Individual tree growth models for eucalyptus in northern Brazil

The diameter and height growth model is one of three submodels used for simulating individual tree growth. In Brazil, there are few studies on the dimensional growth of individual trees be they native or exotic species, despite their potential. This study aimed to evaluate diameter and height growth models for individual trees for eucalyptus stands and to validate the best fitting model. Tree diameter and height data were obtained from 48 permanent plots of unthinned stands of Eucalyptus grandis × Eucalyptus urophylla hybrid located in northern Brazil. The evaluation of the diameter and height growth models was based on adjusted coefficient of determination, standard error of estimate as a percentage, trend, root mean square error and Akaike Information Criterion. Analysis also included distribution of residual percentage, statistical significance and signs of the coefficients. The Lundqvist–Korf model provided the most accurate estimates for diameter and height growth, in comparison with the other models, providing better statistical values, greater proximity to observed values and better distribution of residual percentages. The use of this type of model is feasible and can result in significant improvements in the accuracy of yield estimates.


Introduction
Dimensional growth in terms of diameter and height is one of the three constituents of an individual tree model and is subject to complex interactions (Andreassen and Tomter, 2003;Soares and Tomé, 2002).It is influenced by growth vigor, past growth conditions, microenvironment, genetic traits and competitive status (Tomé and Burkhart, 1989).
In models at the level of the individual tree, growth is often estimated via the potential growth function or growth equations (Davis et al., 2005).In the potential growth function, growth is obtained by multiplying potential growth (Pg) by a modifier function (fm) (Biging and Dobbertin, 1992;Soares and Tomé, 1997).Pg describes the maximum possible growth that a tree can attain, whereas fm describes the decrease in growth potential due to competition (Kiernan et al., 2008).In contrast, growth equations (or functions) use tree attributes (tree size, competition indices, crown ratio, vigor), stand attributes (age, site index, stand basal area) and site characteristics as predictor variables, all combined in a single equation (Uzoh and Oliver, 2006).
Several equations are used to estimate growth, including linear or polynomial equations (Kiernan et al., 2008), the Bertalanffy equation (Vanclay, 1994), the Richards equation (Amaro et al., 1998), the Gompertz equation, the logistic equation and the exponential equation (Zeide, 1993) in addition to nonlinear functions (Zhang et al., 2004).However, the superiority of modifier functions over growth equations (or functions) has not been demonstrated.
Many studies have been conducted on model growth in diameter and height at the individual tree level in forests in the USA and Europe (Biging and Dobbertin, 1992;Lynch and Murphy, 1995;Tomé and Burkhart, 1989;Vospernik et al., 2010).In Brazil, there have been few studies estimating forest growth at this level.Existing studies refer to native species such as Cabralea canjerana (Durlo, 2001), araucaria (Araucaria angustifolia) (Chassot et al., 2011), cedar (Cedrela fissilis) (Durlo et al., 2004) and certain species from the Amazon region (Silva et al., 2002).On the other hand, there are no models that estimate growth at the individual tree level for planted commercial species.Because of the importance of the genus Eucalyptus in Brazil, with more than four million hectares planted (ABRAF, 2011) and the gap in growth modeling at individual tree level, the aim of this study was to evaluate and compare various diameter and height models for individual Eucalyptus grandis × Eucalyptus urophylla trees.

Materials and Methods
The study was conducted in Monte Dourado, in the state of Pará, Brazil (0º53'22" S, 52º36'6" W, 65 m a.s.l.).The region has a tropical monsoon climate (Am), with an average annual precipitation of 2.115 mm and a short dry season between Sept and Nov, according to the Köppen classification.The average annual temperature is 26.4 ºC, and average relative humidity ranges between 80 % and 85 % (Martins et al., 2011).
Data were obtained from 48 permanent plots (1997 to 2003) in a continuous forest inventory of unthinned stands of hybrid Eucalyptus grandis W. Hill ex Maiden x Eucalyptus urophylla S.T. Blake (urograndis).Thirty of these plots were used for model fitting, and the others were used for model testing (simulation).Each plot was 500 m² with spacing between trees of 3 × 3 m.The following measurements were made: diameter 1.3 m above ground level (dbh) of all trees using a caliper, total height (Ht) of the first 15 trees (Table 1) and total height (Hd) of the five dominant trees, using Vertex IV (Campos and Leite, 2009).To estimate Ht for the remaining trees, a hypsometric equation was used, which was fitted to the site (Martins et al., 2011) (eq. 1): 9876 -30.4340 .exp(-0.000499 .(dhb .ln(Hd ln(t)) 1.388275  (1) where dbh = diameter outside bark as measured 1.30 m above ground level (cm); Hd = average height of dominant trees (m); t = age (months); 2 R % = adjusted coefficient of determination (percentage); and S y.x% = standard error of estimate (percentage), both computed in the original units of the dependent variable (m).The heights of dominant trees (Hd) were also used to classify the productive capacity by means of site indices (SI) (eq.2) via the guide curve method (Campos and Leite, 2009), which correlates the height of dominant trees with the stand age at an index age.ln SI = ln(Hd) + 14.8802 .
The thresholds used for plot classifying into productivity classes were: i) low productivity class (SI = 20, which represents the center value of the class): plots with Hd ≤ 23 m with index age of 60 months; ii) average productivity class (SI = 26): plots with Hd between 23 and 29 m; and iii) high productivity class (SI = 32): plots with Hd > 29 m.
Seven models for estimating the diameter and height growth of an individual tree were evaluated (Table 2).As the growth models used in this study have a projection structure (t → t +1 ), the Durbin-Watson test (dw) was applied to verify autocorrelation in these models, which all have a similar autoregressive structure (Gujarati, 2004).This analysis evaluated linear and nonlinear relationships.Models were fitted independently to each productivity class (Martins et al., 2011) (SI = 32, SI = 26 and SI = 20).
Two model fitting tools were used in this study: the SAS software MODEL procedure (Statistical Analysis System, version 8.0, 2001), with maximum likelihood fitting for nonlinear estimation, and the nonlinear estimation procedure from Statistica software (Statsoft, version 7.0, 2008), which uses a variant of the Gauss-Newton method to estimate the parameters of nonlinear regression by the least squares method.
The fit of the equations of the seven models was verified with the following statistics: the adjusted coefficient of determination ( 2 R ), trend (BIAS), standard error of estimate in percent (S y.x% ), root mean square error (RMSE) and Akaike Information Criterion (AIC) (Gujarati, 2004): where y i = i-th observed value for the dependent variable; i y = i-th estimated value for the dependent variable; i y = mean observed value for the dependent variable; n-1 = degrees of freedom of the total in the analysis of variance; n-p-1 = degrees of freedom of the residual from the analysis of variance of the regression, p = number of coefficients and n = number of observations.The model resulting in the greatest 2 R , least BIAS, S y.x% , RMSE and AIC was selected as the best model.In addition to the above statistics, graphs were developed to compare observed diameters and heights with those estimated by models, as well as graphs of the distribution of residual percentages (res%) relating to estimated diameters and heights (Vanclay, 1994).Volume (m³ ha -1 ) 23.8 -353.9 Density (trees ha -1 ) 760 -1180 Ht is the distance between the ground and the top of tree, and Hd is the five trees heights of largest dbh in each plot.

Sci
Pienaar and Schiver (1981)  ( ) Adapted Schumacher equation Campos and Leite ( 2009) Linear / adapted from Bella (1971) and Campos and Leite ( 2009) The best-fitting model to represent diameter and height growth was also tested with independent data.Simulation was performed on 18 permanent plots whose evaluation was based on the RMSE.The homogeneity of variance was tested between the observed values of diameter and height and the values estimated by the best model in each productive capacity class using a Bartlett's test (H 0 = homogeneous variances versus H 1 = heterogeneous variances).This same Bartlett's test can be used to test the absence of normality, and this test was used in this study.The observed mean values of diameter and height were compared, using a t-test, with the mean values estimated by the best model in each productive capacity class.

Results
In this study, all equations referring to the models assessed for the variables diameter and height in all three productivity classes provided values that were very close to each other according to the relevant statistics ( 2R , BIAS, S y.x%, RMSE and AIC) (Table 3).The models with the best fit were those for SI = 32, followed by SI = 26 and finally SI = 20.Individual tree diameter and height growth models are among the basic and essential components of forest growth models (Sánchez-González et al., 2006).
The best estimates of diameter were obtained using models 1, 2 and 6.Only a few equations showed significant autocorrelation according to a dw test (Table 3).In these cases, it is incorrect to compare the 2 R values for the estimators with those reported by other studies because the ordinary least squares estimators are biased (Gujarati, 2004).
Generally, lower estimates of 2 R (0.28 to 0.83) were found by Sterba and Monserud (1997) and by Andreassen and Tomter (2003) for the basal area increment of Pinus sylvestris, Picea abies and Betula sp.Tomé and Burkhart (1989) and Adame et al. (2008) also found lower estimates (0.51 to 0.54) for the diameter increment of Eucalyptus globulus and Quercus pyrenaica (0.44).Lower estimates were also obtained by Soares and Tomé (1997) (0.99) using the Lundqvist-Korf (L-K) model for the quadratic dbh mean of Eucalyptus globulus.
All equations based on the models provided very similar values for height for all the statistics assessed.The best equations involving height were those based on models 2 and 6, as they provided higher 2 R values and  3).The statistic 2 R was higher in model 2 in all productivity classes, with estimates between 0.97 and 0.99 with values referring to S = 20 and 32, respectively.Lynch and Murphy (1995) and Mabvurira and Miina (2002) found similar estimates of 2 R for the height growth of Pinus echinata (0.96 to 0.98) and Eucalyptus grandis (0.94) with different models.Filipescu and Comeau (2007) and Mette et al. (2009) obtained lower estimates for Picea glauca (0.47 to 0.86), Abies alba (0.18 to 0.89) and Picea abies (0.39 to 0.82), using different height increment equations.
Virtually, all coefficients of the equations were significant (p ≤ 0.05) (Table 4).The sign of the coefficient of age was negative (models 1, 5, 6 and 7), indicating that growth increases with increasing age until a certain point is reached (the index age), whereas other variables remain constant.These findings are biologically consistent and similar to results found in other growth studies at the individual tree level (Lee et al., 2004;Subedi and Sharma, 2011).Additionally, the signs of the site index and competition index were positive for diameter and height growth in all productivity classes.These results show that trees will reach greater diameters and greater heights in better sites (Adame et al., 2008) where there is a greater opportunity to compete.Thus, these results are consistent with the literature (Mabvurira and Miina, 2002;Lee et al., 2004;Sánchez-González et al., 2006) and are biologically realistic, reflecting good estimates for all diameter and height growth models evaluated in this study.
The values estimated by the equations based on the seven diameter growth models were concentrated near the 1:1 line (Figure 1), indicating a good estimation capability for all three productivity classes.Models 3, 4 and 5 had a slight tendency to overestimate smaller diameters and underestimate larger diameters.This trend was also observed by Sterba and Monserud (1997), Mabvurira and Miina (2002), Mette et al. (2009) and Vospernik et al. (2010).These trends are common and difficult to explain, with overestimation occurring more often in low-density stands and underestimation more often in high-density stands (Vospernik et al., 2010).
Models 1, 2 and 6 performed well in terms of the assessed statistics (Table 3) and managed to accurately estimate tree diameter in all three productivity classes.Nevertheless, model 1 underestimated the diameters of the larger trees, those in SI = 32.model 6 overestimated the diameters of the larger trees in a narrow range in SI = 20.This result is similar to that reported by Filipescu and Comeau (2007) for Picea glauca.
The residual percentages for diameter growth (Figure 2) were well distributed for models 2, 6 and 7 in all three productivity classes.Despite its good statistical performance, model 1 showed a slight deficiency in its residual distribution, underestimating trees of larger diameter in all three productivity classes.Models 3, 4 and 5 had a marked deficiency in their residual distributions and failed to capture the growth trends for trees of smaller and larger diameters (S = 32, 26 and 20).These three models overestimated the small-diameter region and underestimated the large-diameter region.This trend was similar to that reported by Härkönen et al. (2010).For diameter growth, the equations based on models 2, 6 and 7 succeeded in accurately estimating diameter variation in all three productivity classes and did not exceed a residual percentage of ± 20 %.Nevertheless, model 2 showed better estimates in terms of the assessed statistics and had well distributed residuals.Wykoff (1990), Kiernan et al. (2008) and Monty et al. (2008) obtained similar results, for the basal area increment, diameter growth and circumference, respectively, for different species.Andreassen and Tomter (2003) found variations of more than 20 % for the basal area increment in Picea abies, Pinus sylvestris and Betula sp and Härkönen et al. ( 2010) reported a ± 17% for Pinus sylvestris, ± 26 % for Betula pubescens and ± 35 % for Betula pendula.
For height growth (Figure 3), the points were close to the 1:1 line in all models.As in the case of diameter growth, a tendency of the models to produce incorrect estimates was noted in height growth for trees of smaller and larger sizes.Model 1 overestimated smaller and larger trees in all three productivity classes.Model 2 slightly overestimated larger trees, with a wider range in SI = 32.Models 3, 4 and 5 showed a pattern similar to each other, with little variation among the three productivity classes.SI = 32 and 26 showed overestimates for trees of smaller height, whereas SI = 20 showed overestimates for trees of smaller height and underestimates for taller trees.Models 6 and 7 showed a slight tendency to overestimate smaller trees in all productivity classes and a strong tendency to overestimate larger trees.
Results from Soares and Tomé (2002) corroborated those obtained from most of the models evaluated in this study.Soares and Tomé (2002) state that this response is due to data quality and to equation fitting, with equation accuracy decreasing as the productivity class gradually decreases.Additionally, height is a difficult variable to measure and must be obtained indirectly via hypsometric equations that, in turn, contain intrinsic inaccuracies.
The trends described above were validated by Figure 4.In all three productivity classes, model 1 failed to accurately estimate smaller heights, particularly between 10 and 15 m, and heights greater than 30 m. Model 2 was superior to the other models, providing good fits to the height data.Models 3, 4 and 5 had good performance only at intermediate and greater heights.2010) also found it difficult to estimate height growth, particularly for trees of smaller (10 m) and larger sizes (25 m).It is important to note that trees in all height classes directly influence the overall volume attained (Mette et al., 2009).Therefore, models that are incapable of capturing variation in a given height range should be avoided even if they provide good estimates for the remaining height classes.Only model 2 was capable of estimating height growth for the three productivity classes with a residual percentage not exceeding ±20 %.Except for where SI = 32, model 2 was also accurate in simulating diameter and height growth in individual eucalyptus trees (Figure 5) using independent data.One of the reasons for the superiority of the L-K model is the   functional relationship and flexibility of the equation, whose coefficients have biological significance (Amaro et al., 1998).Thus this model is among the functions most commonly used to estimate growth phenomena (Burkhart and Tomé, 2012).Crescente-Campo et al. (2010) reported serious heteroscedasticity problems in equations for the basal area increment, as well as non-normality of errors in equations for the diameter and height increment, in contrast to the results of this study (Table 5).A Bartlett's test indicated that the assumptions of normality were met for the observed and estimated diameters by the L-K model in all three productivity classes (p > 0.05).The same was true for height except where SI = 26.Additionally, this test confirmed homogeneity of variance for the observed and estimated diameters in all three productivity classes and for the observed and estimated heights in classes SI = 32 and SI = 20, a desirable result.
No difference (p > 0.05) was found between the observed mean values and the values estimated by the L-K model for diameter and height (S = 32 and 20).For diameter, the RMSE was less than 1 cm in all three productivity classes, a result similar to those reported  by Bueno and Bevilacqua (2010).Nevertheless, the simulation of height was less accurate than the simulation of diameter, reflecting the difficulty found in fitting an ideal model to the variable height.A visual analysis of the observed and estimated values (Figure 5 an underestimation of height in trees taller than 30 m (SI = 32) and in trees taller than 22 m (SI = 26).However, RMSE was 0.59 m for SI = 32 and 0.91 m for SI = 26.An error in the range of 0.50 m can be considered low and totally acceptable from the standpoint of height modeling because the top portion of the tree is typically ignored for commercial purposes.However, an error greater than 0.50 m can be harmful, as it could affect the estimation of volume.One approach to the problem of estimating height is to use hypsometric equations with simulated data with regard to diameter, as the simulation of diameter was excellent.The L-K model showed greater consistency between the statistics and biological reality and thus provided the most accurate and best fitting model for diameter and height growth.

Discussion
Although the models in this study include diameter and height explicitly as dependent variables, many studies use alternative forms of diameter and height growth as dependent variables (e.g., diameter and height increment, diameter and height growth rate, square increment and natural logarithm of each), as well as growth modifier functions (Adame et al., 2008).However, modeling the diameter and height increment is not the only alternative for predicting tree growth, and other variables have been modeled, including future diameter and height (Bueno and Bevilacqua, 2010).All are alternative approaches to estimating the increase in stem and height size.They are mathematically related, and few differences in the outcome of the modeling process are expected (Vanclay, 1994) if the assumptions regarding the error term are met (Bueno and Bevilacqua, 2010;Zhang et al., 2004).
In Brazil, the use of diameter and height as dependent variables in models to assess growth at the individual tree level is common (Chassot et al., 2011) and conceivable (Campos and Leite, 2009).Bueno and Bevilacqua (2010) compared the two approaches in modeling the diameter growth of Pinus occidentalis and found that the estimates of future diameters (as used in this study) showed lower errors if directly projected by the model than those resulting from estimates using the increment in diameter.A possible justification is that the periodic increment varies significantly as a function of the environmental conditions of the study site (Garcia, 1988), which are extremely variable in northern Brazil.Additionally, this problem is exacerbated in rapidly growing species such as eucalyptus, as they show large increments in comparison with slowly growing species.
Growth equations can be derived directly from functions correlating diameter and height as a function of independent variables such as competition index, site, stand height and stand density (Davis et al., 2005;Lynch and Murphy, 1995).However, there has been no confirmation of the universal superiority of one dependent/independent variable over another or of the performance of modifier functions relative to that of growth equations.The choice of one function in preference to another and the functional relationship chosen between variables (Sánchez-González et al., 2006;Soares and Tomé, 1997) will depend on the interests and convenience of the researcher (Vanclay, 1994).
A large number of growth models with numerous combinations of variables are continually evaluated and tested for a wide variety of species (Uzoh and Oliver, 2006).It becomes more difficult, however, to select the best model to estimate growth if the modeling unit is an individual tree (Davis et al., 2005) because the high resolution of this type of modeling entails problems caused by cumulative errors (Cao, 2006).Even with such difficulties, it was nevertheless possible to obtain a good estimate of diameter and height growth using model 2.
The reasons for prefering model 2 (L-K) is that this model provided better statistical estimates, as the values estimated by model 2 were close to the observed values in terms of accuracy for diameter and height growth (Figures 1 and 3).This model also showed a good distribution of residual percentages (Figures 2  and 4), and accurate estimating of diameter and height growth in all size classes (smaller, intermediate and larger trees) in all three productivity classes.Model 2 was also found to be accurate in simulating diameter and height growth in individual eucalyptus trees (Figure 5), revealing greater consistency between the statistics and biological reality and thus providing the most accurate and best fitting model for diameter and height growth.
In Brazil, individual tree growth models are still rarely used to model growth and yield.Most applications use whole-stand and size-class models.A major reason for this practice is that models on an individual tree level are considered more complex; because of this belief, users in Brazil have little experience with this type of model.However, the results presented in this study show that the use of this level of modeling is feasible and can offer significant improvements for estimating growth and yield more accurately.Moreover, it is a flexible type of model because it provides detailed information about dynamics and stand structure, including the distribution of volume by size class.From this detailed information, it is possible to correct projections for different uses of timber, sawmill, lumber, paper, plywood, charcoal, pulp and biomass and to understand how competition and the site index impact growth.
7 failed to estimate the endmost values of the variable height, particularly initial values smaller than 18 m.Mette et al. (2009) and Härkönen et al. (

Figure 1 -
Figure 1 -Diameter growth (dbh) as estimated and observed by equations based on the seven models in each productivity class.The solid line is the 1:1 line.

Figure 2 -
Figure 2 -Residual percentages of equations based on the seven diameter growth (dbh) models as a function of estimated diameters in each productivity class.

Figure 3 -Figure 4 -
Figure 3 -Height growth as estimated and observed by equations based on the seven models in each productivity class.The solid line is the 1:1 line.

Figure 5 -
Figure 5 -Diameter and height simulated by the Lundqvist-Korf model in all three productivity classes.

Table 1 -
Characteristics of Eucalyptus grandis × Eucalyptus urophylla located in Monte Dourado, in the state of Pará, Brazil.

Table 2 -
Models used for estimating diameter and height growth for individual eucalyptus trees.

Table 3 -
Statistics used for evaluating the seven diameter and height growth models.S yx % = standard error of estimate in percent, RMSE = root mean square error, AIC = Akaike information criterion, dw = Durbin Watson test.ns = the null hypothesis could be rejected at 5 % level of significance (there is autocorrelation).

Table 4 -
Estimates of coefficients of equations based on the seven models assessed for diameter and height growth in each productivity class.

Table 5 -
Mean and variance related to diameters and heights observed and simulated by the Lundqvist-Korf model in all three productivity classes.Sim = simulated values; ns= not significant; *significant at 5 % by the t-test (compares the observed mean with each estimated mean) and by the Bartlett's test (compares the observed variance with each estimated variance) in all three productivity classes.