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Revista mexicana de ciencias forestales
versão impressa ISSN 20071132
Rev. mex. de cienc. forestales vol.9 no.50 México Nov./Dez. 2018
https://doi.org/10.29298/rmcf.v9i50.261
Articles
Volume equations for Prosopis articulata S. Watson and Lysiloma divaricata (Jacq.) J. F. Macbr. in northwestern Mexico
^{1}Facultad de Ciencias Forestales, Universidad Autónoma de Nuevo León. México.
Allometric models are a tool of great importance for forest management, as they allow reliable estimations of the volume of both individual trees and forest stands, as well as the construction of volume tables in order to facilitate the quantification of timber stocks. In the present work, data of 160 trees were adjusted; 80 trees of each species were cut, covering all the diameter categories; the variables measured were: normal diameter (D), base diameter (DB) total height (H), and crown diameter (DC), as well as the diameters and lengths of each of the branches with a diameter above 6 cm at its base. Linear, multiple and nonlinear regression models with one and two independent variables were tested to estimate the total stem volume with bark. For the selection of the best model, the RMSE, adjusted R^{2} and the level of significance of the parameters were taken into account. For the prediction of the total volume according to the normal diameter, the allometric model of Berkhout presented better adjustments, with R^{2} of 0.9053 and RMSE of 0.0432 for Prosopis articulata and R^{2} of 0.8178, as well as RMSE of 0.1048 for Lysiloma divaricata. Using the variables normal diameter and total height, the best adjustments were obtained with the SchumacherHall model, with values of 0.9139 in R^{2} and RMSE of 0.0411 for Prosopis articulata. For Lysiloma divaricata the best fit model was the Spurr potential with R^{2} of 0.7936 and a value of RMSE of 0.1118.
Key words: Diametric categories; forest management; model; prediction; volume tables; volume
Una herramienta de gran importancia en el manejo forestal son los modelos alométricos, con los cuales se puede estimar de forma confiable el volumen en árboles individuales y masas forestales, además de construirse tablas de volumen que faciliten la cuantificación de existencias maderables. En el presente trabajo se ajustaron datos de 160 árboles; para ello, se derribaron 80 individuos de cada especie que incluyeron todas las categorías diamétricas. Se midieron las variables: diámetro normal (D), diámetro de la base (DB) altura total (H) y diámetro de copa (DC), así como los diámetros y longitudes de cada una de las ramas mayores a 6 cm de diámetro en su base. Se probaron modelos de regresión lineal simple, múltiple y nolineal con una y dos variables independientes, para estimar el volumen fustal total con corteza. La selección del mejor modelo se hizo con base en la RCME, R^{2} ajustado y el nivel de significancia de los parámetros. Para la predicción del volumen total en función del diámetro normal, el modelo alométrico de Berkhout presentó mejores ajustes, con R^{2} de 0.9053 y RCME de 0.0432 para Prosopis articulata; y R^{2} de 0.8178, RCME de 0.1048 para Lysiloma divaricata. A partir del diámetro normal y altura total, los mejores ajustes se obtuvieron con el modelo de SchumacherHall, con valores de 0.9139 en R^{2} y RCME de 0.0411 para Prosopis articulata; para Lysiloma divaricata el modelo de mejor ajuste fue el Spurr Potencial con R^{2} de 0.7936 y un valor de RCME de 0.1118.
Palabras clave: Categorías diamétricas; manejo forestal; modelo; predicción; tablas de volumen; volumen
Introduction
The importance of generating biometric systems lies in the integration of static or dynamic statistical mathematical models for a rational, sustainable management of forests. Today these models are the most widely used analytical tools for generating knowledge about the production and growth of forest masses (^{Weiskittel et al., 2011}; ^{VargasLarreta et al., 2017}). This is achieved by measuring certain variables, such as the normal diameter and the total height; subsequently estimating the volume of each tree, and, finally, extrapolating the information to a whole stand (^{AcostaMireles and CarrilloAnzures, 2008}; ^{CruzCobos et al., 2016}).
Simulation with mathematical models is helpful for making decisions in sustainable forest management. Volume models predict the total volume of trees with and without bark, as well as of the clean trunk, total stem and commercial stem; furthermore, it is the main method for the construction of prices or volume tables (^{Schumacher and Hall, 1933}; ^{Tapia and Návar, 2011}).
Volume equations and their tabulated expressions are one of the main tools for gaining reliable knowledge of the actual stocks and carrying out sustainable management. They are also helpful for forest management, the commercialization of timber products, and research, particularly in the form or seasonal quality studies (^{MuñozFlores et al., 2003}; ^{Velasco et al., 2010}).
The timber exploitation of Prosopis articulata S. Watson (mesquite) and Lysiloma divaricate (Jacq.) J. F. Macbr. (mauto) is a forest activity in the arid and semiarid areas of northern Mexico (^{LeónDe la Luz et al., 2008}; ^{RodríguezSauceda et al., 2014}). Despite the importance of the forest resources of these regions, very little detailed research has been carried out so far to accurately assess the estimates of volume production at species level.
The objective of the present research is to develop a volume system for the two species with the highest forest importance Prosopis articulata and Lysiloma divaricata in Southern Baja California, Mexico. The information to be thus obtained will serve as a basis for the silvicultural management of arid forest masses in the study region.
Materials and Methods
Study area
The study was carried out in plots that belong to a Forest Management Unit (Umafor) 303, located in the south of Southern Baja California, includes two municipalities: La Paz and Los Cabos (Figure 1) and is managed by the Sierra La Laguna Association of Forest Producers, A.C.
According to Köppen’s classification modified by ^{García (1988)}; the dominant climate is very dry, semiarid (BWh), followed by very dry, warm (BW(h'), dry semiarid (BSh), and only the area of Sierra La Laguna has temperate climates, such as temperate subhumid with summer rains with less humidity C(w_{0}), and temperate subhumid with summer rains and medium humidity C(w_{1}).
The physical factors of the environment favor the development of different vegetation types along an altitude gradient (^{Arriaga and Ortega, 1988}), characterized by xerophilous shrubs, represented by sarcocaule shrubs, sarcocrassicaule shrubs, low deciduous forest, low subdeciduous forest, and gallery vegetation.
The topography corresponds to mountain chains, plains, hills, beaches, plateaus and ravines.
Infield assessment
Privately owned plots and communal lands (ejidos) with timber potential were selected as sources of the dasometric data required for the study, using a destructive sampling, which consisted in felling, sectioning and measuring the trees chosen through directed sampling of 80 individuals per species, and in which the qualities of the season and the categories of diameter and height were represented.
The following variables were measured: normal diameter with bark (D), total height (H) and diameters at different heights with bark (d). The height from the ground level (h) was recorded for each of the sections. Diameters were measured at the base of branches with a diameter above 6 cm, with and without bark; smaller diameters were not included because they lack commercial interest. The total height was measured with a Suunto Pm5/360pc clinometer; and to obtain the different diameters a 122450 Ben Meadows diameter tape was used.
The felled trees were cut to a maximum stump height of 0.30 m above the ground level; next, two sections each with a length of 0.30 cm were cut above the stump. The next corresponded to the normal diameter (1.30 m); 0.50 m long sections were subsequently obtained, up to the tip of the tree (Figure 2). Two perpendicular diameters were considered in each, with and without bark, and the mean diameter was estimated.
Based on the characteristics described above, each of the sections of the trunk was cubed, from the stem to the branches, using Smalian’s formula:
Where:
𝑉 = Volume of the section (m³)
d1 = Larger diameter (m)
d2 = Smaller diameter (m)
𝐿 = Length (m)
The total volume of the tree was calculated based on the sum of all the volumes estimated for each section of the tree.
Analysis of the information
25 simple, multiple allometric, linear and nonlinear regression models were adjusted with one (Table 1), two or three predictive variables, respectively (Table 2), by applying the minimum ordinary squares method (MOS) and weighted least squares (WLS) with the SAS statistical software package (^{SAS, 1990}).
Model  Name  Expression 

M1  DissescuStanescu (second degree incomplete polynomial) 

M2  Berkhout (allometric) 

M3  DissescuMeyer 

M4  HohenadlKrenn (second degree complete polynomial) 

M5  DissescuStanescu (second degree incomplete polynomial) (Mod. Var. DB) 

M6  Berkhout (allometric ) (Mod. Var. DB) 

M7  DissescuMeyer (Mod. Var. DB) 

M8  HohenadlKrenn (second degree complete polynomial) (Mod. Var. DB) 

M9  DissescuStanescu (second degree incomplete polynomial) (Mod. Var. H) 

M10  Berkhout (allometric) (Mod. Var. H) 

M11  DissescuMeyer (Mod. Var. H) 

M12  HohenadlKrenn (second degree complete polynomial) (Mod. Var. H) 

M13  DissescuStanescu (second degree incomplete polynomial) (Mod. Var. DC) 

M14  Berkhout (allometric) (Mod. Var. DC) 

M15  DissescuMeyer (Mod. Var. DC) 

Model  Name  Expression 

M16  SchumacherHall 

M17  Spurr 

M18  Spurr potential 

M19  Spurr with independent term 

M20  Generalized incomplete combined variable 

M21  Undefined 

M22  Spurr with independent term (Mod.) 

M23  Undefined 

M24  Undefined 

M25  Undefined 

Source: ^{CruzCobos et al., 2016}; ^{VargasLarreta et al., 2017}.
V = Volume (m^{3}); D = Normal diameter (cm); H = Total height (m); DB = Diameter at base (cm); CD = Crown diameter (m); NB = Number of branches; VC = Squared base diameter ratio (cm) multiplied by the height (m); 𝛽_{0}, 𝛽_{1}, 𝛽_{2}, 𝛽_{3} = Parameters to be estimated.
The quality of the adjustment of the models was assessed using graphic and numeric analysis, based on the highest value for the adjusted coefficient of determination (adjusted R^{2}), the lowest value for the root mean square error (RMSE), the graphic representations, and the level of significance of the parameters according to ^{VargasLarreta et al. (2017)}.
Where:
n = Number of observations
p = Number of parameters
^{TorresRojo and MagañaTorres (2001)}, as well as ^{CruzCobos et al. (2016)}, point out that most volume models exhibit heteroscedasticity issues because the variation in the volume of the trees increases in direct proportion to the values of diameter and height, and corrections with weighted least squares are therefore required.
The graphical analysis of the studentized residuals versus the estimated values using the selected models evidenced heteroscedasticity issues; for this reason, they had to be corrected through weighted regression, with the weighted least squares method. The ordinary least squares method is employed to estimate the regression line, which seeks to minimize the sum of squared errors; i.e. each error receives an equal weighting, whether or not it stems from an observation with a higher or a lower variance. Conversely, the weighted least squares method adds more weight to those observations that exhibit a lower standard deviation than to those with a higher one, allowing a more accurate estimation of the regression line.
Results and Discussion
Table 3 shows the basic descriptive statistics of the variables considered in the models for Prosopis articulata and Lysiloma divaricata; the adjustment of the models was observed to cover a broad interval of the independent variables utilized.
Variable  P. articulata statistics  L. divaricata statistics  

Maximum  Minimum  Mean  SD  Maximum  Minimum  Mean  SD  
D  42.5  7.6  23.96  9.55  41  8.9  24.18  9.43 
H  10.58  3.53  6.50  1.57  12.2  3.5  9.46  2.31 
DB  52  11.5  29.24  10.63  53.5  10  28.91  11.27 
CD  8.5  1  3.65  1.86  13  1.5  5.08  2.93 
NB  7  0  1.17  1.11  1.10  0.02  0.29  0.24 
VT  0.60  0.013  0.18  0.13  3  0  1.77  1.02 
VT = Volume (m^{3}); D = Normal diameter (cm); H = Total height (m); DB = Diameter at base (cm); CD = Crown diameter (m); NB = Number of branches.
Figure 3 shows the tendency of the volumenormal diameter and volumetotal height pairs of data for Prosopis articulata. The sample size for each diameter category varied due to the abundance of trees of certain dimensions in the study area; diameters of 7.6 to 42.5 cm were included, the most frequent being those between 10 and 30 cm. Likewise, the most frequent heights ranged between 5 and 10 m.
Figure 4 shows the tendency of the volumenormal diameter and volumetotal height data pairs for Lysiloma divaricata. Diameters of 8.9 to 41 cm were considered, of which the most frequent ranged between 10 and 30 cm; the heights were of 5 to 12 m.
Table 4 summarizes the goodnessoffit statistics of the volume model of a variable for Prosopis articulata. The highest values for R^{2} and the lowest for RMSE were obtained using those models whose independent variable is the normal diameter (M1M4).
Equation  R^{2}  RMSE  Bias 

M1  0.90527794  0.0428437  0.0064217 
M2  0.90545388  0.04280389  0.0049428 
M3  0.90531866  0.04283449  0.0057673 
M4  0.90558798  0.04305038  0.0064217 
M5  0.67344067  0.07955041  0.0064217 
M6  0.67382001  0.0795042  0.0066161 
M7  0.67386939  0.07949818  0.0067407 
M8  0.67431109  0.07995853  0.0064217 
M9  0.47238585  0.10111594  0.0012141 
M10  0.46996466  0.10134768  0.0032696 
M11  0.47050001  0.10129648  0.0021356 
M12  0.50162254  0.09891053  0.0019025 
M13  0.52686768  0.09575307  0.0064217 
M14  0.53692929  0.09472946  0.0054247 
M15  0.53827914  0.09459129  0.0036454 
The analysis of the residuals of model M2, selected as the best for estimating the volume of Prosopis articulata trees, exhibited a tendency to increase in direct proportion to the predicted volume. The correction was made using weighted regression.
Table 5 shows the goodnessoffit statistics and estimators of the parameters of Berkout’s Allometric Model (M2) for Prosopis articulata.
The twovariable volume models for Prosopis articulata exhibited greater adjustments than those that consider a single independent variable; those involving the diameter and height as independent variables resulted in slight improvements; only a few failed to attain the selected level of significance (5 %). Table 6 shows the statistics of Model 16, which had the highest R^{2}, the lowest RMSE, almost zero bias, and significant parameters; for these reasons it was selected for the estimation of the volume of P. articulata.
Equation  R^{2}  RMSE  Bias 

M16  0.9186  0.0400  0.00022 
M17  0.8838  0.0471  0.01417 
M18  0.8574  0.0525  0.00043 
M19  0.9024  0.0435  0.00032 
M20  0.9075  0.0425  0.00022 
M21  0.8057  0.0617  0.00132 
M22  0.7974  0.0627  0.00003 
M23  0.9050  0.0431  0.05745 
M24  0.9024  0.0435  0.06197 
M25  0.9176  0.0405  0.00005 
The graphic analysis of the residuals of Model 16 for Prosopis articulata evidenced the presence of heteroscedasticity. Furthermore, the residuals were corrected using the weighted least squares method.
Arter the correction of the heteroscedasticity, Table 7 shows the statistics and adjusted parameters of ShumacherHall’s model for the estimation of the volume of Prosopis articulata.
Table 8 shows the goodnessoffit statistics of the one entry volume models for Lysiloma divaricata. The highest R^{2} values and the lowest RMSE values were obtained using those models that include the normal diameter (M1M4), followed by those that considered the diameter at base (M5M8). Conversely, the adjustments of those models that include the total height of the trees and the crown diameter were less efficient.
Model  R^{2}  RMSE  Bias 

M1  0.8287  0.1019  2.0052E10 
M2  0.8294  0.1016  0.000427 
M3  0.8293  0.1017  0.00035825 
M4  0.8294  0.1023  3.9996E11 
M5  0.6795  0.1393  5.649E10 
M6  0.6918  0.1366  0.28615368 
M7  0.6853  0.1381  0.0062446 
M8  0.7150  0.1322  1.7601E10 
M9  0.5821  0.1591  4.627E11 
M10  0.5780  0.1599  0.005714 
M11  0.5799  0.1595  0.0023334 
M12  0.5919  0.0985  8.7213E11 
M13  0.4518  0.1822  8.223E17 
M14  0.4191  0.1876  0.0069449 
M15  0.4208  0.1873  0.01102998 
The graphic analysis of the studentized residuals versus the estimated values of Model 2 showed the absence of homogeneity in the values of the variances, corrected by means of weighted regression.
After the correction of the heteroscedasticity, the final results of the adjustment to Berkout’s Allometric Model (M2) exhibits the adjustment statistics and the parameters shown in Table 9.
Those models that consider two predictive variables e.g. diameter and height (M16M25) evidenced adjustments above those of the models that consider a single independent variable (diameter), with R^{2} values ranging between 0.75 and 0.86,and RMSE values of 0.09 to 0.1. However, some models fail to attain the necessary level of significance (5 %) (Table 10), and therefore Model 18 was selected, due to its good fit and to the significance level of the parameter estimators.
Equation  R^{2}  RMSE  Bias 

M16  0.8336  0.1010  0.0010269 
M17  0.8044  0.1081  0.01737528 
M18  0.7977  0.1107  0.0026761 
M19  0.8162  0.1055  0.00106051 
M20  0.8167  0.1060  0.0005477 
M21  0.8625  0.0919  0.0035862 
M22  0.6786  0.1395  0.0031733 
M23  0.8625  0.0919  0.0035862 
M24  0.6786  0.1395  0.0031733 
M25  0.6786  0.1395  6.7808E10 
Like the previously analyzed models, M18 the Spurr potential model with one combined variable for Lysiloma divaricata exhibited heteroscedasticity, which was corrected using the WLS method. Table 11 shows the final adjustment statistics of the corrected M18.
The best adjustments for one entry models for Prosopis articulata and Lysiloma divaricata were obtained using M2, whose independent variable is the normal diameter. These results agree with those obtained by Rodríguez et. al. (2012), who carried out a study in a cloud forest in Tamaulipas, Mexico, in which they adjusted models for broadleaf and conifer species and concluded that Berkhout’s allometric model has a very good fit and is reliable for the estimation of total stem volumes based on the normal diameter. Furthermore, the results of the research documented herein agree with those of ^{Cruz et al. (2016)}, according to whom Berkhout’s allometric model with the diameter variable and ShumacherHall’s model with the independent variables diameter and height had the best fit for the estimation of volumes for Arbutus spp. in the region of Pueblo Nuevo, Durango, Mexico (Figure 5).
Conversely, ^{HernándezHerrera et al. (2014)} adjusted a series of regression models based on a sample of 38 trees using various predictive variables in the selection and adjustment of their best volume estimation.
^{VargasLarreta et al. (2017)} compared models in order to predict the total volume of several timber species in different states of the Mexican Republic, based on the normal diameter and the total height; their results agree with those obtained in this study, in the sense that ShumacherHall’s model exhibited the best adjustments. This model is widely used to estimate the total volume, both in conifers and in broadleaves. ^{Da Cunha and GuimarãesFinger (2009)} adjusted a series of regression models and selected the Spurr potential model for developing a doubleentry volume table for Pinus taeda L. in southern Brazil.
The best fits for the twoentry models for Prosopis articulata and Lysiloma divaricata were obtained using the independent variables normal diameter and total height. The models selected for the estimations were M16 i.e. ShumacherHall’s, and M18, the Spurr potential model for the simplicity of their implementation and structure (Figure 6).
Conclusions
The independent variable normal diameter produced the best adjustments using Berkhout’s Allometric Model (M2) for Prosopis articulata and Lysiloma divaricata, since this model allows a reliable estimation of the volume; however, ShumacherHall’s combined variable model was the one with the best fit in the estimation of the total volume of Prosopis articulata, while the Spurr potential model produced the best fit for the estimation of the total stem volume of Lysiloma divaricata. These two models yield reliable, accurate estimates, as their structure includes an additional variable.
Acknowledgments
To the Comisión Nacional Forestal Baja California Sur (South Baja California National Forest Commission) for the funding provided to accomplish this project, as well as to ASAMYFOR SC company for the technical support during field work
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Received: March 22, 2018; Accepted: September 24, 2018