Tree height estimation using a stochastic height-diameter relationship
M. Barrio1, U. Diéguez-Aranda1, F. Castedo2, J.G. Álvarez-González1 and A. Rojo1
Unidade de Xestión Forestal Sostible
Web: http://www.lugo.usc.es/uxfs/
1
Departamento de Ingeniería Agroforestal, Universidad de Santiago de Compostela. Escuela
Politécnica Superior, Campus universitario, 27002 Lugo, Spain.
2
Departamento de Ingeniería Agraria, Universidad de León. Escuela Superior y Técnica de
Ingeniería Agraria, Campus de Ponferrada, 24400 Ponferrada, Spain.
Abstract
Measuring the height of a tree takes longer than measuring its diameter at breast height and
often only the heights of a subset of trees of known diameter are measured in forest
inventories. Since trees with the same diameter are not usually of the same height, even within
the same stand, a stochastic component was added to the deterministic height function. This
approach mimics the natural variability of heights and therefore provides more realistic height
predictions, as demonstrated by the results of the Kolmogorov Smirnov test and by visual
analysis of box plot graphs.
Introduction
Individual tree heights and diameters are essential measurements in forest inventories, and are
used for estimating timber volume, site index and other important variables related to forest
growth and yield, succession and carbon budget models (Peng, 2001). The time taken to
measure tree heights takes longer than measuring the diameter at breast height. For this
reason, often only the heights of a subset of trees of known diameter are measured, and
accurate height-diameter equations must be used to predict the heights of the remaining trees
to reduce the costs involved in data acquisition. If stand conditions vary greatly within a forest, a
height regression may be derived separately for each stand, or a generalized function, which
includes stand variables to account for the variability, may be developed (Curtis, 1967; Zhang et
al., 1997; Sharma & Zhang, 2004).
Two trees within the same stand and that have the same diameter are not necessarily of the
same height; therefore a deterministic model does not seem appropriate for mimicking the real
natural variability in height (Parresol & Lloyd, 2004).
The objective of the present study was to evaluate the performance of a stochastic
height-diameter approach in mimicking the observed natural variability in tree height.
Material and methods
We used data from two thinning trials installed in Pinus pinaster and Pinus sylvestris
even-aged stands located in northwestern Spain. The stand conditions within each thinning trial
were similar and thus we considered the data obtained as belonging only to two different
stands.
Table 1. Characteristics of the tree samples used for model fitting.
Thinning trial
P. pinaster
P. sylvestris
Number of trees
1062
488
Mean
11.2
22.4
d (cm)
Min.
Max.
2.9
19.8
7.0
34.8
SD
3.3
7.2
Mean
9.2
19.6
h (m)
Min.
Max.
4.8
12.4
9.6
25.4
SD
1.2
2.8
Diameter at breast height (d) was measured to the nearest 0.1 cm in all the trees. Total tree
height (h) was measured to the nearest 0.1 m in a sample of one third of the trees in each of the
12 plots belonging to each thinning trial. Summary statistics, including the mean, minimum,
maximum and standard deviation (SD) of the above-mentioned variables are shown in Table 1.
A large number of local height-diameter equations have been reported in the forestry literature
(e.g. Curtis, 1967; García, 1974; Wykoff et al., 1982; Huang et al., 1992, 2000; Fang & Bailey,
1998; Peng, 1999; Soares & Tomé, 2002), most of which exhibit S-shaped or concave-shaped
patterns. From among all the models we selected the Schnute growth model (Schnute, 1981)
for the purposes of our study. This function is easy to fit and quick to achieve convergence for
any database (Bredenkamp & Gregoire, 1988; Lei & Parresol, 2001), even with small data sets
(Castedo et al., 2005). The expression of Schnute’s function is as follows:
(
)
1b
⎛
1 − e −ad ⎞
b
⎟ +ε
(1)
h = ⎜⎜1.3 b + h2 − 1.3 b
1 − e −ad 2 ⎟⎠
⎝
where: h is total height of the tree; d is tree diameter at breast height; d2 is diameter at breast
height of a large tree (upper range of data); h2 is parameter representing mean tree height at d2;
a and b are the model parameters to be estimated and ε the residual error.
Since the scatter plots of total tree height against diameter did not show an unequal error
variance for any of the two data sets, parameters of equation (1) were estimated using nonlinear least squares, using the NLIN procedure of SAS/STAT® (SAS Institute Inc., 2004).
The goodness-of-fit of the model was evaluated by means of the coefficient of determination for
non-linear regression (R2) (see Ryan, 1997) and the root mean squared error (RMSE).
The stochastic approach uses this standard error in a similar way as the prediction interval is
obtained for a new individual in a regression model. Instead of using the t value corresponding
to a fixed limit for all the trees, it is substituted by a value randomly generated from the inverse
of the normal distribution function for each individual. Thus, the expression used to assign
stochastically the height to each tree in a sample is:
yˆ i (stoch ) = yˆ i + FU−1s yˆ i (new )
(2)
where ŷ i (stoch ) is the stochastic height estimation, ŷ i is the deterministic height obtained from
equation (1) (the fitted model without considering the stochastic component), FU−1 is the inverse
of the standard normal distribution function for U – a uniform random variable in the interval
(0,1) – and syˆ i (new ) is the standard error of prediction of a new observation and is expressed as:
s yˆ i ( new ) = var ( yˆ i ( new ) ) = RMSE 2 + z (b )' i S 2 (b ) z (b )i
(3)
2
where the scalar RMSE is the regression mean squared error, z (b )i is the partial derivatives
of the least squared estimator b of the parameter vector β .
The performance of the stochastic approach proposed was evaluated for diameter classes of 5
cm width and for each thinning trial. A comparison between the observed height frequencies
and the variability of the deterministic and stochastic approaches was carried out. The
Kolmogorov-Smirnov test was used to compare the observed and the estimated height
frequencies of the two approaches. Box plot graphs were used to visually contrast the means,
medians, and 25/75 percentiles of the height distributions.
Results and discussion
The Schnute function provided a reasonable fit, taking into account the high variability in the
height-diameter relationship in the databases (see Table 1). All the parameters were significant
at the P = 95% level and showed reasonable values. The low variability explained by the local
model indicates that stand variables must be taken into account to improve this relationship.
Deterministic values of the height ŷ i were obtained using equation (1) with the parameter
estimates obtained for each data set (Table 2). The prediction of the stochastic values ŷ i (stoch )
involved the following steps: a) calculation of the standard error of the prediction syˆ i (new ) by
means of equation (3); b) calculation of the inverse of the standard normal distribution function
for one pseudo-random number in the interval (0,1) and c) estimation of the ŷ i (stoch ) from
equation (2) by substituting the values obtained in the previous steps.
Comparison of the observed, deterministic and one random stochastic height distribution per
diameter class showed substantial differences between the two approaches, even though the
mean height values per diameter class obtained by each approach were very similar (Figure 1,
Table 3).
Table 2.- Parameter estimates (standard error in brackets) and goodness-of-fit statistics for the
Schnute (1981) model and the two data sets under analysis.
Thinning trial
P. pinaster
P. sylvestris
Parameter estimates
h2
a
b
0.0896
1.9051
10.9793
(0.0246)
(0.3197)
(0.1536)
0.1069
0.9516
21.9859
(0.0176)
(0.2575)
(0.2805)
R2
RMSE
0.5091
0.828
0.5188
1.918
The observed height distribution generally followed a normal distribution within diameter classes
with a high degree of variability in the two data sets evaluated. The deterministic estimate
provided a height distribution with low variability located around the observed mean value,
whereas the random stochastic approach provided greater variability, consistent with the
observed distribution. This is evident from the results of the Kolmogorov-Smirnov test for each
diameter class (Table 3) and the interquartile and maximum ranges of the distributions
(Figure 1).
Figure 1.- Box plots of height distributions (Y-axis) against diameter classes (X-axis) for Pinus
sylvestris (left) and Pinus pinaster (right) data sets. The triangles represent the means of height
estimates. The boxes represent the interquartile ranges. The maximum and minimum height
estimates are represented by the upper and lower small horizontal lines crossing the vertical
bars, respectively. In white: observed distribution; in light grey: deterministic estimation; in dark
grey: stochastic estimation.
Table 3. Results of the Kolmogorov-Smirnov test for diameter classes in the two data sets.
Pinus sylvestris
Pinus pinaster
n Determ. Stoch. n Determ. Stoch.
2.5
----15 0.334 0.134**
7.5
15 0.334 0.267* 390 0.297 0.049*
12.5
64 0.344 0.219 509 0.261 0.080
17.5
111 0.324 0.189 148 0.263 0.169
22.5
121 0.256 0.074**
----27.5
96 0.229 0.062**
----32.5
81 0.432 0.062**
----** Indicates significance at P = 0.05. * Indicates significance at P = 0.01.
Diameter class (cm)
Conclusions
The relationship between tree height and diameter is one of the most important elements in
natural and managed forest dynamics and structure. However a deterministic model is not
capable of completely describing the nature of the height-diameter relationship. Thus, if real
variability is required, a stochastic component should be added to the deterministic estimation.
The suggested approach improves the realism and accuracy of height predictions at stand level.
This feature is considered very important because stand growth level models used for these
species in northwestern Spain are disaggregated by diameter classes to estimate merchantable
volume and biomass using taper functions and individual tree biomass equations.
The stochastic approach proposed may also be used in most cases where different predictions
are required for the same values of the independent variables. In each situation, the process
should involve the study of the specific distribution function to be used for generating variability
in the simulated data (Parresol & Lloyd, 2004).
Acknowledgments
The fieldwork involved in the collection of the data was financed by research project
PGIDIT02RFO29103PR, entitled “Estudio de las cortas de mejora en las masas de coníferas de
Galicia”. We thank Dr. Christine Francis for correcting the English grammar of the text.
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