Modeling dominant height growth including site attributes in the GADA approach for Quercus faginea Lam . in Spain

Aim of the study: To develop a site index model for Quercus faginea Lam. stands. Area of study: Spain Material and methods: Data from 81 growth series collected in plots where Q. faginea was the main species were used for modelling. Different generalized algebraic difference equations (GADA) were fitted from traditionally used models. Richards model was selected and used to expand the parameters with environmental variables. Research highlights: Winter rainfall (WR), annual potential evapotranspiration (PET) and pH were introduced increasing the prediction ability of the GADA. It is strongly recommended to apply the model with ages lower than 80 years because the lack of data above that age makes bias increase and efficiency decrease.


Introduction
Lusitanian oak (Q.faginea Lam.) is a semi-deciduous species widely extended in the Iberian Peninsula with a fragmented distribution (Costa et al., 2005).Despite their multiple benefits, its stands have been reduced to the current distribution in favour of the holm oak (Quercus ilex L.), because of its better ability to sprout from both root and stump and the sweetness of its acorns.
Dominant tree height at a given age is commonly used as a measure of site quality, which in turn expresses the potential productivity of a site for a particular tree species (Clutter, 1983;Ortega & Montero, 1988).There are many studies about growth equations for Mediterranean species like cork oak (Sánchez González et al., 2005;Sánchez González et al., 2010), stone pine (Calama & Montero, 2004), pyrenean oak (Adame et al., 2006), holm oak (Gea-Izquierdo, et al., 2008), juniper (Alonso Ponce, 2008) and Mediterranean maritime pine (Bravo-Oviedo et al., 2007;Bravo-Oviedo et al., 2008).Nevertheless, there are few studies about site index for Q. faginea (San Miguel, 1986;López-Senespleda & Sánchez-Palomares, 2007), thus the aim of this work was to build a height-age site index model for Q. faginea stands in Spain, based on stem analysis covering the main range of its distribution area.The functions tested in this study were Richards, Gompertz, Levakovic III & Hossfeld IV (Richards, 1959;Kiviste et al., 2002), using the Generalized Algebraic Difference Approach (GADA) (Cieszewski & Bailey, 2000) including environmental variables in the model.each log, from the bottom to the tree top.The TSAP © software (Rinn, 2005) was used combined with a linear positioning digitizer tablet (LINTAB) to measure the tree-rings and age for each disk.For the true height estimation at a given age, the Carmean's algorithm was used (Carmean, 1972).Finally 81 growth series were used to build the site index models with a mean age of 60 years (range 27-120 years), and a mean height of 7.9 m (range 4.9-14.8m).

Modeling approach
To apply the GADA approach (Cieszewski and Bailey, 2000), the growth functions must be expressed via dynamic equations obtained through the generalization of the main parameters.The methodology could be summarized by the following steps (Krumland and Eng, 2005): i) Select a suitable base equation; ii) Identify all the parameters that potentially change for different levels of site productivity and reformulate the base equation by replacing these parameters as function of X; iii) Solve the resulting GADA formulated model for X and f ind a specif ic solution in terms of initial conditions of height and age (H 0 , t 0 ).The GADA approach has desirable attributes when modeling dominant height equations, i.e., polymorphism, inflection point, asymptote, biologically acceptable behavior and base-age invariance (Goelz and Burk, 1992;Álvarez González et al., 2004).In order to fit the models, the dummy variable approach was apply (Cieszewski et al., 2000).

Model selection
The evaluation of the models was based on qualitative and quantitative examinations of the residuals of the fitted models and the plotted curves.The first step was to evaluate the statistics of the fitted models (Amaro et al., 1998): the bias or the deviation of the model with respect to the observed values, the mean square error (MSE), the adjusted coefficient of determination (R 2 adj ) and Akaike's information criterion (AIC) (Akaike, 1974).
The second step was focused on the graphical analyses.The biological sense of the model was evaluated through the shape of the fitted curves and the value of their asymptotes.Once the analysis of the logical and biological consistency of the parameters estimation was made, the selected functions were subjected to Jackknife cross-validation (Davidson and Hincley, 1997).
In order to set the age of reference, the relative error in height and the number of observations were represented by age classes.The site index predictions with age were plotted to characterize their consistency also (Álvarez González et al., 2004).

Parameter expansion
Once a GADA model was selected, the expansion of the parameters was performed.In order to select the environmental variables to be used, a Principal Component Analysis (PCA) was made and the variables which accounted for the maximum amount of variance were chosen and combined in different growth theory assumptions (Bravo-Oviedo et al., 2008).The model selection was carried out with the same criteria than in the previous section.Finally, the model with the expanded parameters and the GADA model where compared to analyze their behavior.

Results and discussion
All the models were fitted with a second order continuous autoregressive error structure (Gregoire et al., 1995;Zimmerman and Núñez-Antón, 2001;Diéguez Aranda et al., 2005), which seems to be more appropriate to account for autocorrelation.Only 6 of the 24 tested equations met the convergence criterion (Table 1).
All the models showed an appropriate graphic performance except models M5 and M6 which had the poorest fitting values.The other models exhibited very small differences among them.
The asymptotes of the M1 model are 7.5-20.6m with t 0 = 50 years and H 0 = 4-12 m as initial conditions.In the M2 model, the asymptotes are 7.7-19.7 m, and in the M3 model are 9.2-27.5 using the same initial Resource Communication.Height group model for Quercus faginea conditions in both models.The M4 model has a unique asymptote of 22.9 m.Finally, model M1 was selected because of its better goodness of fit statistics and biological behavior (Fig. 1a).The age of reference chose is consistent with the results plotted in Fig. 1e.Finally, the site index shows an acceptable stability which increase with age (Fig. 1b).
The environmental variables extracted from the PCA were pH, fine earth fraction (FE in %), winter rainfall (WR in mm) and annual potential evapotranspiration (Thornthwaite, 1948) (PET in mm).Different structures combining the environmental variables were tested to expand the parameters in the M1 model.The values of the fitted parameters, the coefficient of determination, MSE, AIC and bias obtained for each of the four structures are summarized in Table 2.
The inclusion of climatic and edaphic variables into the model corrected the bias showed by the mo-del approach without environmental variables and increase the model efficiency.The model selected was CM1.4 whose final equations are: [1] [2] [3] [4] H and t means height and age respectively and the 0 subscript means initial conditions.
In order to assess the influence of the environmental variables over the model, the curves were plotted with one free variable while the other two remains f ixed (mean value).For pH values of 8 and 5 the asymptotes of the model varies between 11.9-16.3m respectively.If PET varies between 550-750 mm, asymptotes varies between 13.4-12.7 m respectively, and finally for WR values of 75 and 275 mm, asymptotes varies between 11.5-13.8m.
The inclusion of environmental variables into the model allows comparing variations in growth pattern between different sites with different climatic conditions or comparing the growth pattern of the same site under different future climatic conditions.The differences between two sites with the same height at the same age could follow different growth patterns as a result of the climate conditions, that is, the intraregional variations that existed during the observed growing period (Bravo-Oviedo et al., 2010).
Bias and eff iciency were analyzed with data distributed in age classes.The bias increases and the efficiency decreases (Figs.1c and 1d) when ages over 80 years are used mainly due to lack of suff icient data in older trees.Thus, it is strongly recommen- ded to use the model for ages below 80 years, which on the other hand includes the rotation period in Lusitanian oak coppices forests (Serrada et al., 2008).

Figure 1 .
Figure 1.(a) GADA M1 (Richards base model) potential growth curves (81 series) for site indexes 4, 6, 8, 10 and 12 m at a reference age of 50 years.Grey thin lines correspond to the growth series used to fit the model.(b) Site index predictions vs age.(c) and (d) shows the model efficiency and the bias classified in age intervals.Dashed line correspond to M1 model, solid line correspond to CM1.4 model.(e) Relative error (%) in height prediction (solid line) and number of observations n (dotted line).

Table 1 .
Dynamic equations and solutions for X 0 used in the four models or base equations

Table 2 .
Parameter estimates and statistics for the model structures analyzed