Introduction

Nearly 269 600 ha of commercial forest plantations have been established in Mexico (CFP) (^{Semarnat-Conafor, 2014a}), reducing the pressure on the natural forests, promoting private investment and social development (^{Semarnat-Conafor, 2014b}), and favoring the reconversion of degraded or unproductive areas into woodland (^{Probosque, 2019}).

In the State of Mexico, the establishment of forest plantations (FP) has exhibited an upward trend; 65 ha were reported by 2003; 504 ha, by 2008, and approximately 537 ha by 2014, with a total of 3 070 ha of FP established between 2000 and 2014 (^{Semarnat-Conafor, 2015a}; ^{Probosque, 2019}). Nearly 81 % of the forest plantations have been established with species of the Pinus genus (^{Semarnat-Conafor, 2015b}), which must be managed using quantitative forestry tools.

Knowledge of the allometric relationships between the species and the environment in which they grow is essential for an adequate management ofFPs because they influence their growth and development (^{Semarnat- Conafor, 2014b}).

Allometric relationships are a reflection of the adaptations of the species to the environmental conditions of the place where they develop (^{Niklas, 1995}; ^{Gildardo et al., 2011}) and can be quantified through a regression analysis and represented mathematically by means of allometric equations (^{Diéguez et al., 2003}; ^{Corral et al., 2007}; ^{Picard et al., 2012}).

In the forest area, allometric relationships have been modeled between the stump diameter (sd) and the normal diameter (d) (^{Benítez et al., 2004};^{Pompa et al., 2009}), the sd and the total height (h) (^{Diéguez et al., 2009}; ^{Hernández et al., 2017b}; ^{Hernández et al. 2018}), the d and the volume (V) (^{Martínez and Acosta, 2014}; ^{García et al., 2017}), biomass (B) (^{Návar et al., 2013}) or carbon (C) (^{Martin et al., 1998}; ^{Méndez et al., 2011}).

In the State of Mexico, the rate of establishment of CFPs has increased and has played an important role in the development and growth of the forest sector (^{Probosque, 2019}). However, there are certain management problems, such as clandestine felling, loss of plantations due to meteorological causes or health issues, which make it necessary to have quantitative tools for assessing the volumetric losses in these crops. Thus, the objectives of the present study were 1) to adjust allometric models between the variables of commercial interest (sd, d, h and V) for Pinus patula Schiede ex Schltdl. et Cham. at CFPs in Regions VI and VII of the State of Mexico, and 2) to assess the differences between the volumes resulting from the equations used by the forest technicians of the State of Mexico for the Pinus genus.

Materials and Methods

The study area is located on the Neovolcanic Axis, in the forest regions delimited by Probosque: Coatepec Harinas (R-VI) and Valle de Bravo (R-VII), in the State of Mexico, which include Coatepec de Harinas and Ocuilán municipalities, for R-VI, and Amanalco, Villa de Allende and Villa Victoria, for R-VII (^{Cervantes et al., 1990}; ^{Probosque, 2019}). The altitude ranges between 2 000 and 2 850 masl; the climate is temperate (Cw); the precipitation is 1 200 mm, and the mean temperature, 14.5 °C (^{Inegi, 2016}). The soil type is Andosol (^{CCT, 2010}).

The sample was established in 90.8 ha distributed in 11 P. patula CFPs aged 4 to 20 years, with a 3 m × 3 m spacing and a mean density of 950 plants ha-1. Eight CFPs are located in R-VII, and three, in R-VI. The polygons of each CFP were constructed with Google Earth® Pro, version 7.3.2.5491 and QGIS Madeira®, version 3.4.13. The sites were 250 m2 in a rectangular shape, using a systematic sampling and a 100 m grid, at whose intersections they were established with the QGIS tool known as “regular points”. In order to avoid the border effect, only sites located at a distance from the boundaries of the plantation were considered.

65 sites were assessed, in which the altitude and the exposure, as well as the stump diameter (sd), and normal diameter (d) were recorded with a Haglöf Mantax Blue® aluminum caliper; and total height (h) of each individual were recorded with a Pm-5/1520 Pc Opti Height Meter® clinometer. The individual volume was estimated using three methods:

a) Through the expressions of the shape factor:

b) With the method proposed in the Second Dasonomic Study of the State of Mexico (Sedemex) (^{Probosque, 1990}):

c) Using the method generated by ^{Vargas et al. (2017)}:

Where:

V = Stem volume (m3)

d = Normal diameter at 1.30 m (m)

π = Constant of 3.1416

sf = shape factor (0.77)

h = Total (m)

e = Exponential function (2.718281828)

Nl = Natural logarithm

The observed data were captured in a scatter plot of the dispersion between the variables sd-d, sd-h, sd-V1, sd-V2, sd-V3, d-h, d-V1, d-V2, and d-V3; this allowed identifying atypical or aberrant values, which were then deleted from the database. The mathematical models were adjusted with the resulting screened information and after verifying the kurtosis.

13 allometric models were assessed for the sd-d, sd-h, d-h, sd-V, and d-V relationships were reported in the specialized literature (^{Huang et al., 1992}; ^{Pompa et al., 2009}; ^{Hernández et al., 2015}; ^{Hernández et al., 2017b}; ^{García et al., 2017}) (Table 1). The models were adjusted using the full information maximum likelihood technique (FIML) in the SAS® 9.2 software (^{SAS Institute In., 2008}).

The best model was selected based on the significance of the estimators (p> 0.05), the least square mean error (LSME) and the maximum adjusted determination coefficient (R2Adj). Furthermore, the normality and the distribution of the residuals were verified using the Shapiro-Wilk (SW) test and the graphic tendency of the observations, respectively (^{Huang, 1992}; ^{García et al., 2017}). The accuracy of the estimations using the best models for each variable was assessed with the bias (E) (^{García et al., 2017}; ^{Corral et al., 2019}), and the estimations versus the observed data chart (^{García et al., 2017}).

The three estimated volumes (V1, V2 and V3) were compared by the model selected as the best, based on the differences in the obtained biases and statistically with a t test for independent samples at a confidence level of 95 % (p=0.05), contrasting V1-V2, V1-V3, and V2-V3 (^{Di Rienzo et al., 2008}).

Results

The descriptive statistics of the sample indicate that the sd exhibits a range of values between 6 and 42 cm, while the d ranges from 3 to 40 cm, with a maximum height of 25 m; the difference in the volume was 0.979 m3 (V1-V2) when the equations were used. The kurtosis of the registered data (independent variables) indicates a normal distribution, without deviation problems (Table 2).

In a first fit, when verifying the assumptions of the regression in the variables of interest, normality was observed (SW>0.93) for all models; however, the tendency of the residuals in the charts exhibited heteroscedasticity issues. Variance weighting functions were utilized in the solution of this problem (^{Prodan et al., 1997}).

The sd-d, sd-h and d-h relationships were weighted using the formula: Residual/(xφ)0.5, and the sd-V1, sd-V2, sd-V3, d-V1, d-V2 and d-V3 relationships, using the Residual/((1/x)φ)0.5) formula (^{Crecente et al., 2009}; ^{García et al., 2017}; ^{Hernández et al., 2017a}). This function, based on an exponential equation, was applied according to the methodology suggested

by ^{Harvey (1976)}, where the utilized variable is interpreted with x and φ, derived from the linear regression of the natural logarithm (Nl) of the residuals in the dependent variable based on Nl(x); with this procedure, the residuals were homoscedastic in all the adjustments (Figure 1) and SW > 0.93.

Normal diameter; b) Height; c) Volume 1; d) Volume 2; and e) Volume 3 based on the stump diameter; and for f) Height; g) Volume 1; h) Volume 2; and i) Volume 3 (k) based on the normal diameter.

Figure 1 Distribution of the residuals resulting from the statistical adjustment in the allometric relationships for commercial forest plantations of Pinus patula Schiede ex Schltdl. et Cham. in the State of Mexico. 

Once the heteroscedasticity problems were corrected, the results of the fit of the models utilized for the allometric relationship between d, h, V1, V2 and V3 based on sd account for 96.0, 54.7, 86.6, 89.1 and 85.2 % of the variability (R2Adj), respectively, and for 57.0, 91.5, 94.5 and 89.8 %, respectively, in the relationship between d and h, V1, V2 and V3 (Table 3). Significant parameters with a 95 % confidence interval for the best models and standard errors below 1.5 in all cases.

In the results of the relationships, it was observed that the equations with the best statistical adjustment exhibited inconsistencies in the trends, and therefore we performed a graphic analysis (Figure 2), in which models 4, 8 and 14 were found to be the ones that best predict the information, while 10.V2 and 15.V2 in combination with the expression proposed by Probosque are the ones that best adjust to the sample used, as the shape factor turned out to be high, and the expression of ^{Vargas et al. (2017)} yielded results below the trend. However, the fact that the equations were developed for natural forests, and the growth habit is different from that of a CFP, must be considered.

Figure 2 Prediction of the allometric relationships versus observed data of stump diameter (sd), normal diameter (d), total height (h), and volume (V) for Pinus patula Schiede ex Schltdl. et Cham. forest plantations in the State of Mexico. 

In the evaluation of the estimations made using the selected model, and with the sd as the independent variable, the bias was 0.00001 cm, 0.000002 m and 0.00000004 m3 for d, h, and V, respectively, and of 0.000011 m and 0.00252 m3 in the estimation of h and V based on the sd. On the other hand, based on the sd, the bias was similar for models V1 and V3, with 0.000001 m3, and based on the d, the bias was 0.0053 m3 for V1, and 0.0030 m3 for V2. While, according to the statistical test, the estimations between models V1- V2 and V1-V3 are different (p<0.05), but those between models V2-V3 are equal (p=0.03).

Discussion

The results from model 4 explain 96.0 % of d variability based on the dimensions of the sd, and, therefore, the R2adj can be considered high according to ^{Gujarati and Porter (2010)}, who propose that values around 0.8 in the models are synonymous of efficiency when used. When comparing the values thus obtained, it becomes evident that they are similar to those reported by ^{Benítez et al. (2004)}, who used a log-type model that explains more than 95% of the variability of the data by adjusting the d-sd relationship for Casuarina equisetifolia Forst. in La Providencia Camagüey, Cuba, besides obtaining an aggregate difference that overestimated the d by 1.97 cm; for their part, ^{Bava and López (2006)} used a logarithmic model for Nothofagus pumilio (Poepp. & Endl.) Krasser and generated a result that accounts for 97 %, and ^{Quiñonez et al. (2012)} utilized allometric relationships for different Pinus and Quercus species, and obtained coefficients that account for more than 92 % of the variability in the d-sd relationship. However, this percentage is lower than that reported by ^{García et al. (2017)}, who predicted nearly 99 % of the d based on the sd and presented a RSME of 1.777 cm for Abies religiosa (Kunth) Schltdl. et Cham.

Modeling the h based on the sd and the d resulted in an R2Adj of 0.53 and 0.57, respectively, a situation that agrees with that described by ^{Diéguez et al. (2003)}. These values are indicative of the complexity of modeling this radio due to the great variability and distribution of the information. An example of this are the equations proposed by ^{Quiñonez et al. (2012)} for Pinus arizonica Engelm., P. ayacahuite Ehrenb. ex Schltdl., P. durangensis Martínez, P. leiophylla Schiede ex Schltdl. et Cham., P. teocote Schiede ex Schltdl. et Cham. and Quercus sideroxila Bonpl., with values between 0.47 and 0.77; also, ^{García et al. (2017)} reported coefficients between 0.37 and 0.68 for species of commercial interest in the tropical forests of Quintana Roo.

For the volume, the results obtained are acceptable, as they account for 89.1 % of the variation of the data, a value that is similar to that reported by ^{Quiñonez et al. (2012)}, who used an allometric model in their linearized form and obtained an explanation for 90 % of the variation, and results accounting for 86 % to 93 % were obtained by ^{García et al. (2017)}, who infer that y=a·xb is the adequate model type for predicting the volume based on the sd.

Model 10 is observed to be more reliable for estimating the volume (V1, V2 and V3); however, among the equations, we may appreciate that V2 and V3 exhibit a lower variance in the data than V1, which is handled at convenience because it is a high sf (0.77), while the equation proposed by ^{Probosque (1990)} has more stability and is the one most utilized for estimating the individual volume of this species in the region.

The normality tests did not exhibit distribution problems (SW>0.93), and the residuals behaved homoscedastically after the correction, as described by ^{Huang et al. (1992)}, ^{Crecente et al. (2009)} and ^{García et al. (2017)} (Figure 2). The biases and trends were acceptable, as the values of the deviations were lower than those obtained by ^{Quiñonez et al. (2012)} and ^{García et al. (2017)}; furthermore, the trends o1f 2 the estimations agree with the values of ^{Martínez and Acosta (2014)} when estimating d, by ^{García et al. (2017)} when calculating h; ^{Hernández et al. (2018)} when projecting the volume, and by ^{Díaz-Franco et al. (2007)} when adjusting this type of allometric equations for biomass (B) and captured carbon (C).

As a practical way of exemplifying an application of these equations, it is proposed to evaluate a clandestine felling of a hectare, where an average sd of 40 cm was found when measuring the diameters of 260 stumps in the FP (260 trees ha-1), and allometric equations resulted in an average d of 33.69 cm, an average h of 21.67 m, and average volumes of 1.3233 m3, 0.7276 m3 and 0.7957 m3. Therefore, the timber yield can be estimated in 344.06 m3, 189.19 m3 and 206.89 m3, with the expressions of V1, V2 and V3, respectively. The values of V2 based on a basic density of 0.5049 g cm-3 reported by ^{Goche et al. (2011)} for the species allowed projecting a value of 95.52 Mega-grams ha-1 for the extracted biomass, and of 47.76 Mega-grams ha-1 for C0₂, using the values proposed by ^{Acosta et al. (2009)}.

Conclusions

The allometric models proposed as the best between the variables of forest interest -stump diameter (cm), normal diameter (cm), total height (m), and volume (m3)- for the Pinus patula commercial forest plantations in Regions VI and VII of State of Mexico are statistically reliable and precise, and therefore they can be included in growth and yield systems for this cultivated species or for the assessment of the products obtained after legal or clandestine exploitation and, in general, in the development of forest management plans.

The evaluation of the differences between the volumes estimated with the volume equations utilized herein considers that the shape factor 0.77 used is high and differs from the other two equations; however, it is currently used in the forest management programs in the state (V1).

Although numerically speaking there is a difference between the volumes estimated by the equations proposed by ^{Probosque (1990)} and those calculated with the equations set forth by Vargas-Larreta et al. (2019), they are not statistically different; however, they do differ when the estimation is based on the shape factor. Therefore, either of these two options can be utilized until a specific option is developed for estimating the volumetric stock in the plantations.