Distance-independent tree basal area growth models for Norway spruce, Douglas-fir and Japanese larch in Southern Belgium

Abstract

This paper presents new harmonized distance-independent individual tree basal area growth models for Norway spruce, Douglas-fir and Japanese larch in pure even-aged stands in Southern Belgium. The selected model was originally developed for Norway spruce and Douglas-fir in neighboring France. New formulations are proposed for some of the model components in order to lower the number of fitted parameters and facilitate the fitting procedure. The resulting models integrate the most recent corresponding top-height growth models and use four simple and usually collected explanatory variables: stand age, top-height, total basal area and tree girth at breast height. The modified formulations maintain similar fitting performances and make it easier to interpret the influence of the explanatory variables on tree growth. Parameters estimates were fitted on thousands of growth measurements gathered from several monitoring plots, forest management inventories and silvicultural field experiments that represent the wide range of site conditions and of forest management scenarios applied to coniferous stands in Southern Belgium. Cross-validation of the models revealed no bias and highlighted their consistent behavior over the entire range of girth at breast height, age, top-height, site index and density represented in our dataset. Combining utility and robust performances, these models represent useful forest management tools, purposely ideal for forest simulation software development. Moreover, the flexibility and generic capabilities of the model formulation should make it easily adjustable for other species in even-aged stands.

Full text

ORIGINAL PAPER
Distance-independent tree basal area growth models for Norway
spruce, Douglas-fir and Japanese larch in Southern Belgium
Je
´ro
ˆme Perin
1
•Hugues Claessens
1
•Philippe Lejeune
1
•Yves Brostaux
2
•
Jacques He
´bert
1
Received: 4 August 2016 / Revised: 21 November 2016 / Accepted: 30 November 2016
ÓSpringer-Verlag Berlin Heidelberg 2016
Abstract This paper presents new harmonized distance-
independent individual tree basal area growth models for
Norway spruce, Douglas-fir and Japanese larch in pure
even-aged stands in Southern Belgium. The selected model
was originally developed for Norway spruce and Douglas-
fir in neighboring France. New formulations are proposed
for some of the model components in order to lower the
number of fitted parameters and facilitate the fitting pro-
cedure. The resulting models integrate the most recent
corresponding top-height growth models and use four
simple and usually collected explanatory variables: stand
age, top-height, total basal area and tree girth at breast
height. The modified formulations maintain similar fitting
performances and make it easier to interpret the influence
of the explanatory variables on tree growth. Parameters
estimates were fitted on thousands of growth measurements
gathered from several monitoring plots, forest management
inventories and silvicultural field experiments that repre-
sent the wide range of site conditions and of forest man-
agement scenarios applied to coniferous stands in Southern
Belgium. Cross-validation of the models revealed no bias
and highlighted their consistent behavior over the entire
range of girth at breast height, age, top-height, site index
and density represented in our dataset. Combining utility
and robust performances, these models represent useful
forest management tools, purposely ideal for forest simu-
lation software development. Moreover, the flexibility and
generic capabilities of the model formulation should make
it easily adjustable for other species in even-aged stands.
Keywords Tree growth modeling Softwood Belgium 
Picea abies Pseudotsuga menziesii Larix kaempferi
Introduction
Forest growth and yield modeling are used to analyze and
estimate the key relationship linking forest stand develop-
ment to various factors such as species composition, site
characteristics and silvicultural management. There are
various modeling approaches which are often classified
into two main groups: mechanistic or process-based models
which are based on presumed or observed mechanism and
attempt to explain the eco-physiological processes of forest
growth (Bossel 1991; Twery 2004), and empirical models
which are based on measured data and describe the relation
linking several stands or tree characteristics.
Empirical growth and yield models are commonly used
in forestry to predict forest growth and production (Johnsen
et al. 2001) and to compare the likely effects of various
management scenarios on the evolution of forest resources
(Peng 2000; Courbaud et al. 2001; Burkhart and Tome
´
2012). Successful practical applications of such models
range from simple growth and yield curves to more
advanced forest management simulation software. Yield
tables are probably the oldest and best known models used
in forestry science and forest management (Pretzsch 2009).
Communicated by Aaron R. Weiskittel.
&Je
´ro
ˆme Perin
j.perin@ulg.ac.be
1
Forest Resources Management (GRF), Department of
Biosystems Engineering (BIOSE), Gembloux Agro-Bio Tech
(GxABT), University of Liege (ULG), 2 Passage des
De
´porte
´s, 5030 Gembloux, Belgium
2
Applied Statistics, Computer Science and Mathematics
(SIMa), Gembloux Agro-Bio Tech (GxABT), University of
Liege (ULG), 2 Passage des De
´porte
´s, 5030 Gembloux,
Belgium
123
Eur J Forest Res
DOI 10.1007/s10342-016-1019-y

Early yield tables were based on inventory data and were
not able to reflect the effects of changing management
practices and environmental conditions and are therefore
no longer valid in many cases (Pretzsch 2009; Pretzsch
et al. 2014). Modern variable density yield tables and stand
density management diagrams rely increasingly on growth
and yield modeling to estimate the effect of variable
management practices and environmental conditions on
stand evolution (e.g., Longchang et al. 1991; Valbuena
et al. 2008; Vacchiano et al. 2013).
The ever-increasing use of computer technology has
made it easier to use more complex growth model to
improve the resolution scale and to take more explanatory
variables into account. Therefore, whole-stand modeling
approaches are now considered outdated and individual
tree-level modeling is the new standard (Weiskittel et al.
2011). These models are labeled distance-independent or
distance-dependent depending on whether or not they
include spatially explicit explanatory variables. Distance-
dependent models are very useful for research purpose:
They have a high potential for estimating the impact of
silvicultural treatment on individual tree growth and are
well suited for both homogenous and heterogeneous stands
structure and composition (e.g., Courbaud et al. 2001;
Porte
´and Bartelink 2002; Pretzsch et al. 2002). However,
distance-independent models are generally simpler and are
considered as more practically oriented (e.g., Monserud
and Sterba 1996; Andreassen and Tomter 2003; Deleuze
et al. 2004), and their performances are known to generally
only be slightly lower than distance-dependent ones in
even-aged pure stands (e.g., Vanclay 1994; Wimberly and
Bare 1996; Contreras et al. 2011).
Taking advantage of tree-level modeling, traditional
yield tables are progressively replaced by the simulation
software of silvicultural treatment (e.g., Pain 1997; Pret-
zsch et al. 2002; Pauwels et al. 2007; Dufour-Kowalski
et al. 2012) that allows simulation of customized silvicul-
tural management scenario at the tree level to accurately
estimate growth and production based on the forester
preferences. However, it is difficult, if not impossible, to
ensure that the validity limits of the models introduced in
the simulation software are always respected. Conse-
quently, growth models should be designed with a greater
focus on their structure to ensure that they exhibit a rele-
vant behavior not only inside, but also outside of their
validity area (Deleuze et al. 2004).
Although not native to Western Europe, Norway spruce
(Picea abies (L.) Karst) is the most important timber pro-
duction species in Southern Belgium (Alderweireld et al.
2015), where it is estimated that pure spruce stands account
for about 30% of the productive forest area (&140,000 ha)
and approximately 40% of the standing timber volume
(&46 million m
3
). However, for a variety of historical,
socioeconomic and ecological reasons (Claessens 2001),
the area devoted to this species declined steadily since the
early 1990s partly in favor of other softwood species such
as Douglas-fir (Pseudotsuga menziesii (Mirb.) Franco) and
Japanese larch (Larix kaempferi (Lam.) Carrie
`re). These
two species were brought in Belgium over one century ago
(Crahay 1900; Millard 1949), but large-scale plantations
only started after the middle of the twentieth century
(Claessens et al. 1996,2002). Their higher growth rate and
the technological quality of their wood make them inter-
esting alternatives to Norway spruce.
In Southern Belgium, the first yield tables and site index
curves fort these species were built by Dagnelie et al.
(1988) for Norway spruce, by Rondeux et al. (1991) and
Thibaut et al. (1995) for Douglas-fir and by Pauwels et al.
(2007) for larch. More recently, differences were observed
in Norway spruce and Douglas-fir stands between the field
data and the values estimated using these tools, especially
in stands aged 50 and over where the growth rate and the
level of production were significantly underestimated
(Perin et al. 2013). Therefore, the development of new
growth and yield models was required to update the
existing site index curves and yield tables.
In this context, we began the development of new har-
monized growth and yield models that will be integrated in
a simulation software to provide accurate tools for com-
parison and growth simulations in even-aged stands of
Norway spruce, Douglas-fir, and larch. New harmonized
top-height growth and site index models (Perin et al.
2013,2014) constituted the first step, the next being har-
monized girth increment models which are presented
thereafter. Our forest simulation software will primarily be
used to estimate the effect of various thinning regimes on
individual tree growth and stand structure and to predict the
evolution of existing stands using actual forest inventory
data. As a result, we require growth models that offer good
predictive performance while only using simple explana-
tory variables usually collected in forest inventory. We thus
favor a tree-level distance-independent empiric modeling
approach.
Material
The study concerns the Southern Belgium, especially at the
south of the river Meuse where most of the softwood
resource is located. The area is characterized by a subat-
lantic climate with an annual rainfall of 900–1300 mm,
well distributed along the year, and a mean annual tem-
perature from 7.7 to 9.6 °C. Coniferous stands cover a wide
range of sites, but are most commonly planted on well-
drained oligotrophic brown soils, except for spruce that is
also planted on more humid soils. All the sampled plots
Eur J Forest Res
123

(Fig. 1) are located in even-aged and pure stands (Norway
spruce, Douglas-fir or larch account for more than 90% of
the stand basal area). The past silvicultural treatments
applied to sampled stands are not always known, but
coniferous silviculture in Belgium is widely based on even-
aged stands established by planting high density
(1500–3000 stems/ha) of 3- to 4-year-old planting stocks
(Alderweireld et al. 2015). Coniferous stands are normally
thinned every 5–10 years after reaching 13–20 m of
dominant height, and unthinned stands are known to be
fairly uncommon in Southern Belgium. Clearcutting is then
usually applied shortly after those stands reach a top-height
of, respectively, 30 m for Norway spruce and larch stands
and 40 m for Douglas-fir.
Thousands of growth measurements were gathered from
several monitoring plots, forest management inventories
and silvicultural field experiments in pure even-aged stands
of Norway spruce, Douglas-fir or larch. This includes
several silvicultural field experiments installed during the
1970s in Norway spruce stands and monitored for close to
three decades (He
´bert et al. 2002) and an extensive plot
network installed and monitored during the 1990s in larch
stands (Pauwels et al. 2007). As a result, the sampled plots’
design is rather heterogeneous: round and rectangular plot
shapes, sizes ranging from 50 to 2400 square meters and
monitoring durations varying from 3 to 27 years. We then
selected all individual tree girth growth segments of
3–6 years measured over bark in stands of known age
where top-height (Hdom), density (Nha) and total basal
area (Gha) were measured. Following data quality control,
the recovered dataset is composed of 51,159 growth seg-
ments measured on 18,135 trees in 537 plots monitored
between 1979 and 2013. Norway spruce accounts for
33,931 of these data (7220 trees in 181 plots), larch for
14,382 (8988 trees in 289 plots) and Douglas-fir for 2846
(1927 trees in 67 plots).
These data represent all site conditions and the wide
range of forest management scenarios applied to even-aged
Norway spruce, Douglas-fir and larch stands of Southern
Belgium, as well as more unusual silvicultural management
scenario tested in several silvicultural field experiments
(Table 1). Site index (SI, top-height at 50 years since
Fig. 1 Repartition of the sampled stands in the study area. Norway spruce stands are represented by round dots, Douglas-fir by triangular dots
and larch by star dots
Eur J Forest Res
123

planting) was evaluated for every sampled stands by using
the online tool ‘‘H50’’ v1.1 (Perin and De Thier 2014). The
age, top-height and density range of the sampled stands are
representative of what is mostly encountered in coniferous
stands between the first thinning and the clearcutting.
Method
We decided to evaluate the tree-level distance-independent
growth model of Deleuze et al. (2004) on our data because
of its interesting formulation. This model has also already
been tested in similar site conditions (Northeastern France)
for Norway spruce and Douglas-fir stands subject to silvi-
cultural management practices comparable to those
encountered in Southern Belgium. It is a nonlinear hyper-
bolic model that describes annual tree basal area increment
(Ig
i
) as a function of initial circumference (C
i
):
Igi¼0:5PðCimA
þffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
mAþCi
ðÞ
24ACi
qÞð1Þ
The three parameters A,Pand mcan all be expressed as
functions of the stands characteristics, and each affects the
model in very different ways (Fig. 2). The parameter mhas
to be greater or equal to 1 and affects the general shape and
the flexibility of the model. If mis equal to 1, the model is
the segmented linear function of Dho
ˆte (1991) where Ig
i
is
null from C
i
=0 to the threshold Aand then increases with
a slope equal to P. Greater values of mgive more flexibility
to the model around the threshold A, essentially leading to
a more gradual increase in tree basal area increment in
relation to increasing circumference.
Deleuze et al. (2004) proposed the following formula-
tion for parameters Aand P:
A¼ðAa þAb HdomÞð1þAc exp( a
Gha=Hdom)Þð2Þ
P¼ðPa þPb dHdomÞð1þPc expða
Gha=HdomÞÞ ð3Þ
where Gha is the initial stand basal area per hectare in m
2
/
ha, Hdom is the initial stand top-height in meters and
dHdom is the annual top-height increment during the
growth interval in meter per year. As final top-height was
not always measured in the recovered inventory data, the
annual top-height increment is not consistently available.
Thus, we will always use an estimated value for this
variable, calculated with the corresponding top-height
growth model fitted on stem analysis data by Perin et al.
(2013,2014), whose formulation and parameters values are
presented in Eq. (4) and Table 2.
Hdom ¼aðage ageMÞþ HdomM
1exp ageM
c

r
!"#
1exp age
c
hi
r
ð4Þ
where Hdom–age is the predicted height–age couple and
Hdom
M
–age
M
is the measured height–age couple.
First, we tested the parameterized formulations of
Deleuze et al. (2004) for Norway spruce and Douglas-fir
directly on our data by comparing predicted and observed
values to check its accuracy and applicability for South-
ern Belgium data. We then adapted the model to our data
using the nls() nonlinear regression procedure in R (R
Core Team 2012). Several new formulations for param-
eters A,Pand mwere also tested in order to try to
simplify the model while improving its performances on
our dataset.
Table 1 Main attributes of the selected permanent sample plots installed in even-aged pure stands of Norway spruce (181), Douglas-fir (67) and
larch (289)
Norway spruce (Picea abies) Douglas-fir (Pseudotsuga menziesii) Larch (Larix kaempferi)
Min Mean Max SD Min Mean Max SD Min Mean Max SD
Area (m
2
) 50 448 707 128 200 585 1500 346 100 626 2400 409
Age (years) 19 43 92 13 14 39 93 17 9 35 91 13
Hdom (m) 11.1 21.2 34.7 4.3 9.9 26.8 48.9 8.4 8.0 22.2 34.4 5.7
SI (m) 13.1 25.4 32.7 2.3 19.3 36.0 44.7 4.3 20.8 28.4 37.5 2.6
Nha (N/ha) 141 1226 6020 807 60 614 2250 463 98 729 3700 558
Gha (m
2
/ha) 6.6 35.3 61.5 10.3 18.2 36.2 59.6 9.1 9.3 26.4 55.5 7.3
Cg (cm) 26.3 67.3 150.3 20.9 39.1 106.4 265.8 47.6 28.4 80.1 170.8 26.1
Elevation (m) 158 425 654 105 154 318 550 103 36 361 606 117
Slope (°) 0 6 23 4.6 1 6 21 4.8 0 6 29 5.1
Area =sample plot area; Age =total age of the stands since planting; Hdom =top-height; SI =site index; Nha =number of trees per hectare;
Gha =total basal area per hectare; Cg =quadratic mean girth at breast height
Eur J Forest Res
123

The recovered data were quite heterogeneous as they
were obtained from different sources that used different
sampling and measurement methodology. In particular, the
number of trees monitored in each sampled stands ranged
from 10 to 300. This needed to be addressed to avoid that
some sampled stands outweigh too much the others during
the parameterization of A,Pand mformulations. We
applied a simple weighting method to ensure that each
sampled stands contribute to the fitting of the model in
accordance with the length of the period it was monitored
rather than the number of growth segments which were
measured: Each of the growth segment data was weighted
by its length (in years) divided by the number of growth
segments measured at the same time in the permanent
sample plots (PSP). In this way, every sampled plot has a
total weight equal to the total duration of its monitoring.
To take heteroscedasticity into account, Deleuze et al.
(2004) weighted observations by 1/C
i
2
, but this does not
solve the issue for our dataset. We instead choose to fit the
model on girth increment value (Ic
i
) rather than tree basal
area increment data (Ig
i
obtained from Eq. 1):
Ici¼C2
iþ4pIgi

0:5Cið5Þ
We then validated the model by using a K-fold cross-val-
idation procedure (Kohavi 1995). For each species, the
sampled stands were sorted according to their size in the
dataset (number of growth segments) and then spread in
four independent and roughly equal-sized parts. The model
was then fitted on three parts (training dataset) and then
applied on the remaining part (validation dataset); this was
done four times to ensure that every part was used as
validation dataset. This way, the validation mean error and
validation root-mean-square error can be used as the
Fig. 2 Influence of the variation of one of the parameters A,Por
mvalue on the tree growth model shape, other being constant:
avariation of A(50–150); bvariation of P(0.5–1.5); cvariation of
m(1–1.1). Upper panels represent tree basal area increment (cm
2
/
year); lower ones represent tree girth increment (cm/year)
Table 2 Corresponding parameters values of the top-height growth
models fitted by Perin et al. (2014) for Southern Belgium Norway
spruce, Douglas-fir and larch pure even-aged stands
Norway
spruce (Picea
abies)
Douglas-fir
(Pseudotsuga
menziesii)
Larch
(Larix
kaempferi)
a0.1299 0.2418 0.1449
c22.3659 31.1379 14.9904
r2.0464 1.4668 1.6818
Eur J Forest Res
123

estimator of the predictive performance of the model when
applied on an independent dataset.
All confidence intervals presented thereafter are calcu-
lated with a significance level of a=0.05 (95% confidence
level) unless stated otherwise.
Results
At first glance (Table 3), the original model and the fitted
parameters values proposed by Deleuze et al. (2004)
seemed to perform quite well on our data with an RMSE on
the annual tree girth increment estimation of 0.60 cm/year
for Norway spruce and 1.00 cm/year for Douglas-fir.
Nevertheless, a bias was observed: On average, tree girth
increment is overestimated by 0.18 cm/year for Norway
spruce and by 0.23 cm/year for Douglas-fir. Moreover, we
observed significant correlation between the residuals val-
ues and the girth of the measured trees (Fig. 3): Increments
are generally rightly estimated for girth under 90 cm (dbh
under 30 cm), but are noticeably overestimated for bigger
trees.
We then parameterized the original model formulation
on our dataset (Table 4), but were unable to meet the
convergence criteria for Douglas-fir and Japanese larch
(step-size factor was reduced below minFactor). This type
of nonconvergence issue can often be explained by the use
of an overparameterized model, and the correlation matrix
of the parameters showed that the formulation for Aand
Pwas characterized by highly correlated parameters. Fur-
ther test also highlighted that the suppression of either
parameter Aa,Ac or Pc greatly facilitated convergence
while having little impact on the fitting performance of the
models for all three species. Thus, it seemed relevant to
seek for a simpler formulation of Aand P.
We determined that the model could be significantly
simplified while keeping most of its fitting performances
by transferring the expression of the stand density effect
from the threshold and slope parameters (Aand P) to the
shape parameter (m). In this way, we found that the
threshold parameter A could be expressed as a simple
power function of dominant height and the slope parameter
Pas a simple linear function of annual top-height growth.
Shape parameter mwas best expressed as an exponential
function of top-height and total basal area that ensure that
its value is always greater than 1. The proposed formula-
tions for parameters A,Pand mare presented below:
A¼Aa HdomAb ð6Þ
P¼Pa þPb dHdom ð7Þ
m¼1þexp ma Hdom mb GhaðÞð8Þ
These formulations reduce the total number of parameters
that need to be fitted to six instead of eight in the original
formulation. Moreover, the formulations for the threshold
and slope parameters (Aand P) are simpler and no longer
share the same explanatory variable.
The fitting of the new formulations easily met the con-
vergence criteria for all three species and provided
parameters estimates that were at least significant to the
0.05 level (Table 5). Compared to the original, the new
formulation led to a small increase in the validation RMSE
on annual tree girth increment estimation for Norway
spruce (?1.1%) and larch (?2.1%) and to a minor decrease
for Douglas-fir (-0.5%). Further graphical analysis of the
training and validation errors distributions according to
each explanatory variable showed that the fitted models
were unbiased and characterized by robust performance
over the entire range of girth at breast height, age, top-
height, site index and density represented in our dataset. As
an example and for comparison with Fig. 3, the distribution
of validation errors according to girth at breast height is
represented in Fig. 4.
Discussion
The dataset representativity is always the most critical
component in empirical growth modeling as it is essential
that it covers all the conditions for which the model will
have to be valid. We worked in collaboration with other
researchers and the Nature and Forest Department of
Wallonia (DNF) to collect measured data from all known
observation networks, field experiments and forest inven-
tories relevant to our study and representative of the wide
diversity of conditions (sites, age, density, etc.)
Table 3 Evaluation on our dataset of the prediction errors of the
models parameterized by Deleuze et al. (2004) for Norway spruce and
Douglas-fir: root-mean-square error (RMSE) and mean error (ME) for
the annual tree girth increment estimation (cm/year) and the annual
tree basal area increment estimation (cm
2
/year)
Norway spruce (Picea abies) Douglas-fir (Pseudotsuga menziesii)
Ic (cm/year) Ig (cm
2
/year) Ic (cm/year) Ig (cm
2
/year)
Prediction RMSE 0.600 8.527 0.997 26.648
Prediction ME 0.183 2.696 0.231 8.389
Eur J Forest Res
123

encountered in coniferous stands of Southern Belgium. The
site index estimated in the sampled stands ranges from 13
to 32.5 for Norway spruce, 19.5–44.5 for Douglas-fir and
21–37.5 for larch. These amplitudes are generally wider
than those previously estimated by Perin et al. (2014)
thanks to data recovered from some unusually low-pro-
ductivity stands located in site conditions where these
species are no longer planted (e.g., peatland and humid
clay soils). We also recovered data from several silvicul-
tural field experiments where a wide variety of planting
spacing and thinning intensity were tested (e.g., He
´bert
et al. 2002; Pauwels et al. 2007). Thus, density and total
basal area variability are also greater in our dataset than
what is found in most Southern Belgium young and mature
coniferous stands managed for timber production. The
number of sampled stands, trees and measured growth
segments in our dataset is consistent with what is typically
used in this type of research (e.g., Monserud and Sterba
1996; Andreassen and Tomter 2003; Monty et al. 2007).
To fit on our dataset, we selected a model which had
recently been proven adequate for Norway spruce and
Douglas-fir in neighboring France (Deleuze et al. 2004)in
sites and management conditions similar to those encoun-
tered in our study area, for example in the wooded plateau
of the Ardennes which covers both part of Southern Bel-
gium and Northern France. However, further test showed
that the original parameterized models were not directly
applicable in Southern Belgium as they lead to significantly
overestimate Norway spruce and Douglas-fir growth when
applied on our dataset (Fig. 3). It is possible that French
Fig. 3 Distribution of the
residuals (predicted—measured)
on the annual girth increment
estimation (in cm/years)
obtained by applying the
parameterized models of
Deleuze et al. (2004) for
Norway spruce and Douglas-fir
on our dataset
Table 4 Parameter values and fitting statistics of the original model
formulation after parameterization on our data for Norway spruce,
Douglas-fir and larch; the confidence interval (1-a=95%) is
presented in italics next to each parameter value, and nonsignificant
parameters values are followed by a hash symbol (
#
); Akaike’s
Information Criterion (AIC), training and cross-validation values of
the root-mean-square error (RMSE), adjusted R
2
and mean error (ME)
are also presented for the annual tree girth increment estimation (cm/
year) and between bracket for the annual tree basal area increment
estimation (cm
2
/year)
Norway spruce (Picea abies) Douglas-fir (Pseudotsuga menziesii) Larch (Larix kaempferi)
Aa -2.7140 ±0.8053 -25.6905 ±36.2994
#
-41.3189 ±15.9502
Ab 2.2520 ±0.0787 5.5801 ±7.4928
#
8.0782 ±2.9599
Ac -0.8097 ±0.09 -0.6923 ±0.3444 -0.8739 ±0.0464
Pa 0.1685 ±0.0094 0.3124 ±0.1718 0.0306 ±0.1203
#
Pb 0.5931 ±0.0298 0.8482 ±0.4304 0.0866 ±0.3434
#
Pc 1.0827 ±0.1938 0.0521 ±0.7005
#
11.0773 ±47.6943
#
a1.0228 ±0.0892 0.2306 ±0.6124
#
0.2216 ±0.1103
m1.0077 ±0.002 1.0287 ±0.0121 1.0361 ±0.0067
AIC 4 737 673 837 15 278 967 480 512 653
Training RMSE 0.530 (7.141) 0.757 (16.417) 0.608 (8.918)
Adjusted R
2
0.620 (0.691) 0.505 (0.726) 0.630 (0.677)
Validation RMSE 0.539 (7.250) 0.775 (16.810) 0.614 (8.997)
Validation ME -0.005 (-0.117) 0.017 (0.089) 0.000 (-0.134)
Eur J Forest Res
123

coniferous stands are generally more productive than their
Belgian counterpart, but these divergences are more likely
related to dataset and methodological differences. In par-
ticular, these models were fitted on data from experimental
permanent plots of the AFOCEL’s network which,
according to a report of Gastine et al. (2003), were almost
all installed in stands under the age of 45 years at the time
of the study. The parameters estimates fitted by Deleuze
et al. (2004) are thus probably less valid for coniferous
stands over the age of 45 years which are quite common in
Southern Belgium (Alderweireld et al. 2015) and represent
45 and 20% of the sampled Norway spruce and Douglas-fir
stands in our dataset.
Deleuze et al.’s general model formulation Eq. (1)is
interesting as it allows to distribute the effect of the stand-
level explanatory variables in three different parameters (A,
Pand m) that influence the shape of the relation between
individual girth and tree basal area increment in very
contrasted way (Fig. 2), making this model very flexible
while always ensuring a biologically plausible constrained
Table 5 Parameters values and fitting statistics of the proposed six
parameters model formulation after parameterization on our data for
Norway spruce, Douglas-fir and larch; the confidence interval
(1-a=95%) is presented in italics next to each parameter value;
Akaike’s Information Criterion (AIC), training and cross-validation
values of the root-mean-square error (RMSE), adjusted R
2
and mean
error (ME) are also presented for the annual tree girth increment
estimation (cm/year) and for the annual tree basal area increment
estimation (cm
2
/year, between parenthesis)
Norway spruce (Picea abies) Douglas-fir (Pseudotsuga menziesii) Larch (Larix kaempferi)
Aa 3.9825 ±0.3010 1.9987 ±0.4563 4.7358 ±0.5610
Ab 0.7802 ±0.0239 1.0544 ±0.0619 0.8241 ±0.0361
Pa 0.2160 ±0.0089 0.3725 ±0.0595 0.3435 ±0.0182
Pb 0.8014 ±0.0258 0.8500 ±0.1103 1.0514 ±0.0428
ma 0.1345 ±0.0107 0.0415 ±0.0256 0.0522 ±0.0081
mb 0.1853 ±0.0081 0.1225 ±0.0230 0.1328 ±0.0067
AIC 4 737 674 321 15 278 788 480 504 651
Training RMSE 0.537 (7.173) 0.758 (16.334) 0.622 (9.065)
Adjusted R
2
0.607 (0.685) 0.499 (0.729) 0.612 (0.666)
Validation RMSE 0.545 (7.284) 0.771 (16.619) 0.627 (9.125)
Validation ME -0.005 (-0.141) 0.009 (-0.067) -0.001 (-0.145)
Fig. 4 Distribution of the
validation residuals (predicted–
measured) on the annual girth
increment estimation (in cm/
years) obtained for Norway
spruce, Douglas-fir and larch
with our new formulation
Eur J Forest Res
123

form. In addition, individual girth being the only tree-level
variable makes it really easy to represent and understand
the influence of the explanatory variables on growth esti-
mates, to evaluate the model robustness and finally to
integrate it in a simulation software.
The parameter A determines the girth value below
which tree growth is near zero and therefore represents a
threshold below which trees could be considered as heavily
suppressed. Our analysis showed that, consistent with the
conclusion of Deleuze et al. (2004), this parameter was best
expressed using variable related to the stand development
stage (top-height, age, mean dbh, etc.), and the most suit-
able formulation appeared to be a power function of top-
height. Combining height or top-height with tree girth
value to identify suppressed trees can be linked to the tree
height to diameter ratio (H/D) which is an already well-
known indicator of individual tree stability (e.g., Bruchert
et al. 2000; Wonn and O’Hara 2001; Slodicak and Novak
2006) and of crown dimension (Dyer and Burkhardt 1987;
Hasenauer and Monserud 1996) and has already been
investigated as a potential competition index (Opio et al.
2000; Bachofen and Zingg 2001). Therefore, dividing tree
girth by the corresponding A parameter, calculated value
could provide an interesting indicator which would be
inversely proportional to the past cumulative level of
competition experienced by the tree and proportional to its
present potential for utilizing growing space. In particular,
estimates of this ratio close to (or less than) unity would
probably indicate highly unstable trees with small crown
ratio and low potential vigor that were heavily suppressed
during a significant part of their lifetime.
The parameter Phas a simple multiplicative effect on the
increment estimation that is independent of the dominance
status of the trees. In accordance with Deleuze et al. (2004),
we obtained excellent results by expressing this parameter as
a simple linear function of the estimated top-height annual
growth (dHdom). In our data, dHdom was always calculated
using the corresponding top-height growth models (Perin
et al. 2013,2014) which are nonlinear function of age, site
index and species. Thus, the parameter Pvalue is propor-
tional to the estimated site index and increases to a maximum
around the age of 10 years for larch and 20 years for Norway
spruce and Douglas-fir, slowly decreasing thereafter. In
practice, top-height annual growth also depends on annual
climate variability, and thus, parameter Pwould probably be
the most suitable to integrate weather variables in its
formulation.
The parameter mdetermines the model flexibility
around the threshold A, and its value has an important
positive influence on the growth estimation of trees with a
girth at breast height inferior or close to A, but almost none
for bigger ones (Fig. 2). We identified mas the ideal
parameter to express the effect of density on growth. With
the proposed formulation (Eq. 8), mconverges to 1 in
denser stands, bringing the model closer to a segmented
shape that ensures very low basal area growth estimation
value for small suppressed trees which suffer from the
intense competition for resources. This is consistent with
the fact that dominant trees are less affected by competition
than dominated ones (Schu
¨tz et al. 2015). The ‘‘competi-
tion effect’’ simulated by the parameter mis also inversely
proportional to the dominant height which indicates that a
given total basal area value accounts for a higher level of
competition in younger forests stands than in older ones.
Thereby, the stand density variable (Gha) was trans-
ferred into the formulation of the parameter m, allowing us
to greatly simplify the formulations of Aand Pso that they
no longer share the same explanatory variables (Hdom and
Gha). This allowed us to reduce the number of fitted
parameters to six by species (instead of eight in the original
formulation) and to significantly facilitate the fitting pro-
cess convergence. Unlike the original formulation of
Deleuze et al. (2004), this new formulation converged
easily without requiring an initial estimation of the starting
values for the model parameters (or a self-starting func-
tion): Using a starting value of 1 for each parameter proved
to be perfectly appropriate on our dataset. We thus consider
that the benefits of this new formulation more than offset
the negligible loss of flexibility and precision.
The calculated adjusted R
2
shows that our parameterized
models explain 61, 50 and 61% of the annual girth incre-
ment variance and 68, 73 and 67% of the annual basal area
increment variance for Norway spruce, Douglas-fir and
larch. Their level of performance (Table 5) seems rela-
tively good (e.g., Monserud and Sterba 1996; Andreassen
and Tomter 2003; Monty et al. 2007; Pauwels et al. 2007)
and is probably close to the maximum that is possible to
obtain with a distance-independent growth model that does
not take climate annual variability into account. Thus, this
model represents an interesting compromise between per-
formance and utility as it only uses simple explanatory
variables (age, Hdom, Gha, Ci) that are usually collected in
forest management inventories. This greatly facilitates the
collection of an appropriate training dataset and allows a
direct application of the parameterized model on actual
forest inventory data in order to predict the growth of
existing trees.
A theoretical application of these new growth models is
presented in Fig. 5: Individual tree girth growth curves
were built for each species in order to compare the
development of dominant and suppressed trees in high- and
low-density stands. We used values for the explanatory
variables (Table 6) that are consistent with what can be
encountered in Norway spruce, Douglas-fir and Japanese
larch stands of average site index from Southern Belgium
(Dagnelie et al. 1988; Rondeux et al. 1991; Pauwels et al.
Eur J Forest Res
123

2007; Perin et al. 2014). The initial age was selected to
match a dominant height of 14 m because it is consistent
with the model validity range and usually corresponds to
the first thinning stage in Southern Belgium coniferous
stands (Alderweireld et al. 2015). The initial tree girth
corresponds to a dominant tree juvenile growth of 2.5 cm/
years for Norway spruce and 3.5 cm/years for Douglas-fir
and Japanese larch and to 60% of those values for sup-
pressed trees. In those examples, the stand total basal area
(density) is initially fixed for low- and high-density stands
at, respectively, 25 and 35 m
2
/ha and then increases by
0.25 m
2
/ha each year.
These growth curves (Fig. 5) show that Douglas-fir is
characterized by a substantially faster growth rate
(&?50%) than that of Norway spruce and Japanese larch
which is consistent with the general knowledge about these
species in Southern Belgium. It also highlights that density
has a much greater effect on the growth rate of suppressed
trees than that of dominant ones which is an important
feature of this model that was already discussed before.
The differences between the mean annual girth increment
estimated in low- and high-density stands are equal to 8, 10
and 18% for dominant trees compared to 73, 65 and 74%
for suppressed trees, respectively, for Norway spruce,
Fig. 5 Simulation with the new growth models of the evolution of the individual girth at breast height of dominant and suppressed trees of
Norway spruce, Douglas-fir and Japanese larch in high- and low-density forest stands
Table 6 Parameters of the growth simulation presented in Fig. 5
Norway spruce
(Picea abies)
Douglas-fir
(Pseudotsuga menziesii)
Larch (Larix
kaempferi)
Site index (top-height at age 50) (m) 27 36 27
Initial age (14 m top-height) (year) 23 18 17
Initial tree girth (dominant) (cm) 57.5 63 59.5
Initial tree girth (Suppressed) (cm) 34.5 37.8 35.7
Initial stand density (high) (m
2
/ha) 35 35 35
Initial stand density (low) (m
2
/ha) 25 25 25
Annual evolution of stand density (m
2
/ha/year) ?0.25 ?0.25 ?0.25
Eur J Forest Res
123

Douglas-fir and larch. It also indicates that Japanese larch
growth rate appears to be particularly sensitive to stand
density variation as already highlighted by Pauwels et al.
(2007) who recommended lower stand density for larch
than for Norway spruce and Douglas-fir.
Conclusion and perspectives
The new girth growth parameterized models presented in
this study allow for unbiased increment prediction over the
entire range of girth at breast height, age, top-height, site
index and density encountered in most monospecific Nor-
way spruce, Douglas-fir and larch stands managed for
timber production in Southern Belgium.
The model formulation is purposely ideal for forest
simulation software as the explanatory variables needed are
simple and usually collected in forest inventory: individual
girth at breast height, total basal area, top-height and
estimated annual top-height growth (calculated from the
top-height and the age of the stand by using the corre-
sponding site index model).
Future uses of these models concern the development of
a forest simulation software to help optimize silvicultural
management techniques and to predict medium- and long-
term changes in softwood forest resources of Wallonia. We
also intend to further test the generic capabilities of this
model formulation with other species in regular stands,
especially with hardwood.
Acknowledgements We appreciate the support provided by the
Walloon Region—DGRNE through the ‘‘Accord-Cadre/Forest
Research’’ project. We would like to address a special thanks to Ir.
Hugues Lecomte from the Permanent Regional Forest Inventory of
Wallonia (IPRFW) as well as to Dr. Quentin Ponette and Dr. Jose
´
Genon of the Earth and Life Institute of the Catholic University of
Louvain (UCL—ELIE) who provided some of the data used in this
study.
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