Height curve

Forest mensuration lecture notes

[+] Definitions in forest inventory

[+] Functions and models in forest inventory

[+] Measurement devices

[+] Single tree variables

[+] Stand variables

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General observations

Figure 1 Height curves in even aged stands exhibit typical changes over time which are depicted here simplified and schematically: they get flatter, shift right on the dbh axis and up on the height axis and cover a wider range of diameters (“become longer”).

Figure 2 Simple linear regression with ln(dbh) as sole independent variable. In addition to the data points, the mean heights per 10cm dbh-class are given. In this case, the model is obviously not flexible enough to adjust well to the height values for very large diameters.

The height curve describes the relationship between height and dbh, where it needs to be defined first which height is meant; standard height curves refer to total tree height. While regression is the standard to determine a height curve, there are other techniques which are less statistically rigorous. A visual (or manual) fit can be done when the dbh/height points are drawn in a grid system. From that drawn curve, height values can directly be read without the requirement to define a mathematical model and to calculate its coefficients. For a so-called half-graphical fitting, in a first step, mean height values are calculated for diameter classes (the width of these classes needs to be determined); the manual fit is then done for the set of mean values and not for all measurement points; by that, the fit can graphically easier be done (which is illustrated in Figure 2).

Height curves are nowadays usually given as mathematical functions that can be integrated into analysis software. In former times, they were given in tabular form so that for each diameter (or diameter class) the height could be read directly.

The shape of the height curve as given in Figure 2 is typical for even-aged stands. The curve is steeper in the range of smaller diameters and flattens for larger diameters. It may even be that the curve goes down a little bit for very large trees; this can be the case in regions with site differences where the very large trees on poorer sites are less tall then slightly smaller trees on better sites.

It must be clearly understood that the height curve describes the status of the relationship between dbh and height for a given point in time. It is not a growth curve. It can, therefore, not be used to predict the height of a particular tree for later points in time! How height curves in even-aged stands do commonly change over time is depicted in Figure 1.

Modeling height curves

Figure 3 The same height curve as in Figure 2 but drawn in a grid with ln(dbh) on the abscissa instead of dbh only.

There are many different models that have been used to fit height curves, some of them are given in Table 1. The simple quadratic model (model (1) in Table 1) is well suited in most cases. Model (2) in Table 1 is interesting because it is a simple linear model with only one independent variable, the logarithm of dbh; this model produces a straight line when drawn in a half-logarithmic grid but produces a curve well suited for height curves when drawn in normal linear grids (compare Figure 2 and Figure 3).

Examples of models of height curves

Linear model	\(h=b_o+b_1dbh+b_2dbh^2\)
Transormed linear model	h=b_0+b_1ln(dbh)
Nonlinear model	h=b_1*(1-e^{-b_2dbh)