Study Area

The E. urophylla PFCs evaluated are located in the municipality of Huimanguillo, Tabasco, Mexico. The climate is hot and humid (Am): the mean annual temperature is 26 °C and the mean annual precipitation is 2500 mm. The soils belong to the Phaeozem group and the terrain is made up of low hills (^{INEGI, 2005}).

Description of the sample and estimation of the total volume

The sample was taken from 727 E. urophylla trees, out of which: 175 belonged to clones measured in 2007 (P1), 93 to clones measured in 2014 (P2), and 459 to trees grown from seeds measured in 2007 (P3). The clones’ age (P1 and P2) fluctuated from one to seven years, while trees grown from seeds (P3) ranged from two to eleven years.

The sample trees were chosen according to: their morphological condition (to cover the greatest possible phenotypic variability within the plantations); the variability between their dn and A dimensions; and their age distribution. In each tree, the diameter at breast height (dn) and total height (A) were measured, and then they were felled and cut at different sections (di and Ai) along the bole, in order to measure their diameter and height; measures were taken 1 m distance from each other, starting with the diameter and height of the tree stump (dt and ht, respectively). The Newton and the cone formulas were used to find out the volume of each log (V_troza) and each tip (V_punta), respectively. The total volume (Vt) was calculated using ^{Bailey’s overlapping bolts method (1995)}.

Interpopulation comparison of trees

In order to determine differences between clones’ populations (P1 and P2) with regard to trees grown from seeds (P3), the following hypotheses were tested on the dn and A variables. The analysis was performed contrasting P1 vs. P2, P1 vs. P3, and P2 vs. P3.

Null hypothesis (Ho): populations are the same.

Alternative hypothesis (Ha): populations are different.

In order to verify if trees of the three populations have differences in Vt -and on the assumption that two trees can have the same volume without necessarily presenting the same dn and A dimensions-, the Vt of the two types of clones and plantations grown from seed was fitted. In addition, to quantify the effect of each type of tree (either clone or grown from seeds), an indicator variable was inserted into the Schumacher-Hall (^{Draper and Smith, 1966}) volume model (Table 1). Analyzes were carried out with SAS 9.2. (^{Institute Inc. Statistical Analysis System, 2008}).

Table 1 Models used to estimate Vt of trees in E. urophylla PFC. 

The indicator variables (W_n) were programmed in the Schumacher-Hall volume model (1), according to the contrast performed for each population.

The model is expressed as follows:

(1)

where a₀, a₁, and a₂ are the regression parameters, whereas a _0c and a_1c are the addition parameters for the different populations.

Estimation of slenderness ratio and form factor

The slenderness ratio (IE) was calculated to verify the trees’ degree of mechanical stability against winds or hurricanes, dividing the A average by the dn average, as reported by ^{Arias (2004} and ²⁰⁰⁵⁾ and ^{Nájera and Hernández (2008)}. In order to verify if these are statistically different from each other, a one-factor ANOVA was carried out for the average IE between the populations at a p=0.05 level.

The form factor (ff) was calculated with the slope (p) of the bole volume and the combined variable (dn² A), and, then, the factor was calculated with regard to a theoretical cylinder, using the following formula:

(2)

The adjusted total volume models are those of ^{Da Cunhaa y Guimaraes (2009)}, ^{Corral-Rivas and Návar-Cháidez (2009)}, ^{Tschieder et al. (2011)}, and ^{Casnati et al. (2014)}. Meanwhile, the standard type or double-entry types (^{Prodan et al., 1997)} only differ in the form and number of parameters to be estimated (Table 1).

The best model was selected on the basis of the highest value of the determination coefficient, adjusted by the number of parameters (R²_aj) and the lower values in the sum of squares error (SCE) and the root mean square error (RCME), as well as the best graphic distribution of residuals. With a model that complied with the above-described conditions, and, in order to avoid increasing the variance as the diameter increases (heteroscedasticity), weighted regression was used (^{Cailliez, 1980}; ^{Tschieder et al., 2011}), with the following weight variables: 1/dn, 1/dn², 1/dnA y 1/d ² A. The bole volume estimates for the three populations were made with the corrected models.

Volumetric ratio models

The volumetric ratio (r) was calculated as the volume (at different heights and diameters) and the total volume (Vi/Vt) quotient. With this purpose, volume ratio models using dn, di, A, and Ai as independent variables were successfully utilized in other studies (^{Pece, 1994}; ^{Chauchard and Sbrancia, 2005}; ^{Barrio et al., 2007}; ^{Gilabert and Paci, 2010}; ^{Barrios et al., 2014}) (Table 2).

Table 2 Volume ratio models used in E. urophylla PFC. 

R_d is the volume ratio that uses the following as independent variables: the diameter at breast height (dn) and the diameter at different heights (di); R_h, which includes in the model the total height (A) and height at different sections along the bole (Ai); b₀, b₁, b₂, b₃, b₅, and b ₆ are the parameters to be estimated.

To fit these models, 1958, 2113 and 6060 pairs of di-Ai data were used for the P1, P2 and P3 populations, respectively. As a first step, the volume ratio models were adjusted without including the error term with the purpose of verifying the residuals trends and the value of the Durbin-Watson (DW) autocorrelation statistic, as ^{Barrios et al. (2014)} indicate.

In order to fit this type of models, the volume is calculated at different heights and diameters to obtain a volumetric ratio. Therefore, the longitudinal structure of the information used is inevitable; its measures are different along the bole and they have a close correlation, because they belong to the same tree. As a result of this, a continuous-time autoregressive (CAR) model was used to correct the error structure (^{Zimmerman and Nuñez-Antón, 2001}), and the model is expressed as follows:

(22)

(23)

Where Y_ij is the vector of the dependent variable. X_ij is the matrix of independent variables. B is the vector of the parameters to be estimated. e_ij is the j-th residual of i tree. I_k =1 for j>k and I _k =0 for j≤k. ρ_k is the autoregressive parameter of the k order to be estimated. h_ij -h _ij-k is the distance that separates the j-th measurement height from the j-th-k measurement height in each tree (h_ij >h _ij-k). ε_ij is the random error (^{Barrios et al., 2014}). The number of lags applied in the CAR(X) model was established at the moment that the DW statistic was evaluated, which should have been close to 2 as mentioned by ^{Verbeek (2004)}.

Model adjustment and evaluation criteria

The statistical fit of Vt and r models was carried out using the MODEL procedure in SAS 9.2 and the full information maximum likelihood (FIML) technique (^{Institute Inc. Statistical Analysis System, 2008}).

The evaluation and selection of the best model was performed with the highest value of R²_aj and the lowest values in the SCE and RCME, along with the tests of independence (DW) and normality of the frequency of the residuals (^{Da Cunha and Guimaraes, 2009}). The homogeneity of variance was subject to a graphical evaluation (^{Tschieder et al., 2011}), while the predictive capacity of the best models was evaluated estimating the absolute bias (Ē) and the aggregate deviation (DA %) for each population.

Determination of merchantable volume (Vc)

In order to estimate the merchantable volume at any point along the bole, a scale system consisting of the best Vt and r model was formed for each of the populations. With the tree’s Vt and applying the volumetric ratio model, the merchantable volume is the result of multiplying the two equations results (^{Chauchard and Sbrancia, 2005}). The evaluation of the estimation accuracy was carried out with a linear trend chart of predicted against observed values (^{Pece, 1994}).

Interpopulation comparison of Vt, dn and A in trees

The fitting of the Schumacher-Hall model was acceptable (Table 3) and the values of the parameters in the W _n indicator variable were significant (p≤0.05), which confirms that the three populations have different dn and A averages, as well as different Vt averages, and that their growth rates are different. This difference between the 2007 and 2014 clones is the result of the reproduction selection of the latter, because it was made on the basis of their larger dn or A dimensions, without taking into account the Vt or the form factor.

Table 3 Estimated parameters and goodness of fit statistics of the Schumacher-Hall Vt model with indicator variables in the three populations of E. urophylla, in Huimanguillo, Tabasco, Mexico. 

SCE: Squared sum of errors; RCME: Root mean square error; R²_aj: Adjusted coefficient of determination; Eea: Approximate standard error; a_n: Parameters to be estimated.

Estimation of slenderness index, form factor, and total volume

Once it was identified that the three populations should be analyzed separately, IE was calculated. The mean values by population were 114, 122, and 118 cm cm^-1, for P1, P2 and P3, respectively. The lower average IE value suggests that the clones measured in 2007 were selected for their growth in diameter, rather than for their growth in height, because it is more practical and simple to take only the diameter at breast height as a production indicator. In the ANOVA test, the minimum significant IE difference was 4.77; in addition, the means of the slenderness ratio between the populations were statistically different (P1=122.571, P2=112.168, and P3=93.804), because the error probability value was low (Pr>F=<0.0001).

These indexes are similar to those for fast-growing tropical species such as E. Nitens (IE=124) (^{Díaz et al., 2012}), Hieronyma alchorneoides (IE=111), and Terminalia amazonia (IE=106) (^{Arias, 2005}). However, because IE is higher than the 1:1 growth ratio between dn and A, the trees are assumed to be thin and strong thinning intensities must be applied with care (^{Arias, 2005}; ^{Díaz et al., 2012}), due to the PFC’s susceptibility to the mechanical damages caused by winds (^{Wilson and Oliver, 2000}).

With regard to ff, the shape of trees in the P2 and P3 populations was the same (conical type with a value of 0.34), while it was different in the P1 population (ff=0.45), which indicates that the shape of trees is more similar to a paraboloid.

As a result of the fit statistics of the best Vt models -determined by the higher R²_aj values, the lower SCE and RCME values and the significance of the parameter value (p<t)-, the Schumacher-Hall model was best fitted to P1 and P3 data, and the Spurr model for P2 (Table 4).

Table 4 Estimated parameters and goodness of fit statistics of the Vt models for the three populations of E. urophylla, in Huimanguillo, Tabasco, Mexico. 

SCE: sum of squared errors; RCME: Root mean square error; R²_aj: Adjusted coefficient of determination; Eea: Approximate standard error; b_n: Parameters to be estimated.

In the three fittings, all the resulting parameters were significant (95 % confidence), and the Shapiro-Wilk normality test showed values higher than W=0.92 and a level of Pr<W=0.0001, whereas the residual plots showed a tendency towards a straight line in the form of a Gaussian bell curve, which indicates that the data are normal. However, in the variance homogeneity test, very marked trends were observed; therefore, heteroscedasticity problems were assumed, and the weighted variable with best corrective results was 1/dn²A, with which a homogenous distribution of residuals was obtained (Figure 1). The estimation of the parameters and the goodness of fit indicators of the bole volume equations for the P1, P2, and P3 populations are shown in Table 5.

Figure 1 Distribution of Vt residuals with respect to the diameter at breast height (dn) of the best fitted models, corrected by heteroscedasticity, for the P1, P2, and P3 plantations of E. urophylla, in Huimanguillo, Tabasco, Mexico. 

Table 5 Estimated parameters and goodness of fit statistics of the Schumacher-Hall (P1 and P3) and Spurr (P2) models, corrected for heteroscedasticity, for the three populations of E. urophylla, in Huimanguillo, Tabasco, Mexico. 

SCE: sum of squared errors; RCME: Root mean square error; R²_aj: Adjusted coefficient of determination; Eea: Approximate standard error; b_n: Parameters to be estimated.

In the estimates made with the corrected models, as well as their projections, the average IE was considered when calculating the height of each population. As it is observed, the selection of the vegetative material to produce the clones measured in 2014 was carried out taking the samples of highest dn and A, without considering the Vt, because the line that corresponds to this population is the lowest of the three and makes reference to the more slender trees (conical form) and have less volume than the 2007 clones and the trees grown from seeds (Figure 2).

Figure 2 Estimation and projection of Vt with the Schumacher-Hall model for P1 and P3 and Spurr model for P2 of E. urophylla populations, in Huimanguillo, Tabasco, Mexico. 

Taking as reference the 2007 clone plantations -which have more Vt than the rest-, it was observed that, up to the 0.30 diameter range -which on average is the maximum in the three populations-, the 2014 plantations have 3.60 % less Vt than those grown in 2007, and 1.47 % less than trees grown from seeds (Figure 2). Projections outside the sample up to 45 cm diameter indicate that the differences increase to 11.35 % and 4.72 % in the P2 and P3 populations, respectively, with regard to P1’s projected Vt.

Volumetric ratio models

When the first fitting was made, the values of the parameters and the statistics obtained were relevant, but the residuals showed a trend towards heteroscedasticity, and the value of DW for P1 varied from 0.59 to 1.03, for P2 from 0.13 to 0.82, and for P3 from 0.44 to 1.09, which shows that the data are correlated. Therefore -after a graphical analysis of the autocorrelation function (ACF) was carried out in the three populations-, a auto-regressive structure of first, second, and third order was used to correct the autocorrelation of the errors (CAR (X)) and the one that had best results was selected (Table 5).

After the autocorrelation was corrected for the R_d model (11) of P2, we decided that this was the best one, based on ^{Fuentes et al. proposal (2001a} and ^2001b) about the absence of correlation for values in this statistic that are higher and equal to 1 and the greater precision that this model generated with regard to the rest. The DW values demonstrate an autoregressive type CAR(1) serial correlation, due to the existence of logs with bole diameters at a given height, related to a similar or equal diameter to a different Ac (^{Pérez, 1996}; ¹⁹⁹⁸), and that -when another delay was applied to the errors and the DW value was improved- we observed that quality was lost in the fit, because no significant parameters were evident (i.e., R²_aj lower or biased estimations with regard to the observed data).

In the three populations analyzed, one R_d -type and one R_h -type volume ratio models were selected in order to use either of these two variants to estimate Vc. This selection was based on the highest R²_aj values, and the lowest SCE and RCME values, as reported by ^{Trincado et al. (1997)} and ^{Barrio et al. (2007)}. The fit of best models had satisfactory results and all the parameter values were different than zero (p≤0.05) (Table 6).

Table 6 Estimated parameters and goodness of fit statistics of the volume ratio models that best fit each population in E. urophylla PFC, in Huimanguillo, Tabasco, Mexico. 

SCE: Sum of squared errors; RCME: Root mean square error; R²_aj: Adjusted coefficient of determination; Eea: Approximate standard error; b_n: Parameters to be estimated; p1r, p2r and p3r indicate the order and number of delays applied in the CAR (X) type model applied.

When the residuals’ distribution was verified (Figure 3) for the R_d models in the P1 and P3 populations, the results matched those mentioned by ^{Barrios et al. (2014)} for E. grandis, which has an homoscedastic distribution (a and c). However, there is a slight presence of heteroscedasticity in P2; but it does not affect the estimations made with this model, because when the volume ratio was calculated with better distribution models, the absolute error was lower in the selected model (b). In no case, the R _h models show a trend (d, e, and f), which matches what ^{Pece (1994)} described when he used this type of model for E. pellita.

Figure 3 Residuals of the best volumetric ratio models for trees of the three E. urophylla populations evaluated in Huimanguillo, Tabasco, Mexico. 

When the absolute bias (Ē) and the aggregate deviation (DA %) for each population were estimated -and with the purpose of verifying the precision of the estimations with the R_d y R_h models- we observed that their values were low for the three populations, and in no situation did DA without exceed 1.29 % (Table 7).

Table 7 Absolute bias (Ē) and aggregate difference % (DA%) of the volumetric ratio models in the three populations. 

Estimation of the merchantable volume of trees

The fitted models allow us to find the volumetric accumulation ratio, as the total height is reached, starting from the tree stump height, as well as the relation between the total volume and the volume of a limited diameter or height. The expressions used to obtain the Vc of individual trees in each population consist of a Vt and r model; when multiplied, they will result in the Vc of any diameter or height of defined exploitation. The resulting equations are the following:

Vc equations for the P1 population.

(24)

(25)

Vc equations for the P2 population.

(26)

(27)

Vc equations for the P3 population.

(28)

(29)

When the Vc predictions obtained with the volume ratio method are subject to a graphic comparison with the accumulated volume, the result is close to a straight line (Figure 4); and, when a linear regression is applied to these data the use of the R _d -type models shows -by the value of R²- that the sample estimation is higher than 94 %, while the use of R_h -type models has 96 % results. When the accuracy and absolute bias are verified, taking as reference the R ² value and the difference in its approximation to the unit, the difference for the R_d models is 3.0, 5.4 and 5.5 % in the P1, P2 and P3 populations, respectively; while it is 1.1, 3.1, and 2.6 % in R_h models (Figure 4). The trend observed and the values of the fittings between the predicted and the observed values match those indicated in the studies carried out by ^{Pece (1994)} in Eucalyptus pellita, by ^{Chauchard and Sbrancia, (2005)} in P. radiata, and by ^{Barrios et al. (2014)} in E. grandis.

Figure 4 Estimated merchantable volume (Vc) vs. the predicted Vc for P1, P2, and P3 populations, using the models selected. 

In order to estimate a merchantable diameter at a given commercial height for P1 and P2 populations, we solve for di from equation (11) and R _d is replaced by R_h (equation 30), according to the procedure set forth by ^{Trincado et al. (1997)}, using the parameters estimated for each model and replacing them in the resulting equation. If the inverse calculation was necessary (i.e., to estimate Ac for a specific diameter in the three populations), Ai would be solved from equation (21), and R_h would be replaced by R_d (equation 31).

(30)

(31)

For P3, in model 15 of R_d, di at any height is calculated by a numerical approximation with the Excel tool SOLVER, because when this variable is solved from the equation, the roots can be mathematically reduced until they disappear.

The models selected show consistency in the total and merchantable volume determination, their values do not crossover, and they enable a direct estimation of the bole diameter at an established height, or the bole height at a limit diameter. As a whole, these models are highly trustworthy and allow to carry out estimations of product distribution, to create tables of merchantable volumes for any useful diameter or height established by the sawmill industry or to calculate the economic valuation of plantations. Stock and operational harvest data are required to validate the models generated.