Modeling dominant height growth of eucalyptus plantations with parameters conditioned to climatic variations

Forest Ecology and Management 380 (2016) 182–195

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Modeling dominant height growth of eucalyptus plantations with parameters conditioned to climatic variations Henrique Ferraco Scolforo a, Fernando de Castro Neto b, Jose Roberto Soares Scolforo b,⇑, Harold Burkhart c, John Paul McTague a, Marcel Regis Raimundo b, Rodolfo Araújo Loos d, Sebastião da Fonseca d, Robert Cardoso Sartório d a

Department of Forestry and Environmental Resources, North Carolina State University, 2820 Faucette Dr., Campus Box 8001, Raleigh, NC 27695, United States Department of Forest Science, Federal University of Lavras, Campus Universitario, Campus Box 3037, LEMAF, Lavras, Minas Gerais 37200-000, Brazil Department of Forest Resources and Environmental Conservation, Virginia Polytechnic Institute and State University, 310 W Campus Dr, Campus Box 169, Blacksburg, VA 24061, United States d Fibria Celulose S.A., Rod. Aracruz/B. Riacho, Aracruz, 29192000, Brazil b c

a r t i c l e

i n f o

Article history: Received 15 April 2016 Received in revised form 31 August 2016 Accepted 4 September 2016

Keywords: Precipitation Clear-cut Coppice Brazil

a b s t r a c t Dominant height growth equations, which given at some base age is defined as site index, is usually used to assess site quality. A flexible and accurate way to represent the potential productive capacity of forest stands of Eucalyptus spp. was developed. The generalized algebraic difference method was used, in which 15 dynamic equations were tested for modeling dominant height growth. The models were fitted to a data set derived from permanent plots located in the states of Bahia (BA) and Espirito Santo (ES), Brazil, with clonal eucalyptus plantations. The database was analyzed separately for the clear-cut and coppice regimes. The selection of the best-fitting model for each management regime was based on statistical fitting, predictive validation, and graphical analysis. After selection of the best model, one of its parameters were expanded with the addition of climatic variables that allowed for the creation of scenarios. The polymorphic modified Von Bertalanffy-Richards model with a single asymptote performed the best for the two management regimes. For clear-cut management, conditioning the slope parameter by the mean monthly precipitation obtained the best performance. For coppice management, the asymptote parameter conditioned by the mean monthly precipitation and its distribution throughout the year provided the best performance. The inclusion of the climate modifiers added flexibility for the models, which was represented by the interannual variations of precipitation. Expansions of the parameters did not mischaracterize the behavior of the modified Von Bertalanffy-Richards model for the management regimes studied. Climatic conditioning of the parameters of the slope and asymptote for the two management regimes led to accuracy gains in the estimates. Additionally, this enabled the generation of productivity scenarios based on the amount and distribution of the total precipitation for the areas under study. Ó 2016 Published by Elsevier B.V.

1. Introduction The area of planted forests in the world is equivalent to 264 million hectares. Of this total, 61% of the forest stands are located in China, India, and the United States. In Brazil, the plantations of Eucalyptus and Pinus sp. reached 7.60 million hectares in 2013, with 71.2% of this being plantations of Eucalyptus sp. Brazil contributes 17% of the total timber harvested in the world, much of

⇑ Corresponding author. E-mail address: [email protected] (J.R.S. Scolforo). http://dx.doi.org/10.1016/j.foreco.2016.09.001 0378-1127/Ó 2016 Published by Elsevier B.V.

it due to the high productivity of its forests, especially those of the eucalyptus genus (IBÁ, 2014). Brazilian timber is used as raw material for the production of pulp, paper, wood panels, solid products, coal, industrial wood, treated wood, and wood chips; however, pulp is the product of greatest interest. Among the Brazilian states, 15.5% of the eucalyptus forest stands established for pulp production are located in the states of Bahia (BA) and Espirito Santo (ES) (IBÁ, 2014). In 2014, the forestry sector accounted for 1.1% of the Brazilian GDP and 5.5% of its industrial GDP (Cirillo, 2015). Due to this importance to the national economy, there is a need to periodically monitor forest stands as a way of fostering information for planning and

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decision-making of a strategic, tactical, and operational nature (Guedes et al., 2012). This periodic monitoring is performed by means of sample delineations that efficiently capture the changes in forest stands over time. Sampling consists of a network of permanent plots, and the procedure is known as Continuous Forest Inventory (CFI), which provides information regarding stock and annual growth (Mello et al., 2009). CFI information is fundamental for the building of mathematical models that, by forecasting future timber production, are a key to guiding forest planning. In this context, the classification of the potential productive capacity or site index of different plantation locations is the key to the process (Pokharel and Dech, 2011). Timber prognosis in forest stands can be performed in two different ways, which derive from different philosophies of the tree growth. The first is based on ecophysiological principles in which a conceptual model is used to identify the cause and effect factors in biomass production. The second approach is called descriptive and aims to identify and describe the patterns of growth and production for the management practices of forest resources (Burkhart and Tomé, 2012). The descriptive method of predicting timber production is based on the dendrometric data from CFI; that is, it is based on repeated measurements for making estimates of growth. The sampling intensity enables inference per plot or set of plots with the same characteristics to guide medium - and long-term forest planning. An overview of forest site productivity assessment can be found in Skovsgaard and Vanclay (2008). In addition Burkhart and Tomé (2012) provide detailed coverage of methods for even aged stands. The combination of the second approach with climate variables can provide greater information accuracy, given that it can fill in the gaps of descriptive models, which are not sensitive to interannual climatic variations (Scolforo et al., 2013). Ferraz-Filho et al. (2011) and Scolforo et al. (2013) studying Eucalyptus grandis in southeastern Brazil showed that combining descriptive models with climate variables facilitates the application and understanding by forest users. Since, descriptive models have been used to describe the patterns of growth and production for forest management practices, these authors claimed that this approach would be an useful tool for updating forest inventory in the short term while accounting for climate variation. Combining descriptive models to climate variables has been applied with success in site classification, as shown by González-García et al. (2015), where the authors predicted site index for Eucalyptus nitens and found appropriate results when applying the fitted function for areas without inventory information. Remy de Perthuis de Laillevault, in the 18th century in France, was the first author to assess site quality by height growth (Batho and García, 2006). Subsequently, using dominant height of trees of the stand became popular to assess site quality. The concept of site index to assess site quality was first introduced for species with long rotations (Alemdag, 1991). Probably the most recognized technique to site index modeling was developed by Bailey and Clutter (1974), where the authors presented and applied the idea of dynamic, base age invariant equations for Pinus radiata in New Zealand. This approach was later generalized by Cieszewski and Bailey (2000). These dynamic equations are commonly used to compute predictions directly from any age-height pair, excluding the needs to incorporate the climate effect. Several successful examples of using these approaches are reported in the literature, including, Monserud (1984) and Diéguez-Aranda et al. (2005). In the late 1990s, authors such as Woollons et al. (1997), started to include climatic variables in the classical dynamic equations. The authors, however, found no significant improvement by incorporating climatic variables in analyses for Pinus radiata in New Zealand. Bravo-Oviedo et al. (2008) and Nunes et al. (2011) study-

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ing Pinus pinaster Ait in Spain and Portugal, respectively, also related dominant height growth to climate variables. These authors concluded that inserting climate variables in the site index equations provided an improvement of the estimates by increasing efficiency on a regional scale and reducing bias. Although regional climate differences have been incorporated in some studies, interannual climatic variation also has a great impact on the development of Brazilian eucalyptus cultivation (Stape et al., 2004). For example, considering a rotation of seven years for a eucalyptus stand, a dry period in one of the years can dramatically affect its final yield (Almeida et al., 2004). In other words, after data collection and model update, the observed pattern of climatic variation is key to understanding forest growth in following years. Thus, the development of accurate tools that are also sensitive to these variations has great potential for use in the classification of the potential production capacity for eucalyptus in Brazil, as responsiveness to changes in climate is essential for the production of plantations, due to the short rotation. Therefore, an approach of incorporating climatic variables in the descriptive models used for classification of the potential productive capacity of eucalyptus plantations in Brazil is necessary—a model country in the production of rapid-growth timber but which has a gap in the observation of mechanisms for understanding impacts on the growth of forest stands caused by the climate. The addition of interannual climatic variation allows the insertion of sensitivity into mathematical models. Additionally, this sensitivity in the model allows forest managers to assess the impact on local climate variability, either for the full rotation or for 1 year ahead. This feature is important for enabling more accurate planning. The objectives of this study were to represent the potential productive capacity (site index) of a eucalyptus clone located in the states of Bahia and Espirito Santo; increase the performance of the descriptive model via climatic variables; and generate productivity scenarios in accordance with possible interannual climatic variations. 2. Materials and methods 2.1. Physiographic and socio-economic characterization of the study area The eucalyptus stands are located in the states of Bahia (BA) and Espirito Santo (ES), with latitude ranging from 17°150 S to 20°150 S and longitude from 39°050 W to 40°200 W. BA is located in the northeast region of Brazil and is the fifth largest by area, the fourth most populous, and has the eighth largest GDP in Brazil (3.8% of national GDP). ES is located in the southeast region of Brazil and is the fourth smallest by area, the fourteenth most populous, and has the eleventh largest GDP in Brazil (2.4% of national GDP) (IBGE, 2014). To get an idea of the size of these states, the sum of their areas exceeds the area of France and Belgium together. The area’s climate classification, in accordance with the Köppen classification, ranges from Aw (tropical climate with precipitation greater than 1500 mm and dry winter) to Cwb (humid subtropical climate with dry winter and temperate summer) in ES and Af (tropical climate without dry season with mean air temperature above 18 °C) in BA (Alvares et al., 2013). In accordance with the climatic conditions in BA and ES and, more specifically, in the study area, there is climatic similarity but with a significant precipitation difference. 2.2. Sampling and data acquisition The database was composed of forest inventory data derived from the CFI as well as climatic data from weather stations. The

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plantation areas were composed of a single clone of Eucalyptus spp., focusing pulpwood production, planted at a 3  3 m spacing. The sample plots used had a circular configuration with an area of 500 m2, and they were systematically distributed (Cochran, 1977). The CFI measurements are for the period between 1994 and 2014, and for each plot, circumferences were measured at 1.30 m from the ground level for all trees, together with the height of 30% of the trees within the plot plus the dominant trees. Utilizing the definition of mean top height, dominant trees were determined based on the one-hundred largest DBH trees per hectare, i.e., on the five largest DBH trees per plot. The areas were managed by regimes involving regeneration of the stand (clear-cut) and management of the sprouting (coppice). Measurements of the clear-cut regime began in the 1990s, with the introduction of improved genetic material (Gonçalves et al., 2013). For this regime, 144 plots were used (Fig. 1). Their descriptive statistics are presented in Table 1. After the global financial crisis in 2008, the production of timber via the coppice system began to receive more attention from foresters and researchers (Gonçalves et al., 2014). For this management, which represents one rotation then followed by clear-cut, 155 plots were used (Fig. 1). Their descriptive statistics are also presented in Table 1.

The climatic data came from 31 meteorological stations, which were installed by the company in the past in order to cover the entire range of interest, including possible new areas for expansion. Daily values for precipitation, temperature, vapor pressure deficit and solar radiation were taken during the same observation period as the CFI (Fig. 1). Table 2 shows statistics related to the climatic variables of the study area. There is relatively high variation in precipitation and in the distribution of the total precipitation in the areas throughout the year (rainy days, which is defined as the number of days with rainfall exceeding 1 mm). The areas are characterized by wet summers (high amount of rainfall between October and March) and a pronounced dry winter (between June and August), except for the Af climate, which does not present a dry period, although it presents less rainfall during the winter. 2.3. Evaluation of the potential productive capacity of the areas 2.3.1. Dominant height projection The generalized algebraic difference approach (GADA) was used to model the dominant height growth. The GADA approach allows various parameters to be related to the quality of the site by means of an algebraic transformation. According to Cieszewski et al.

Fig. 1. Distribution of the plots and meteorological stations for the study area. Table 1 Descriptive statistics (age in years, dominant height in meters, annual dominant height growth in meters year1, survival in %, measurements – average number of measurements per plot) of the clear-cut and coppice management regimes stratified by climatic classification and region. Region

Climatic classification

Manag

Age

Dominant height

Annual dominant height growth

Min

Avg

Max

Min

Avg

Max

Min

Avg

Max

Survival (%)

Measurements

BA ES ES

Af Cwb Aw

Clear-cut

1.2 0.9 1.1

3.8 3.8 4.0

7.6 7.3 7.9

5.9 5.3 7.4

19.9 18.4 19.4

29.7 27.7 30.3

0.1 0.1 0.1

3.5 3.4 3.1

9.4 10.6 11.0

96 97 95

4 5 4

BA ES ES

Af Cwb Aw

Coppice

1.0 1.0 1.1

3.6 3.6 3.6

6.7 6.4 7.6

6.1 6.0 6.6

18.6 18.3 17.6

31.1 27.5 28.8

0.1 0.2 0.1

3.8 3.7 3.4

10.4 11.5 9.3

90 94 94

3 4 5

Manag: management regime; Min: minimum; Avg: average; Max: maximum.

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H.F. Scolforo et al. / Forest Ecology and Management 380 (2016) 182–195 Table 2 Descriptive statistics of the climatic variables (values of the variables are mean annual data). Variables

Climatic classification Af

Cwb

Aw

Precipitation (mm year1)

Minimum Average Maximum Standard deviation

681.80 1598.99 3710.96 84.56

425.46 1334.10 2670.13 87.60

435.11 1237.66 2465.58 80.24

Rainy days (days year1)

Minimum Average Maximum Standard deviation

99.32 182.12 435.35 4.10

77.28 155.03 286.06 4.80

91.62 160.09 230.17 4.21

Temperature (°C)

Minimum Average Maximum Standard deviation

21.17 23.52 25.58 1.51

21.55 23.64 25.63 1.63

20.82 23.63 25.91 1.61

Radiation (MJ m2)

Minimum Average Maximum Standard deviation

13.26 18.08 22.76 2.99

14.27 17.47 21.59 2.74

13.96 17.64 22.35 2.68

Vapor pressure deficit (kPa)

Minimum Average Maximum Standard deviation

0.27 0.53 0.76 0.08

0.33 0.56 0.82 0.07

0.27 0.49 0.75 0.07

(2007), the method consists of (i) choosing an equation to model the variable of interest; (ii) deciding which parameters will depend on the theoretical quality of the site (X) and expressing the relationship through a mathematical equation; (iii) solving the equation for X; and (iv) inserting the solution of X to obtain the dynamic equation h = f(t, X) for the initial conditions of t0 and h0. We considered the situation of one and/or two parameters of the models related to X. When only one parameter is related to X, GADA is equivalent to the algebraic difference approach (ADA).

The GADA structure is an evolution of the method developed by Bailey and Clutter (1974), which enables construction of polymorphic curves with multiple asymptotes, as shown in the studies conducted by Diéguez-Aranda et al. (2005) and Scolforo et al. (2015). Fifteen dynamic equations were evaluated for modeling dominant height growth of eucalypts for each of the management regimes (clear-cut and coppice). The candidate models are shown in Table 3. An overview of growth functions can be found in

Table 3 Base models and dynamic equations obtained by the GADA approach. Base model

Parameters related to site quality

Initial solution for X with (h0 e t0)

Dynamic equation

Richards (1959) h = a(1  exp(bt))c

c=X

lnðh0 =aÞ X 0 ¼ lnð1expðbt 0 ÞÞ

h ¼ aðh0 =aÞ
c 1expðbtÞ h ¼ h0 1expðbt 0Þ

ðlnð1expðbtÞÞÞ=ðlnð1expðbt 0 ÞÞÞ

a=X

X0 ¼

a = exp(X) c = c1 + c2X

X 0 ¼ ðln h0  c1 F 0 Þ=1 þ c2 F 0 F 0 ¼ lnð1  expðbt 0 ÞÞ

h ¼ expðX 0 Þð1  expðbtÞÞ

a¼X

X 0 ¼ h0 ð1 þ bexpct0 Þ

h ¼ X 0 =ð1 þ bexpct Þ

b¼X

X 0 ¼ ða  h0 Þ=ðh0 expct0 Þ

h ¼ a=ð1 þ X 0 expct Þ

Modified logistic c h ¼ a=1 þ bt

a ¼ b1 X b ¼ b2 =X

X 0 ¼ 0:5ðh0  b1 þ

Schumacher

a¼X b¼X a¼X b ¼ b1 X a¼X

Logistic h = a/(1 + b expct)

ln h ¼ a þ bt

1

Korf c h ¼ a expðbt Þ

h0 c ð1expðbt 0 ÞÞ

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 ðh0  b1 Þ þ 4b2 h0 tc 0 Þ

c1 þc2 X 0

References

Id

Scolforo (2006)

M1

Scolforo (2006)

M2

Scolforo et al. (2015)

M3

Cieszewski and Strub (2008) Cieszewski and Strub (2008)

M4 M5

0 h ¼ 1þðbb12þX =X 0 Þt c

Cieszewski (2004)

M6

X 0 ¼ ln h0  b=t0 X 0 ¼ ðln h0  aÞt0 X 0 ¼ ln h0 =ððt0 þ b1 Þ=t 0 Þ

ln h ¼ X 0 þ b=t ln h ¼ a þ ðX 0 =tÞ ln h ¼ X 0 þ X 0 ðb1 =tÞ

Scolforo (2006) Scolforo (2006) Cieszewski (2004)

M7 M8 M9

h0 X 0 ¼ expðbt c Þ

c h ¼ h0 expðbðtc 0  t ÞÞ

Tomé (1989)

M10

Tomé (1989)

M11

h ¼ expðX 0  ðb1 =X 0 Þ lnð1  expðt ÞÞ

Cieszewski (2004)

M12

1 h0 b2 F 0 h ¼ F 1 ðb2 þhF0 Þb 0 b1

Cieszewski et al. (2006)

M13

h ¼ h0 tc ðt0c R00 þexpðb11 ÞÞ

Cieszewski (2001)

M14

ln h ¼ X 0 þ ðb1 þ b2 X 0 Þ lnð expðtc ÞÞ

Cieszewski (2004)

M15

0

b¼X

ðt 0 =tÞc

0 =aÞ X 0 ¼  lnðh t c

h ¼ aðh0 =aÞ

0

Cieszewski and Bella (1989) c h ¼ a=1 þ bt

b¼X

Modified Gompertz h ¼ a expðb expðctÞÞ þ d

a¼X d ¼ b1 X  b2

F 1 ¼ expðb expðctÞÞ F 0 ¼ expðb expðct 0 ÞÞ

Hossfeld c h ¼ bt =t c þ a

b ¼ b1 þ X a ¼ a1 =X

R0 ¼ h0  a1 þ ððh0  a1 Þ þ 2h0 expðb1 Þ=t c0 Þ

Modified Weibull ln h ¼ a þ b lnð1  expðtc ÞÞ

a¼X b ¼ b1 þ b2 X

X0 ¼

2

X 0 ¼ 0:5ðln h0 þ ððln h0 Þ  4b1 tc0 Þ

2

ln h0 b1 lnð1expðt c0 ÞÞ 1þb2 lnð1expðt c0 ÞÞ

1=2

c

Þ

1=2

t c ðt c R þexpðb ÞÞ

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Zeide (1993). Unlike static site equations, the dynamic equations have an invariant property; that is, they make it possible to estimate the dominant height and, consequently, the site index, without the shape of the curve being changed, independent of the base age adopted (Scolforo, 2006). Forest biometricians have sought methods that increase the flexibility of site equations; therefore, the algebraic difference and its subsequent generalization introduced by Cieszewski and Bailey (2000) was considered to be an important step for the versatility of the models. The M2, M4, M7, and M10 models were defined as anamorphic with multiple asymptotes; the M1, M5, M8, M11, and M12 models were polymorphic with a single asymptote; the M3, M6, M9, M13, M14, and M15 models were polymorphic with multiple asymptotes. The structure of the data used to build the models included height-age pairs for each plot. In these situations, the data did not behave in a linear fashion; therefore, the non-linear models were employed to express the behavior of dominant height. The number of tested equations was extended, since Schumacher and modified Von Bertalanffy-Richards equations, both anamorphic, are usually fitted to assess site quality of eucalyptus plantations in Brazil. Thus, we decided for testing several equations given by the literature in order to define the best for dominant height and consequently site index estimation. Each equation was fitted for each management regime. The equations were separately fitted, since even for same site index, the clear-cut regime is more productive compared to coppice. In addition, the coppice dataset presented greater variation than clear-cut, which could create a bias for some clear-cut stands. Finally, when one stand was harvested and the forest products company decided to apply a coppice regime, a permanent plot was not installed in the same location, which again highlights possible source of bias. For fitting of the models, the nls package of R version 3.1.2 was used (R Development Core Team, 2014). This package makes use of the Gauss-Newton algorithm for estimation of the parameters of the models. 2.3.2. Evaluation of the dynamic equations The performance evaluation of the dynamic equations for modeling dominant height growth of eucalypts was conducted in two phases. In the first phase, the statistical quality of the fittings was based on AIC and BIC. However, the quality of the fitting does not necessarily reflect the quality of the prediction; therefore, in the second phase of the evaluation, the equations were tested in an independent database (Kozak and Kozak, 2003). A total of 44 and 47 plots (30% of the total numbers of plots) were used to validate the clear-cut and coppice management regimes, respectively. For this predictive validation, the statistics for the mean absolute error (MAE, %), mean error (T, %), and R2 fitted into linear regression (MEF) were calculated:

0 B MAE ð%Þ ¼ B @

 1 yi  b  yi yi i¼1 C C100 n A

Pn

0P


MEF ¼ 1 

b

yi  yi yi

n

T ð%Þ ¼ @

i¼1

n

1 A100

2 Pn  yi  ybi Pi¼1 n 2 np ðyi yi Þ i¼1

n1

where n is the number of observations; yi is the observed dominant height; ybi is the predicted dominant height; p is the number of parameters of the equation; and yi is the observed mean dominant height.

MAE represents a measurement of the mean error, T indicates a possible trend of the fitting (i.e., it must be close to zero), and MEF is used to assess the accuracy of the model. Graphical analysis is essential for choosing the most efficient model because the curves may show considerable differences, even when fittings statistics are similar (Neter et al., 1990). Thus, the behavior of the residuals and the generation of the site curves were used to evaluate the fitting of the candidate equations. In addition to the site curves, the stability of the plots in each site was also evaluated via the fitted models, in order to confirm the veracity of the previous analyses. 2.3.3. Expansion of model parameters by insertion of climatic variables According to Scolforo et al. (2013), for correct interpretation and modeling of the effect of climatic variations on the forest stand growth, there must be a perfect space-time fit between the measurements of the forest inventory and the climatic variables. In temporal terms, the climatic variables evaluated corresponded to the interval between the first and second measurement of the forest inventory of the same plot and so on until the last measurement. Thus, the climatic variation between each measurement interval for each plot was observed in order to capture and better describe why plots presented lesser or higher growth. In spatial terms, the climatic variables were interpolated from the weather stations for the plots of the forest inventory, via the Inverse Square Distance method. Inverse Square Distance is a local interpolator with the weights being determined as function of the inverse square Euclidean distance, i.e., closer the neighbor greater is the weight. The weather stations were well distributed in the area, however, to evaluate growth variation regarding climatic variation, a match is necessary between climate variables and inventory plots, which were performed by interpolation technique considering each inventory measurement. Although, interpolation could be a problem, the distance between weather stations and plots were not greater than 50 km and the weather stations were well distributed in the study area. In addition, Inverse Square Distance method did not present bias when interpolating the variables, since the study area is located in flat and coastal lands with predominantly same soil (yellow argisol). It is important to mention that the period of measurements for both regions (BA and ES) were the same. Climatic variations are among the main factors that explain the growth capacity of forest species (Watt et al., 2014). Therefore, the possible increase in the performance of the estimate of dominant height was evaluated by relating the parameters from the selected models for each management regime to the climatic variables. Statistical evaluations of the fitting (AIC, BIC, MEF, MAE and T) as well as graphical analyses were performed. This method has been successfully applied to the modeling of forestry variables - for example, see Ferraz-Filho et al. (2011) and Scolforo et al. (2013). The parameters from the best models selected for each management regime were then expanded, beginning with the insertion of the climatic variables, in which the conditioning of the parameters that had the most significant impact on the development of the dominant height variable was sought. 2.3.4. Productivity scenarios based on climatic variations The insertion of climatic variables provides sensitivity to the descriptive models, which consequently allows for the creation of growth scenarios for forest stands as a result of the interannual climate fluctuations. Site curves were generated, considering three plausible occurrence scenarios: (i) low, which was represented by the mean minus one standard deviation of the climatic variable of interest; (ii) normal, which was defined as the mean occurrence of the climatic variable of interest in the areas; and (iii) high, which was represented by the mean plus one standard deviation of the climatic variable of interest. Finally, the descriptive model without

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superior to the anamorphic models, indicating that the dominant height growth for the eucalyptus in this region exhibits obvious polymorphism. Both for the clear-cut and the coppice management regimes, the models resulting from the modified Von BertalanffyRichards equation (M1 and M3) had statistical indices superior to the other models. This shows the great applicability of this model for representing the sigmoidal behavior of biological growth. Unlike the other biological growth functions—represented by the monomolecular, logistic, and Gompertz models— the modified Von Bertalanffy-Richards equation has no fixed inflection point, which contributes to the increased flexibility of the function (Scolforo, 2006). McTague (1985) sucessfully used the M1 model for dominant height and site index, however it was for loblolly (P. taeda) and slash pine (P. elliottii) in southern Brazil. Although model M1 was slightly superior to model M3 for both management regimes, graphical analysis was important for defining the most efficient model (Figs. 2 and 3). To prove the nonexistence of a trend in the models, Figs. 2 and 3 show the behavior of the residuals in mean and absolute terms as a function of the age classes. Again, the similarity between the fittings can be seen, highlighting the major differences between the observed and estimated dominant height in the classes with fewer observations, as expected. Thus, the slight superiority of the M1 model for expressing the dominant height growth was confirmed for both management regimes. Site curves were generated considering the base age as seven years. This analysis was performed to verify whether the M1 model was actually superior to the M3 model, in view of the similarity of the quality of the fit of both models and also because model M3 had multiple asymptotes, unlike model M1. This analysis was performed because, given that eucalyptus develops fast, the convergence to a common asymptote at a point early in time may

climatic variables (Traditional or classic model without insertion of climatic variables) was used to assess the proximity between its estimate with the normal growth scenario. 3. Results and discussion 3.1. Selection of the dynamic equation for each management regime For the clear-cut management regime, Table 4 presents the estimated parameters for the 15 models, their statistical results, and the statistics computed in the validation phase. Among the five best-fitting models, four were associated with polymorphic equations with a single asymptote (M1, M8, M11, and M12) and one was associated with a polymorphic equation with multiple asymptotes (M3). For all the models, all the parameters had high statistical significance (p < 0.0001) as well as consistent magnitude and suitable character for modeling of biological growth. Among the models with the best statistical indices for the coppice management regime were three polymorphic equations with a single asymptote (M1, M11, and M12) and two polymorphic equations with multiple asymptotes (M3 and M6). The fitting and validation statistics and the respective coefficients of the equations are presented in Table 5. Similar to the clear-cut management regime, all the parameters were significant and appropriate for modeling of the biological growth. It is interesting to mention that for most of the models in either management regimes, the mean error when applying the equations to the validation data set are below the measurement error of tree height, which for eucalyptus is between 0.5 and 1.5 m. For the two management regimes studied, the polymorphic model with a single asymptote (M1) and the polymorphic model with multiple asymptotes (M3) showed similar results, though model M1 was always slightly superior for both management regimes. As shown in Tables 4 and 5, the polymorphic models were

Table 4 Estimated parameters with their standard errors in parenthesis, fitting statistics, and validation statistics of the candidate models for the clear-cut management regime. Model

M1 M2 M3 M4 M5

Parameters

Fitting statistics b

c

b1

b2

AIC

BIC

MAE %

T%

MEF

29.076 (0.725) –

0.344 (0.027) 0.288 (0.033) 35.000 (0.628) 4.466 (0.240) –

–

–

–

1247

1260

4.02

0.01

0.90

1.061 (0.064) 10.117 (0.063) 0.729 (0.029) 0.697 (0.026) 3.832 (0.252) –

–

–

1411

1424

4.87

0.30

0.87

–

–

1357

1390

4.08

0.03

0.90

–

–

1409

1421

5.17

0.78

0.86

–

–

1360

1382

4.39

0.28

0.88

0.215 (0.109) –

1.130 (1.359) –

1409

1421

4.7

0.25

0.88

1488

1497

5.16

1.36

0.87

–

–

–

1399

1407

4.29

0.32

0.90

–

1432

1446

5.42

2.12

0.85

0.481 (0.057) –

0.461 (0.080) –

–

1418

1430

4.86

0.26

0.88

–

–

1389

1400

4.27

0.04

0.90

–

–

–

1407

1419

4.27

0.02

0.89

0.250 (0.090) –

36.000 (0.859) 4.623 (0.320) 0.038 (0.008)

12.103 (0.843) 1.176 (0.065) 0.787 (0.039)

1482

1489

4.72

1.14

0.88

1415

1429

4.86

0.28

0.88

1408

1419

5.25

0.30

0.90

0.397 (0.030) –

M6

26.292 (0.340) –

M7

–

M8 M9

3.486 (0.015) –

M10

–

M11

M13

32.265 (2.517) 31.821 (1.065) –

M14

–

2.265 (0.147) 0.481 (0.060) 1.486 (0.065) 2.000 (0.140) –

M15

–

–

M12

Validation statistics

a

– 1.679 (0.039) – –

0.709 (0055)

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Table 5 Estimated parameters with their standard errors in parenthesis, fitting statistics, and validation statistics of the candidate models for the coppice management regime. Model

M1 M2 M3 M4 M5

Parameters

Fitting statistics

a

b

c

b1

b2

AIC

BIC

MAE %

T%

MEF

34.309 (1.659) –

0.206 (0.026) 0.337 (0.032) 10.453 (4.274) 4.736 (0.247) 0.606 (0.026) –

–

–

–

1058

1070

5.16

0.20

0.93

1.124 (0.062) 2.768 (1.263) 0.784 (0.028) –

–

–

1087

1099

5.31

0.85

0.92

–

–

1056

1072

5.15

0.33

0.93

–

–

1086

1098

5.41

1.30

0.92

–

–

1061

1073

5.6

0.52

0.92

0.099 (0.091) –

3.709 (0.214) –

1138

1146

5.24

0.66

0.93

1.538 (0.034) –

0.332 (0.083) –

1113

1121

5.72

2.14

0.91

–

–

–

1121

1130

5.74

0.99

0.92

–

–

–

1127

1135

5.93

2.57

0.90

0.556 (0.057) –

0.450 (0.007) –

–

1108

1116

5.37

0.83

0.92

–

–

1099

1108

5.21

0.31

0.93

–

–

–

1090

1098

5.16

0.31

0.93

0.110 (0.018) –

34.000 (1.056) 4.088 (0.263) 0.005 (0.006)

12.503 (0.965) 1.241 (0.062) 1.105 (0.075)

1109

1118

5.54

1.25

0.93

1096

1103

5.32

0.84

0.92

1103

1111

5.45

0.28

0.93

0.309 (0.034) –

M6

28.139 (0.535) –

M7

–

M8 M9

3.399 (0.019) –

M10

–

M11

39.671 (6.152) 37.720 (1.959)

M14

–

2.040 (0.115) 0.545 (0.051) 1.199 (0.055) 2.001 (0.243) –

M15

–

–

M12

Validation statistics

M13

0.407 (0.060)

Fig. 2. (a) Residuals in mean terms; and (b) residuals in absolute terms from the projection of dominant height, as a function of the age classes for the clear-cut management regime.

become an issue in the use of the M1 model. For this reason, a new calibration of the parameters was conducted for each management regime with the use of the full dataset from the CFI. Fig. 4 presents the curves generated by models M1 and M3 in which differences in the dominant height growth as a function of time can be seen for the clear-cut management regime for both of the fitted models. For this management regime, although the multiple asymptotes of M3 model could provide an advantage (Kitikidou et al., 2012), the M1 model indicated an asymptotic convergence only after 20 years, far beyond what is usually defined as the rotation age for eucalypts in Brazil. This delayed asymptotic

convergence displayed by the M1 model indicates that for these eucalyptus areas under the clear-cut management regime, there is no illogical behavior associated with the estimates. Additionally, the M1 model showed greater stability in terms of site classification, as 63% of the plots did not change class, 24% changed once, and 13% changed twice. For the M3 model, 56% of the plots did not change class, 33% changed only once, and 11% changed twice or more. Thus, the choice of the M1 model is reinforced as the most appropriate for this management regime. Moreover, in this case, it has the advantage of simplifying the interpretation.

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189

Fig. 3. (a) Residuals in mean terms; and (b) residuals in absolute terms from the projection of dominant height as a function of the age classes for the coppice management regime.

Fig. 4. Site index curves with a base age of seven years for models (a) M1 and (b) M3 in the clear-cut management regime.

For the coppice management regime, the M3 and M1 models showed similar curves (Fig. 5), however, the highest class presented a greater variation. Additionally, the M1 model exhibited an asymptotic convergence only after 40 years. The M1 model again showed greater stability—90% of the plots did not change class, 10% changed once, and in only 1% was more than one change observed. For the M3 model, 76% of the plots did not change class, 18% changed once, and 6% changed more than once. Thus, the M1 model was selected to represent the productive capacity of the locations for the coppice regime. It is worth noting the difference in the asymptotic value between the clear-cut and coppice management regimes, which indicates the existence of a smaller number of observations in the older age classes for the coppice management regime; however, this does not affect the quality of the results. Although the ADA site index curves seem to be less flexible than GADA curves, the tree growth was better assessed by the first approach, which captured the high growth rate for both management regimes especially at ages 2–5. As shown in the study conducted by Cieszewski et al. (2007), the adoption of multiple asymptotes is justified when they con-

verge very early; however, this was not the case for the polymorphic curves with single asymptotes in this study. Bailey and Clutter (1974) arrived at this conclusion also. Given the analyses conducted so far, the M1 model was the one indicated for the sequence of assessments both for the clear-cut management regime and for the coppice management regime.

3.2. Expanding the parameters based on climatic variables for the models selected for each management regime The increase in the predictive ability of the M1 models fitted for the two management regimes was investigated, starting from the expansion of the parameters of the models as a function of climatic variables. Thus, for each management regime, the climatic variables that showed greater significance in the dominant height growth variable were tested. Given that the site classification was appropriate and showed good stability for the study area, the site index for each plot in each management regime was defined (predicted based on the closest age-height pair to 7 years old) at base age 7. Consequently, in accordance with the 20-year climate historic data of the area, it was evaluated which climatic

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Fig. 5. Site index curves with a base age of seven years for the models (a) M1 and (b) M3 in the coppice management regime.

Table 6 b 7 , and the climatic variables. Pearson’s correlation coefficient between predicted site index, SI

*

Climatic variable (monthly mean)

Coppice

p-Value

Clear-cut

p-Value

Rainy days in the year Precipitation Radiation Maximum temperature Mean temperature Minimum temperature Vapor pressure deficit

0.38 0.38 0.14 0.08 0.09 0.17 0.11

0.0000* 0.0000* 0.0958 0.3808 0.2946 0.0678 0.0606

0.23 0.30 0.04 0.16 0.05 0.23 0.07

0.0223 0.0014* 0.5823 0.0852 0.0580 0.0212 0.7615

Significant at the 1% level.

variables that were significantly correlated with the potential productive capacity of eucalyptus (Table 6). The precipitation variable had the greatest impact on the dominant height growth for both management systems; that is, it has a greater impact on the determination of the local potential production capacity. However, while for the clear-cut management regime the mean monthly precipitation was the most significant factor for the dominant height growth, for the coppice management regime, both monthly precipitation and the distribution of the total precipitation throughout the year (rainy days) were highly significant. The addition of the distribution of the total precipitation throughout the year for the coppice management regime makes sense, considering that as a sprouting regime, it already exhibits a decline in productivity in relation to the clear-cut management regime, so any irregularities in the rainy season or dry season end up having a greater impact on the potential capacity of the site to produce timber (Miranda et al., 2015). For the other climatic variables, although they had an impact on the dominant height growth, they had a diminished impact when compared to the effect of precipitation, as already observed by Stape et al. (2004) for a similar region. From the definition of these variables, the next step was to check where the insertion of these variables in the M1 model would make most sense from a biological and statistical point of view. Given this, the expansion of the parameter that represents the asymptote (a), which was called model M1_A, as well as the parameter related to the slope (b), which was called M1_B, were tested. Thus, the expansion proposed for each parameter in the clear-cut management regime was:

M1 A

a ¼ c0 þ c1  Mppt

M1 B

b ¼ c0 þ c1  Mppt

where a and b are the parameters to be estimated; c0 and c1 represent the regression coefficients; and Mppt is the mean monthly precipitation. The expansion proposed for each parameter in the coppice management regime, where M1_C (expansion of asymptote and slope parameters) was included, was:

M1 A

a ¼ c0 þ c1  Mppt þ c2  RainyDays

M1 B

b ¼ c0 þ c1  Mppt þ c2  RainyDays

M1 C

a ¼ c0 þ c1  Mppt and b ¼ d0 þ d1  RainyDays

where a and b are the parameters to be estimated; c0, c1, c2, d0 and d1 are the regression coefficients; Mppt is the mean monthly precipitation, and RainyDays is the distribution of the total precipitation throughout the year. Since dominant height growth is highly correlated with the rainfall pattern, it can be inferred that either the slope or asymptote parameters are already indirectly influenced by rainfall. Therefore, for the coppice management regime this was the reason for decomposing only one of the parameters as a function of both climate variables, i.e., changing slope by climatic variation also changes the asymptote and vice versa. Finally, when decomposing the slope and asymptote parameters as function of both climatic variables may cause multicollinearity or non-statistical significance and/or the degeneration of the original equation, which is based on strong differential calculus properties. Table 7 shows the parameters resulting from the expanded versions of the M1 models as well as the respective precision statistics for the clear-cut management regime. The parameters were significant (p < 0.001) and appropriate for modeling biological growth.

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Table 7 Coefficients and their standard errors in parenthesis of the M1 model with parameters expanded as a function of the climatic and statistical variables of the fit for the clear-cut management regime. Model

M1 M1_A M1_B

Parameters

Fitting statistics

a

b

c0

c1

AIC

BIC

MAE %

T%

MEF

29.004 (0.653) –

0.340 (0.024) 0.33 (0.026) –

–

–

1808

1821

5.89

0.06

0.94

22.922 (0.603) 0.074 (0.026)

0.056 (0.005) 0.002 (0.0002)

1727

1745

5.50

0.10

0.95

1712

1730

5.44

0.05

0.95

29.461 (0.632)

The inclusion of the mean monthly precipitation variable increased the efficiency of the M1 model by 7% for the clear-cut management regime; however, it is important to emphasize that the best performance was in the slope parameter (b). The conditioning of the slope parameter to the expansion, based on climatic modifiers, was also successfully applied in the works of FerrazFilho et al. (2011) and Lopez-Senespleda et al. (2014) for Quercus faginea Lam. in Spain. The M1 model fitted in the traditional manner, and this same fitted model with the inclusion of the climatic variable mean monthly precipitation in the slope parameter (M1_B) behaved similarly among the age classes. The M1_B model was slightly better, especially at the extremes (Fig. 6). For the coppice, the M1 model was expanded by the addition of the mean monthly precipitation variable and the distribution of the total precipitation throughout the year. It can be seen in Table 8 that once again, the expansion of the parameters with these two variables was statistically significant (p < 0.001). Like the clear-cut management regime, the M1 model plus climatic variables (mean monthly precipitation and distribution of the total precipitation throughout the year) increased the efficiency of the estimate by 7%, as also shown by González-García et al. (2015). It is worth mentioning the importance of conducting the tests on the two parameters (a and b) because the modified Von Bertalanffy-Richards model, as a biomathematical model, allows for the consistent representation of the parametric expansion in its asymptote or slope or slope/asymptote—it is up to the researcher to note where this expansion has the greatest impact. Fig. 7 indicates that model M1_A performed better than model M1.

Comparing the predictive ability of the M1 model, we found that the expansion of the parameters with climatic variables resulted in improved performance of both management regimes, without mischaracterizing the expected biological growth behavior. The studies of Bravo-Oviedo et al. (2008), Nunes et al. (2011) and Thapa and Burkhart (2015) – also based on the relationships of climatic variables with the parameters of dynamic equations— indicated an increase in the performance of the models. 3.3. Creation of interannual climate variation scenarios with the models conditioned to the insertion of climatic variables The descriptive models, together with climatic variables, can play an important role in the evaluation of the effects of climatic alterations on forest growth. The expansion of the parameters of the modified Von Bertalanffy-Richards model resulted in an increase in precision for the two management regimes studied. However, much more than the increase in performance of the fitted models, the inclusion of climatic variables makes it possible to create scenarios, that is, to observe how the interannual climatic variation can affect the local potential productive capacity for producing timber. For species such as Pinus sp. or Douglas fir, interannual climatic oscillations do not have a great impact on site index classification because these species—with long rotations—end up not being so sensitive to short climatic variations (Scolforo et al., 2013). Even by reducing their growth in drier periods, those species have the possibility of recovering their regular growth in the following years. However, as eucalyptus grows fast in Brazil and has a short rotation of approximately 7 years, it ends up having

Fig. 6. (a) Residuals in mean terms and (b) residuals in absolute terms from the projection of the dominant height via the normal (traditional) fit (M1) and the M1 model expanded by the addition of the mean monthly precipitation variable (M1_B) for the clear-cut management regime.

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Table 8 Coefficients and their standard errors in parenthesis of the M1 model with parameters expanded as a function of the climatic and statistical variables of the fit for the coppice management regime. Model

M1 M1_A M1_B M1_C

Parameters

Fitting statistics

a

b

c0

c1

c2

AIC

BIC

MAE %

T%

MEF

33.953 (1.340) –

0.206 (0.022) 0.219 (0.021) –

–

–

–

1564

1577

5.18

0.11

0.96

24.039 (1.514) 0.052 (0.002) 22.110 (1.339)

0.102 (0.006) 0.002 (0.0002) 0.097 (0.014)

0.013 (0.0005) 2.39E04 (0.00007)

1499

1520

4.76

0.07

0.97

1501

1522

4.81

0.17

0.97

1499

1520

4.82

0.15

0.97

33.810 (1.184) –

–

d0

0.259 (0.026)

d1

2.39E04 (0.00007)

Fig. 7. (a) Residuals in mean terms and (b) residuals in absolute terms, from the projection of the dominant height via the normal fit (M1) and through model M1 expanded by the addition of the mean monthly precipitation variable and distribution of the total precipitation throughout the year (M1_A), for the coppice management regime.

its growth considerably influenced by climatic variations (Almeida et al., 2004). In other words, a period in one of the years of its rotation that is drier or rainier compared to normal can significantly affects its final yield. Thus, the conditioning of parameters of the fitted model, M1, for the two management regimes studied sought to quantify the response of forest growth under possible climatic conditions occurring due to interannual variations that have been observed in Brazil - specifically for ES and BA in recent years. Figs. 8 and 9 show the flexibility of these fitted models (M1_B for the clear-cut management regime and M1_A for the coppice management regime) with the inclusion of the precipitation variable. This flexibility is important when forecasting the future resources expected from the eucalyptus forests. Fig. 8 and Table 9 corroborate the statements made by Almeida et al. (2004) and Stape et al. (2004) for the clear-cut management regime. They observed a strong correlation between the total precipitation in the area and the dominant height growth. Greater variation was noted in the least productive site index class (23.0 m); that is, poorer site qualities are impacted more by climatic fluctuations. In other words, atypical years may result in large differences in the growth expectation for these sites, if not accounted for in the modeling of the dominant height growth for variable precipitation levels. Additionally, it can be seen that the most productive areas (27.0 m) are less sensitive to climatic variations, explained by the fact that these locations (at least at the present time) have greater availability of resources (nutrient and water availability) for the tree growth.

From Fig. 8, it can be seen that the use of the descriptive model without climatic variables (Traditional or classic model without insertion of climatic variables) indicates close proximity of the estimate of the model with parameter expansion through the mean monthly precipitation variable for the normal scenario in the three site classes. This is an indication of the reliability of the expanded model with climatic variables; that is, in addition to its performing better in predictive terms, this result indicates that the parametric expansion did not alter the biological characteristic of the original model. As in the clear-cut management regime, the behavior of the dominant height growth was analyzed for the coppice management regime in accordance with the climatic scenarios. The scenarios were constructed following the same logic; however, for this management regime, the variables selected were mean monthly precipitation and distribution of precipitation throughout the year (Rainy Days). This combination has the advantage of relating the amount of rainfall to the distribution of rainfall during the year. Once again, it was possible to note the strong response of dominant height growth to the amount and distribution of the total precipitation over the forest stand (Fig. 9). Table 10 and Fig. 9 show the impact of climatic fluctuations on the local site index for the coppice management regime. The least productive site index class experienced a major impact from climatic variations. Additionally, the parametric expansion for the coppice management regime did not change the biological characteristic of the original model.

H.F. Scolforo et al. / Forest Ecology and Management 380 (2016) 182–195

193

Fig. 8. Site curves generated considering the three climatic scenarios for the clear-cut management regime for sites (a) III, (b) II, and (c) I.

Regardless of expanding the slope or asymptote parameter, the important point is the precise representation of dominant height growth variation regarding climatic variation. Choosing one parameter to expand in relation to the other was guided by which provided a superior fit and explanation of the effect of environment on the response variable. As already mention before, expanding the slope or asymptote parameter also affects the parameter not expanded, since all the model parameters are well linked. The great advantage of using the modified Von-Bertallanfy Richards equation is its capacity to simplify and generalize a complex system, i.e., the equation well represents the abiotic and biotic factors that impact in site quality, although they are represented by one response variable - site index. When incorporating the climatic effect, we may use this method for silviculture planning. High or low rainfall scenarios indicate different management pathways and now the expanded model can instantaneously explain this effect in a short time. For instance, for the same soil with different rainfall regimes, eucalyptus growth will be different. Since a eucalyptus tree is highly dependent on water availability, even on a high site quality a decrease in rainfall will reduce eucalyptus growth, since transpiration rates are usually high in Brazil (a tropical country), and therefore, with water limitation the tree will not use the available soil resources because it needs water for survival. Thus, forecasting low or high scenarios may be helpful to plan fertilization or even corrective fertilization at specific times for better tree growth, by avoiding extreme dry or wet moments.

The results also demonstrate that the proposed models with climatic variables for each management regime are capable of updating the local potential timber production capacity, a factor that contributes to more adequate planning of eucalyptus cultivation. For example, Figs. 8 and 9 presented three averaged scenarios for a full eucalyptus rotation. Therefore, it is possible to assess how site index on average vary with different rainfall regimes and the financial risk of that area. These examples illustrate generically how site index changes with different climate patterns when observing the forest in a macro scale. However, it is also possible to observe interannual climatic variation not only on average assuming a full rotation, but by each year. Since the dataset is originating from permanent plots re-measured each year, it is possible to update the dominant height growth for the next year considering the past climatic variation and then propose feasible scenarios that may happen in the year ahead. Therefore, different ways to assess how climatic variation affect eucalyptus in the short run (consistent with a eucalyptus rotation) can be analyzed in order to provide for a more accurate and flexible medium-long term planning. However, potential bias sources need to be investigated, for example, since it is unfeasible to have weather stations in the plots, a well distributed network of these stations are required. Additional tests for possible new scenarios that will be carried out in the next few years will serve as a basis for reinforcing the advantages of using this method as a way of making the mathematical models flexible for possible interannual climatic variations that may greatly affect eucalyptus production in short time.

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Fig. 9. Site curves generated considering the three climatic scenarios for the coppice management regime for sites (a) III, (b) II, and (c) I.

Table 9 Impact of the climate scenarios on the site index for the clear-cut management regime with their confidence intervals in parenthesis. Site class III (23.0 m)

SI7 V

Site class II (25.0 m)

Site class I (27.0 m)

L

N

H

L

N

H

L

N

H

17.8 (15.8, 19.7) 77%

22.6 (20.5, 24.3) 98%

25.6 (24.5, 26.8) 111%

21.3 (19.8, 22.7) 85%

24.9 (23.4, 25.6) 99%

26.9 (25.6, 27.8) 108%

25.2 (24.3, 26.0) 93%

27.1 (26.3, 27.7) 100%

28.2 (27.7, 28.7) 104%

V: Variation; L: Low; N: Normal; H: High.

Table 10 Impact of the climatic scenarios on the site index for the coppice management regime with their confidence intervals in parenthesis. Site class III (19.0 m)

SI7 V

Site class II (23.0 m)

Site class I (27.0 m)

L

N

H

L

N

H

L

N

H

17.5 (17.0, 18.1) 92%

19.6 (18.8, 20.5) 103%

21.7 (20.7, 22.9) 114%

20.9 (20.3, 21.6) 91%

23.4 (22.5, 24.4) 102%

25.9 (24.6, 27.3) 113%

24.2 (23.5, 25.0) 90%

27.1 (26.0, 28.3) 100%

30.0 (28.6, 31.6) 111%

V: Variation; L: Low; N: Normal; H: High.

4. Conclusion The modified polymorphic Von Bertalanffy-Richards model with a single asymptote was found to be the most accurate for predicting the dominant height growth variable with the consequent classification of the productive capacity (site index) of the areas for the clear-cut and coppice management regimes.

The present study indicates that the performance of the models for predicting the dominant height growth can be enhanced by expanding the parameters in accordance with the climatic conditions. The modified biomathematical Von Bertalanffy-Richards model with inclusion of the climatic variable precipitation did not jeopardize its biological behavior, and the insertion of the climatic component in this model provided it with flexibility for updating in accordance with interannual climatic variations.

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