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Nonlinear versus linearised model on stand density model fitting and stand density index calculation: analysis of coefficients estimation via simulation

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  • Italian National Research Council - Institute of Biosciences and BioResources (IBBR) - Florence division

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The stand density index, one of the most important metrics for managing site occupancy, is generally calculated from empirical data by means of a coefficient derived from the “self-thinning rule” or stand density model. I undertook an exploratory analysis of model fitting based on simulated data. I discuss the use of the logarithmic transformation (i.e., linearisation) of the relationship between the total number of trees per hectare (N) and the quadratic mean diameter of the stand (QMD). I compare the classic method used by Reineke (J Agric Res 46:627–638, 1933), i.e., linear OLS model fitting after logarithmic transformation of data, with the “pure” power-law model, which represents the native mathematical structure of this relationship. I evaluated the results according to the correlation between N and QMD. Linear OLS and nonlinear fitting agreed in the estimation of coefficients only for highly correlated (between − 1 and − 0.85) or poorly correlated (> − 0.39) datasets. At average correlation values (i.e., between − 0.75 and − 0.4), it is probable that for practical applications, the differences were relevant, especially concerning the key coefficient for Reineke’s stand density index calculation. This introduced a non-negligible bias in SDI calculation. The linearised log–log model always estimated a lower slope term than did the exponent of the nonlinear function except at the extremes of the correlation range. While the logarithmic transformation is mathematically correct and always equivalent to a nonlinear model in case of prediction of the dependent variable, the difference detected in my studies between the two methods (i.e., coefficient estimation) was directly related to the correlation between N and QMD in each simulated/disturbed dataset. In general, given the power law as the “natural” structure of the N versus QMD relationship, the nonlinear model is preferred, with a logarithmic transformation used only in the case of violation of parametric assumptions (e.g. data distributed non-normally).
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ORIGINAL PAPER
Nonlinear versus linearised model on stand density model fitting
and stand density index calculation: analysis of coefficients
estimation via simulation
Maurizio Marchi
1,2
Received: 20 August 2018 / Accepted: 25 January 2019 / Published online: 4 May 2019
The Author(s) 2019
Abstract The stand density index, one of the most
important metrics for managing site occupancy, is gener-
ally calculated from empirical data by means of a coeffi-
cient derived from the ‘‘self-thinning rule’’ or stand density
model. I undertook an exploratory analysis of model fitting
based on simulated data. I discuss the use of the logarith-
mic transformation (i.e., linearisation) of the relationship
between the total number of trees per hectare (N) and the
quadratic mean diameter of the stand (QMD). I compare
the classic method used by Reineke (J Agric Res
46:627–638, 1933), i.e., linear OLS model fitting after
logarithmic transformation of data, with the ‘‘pure’’ power-
law model, which represents the native mathematical
structure of this relationship. I evaluated the results
according to the correlation between N and QMD. Linear
OLS and nonlinear fitting agreed in the estimation of
coefficients only for highly correlated (between -1 and
-0.85) or poorly correlated ([-0.39) datasets. At
average correlation values (i.e., between -0.75 and
-0.4), it is probable that for practical applications, the
differences were relevant, especially concerning the key
coefficient for Reineke’s stand density index calculation.
This introduced a non-negligible bias in SDI calculation.
The linearised log–log model always estimated a lower
slope term than did the exponent of the nonlinear function
except at the extremes of the correlation range. While the
logarithmic transformation is mathematically correct and
always equivalent to a nonlinear model in case of predic-
tion of the dependent variable, the difference detected in
my studies between the two methods (i.e., coefficient
estimation) was directly related to the correlation between
N and QMD in each simulated/disturbed dataset. In gen-
eral, given the power law as the ‘‘natural’’ structure of the
N versus QMD relationship, the nonlinear model is pre-
ferred, with a logarithmic transformation used only in the
case of violation of parametric assumptions (e.g. data dis-
tributed non-normally).
Keywords Ordinary least squares Power law Reineke
function Silviculture Ecological mathematics Forest
mathematics
Introduction
The maximum degree of competition that forest popula-
tions of a species can sustain is described by the principle
of self-thinning (Yoda et al. 1963; Westoby 1984). This
rule, a power function linearised by a logarithmic trans-
formation, states that environmental resources can satisfy
only a limited number of trees within a stand, and this
number grows progressively smaller as tree age and size
Project funding: The project was fully funded by the Italian National
Rural Network project, in the framework of the European Network for
Rural Development (ENRD) and also by EU, in the framework of the
Horizon 2020 B4EST project ‘Adaptive BREEDING for productive,
sustainable and resilient FORESTs under climate change’’, UE Grant
Agreement 773383 (http://b4est.eu/).
The online version is available at http://www.springerlink.com
Corresponding editor: Zhu Hong.
&Maurizio Marchi
maurizio.marchi85@gmail.com
1
CREA - Research Centre for Forestry and Wood, Arezzo,
Italy
2
geoLAB - Laboratory of Forest Geomatics, Department of
Agriculture, Food, Environment and Forestry, Universita
`
degli Studi di Firenze, Florence, Italy
123
J. For. Res. (2019) 30(5):1595–1602
https://doi.org/10.1007/s11676-019-00967-0
increase. The proximity of a population to this asymptotic
boundary indicates the intensity of intraspecific competi-
tion in a monospecific population. After the boundary is
exceeded, competition between individuals begins and, in
the absence of disturbance, growth trends of dominated
individuals progressively slow in favour of more compet-
itive species. This process leads the population to a max-
imum number of plants of a given size that can coexist
within a given unit area of land (Liira et al. 2011; Vos-
pernik and Sterba 2015).
Maximum biomass stocking is a fundamental indicator
to balance forest management. Growth trends, timber
quality and provision of ecosystem services from forests
are highly influenced by tree density and competition for
resources (Solomon and Zhang 2002; Corona 2015; Mason
and Connolly 2016; Marchi et al. 2018). Competition
indices ofplant interactions (Pommerening and Sa
¨rkka
¨
2013; Cabon et al. 2018) or allometric relationships (An-
fodillo et al. 2013; Marchi et al. 2017a,b) are valuable
tools that support decision-makers. Forest management
options that promote sustainable use (Fabbio et al. 2018)
can reduce competition between trees and aid selection of
preferred trajectories for forest stands in view of predicted
climate change effects (Wang et al. 1998; Ray et al. 2017).
The stand density index (SDI), widely accepted as a
powerful tool for evaluating growth sustainability of even-
aged forest systems, was proposed by Reineke (1933) for
tree species in the United States and later was adopted in
many other countries (Pretzsch and Biber 2005;Shaw
2006; Castagneri et al. 2008; Poschenrieder et al. 2018).
SDI is calculated as:
SDI ¼NQMD
a

k
;ð1Þ
where N is tree density, QMD is the quadratic mean
diameter, ais a constant whose value can be 10 or 25
according to the use of imperial or metric measuring sys-
tem, respectively, and kis a coefficient. The biological
meaning of this index is the maximum number of trees per
unit area a forest population (stand) can sustain for a pre-
determined QMD. For instance, an SDI of 400 means that
400 trees per hectare are expected when the QMD of the
stand is 25 cm. Although originally implemented for
monospecific, even-aged stands, generalized use in uneven-
aged stands and in multi-species stands has often been
proposed (Rivoire and Le Moguedec 2012). The core of the
function (1) is represented by the coefficient k. The value
of this parameter is the relationship between the total
number of trees per unit area (N) and the mean tree volume
or mass of the stand. This rule is parameterized for forests
as the stand density model where N is related to the QMD
of the stand. Even when approximated with a negative
exponential function (Fonseca and Duarte 2017), this
relationship follows a power law (PWL), which represents
the ‘‘pure’’ and mathematically-corrected shape of an N
versus QMD relationship that is expressed as:
N¼bQMDk
;ð2Þ
when QMD is either large or small, being asymptotic to
zero and ??, respectively. The use of PWL is generally
dropped in favour of logarithmic transformation of the data
fitted using an ordinary least squares (OLS) function
(Reineke 1933; Pretzsch and Biber 2005;Shaw2006;Ge
et al. 2017). In this case, Eq. (2) becomes:
log N ¼logbþklog QMD:ð3Þ
This basic equation, first used by Reineke (1933),
enabled the analysis of the size–density relationship for
many forest species in both pure and mixed stands. Using
Eq. 3, Reineke (1933) derived a slope (kcoefficient) equal
to -1.605 that was consistent across 15 datasets repre-
senting 14 species (13 coniferous) and was identical for 12
species. Although the slope was taken as constant, the
intercept term bwas considered to be species-specific. This
rule was supported by Yoda et al. (1963) but researchers in
Europe have often questioned it (Pretzsch and Biber 2005;
Vacchiano 2005; Castagneri et al. 2008). Recent studies
demonstrated the need to evaluate the slope of plotted
curves according to site quality (Ge et al. 2017), and biases
could result from the log-transformation of the data (Smith
1993), and questions have been raised on its biological
validity (Packard 2014). Despite this, the use of log-
transformed data versus nonlinear regression for analysing
biological power laws remains a common approach. Other
methodologies have been tested for estimating coefficients,
including stochastic frontier functions (Zhang et al. 2005;
Weiskittel et al. 2009), reduced major axis regression
(Solomon and Zhang 2002; Zhang et al. 2005), quantile
regression (Vospernik and Sterba 2015; Ducey et al. 2017),
bisector regression (Newton 2006), hierarchical Bayesian
models (Zhang et al. 2015) and mixed modelling (Zhang
et al. 2005).
The aims of this study were to evaluate the performance
of OLS and PWL methods for estimating kin SDI calcu-
lations and to analyse the discrepancies between them.
Comparison between these two methods for stand density
model fitting is based on simulated data with a known
degree of correlation between N and QMD. The subsequent
stand density management trajectories that were generated
were evaluated according to the results of these
comparisons.
1596 M. Marchi
123
Materials and methods
To analyse OLS and PWL robustness in coefficient esti-
mation, I generated an artificial dataset composed of
100,000 replicates of 200 records each. The aim was to
simulate empirical records derived from a hypothetical
distribution of monitored stands with a known degree of
correlation between N and QMD. A five-step procedure
was followed for this simulation (Fig. 1).
First, a ‘‘perfect’’ dataset with nonparametric correlation
between N and QMD of -1.0 was built with the normally
distributed dependent variable N by extracting the 200
records representing the total number of trees per hectare
(N) from a normal distribution with mean = 2700 tree-
sha
-1
and a standard deviation = 685 treesha
-1
. Those
starting values were judged as adequate to obtain a broad
range of simulated stands of density from 300 (mature
stand) to 5000 (young high forest stand before self-thinning
starts) and a theoretical SDI around 680. Then, the
corresponding quadratic mean diameter (QMD) for each of
the 200 normally distributed N values was calculated with
the following equation, derived from Eq. (2) and assuming
k=-1.605 and b= 1.210
5
:
QMD ¼ffiffiffi
N
b
k
s:ð4Þ
This first simulated dataset represented the hypothetical
and perfect situation of a forest stand in self-thinning
mode. Afterward, this original vector of QMD values (200
records) was iteratively modified to generate 99,999
alternative artificial stands. A series made by 200 random
multipliers was then generated 99,999 times, to increase or
reduce QMD with an artificial degree of noise. Such mul-
tipliers were derived from a second normal distribution
with mean = 1 and with a variable and randomly generated
standard deviation between 0.01 and 0.3. As a result, the
Spearman correlation coefficient (Spearman 1987) between
the original vector N and the 100,000 artificial vectors of
5
Fig. 1 Flowchart of the simulation process
Nonlinear versus linearised model on stand density model fitting and stand density index1597
123
QMD values ranged from -1.0 to -0.2 (Fig. 2). The
main idea behind this operation was to control the degree
of correlation between N and QMD to determine whether
this information might be used as ancillary data to evaluate
the coefficients estimated by the two methods. Summary
statistics of 100,000 generated plots are reported in
Table 1.
With the generated and log-transformed data, the stand
density model of the simulated 100,000 records was fitted
using the classical linear ordinary least squares (OLS) fit and
a power law was fitted using the Gauss–Newton algorithm
(PWL). The degree of equivalence between the estimated b
and kcoefficients was evaluated using a simple linear model
analysis (expected results were slope =1 and inter-
cept =0). The difference between coefficients, i.e. kor b
estimated by PWL minus kor bestimated by OLS, was also
evaluated in light of the correlation between N and QMD in
each of the simulated plots. A parametric ANOVA based on
correlation coefficient classes was finally performed to
assess differences within the estimated kvalues using the
Duncan test post hoc to identify possible groups. The degree
of correlation between N and QMD was the factor targeted
for evaluation, and eight classes were drawn dividing the
dataset into eight groups. Every class was defined including
all the plots with a qvalue between xand x?0.1 (e.g., class
7 included all the records with -0.4 Bq\-0.3, while
class 6 was -0.5 Bq\-0.4).
Results
Given the nature of generated datasets, comparison
between fitted models was possible in the absence of ref-
erence values. Indeed, the original kand bvalues were
assumed to be starting points that were only calculable with
the initial perfect dataset that contained no random noise.
Based on this assumption, the introduced changes to the
QMD vector brought the two models to calculate a wide
range of kand bvalues. The kcoefficient for OLS ranged
from -1.61 to -0.11 while branged from 810
3
to
1.310
5
. Similarly, with PWL, kranged from -1.62 to
-0.08, while branged from 8.510
3
to 1.410
5
. With the
linear model analysis, a common trend resulted between
the estimated kand bvalues. Models appeared to be similar
but with very high (-1.0 Bq\-0.8) and very low
correlations (-0.3 \q) as shown in Fig. 3. The OLS
model generally estimated higher k-values (Fig. 3, left
side) and lower bvalues (Fig. 3, right side) than did PWL,
especially when correlation coefficients ranged from
-0.75 to -0.4.
The relationship between the difference inkestimates
from OLS versus PWL and the correlation between N and
QMS in each artificial plot is shown in Fig. 4. Here, a
simple local polynomial regression fitting was also added
as a trend line and to characterize the behaviour of the
phenomenon. Concerning k(on the left), the core of the
stand density index calculation, the difference increased
quickly, when the correlation coefficient was between
-1.0 and -0.75, becoming almost flat at -0.6. Then the
difference decreased, becoming smaller until correlation
values exceeded -0.3. The difference in calculated max-
imum SDI values was also evaluated and is reported in the
right panel of Fig. 4.
Seven statistically different correlation groups were
detected, including border crossing classes (e.g., ‘‘ab’’,
‘bc’’ and ‘‘abc’’) (Table 2). The smallest differences sup-
ported by statistical evidence in kestimation were detected
for the class 8 (-0.3 Bq), class 7 (-0.4 Bq\-0.3)
and class 6 (-0.5 Bq\-0.4), all of which differed
from each other and from the other groups. Then class 1
(-1.0 Bq\-0.9) was intermediate between classes 6
Fig. 2 Histogram of Spearman correlation coefficients between the
generated number of trees per hectare (N), and the 100,000 quadratic
mean diameter (QMD) simulations
Table 1 Summary statistics of the 100,000 randomly generated
forest monitoring plots
Dataset N QMD
Original
Minimum 522 10.3
Mean 2512 25.2
Maximum 5097 35.7
SD used to
modify QMD
Correlation
coef. (q)
Adjusted
Minimum 522 0.01 -0.99
Mean 2512 0.15 -0.71
Maximum 5097 0.35 -0.22
N is number of trees per hectare; QMD is the quadratic mean diameter
of the stand; SD is standard deviation of the normal distribution used
to generate multipliers
1598 M. Marchi
123
and 2 (-0.9 Bq\–0.8), 3 (-0.8 Bq\-0.7) and 4
(-0.7 Bq\-0.6), and was ranked with an average
difference in kbetween 0.2769 and 0.2968. An ‘‘admixture
zone’’ was comprised of class 2 (‘‘abc’’ group) and classes
3 and 4 (‘‘ab’’ group) where differences were less pro-
nounced. The highest values were recorded for class 5
(-0.6 Bq\-0.5) with an average difference around
0.3. Class 5, which might also represents the most common
case in empirical data, was used to generate Fig. 5, where a
simulation of an elementary stocking chart was drawn with
two SDI values (1000 and 500) and two possible kvalues
differing from ±0.3. As a result, a discrepancy between
the two lines was evident (Fig. 5).
Fig. 3 Relationship between k(left) and b(right) coefficients
calculated by OLS and PWL models in each of the 100,000 simulated
plots. Each dot represents a single record and is coloured according to
the correlation class between N and QMD. N = number of trees/ha;
OLS = ordinary least squares; PWL = power law; QMD = quadratic
mean diameter of the stand b
Fig. 4 Relationship between the difference in k(left) and maximum
SDI (right) estimated with ordinary least squares and power law
models in each of the 100,000 simulated plots (y-axis) and the
correlation between N and quadratic mean diameter of the stand
(QMD) in each (x-axes). Each dot represents a single record and is
coloured according to the correlation class between N and QMD. The
black line is a local polynomial regression model and describes the
average trend
Nonlinear versus linearised model on stand density model fitting and stand density index1599
123
Discussion
The degree of equivalence between OLS and PWL coef-
ficients were highly dependent on the ‘‘quality’’ of the data,
i.e. on the degree of random noise during QMD and N
calculation for each plot. Even if logarithmic transforma-
tion is a primary tool for OLS linear regression of power
law relationships (Anfodillo et al. 2013), the results showed
that this mathematical adjustment should be carefully
evaluated when the primary goal is to study the model’s
coefficients rather than its predicted values. Transformation
might introduce biases into the fitting procedure, resulting
in quite different values for kand b. This bias could then
directly be transmitted to the stand density index calcula-
tion where kis the only coefficient.
Field data collection often is the most expensive com-
ponent of research projects but is necessary to obtain high
quality data. Similarly, the contributions of robust models,
analytical procedures, and statistics must not be underes-
timated (Fassnacht et al. 2014; Ferrara et al. 2017; Marchi
et al. 2017a,b). Important adjustments are sometimes used
to handle biases introduced to datasets, especially when
using inverse functions (Smith 1993). For SDI calculation,
the use of a log-transformed version of the canonical SDI
formula, Eq. (1), cannot be a possible solution. This would
be true even if Eq. (1) could be written as:
log SDI ¼log N þklog QMD
a

:ð5Þ
Using Eq. (5) with a value for kestimated from Eq. (3)
is worthy of note. An additional test highlighted that the
difference in SDI curves shown in Fig. 5remains. It is well
known that the logarithmic transformation of an arithmetic
dataset often results in slightly biased estimates when
values are predicted and transformed back to arithmetic
units from OLS (Baskerville 1972; Newman 1993; Packard
2013). This issue is mainly due to an estimation based on a
geometric mean of the dependent variable rather than on
the arithmetic mean at that value of the independent vari-
able (Smith 1993).
Accurate determination of the self-thinning trajectory
for any population is a difficult task, whatever the data
source and for many biological issues. Despite taking the
appropriate analytical steps, some problems are inherent in
the data where the actual area of a sampled stand that is in
self-thinning mode is often poorly defined. Indeed, despite
the intensity of sampling efforts, only some stands are
actually in self-thinning mode (Weiskittel et al. 2009;
Vospernik and Sterba 2015). Pests, insects, disease, and
wind storms can reduce tree densities more quickly than
natural mortality (Ray et al. 2017). Forests are already
adapting to a changing climate with a plastic reaction
across different ecological regions (O’Neill et al. 2008).
Many forest tree species are still growing under different
ecological regimes than those observed in the past (Pecchi
et al. 2019), shaping their spatial distribution and influ-
encing their ecological dynamics. This could generate
unexpected stresses which might alter the canonical self-
thinning rule. In all the cases, it is necessary to monitor
forest resources carefully as well as to adjust statistical
analyses and mathematical models that are insensitive to
observations in under-stocked conditions. Although some
efforts have been made to address these problems (Bi et al.
2000), limitations still persist. Stochastic natural impacts to
stands might be represented by introduction of random
noise to the artificial dataset. Most recommended analytical
Fig. 5 SDI curves calculated with different kvalues for the same N
and QMD dataset; kvalues here were adopted according to detected
difference for class 5 (±0.03). N = number of trees/ha; QMD =
quadratic mean diameter of the stand; SDI = stand density index
Table 2 Results of Duncan test where the difference between Power
Law and Ordinary Least Squares in k estimation was statistically
evaluated
Correlation class Mean Group
5(-0.6 Bq\-0.5) 0.3013 a
4(-0.7 Bq\-0.6) 0.2968 ab
3(-0.8 Bq\-0.7) 0.2876 ab
2(-0.9 Bq\-0.8) 0.2810 abc
1(-1.0 Bq\-0.9) 0.2769 bc
6(-0.5 Bq\-0.4) 0.2627 c
7(-0.4 Bq\-0.3) 0.1496 d
8(-0.3 Bq) 0.0339 e
1600 M. Marchi
123
approaches would suggest cleaning the dataset prior to
modelling by removing the non-asymptotic size-density
values from the data sets. However, this issue is irrelevant
to this study where the equivalence between methods was
tested rather than the ability of a method to yield a real
value.
Concerning the use of classic modelling techniques (i.e.,
OLS or PWL) on longitudinal studies such as SDI calcu-
lation from long-term monitoring plots data, a possible
solution might call for mixed models, linear and nonlinear.
Such techniques, which uses the restricted maximum
likelihood algorithm, are known to be much more reliable
than OLS and the Gauss–Newton fitting procedure when
autocorrelated data and large datasets are analysed, such as
those derived from long-term forest time series. Some
authors have tested this possibility (Solomon and Zhang
2002; Pourmajidian et al. 2010). In the current case, an
interesting solution could include the correlation class as a
random term with the QMD as a fixed effect. In this case,
the mixed models ‘‘gain’’ could refer to the reduction of
mean squared error from estimating coefficients rather than
from the use of separate models. However, possible biases
in kestimation should also be evaluated as stressed by our
results.
As regards possible solutions for the detected problem,
even if a simple mathematical correction could be thought
to be applied, no general rules could be provided. The bias
correction for log transformation is generally simple when
data are available. Anyway, this correction is aimed to
adjust predicted values and not for coefficients. In addition,
there is a chance, probably quite high, that the findings will
not hold, meaning they are as they are just because of the
bias. In this view, the use of PWL is preferred, with a
logarithmic transformation just in case of violation of
parametric assumptions (non-normality of data) to deal
with this.
Conclusions
Understanding long-term dynamics is fundamental for
sustainable forest management. The stand density model is
a well-known way to evaluate stand loading and to derive
the maximum stand density index in even-aged forests.
Even if the 100,000 simulated plots could include a wide
spectrum of study cases, empirical results from long-term
monitoring networks are necessary to support the evidences
obtained via simulation. In conclusion, given the power
law as the ‘‘natural’’ structure of the N versus QMD rela-
tionship, the nonlinear method is preferred, thus avoiding
the logarithmic transformation.
Open Access This article is distributed under the terms of the
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link to the Creative Commons license, and indicate if changes were
made.
References
Anfodillo T, Carrer M, Simini F, Popa I, BanavarJR MaritanA (2013)
An allometry-based approach for understanding forest structure,
predicting tree-size distribution and assessing the degree of
disturbance. Proc Biol Sci 280:20122375. https://doi.org/10.
1098/rspb.2012.2375
Baskerville GL (1972) Use of logarithmic regression in the estimation
of plant biomass. Can J For 2:49–53. https://doi.org/10.1139/
cjfr-2018-0119
Bi H, Wan G, Turvey ND (2000) Estimating the self-thinning
boundary line as a density-dependent stochastic biomass frontier.
Ecology 81:1477–1483. https://doi.org/10.1890/0012-
9658(2000)081%5b1477:ETSTBL%5d2.0.CO;2
Cabon A, Mouillot F, Lempereur M, Ourcival J, Simioni G, Limousin
J (2018) Thinning increases tree growth by delaying drought-
induced growth cessation in a Mediterranean evergreen oak
coppice. For Ecol Manag 409:333–342. https://doi.org/10.1016/j.
foreco.2017.11.030
Castagneri D, Vacchiano G, Lingua E, Motta R (2008) Analysis of
intraspecific competition in two subalpine Norway spruce (Picea
abies (L.) Karst.) stands in Paneveggio (Trento, Italy). For Ecol
Manag 255:651–659. https://doi.org/10.1016/j.foreco.2007.09.
041
Corona P (2015) Forestry research to support the transition towards a
bio-based economy. Ann Silvic Res 38:37–38. https://doi.org/10.
12899/asr-1015
Ducey MJ, Woodall CW, Bravo-Oviedo A (2017) Climate and
species functional traits influence maximum live tree stocking in
the Lake States, USA. For Ecol Manag 386:51–61. https://doi.
org/10.1016/j.foreco.2016.12.007
Fabbio G, Cantiani P, Ferretti F, Di Salvatore U, Bertini G, Becagli C,
Chiavetta U, Marchi M, Salvati L (2018) Sustainable land
management, adaptive silviculture, and new forest challenges:
evidence from a latitudinal gradient in Italy. Sustainability
10:2520. https://doi.org/10.3390/su10072520
Fassnacht FE, Hartig F, Latifi H, Bergerd C, Herna
´ndez J, Corvala
´nP,
Koch B (2014) Importance of sample size, data type and
prediction method for remote sensing-based estimations of
aboveground forest biomass. Remote Sens Environ
154:102–114. https://doi.org/10.1016/j.rse.2014.07.028
Ferrara C, Marchi M, Fares S, Salvati L (2017) Sampling strategies
for high quality time-series of climatic variables in forest
resource assessment. iForest - Biogeosci For 10:739–745. https://
doi.org/10.3832/ifor2427-010
Fonseca T, Duarte J (2017) A silvicultural stand density model to
control understory in maritime pine stands. iForest - Biogeosci
For 10:829–836. https://doi.org/10.3832/ifor2173-010
Ge F, Zeng W, Ma W, Meng J (2017) Does the slope of the self-
thinning line remain a constant value across different site
qualities? An implication for plantation density management.
Forests 8:355. https://doi.org/10.3390/f8100355
Liira J, Sepp T, Kohv K (2011) The ecology of tree regeneration in
mature and old forests: combined knowledge for sustainable
forest management. J For Res 16:184–193. https://doi.org/10.
1007/s10310-011-0257-6
Nonlinear versus linearised model on stand density model fitting and stand density index1601
123
Marchi M, Chiavetta U, Cantiani P (2017a) Assessing the mechanical
stability of trees in artificial plantations of Pinus nigra J.
F. Arnold using the LWN tool under different site indexes. Ann
Silvic Res 41:48–53. https://doi.org/10.12899/asr-1312
Marchi M, Ferrara C, Bertini G, Fares S, Salvati L (2017b) A
sampling design strategy to reduce survey costs in forest
monitoring. Ecol Indic 81:182–191. https://doi.org/10.1016/j.
ecolind.2017.05.011
Marchi M, Paletto A, Cantiani P, Bianchetto E, De Meo I (2018)
Comparing thinning system effects on ecosystem services
provision in artificial black pine (Pinus nigra J. F. Arnold)
forests. Forests 9:188. https://doi.org/10.3390/f9040188
Mason B, Connolly T (2016) Long-term development of experimental
mixtures of Scots pine (Pinus sylvestris L.) and silver birch
(Betula pendula Roth.) in northern Britain. Ann Silvic Res
40:11–18. https://doi.org/10.12899/ASR-1119
Newman MC (1993) Regression analysis of log-transformed data:
statistical bias and its correction. Environ Toxicol Chem
12:1129–1133. https://doi.org/10.1002/etc.5620120618
Newton PF (2006) Asymptotic size–density relationships within self-
thinning black spruce and jack pine stand-types: parameter
estimation and model reformulations. For Ecol Manag
226:49–59. https://doi.org/10.1016/J.FORECO.2006.01.023
O’Neill GA, Hamann A, Wang TL (2008) Accounting for population
variation improves estimates of the impact of climate change on
species’ growth and distribution. J Appl Ecol 45:1040–1049.
https://doi.org/10.1111/j.1365-2664.2008.01472.x
Packard GC (2013) Is logarithmic transformation necessary in
allometry? Biol J Linn Soc 109:476–486. https://doi.org/10.
1111/bij.12038
Packard GC (2014) On the use of log-transformation versus nonlinear
regression for analyzing biological power laws. Biol J Linn Soc
113:1167–1178
Pecchi M, Marchi M, Giannetti F, Bernetti I, Bindi M, Moriondo M,
Maselli F, Fibbi L, Corona P, Travaglini D, Chirici G (2019)
Reviewing climatic traits for the main forest tree species in Italy.
iForest - Biogeosci For 12:173–180. https://doi.org/10.3832/
ifor2835-012
Pommerening A, Sa
¨rkka
¨A (2013) What mark variograms tell about
spatial plant interactions. Ecol Modell 251:64–72. https://doi.
org/10.1016/j.ecolmodel.2012.12.009
Poschenrieder W, Biber P, Pretzsch H (2018) An inventory-based
regeneration biomass model to initialize landscape scale simu-
lation scenarios. Forests 9:212. https://doi.org/10.3390/f9040212
Pourmajidian MR, Jalilvand H, Fallah A, Hosseini SA, Parsakhoo A,
Vosoughian A, Rahmani A (2010) Effect of shelterwood cutting
method on forest regeneration and stand structure in a Hyrcanian
forest ecosystem. J For Res 21:265–272. https://doi.org/10.1007/
s11676-010-0070-7
Pretzsch H, Biber P (2005) A re-evaluation of Reineke’ s rule and
stand density index. For Sci 51:304–320
Ray D, Petr M, Mullett M, Bathgate A, Marchi M, Beauchamp K
(2017) A simulation-based approach to assess forest policy
options under biotic and abiotic climate change impacts: A case
study on Scotland’s National Forest Estate. For Policy Econ.
https://doi.org/10.1016/j.forpol.2017.10.010
Reineke LH (1933) Perfecting a stand-density index for even-aged
forests. J Agric Res 46:627–638
Rivoire M, Le Moguedec G (2012) A generalized self-thinning
relationship for multi-species and mixed-size forests. Ann For
Sci 69:207–219. https://doi.org/10.1007/s13595-011-0158-z
Shaw JD (2006) Reineke’s stand density index: where are we and
where do we go from here? Proceedings: Society of American
Foresters 2005 National Convention. October 19–23, 2005, Ft.
Worth, TX [published on CD-ROM]: Society of American
Foresters, Bethesda, p 13
Smith RJ (1993) Logarithmic transformation bias in allometry. Am J
Phys Anthropol 90:215–228. https://doi.org/10.1002/ajpa.
1330900208
Solomon DS, Zhang L (2002) Maximum size-density relationships for
mixed softwoods in the northeastern USA. For Ecol Manag
155:163–170. https://doi.org/10.1016/S0378-1127(01)00556-4
Spearman C (1987) The proof and measurement of association
between two things. By C. Spearman, 1904. Am J Psychol
100:441–471. https://doi.org/10.1037/h0065390
Vacchiano G (2005) Valutazione dello Stand density index in
popolamenti di abete bianco (Abies alba Mill.). Ital For e Mont
3:269–286
Vospernik S, Sterba H (2015) Do competition-density rule and self-
thinning rule agree? Ann For Sci 72:379–390. https://doi.org/10.
1007/s13595-014-0433-x
Wang Y, Titus SJ, LeMay VM (1998) Relationships between tree
slenderness coefficients and tree or stand characteristics for
major species in boreal mixedwood forests. Can J For Res
28:1171–1183. https://doi.org/10.1139/x98-092
Weiskittel A, Gould P, Temesgen H (2009) Sources of variation in the
self-thinning boundary line for three species with varying levels
of shade tolerance. For Sci 55:84–93
Westoby M (1984) The self-thinning rule. Adv Ecol Res 14:167–225.
https://doi.org/10.1016/S0065-2504(08)60171-3
Yoda K, Kira T, Ogawa H, Hozumi K (1963) Self-thinning in
overcrowded pure stands under cultivated and natural conditions
(intraspecific competition among higher plants XI). J Biol
14:107–129
Zhang L, Bi H, Gove JH, Heath LS (2005) A comparison of
alternative methods for estimating the self-thinning boundary
line. Can J For Res 35:1507–1514. https://doi.org/10.1139/x05-
070
Zhang X, Zhang J, Duan A, Deng Y (2015) A hierarchical Bayesian
model to predict self-thinning line for Chinese fir in Southern
China. PLoS ONE 10:e0139788. https://doi.org/10.1371/journal.
pone.0139788
Publisher’s Note Springer Nature remains neutral with regard to
jurisdictional claims in published maps and institutional affiliations.
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... Additionally, DBH data were log-transformed, a common procedure in spatial modelling and forest sciences in general (Benito , Hallingbäck et al. 2021. Even if known to be able to generate biased coefficients in some circumstances (Marchi 2019) this transformation does not impact on variables importance and model predictions. All computations were done in the R environment (R Development Core Team 2020) with the "lme4" package (Bates et al. 2015). ...
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The cultivation of hybrid poplar clones is increasing worldwide. Hundreds of hectares of plantations now occur across Europe and other continents such as North America, using tested clones and novel genotypes. Research effort aims are to develop fast growing disease-and pest-resistant clones to improve production quality and quantity. In this study the phenotypic plasticity of poplar clones was tested across environmental and temporal gradients. The growth performance of 49 hybrid poplar clones recorded between 1980 and 2021 was analysed using a mixed-effects model with climatic data as a predictor variable. Clones were aggregated into two groups according to their breeding protocol (i.e., standard clone, and improved material) and their growth modelled for future climate scenarios of RCPs 2.6 and 8.5 using a downscaled version of the variants 01 and 21 of UKCP18 climate projections dataset for three 30-year normal period time-slices: 2030s, 2040s, 2050s. The fitted growth models showed highly significant results, explaining more than 85% of the variance, with a mean relative absolute error of approximately 2%. Improved material showed more resistance to warmer and drier climates and less sensitivity to the changing climate. While no unique pattern was found when comparing growth performances, new improved clones were more productive than older clones (e.g., 'I-214') with an additional benefit of resistance to rust and pests. Spatial predictions confirmed the Po valley as the most important geographic area for poplar cultivation in Italy, but zones in Central and Southern Italy show potential. However, the Po Valley is also where poplars are predicted to be suitable in the next decades with large uncertainties. The analysis identified the need for more research on the topic of poplar breeding. For example, models using the most extreme (warm and dry) climate projection, variant 01 of RCP8.5 of the UKCP18, exceeded the historic climate threshold, and predictions used model extrapolation, with associated statistical uncertainty. Therefore , predictions should be considered with care and more research effort is required to test clones over wider environmental conditions.
... It has been used frequently since it was first proposed, and this usage has continued in recent years (e.g. Comacho-Montoya et al. 2018;Quiñonez-Barraza et al. 2018;Salas-Eljatib and Weiskittel 2018;Zhang et al. 2018Zhang et al. , 2019Cabrera-Pérez et al. 2019;Lee and Choi 2019;Marchi 2019;Quiñonez-Barraza and Ramirez-Maldonado 2019;De Prado et al. 2020;Walker et al. 2020;Dean et al. 2021;Mäkinen et al. 2021). Reineke's original belief was that the slope, s*, of his model took close to the same value, −1.605, across a wide range of tree species. ...
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Data were available from two measurements of each stand of a set of stands in native regrowth and plantation forests of Eucalyptus pilularis (blackbutt) in subtropical New South Wales and Queensland. Measurements were made after competition-induced mortality had started in each stand. Most of the regrowth forest had been thinned some time before the measurements. Based on the Reineke model that relates stand stocking density to quadratic mean diameter, a maximum density line was determined for each stand based on methods developed by Vanclay and Sands. It was found that the intercept of the maximum density line increased both with increasing stand site productive capacity and in the thinned regrowth stands. The latter finding does not seem to have been reported for forests before; it is hypothesised to be a result of the delayed redevelopment of inter-tree competitive processes following the major reduction in stand density resulting from thinning. Both effects will have consequences for the structure of density management diagrams commonly used to assist in forest management practice.
... Parametric and nonparametric models are nowadays seen as mandatory in many research fields, able to provide estimations and confidence intervals at the same time (Aertsen et al. 2010;Petr et al. 2014;Pecchi et al. 2019). The simple multipliers obtained averaging raw data showed to be almost unable to detect the variability of the analysed phenomena as well as to provide inferences on the output (Bayer et al. 2013;Marchi 2019). The use of GAMM as modelling techniques allowed us to fit L 0 and r cb more properly than using simple coefficients as proposed by Pretzsch (2009). ...
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Key message The crown volume of Pinus nigra trees can be modelled as a function of the total crown length and the crown radius at crown base to support forest management practices. ContextThe crown volume of trees is rarely considered in forest management practices and is often approximated to a simple cone or paraboloid whose volume can be broadly derived from costly measurements.AimsWe developed two equations to predict single-tree crown volumes for Pinus nigra plantations in Italy based on the analysis of a database with 3578 trees.Methods Two key crown parameters (total length of the crown and crown radius at crown base) were here modelled using directly measurable mensurational data using Generalise Additive Mixed-effect Model. Afterwards, two functions were then proposed to predict single-tree crown volumes.ResultsThe fitted models were statistically significant and explaining 57.6% for crown radius at crown base and 87.1% for crown length of the total variance. The power model for single-tree crown volumes calculation showed a mean absolute error around 4.1 m3 for the upper portion of the crown and 12.1 m3 for the lower part for a mean absolute relative error of 12.5% and 32.0% respectively for a global value of 16.4%.Conclusion Single-tree and stand-level data are fundamental to balance forest management trajectories. The provided functions may be used in external dataset to derive indication on the single-tree or stand-level crown volume to be used as indicators of ground coverage.
... Unfortunately, this method does not properly scale the underlying data in geometric (logarithmic) space (see Glossary; Kerkhoff and Enquist, 2009). As a result, power functions derived from arithmetic versus logarithmic data are often quite different (Zar, 1968;Hui and Jackson, 2007;Packard et al., 2011;Xiao et al., 2011;Lai et al., 2013;Starostová et al., 2013;Marchi, 2019;Chen et al., 2020). In my opinion, sizescaling analyses should involve two steps. ...
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The magnitude of many biological traits relates strongly and regularly to body size. Consequently, a major goal of comparative biology is to understand and apply these ‘size-scaling’ relationships, traditionally quantified by using linear regression analyses based on log-transformed data. However, recently some investigators have questioned this traditional method, arguing that linear or non-linear regression based on untransformed arithmetic data may provide better statistical fits than log-linear analyses. Furthermore, they advocate the replacement of the traditional method by alternative specific methods on a case-by-case basis, based simply on best-fit criteria. Here, I argue that the use of logarithms in scaling analyses presents multiple valuable advantages, both statistical and conceptual. Most importantly, log-transformation allows biologically meaningful, properly scaled (scale-independent) comparisons of organisms of different size, whereas non-scaled (scale-dependent) analyses based on untransformed arithmetic data do not. Additionally, log-based analyses can readily reveal biologically and theoretically relevant discontinuities in scale invariance during developmental or evolutionary increases in body size that are not shown by linear or non-linear arithmetic analyses. In this way, log-transformation advances our understanding of biological scaling conceptually, not just statistically. I hope that my Commentary helps students, non-specialists and other interested readers to understand the general benefits of using log-transformed data in size-scaling analyses, and stimulates advocates of arithmetic analyses to show how they may improve our understanding of scaling conceptually, not just statistically.
... Thanks to the tree-level raw data provided by the NFI management team, summary statistics of the measured trees are provided in this paper and compared with the native trees growing in the same stands. In this paper, a threshold of 85% of the total standing volume was used to classify the degree of admixture [56,57]; therefore, all the stands where the standing biomass was mainly allocated on Douglas-fir trees were used for the calculation of species-specific indicators, such as the stand density index [58] and the site index [59]. A known shortcoming of INFC2005 is that the spatial coordinates of NFI plots have an uncertainty of about 1 km. ...
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The introduction of Douglas-fir [Pseudotsuga menziesii (Mirb.) Franco] in Europe has been one of the most important and extensive silvicultural experiments since the 1850s. This success was mainly supported by the species' wide genome and phenotypic plasticity even if the genetic origin of seeds used for plantations is nowadays often unknown. This is especially true for all the stands planted before the IUFRO experimentation in the 1960s. In this paper, a methodology to estimate the Douglas-fir provenances currently growing in Italy is proposed. The raw data from the last Ital-ian National Forest Inventory were combined with literature information to obtain the current spatial distribution of the species in the country representing its successful introduction. Afterwards, a random forest classification model was run using downscaled climatic data as predictors and the classification scheme adopted in previous research studies in the Pacific North West of America. The analysis highlighted good matching between the native and the introduction range in Italy. Coastal provenances from British Columbia and the dry coast of Washington were detected as the most likely seed sources, covering 63.4% and 33.8% of the current distribution of the species in the country, respectively. Interior provenances and those from the dry coast of Oregon were also represented but limited to very few cases. The extension of the model on future scenarios predicted a gradual shift in suitable provenances with the dry coast of Oregon in the mid-term (2050s) and afterwards California (2080s) being highlighted as possible new seed sources. However, only further analysis with genetic markers and molecular methods will be able to confirm the proposed scenarios. A validation of the genotypes currently available in Italy will be mandatory as well as their regeneration processes (i.e., adaptation), which may also diverge from those occurring in the native range due to a different environmental pressure. This new information will also add important knowledge, allowing a refinement of the proposed modeling framework for a better support for forest managers.
... The observed data were plotted against the fitted curve, in which the value of the standard normal variable is used as the horizontal axis to linearize the plot (Beaulieu and Xie 2004;Marchi 2019). Data shows that their fitted lines are consistent with the observed data. ...
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Efficient rainfall/runoff data modeling necessitates field data availability. Remote and rough terrain areas restrict data collection that leads to less reliable simulated models. Consequently, complete geographic databases are the quest to conduct over the catchment under investigation. The hydrologic model developed for this research based on different return periods (2, 5, 10, 25, 50, 100, and 200 years) gave significant discharge outputs. It was found that a basin average precipitation having a return period of 5 years yields a peak discharge of 1032.7 m3/s with the time of peak occurring 23.25 h after the event has started. It results in a volume of 79.9 × 106 m3. A storm event having a return period of 200 years, with basin average rainfall approximately two times more intense than the above yields an enormous discharge of 2191.1 m3/s and an accumulative volume of water of 158.8 × 106 m3. Accordingly, the catchment cannot accommodate such significant volumes of water and flooding becomes unavoidable. Therefore, hydrological, and hydraulic models can support decision-makers in correspondence to the catchment management problems for the sustainable and economic development of the wider area, by providing systematic and consistent information.
... Only the selected data are then used to formally delineate the perceived boundary line through traditional methods such as least squares regression or principle component analysis (e.g., Bi and Turvey 1997;Zhang et al. 2005;Marchi 2019). However, no matter how data selection is done, some degree of subjectivity cannot be avoided, resulting in a certain lack of objectivity in the estimated self-thinning boundary line. ...
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Stochastic frontier analysis and quantile regression are the two econometric approaches that have been commonly adopted in the determination of the self-thinning boundary line or surface in two and higher dimensions since their introduction to the field some 20 years ago. However, the rational for using one method over the other has, in most cases, not been clearly explained perhaps due to a lack of adequate appreciation of differences between the two approaches for delineating the self-thinning surface. Without an adequate understanding of such differences, the most informative analysis may become a missed opportunity, leading to an inefficient use of data, weak statistical inferences and a failure to gain greater insight into the dynamics of plant populations and forest stands that would otherwise be obtained. Using data from 170 plot measurements in even-aged Larix olgensis (A. Henry) plantations across a wide range of site qualities and with different abundances of woody weeds, i.e. naturally regenerated non-crop species, in northeast China, this study compared the two methods in determining the self-thinning surface across eight sample sizes from 30 to 170 with an even interval of 20 observations and also over a range of quantiles through repeated random sampling and estimation. Across all sample sizes and over the quantile range of 0.90 ≤ τ ≤ 0.99, the normal-half normal stochastic frontier estimation proved to be superior to quantile regression in statistical efficiency. Its parameter estimates had lower degrees of variability and correspondingly narrower confidence intervals. This greater efficiency would naturally be conducive to making statistical inferences. The estimated self-thinning surface using all 170 observations enveloped about 96.5% of the data points, a degree of envelopment equivalent to a regression quantile estimation with a τ of 0.965. The stochastic frontier estimation was also more objective because it did not involve the subjective selection of a particular value of τ for the favoured self-thinning surface from several mutually intersecting surfaces as in quantile regression. However, quantile regression could still provide a valuable complement to stochastic frontier analysis in the estimation of the self-thinning surface as it allows the examination of the impact of variables other than stand density on different quantiles of stand biomass.
... Second, to obtain expected fit, in many instances the variables are transformed [2,3]. However, variable transformations did not necessarily supply the desired results even when the assumptions are met, particularly when the predicted values were changed [12][13][14]. The bias induced by the transformation of the dependent variable was noted early by Williams [15] and Cochran [16], but its formal removal was only developed for few transformations, the most popular being the one for the logarithm function of Finney [17]. ...
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: The parameters of nonlinear forest models are commonly estimated with heuristic techniques, which can supply erroneous values. The use of heuristic algorithms is partially rooted in the avoidance of transformation of the dependent variable, which introduces bias when back-transformed to original units. Efforts were placed in computing the unbiased estimates for some of the power, trigonometric, and hyperbolic functions since only few transformations of the predicted variable have the corrections for bias estimated. The approach that supplies unbiased results when the dependent variable is transformed without heuristic algorithms, but based on a Taylor series expansion requires implementation details. Therefore, the objective of our study is to investigate the efficient expansion of the Taylor series that should be included in applications, such that numerical bias is not present. We found that five functions require more than five terms, whereas the arcsine, arccosine, and arctangent did not. Furthermore, the Taylor series expansion depends on the variance. We illustrated the results on two forest modeling problems, one at the stand level, namely site productivity, and one at individual tree level, namely taper. The models that are presented in the paper are unbiased, more parsimonious, and they have a RMSE comparable with existing less parsimonious models.
... Optimal assortment allocation is a key element in the wood products supply chain [12], particularly artificial stands. The use of spatial indices and models to describe the structure of a forest and to support forest management trajectories [13][14][15][16] are nowadays acknowledged as compulsory in a precision forestry framework [17,18] but research in this area, in Italy, is in serious delay [19]. Recent studies also raised the interest in the ability of Pinus nigra spp. to provide a wide range of ecosystem services [20,21] creating an ecologically dynamic system where biodiversity level increased quickly, thanks to all the ecological processes restored due to the artificial stands [22]. ...
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Stem tapers are mathematical functions modelling the relative decrease of diameter (rD) as the relative height (rH) increase in trees and can be successfully used in precision forest harvesting. In this paper, the diameters of the stem at various height of 202 Pinus nigra trees were fully measured by means of an optical relascope (CRITERION RD 1000) by adopting a two-steps non-destructive strategy. Data were modelled with four equations including a linear model, two polynomial functions (second and third order) and the Generalised Additive Model. Predictions were also compared with the output from the TapeR R package, an object-oriented tool implementing the β-Spline functions and widely used in the literature and scientific research. Overall, the high quality of the database was detected as the most important driver for modelling with algorithms almost equivalent each other. The use of a non-destructive sampling method allowed the full measurement of all the trees necessary to build a mathematical function properly. The results clearly highlight the ability of all the tested models to reach a high statistical significance with an adjusted-R squared higher than 0.9. A very low mean relative absolute error was also calculated with a cross validation procedure and small standard deviation were associated. Substantial differences were detected with the TapeR prediction. Indeed, the use of mixed models improved the data handling with outputs not affected by autocorrelation which is one of the main issues when measuring trees profile. The profile data violate one of the basic assumptions of modelling: the independence of sampled units (i.e., autocorrelation of measured values across the stem of a tree). Consequently, the use of simple parametric equations can only be a temporary resource before more complex built-in apps are able to allow basic users to exploit more powerful modelling techniques.
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A stand density index was derived from three hypotheses, of which two4 concern growth and mortality up to the age at which there is an inflection point in growth,5 and the third relates mean tree volume and dbh up to that same age. The equations are6 log (no. of trees) = K − λ 1−β log dbh, or SDI = T rees/unit area × (dbh reference dbh) λ 1−β ,7 where reference dbh = 25.4 cm (or 10 in.). The parameters λ and β relate mean tree volume8 to dbh and stand volume to number of trees. Either equation provides a simple alternative to determining the self-thinning line and finding its slope, and also provides a direct comparison to the Reineke equation. Comparison of SDI values and exponents from9 the above formula to others’ results indicates it produces smaller10 exponents, and larger SDI values, for dbh values smaller than 25.4 cm. Val-11 ues of λ 1−β varied with species and site quality, with smaller values than 1.605 for12 conifers, and larger values for angiosperms, when these were determined from yield13 tables, while values from experimental plots were always smaller. 14 For these formulas to be valid, parameter estimates must be from data that is15 not significantly beyond the inflection age, and β and the ratio of growth to mortality16 parameters must be comparable.17 18
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The future dynamics of forest species and ecosystems depend on the effects of climate change and are related to forest management strategies. The expected impacts of climate change are linked to forest growth and productivity. An increase in the length of the growing season and greater productivity are likely as well as shifts in average climatic values and more variable frequencies, intensities , durations and timings of extreme events. The main aim of this work is to assess and describe the climatic requirements for Italian forest tree species. We used 7,272 field observations from Italian National Forest Inventory plots and average annual temperatures and precipitation as interpolated from raster maps with 1 km spatial resolution. On this basis we evaluated the current observed distributions of the 19 most important tree species in Italy with respect to potential climatic limits based on expert knowledge and the available literature. We found that only 46% of the observations fall within the potential joint temperature and precipitation limits as defined by expert knowledge. For precipitation alone, 70% of observations were within the potential limits, and for temperature alone, 80% of observations were within the potential limits. Similarity between current observed and potential limits differ from species-to-species with broadleaves in general more frequently distributed within the potential climatic limits than conifers. We found that ecological requirements and potential information should be revised for some species, particularly for the Pinus genus and more frequently for precipitation. The results of the study are particularly relevant given the threat of climate change effects for Italian forests which are broadly acknowledged to be a biodiversity hotspot. Further investigations should be aimed at modelling the effects of climate changes on Italian forests as a basis for development of mitigation and adaptation forest management strategies.
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Aimed at reducing structural homogeneity and symmetrical competition in even-aged forest stands and enhancing stand structure diversity, the present study contributes to the design and implementation of adaptive silvicultural practices with two objectives: (1) preserving high wood production rates under changing environmental conditions and (2) ensuring key ecological services including carbon sequestration and forest health and vitality over extended stand lifespans. Based on a quantitative analysis of selected stand structure indicators, the experimental design was aimed at comparing customary practices of thinning from below over the full standing crop and innovative practices of crown thinning or selective thinning releasing a prefixed number of best phenotypes and removing direct crown competitors. Experimental trials were established at four beech forests along a latitudinal gradient in Italy: Cansiglio, Veneto; Vallombrosa, Tuscany; Chiarano, Abruzzo; and Marchesale, Calabria). Empirical results indicate a higher harvesting rate is associated with innovative practices compared with traditional thinning. A multivariate discriminant analysis outlined significant differences in post-treatment stand structure, highlighting the differential role of structural and functional variables across the study sites. These findings clarify the impact of former forest structure in shaping post-treatment stand attributes. Monitoring standing crop variables before and after thinning provides a basic understanding to verify intensity and direction of the applied manipulation, the progress toward the economic and ecological goals, as well as possible failures or need for adjustments within a comprehensive strategy of adaptive forest management.
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Dynamic landscape simulation of the forest requires an initial regeneration stock specific to the characteristics of each simulated stand. Forest inventories, however, are sparse with regard to regeneration. Moreover, statistical regeneration models are rare. We introduce an inventory-based statistical model type that (1) quantifies regeneration biomass as a fundamental regeneration attribute and (2) uses the overstory’s quadratic mean diameter (Dq) together with several other structure attributes and the Site Index as predictors. We form two such models from plots dominated by European beech (Fagus sylvatica L.), one from national forest inventory data and the other from spatially denser federal state forest inventory data. We evaluate the first one for capturing the predictors specific to the larger scale level and the latter one to infer the degree of landscape discretization above which the model bias becomes critical due to yet unquantified determinants of regeneration. The most relevant predictors were Dq, stand density, and maximum height (significance level p < 0.0001). If plot data sets for evaluation differed by the forest management unit in addition to the average diameter, the bias range among them increased from 0.1-fold of predicted biomass to 0.3-fold.
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Provision of forest ecosystem services is influenced by site and stand characteristics as well as forest management practices. In order to evaluate the influence of forest management on ecosystem services provision, two artificial black pine forests located in Central Italy were studied where two different thinning approaches (traditional and selective) were applied under the SelPiBio LIFE project. Four main ecosystem services were selected and assessed: timber and bioenergy production, carbon sequestration, forest stand stability-protection, and biodiversity conservation. Even if not supported by statistical evidence, results highlighted an interesting trend just 2 years after treatment. The selective thinning was able to enhance the majority of ecosystem services compared to the traditional one. A higher growth rate of selected crop trees was measured (i.e., carbon sequestration). The slenderness ratio was sensibly reduced (i.e., mechanical stability) with a positive implication on soil retention and the prevention of landslides. Moreover, valuable and interesting commercial assortments have been proven to be retrieved from the stands with the selective approach. Larger and also better formed trees were harvested, given the impact of selective thinning on the co-dominant class. The Shannon index increased only with the selective thinning intervention. In conclusion, the provided results and methods are encouraging and might represent the basis for novel and longer monitoring efforts
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The aim of this study was to provide optimal silvicultural guidelines for the maintenance of low understory vegetation cover in maritime pine (Pinus pinaster Ait.) stands in Mediterranean areas prone to the occurrence of forest fires. An extensive data set from maritime pine stands of northern Portugal was used to assess the effect of stand density on the understory cover. A statistically significant relationship between the spacing-top height factor (Fw) and the understory cover was found. An ecologically-based density regulation model was developed based on Fw = 0.21, which provided the optimal stand density and canopy cover to prevent the understory growth and proliferation, thereby reducing the vulnerability to forest fire and ensuring at the same time the highest values of stand yield. The developed model represents a supporting tool for density regulation of maritime pine stands in areas prone to forest fires. The representativeness of the supporting data set (in terms of number of sample plots and variability of the stands characteristics) provides confidence in the generalization of our results to different maritime pine stands in the Mediterranean area. This study suggests that managing stand density may be an effective adaptive management procedure which can help reducing the forest fire hazard.
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The self-thinning rule is regarded as one of the most important principles in plantation management. This rule, involving the assumption of a constant slope coefficient, has been universally applied when regulating stand density. In this study, we hypothesized that the slope coefficient can change significantly with changes in site quality. To test this hypothesis, we first grouped forest plots into 5 categories based on site index. Second, we produced the self-thinning line represented by the Reineke function for each of the 5 site categories, selecting fully stocked plots using reduced major axis regression. Third, the slope coefficients for the different categories were tested for significant differences. The results indicated that in general, the slope was significantly different with different site quality. In addition, we observed that the slope of the self-thinning line exhibited a steeper trend for sites of lower quality, which indicated increased self-thinning or reduced self-tolerance. Finally, we concluded that it is imperative to produce specific self-thinning lines for different site quality categories.
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