227
Silva Fennica 39(2) research articles
Functions for Estimating
Stem Diameter and Tree Age Using
Tree Height, Crown Width and
Existing Stand Database Information
Jouni Kalliovirta and Timo Tokola
Kalliovirta, J. & Tokola, T. 2005. Functions for estimating stem diameter and tree age using
tree height, crown width and existing stand database information. Silva Fennica 39(2):
227–248.
The aim was to investigate the relations between diameter at breast height and maximum
crown diameter, tree height and other possible independent variables available in stand
databases. Altogether 76 models for estimating stem diameter at breast height and 60
models for tree age were formulated using height and maximum crown diameter as
independent variables. These types of models can be utilized in modern remote sensing
applications where tree crown dimensions and tree height are measured automatically.
Data from Finnish national forest inventory sample plots located throughout the country
were used to develop the models, and a separate test site was used to evaluate them. The
RMSEs of the diameter models for the entire country varied between 7.3% and 14.9%
from the mean diameter depending on the combination of independent variables and spe-
cies. The RMSEs of the age models for entire country ranged from 9.2% to 12.8% from
the mean age. The regional models were formulated from a data set in which the country
was divided into four geographical areas. These regional models reduced local error and
gave better results than the general models.
The standard deviation of the dbh estimate for the separate test site was almost 5 cm
when maximum crown width alone was the independent variable. The deviation was
smallest for birch. When tree height was the only independent variable, the standard devia-
tion was about 3 cm, and when both height and maximum crown width were included it
was under 3 cm. In the latter case, the deviation was equally small (11%) for birch and
Norway spruce and greatest (13%) for Scots pine.
Keywords forest inventory, crown diameter, stem diameter, modeling
Authors´ address University of Helsinki, Dept. of Forest Resource Management,
P.O.Box 27, FI-00014 University of Helsinki, Finland
E-mail timo.tokola@helsinki.fi
Received 27 September 2004 Revised 17 February 2005 Accepted 18 May 2005

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Silva Fennica 39(2) research articles
1 Introduction
The development of modern remote sensing sen-
sors has increased the need to create new forest
models (Maltamo et al. 2003). One of the most
promising methods is to use high resolution digi-
tal aerial photographs (Pollock 1996, Gong et al.
2002, Korpela 2004, Wang et al. 2004) or laser
scanning (Hyyppä et al. 2001, Holmgren 2003,
Næsset 2004) to measure individual trees. As
early as the 1970s, Jakobsons (1970) and Talts
(1977) described the possibility of measuring the
height of a tree, the crown diameter or even the
diameter at breast height on aerial photographs
by photogrammetry. However, these measure-
ments usually only represent the dimensions of
the crown as visible on the aerial photographs,
the resolution and visibility of small branches and
irregular crown parameters being dependent on
the scale of the photograph. In theory, however,
a close correlation exists in principle between
crown diameter and stem characteristics, such
as diameter at breast height, and the latter is also
highly correlated with the photogrammetrically
measured crown diameter, a relation for which
Petlewitz (1976) observed a correlation coef-
ficient of 0.9 in Pinus silvestris and a standard
deviation of the regression of 2.5 cm. Klier (1970)
emphasized the influence of scale, image quality,
species and species mixture, while the close rela-
tionship between these variables motivated many
researchers (e.g. Sayn-Wittenstein et al. 1967)
to construct aerial tree and stand volume tables
based on crown diameter. Such tables, based on
stand height, crown closure and crown diameter
as independent variables or in a modified form
(Eid and Næsset 1998, Gingrich et al. 1955, Avery
and Meyer 1959), are today in common use in
North America and Norway.
Krajicek et al. (1961) studied relations of crown
and diameter at breast height in open-grown trees
not confounded by competition, measuring 340
such trees in eastern Iowa. The crown width of
a tree in an open stand is closely related to its
diameter at breast height, the correlation coeffi-
cient for every species being over 0.98. This rela-
tion was found to be independent of age and site
quality, but differed slightly between tree species.
Open-grown trees were shorter than forest-grown
ones of the same diameter on similar soils and
under similar conditions. This is attributed to
competition between adjacent trees under forest
conditions, a factor which also tends to reduce
the size of the live crown, and especially the
crown width.
Ilvessalo (1950) and Jakobsons (1970) studied
the correlation between tree crown diameter and
diameter at breast height under boreal managed
forest conditions. Ilvessalo (1950) found that as
branches are cloaked by adjacent trees, measure-
ments of maximum crown diameter on photo-
graphs are generally smaller than those made on
the ground. Also, crown diameter varies with tree
species, tree height, site and stand density. The
correlation between crown diameter and diameter
at breast height was best for Scots pine and much
weaker for Norway spruce. Jakobsons (1970)
studied this correlation for pine, spruce and birch
separately and reached the following conclusions
for trees belonging to the same diameter (at breast
height) class. Conifers have smaller crown diam-
eters than deciduous trees, but the location of the
tree is also important, such that trees in southern
Sweden have greater crown diameters than those
in the north. Meanwhile, trees on poor sites or in
open stands have greater crown diameters than
those on nutrient-rich sites or in denser stands.
Jakobsons (1970) also found that an almost
linear relation exists between crown diameter and
diameter at breast height, although this differed
between tree species and between geographically
distant trees. The crown diameter of young trees
was wider than that of older trees. The relation
was also confounded by competition between
trees, the availability of light and site factors.
Jakobsons (1970) nevertheless maintained that it
was possible to estimate diameter at breast height
as a function of crown diameter. Talts (1977), by
contrast, concluded that also other independent
variables in addition to crown diameter were
necessary.
Nash (1949) and Nyyssönen (1955) found a
standard error of 0.6 m in crown diameter esti-
mates on photographs, and Worley et al. (1955)
obtained a standard error between 0.9 m and 1.2
m on 1:12 000 photographs. More recently, Hilde-
brandt (1996) reconstructed the dbh distribution
of beech stands from the observed distribution of
crown widths. Stand age can also be estimated

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Kalliovirta and Tokola Functions for Estimating Stem Diameter and Tree Age Using Tree Height, Crown Width and …
from a regression equation with photogrammetri-
cally determined stand height and crown size as
the predictor variables, although because of the
inherent uncertainties, a given stand is usually
assigned to one of 20 year classes. Studies in
Germany (see Van Laar and Akca 1997) have
indicated that the age class of a stand can be esti-
mated from photographic measurements.
New measuring methods, such as laserscanning
(Hyyppä et al. 2001, Holmgren 2003, Næsset
2004) or digital photogrammetry (Korpela 2000,
2004); have specific characteristics and measure-
ment techniques. Because imaging condition and
applicability of tree measurements differ accord-
ing to the distance to objects, the relative position
of the tree and other similar factors, traditional
photography-based crown diameter measure-
ments are not a good basis for modelling. When
allometric tree models are created using field
measurement, separate calibration models can be
used to relate photography-based measurements
and ground measurements with improved accu-
racy. When models are applied directly without
calibration using automatic segmentation, small
trees are easily overestimated and large trees
are underestimated (Ikonen 2004). This type of
error can be reduced using calibration techniques
which utilize imaging parameters and few field
observations (Mäkinen 2004). The models can
be directly applied, when laser scanning is used
as a remote sensing technique. Tree volume can
then be derived from these variables using a chain
of models in which diameter at breast height is
estimated first. The aim of this study was to inves-
tigate the relations between diameter at breast
height and maximum crown diameter, tree height
and other possible independent variables and to
formulate models for estimating the diameter at
breast height using different independent vari-
ables and chains of models. Models for tree age
were also formulated, with height and maximum
crown diameter as independent variables.
2 Material
The main material used in the present work was
based on the 1889 permanent sample plots estab-
lished throughout Finland for the purposes of the
Finnish National Forest Inventory (NFI). Plot size
varied according to diameter at breast height of
a tree. Plot size was 100 m2, when diameter was
under 10.5 cm and otherwise 300 m2. An addi-
tional data set (Korpela 2004), comprising 346
Scots pines, 245 Norway spruces and 120 birches
on a site near the Hyytiälä Research Station, was
used to validate the models.
The NFI sample plot network is based on clus-
ter sampling, where each cluster in southern Fin-
land includes four sample plots and each cluster
in northern Finland three. The distance between
two clusters is also greater in the north than in the
south, as is the sample plot interval. The mate-
rial contains data from the 1st and 3rd rounds of
measurements made on the permanent sample
plots (in 1985–86 and 1995).
The material includes only trees for which
crown diameter measurements are available, and
only the data for 1995 were used to formulate the
models. The crown diameters in the NFI material
were measured according to field instructions,
i.e. by taking the widest dimension of the crown.
Any obvious mistakes in measuring and recording
the data were removed, leaving a total set of 11
246 trees. Trees have been classified according
to their position in the stand into the following
categories: dominant (63%), intermediate (33%),
and suppressed (4%), which refer to determined
relative height of tree, over 80 %, 50–80% and
less than 50%, respectively. The locations of the
clusters are presented in Fig. 1.
The material also includes damaged and dis-
eased trees, which can exhibit a highly abnormal
relation between diameter at breast height and
either height or crown diameter, causing bias in
the models. It is assumed that living trees can be
identified by remote sensing material. This may
not be the case if the top of the tree is broken or
the tree is dying (barely any living canopy left).
After removing these abnormal trees, the data
used for the diameter at breast height and the
age models comprised 5303 Scots pines, 3661
Norway spruces and 2282 birches. The average
values for the sample tree and stand variables
are presented in Table 1. A caliper was used to
measure the diameter at breast height, a Suunto
hypsometer to measure tree height, an increment
borer to measure tree age and a Kajanus tube to
measure crown width.

230
Silva Fennica 39(2) research articles
Fig. 1. Models were constructed for all of Finland (right side) and for four separate
regions (left side). Geographical regions are defined by the forest flora and
climatic conditions (1 = Hemiboreal, 2 = South boreal, 3 = Middle boreal,
4 = North boreal). The entire area is covered by clusters. The locations of
the clusters are shown on the right side of the figure.
As the relations between tree variables may
vary depending on the location (see Jakobsons
1970), the material for the entire country was
divided into four geographical areas defined
according to the forest flora and climatic condi-
tions (Fig. 1). The resulting distribution is pre-
sented in Table 2.
3 Methods
Due to the hierarchical nature of the data, a
mixed effect method with iterative general-
ized least squares (IGLS) was used for linear-
ized regression. The independent variables were
selected according to the requirements defined for
the new forest inventory procedure, i.e. that all
independent variables should be accessible from
high resolution aerial photographs or existing
databases. The photogrammetric variables were
height, maximum crown diameter, stem number
of dominant trees per hectare and relative tree
height class. The photogrammetric variables and
variables from the stand database were treated as
independent variables in the regression model.
The intercept was the only fixed effect of the basic
model. Clusters and plots were treated as random
effects. The form of model is
y = Xb + Zc + e
⇔ ykji = xkji´b + ck +dkj+ ekji,
where y is an n × 1 vector of observed values of
the dependent variable, b is a p × 1 vector of fixed
parameters, X is an n × p matrix of independent
variables associated with fixed parameters, c is a
q × 1 vector of random parameters with expecta-
tion zero, Z is an n × q matrix of explanatory
variables associated with random parameters and
e is an n × 1 vector of error terms, e ~ N(0,σ2).
Furthermore, in this case, k is the cluster to which
the tree i in the plot j belongs, ck is the random
parameters of cluster k and dkj is the random
parameters of plot j.
The variables (α) from existing stand databases

231
Kalliovirta and Tokola Functions for Estimating Stem Diameter and Tree Age Using Tree Height, Crown Width and …
that were tested were similar to variables which
can be found in the forest planning databases
provided by private forest owners in Finland,
together with a few generally accepted variables:
x co-ordinate, y co-ordinate, height above sea
level, temperature sum, mean diameter, mean age,
tree class, basal area, land-use class, site class and
soil type. Stand variables, which could be derived
from an aerial photograph, such as stem number
of dominant trees per hectare and relative tree
height, were also tested. In general, dominant
height is defined as the mean height of the 100
thickest trees at breast height in one hectare. In
the context of this study only tree heights can be
used to define dominant height because diameters
are not known. Dominant tree is defined as a tree
which height is more than 80 % from dominant
height. Relative tree height could be estimated
by comparing the height of the recognized tree
to the dominant height of the recognized trees of
the remote sensing material on a site. Relative tree
height class is used as a dummy variable (D9). It
indicates that a tree is suppressed or dominated
defined as a tree which height is under 80 percent
Table 1. Mean statistics of field material (NFI) by species.
N Mean Sd Min Max
D1,3, mm
Pine 5303 145 69 4 574
Spruce 3661 148 78 4 515
Birch 2282 115 56 6 532
H, dm
Pine 5303 113 47 14 286
Spruce 3661 123 59 14 318
Birch 2282 113 44 16 310
Dcrm, dm*
Pine 5303 31 12 4 101
Spruce 3661 33 12 6 95
Birch 2282 33 12 7 104
Age, years
Pine 5303 59 34 11 297
Spruce 3661 66 31 12 278
Birch 2282 48 20 3 148
x, km 11246 3452 122 3117 3725
y, km 11246 7015 208 6650 7725
Altitude (alt), m 11246 127 65 0 410
Temperature sum (ts), ° 11246 1100 164 531 1425
Basal area (ba), m2/ha 11246 20.9 7.9 1 48
Mean diameter (d1,3m), cm 11246 17.4 6.5 6 46
Mean age (agem), years 11246 71.9 40.3 12 334
Number of trees/ha (n) 11246 1554 934 33 7067
Relative tree height class (dummy) 11246 0 1
Site class (dummy) 11246 0 1
Soil type (dummy) 11246 0 1
Land-use class (dummy) 11246 0 1
* Dcrm refers to maximum crown diameter
Table 2. Number of trees of NFI field plots in different
geographical areas.
Pine Spruce Birch Total
Area 1 129 104 39 272
Area 2 1840 2180 871 4891
Area 3 2641 1200 1190 5031
Area 4 693 177 182 1052

232
Silva Fennica 39(2) research articles
from dominant height, therefore differing from a
dominant or emergent tree. A model with three
variables (h, dcrm, _) was chosen for each tree spe-
cies and area based on a log likelihood ratio test
(Goldstein 1995) achieving the best coefficient
of determination.
To meet the normality and homoscedaticity
assumptions, square root and logarithm trans-
formations were used for the independent and
dependent variables.
The models for diameter at breast height were
of the forms:
d f h1 3, = ( )+ ε (1)
d f dcrm1 3, = ( )+ ε (2)
d f h dcrm1 3, ,= ( )+ ε (3)
d f h dcrm1 3, , ,= ( )+α ε (4)
and the age models of the forms:
ln( ) ln( )age f h= ( )+ ε (5)
ln( ) ln( )age f dcrm= ( )+ ε (6)
ln( ) ln( ),ln( )age f h dcrm= ( )+ ε (7)
where
d1,3 = diameter at breast height (mm),
h = height (dm),
dcrm = crown diameter, maximum (dm)
_ = stand variable from database or aerial
photograph
The models were used to estimate the value of
the variable in its original unit of measurement.
As non-linear transformations were used for the
dependent variables, such an estimate will be
biased (Lappi 1993), an effect that can be reduced
by bias correction. Taking this into account, the
model for diameter at breast height assumes the
form
d f1 3
2
, var( )= ( ) + ε
and the age model the form
age f= ( ) ∗ +


exp var( )1
1
2
ε
R2 was calculated separately to cluster, plot and
tree effects, e.g. R2 for plot indicates the propor-
tion of variance between plots, that is explained
by a model. Proportion of total variance between
clusters and between plots are also presented.
R2 was calculated using a method described in
Lappi (1997), where relation of estimated full
mixed model variance and initial variance of
random effect model of clusters and plots (the
fixed part includes only a constant) were utilized
as follows:
R
(estimated variance of full model)
(ini
2 1= −
tial variance of model)
The non-linear extra sum of squares method
(Bates and Watts 1988) was used to evaluate the
differences between the geographical areas. The
method requires the fitting of full and reduced
models. The full model corresponds to different
sets of parameters for each of the geographical
areas involved. The reduced model corresponds
to the same set of parameters for all regions.
The suitability of the division and the need for
any division at all were assessed on the basis of
the test results. The appropriate test statistic is
described in Bates and Watts (1988).
4 Results
4.1 Data Analysis for Modelling
The normality and homoscedasticity of models
were tested. As an example of a model that meets
these assumptions well, the diameter of Scots
pines at breast height in area 3 is presented in Fig.
2. There were about 2640 pines in the area.
Altogether 136 models were constructed. These
were numbered using a system in which the first
digit for a model defines the geographic area
(Fig. 1) in question (number of the area or 9 as an
indication of the entire country), the second digit
the form of the model and the last the tree species.
For example, model 2.2.3 applies to diameter

233
Kalliovirta and Tokola Functions for Estimating Stem Diameter and Tree Age Using Tree Height, Crown Width and …
model for birch in area 2 (tree species = 3), with
the maximum diameter of the crown as the only
independent variable (form of the model = 2). It
should be noted that tree height and maximum
crown diameter are expressed in decimetres in all
the models, yielding the diameter at breast height
in millimetres.
4.2 Models for Diameter at Breast Height
The data for all sample plots in the country were
used to formulate the first set of models for
diameter at breast height. General information on
these models is given by tree species in Table 3.
As it can be seen, even the best third independ-
ent variable, y co-ordinate for Scots pine and
temperature sum for Norway spruce and birch,
was of minor significance. The models for the
diameter at breast height for the entire country
are presented in Table 4.
Further models for diameter at breast height
were formulated after dividing the data into four
geographical areas. General information on these
regional models is presented in Table 5. The
RMSEs of the models for the ecoregions varied
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Fig. 2. Diagnostic testing of the model d1,3 = f(h, dcrm) for Scots pine in area 3. Residual plot
in the left side and normality plot of residuals in the right side.
Table 3. Statistical properties of the models for the entire country. R2 is divided into cluster (Clus), plot (Plot) and
tree (Tree) effects. Proportion of total variance (VAR%) is calculated for clusters and plots. The first digit in
number of model refers to the geographic area (Fig. 1) in question (number of the area or 9 as an indication
of the entire country), the second digit the form of the model and the last digit the tree species.
Model No. of Predictor RMSE R2 VAR% VAR
model % mm Clus Plot Tree Clus Plot
All 9.1.0 h 12.5 17.5 0.53 0.85 0.77 0.18 0.23 2.058
Pine 9.1.1 h 12.3 17.8 0.76 0.85 0.68 0.18 0.26 2.057
Spruce 9.1.2 h 10.1 15.0 0.24 0.94 0.86 0.26 0.23 1.408
Birch 9.1.3 h 13.1 15.0 0.31 0.83 0.73 0.30 0.26 1.854
All 9.2.0 dcrm 14.8 20.7 0.70 0.78 0.64 0.08 0.25 2.862
Pine 9.2.1 dcrm 13.0 18.8 0.75 0.75 0.72 0.16 0.39 2.309
Spruce 9.2.2 dcrm 14.9 22.1 0.39 0.84 0.63 0.10 0.28 3.056
Birch 9.2.3 dcrm 12.8 14.7 0.72 0.88 0.60 0.13 0.19 1.770
All 9.3.0 h, dcrm 9.8 13.8 0.67 0.91 0.86 0.21 0.22 1.269
Pine 9.3.1 h, dcrm 8.0 11.6 0.87 0.96 0.85 0.23 0.16 0.869
Spruce 9.3.2 h, dcrm 8.3 12.3 0.38 0.96 0.91 0.32 0.21 0.948
Birch 9.3.3 h, dcrm 9.6 11.0 0.65 0.93 0.82 0.28 0.19 1.000
All 9.4.0 h, dcrm, D9 9.3 13.0 0.73 0.92 0.87 0.19 0.21 1.141
Pine 9.4.1 h, dcrm, y 7.7 11.1 0.91 0.96 0.86 0.17 0.17 0.806
Spruce 9.4.2 h, dcrm, ts 7.3 10.8 0.80 0.96 0.91 0.13 0.26 0.738
Birch 9.4.3 h, dcrm, ts 8.8 10.1 0.84 0.92 0.84 0.16 0.25 0.838

235
Kalliovirta and Tokola Functions for Estimating Stem Diameter and Tree Age Using Tree Height, Crown Width and …
Table 5. continued
Model No. of Predictor RMSE R2 VAR% VAR
model % mm Clus Plot Tree Clus Plot
Area 2
All 2.1.0 h 11.6 17.4 0.40 0.88 0.79 0.07 0.30 1.888
Pine 2.1.1 h 11.6 18.1 0.82 0.89 0.64 0.09 0.31 1.986
Spruce 2.1.2 h 9.6 14.8 0.66 0.95 0.86 0.17 0.24 1.316
Birch 2.1.3 h 10.7 13.8 – 0.85 0.78 0 0.46 1.390
All 2.2.0 dcrm 14.9 22.3 –0.09 0.81 0.65 0.08 0.29 3.091
Pine 2.2.1 dcrm 13.3 20.8 0.59 0.78 0.71 0.16 0.47 2.597
Spruce 2.2.2 dcrm 14.7 22.6 0.53 0.87 0.67 0.10 0.28 3.071
Birch 2.2.3 dcrm 12.5 16.1 – 0.91 0.63 0.14 0.21 1.890
All 2.3.0 h, dcrm 9.1 13.6 0.71 0.93 0.87 0.06 0.31 1.153
Pine 2.3.1 h, dcrm 7.4 11.6 0.89 0.97 0.84 0.13 0.20 0.811
Spruce 2.3.2 h, dcrm 7.2 11.1 0.88 0.97 0.92 0.11 0.24 0.742
Birch 2.3.3 h, dcrm 8.0 10.3 – 0.94 0.86 0.06 0.35 0.780
Pine 2.4.1 h, dcrm, D9 7.4 11.5 0.89 0.97 0.84 0.14 0.21 0.801
Spruce 2.4.2 h, dcrm, D9 7.0 11.1 0.88 0.98 0.92 0.11 0.22 0.694
Birch 2.4.3 h, dcrm, D9 7.5 9.6 – 0.95 0.88 0.10 0.32 0.681
Area 3
All 3.1.0 h 12.2 15.6 0.64 0.80 0.77 0.13 0.24 1.796
Pine 3.1.1 h 12.0 16.2 0.71 0.85 0.72 0.17 0.25 1.833
Spruce 3.1.2 h 8.9 12.1 0.61 0.97 0.87 0.23 0.11 1.008
Birch 3.1.3 h 12.4 12.9 0.44 0.78 0.72 0.26 0.22 1.524
All 3.2.0 dcrm 14.6 18.6 0.74 0.75 0.63 0.07 0.21 2.548
Pine 3.2.1 dcrm 12.8 17.3 0.73 0.74 0.74 0.13 0.39 2.088
Spruce 3.2.2 dcrm 15.2 20.5 0.76 0.74 0.62 0.05 0.28 2.911
Birch 3.2.3 dcrm 12.2 12.7 0.85 0.82 0.60 0.07 0.18 1.485
All 3.3.0 h, dcrm 9.5 12.1 0.77 0.90 0.86 0.14 0.21 1.078
Pine 3.3.1 h, dcrm 7.5 10.2 0.88 0.96 0.87 0.17 0.17 0.723
Spruce 3.3.2 h, dcrm 7.7 10.4 0.63 0.97 0.92 0.31 0.13 0.741
Birch 3.3.3 h, dcrm 8.9 9.3 0.79 0.91 0.82 0.19 0.17 0.787
Pine 3.4.1 h, dcrm, D9 7.3 9.9 0.90 0.96 0.88 0.14 0.20 0.684
Spruce 3.4.2 h, dcrm, ts 7.0 9.4 0.87 0.96 0.92 0.13 0.19 0.611
Birch 3.4.3 h, dcrm, D9 8.1 8.4 0.91 0.91 0.84 0.10 0.21 0.648
Area 4
All 4.1.0 h 12.3 17.4 – 0.82 0.75 0.18 0.14 2.036
Pine 4.1.1 h 12.7 18.7 0.44 0.77 0.70 0.21 0.21 2.267
Spruce 4.1.2 h 9.4 13.9 0.81 0.82 0.82 0.11 0.27 1.247
Birch 4.1.3 h 13.2 15.0 – 0.75 0.67 0.13 0.33 1.893
All 4.2.0 dcrm 14.0 19.8 – 0.75 0.64 0.12 0.14 2.642
Pine 4.2.1 dcrm 11.9 17.4 0.42 0.92 0.70 0.25 0.08 1.966
Spruce 4.2.2 dcrm 14.5 21.4 –0.13 1.00 0.50 0.27 0 2.966
Birch 4.2.3 dcrm 13.8 15.7 – 0.93 0.50 0.17 0.09 2.061
All 4.3.0 h, dcrm 9.4 13.3 – 0.95 0.85 0.26 0.06 1.198
Pine 4.3.1 h, dcrm 8.7 12.7 0.58 0.99 0.85 0.34 0.02 1.050
Spruce 4.3.2 h, dcrm 7.8 11.5 1.00 0.84 0.87 0 0.36 0.858
Birch 4.3.3 h, dcrm 10.5 11.9 – 0.89 0.78 0.17 0.24 1.185
Pine 4.4.1 h, dcrm, agem 8.3 12.2 0.70 0.99 0.84 0.27 0.03 0.964
Spruce 4.4.2 h, dcrm, ba 7.0 10.4 1.00 0.97 0.85 0 0.09 0.699
Birch 4.4.3 h, dcrm, D9 9.9 11.2 – 0.96 0.79 0.27 0.10 1.052

236
Silva Fennica 39(2) research articles
Table 6. Parameter estimates and t-test statistics (t) of regional models for diameter at breast height. The first digit
in number of model refers to the geographic area (Fig. 1) in question (number of the area or 9 as an indication
of the entire country), the second digit the form of the model and the last digit the tree species.
No. of Constant H Dcrm Age, ba, d1,3m, D9 or ts
model Estimate t Estimate t Estimate t Estimate t
1.1.0 –2.145 –3.85 1.291 28.07 – – – –
1.1.1 –1.775 –2.10 1.314 18.25 – – – –
1.1.2 –1.740 –2.36 1.228 20.13 – – – –
1.1.3 –5.533 –4.03 1.475 13.53 – – – –
1.2.0 –0.805 –1.36 – 2.327 24.24 – –
1.2.1 2.628 3.34 – 1.796 14.37 – –
1.2.2 –3.444 –3.52 – 2.774 16.81 – –
1.2.3 –1.867 –1.30 – 2.549 11.18 – –
1.3.0 –4.765 –11.03 0.846 19.67 1.321 16.31 – –
1.3.1 –3.324 –5.15 0.910 13.79 1.029 10.19 – –
1.3.2 –5.512 –9.57 0.800 14.81 1.512 11.91 – –
1.3.3 –6.978 –6.87 0.972 7.65 1.271 5.23 – –
2.1.0 –1.049 –11.16 1.159 144.86 – – – –
2.1.1 –0.785 –4.49 1.177 78.47 – – – –
2.1.2 –0.960 –8.65 1.168 129.78 – – – –
2.1.3 –2.938 –12.50 1.226 61.30 – – – –
2.2.0 –2.387 –16.81 – 2.474 103.08 – –
2.2.1 –0.444 –2.36 – 2.226 69.56 – –
2.2.2 –4.088 –17.93 – 2.769 74.84 – –
2.2.3 –1.492 –5.72 – 2.106 48.98 – –
2.3.0 –3.733 –42.42 0.807 89.67 1.144 54.48 – –
2.3.1 –3.524 –28.42 0.729 56.08 1.345 49.81 – –
2.3.2 –3.835 –33.94 0.860 78.18 1.079 38.54 – –
2.3.3 –4.250 –23.10 0.804 36.55 1.028 25.70 – –
3.1.0 –1.187 –12.24 1.212 134.67 – – – –
3.1.1 –0.948 –6.72 1.218 87.00 – – – –
3.1.2 –0.457 –3.63 1.161 96.75 – – – –
3.1.3 –1.984 –8.74 1.206 52.43 – – – –
3.2.0 –1.193 –9.32 – 2.260 98.26 – –
3.2.1 –0.547 –3.80 – 2.221 85.42 – –
3.2.2 –2.841 –9.63 – 2.594 48.94 – –
3.2.3 –0.363 1.76 – 1.896 51.24 – –
3.3.0 –3.501 –40.71 0.838 83.80 1.125 56.25 – –
3.3.1 –3.306 –34.08 0.743 67.55 1.334 60.64 – –
3.3.2 –2.739 –19.02 0.920 61.33 0.868 22.84 – –
3.3.3 –3.420 –19.66 0.741 33.68 1.107 31.63 – –
4.1.0 –1.717 –6.66 1.378 53.00 – – – –
4.1.1 –1.854 –5.21 1.389 39.69 – – – –
4.1.2 –0.343 –0.77 1.275 28.33 – – – –
4.1.3 –1.480 –2.31 1.327 18.96 – – – –
4.2.0 0.690 2.60 – 1.999 42.53 – –
4.2.1 1.177 4.51 – 1.957 42.54 – –
4.2.2 –1.005 –1.17 – 2.422 15.23 – –
4.2.3 –0.721 –1.08 – 1.982 17.09 – –
4.3.0 –3.432 16.74 0.941 37.64 1.087 27.87 – –
4.3.1 –2.734 –11.34 0.797 25.71 1.230 28.60 – –
4.3.2 –2.948 –6.25 1.013 21.55 0.962 8.83 – –
4.3.3 –3.770 –6.77 0.886 12.48 1.108 10.17 – –
1.4.1 –3.913 –6.30 0.890 14.59 0.985 10.26 0.016 4.00
1.4.2 –4.201 –6.28 0.802 15.73 1.490 12.31 –0.047 –3.13
1.4.3 –6.863 –6.88 0.965 7.72 1.051 4.29 0.058 2.15
2.4.1 –2.978 –19.72 0.721 55.46 1.279 45.68 –0.379 –6.32
2.4.2 –2.836 –19.97 0.803 73.00 1.059 39.22 –0.493 –10.96
2.4.3 –2.859 –13.55 0.731 33.23 0.981 25.82 –0.738 –11.35
3.4.1 –2.311 –18.79 0.712 64.73 1.233 56.05 –0.538 –12.81
3.4.2 1.469 4.33 0.934 66.71 0.888 24.67 –0.00430 –13.44
3.4.3 –1.688 –8.75 0.666 33.33 0.996 30.18 –0.789 –14.89
4.4.1 –2.728 –11.56 0.751 23.47 1.234 28.70 0.005 5.00
4.4.2 –2.287 –5.02 1.042 23.16 0.949 9.40 –0.053 –5.30
4.4.3 –2.141 –3.37 0.745 10.21 1.134 11.01 –0.806 –4.63

237
Kalliovirta and Tokola Functions for Estimating Stem Diameter and Tree Age Using Tree Height, Crown Width and …
between 8.4 and 23.4 mm depending on the com-
bination of independent variables and species.
Negative R2-values in the table indicate that esti-
mated variances may not change logically, e.g.
because of correlated regressors.
The third variable for Scots pine in area 1 was
the mean age of the growing stock (in years),
for Norway spruce the basal area (m2/ha) and
for birch the mean diameter (cm). In area 2, the
third variable for all tree species was relative tree
height class (D9). The third variable for Norway
spruce in area 3 was the temperature sum (°) and
for Scots pine and birch the relative tree height
class (D9), while in area 4 it was for Scots pine
the relative tree height class (D9), for Norway
spruce the basal area (m2/ha) and for birch the
mean age of the growing stock (in years). The
regional models for diameter at breast height are
presented in Table 6.
4.3 Validation of the Models for Diameter at
Breast Height
The functionality of the models was tested with
data collected from a site near the Hyytiälä
Research Station (in area 2). One aim was to
evaluate the convenience of the division into
regions, i.e. to determine whether the predicted
values differed between the models for the areas
and between the models for area 2 and those for
the entire country. This implies that the models
for area 2 were compared in terms of functionality
with those for the other areas, taking into account
the differences between tree species.
The test results by tree species are presented in
Table 7. When evaluating these results, it should
be noted that the test data for all models are the
same.
The average diameter at breast height for all
three tree species is overestimated when the
height of the tree is the only independent variable,
whereas the models with maximum crown diam-
eter as the independent variable always underes-
timate the diameter at breast height. When both
variables (h, dcrm) are included, the prediction is
virtually unbiased.
The average standard deviation when maximum
crown width alone was the independent variable
was 4.9 cm (about 22% from mean dbh), being
smallest for birch. When tree height was the only
independent variable, the standard deviation was
3.2 cm, which is about 14% from the mean dbh
(smallest for Norway spruce), and when both vari-
ables (h, dcrm) were included, it was 2.7 cm (about
12% from mean dbh). The standard deviation for
the latter model was equally small for birch and
Norway spruce if evaluated in a relative unit of
measure, and largest for Scots pine. The third
variable models were also tested. In all cases, the
effect of the third variable was minor.
The models for the entire country based on
the test data predict the diameter at breast height
equally well. Only a slight difference existed
between the predictions given by the models for
the entire country and for area 2, but it is note-
worthy that 85% of the trees in the data set for
the entire country were located in areas 2 and 3.
Had the test data been taken from area 1 or area
4, the differences would undoubtedly have been
more marked.
The influence of tree species was studied by
comparing models formulated for all tree spe-
cies with species-specific models. This was done
again with the test data from area 2. As might be
expected, the latter models predicted the diameter
at breast height better than the former, the differ-
ences being small for the conifers but consider-
able for birch (Fig. 3).
The need for ecoregions was tested using the
combined model in which the observations from
all regions were included. Because the results of
F-tests revealed that differences existed among
Fig. 3. Averages and standard deviations for predicted
values of d1,3 = f(h, dcrm) in models for area 2 with
and without information on tree species.
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238
Silva Fennica 39(2) research articles
Table 7. Test statistics of the models for dbh using external data from the Hyytiälä Research Station. Bias refers to
the mean of differences between observed and predicted diameters in absolute terms (mm) and proportional
terms (%) per cent from mean diameter. S.E. refers to the standard deviation for the differences.
f(h) f(dcrm) f(h, dcrm)
n Bias S.E. Bias S.E. Bias S.E.
mm (%) mm (%) mm (%) mm (%) mm (%) mm (%)
Scots pine
Entire coutry 346 –26(11) 36(15) 66(28) 51(22) 9(4) 31(13)
Area 1 346 –45(19) 37(16) 52(22) 51(22) –10(4) 27(12)
Area 2 346 –15(6) 34(15) 65(28) 52(22) 16(7) 31(13)
Area 3 346 –28(12) 36(15) 70(30) 51(22) 6(3) 30(13)
Area 4 346 –79(34) 35(15) 65(28) 51(22) –16(7) 29(12)
Norway spruce
Entire coutry 245 –17(8) 30(13) 48(22) 48(22) –4(2) 24(11)
Area 1 245 –16(7) 30(13) 35(16) 48(22) –1(0) 24(11)
Area 2 245 –14(6) 30(13) 52(23) 48(22) 3(1) 24(11)
Area 3 245 –26(12) 30(13) 48(22) 49(22) –16(7) 24(11)
Area 4 245 –83(37) 32(14) 28(13) 49(22) –70(31) 25(11)
Birch
Entire coutry 120 –29(15) 32(17) 55(29) 32(17) 4(2) 21(11)
Area 1 120 –48(25) 33(18) 14(7) 42(22) –24(13) 24(13)
Area 2 120 –19(10) 32(17) 54(29) 33(18) 8(4) 20(11)
Area 3 120 –39(21) 32(17) 59(31) 31(16) –4(2) 21(11)
Area 4 120 –111(59) 34(18) 54(29) 32(17) –54(29) 23(12)
All tree species
Entire coutry 711 –22(10) 32(14) 59(27) 49(22) 1(0) 27(12)
Area 1 711 –34(15) 32(14) 41(18) 50(22) –7(3) 26(12)
Area 2 711 –10(4) 33(15) 60(27) 49(22) 11(5) 27(12)
Area 3 711 –29(13) 32(14) 62(28) 49(22) –5(2) 26(12)
Area 4 711 –89(40) 35(16) 55(25) 49(22) –45(20) 25(11)
the models from different geographical areas,
the differences between pairs of ecoregions were
tested. Results of these tests for model d1,3 = f(h,
dcrm) by tree species are presented in Table 8.
The differences between the areas were mostly
statistically significant for the models d1,3 = f(h),
d1,3 = f(dcrm) and d1,3 = f(h, dcrm). Only a few
combinations of model form and tree species
formed exceptions on some pairs of areas. Only
minor differences were present between the trees
species. The main features of the phenomenon
are easily perceived by examining the means of
the prediction errors in Table 7. The tests indicate
that the division into areas is helpful and can be
recommended for use in the context of the models
formulated here for diameter at breast height.
The need for regional models can also be seen
in Fig. 4, where the residuals (+/–) of the diam-
eter models are presented as interpolated sur-
faces, using the inverse distance weighted (IDW)
method. The residuals of the model for the entire
country were quite large and unevenly distributed
for all tree species. For example, for Scots pine,
the model underestimated the diameter on aver-
age in northern Finland but overestimated it in
southern Finland. With the regional models, the
residuals were lower and distributed more evenly
over the whole country. It should be noted that
the residual surfaces in the most northern part of
country could be misleading because of interpola-
tion problems arising from the small number of
observations.

239
Kalliovirta and Tokola Functions for Estimating Stem Diameter and Tree Age Using Tree Height, Crown Width and …
4.4 Models for Tree Age
The models for the age of the tree were formu-
lated with the same procedure as diameter models,
using height of the tree or maximum crown width
or both as independent variables. General infor-
mation on the age models for entire country is
presented in Table 9, and the models are listed
in Table 10.
Further age models were formulated for four
ecoregions. General information on these regional
models are presented in Table 11. The RMSEs
of the models for the ecoregions varied between
2.8 and 9.7 years depending on the combination
of independent variables and species. Negative
R2-values in the table indicate that estimated
variances may not change logically, e.g. because
of correlated regressors. The regional age models
are presented in Table 12.
For the all species, the age of the tree was
dependent most on its height, and inclusion of
the maximum crown diameter increased the coef-
ficient of determination only slightly. For birch,
however, the maximum crown diameter was more
important independent variable, than for conifers.
In some combinations of regions and tree species
maximum crown diameter was not statistically
significant as independent variable in f(h, dcrm)
models. However, the coefficient of determination
was quite low in all cases.
4.5 Validation of the Models for Tree Age
A validation data set from a site near the Hyytiälä
Research Station was also used to evaluate the
models and ensure reliability in the prediction for
tree age. The growing stock of the site was quite
homogenous and only some age measurements
were done. So, mean age of the stratifications
were used as tree age. This should be noted when
evaluating the test results.
Table 8. F-tests of the regional differences of diameter models: d=f(h, dcrm) by tree species.
Ecoregion pair Full model Reduced model n F-value
dfF SSEF MSEF dfR SSER MSER
Pine
Combined 5291 4253.44 0.803901 5300 4599.43 0.867817 5303 47.821*
Area1–Area2 1963 1629.158 0.829933 1966 1647.284 0.837886 1969 7.280*
Area1–Area3 2764 2050.791 0.741965 2767 2054.066 0.742344 2770 1.471
Area1–Area4 816 853.907 1.046455 819 885.990 1.081795 822 10.220*
Area2–Area3 4475 3395.12 0.758686 4478 3478.022 0.776691 4481 36.424*
Area2–Area4 2527 2203.445 0.871961 2530 2534.028 1.001592 2533 126.375*
Area3–Area4 3328 2628.285 0.789749 3331 2794.609 0.83897 3334 70.201*
Spruce
Combined 3649 2766.84 0.758246 3658 3456.967 0.945043 3661 101.129*
Area1–Area2 2278 1674.423 0.735041 2281 1681.149 0.737023 2284 3.050*
Area1–Area3 1298 990.151 0.762828 1301 1009.256 0.775754 1304 8.348*
Area1–Area4 275 240.846 0.875803 278 411.107 1.478802 281 64.802*
Area2–Area3 3374 2508.124 0.743368 3377 2773.908 0.821412 3380 119.180*
Area2–Area4 2351 1751.189 0.74487 2354 2306.113 0.979657 2357 248.331*
Area3–Area4 1371 1048.793 0.764984 1374 1264.257 0.920129 1377 93.886*
Birch
Combined 2270 1896.433 0.835433 2279 2275.035 0.99826 2282 50.353*
Area1–Area2 904 738.016 0.816389 907 760.215 0.838164 910 9.064*
Area1–Area3 1223 996.492 0.814793 1226 1000.365 0.815958 1229 1.584
Area1–Area4 215 252.360 1.173769 218 287.676 1.319613 221 10.029*
Area2–Area3 2055 1614.348 0.785571 2058 1763.908 0.857098 2061 63.461*
Area2–Area4 1047 893.412 0.853307 1050 1205.01 1.147629 1053 121.722*
Area3–Area4 1366 1155.881 0.846179 1369 1286.066 0.93942 1372 51.284*
* Significant F-value.

240
Silva Fennica 39(2) research articles
Fig. 4. Interpolated residual surfaces obtained from the dbh models for Scots pine, Norway spruce and
birch formulated over the entire country (left side) and for the four geographical areas (right side).

241
Kalliovirta and Tokola Functions for Estimating Stem Diameter and Tree Age Using Tree Height, Crown Width and …
Table 9. Statistical properties of the age models for the entire country. R2 is divided into cluster (Clus), plot (Plot)
and tree (Tree) effects. Proportion of total variance (VAR%) is calculated for clusters and plots. The first digit
in number of model refers to the geographic area (Fig. 1) in question (number of the area or 9 as an indication
of the entire country), the second digit the form of the model and the last digit the tree species.
Model No. of Predictor RMSE R2 VAR% VAR
model % Years Clus Plot Tree Clus Plot
All 9.5.0 h 10.8 6.4 –0.29 0.42 0.42 0.32 0.51 0.181
Pine 9.5.1 h 11.0 6.5 0.11 0.45 0.41 0.39 0.52 0.189
Spruce 9.5.2 h 9.2 6.1 – 0.55 0.55 0.41 0.47 0.142
Birch 9.5.3 h 10.4 5.0 –0.19 0.42 0.41 0.55 0.32 0.155
All 9.6.0 dcrm 11.8 7.0 –0.13 0.20 0.25 0.24 0.58 0.217
Pine 9.6.1 dcrm 12.8 7.5 0.04 0.15 0.31 0.31 0.61 0.252
Spruce 9.6.2 dcrm 10.2 6.7 –0.83 0.31 0.32 0.25 0.60 0.173
Birch 9.6.3 dcrm 10.1 4.8 0.06 0.36 0.29 0.47 0.37 0.146
All 9.7.0 h, dcrm 10.8 6.4 –0.29 0.41 0.42 0.32 0.51 0.182
Pine 9.7.1 h, dcrm 11.1 6.6 0.10 0.44 0.41 0.39 0.53 0.192
Spruce 9.7.2 h, dcrm 9.3 6.1 – 0.54 0.55 0.41 0.48 0.143
Birch 9.7.3 h, dcrm 10.0 4.8 –0.08 0.44 0.44 0.54 0.33 0.144
Table 10. Parameter estimates and t-test statistics (t) of the age models for the entire country. The first digit in
number of model refers to the geographic area (Fig. 1) in question (number of the area or 9 as an indication
of the entire country), the second digit the form of the model and the last digit the tree species.
No. of Constant H Dcrm
model Esimate t Estimate t Estimate t
9.5.0 1.684 58.07 0.490 81.67 – –
9.5.1 1.376 30.58 0.556 61.78 – –
9.5.2 2.085 56.35 0.429 53.63 – –
9.5.3 1.252 15.46 0.544 32.00 – –
9.6.0 2.540 94.07 – – 0.420 52.50
9.6.1 2.840 94.67 – – 0.332 36.89
9.6.2 2.459 52.32 – – 0.474 36.46
9.6.3 2.225 39.73 – – 0.454 28.38
9.7.0 1.639 56.52 0.436 54.50 0.087 9.67
9.7.1 1.407 31.27 0.520 40.00 0.040 3.64
9.7.2 2.019 49.24 0.398 36.18 0.061 3.81
9.7.3 1.264 16.00 0.371 16.86 0.232 12.21
The functionality of the models was different
depending on the combination of independent
variables and species. For conifers, the prediction
of tree age was almost equal when using models,
f(h) or f(h, dcrm). Maximum crown diameter as
the independent variable seems not to be suitable
independent variable of its own. However, maxi-
mum crown diameter as the independent variable
was the best age model for birch. For Scots pine,
it seems that the models for ecoregion 3 were the
best although the test site is in area 2. It seems that
only height or both height and maximum crown
diameter as independent variables for conifers
can be used. Maximum crown diameter as the
only independent variable worked well for birch.
The test results by tree species are presented in
Table 13. When evaluating these results, it should
be noted that the test data for all models are the
same.
The average standard deviation of age when
maximum crown width alone was the independ-
ent variable was about 30 years (41% from mean
age). When tree height was the only independent
variable or both variables (h, dcrm) were included,
the standard deviation was about 27 years (37%
from mean age). For all models, the standard

242
Silva Fennica 39(2) research articles
Table 11. Statistical properties of regional age models. R2 is divided into cluster (Clus), plot (Plot) and tree (Tree)
effects. Proportion of total variance (VAR%) is calculated for clusters and plots. The first digit in number of
model refers to the geographic area (Fig. 1) in question (number of the area or 9 as an indication of the entire
country), the second digit the form of the model and the last digit the tree species.
Model No. of Predictor RMSE R2 VAR% VAR
model % Years Clus Plot Tree Clus Plot
Area 1
All 1.5.0 h 8.0 4.6 – 0.32 0.38 0 0.82 0.100
Pine 1.5.1 h 7.8 4.8 – 0.21 0.41 0 0.94 0.100
Spruce 1.5.2 h 7.9 4.5 – 0.36 0.46 0 0.88 0.098
Birch 1.5.3 h 6.2 2.8 – 0.66 0.49 0 0.94 0.053
All 1.6.0 dcrm 8.6 4.9 – 0.22 0.24 0 0.81 0.116
Pine 1.6.1 dcrm 8.3 5.1 – 0.10 0.26 0 0.94 0.114
Spruce 1.6.2 dcrm 8.3 4.7 – 0.32 0.26 0 0.85 0.108
Birch 1.6.3 dcrm 8.1 3.6 – 0.40 0.50 0 0.96 0.091
All 1.7.0 h, dcrm 7.9 4.5 – 0.33 0.41 0 0.83 0.098
Pine 1.7.1 h, dcrm 7.8 4.8 – 0.21 0.45 0 0.95 0.099
Spruce 1.7.2 h, dcrm 7.7 4.4 – 0.40 0.47 0 0.87 0.093
Birch 1.7.3 h, dcrm 6.1 2.8 – 0.67 0.48 0 0.93 0.052
Area 2
All 2.5.0 h 9.7 5.1 – 0.49 0.42 0.12 0.71 0.139
Pine 2.5.1 h 10.7 5.4 0.34 0.51 0.30 0.15 0.78 0.163
Spruce 2.5.2 h 7.3 4.3 – 0.54 0.53 0 0.82 0.085
Birch 2.5.3 h 9.4 4.2 – 0.57 0.39 0.29 0.59 0.121
All 2.6.0 dcrm 11.1 5.9 –0.63 0.28 0.26 0.07 0.76 0.184
Pine 2.6.1 dcrm 13.6 6.9 0.10 0.15 0.25 0.13 0.83 0.264
Spruce 2.6.2 dcrm 8.6 5.0 – 0.35 0.36 0 0.82 0.118
Birch 2.6.3 dcrm 10.0 4.5 – 0.48 0.26 0.24 0.63 0.138
All 2.7.0 h, dcrm 9.7 5.1 –0.88 0.49 0.42 0.11 0.71 0.139
Pine 2.7.1 h, dcrm 10.7 5.4 0.35 0.51 0.30 0.15 0.78 0.162
Spruce 2.7.2 h, dcrm 7.3 4.3 – 0.53 0.55 0 0.82 0.085
Birch 2.7.3 h, dcrm 9.2 4.1 – 0.60 0.42 0.31 0.57 0.117
Area 3
All 3.5.0 h 10.3 6.2 0.20 0.33 0.41 0.24 0.55 0.167
Pine 3.5.1 h 9.8 5.9 0.41 0.40 0.44 0.28 0.60 0.152
Spruce 3.5.2 h 8.7 6.5 –0.73 0.51 0.58 0.36 0.48 0.133
Birch 3.5.3 h 9.2 4.4 0.18 0.24 0.42 0.50 0.31 0.121
All 3.6.0 dcrm 11.5 7.0 0.04 0.14 0.25 0.23 0.56 0.211
Pine 3.6.1 dcrm 11.8 7.1 0.09 0.14 0.32 0.30 0.60 0.220
Spruce 3.6.2 dcrm 10.0 7.4 –0.47 0.20 0.38 0.24 0.59 0.175
Birch 3.6.3 dcrm 9.4 4.5 0.20 0.19 0.29 0.47 0.32 0.127
All 3.7.0 h, dcrm 10.3 6.2 0.20 0.33 0.42 0.24 0.56 0.167
Pine 3.7.1 h, dcrm 9.9 5.9 0.41 0.40 0.44 0.28 0.60 0.153
Spruce 3.7.2 h, dcrm 8.7 6.5 –0.74 0.50 0.59 0.36 0.49 0.135
Birch 3.7.3 h, dcrm 9.0 4.3 0.24 0.25 0.45 0.49 0.32 0.116
Area 4
All 4.5.0 h 9.5 7.8 0.05 0.44 0.50 0.55 0.21 0.165
Pine 4.5.1 h 9.1 7.2 0.29 0.49 0.51 0.61 0.24 0.150
Spruce 4.5.2 h 8.1 8.8 0.03 0.01 0.64 0.55 0.30 0.138
Birch 4.5.3 h 8.3 5.6 0.05 0.31 0.53 0.50 0.31 0.118
All 4.6.0 dcrm 10.4 8.5 0.02 0.21 0.29 0.47 0.24 0.199
Pine 4.6.1 dcrm 10.3 8.2 0.11 0.32 0.36 0.59 0.25 0.192
Spruce 4.6.2 dcrm 8.9 9.7 0.09 –0.31 0.27 0.42 0.32 0.168
Birch 4.6.3 dcrm 8.2 5.5 0.02 0.65 0.27 0.52 0.16 0.115
All 4.7.0 h, dcrm 9.4 7.7 0.05 0.44 0.51 0.55 0.21 0.164
Pine 4.7.1 h, dcrm 9.1 7.2 0.27 0.51 0.54 0.62 0.24 0.150
Spruce 4.7.2 h, dcrm 8.1 8.8 0.05 –0.04 0.64 0.54 0.31 0.139
Birch 4.7.3 h, dcrm 8.0 5.3 0.07 0.47 0.55 0.53 0.26 0.108

244
Silva Fennica 39(2) research articles
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Fig. 5. Averages and standard deviations for predicted
values of age = f(h) for area 2 with and without
information on trees species.
Table 13. Testing of the models for tree age using external data from the Hyytiälä Research Station. Bias refers
to the mean of differences between observed and predicted ages in absolute terms (years) and proportional
terms (%) per cent from mean age. S.E. refers to the standard deviation for the differences.
f(h) f(dcrm) f(h, dcrm)
n Bias S.E. Bias S.E. Bias S.E.
years (%) years (%) years (%) years (%) years (%) years (%)
Scots pine
Entire coutry 346 3(4) 29(34) 21(25) 35(41) 4(5) 29(34)
Area 1 346 11(13) 31(37) 20(24) 35(41) 11(13) 32(38)
Area 2 346 21(25) 31(37) 32(38) 35(41) 20(24) 30(36)
Area 3 346 –3(4) 28(33) 19(23) 35(41) –2(2) 28(33)
Area 4 346 –40(47) 24(28) –1(1) 35(41) –33(39) 26(31)
Norway spruce
Entire coutry 245 –9(12) 25(34) 3(4) 26(36) –5(7) 25(34)
Area 1 245 7(10) 25(34) 13(18) 27(37) 8(11) 25(34)
Area 2 245 5(7) 26(36) 14(19) 27(37) 6(8) 25(34)
Area 3 245 –25(34) 24(33) –10(14) 26(36) –25(34) 23(31)
Area 4 245 –89(122) 23(31) –51(70) 26(36) –89(122) 23(31
Birch
Entire coutry 120 –27(67) 20(49) –11(27) 20(49) –22(54) 19(47)
Area 1 120 –16(39) 20(49) –8(20) 20(49) –15(37) 20(49)
Area 2 120 –17(42) 20(49) –5(12) 20(49) –14(34) 20(49)
Area 3 120 –33(81) 20(49) –12(30) 20(49) –28(69) 19(47)
Area 4 120 –82(202) 21(52) –33(81) 21(52) –69(170) 20(49)
All tree species
Entire coutry 711 –5(7) 26(36) 9(12) 30(41) –3(4) 27(37)
Area 1 711 3(4) 27(37) 13(18) 30(41) 5(7) 27(37)
Area 2 711 9(12) 27(37) 18(25) 30(41) 10(14) 27(37)
Area 3 711 –13(18) 26(36) 6(8) 30(41) –11(15) 26(36)
Area 4 711 –63(86) 23(31) –18(25) 30(41) –57(78) 24(33)
deviation was smallest for Norway spruce and
largest for birch when evaluated in a relative unit
of measure (Table 13).
The influence of tree species was studied by
comparing models formulated for all tree species
with species-specific models using the test data
from area 2. The predictions of the latter models
differed only slightly from the former for Scots
pine, whereas the differences were considerable
for Norway spruce and birch (Fig. 5). The need
for ecoregions was tested using the combined
model in which the observations from all regions
were included as with diameter models. Test
results for model age = f(h) by tree species are
presented in Table 14. The differences between
the areas were mostly statistically significant for
the models age = f(h), age = f(dcrm) and age =
f(h, dcrm). Only a few combinations of model

245
Kalliovirta and Tokola Functions for Estimating Stem Diameter and Tree Age Using Tree Height, Crown Width and …
form and tree species formed exceptions on some
pairs of areas. There were only minor differences
between the trees species.
5 Discussion
The primary aim of the modelling was to develop
a part of the chain of models required for a new
inventory method based on measurements of tree
height and maximum crown diameter obtained
from high-resolution aerial photographs by digital
photogrammetry (Korpela 2000, 2004) combined
with information available from existing stand
databases and forest plans. The models could also
be utilized when airborne laser scanning data is
available. The idea is to predict the diameter at
breast height for a single tree by using information
derived from aerial photographs and forest plans,
which will in turn enable its volume to be calcu-
lated. This will mean that the volume of growing
stock for a sample plot can be derived from an
aerial photograph. Number of independent vari-
ables were tested during the study. For example,
the number of dominant trees per hectare could
be derived from remote sensing data, but it didn’t
improve the estimation results. According to the
tests, the best third variable in the models was
basal area. The coefficients of the determination
for models with three variables were only slightly
better than for those with two variables; thus the
benefit achieved with a third variable is negligible.
The effect of the third variable was minor also in
validation phase of study.
Models for predicting the diameter at breast
height for a single tree were formulated here
based on field data only. Traditionally, aerial
photography based volume models are constructed
using photogrammetric height and crown width
measurements for specific image material. How-
ever, the imaging condition and visibility of tree
Table 14. F-tests of the regional differences of age models: Age=f(h) by tree species.
Ecoregion pair Full model Reduced model n F-value
dfF SSEF MSEF dfR SSER MSER
Pine
Combined 5291 803.1368 0.151793 5300 1008.076 0.190203 5303 150.014*
Area1–Area2 1963 312.6647 0.159279 1966 319.9468 0.16274 1969 15.240*
Area1–Area3 2764 414.8875 0.150104 2767 416.0185 0.15035 2770 2.512
Area1–Area4 816 117.2568 0.143697 819 134.5371 0.16427 822 40.085*
Area2–Area3 4475 689.9734 0.154184 4478 789.9237 0.176401 4481 216.084*
Area2–Area4 2527 387.7959 0.153461 2530 569.5738 0.225128 2533 394.841*
Area3–Area4 3328 505.7562 0.15197 3331 552.7361 0.165937 3334 103.047*
Spruce
Combined 3649 372.1469 0.101986 3658 522.388 0.142807 3661 163.684*
Area1–Area2 2278 184.5613 0.081019 2281 185.2537 0.081216 2284 2.849*
Area1–Area3 1298 170.6416 0.131465 1301 183.8495 0.141314 1304 33.489*
Area1–Area4 275 33.6570 0.122389 278 75.4370 0.271356 281 113.790*
Area2–Area3 3374 338.3279 0.100275 3377 411.4739 0.121846 3380 243.152*
Area2–Area4 2351 199.0333 0.084659 2354 299.6171 0.12728 2357 396.035*
Area3–Area4 1371 184.9698 0.134916 1374 215.3127 0.156705 1377 74.967*
Birch
Combined 2270 270.6566 0.119232 2279 351.08 0.15405 2282 74.946*
Area1–Area2 904 106.2435 0.117526 907 106.6052 0.117536 910 1.026
Area1–Area3 1223 146.1228 0.119479 1226 148.8303 0.121395 1229 7.553*
Area1–Area4 215 22.8635 0.106342 218 37.6083 0.172515 221 46.218*
Area2–Area3 2055 248.281 0.120818 2058 276.6014 0.134403 2061 78.135*
Area2–Area4 1047 125.0935 0.119478 1050 195.2979 0.185998 1053 195.864*
Area3–Area4 1366 166.066 0.121571 1369 200.2231 0.146255 1372 93.655*
* Significant F-value.

246
Silva Fennica 39(2) research articles
dimensions differ according to the scale of pho-
tograph and the relative position of the tree in the
aerial photograph. When multiple photographs are
utilized, crown dimensions can be measured from
several sources, improving the process (Korpela
2004). Laser scanning is one of the most promis-
ing technologies in remote sensing-based forest
inventories. Stand mean tree height and crown
dimensions can be measured relatively accurately
from airborne laser scanning data (Hyyppä et al.
2001, Næsset 2004), but further estimation of tree
parameters is still required. Mainly these models
are planned to be utilised with tree specific pro-
cedures, although stand specific procedures could
utilise models to estimate mean size of trees.
When allometric tree models are created using
field measurements, like in this study, separate
calibration models can be used to relate remote
sensing-based measurements and ground meas-
urements with improved accuracy.
Because the data set used for modelling con-
tains random measurement errors, the estimated
coefficients are biased (Kangas 1998). The statis-
tical tests of the coefficients may also be invalid.
However, the coefficients, that are clearly signifi-
cant remain significant even when measurement
errors are taken into account. If the significance
is less clear, changes in significance may occur.
The effect of random measurement errors on the
models can be evaluated by using, for example,
the simulation extrapolation method (Carroll et al.
1995). Because no measurement error informa-
tion is available in the data set, the error effect
here is evaluated based on existing studies. The
standard error of height using a Suunto hypsom-
eter is, for instance, according to Päivinen (1992)
7.1 dm (3.4%) and Hyppönen and Roiko-Jokela
(1978) 8.0 dm (5.7%). No crown diameter meas-
urement error information is available for using
the Kajanus tube. If the error of height measure-
ment is assumed to be 5% and the error of crown
diameter measurement to 10%, both of which
are reasonable, it would be possible to estimate
the effect of the maximum error of diameter at
breast height.
The models for Norway spruce being the best
in terms of RMSE was somewhat unexpected,
as according to Ilvessalo (1950), the diameter at
breast height can be determined most accurately
for Scots pine, the predictions for Norway spruce
and birch being much weaker. Scots pines and
birches also grow on poor sites, especially on
the coast and in northern Finland, where Norway
spruce is not found, and seem to produce rather
abnormal stem forms there. This could explain the
superiority of the Norway spruce models.
The small, young trees (height < 3 m) are a
weak point in the models formulated here, and
prediction of their diameter at breast height is not
necessarily always reliable. On the other hand,
these small trees will not be a problem when using
the models in an inventory chain if only because
they tend to be obscured by the older growing
stock in aerial images. An inventory of sapling
stand is, of course another matter. The difference
in the case of small trees is obviously due to their
not having had to compete with adjacent trees
for growing space and light, so that the relations
between tree variables are slightly different from
those for a tree at a later stage of development
(Jakobsons 1970). Young trees should therefore
have models of their own. Damaged and diseased
trees were not included in the modelling. The
allometric characteristics do not work well with
broken or damaged trees, which mean that these
objects should be identified somehow from the
remote sensing material. The identification could
be based on exceptional allometric features or
spectral features in aerial photography.
The applicability and validity of the models was
tested with small data set collected from subarea.
The conclusion with regard to the modelling of
diameter at breast height was the same as that
reached by Talts (1977): that crown width is not
very reliable as the only independent variable.
For example, the heights of the Norway spruces
defined the diameter very well, although the
crown diameter was not such a particularly good
independent variable, at least partly on account
of the shaded character of spruces. Tree height
was better for this purpose, but it was only when
both were used that a reasonable prediction was
obtained. This also increased the flexibility of
the models, allowing them to take into account
the state of competition in the growing stock and
its density. Use of models that have at least tree
height and maximum crown diameter as inde-
pendent variables is therefore recommended. To
ensure reliability, a division of the country into
areas, i.e. regional models, should also be used.

247
Kalliovirta and Tokola Functions for Estimating Stem Diameter and Tree Age Using Tree Height, Crown Width and …
The test results of the models indicated the same.
The prediction of tree age proved to be challeng-
ing task. For all tree species, the standard devia-
tion of age prediction was large.
The age models were constructed because tree
age is an important criteria in defining need for
silvicultural treatment. It is important that age
estimated are also available in addition to tree size
and stand density estimation, when forest infor-
mation system is used for silvicultural planning.
For conifers, the age of the tree was dependent
most on its height, and for birch, the maximum
crown diameter was the most important inde-
pendent variable. Relative RMSE of age models
for entire country was about 10%. Precision of
models was improved significantly when ecore-
gion specific models were applied. Age prediction
for birch was especially difficult. According to the
tests, only maximum crown diameter should be
used as an independent variable.
Although it is technically possible to measure
crown width, crown projection area and crown
length on aerial photographs, only the proportion
of the crown which is visible can be measured,
and the actual maximum crown width can not
always be seen because of neighbouring trees.
The resolution and visibility of small branches
and irregular crown parameters are also dependent
on the scale of photograph. One important issue
is thus to examine the difficulties encountered in
measuring crowns in different stand structures
and under varying imaging conditions, involving
at least changes in sun-target angle, wind, film
and scanning quality. The final estimates can
also be affected by local topographical variation.
Thus, numerous factors can potentially cause
error in photogrammetric forest inventories. The
models might behave wrongly when those are
applied with unexpected combination of inde-
pendent variables. Still, the modelling data set is
covering entire area of Finland and measurement
of permanent sample plots of NFI are carefully
collected, which should ensure that most of exist-
ing variation of target area is modelled properly.
However, the models constructed here serve the
need to estimate tree characteristics from crown
dimensions from different remote sensing materi-
als and will reduce the need for fieldwork in single
tree-based forest inventory procedures.
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