T11:44:50Z<p>Fehrmann: </p>
<hr />
<div>__NOTITLE__<br />
<div style="float:left; width:91%;max-width:1340px"><br />
{{wide image-noborder|AWF-Wiki portal banner4.jpg|1340px}}<br />
<!--####################Box top middle###################--><br />
<div style="margin-bottom:1em;"><br />
{|<br />
|+align="middle" style="background:#163776; color: white"|<br />
'''About {{SITENAME}}'''<br />
|<br />
[[image:Logo_GAUG.gif|300px|left|link=http://www.uni-goettingen.de/de/1.html]]<br />
<br />
{{SITENAME}} is a platform for sharing information, knowledge and expertise in the fields of forest mensuration, forest inventory and remote sensing. It was initiated by the [http://www.uni-goettingen.de/en/67094.html Chair of Forest Inventory and Remote Sensing] at the Georg-August-Universität Göttingen, Germany to extend academic offers to students and interested scientists. At the beginning it was planned to implement [[:category:Lecture notes|lecture notes]] in the mentioned scientific fields and to cultivate an appropriate structure of this Wiki for additional content in other categories.<br />
|}<br />
</div><br />
<br />
<!--####################Box bottom left###################--><br />
<div style="float:left; width:48%; margin-bottom:1em; "><br />
{| <br />
|+align="middle" style="background:#5C739F; color:white"|<br />
'''Categories in {{SITENAME}}'''<br />
|<br />
Content in {{SITENAME}} is organized in different main categories. You can browse to the different main topics in the given categories below, use the search function in the tool bar to search for special articles or keywords or look up the [[Forest Inventory Glossary|glossary]].<br />
<br />
*'''[[:Category:Forest inventory|Forest Inventory lecture notes]]'''<br />
*'''[[:Category:Forest mensuration|Forest Mensuration lecture notes]]'''<br />
*'''[[:Category:Remote sensing and image processing with open source software|Remote sensing and image processing with open source software]]'''<br />
*'''[[:Category:projects|Projects & Workshops]]'''<br />
<br />
===News:===<br />
* '''Video Tutorials on YouTube: we have launched a [https://www.youtube.com/channel/UCLbaMjH2JxfBogxvmmAzmVg AWF-YouTube channel] to share video tutorials as supplementary material to our lecture "Monitoring of Forest Resources"'''<br />
* '''For self-teaching purposes in times of the current Corona crisis, please make use of our [[:Category:Forest inventory|Forest Inventory lecture notes]] and other resources!'''<br />
* '''A [[Forest Inventory Glossary|glossary]] for forest inventory and statistical sampling.'''<br />
<br />
<br />
[[File:studienbeiträge.jpg|left|60px|link=AWF-Wiki:About]]<br />
Some contributions to {{SITENAME}} are only possible because of [[AWF-Wiki:About|financial support from tuition fees]] of students allocated by the Georg-August-Universität Göttingen, Germany in order to enhance the quality of teaching.<br />
<br />
Did you know that we implemented an extension that allows you to compile articles in a collection, to structure and download them as PDF?<br />
You can find the function in the left toolbar ("Create a book"). Please feel free to create your own lecture notes! [[Help:books|read more]]<br />
|}<br />
<br />
<!--#######################New Page Box bottom left##############--><br />
{| <br />
|+align="middle" style="background:#7387AD; color:white"|<br />
'''Start a new page!'''<br />
| <br />
<!--<inputbox><br />
type=create<br />
width=35<br />
</inputbox>--><br />
<center><br />
Input a title for your new page and create an article! <br />
You can learn how to create articles [[Help:editing|here]]. Before you start please check whether the topic already exists by searching for keywords!<br />
</center><br />
|}<br />
</div><br />
<br />
<!--####################Box bottom right###################--><br />
<div style="float:right; width:48%; margin-bottom:1em;"><br />
{| <br />
|+align="middle" style="background:#5C739F; color:white"|<br />
'''How to participate'''<br />
|{{picture of the week}}<br />
To contribute to {{SITENAME}} (actively write or edit articles) you need to have an account. You don't need an account to read our content and everybody is invited to do so and make use of our resources! It is not possible for external users to create an account on their own. If you like to contribute to this project, please [[Help:Account|ask for an account]]!<br />
<br />
*[[Help:Account|Account]]<br />
*[[{{SITENAME}}:About|About {{SITENAME}}]]<br />
*[[{{SITENAME}}:Current events|Current events]]<br />
{{Statistic}}<br />
|}<br />
</div><br />
<!--###################Box Tagcloud#####################--><br />
{{tagcloud}}<!--Tagcloud is transcluded from template:tagcloud--><br />
</div><br />
__NOEDITSECTION__</div>Fehrmannhttp://wiki.awf.forst.uni-goettingen.de/wiki/index.php/AWF-Wiki:Community_PortalAWF-Wiki:Community Portal2020-11-09T11:09:13Z<p>Fehrmann: </p>
<hr />
<div>__NOTITLE__<br />
<div style="float:left; width:91%;max-width:1340px"><br />
{{wide image-noborder|AWF-Wiki portal banner4.jpg|1340px}}<br />
<!--####################Box top middle###################--><br />
<div style="margin-bottom:1em;"><br />
{|<br />
|+align="middle" style="background:#163776; color: white"|<br />
'''About {{SITENAME}}'''<br />
|<br />
[[image:Logo_GAUG.gif|300px|left|link=http://www.uni-goettingen.de/de/1.html]]<br />
<br />
{{SITENAME}} is a platform for sharing information, knowledge and expertise in the fields of forest mensuration, forest inventory and remote sensing. It was initiated by the [http://www.uni-goettingen.de/en/67094.html Chair of Forest Inventory and Remote Sensing] at the Georg-August-Universität Göttingen, Germany to extend academic offers to students and interested scientists. At the beginning it was planned to implement [[:category:Lecture notes|lecture notes]] in the mentioned scientific fields and to cultivate an appropriate structure of this Wiki for additional content in other categories.<br />
|}<br />
</div><br />
<br />
<!--####################Box bottom left###################--><br />
<div style="float:left; width:48%; margin-bottom:1em; "><br />
{| <br />
|+align="middle" style="background:#5C739F; color:white"|<br />
'''Categories in {{SITENAME}}'''<br />
|<br />
Content in {{SITENAME}} is organized in different main categories. You can browse to the different main topics in the given categories below, use the search function in the tool bar to search for special articles or keywords or look up the [[Forest Inventory Glossary|glossary]].<br />
<br />
*'''[[:Category:Forest inventory|Forest Inventory lecture notes]]'''<br />
*'''[[:Category:Forest mensuration|Forest Mensuration lecture notes]]'''<br />
*'''[[:Category:Remote sensing and image processing with open source software|Remote sensing and image processing with open source software]]'''<br />
*'''[[:Category:projects|Projects & Workshops]]'''<br />
<br />
===News:===<br />
* '''Video Tutorials on YouTube: we have launched a [https://www.youtube.com/channel/UCLbaMjH2JxfBogxvmmAzmVg AWF-YouTube channel] to share video tutorials as supplementary materal to our lecture "Monitoring of Forest Resources"'''<br />
* '''For self-teaching purposes in times of the current Corona crisis, please make use of our [[:Category:Forest inventory|Forest Inventory lecture notes]] and other resources!'''<br />
* '''A [[Forest Inventory Glossary|glossary]] for forest inventory and statistical sampling.'''<br />
<br />
<br />
[[File:studienbeiträge.jpg|left|60px|link=AWF-Wiki:About]]<br />
Some contributions to {{SITENAME}} are only possible because of [[AWF-Wiki:About|financial support from tuition fees]] of students allocated by the Georg-August-Universität Göttingen, Germany in order to enhance the quality of teaching.<br />
<br />
Did you know that we implemented an extension that allows you to compile articles in a collection, to structure and download them as PDF?<br />
You can find the function in the left toolbar ("Create a book"). Please feel free to create your own lecture notes! [[Help:books|read more]]<br />
|}<br />
<br />
<!--#######################New Page Box bottom left##############--><br />
{| <br />
|+align="middle" style="background:#7387AD; color:white"|<br />
'''Start a new page!'''<br />
| <br />
<!--<inputbox><br />
type=create<br />
width=35<br />
</inputbox>--><br />
<center><br />
Input a title for your new page and create an article! <br />
You can learn how to create articles [[Help:editing|here]]. Before you start please check whether the topic already exists by searching for keywords!<br />
</center><br />
|}<br />
</div><br />
<br />
<!--####################Box bottom right###################--><br />
<div style="float:right; width:48%; margin-bottom:1em;"><br />
{| <br />
|+align="middle" style="background:#5C739F; color:white"|<br />
'''How to participate'''<br />
|{{picture of the week}}<br />
To contribute to {{SITENAME}} (actively write or edit articles) you need to have an account. You don't need an account to read our content and everybody is invited to do so and make use of our resources! It is not possible for external users to create an account on their own. If you like to contribute to this project, please [[Help:Account|ask for an account]]!<br />
<br />
*[[Help:Account|Account]]<br />
*[[{{SITENAME}}:About|About {{SITENAME}}]]<br />
*[[{{SITENAME}}:Current events|Current events]]<br />
{{Statistic}}<br />
|}<br />
</div><br />
<!--###################Box Tagcloud#####################--><br />
{{tagcloud}}<!--Tagcloud is transcluded from template:tagcloud--><br />
</div><br />
__NOEDITSECTION__</div>Fehrmannhttp://wiki.awf.forst.uni-goettingen.de/wiki/index.php/AWF-Wiki:Community_PortalAWF-Wiki:Community Portal2020-11-04T11:13:18Z<p>Fehrmann: </p>
<hr />
<div>__NOTITLE__<br />
<div style="float:left; width:91%;max-width:1340px"><br />
{{wide image-noborder|AWF-Wiki portal banner4.jpg|1340px}}<br />
<!--####################Box top middle###################--><br />
<div style="margin-bottom:1em;"><br />
{|<br />
|+align="middle" style="background:#163776; color: white"|<br />
'''About {{SITENAME}}'''<br />
|<br />
[[image:Logo_GAUG.gif|300px|left|link=http://www.uni-goettingen.de/de/1.html]]<br />
<br />
{{SITENAME}} is a platform for sharing information, knowledge and expertise in the fields of forest mensuration, forest inventory and remote sensing. It was initiated by the [http://www.uni-goettingen.de/en/67094.html Chair of Forest Inventory and Remote Sensing] at the Georg-August-Universität Göttingen, Germany to extend academic offers to students and interested scientists. At the beginning it was planned to implement [[:category:Lecture notes|lecture notes]] in the mentioned scientific fields and to cultivate an appropriate structure of this Wiki for additional content in other categories.<br />
|}<br />
</div><br />
<br />
<!--####################Box bottom left###################--><br />
<div style="float:left; width:48%; margin-bottom:1em; "><br />
{| <br />
|+align="middle" style="background:#5C739F; color:white"|<br />
'''Categories in {{SITENAME}}'''<br />
|<br />
Content in {{SITENAME}} is organized in different main categories. You can browse to the different main topics in the given categories below, use the search function in the tool bar to search for special articles or keywords or look up the [[Forest Inventory Glossary|glossary]].<br />
<br />
*'''[[:Category:Forest inventory|Forest Inventory lecture notes]]'''<br />
*'''[[:Category:Forest mensuration|Forest Mensuration lecture notes]]'''<br />
*'''[[:Category:Remote sensing and image processing with open source software|Remote sensing and image processing with open source software]]'''<br />
*'''[[:Category:projects|Projects & Workshops]]'''<br />
<br />
===News:===<br />
* '''For self-teaching purposes in times of the current Corona crisis, please make use of our [[:Category:Forest inventory|Forest Inventory lecture notes]] and other resources!'''<br />
* '''A [[Forest Inventory Glossary|glossary]] for forest inventory and statistical sampling.'''<br />
<br />
<br />
[[File:studienbeiträge.jpg|left|60px|link=AWF-Wiki:About]]<br />
Some contributions to {{SITENAME}} are only possible because of [[AWF-Wiki:About|financial support from tuition fees]] of students allocated by the Georg-August-Universität Göttingen, Germany in order to enhance the quality of teaching.<br />
<br />
Did you know that we implemented an extension that allows you to compile articles in a collection, to structure and download them as PDF?<br />
You can find the function in the left toolbar ("Create a book"). Please feel free to create your own lecture notes! [[Help:books|read more]]<br />
|}<br />
<br />
<!--#######################New Page Box bottom left##############--><br />
{| <br />
|+align="middle" style="background:#7387AD; color:white"|<br />
'''Start a new page!'''<br />
| <br />
<!--<inputbox><br />
type=create<br />
width=35<br />
</inputbox>--><br />
<center><br />
Input a title for your new page and create an article! <br />
You can learn how to create articles [[Help:editing|here]]. Before you start please check whether the topic already exists by searching for keywords!<br />
</center><br />
|}<br />
</div><br />
<br />
<!--####################Box bottom right###################--><br />
<div style="float:right; width:48%; margin-bottom:1em;"><br />
{| <br />
|+align="middle" style="background:#5C739F; color:white"|<br />
'''How to participate'''<br />
|{{picture of the week}}<br />
To contribute to {{SITENAME}} (actively write or edit articles) you need to have an account. You don't need an account to read our content and everybody is invited to do so and make use of our resources! It is not possible for external users to create an account on their own. If you like to contribute to this project, please [[Help:Account|ask for an account]]!<br />
<br />
*[[Help:Account|Account]]<br />
*[[{{SITENAME}}:About|About {{SITENAME}}]]<br />
*[[{{SITENAME}}:Current events|Current events]]<br />
{{Statistic}}<br />
|}<br />
</div><br />
<!--###################Box Tagcloud#####################--><br />
{{tagcloud}}<!--Tagcloud is transcluded from template:tagcloud--><br />
</div><br />
__NOEDITSECTION__</div>Fehrmannhttp://wiki.awf.forst.uni-goettingen.de/wiki/index.php/Template:Picture_of_the_weekTemplate:Picture of the week2020-04-29T08:08:09Z<p>Fehrmann: </p>
<hr />
<div><noinclude>This template is the "picture of the week"-box for the portal page</noinclude><br />
<br />
[[File:Valdivia.jpg|right|thumb|none|200px]]</div>Fehrmannhttp://wiki.awf.forst.uni-goettingen.de/wiki/index.php/Template:Picture_of_the_weekTemplate:Picture of the week2020-04-29T07:18:54Z<p>Fehrmann: </p>
<hr />
<div><noinclude>This template is the "picture of the week"-box for the portal page</noinclude><br />
<br />
[[File:Valdivia.jpg|right|thumb|none|230px]]</div>Fehrmannhttp://wiki.awf.forst.uni-goettingen.de/wiki/index.php/AWF-Wiki:Community_PortalAWF-Wiki:Community Portal2020-04-29T07:18:04Z<p>Fehrmann: </p>
<hr />
<div>__NOTITLE__<br />
<div style="float:left; width:91%;max-width:1340px"><br />
{{wide image-noborder|AWF-Wiki portal banner4.jpg|1340px}}<br />
<!--####################Box top middle###################--><br />
<div style="margin-bottom:1em;"><br />
{|<br />
|+align="middle" style="background:#163776; color: white"|<br />
'''About {{SITENAME}}'''<br />
|<br />
[[image:Logo_GAUG.gif|300px|left|link=http://www.uni-goettingen.de/de/1.html]]<br />
<br />
{{SITENAME}} is a platform for sharing information, knowledge and expertise in the fields of forest mensuration, forest inventory and remote sensing. It was initiated by the [http://www.uni-goettingen.de/en/67094.html Chair of Forest Inventory and Remote Sensing] at the Georg-August-Universität Göttingen, Germany to extend academic offers to students and interested scientists. At the beginning it was planned to implement [[:category:Lecture notes|lecture notes]] in the mentioned scientific fields and to cultivate an appropriate structure of this Wiki for additional content in other categories.<br />
|}<br />
</div><br />
<br />
<!--####################Box bottom left###################--><br />
<div style="float:left; width:48%; margin-bottom:1em; "><br />
{| <br />
|+align="middle" style="background:#5C739F; color:white"|<br />
'''Categories in {{SITENAME}}'''<br />
|<br />
Content in {{SITENAME}} is organized in different main categories. You can browse to the different main topics in the given categories below, use the search function in the tool bar to search for special articles or keywords or look up the [[Forest Inventory Glossary|glossary]].<br />
<br />
*'''[[:Category:Forest inventory|Forest Inventory lecture notes]]'''<br />
*'''[[:Category:Forest mensuration|Forest Mensuration lecture notes]]'''<br />
*'''[[:Category:Remote sensing and image processing with open source software|Remote sensing and image processing with open source software]]'''<br />
*'''New: [[:Category:projects|Projects & Workshops]]'''<br />
<br />
===News:===<br />
* '''For self-teaching purposes in times of the current Corona crisis, please make use of our [[:Category:Forest inventory|Forest Inventory lecture notes]] and other resources!'''<br />
* '''A [[Forest Inventory Glossary|glossary]] for forest inventory and statistical sampling.'''<br />
<br />
<br />
[[File:studienbeiträge.jpg|left|60px|link=AWF-Wiki:About]]<br />
Some contributions to {{SITENAME}} are only possible because of [[AWF-Wiki:About|financial support from tuition fees]] of students allocated by the Georg-August-Universität Göttingen, Germany in order to enhance the quality of teaching.<br />
<br />
Did you know that we implemented an extension that allows you to compile articles in a collection, to structure and download them as PDF?<br />
You can find the function in the left toolbar ("Create a book"). Please feel free to create your own lecture notes! [[Help:books|read more]]<br />
|}<br />
<br />
<!--#######################New Page Box bottom left##############--><br />
{| <br />
|+align="middle" style="background:#7387AD; color:white"|<br />
'''Start a new page!'''<br />
| <br />
<!--<inputbox><br />
type=create<br />
width=35<br />
</inputbox>--><br />
<center><br />
Input a title for your new page and create an article! <br />
You can learn how to create articles [[Help:editing|here]]. Before you start please check whether the topic already exists by searching for keywords!<br />
</center><br />
|}<br />
</div><br />
<br />
<!--####################Box bottom right###################--><br />
<div style="float:right; width:48%; margin-bottom:1em;"><br />
{| <br />
|+align="middle" style="background:#5C739F; color:white"|<br />
'''How to participate'''<br />
|{{picture of the week}}<br />
To contribute to {{SITENAME}} (actively write or edit articles) you need to have an account. You don't need an account to read our content and everybody is invited to do so and make use of our resources! It is not possible for external users to create an account on their own. If you like to contribute to this project, please [[Help:Account|ask for an account]]!<br />
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*[[Help:Account|Account]]<br />
*[[{{SITENAME}}:About|About {{SITENAME}}]]<br />
*[[{{SITENAME}}:Current events|Current events]]<br />
{{Statistic}}<br />
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__NOEDITSECTION__</div>Fehrmannhttp://wiki.awf.forst.uni-goettingen.de/wiki/index.php/AWF-Wiki:Community_PortalAWF-Wiki:Community Portal2020-04-29T07:17:37Z<p>Fehrmann: </p>
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'''About {{SITENAME}}'''<br />
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[[image:Logo_GAUG.gif|300px|left|link=http://www.uni-goettingen.de/de/1.html]]<br />
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{{SITENAME}} is a platform for sharing information, knowledge and expertise in the fields of forest inventory and remote sensing. It was initiated by the [http://www.uni-goettingen.de/en/67094.html Chair of Forest Inventory and Remote Sensing] at the Georg-August-Universität Göttingen, Germany to extend academic offers to students and interested scientists. At the beginning it was planned to implement [[:category:Lecture notes|lecture notes]] in the mentioned scientific fields and to cultivate an appropriate structure of this Wiki for additional content in other categories.<br />
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Content in {{SITENAME}} is organized in different main categories. You can browse to the different main topics in the given categories below, use the search function in the tool bar to search for special articles or keywords or look up the [[Forest Inventory Glossary|glossary]].<br />
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*'''[[:Category:Forest inventory|Forest Inventory lecture notes]]'''<br />
*'''[[:Category:Forest mensuration|Forest Mensuration lecture notes]]'''<br />
*'''[[:Category:Remote sensing and image processing with open source software|Remote sensing and image processing with open source software]]'''<br />
*'''New: [[:Category:projects|Projects & Workshops]]'''<br />
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===News:===<br />
* '''For self-teaching purposes in times of the current Corona crisis, please make use of our [[:Category:Forest inventory|Forest Inventory lecture notes]] and other resources!'''<br />
* '''A [[Forest Inventory Glossary|glossary]] for forest inventory and statistical sampling.'''<br />
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[[File:studienbeiträge.jpg|left|60px|link=AWF-Wiki:About]]<br />
Some contributions to {{SITENAME}} are only possible because of [[AWF-Wiki:About|financial support from tuition fees]] of students allocated by the Georg-August-Universität Göttingen, Germany in order to enhance the quality of teaching.<br />
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Did you know that we implemented an extension that allows you to compile articles in a collection, to structure and download them as PDF?<br />
You can find the function in the left toolbar ("Create a book"). Please feel free to create your own lecture notes! [[Help:books|read more]]<br />
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Input a title for your new page and create an article! <br />
You can learn how to create articles [[Help:editing|here]]. Before you start please check whether the topic already exists by searching for keywords!<br />
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{| <br />
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'''How to participate'''<br />
|{{picture of the week}}<br />
To contribute to {{SITENAME}} (actively write or edit articles) you need to have an account. You don't need an account to read our content and everybody is invited to do so and make use of our resources! It is not possible for external users to create an account on their own. If you like to contribute to this project, please [[Help:Account|ask for an account]]!<br />
<br />
*[[Help:Account|Account]]<br />
*[[{{SITENAME}}:About|About {{SITENAME}}]]<br />
*[[{{SITENAME}}:Current events|Current events]]<br />
{{Statistic}}<br />
|}<br />
</div><br />
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__NOEDITSECTION__</div>Fehrmann