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}; ^{Vargas-Larreta 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 (^{Acosta-Mireles and Carrillo-Anzures, 2008}; ^{Cruz-Cobos 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 (^{Muñoz-Flores 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 (^{León-De la Luz et al., 2008}; ^{Rodríguez-Sauceda 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.

Figure 1 Location of the study area. 

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(w₀), and temperate subhumid with summer rains and medium humidity C(w₁).

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.

In-field 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.

Sección = Section; Punto de medición = Measuring point; Base = Base.

Figure 2 Graphic representation of the measuring points at the various sections of the tree. 

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:

V=π4d12+π4d222L

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.

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.

Figure 3 shows the tendency of the volume-normal diameter and volume-total 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 3 Dispersion graphs of the volume-diameter (left) and volume-height (right) data pairs for Prosopis articulata S.Watson. 

Figure 4 shows the tendency of the volume-normal diameter and volume-total 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.

Figure 4 Dispersion graphs of the volume-diameter and volume-height for Lysiloma divaricata (Jacq.) J.F.Macbr. 

Table 4 summarizes the goodness-of-fit statistics of the volume model of a variable for Prosopis articulata. The highest values for R² and the lowest for RMSE were obtained using those models whose independent variable is the normal diameter (M1-M4).

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 goodness-of-fit statistics and estimators of the parameters of Berkout’s Allometric Model (M2) for Prosopis articulata.

The two-variable 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 R², the lowest RMSE, almost zero bias, and significant parameters; for these reasons it was selected for the estimation of the volume of P. articulata.

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 Shumacher-Hall’s model for the estimation of the volume of Prosopis articulata.

Table 8 shows the goodness-of-fit statistics of the one entry volume models for Lysiloma divaricata. The highest R² values and the lowest RMSE values were obtained using those models that include the normal diameter (M1-M4), followed by those that considered the diameter at base (M5-M8). Conversely, the adjustments of those models that include the total height of the trees and the crown diameter were less efficient.

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 (M16-M25)- evidenced adjustments above those of the models that consider a single independent variable (diameter), with R² 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.

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 Shumacher-Hall’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).

Figure 5 Graphic representation of the volume estimated using Berkhout’s one entry model for Prosopis articulata S. Watson (left) and Lysiloma divaricata (Jacq.) J. F. Macbr. (right). 

Conversely, ^{Hernández-Herrera 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.

^{Vargas-Larreta 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 Shumacher-Hall’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 Guimarães-Finger (2009)} adjusted a series of regression models and selected the Spurr potential model for developing a double-entry volume table for Pinus taeda L. in southern Brazil.

The best fits for the two-entry 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. Shumacher-Hall’s, and M18, the Spurr potential model- for the simplicity of their implementation and structure (Figure 6).

Figure 6 Graphic representation of the volume estimated for Prosopis articulata S.Watson using Schumacher-Hall’s model (left) and the Spurr Potencial model for Lysiloma divaricata (Jacq.) J. F. Macbr. (right). 

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, Shumacher-Hall’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.