Information provided by traditional growth models is an essential input in decision making processes for managing planted forests. They are generally fitted using inventory data guaranteeing robustness and simplicity. The introduction of explanatory factors affecting tree development in agebased sigmoidal growth and yield equations attempts not only to improve the quality of predictions, but also to add useful information underpinning forest management decisions. This study aimed to assess the use of the following soilbased and physiographic predictors: potentially available soil water (PASW), elevation (Elev), aspect (α) and slope (β) in a system of empirical stand equations comprising: dominant height (h_{dom}), basal area (G), maximum diameter at breast height (d_{max}), and standard deviation of diameters (SD_{d}). Augmented models were compared with the base models through precision and bias of estimations for two contrasting species:
The afforested area in Uruguay grew at a fast rate since the Forestry Law of 1987 provided legal and tax incentives to the sector. Soils prioritized by law occupy 4,420,000 ha currently, and almost a quarter of this area has been planted.
To manage plantations efficiently, it is necessary to use tools that provide accurate growth and yield information, as well as an understanding of the main factors that help differentiate growth and yield between and within regions. It is of special interest to include information that would allow simulations on climate changing scenarios. Agebased sigmoidal equations representing tree growth and yield, fitted using sample plot data usually guarantee robustness and simplicity to these models, but at the expense of some explanatory ability. In this sense, the introduction of factors affecting tree development in growth and yield equations attempts not only to improve the quality of stand projections, but also to add useful information underpinning forest management decisions.
A selection of sitespecific climatic, physiographic and/or soilsbased characteristics, are often considered in assessing productivity (
The objective of this work was to assess the contribution of soilbased and physiographic attributes such as: potentially available soil water (PASW), elevation (Elev), aspect (α), and slope (β), to a system of mensurational stand equations. Those variables are surrogates of key growth drivers such as radiation, soil moisture, and temperature, and have the advantage to be readily available for their use in forestry planning. Based on this background, we hypothesize that the inclusion of this information would improve precision and bias of predictions of height, basal area, and diameter dynamics for
Growth and yield equations were fitted for predicting mean top height or dominant height (h_{dom}), and basal area (the sum of basal area of all trees in a hectare, G) for two species commercially planted in Uruguay, namely:
Based on the KoppenGeigen classification system, the climate in Uruguay is tropicalsub humid without marked variations across the country. This is caused by its position respect to both the Pacific and Atlantic Oceans, and the absence of prominent mountain ranges (
The study was developed in the northern Uruguay, between 30° 50′ and 32° 49′ N and 53° 43′ and 58° 21′ W in three of the zones prioritized for forestry according to the Forestry Law of 1987 (no. 15939). Those areas were defined on production characteristics and limitations of the soils mainly oriented to cattle and wool. Soils with similar production characteristics are grouped in broad categories defined by a number, and plantations occur mainly in groups 7, 8, 9, and 2 (
Information from 974 permanent sample plots (PSP) in georeferenced locations in areas prioritized for forestry was used to fit the equations (
PSPs were linked to the following site characteristics: aspect (α), slope (β), elevation (Elev), and potentially available soil water (PASW), using information publicly available through the Ministry of Cattle, Agriculture, and Fisheries (
Aspect was decomposed to NorthSouth and EastWest components by calculating the sine and cosine of the azimuth angle, respectively. In this way sine values range from 1 to 1 from East to West, and cosine values range from 1 to 1 from North to South. Both components were weighted by the slope using the method proposed by
Procedures involving georeferenced information were developed using QuantumGis (
Datasets for both species were screened by graphical methods before and after computing the required variables, to assess the relationship between variables. The variable h_{dom} was calculated as the mean height of the 100 trees with the largest
For modelling growth of h_{dom}, G, d_{max}, and SD_{d}, several differential equations in polymorphic and anamorphic form were tested (see Tab. S2 and Tab. S3 in Supplementary material). Those were fitted using nonlinear leastsquares as applied by
Candidate equations were compared through the root of the mean square error (RMSE), as a measure of precision; and the mean residual (MR) and mean absolute bias (MAB) as a measure of bias. All three statistics calculated using the modelling dataset were ranked and an overall rank for each model was computed by summing the rank values for all the statistics. The best ranked model for each state variable (h_{dom}, G, d_{max}, and SD_{d}) was selected. Normality was analysed graphically through histograms and QQ plots, and plots of residuals against the variables fitted and the independent variables were also assessed. After selecting the equation for each state variable, the inclusion of predictors was tested using the hypothesis testing dataset. Once the predictors to include were known all the equations were refitted using the modelling dataset. The behaviour of augmented models was assessed by plotting projections for contrasting growth conditions.
For the validation stage, plots of residuals (predicted
Correlations between the soilbased and physiographic variables and site index were initially assessed using the hypothesis testing dataset. High correlation between explanatory variables could interfere in determining the precise effect of each predictor and lead to large standard errors of the parameters. For
For both species the inclusion of a dummy variable (Z7) to distinguish growth in Zone 7 compared to the rest (Zones 9 and 8) was assessed in base models as applied by
In general, models that showed the best fit were polynomial forms of von BertalanffyRichards and Schumacher. Base models selected for
whereas base models adjusted for
For
Augmented equations for h_{dom} and G for
where ω =
An augmented form for d_{max} and SD_{d }did not improved error for the pine species; hence those were not further tested. However, site variables would be indirectly introduced in d_{max} (
For
PASW was significant for all state variables except for SD_{d}, whereas aspect modified by slope was not significant for G. The dominant height curves fitted corresponded to the trajectories of the entire dataset for each species (
An analysis of the influence of the site variables included in
For d_{max}, site variables PASW and
Residual plots using the validation datasets did not show any strong patterns or bias (Fig. S1 to Fig. S5 in Supplementary material). The
PASW was significant for most of the state variables modelled for
The variable PASW used in this analysis is rather theoretical based on a coarse resolution 1:1.000.000 soil map, however it synthesises solidly a series of essential soil characteristics, yielding consistent results with respect to forest productivity. Furthermore, this variable represents potential water availability without interfering on the path invariance property of differential equations, which is a fundamental characteristic to provide robustness to mensurational models.
The effect of slope and aspect on growth varies depending on the species requirements and site characteristics that result from the interaction of mean annual temperature, rainfall regimes, altitude, and velocity and winddirection (
Elevation has no explicit effect on growth but influences key growth factors such as temperature and soil moisture at a local scale. It has been used specifically in mountainous areas for forecasting site productivity (
The augmented models represent an increase in flexibility of curves by including predictors on the shape parameters, the asymptote, or both. Although reductions of the prediction error (calculated using an independent dataset) were modest (2 to 4%), bias was reduced by augmentation of most equations.
To date, models used in Uruguay do not consider site characteristics explicitly. Prognosis systems for
Explanatory components of the models add utilities for forest management, and the information needed is readily available with its quality tending to improve over time. However, care must be taken when using the equations when site variables border on the extremes of the range of values used in this study, especially when using a combination of extreme values. Representation of all possible combinations of explanatory variables’ values is a common issue in forest modelling since the set of PSPs used is restricted. To take a closer look to this potential problem, the extreme values of site variables were plotted against all the values of each of the remaining site variables used, to search for information gaps. Results showed that PSPs combining extreme values of the predictors were rare for the eucalypt species (
Models for both species were adjusted using a dataset comprising a diverse genetic base, covering a large part of the variability for the country. This contributes to the generalization capacity of the models, however adaptations of certain genotypes to sites can occur. Interactions between genotypes and environments were not assessed and are not contemplated in the models.
An information gap was found for
Augmented stand level growth equations have been developed and can be recommended to be incorporated in a prognosis system to simulate height, basal area, and diameter dynamics for
The use of explanatory variables (physiographic and soil variables) decreased the fitting error in a range from 3 to 10.5%, however decreases in the prediction errors calculated with the independent dataset were much lower ranging from 1.6 to 4%. In this sense, main advantage of the augmented models relies on the better description of site differences and their effect on tree growth, consequently adding new possibilities to the use of those models for plantations’ management.
The authors are indebted with the companies that generously have provided data for this study: Global Forest Partners LP and Cambium S.A., Cloverly S.A., Bosques del Sur S.A., and UPMForestal Oriental. We also acknowledge the Directorate of Renewable Resources (RENARE) of the Ministry of Cattle, Agriculture, and Fisheries (MGAP) for contributing with physiographic and soil information. Finally, we thank the reviewers whose valuable comments greatly improved this manuscript. Funding for this study was provided by the Ministry of Foreign Affair and Trade of New Zealand, through NZAid Scholarship, and INIA Uruguay.
Prioritized soils for forestry and plots localization. Produced with spatial information of soils prioritized for forestry (
Range of water potentially available for PSPs of both species. Produced with spatial information of potentially available soil water (1:1.000.000 scale map 
Dominant height curves using base equations (continued black lines), with original plot trajectories (grey lines), and a comparison with augmented equations (dashed lines), for
Effect of site variables on growth curves for
Effect of site variables on growth curves for
Climate characteristics: values corresponding to the period 19802009 (
Descriptor  Rainfall(mm)  Temp.(°C)  A_{f}(days)  Radiation(h day^{1})  RH_{y}(%)  ETP(mm month^{1}) 

Mean  1400  17.7  30  7  74  1100 
Maximum  1600  22.6  40    78  1200 
Minimum  1200  12.9  20    70  1000 
Summary of the dataset used for modelling. (N): total number of sites; (n): mumber of plot measurements; (PSP): permanent sample plots; (G): basal area; (SD): standard deviation.
Variable 




Mean  Min  Max  SD  Mean  Min  Max  SD  
N      669        305   
n  4.00  2.00  11.00    4.00  2.00  11.00   
t (years)  7.13  2  25.9  3.2  6.95  1.18  18.7  3.47 
h_{dom }(m)  10.6  2.2  27  4.65  21.2  4.4  46.6  7.86 
d_{m} (cm)  17.2  2.3  41.9  7.56  17.6  3.1  45.2  7.18 
d_{max} (cm)  21.1  4  46.7  8.71  24.3  5  62.6  8.77 
d_{min} (cm)  12.8  0.48  36.6  6.92  10.1  0.1  41.1  6.64 
SD_{d} (cm)  2.2  0.11  8.42  0.96  3.42  0.64  10.5  1.43 
G (m^{2 }ha^{1})  15.6  0.1  53.6  10.9  19.2  0.78  58.1  8.95 
N (stems ha^{1})  624  100  1667  180  886  87  1775  393 
PSP size (m^{2})  338  200  500  84  682  400  2250  315 
Comparison of RMSE for base and augmented models for each state variable, using the modelling dataset and an independent dataset.
Species  Variable  Modelling dataset  Independent dataset  

Base  Augmented  Base  Augmented  

h_{dom}(m)  0.894  0.867  3.0  0.860  0.850  1.0 
G (m^{2 }ha^{1})  3.151  2.929  7.0  3.230  3.130  3.1  
d_{max }(cm)  1.789      2.080      
SD_{d} (cm)  0.484      0.470      

h_{dom }(m)  1.785  1.685  5.6  1.690  1.660  1.8 
G (m^{2 }ha^{1})  2.847  2.677  6.3  2.760  2.650  4.0  
d_{max }(cm)  2.179  1.951  10.5  2.220  2.180  1.8  
SD_{d }(cm)  0.516  0.494  4.3  0.556  0.560  0.7 
Comparison of MAB for base and augmented models for each state variable, using the modelling dataset and an independent dataset.
Species  Variable  Modelling dataset  Independent dataset  

Base  Augmented  Base  Augmented  

h_{dom}(m)  0.69  0.67  0.67  0.66 
G (m^{2 }ha^{1})  2.25  2.13  2.53  2.21  
d_{max }(cm)  1.32    1.52  1.49  
SD_{d} (cm)  0.34    0.32  0.32  

h_{dom }(m)  1.32  1.28  1.29  1.27 
G (m^{2 }ha^{1})  2.00  1.90  2.05  2.05  
d_{max }(cm)  1.61  1.45  1.58  1.55  
SD_{d }(cm)  0.38  0.37  0.41  0.41 
Fig. S1  Residuals using the validation dataset for h_{dom} for
Fig. S2  Residuals using the validation dataset for
Fig. S3  Residuals using the validation dataset for
Fig. S4  Residuals using the validation dataset for
Fig. S5  Residuals using the validation dataset for
Tab S1  Distribution of plots across predictors’ ranges.
Tab S2  Polymorphic forms of equations tested.
Tab S3  Anamorphic forms of equations tested.
Tab. S4  Parameters of the equations selected for modelling h_{dom}.
Tab. S5  Parameters of the equations selected for modelling G.
Tab. S6  Parameters of the equations selected for modelling d_{max}.
Tab. S7  Parameters of the equations selected for modelling SD_{d}.