View source for Estimating forest area

==Area estimation by lines or clusters of points==

[[File:6.2.3-fig109.png|right|thumb|300px|'''Figure 2''' Two approaches of area estimation when line sampling is used: (1) either the entire line is used for line intercept sampling, or (2) only the end points are observed, so that the observation unit is actually a cluster of two points at a defined distance away from each other (Kleinn 1991<ref    name="kleinn1991">Kleinn C. 1991. Der Fehler von Flächenschätzungen   mit Punkterastern und linienförmigen  Stichprobenelementen. Dissertation. Mitteilungen der Abteilung für  Forstliche Biometrie 91‑1, Universität Freiburg, 128 S.</ref>).]]
[[File:6.2.3-fig110.png|right|thumb|300px|'''Figure 3''' Line sampling with different sampling elements: straight line, L-shape and square in an area that consists to 50% of forest. Total line length is fixed in the left figure and adjusted to the buffer strip (that is: the maximum extension of the line is equal to the width of the buffer) in the right figure (Kleinn 1991<ref name="kleinn1991">Kleinn C. 1991. Der Fehler von Flächenschätzungen mit Punkterastern und linienförmigen  Stichprobenelementen.  Dissertation. Mitteilungen der Abteilung für  Forstliche Biometrie 91‑1,  Universität Freiburg, 128 S.</ref>).]]
Area estimation can be done by [[Line sampling#Line intercept sampling|line intercept sampling]] where sample lines are used as observation units and on each sample line it is observed which proportion comes to. 

In Figure 2 two basic approaches are described for area estimation from such sample lines:

#standard line intercept sampling: in this case, the observation that is made on the line can take on all values between <math>y=0.0</math> and <math>y=1.0</math>;
#observing only the end points of the line, so that clusters of two points are formed that are a defined distance away from each other: in this case, the observation that is made on one cluster can take on only three values <math>y=0.0,</math> <math>y=0.5</math> and <math>y=1.0</math>.

Obviously, line intercept sampling takes more time to implement, because the measurement of the proportion of the line inside forest is time consuming. This is much faster when only the end points are observed. The effort is the same for all observation units completely outside or completely inside the forest (regardless whether it is line intercept sampling or sampling with clusters of points.  

The comparison of [[Accuracy and precision|precision]] of estimation is now of interest. In order to research into that, we may imagine simple situation as depicted in Figure 3 (from Kleinn 1991): the area consists of 50% forest and 50% non-forest, a straight line separates the two halves and we now wish to estimate the forest area with a sample line. In Figure 3, there are two other line sampling units depicted (a square and a half-square) which are not further discussed here. Because of the symmetry of this „map” it is sufficient to look at all positions of the sample line along a line that is perpendicular to the forest edge and extends through the buffer (between the dashed lines in Figure 3) where the buffer is defined such that a line has an intersection probability larger than 0 if its center point falls into it. All lines outside this buffer are either completely outside the forest <math>(y=0)</math> or completely inside the forest <math>(y=1)</math>. '''Observe:''' the buffer as drawn Figure 3 is for the sample '''line'''; the buffer would be smaller for the L-shape and the square line sampling unit which are also depicted.

With geometric probabilities the true parametric mean and error variance of the area estimation can be determined. The mean, obviously, is always <math>p=0.5</math>. The error variances are given in Table 1, for both the estimation from line intercept sampling (observing line sections) and from observing the cluster of end-points. For the line, this cluster consists of two points, for the L-shape (the half square) of three points and for the square of four points.


<blockquote>
{| 
|'''Table 1.''' Result of precision comparison of 2 area estimation approaches using sample lines. The error variances<br>refer to the population of all observation units within the buffer strip between the dashed lines in Figure 3 (Kleinn 1991<ref name="kleinn1991">Kleinn C. 1991. Der Fehler von  Flächenschätzungen mit Punkterastern und linienförmigen   Stichprobenelementen.  Dissertation. Mitteilungen der Abteilung für Forstliche Biometrie 91‑1, Universität Freiburg, 128 S.</ref>).<br>Variance would be <math>\frac{1}{4}(\mapsto\sigma^2=p*q)</math> for single independently selected sample points.
{| class="wikitable"
! Observation unit
! Line intercept sampling
! Clusters of points
|- 
|Line<br>(2 points)
|align="center"|<math>\frac{1}{4}\approx0.14393\,</math>
|align="center"|<math>\frac{1}{4}-\frac{1}{2\pi}\approx0.09085\,</math>
|-
|L-shape = half square<br>(3 points)
|align="center"|<math>\frac{1}{4}\frac{(1+\sqrt{2})}{6\pi\sqrt{2}}\approx0.12693\,</math>
|align="center"|<math>\frac{1}{4}-(1+\sqrt{2})\frac{2}{9\pi}\approx0.07923\,</math>
|-
|Square<br>(4 points)
|align="center"|<math>\frac{1}{4}-\frac{3+\sqrt{2}5\ln(1+\sqrt{2})}{12\pi\sqrt{2}}\approx0.08455\,</math>
|align="center"|<math>\frac{1}{4}-(1+\sqrt{2})\frac{1}{4\pi}\approx0.05788\,</math>
|}
|}
</blockquote>


It may be surprising, but the estimation from clusters of points is more precise: for the line shape, the line intercept sampling approach produces an error variance of

:<math>\sigma_{\hat{p}}^2=\frac{1}{4}-\frac{1}{3\pi}=0.14393\,</math>

while the one for the clusters of two points is

:<math>\sigma_{\hat{p}}^2=\frac{1}{4}-\frac{1}{2\pi}=0.09085\,</math>

It is more precise and less costly to do the area estimation from observations of points only rather then observing along the entire line. This holds for all three compared observation units: the simple sample line, the L-shape or half square and the square.

However, we must not generalize this observation, but need to restrict it to cases for which the simple model from Figure 3 applies, and this is cases in which one observation unit does not intersect with more than one forest boundary; that means: the observation units (the line sample elements used) are relatively short in comparison to the forest patches in the area of interest. In many large area [[forest inventory|forest inventories]], this condition is fulfilled; for example, in the German NFI, square clusters of 150 m side length are used and the connecting line can be used as sample line. While in the first German NFI, line intercept sampling was applied to estimate the area proportion for the different forest types. However, it was found that the estimation via observation of the cluster corners only does yield equally precise area estimations so that line intercept sampling was abandoned for the second German NFI where only a cluster of four points was analyzed for area estimation.