Study area

The mensuration information comes from 44 permanent sampling plots of 400 m² (20 m × 20 m) systematically distributed in ages of 3 to 8 years in 87 ha^-1 planted with Pinus chiapensis in the Tlatlauquitepec municipality of Puebla State. The seedlings were obtained from seeds of selected trees of the region and were established in lands that were previously used for fruit and coffee production. The plantation was established at a density of 1 100 trees ha^-1, at real frame with a spacing of 4 m between rows and 2.25 m between plants. These plantations are located at 19°36'24'' N and 97°14'42'' W and average altitude of 1 900 m (Figure 1). The climate of the place is semi-warm with rain all year round, with an average annual temperature of 17 °C and an average annual rainfall of 2 350 mm (^{Inegi, 2009}).

Área de studio = Study area; Simbología = Symbology

Figure 1 Location of the study area in Tlatlauquitepec municipality, Puebla State. 

Mensuration data

The establishment and measurement of the 44 plots (n) was carried out in 2014 with ages of 3, 5, 6 and 7 years, and for the year 2015 the first re-evaluation of the same plots was obtained, with ages of 4, 6, 7 and 8 years. Of all the trees in each plot the following variables were measured: number of trees (NA), normal diameter (D in cm) with bark at the height of 1.30 m above the soil surface with a Haglöf ^® caliper; and for the total height (H), 10 dominant and co-dominant trees were measured with a Vertex Laser Haglöf ^® . To estimate the total height of each tree from the 2014 and 2015 inventory, a two-parameter Chapman-Richards model proposed and discussed by ^{Fierros et al. (2017)}, for the same database described was used.

The mathematical structure of the model is the following:

Where:

D = Normal diameter (cm)

H = Total height (m)

The H function had a fitted coefficient of determination ( Radj2 ) equal to 88.50 % and 2.328 as root mean square error (RMSE).

The mensuration data (D and H) from the 2014 and 2015 inventories served as a basis to estimate the basimetric area (AB ₁ , AB ₂ ), for such years as well as the volume from the barked total stem at the tree level (V₁, V₂). After it, the AB ₁ , AB ₂ ,V₁ y V₂ values of each tree were summed to find the numbers per site. AB was calculated with the following equation:

Where:

AB = Basimetric area (m²)

D = Normal diameter (cm)

π = Constant with a value of 3.1416

V was estimated with the equation of local volume of the ^{Schumacher and Hall (1933)} type formulated by ^{Martínez (2016)} for P. chiapensis, which has the following structure:

Where:

V = Volume of the total barked stem (m³)

D = Normal diameter (cm)

H = Total height (m)

The V function had a fitted coefficient of determination ( Radj2 ) equal to 99.66 % and 0.0058 as root mean square error (RMSE).

Sampling estimators

To determine the timber stocks of V₁ and V₂ in the P. chiapensis plantations, we two classical estimators based on designs were used: Simple Random Sampling (MSA) and Stratified Sampling (ME), and with two based on models: Ratio estimators (ERaz) and Regression Estimators (EReg); the latter uses a population mean (μ_x = AB and E) that is estimated under ME, which is assumed to be the true population value. For the statistical analysis the statistical program R version 3.4.3 was used. (https://cran.r-project.org/bin/windows/base/). Table 1 shows the mathematical structure of the equations that describe the MSA, ME, ERaz and EReg estimators.

Where:

y- = Sampling mean of the main variable V (m³ ha^-1) observed in the i ^-th sampling site and extrapolated per ha^-1

 x- = Sampling mean of the auxiliary variable

Nh = Sampling frame in the h ^-th h stratum

L = Total number of strata in the population

n _h = Total number of sampling units in total number of included in the sample

n _hi = Observed value of the main variable V (m³ ha^-1) in the i ^-th sampling unit in the stratum h^-th

μ _x = Population mean of the auxiliary variable

n = Sample unit

N = Population size

 β^ = Parameters of the model estimated from the data of the sample through ordinary least squares (MCO, acronym in Spanish)

t = Parameter that represents the Student distribution at 95 % of confidence with n-1 degrees of freedom (gl, acronym in Spanish)

The upper limit and the lower limit (L.S. and L.I.) for the mean ( 𝑦 ) of the volume (m³ ha^-1) were calculated, with their respective accuracy (P in %) and the total inventory (m³ ha^-1), corresponding to the sampling estimators (MSA, ME, ERaz and EReg) used in this study. L.S. and L.I. were determined by the following expression:

Where:

L.S = Upper limit of V expressed (m³ ha^-1)

L.I. = Lower limit of V expressed (m³ ha^-1)

Sy-2 = Variance of the V mean (m³ ha^-1)

y- = Sampling mean of V (m³ ha^-1)

t _{n, gl} = Level of confidence at 95% (1− α=0.95)

(gk)= Degrees of freedom which are equal to the number of sampling sites (n=44 sites)

The accuracy is described as:

Where:

P = Accuracy of the mean (%)

Sy-2 = Variance of the V mean (m³ ha^-1)

y- = Sampling of the V mean (m³ ha^-1)

Finally, the total inventory was obtained with the following expression:

Where:

y^ = Total inventory in volume (m³)

N = Population size

y- = V sampling mean (m³ ha^-1)

Under the ME estimator, the age of the plantation (E) was used as an auxiliary variable, from which four age strata (3, 5, 6 and 7) were obtained, for the 2014 inventory, and four strata (4, 6, 7 and 8) for the 2015 inventory, which are known variables without error (^{Roldán et al., 2013}). In each stratum, the sample of inventory sites was systematically selected. According to ^{Cochran (1977)} and ^{Thompson (2002)}, stratification reduces the variance of the mean in the stratum from an inventory design that groups and weights the variances.

ERaz and EReg use auxiliary information that provides estimates of greater reliability and accuracy than simple estimation methods (MSA and ME). This is the best method to estimate the population mean of the main Y_i (V₁ and V₂), with greater accuracy, when the auxiliary variable X _i (AB and E) has a high correlation with the main variable Y _i . In this context, to estimate V₁ the auxiliary variable that was used were AB₁ and E₁, while for V₂ it was AB₂, E₂ and V₁, auxiliary parameters estimated under ME that assume as true known value at the population (N) level (µ_x) without sampling error.

ERaz offers the advantage that when using the AB as an auxiliary variable there is an R (ratio) that describes the amount of V in m³ ha^-1 standing for each m² ha^-1 of AB. While with E (age) a measure of expected annualized growth is obtained in m³ ha^-1. It is also interesting to evaluate V ₂ as a function of V ₁ , an R that involves the main variable measured at time two (V ₂ ) and its same variable measured at time one (V₁); in this way, a reading is obtained on the percentage increase of V₂ according to the initial volume (V ₁ ).

The comparison of the four estimators was based on a numerical analysis: lower mean of the inventory (m³ ha^-1), greater accuracy (P, %) and lower amplitude (A, m³ ha^-1) of the confidence intervals. This allows to propose a precise and practical sampling strategy (sampling estimators), which during the survey of field sites is efficient in terms of time and cost.

Correlations of the variables

Through an analysis of Pearson's Correlation Matrix, the associated correlations between V, AB and E, which is a statistical measure to evaluate if two quantitative variables have a linear relationship, were studied. The information from this analysis made it possible to observe the trends of the data, which suggests that V and AB had a higher correlation (99.52 and 99.58 %, for the measurement of 2014 and 2015, respectively); secondly, because of the correlation (98.5 %) between the V ₂ of 2015 and the V ₁ 2014, while the E showed the lowest correlation (64.37 and 68 % for 2014 and 2015, respectively). ^{Roldán et al. (2013)} mention that the previous trends suggest that AB is a potential variable that can be used as an auxiliary variable in the estimation of the inventory, either through ERaz or EReg.

Results of the samplings made

The values of the parameters of the mean of the volume (m³) of each evaluated sampling estimator and the comparative in terms of gain in precision and amplitude of the confidence intervals are shown in Table 2. The estimators showed a greater variability in terms of P (statistical precision), with values that ranged from 1.55 to 21.75 % for 2014 and from 1.35 to 20.30 % for 2015. EReg (V/AB) under ME was comparatively better, followed by ERaz (V/AB) under ME, both showed the best accuracies, higher values of gain in precision and amplitudes of small confidence intervals.

Bailes and Brooks (2004) indicate that the consequence of using auxiliary variables (AB) with a greater correlation with the main variable (V) leads to obtain more efficient estimators; in addition, it offers the advantage that it is easy, fast and cheap to measure in the field, which leads to optimize the investment in time and cost during the execution of inventories. The answer to the above is explained in the statistics obtained in the EReg (V/AB) when using the AB estimated under ME, with AB with the highest correlation with V, a P of 1.55 % for 2014 and 1.35 % for 2015, and smaller intervals amplitude (A, m³ ha^-1) equal to 1.72 for 2014 and 2.02 for 2015, followed by ERaz (V/AB) under ME. MSA showed the lowest accuracy (P) with 21.75 % for 2014 and 20.30 % for 2015.

However, ME produces results lightly conservative in terms of total inventories compared to EReg (V/AB) under ME (optimistic inventory); undoubtedly, it is feasible to ascribe to stratification, and when using estimators of ME, smaller variances to the estimated mean result; in addition, when the variances of the interest variable are weighed, a lightly conservative inventory is achieved, but such a variation is minimal. For future projects, it would be attractive to continue exploring in EReg (V/AB) a real estimated population value under ME in order to get more precise estimators, a conservative inventory, more accuracy a confidence intervals of greater amplitude.

It is also important to note that the value of the V AB^-1 ratio from that obtained with the ERaz (V/AB) under ME provides punctual information on the average volume (5.10 and 5.35 m³ ha^-1 for 2014 and 2015, respectively) in existing foot for each m² of AB. This information becomes relevant when fast inventories are made by the relascopy technique, whose aim is to estimate the AB inventory in sites of varying dimensions.

On the other hand, V E^-1 ratio suggests that the average annual increase in plantations of P. chiapensis is 9.58 m³ ha^-1 year^-1 for 2014 and 11.05 m³ ha^-1 year^-1 for 2015. Despite the low correlation of volume with age, it is possible to use it as an auxiliary variable to quantify total inventories. When this happens, ERaz (V/E) works as a simplified growth and yield system, in which, based on the weighted age (5.8 years), which is calculated upon the age classes and the size of each stratum, the inventory can be estimated on an annualized basis (^{Roldán et al., 2013}).

ERaz (V ₂ /V ₁ ) which involves volume two (V ₂ ) of 2015 as the main variable, and volume one (V ₁ ) of 2014 as auxiliary variable, suggests that for every m³ that was in 2014 (initial volume), the volume grew by approximately 35 %. However, the EReg (V ₂ /V ₁ ) assumes a more conservative value of 25 % percentage increase.

When comparing the traditional estimators based on design (ME vs. MSA), it is observed that ME presented a substantial gain in precision (11.13 %) and confidence intervals (11.7 m³ ha^-1) when stratifying by age, in addition it offers the advantage of that generates total inventories more conservative with respect to the MSA. Several authors concluded that when stratifying by age, the accuracy increases as the variance decreases, which allows obtaining values of the most reliable and precise parameters (^{Achard et al., 1998}; ^{Roldán et al., 2013}; ^{Tamarit, 2013}). One of the strategies to optimize an inventory is to resort to stratification with some variable strongly related to the variable of interest, in this case it was the volume with the age of the plantations (^{Lencinas and Mohr, 2007}).

Of the four estimators studied and under the evaluation mentioned criteria, EReg (V/AB) presented the best sample estimators. The inventory in estimated volume with EReg is 4 806.38 m³ for 2014 and 6 496.01 m³ for 2015 and, when age is used as auxiliary variable, the inventory is 4 859.5 for 2014 and 6 562.94 m³ for 2015.