Amaro A., Reed D., Soares P. (editors) Modelling Forest Systems

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16Amaro Forests - Chap 14 25/7/03 11:06 am Page 166

166 R.A. Fleming and T.R. Burns

length) outweighs the effect of reduced light availability limiting foliage production

in the lower crown. (Stand thinning probably weakened this latter effect.)

Acknowledgements

We thank Bo Långström for organizing accommodation at Jädraås and Stockholm

through the Swedish University of Agricultural Sciences, and for arranging ﬁnancial

support through the Swedish Royal Academy of Agriculture and Forestry (KSLA).

We thank STORA Skog for permission to work at their premises.

References

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Draper, N.R. and Smith, H. (1981) Applied Regression Analysis. Wiley, New York, 709 pp.

Ericsson, A., Larsson, S. and Tenow, O. (1980) Effects of early and late season defoliation on

growth and carbohydrate dynamics in Scots pine. Journal of Applied Ecology 17, 747–769.

Ericsson, A., Hellqvist, C., Långström, B., Larsson, S. and Tenow, O. (1985) Effects on growth of

simulated and induced shoot pruning by Tomicus piniperda as related to carbohydrate and

nitrogen dynamics in Scots pine. Journal of Applied Ecology 22, 105–124.

Fleming, R.A. (2001) The Weibull model and an ecological application: describing the dynam-

ics of foliage biomass on Scots pine. Ecological Modelling 138(1–3), 309–320.

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balsam ﬁr: period models of needle survivorship for spaced trees. Forest Science 38(2),

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16Amaro Forests - Chap 14 25/7/03 11:06 am Page 168

17Amaro Forests - Chap 15 25/7/03 1:54 pm Page 169

15 Diameter Distribution Models

and Height–Diameter Equations

for Estonian Forests

A. Kiviste,

A. Nilson,

M. Hordo

and M. Merenäkk

Abstract

To study the structure and growth of Estonian forest stands, a network of 501 permanent sam-

ple plots was established in 1995–2001. The plots were placed randomly using the grid of the

European forest monitoring program ICP FOREST. The network of permanent plots covers the

main forest types and actual age interval of commercial forests. The plots are to be re-mea-

sured at 5-year intervals.

The plots are circles holding a minimum of 100 trees each. On the plots, the polar coordi-

nates and breast height diameters of all trees are measured. Additionally, the total height and

crown length of selected sample trees are also measured. To date, the measurement data for

90,209 trees have been recorded in the database.

The aim of the permanent plot network is to create and recalibrate Estonian forest-growth

models. However, to achieve this, long-term measurement series on the permanent plots are

still required. Nevertheless, on the basis of the existing material, some forest structure models

have been tested and developed for Estonia, such as: (i) diameter distribution models and (ii)

height–diameter models.

The Johnson’s SB distribution was ﬂexible enough to describe the diameter distributions

of Estonian forests. Regression methods of parameter estimation represented a better ﬁt than

percentile methods. A diameter distribution model, the parameters of which were predicted by

stand variables, has been developed for Estonian forests.

A standardized height–diameter equation has been created for Estonian forests.

Depending on the number of height–diameter measurements, the equation can be used as a

one-parameter or two-parameter model. Model parameters can be estimated by solving a sys-

tem of linear equations.

Introduction

Models of forest stand growth and structure are the basis for planning forest man-

agement activities such as cutting, thinning and regeneration. At the Estonian State

Forest Management Centre, a computer-based decision-support system is being

Estonian Agricultural University, Estonia

Correspondence to: akiviste@eau.ee

17Amaro Forests - Chap 15 25/7/03 1:54 pm Page 170

170 A. Kiviste et al.

developed. Both adequate forest stand descriptions and stand growth and structure

models are needed for the effective use of the system.

The current Estonian forest structure and growth models are poor (Nilson,

1999). T

o make the computer-based decision-support system work, many tempo-

rary models based on old tables generalized for large regions (e.g. the Orlov site

index tables, the Tretyakov standard tables) or on limited empirical data have

been used. It is expected that more accurate models will soon be produced by

Estonian forest researchers to replace the incomplete models in the system.

The stand growth models based on the distance-independent individual tree

owth models used in the Finnish

MELA (Hökka, 1997; Hynynen et al., 2002) and

Swedish

HUGIN (Hägglund, 1981) would be acceptable types of growth model in the

computer-based decision-support system. However, much effort is required to

establish and maintain long-term observations on permanent forest growth plots.

An all-Estonian network of permanent forest growth plots has been designed after a

similar Finnish system (Gustavsen et al., 1988) to provide empirical data for the

models. At present, the network consists of 501 plots, 193 of which have already

been re-measured.

According to Finnish results (Hynynen et al.

, 2002), for reliable forest growth

modelling, data from at least 3–4 plot re-measurements (over 15–20 years) are

required. Nevertheless, the collected plot data can be used for the modelling of

Estonian forest structure. This chapter presents our ﬁrst attempts to use the collected

material for modelling of diameter distribution and height–diameter relationships

for Estonian forests.

Materials and Methods

Data collection

Thousands of plots have been established in Estonian forests over many decades.

Most of them ar

e subjectively located temporary plots. However, construction of

forest growth models using data from subjectively located temporary plots is prob-

lematical. In 1999, the National Forest Inventory (NFI) was started in Estonia. The

NFI permanent plots have been optimized to estimate Estonian forest area and vol-

ume. Having a radius of only 10 m, they are, however, too small to reliably charac-

terize stand regularities for research purposes. A new method of establishing a

network of permanent forest growth plots has been developed by the Institute of

Forest Management of the Estonian Agricultural University. The following princi-

ples were used for designing the network of permanent forest growth plots.

1. A long series of re-measured permanent plots would provide the best data for the

modelling of for

est stand growth.

2. All basic forest types and age and density classes should be represented through-

out Estonia.

3. Plots should be located randomly.

4. In addition to measurements of trees, tree coordinates on plots should be deter-

mined in or

der to ﬁnd the same trees at the next re-measuring and to enable the con-

struction of both distance-independent and distance-dependent models.

5. Forest growth plots should be large enough to clearly characterize stand regularities.

The method of establishing permanent forest growth plots is mainly based on the

experience of the Finnish For

est Research Institute (Gustavsen et al., 1988). Some rec-

ommendations from Curtis (1983) were taken into consideration. The codes of species,

17Amaro Forests - Chap 15 25/7/03 1:54 pm Page 171

171 Diameter Distribution Models

site types, faults and measurement units were taken from the Estonian NFI guidelines

(Estonian Forest Survey, 1999). The following procedures characterize the method.

1. The compartments for plots were selected before the beginning of ﬁeldwork

accor

ding to the desired plot distribution. For the placement of plot regions, the grid

of the European forest-monitoring program

ICP FOREST 1 was used. Also, the grid of

the Estonian National Forest Inventory was used to place plot centres. To save on

transport costs, two or three survey pairs worked together in the same plot region.

2. All trees were measured on circular plots with a radius of 15, 20, 25 or 30 m to get

at least 100 ﬁrst-stor

ey trees. Smaller trees (second-storey and understorey trees)

were measured in an inner circle with a radius of 8 m (the surrounding circle 15 m)

or 10 m (the surrounding circle more than 15 m).

3. Measurements on plots were carried out by pairs of researchers. The recorder at

the plot centr

e measured the azimuth and the measurer measured the distance from

the plot centre, breast height diameter in two directions, and the faults of each tree.

The measured trees were marked with a coloured spot at a height of 1.3 m. Every

ﬁfth tree, dominant trees, trees of rare species and trees in the inner circle were mea-

sured as sample trees. For sample trees, the total height, the crown height and the

height of dead branches were also measured. Crown height was deﬁned as height of

the lowest live contiguous whorl. In addition to living trees, the coordinates and

diameters of standing dead trees and fresh stumps were measured.

4. The main instruments for tree measurements were a compass, a calliper and a

‘For

estor Vertex’ hypsometer.

5. The age of stand components was determined by counting tree rings from cores

extracted fr

om sample trees. The thickness of the soil organic layer was measured at

several places on the plot.

6. A metal stick was used to mark the plot centre. In addition, a couple of trees near

the plot centr

e were marked with a coloured circle for easier location in 5 years time.

The geographical coordinates of plot centres were determined using a GPS device.

The plots and their neighbouring objects (roads, ditches, rocks, etc.) were mapped.

In Estonia, the ﬁrst permanent forest growth plots, with tree coordinates, were

established by Urmas Peterson in the Pikknurme and

Aakre forest districts (Central

Estonia) in 1995/96. In 1997/98, plots of the same type were established in the Sagadi

forest district (North Estonia) and in several areas of South Estonia. During those years,

various instruments and methods were tried and, as a result, an optimal ﬁeldwork

technique was developed. In 1999, a network of permanent forest growth plots to cover

the whole of Estonia was designed. Figure 15.1 shows that most of Estonia is already

covered by plots. In the summer of 2002, more than 100 forest growth plots were estab-

lished in north-west Estonia and on some Estonian islands. With that, the ﬁrst round of

measuring was ﬁnished. At present, a total of 501 forest growth plots have been estab-

lished in the network, 193 of which have already been re-measured. Figure 15.2 shows

the distribution of plots by dominant species and age classes. Our database includes a

total of 90,209 tree measurements, 70,002 of which pertain to the ﬁrst layer. The number

of model trees is 28,441. Most of them (22,993) are ﬁrst-layer trees.

Analysis

Diameter distribution

Various density functions have been used for the modelling of diameter distribu-

tions. Most often, normal, log-normal, W

eibull, Gram–Charlier, γ, β and Johnson’s

distributions have been ﬁtted. In this study, we selected Johnson’s SB distribution as

17Amaro Forests - Chap 15 25/7/03 1:54 pm Page 172

172 A. Kiviste et al.

20 30 40 50 60 70 80 90 >100

Age class

120

100

Number of plots

Other Birch Spruce Pine







⋅−







⋅







)

(

)

−

+−

⋅+ −

εελ

ελ







−







(







⋅













Fig. 15.1. Network of permanent forest growth plots in Estonia established in 1995–2001. Each circle on

the map marks a group of plots.

Fig. 15.2. Distribution of permanent forest growth plots by dominant species and age classes.

one of the most ﬂexible among the classical distribution functions (Tarve, 1986;

Kamziah et al., 1999).

The density function of Johnson’s SB distribution (1949) is

γδ

where z =+⋅ ln

and δ are shape parameters, ε and λ are location and scale parameters, d is tree

breast height diameter (cm).

In this study, six methods were tested for the estimation of Johnson’s SB distrib-

ution parameters.

1. Four percentile (4PR) method (Slifker and Shapiro, 1980). According to this method,

parameter estimates ar

e calculated from four symmetrical points (−3z, −z, z, 3z).

exp

x d

fd( )

17Amaro Forests - Chap 15 25/7/03 1:54 pm Page 173

173 Diameter Distribution Models

On the basis of 436 Estonian forest plot data points, the optimal value z = 0.972 was

used.

2. Knoebel and Burkhart’s (1991) (KB) method. According to this method, the lower

limit, range, median and 95th per

centile of the observed diameter distributions were

used for parameter estimation.

3. Moments (MOM) method (Bowman and Shenton, 1983). According to this

method, the lower limit, range, arithmetic mean and standar

d deviation of the

observed diameter distributions were used for parameter estimation.

4. Maximum likelihood (MLE) method (Zhou and McTague, 1996) with predeter-

mined

ε and λ using the Knoebel and Burkhart (1991) method.

5. Linear regression (LIN) method (Zhou and McTague, 1996). According to this

method, the parameters

ε and λ were predetermined using Knoebel and Burkhart’s

(1991) method, and the shape parameters γ and δ were calculated from 10%, 20%,

…,90% percentiles using linear regression.

6. Non-linear regression (NLIN) method (Kamziah et al.

, 1999). According to this

method, the parameters ε and λ were predetermined using Knoebel and Burkhart’s

(1991) method, and the shape parameters γ and δ were calculated from all points

using Marquardt non-linear regression procedure

NLIN (SAS, 1989).

Kolmogorov–Smirnov (KS) statistics were calculated and goodness-of-ﬁt tests

(Zar

, 1999) at the signiﬁcance level α = 0.05 were performed for each distribution

and parameter estimation method.

Height–diameter relationships

In this study we wanted to develop a standard equation for use in the procedure of

appr

oximation of plot height–diameter data in the Estonian forest-management

decision-support system. A considerable number of different equations have been

used for the approximation of height–diameter relationships (e.g. Huang et al., 2000).

The equations having parameters that can be estimated using the linear regression

technique are preferred for practical use. Root mean square errors (RMSEs) of 27 dif-

ferent height–diameter equations used on the Estonian forest plot data (Mängel,

2000) were compared. Several equations had almost the same (minimum) residual

standard error. Finally we settled on the model developed by Professor A. Nilson:

h =



 D 



1 −⋅

−

 













where h is the tree height in m, d is the tree breast height diameter in cm, D is the

quadratic mean diameter in cm, and c is a constant (1.31 for pine, 1.47 for spruce

and 1.38 for other species).

Equation parameters H and b can be calculated fr

om the following system of

linear equations:

n n





b⋅+⋅

∑

z ⋅ =

∑



i=1 i=1



n n n



H ⋅

∑

z +⋅

∑

⋅ =

∑

⋅hb

i i





i=1 i=1 i=1





where

=−

 

and h

are the diameter and height of the ith sample tree, and





D is the quadratic mean diameter (cm).

17Amaro Forests - Chap 15 25/7/03 1:54 pm Page 174

174 A. Kiviste et al.

Results

Diameter distribution

For the modelling of diameter distributions, ﬁrst-storey trees of the dominant

species wer

e selected from each plot. The plots holding less than 56 ﬁrst-storey trees

of the dominant species were excluded from the analysis of distributions. As a

result, 436 diameter distributions, with 52,608 trees, were selected. Table 15.1 pre-

sents summary statistics of the 436 diameter distributions used in this study.

Kolmogorov–Smirnov (KS) statistics were calculated and goodness of ﬁt tests

wer

e performed for each diameter distribution and parameter estimation method.

Table 15.2 presents the means and standard deviations of parameter estimates and the

KS statistics with the numbers of passes of goodness-of-ﬁt tests. Figure 15.3 shows

box-and-whiskers plots of KS statistics by different parameter estimation methods

A comparison of methods applied to Estonian forest plot data showed that the

egression methods (LIN and NLIN) for parameter estimation represented the best

ﬁt. The methods based on certain percentiles (KB and 4PR) were unstable. For both

methods, about 10% of plots did not pass the Kolmogorov–Smirnov goodness-of-ﬁt

test, and for the four percentile (4PR) method the parameter estimation failed in

four cases. The moments (MOM) method and the maximum likelihood (MLE) meth-

ods represented a slightly worse ﬁt than the regression methods; however, they are

simpler to apply.

Table 15.1. Summary statistics of 436 diameter distributions used in this study.

Characteristic Mean Minimum Maximum

Number of trees 121 56 378

Average diameter 17.1 4.6 37.1

Standard deviation 4.7 1.6 9.4

Coefﬁcient of variation 29.9 13.8 72.1

Skewedness 0.51 −0.59 4.1

Kurtosis 0.23 −1.13 23.5

Table 15.2. Arithmetic means and standard deviations (

SD) of Johnson’s SB distribution parameters and

Kolmogorov–Smirnov (KS) statistics on the basis of data from 436 plots. A plot passes the test when the

null hypothesis (H

) fails to be rejected.

Method ε

Parameters

λ γ

KS test (α = 0.05)

KS Number of

δ statistic passes (H

) %

4PR

MOM

MLE

LIN

NLIN

Mean

7.39

4.50

6.89

4.47

5.73

3.12

5.73

3.12

5.73

3.12

5.73

3.12

28.67

21.65

26.31

7.20

30.70

9.97

30.70

9.97

30.70

9.97

30.70

9.97

0.87

0.86

0.64

0.41

0.82

0.35

0.81

0.35

0.85

0.41

0.87

0.42

1.18

0.40

1.20

0.20

1.35

0.25

1.36

0.24

1.41

0.28

1.41

0.29

0.079

0.029

0.084

0.032

0.064

0.019

0.067

0.020

0.060

0.017

0.056

0.015

393

399

434

436

90.0

91.5

99.5

100

17Amaro Forests - Chap 15 25/7/03 1:54 pm Page 175

175 Diameter Distribution Models

4PR

MOM

MLE

LIN

NLIN

0.00 0.04 0.08 0.12 0.16 0.20 0.24

Kolmogorov–Smirnov (KS) statistic

Fig. 15.3. The box and whiskers plot of Kolmogorov–Smirnov (KS) statistics for different methods.

Using the general linear method procedure (GLM) of the SAS software (SAS,

1989), equations to predict Johnson’s SB distribution parameters from stand vari-

ables were developed. Initially, all main stand variables (site type, site index, stand

age, dominant species, average height, quadratic mean diameter, basal area, relative

density, number of trees per hectare and indicator of thinning) were analysed as

independent variables in the model. Finally, only signiﬁcant variables at a signiﬁ-

cance level of α = 0.05 were included. Table 15.3 presents the variables and parame-

ter estimates of linear models for the prediction of the minimum and the maximum

diameters of ﬁrst-layer trees in stands.

The standard deviation of diameter STD can be predicted using the following

equation:

STD = 0.006 + 0.294·MAX – 0.233·D

where MAX is the maximum diameter (cm), and D is the quadratic mean diameter

(cm).

Table 15.3. Parameter estimates of linear models for the prediction of the minimum (MIN) and the

maximum (MAX) diameters of ﬁrst-layer trees from stand variables.

Stand variable MIN MAX

Intercept according to dominant species

Pine −3.88 7.78

Spruce −4.02 8.59

Birch −3.43 7.68

Aspen −3.21 6.71

Common alder −1.87 5.26

Grey alder −2.66 6.10

Site index H

100

(m), base age 100 years 0.0730 −0.1085

Quadratic mean diameter D (cm) 0.6553 1.2499

Number of trees per hectare (N) 0.000285 −0.000577

Relative density of stand T (by Tretyakov) −1.67 3.05

Indicator of thinning (thinned 1, otherwise 0) 0.31 0

Coefﬁcient of determination (R

) 0.89 0.92

RMSE 1.56 2.60