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Revista mexicana de ciencias forestales
versão impressa ISSN 2007-1132
Rev. mex. de cienc. forestales vol.12 no.68 México Nov./Dez. 2021 Epub 28-Fev-2022
https://doi.org/10.29298/rmcf.v12i68.1074
Scientific article
Optimal sampling strategy for timber inventory planning in commercial plantations of Tectona grandis L.f.
1Instituto Nacional de Investigaciones Forestales, Agrícolas y Pecuarias. Campo Experimental San Martinito, CIR Golfo Centro. México.
2Posgrado en Ciencias Forestales, Colegio de Postgraduados. México.
3División de Ciencias Forestales, Universidad Autónoma Chapingo. México.
4Instituto Nacional de Investigaciones Forestales, Agrícolas y Pecuarias. Sitio Experimental Tlaxcala, CIR Centro. México.
The objective of this study was to evaluate the statistical efficiency of six sampling estimators to propose an optimal sampling strategy in terms of precision and time that allows conducting operational timber inventories that support decision-making aimed at improving the technical management of commercial forest teak plantations (Tectona grandis) established in Campeche, Mexico. Data used were from 8 830 sampling sites of a planted area of 2 207.5 hectares. Each sampling site was rectangular of seventy-two m2 included nine stocks, the number of living trees was counted and their diameter at breast height was measured. The total height and volume of each tree were estimated with Chapman-Richards and Schumacher-Hall models, respectively. Basimetric area and total volume per site were obtained and extrapolated at hectare. Plantations were stratified by age classes; the basimetric area and the age of the plantation were used as auxiliary variables. The sampling strategy to estimate the mean volume was formed by associating simple random sampling as the sampling design with the specific ratio estimator in stratified sampling, with a stratification by age classes of one year and basimetric area as auxiliary variable; this gave the accuracy of 0.21 %. The sample size in stratified sampling could be reduced to 68.3 % with an accuracy of 2.5 % of the original sample. This means less sampling effort and economies by reducing the time for forest inventory.
Key words Sampling estimators; stratification; forest inventory; sample size; teak; auxiliary variables
El presente estudio tuvo como objetivo evaluar la eficiencia estadística de seis estimadores de muestreo para plantear una estrategia de muestreo óptima en términos de precisión y tiempo que permita realizar inventarios maderables operativos que apoyen la toma de decisiones orientadas a mejorar el manejo técnico de plantaciones forestales comerciales de Tectona grandis (teca), establecidas en Campeche, México. Para ello, se evaluaron 8 830 sitios de muestreo ubicados en 2 207.5 hectáreas. Cada sitio rectangular de 72 m2 incluyó nueve cepas, se contabilizó el número de árboles vivos y se midió el diámetro normal. La altura total y volumen por árbol se estimaron con modelos tipo Chapman-Richards y Schumacher-Hall, respectivamente. El área basal y el volumen total por sitio se proyectó a nivel de hectárea. Se estratificó por clase de edad; el área basal y la edad de la plantación se utilizaron como variables auxiliares. La estrategia de muestreo para estimar el volumen medio se conformó al asociar como diseño de muestreo al muestreo simple al azar con el estimador de razón específica en muestreo estratificado, con una estratificación por clases de edad de un año y el área basal como variable auxiliar, esto dio la precisión de 0.21 %. El tamaño de muestra en el muestreo estratificado se redujo 68.3 %, con precisión de 2.5 % del muestreo original, lo cual implica que el esfuerzo de muestreo y el tiempo de ejecución para realizar el inventario operativo puede reducirse, con la consecuente disminución de los costos implícitos.
Palabras clave Estimadores de muestreo; estratificación; inventario forestal; tamaño de muestra; teca; variables auxiliares
Introduction
Sampling and inventory methods with statistical validity are very useful for generating reliable and scientifically defensible estimates (Schreuder et al., 2004; Fattorini et al., 2015). In a commercial forest plantation (CFP) it is important to accurately estimate timber stocks, which allows having information to plan actions and make informed decisions about technical management and investment (Roldán et al., 2014). When the CFP area is extensive, the inventory is carried out by sampling to obtain parameters of interest from the population in a correct, precise way and at minimal cost (van Laar and Akça, 2007; Köhl and Magnussen, 2016).
The aim of the timber forest inventory is to provide reliable quantitative information for the operational management of the CFP, for which accurate knowledge is demanded at the minimum management unit level on means and totals of the number of trees (N), basimetric area (BA) and volume (V) per hectare. The foregoing makes it necessary to seek a balance between the available financial resources and the required statistical precision (Roldán et al., 2014). In this sense, the best approach to optimize resources is to design a sampling strategy, which must combine the method to select the sample with the procedure to estimate the population parameters (Gregoire and Valentine, 2008; Grafström et al., 2014).
In order to carry out a multipurpose inventory, the single, simple and flexible sampling scheme is normally adopted, as it favors the assessment of all the mensuration variables of interest (Schreuder et al., 1993; Corona and Fattorini, 2006). Whatever the sampling design and sample size, there are implications in terms of cost and efficiency in the reliability of estimating inventory, which can be quantified based on precision, bias, and mean square error (Köhl et al., 2011; Roldán et al., 2014). Precision is a function of the homogeneity of the population and the number of sampling sites surveyed, so the inventory must be carried out to meet the predetermined level of precision and to optimize both, the time and the costs, to carry it out (Schreuder et al., 2004; van Laar and Akça, 2007; Marchi et al., 2017).
The specialized literature suggests different sampling estimators to make a CFP timber inventory oriented to technical-operational management. Among them, those that make the inference about the population based on the design are simple random sampling (MSA), systematic sampling and stratified sampling (ME), as well as those in which the inference is made based on to models (Raj, 1980; Scheaffer et al., 1987; Cochran, 1993; Gregoire and Valentine, 2008). The latter use auxiliary information through variables that are highly and positively correlated with the variable of interest; among these types, the ratio and regression estimators that can be applied in MSA and ME stand out (Schreuder et al., 1993; Shiver and Borders, 1996; Pérez, 2005; Grafström et al., 2014). However, there is little recent empirical evidence derived from analysis of specific practical cases that allows determining the best sampling estimators or their possible combinations that lead to the design of an optimal sampling strategy.
In this context, the objectives of this study were: 1) To evaluate the statistical efficiency of six sampling estimators to propose an optimal sampling strategy in terms of precision and time that allows carrying out operational timber inventories that support decision-making aimed at improving the CFP technical handling of Tectona grandis L.f. (Teak) established in the state of Campeche, Mexico; and 2) To determine the optimal sample size for the mean volume in MSA and ME that ensures a 2.5 % precision and ( = 0.05.
Materials and Methods
The study was carried out in a commercial forest plantation with an area of 2 207.5 hectares planted with Tectona grandis at a 4 m × 2 m spacing (1 250 plants ha-1), which is located in the Edzná Valley, Campeche municipality, state of the same name, in southeastern Mexico. The prevailing climate is Aw”0(i´)g, which corresponds to the warm subhumid type with rains in summer, average annual precipitation of 1 094.7 mm with six months of drought from December to May; 26.6 °C average annual temperature, with prevailing winds in winter and summer, with maximum gusts of up to 60 km h-1 (Breña, 2004).
A sample size (
At each site, nine strains were evaluated, each one referring to the specific point where a specimen of Tectona grandis was planted; in this way, it was possible to count the number of living trees, each of which had their normal diameter (Dn) measured with a 283D / 5m-CSE diametric tape. In addition, the age (E) of the plantation was recorded in years. The total height (A) of each tree in meters was estimated based on Dn with a Chapman-Richards model used by Tamarit (2013) that presented the following mathematical structure:
Where:
The stem volume (
Based on Fierros et al. (2018), the BA of each tree in m2 was estimated with the formula:
Where:
The number of living trees, the basimetric area and the total volume estimated by site were projected at the hectare level, in order to obtain the N, BA and V variables scaled to conventional units.
To estimate the mean wood volume, the estimators and parameters of the MSA and the ME were applied, in which the inference about the population is based on the design (Cochran, 1993; Gregoire and Valentine, 2008). The ratio (R) and regression (Rg) estimators were also assessed within MSA, whose inference is based on models, in addition to the combined ratio (Rc) and the specific ratio (Re) within the ME (Shiver and Borders, 1996; Pérez, 2005) (expressions 1 to 14 of Table 1).
Estimator | Parameter | Equation | |
---|---|---|---|
MSA | Mean |
|
(1) |
Variance of the mean |
|
(2) | |
Sample size |
|
(3) | |
ME | Mean |
|
(4) |
Variance of the mean |
|
(5) | |
Sample size |
|
(6) | |
Ratio under MSA | Mean |
|
(7) |
Variance of the mean |
|
(8) | |
Regression under MSA | Mean |
|
(9) |
Variance of the mean |
|
(10) | |
Combined ratio in ME | Mean |
|
(11) |
Variance of the mean |
|
(12) | |
Specific ratio in ME | Mean |
|
(13) |
Variance of the mean |
|
(14) |
y
i
= Volume with bark observed at the i-th
sampling site and extrapolated to hectare (m3
ha-1); f = Correction factor for
finitude 1-n/N;
t = Student's t-distribution value at 95 %
reliability (1-( = 0.95) and with n-1 degrees
of freedom (gl); B = Magnitude
of the sampling error acceptable at the specified 1-( confidence
level and the result of multiplying the required precision (
In all cases, the MSA was used as the reference sampling design. To stratify, the age of the plantation was used as an auxiliary variable. Six age classes (CE) were defined with one-year intervals (Table 2).
CE | LI | LS | EP | n | Surface (ha) |
---|---|---|---|---|---|
3 | 2 | 2.99 | 2.92 | 100 | 25.00 |
4 | 3 | 3.99 | 3.62 | 897 | 224.25 |
5 | 4 | 4.99 | 4.37 | 3 436 | 859.00 |
6 | 5 | 5.99 | 5.46 | 2 503 | 625.75 |
7 | 6 | 6.99 | 6.58 | 874 | 218.50 |
8 | 7 | 7.99 | 7.21 | 1 020 | 255.00 |
Total | 8 830 | 2 207.5 |
CE = Age class; LI = Lower limit; LS = Upper limit; EP = Average age; n = Sample size.
In each stratum of the ME, the same estimators referred to for the MSA were applied. In the Ratio and Regression estimators, the BA and the age of the plantation were used as auxiliary variables, the value of the correlation coefficient (ρ) between the V with the BA and age was determined, the auxiliary variable is better when ρ is close to 1. In the estimator of R under MSA, the sample mean of the BA obtained under ME was assumed as the true population value. With the estimator of R in ME, the age was known without sampling error because its exact record was present.
The way in which the confidence interval (
According to Freese (1976), Schreuder et al. (1993)
and Schreuder et al.
(2004), the theoretical approach referred to estimate the mean has
greater meaning when an
The best sampling estimator was selected based on the lowest value of the variance of the mean, the highest precision, and the lowest width of the confidence intervals. The MSA estimators were taken as the reference baseline against which the rest of the estimators were compared (West, 2017). The sampling strategy was formed by combining the sampling design with the best estimator. The calculations to apply the estimators were carried out in the Microsoft Office® Excel 2013 spreadsheet.
The optimal sample size for the mean volume in MSA and ME was determined by
simulating a repeated sampling using the Bootstrap technique with the free
statistical program R version 3.6.2 (https://www.r-project.org, R
Development Core Team, 2020) for Windows. The total sample (8 830
sites = 100 %) was gradually reduced in 5 % intervals; at each level of
reduction 30 samples were obtained without replacement (Marchi et al., 2017) to which the mean,
the variance of the mean, the
Results and Discussion
All the evaluated estimators had accuracies below 2 %. The Specific Ratio
estimator in ME had the best statistical efficiency followed by
the Regression estimator in MSA and the combined Ratio in
ME. These three estimators had the best precisions, the
lowest variances, narrow
Estimator | Parameter |
GP (%) |
A | |||||
---|---|---|---|---|---|---|---|---|
Mean | VM | LI | LS |
(%) |
Inventory
(m3) |
|||
MSA | 29.98 | 0.0609 | 29.49 | 30.46 | 1.61 | 66177 | - | 0.97 |
ME | 29.98 | 0.0432 | 29.56 | 30.39 | 1.39 | 66177 | 0.23 | 0.83 |
R V/BA MSA | 29.98 | 0.0024 | 29.96 | 29.99 | 0.33 | 66177 | 1.29 | 0.03 |
R V/E MSA | 33.32 | 0.0495 | 33.24 | 33.40 | 1.34 | 73560 | 0.28 | 0.16 |
Rg V/BA MSA | 29.98 | 0.0014 | 29.90 | 30.05 | 0.25 | 66177 | 1.36 | 0.15 |
Rg V//E MSA | 29.98 | 0.0475 | 29.54 | 30.41 | 1.45 | 66177 | 0.16 | 0.87 |
Rc V/BA ME | 29.98 | 0.0024 | 29.88 | 30.08 | 0.33 | 66177 | 1.29 | 0.20 |
Re V/BA ME | 29.98 | 0.0010 | 29.92 | 30.04 | 0.21 | 66177 | 1.40 | 0.13 |
Rc V/E ME | 29.98 | 0.0499 | 29.53 | 30.43 | 1.49 | 66177 | 0.12 | 0.89 |
Re V/E ME | 29.98 | 0.0423 | 29.57 | 30.39 | 1.37 | 66177 | 0.24 | 0.82 |
VM = Variance of the sample mean in
m3 ha-1; LS and
LI = Upper and lower limit of the mean;
The favorable effect when stratification by age class is made is outstanding. In this regard, Lencinas and Mohr-Bell (2007) point out that a strategy to optimize an inventory is to stratify with some variable strongly related to the variable of interest, as in this case was the age of the CFPs. From the foregoing, it can be deduced that by implementing stratification by age class in practice, the sampling effort can be achieved to be less, since a smaller sample size is required, which coincides with that determined by Roldán et al. (2014). Furthermore, the sampling error is favored when estimating the population parameter. In contrast, when age was used only as an auxiliary variable, particularly in the case of the R estimator in MSA, poor values were obtained in the referred statistical efficiency parameters and the mean was overestimated.
When BA was used as an auxiliary variable, the ratio estimators within the ME were superior to the ratio estimators in the MSA, based on Freese (1976), Scheaffer et al. (1987) and Corona and Fattorini (2006). This is due to the fact that by dividing the population into strata, homogeneity is achieved within them, thereby reducing the total variance and thereby increasing the accuracy of the estimator; thus, for the same sampling intensity, the ME generates more precise estimates than the MSA. Based on this experience and on Köhl and Magnussen (2016), other variables, in addition to age, that can be considered as logical and feasible to stratify CFPs are the classes by site index, by diameter category or by increment in BA.
BA as an auxiliary variable was better than age, a more pronounced effect for the Rc and Re estimators within the ME; in this regard Fattorini et al. (2015), Vallée et al. (2015) and Adichwal et al. (2019) indicate that the auxiliary variables used, in addition to having a high correlation with the variable of interest, must be easy, fast and cheap to measure, which makes them highly efficient. Scheaffer et al. (1987) report that, when the correlation coefficient (ρ) between the variable of interest and the auxiliary is greater than 0.5, the estimator of the ratio for the population mean is more precise than the MSA; in this case, the superiority of BA as an auxiliary variable is due to the fact that the value of ρ was 0.99 and that of age 0.47.
The volume versus the BA showed a linear relationship through the origin (Figure 1a), the variance of the volume maintained a proportional trend with respect to the BA, favorable conditions for which the ratio and regression estimators had a lower variance for the mean population and consequently greater precision than the simple mean obtained from the MSA. This coincides with what was reported by Roldán et al. (2014) for Eucalyptus urophylla S. T. Blake CFPs, who for the volume mean determined that the AB with ρ = 0.97 was more accurate for the Rc estimator in ME. It also agrees with that recorded by Fierros et al. (2018) when noting that the Rg estimator, with BA as an auxiliary variable, was the most accurate to estimate the timber inventory in CFPs of Pinus chiapensis (Martínez) Andersen.
Ratio estimators have the additional advantage that the auxiliary variable offers practical implication when estimating inventory. In this case, the value of the ratio indicates that for each m2 of BA there is a volume of 4.7 m3 ha-1, which allows estimating the inventory in volume immediately, just by knowing precisely the population BA, which it could be determined by surveying sampling points with a relascope. When age was the auxiliary variable, it was determined that for each passing year, the volume increases on average 5.8 m3 ha-1; this makes it possible for the estimator to function as a simplified growth and yield system, because by knowing the age weighted by strata (5.7 years), the inventory can be estimated in an annualized way as the plantation grows; thus, it is assumed that the planted area will remain constant over time and that plantation areas are not incorporated or removed.
The V/AB ratio value in this study was slightly lower than that determined with a growth system developed by Tamarit et al. (2019) for the same species and study region, which was 5.3 m3 ha-1 for the average condition (site index 15 m). Meanwhile, the value of the V/age ratio was less than the 9.3 m3 ha-1 estimated with the growth system for the same average condition referred to.
Even though the ratio estimators are biased, the bias is minimal and can be considered null as long as the sample size per stratum is equal to or greater than 30 sampling units (Freese, 1976; Scheaffer et al., 1987; Velasco et al., 2003), a condition that is fulfilled in this analysis for each stratum because the smallest stratum was 100 sites. This helps to explain the greater efficiency of the estimator of Re, which has the additional advantage over that of Rc, the fact that when obtaining the estimator for each stratum and then adding them, it causes the ratio to vary between strata, which gives separate estimations and information on the population at the stratum level.
When the relationship between the main variable and the auxiliary variable is linear through the origin (Figure 1a), the estimators of Rc and Re by stratum are practically unbiased and in practice the bias can be considered null. In this context, according to Raj (1980) and Pérez (2005), the estimator of Re was higher than that of Rc (in both the BA was an auxiliary variable), because the number of strata was small and each stratum was relatively large, which implied having few addends and the accumulation of bias was minimal; furthermore, the ratios per stratum were different and ascending as the age class increased.
With reference to the estimator of Rc in the ME with the AB as an auxiliary variable, Pérez (2005) indicates that its advantage over that of Re is that it does not present accumulation of biases in the strata, so that the total bias, when it is present, is reduced to the minimum. However, it has the disadvantage that it does not offer estimates separated by strata; in practice, the Rc should be used when the number of strata is large and the size of each one is small.
The analysis of the behavior of the sample size (
The trend of the averages of the estimated precisions, when varying the sample size, was very similar when it was determined with the Bootstrap technique for the MSA than when it was obtained by the formula (Figure 2b). This behavior leads to the assumption that similar patterns of correspondence would be defined in the rest of the assessed estimators.
To achieve the precision set at 2.5 %, it was determined that the optimal
The Bootstrap procedure also allowed a better appreciation of the effect of the
sample size reduction on the variance, which increases up to a certain limit as
the sample decreases (Figure 3a).
Furthermore, the upper and lower limits of the
The sample size could be further reduced as precision relaxes, especially in
ME; however, it must be borne in mind that in order to put
into practice the best referred estimators, it must be ensured that the sample
has at least 30 sites per stratum; thus, the population parameters will have the
best precision. In this regard, the distribution of the sample size calculated
by stratum and total for the 2.5% precision (( = 0.05) in the proportional
allocation and the Neyman is shown in Table
4. From a total of
Distribution | Stratum (age class) | Total | |||||
---|---|---|---|---|---|---|---|
3 | 4 | 5 | 6 | 7 | 8 | ||
Proportional | 32 | 284 | 1 089 | 793 | 277 | 323 | 2 799 |
Neyman | 6 | 130 | 906 | 975 | 349 | 434 | 2 799 |
When the objective is to obtain separate estimators for each stratum and the global estimator is considered to be of secondary importance, Freese (1976) points out that intense sampling should be carried out in the strata that have material of high commercial value, then the referred assignments go to background. Under this scenario, the number of necessary sites must be raised in such a way as to achieve the desired degree of precision for the strata of interest. This is relevant to apply in CFPs whose ages are close to the shift and in which commercial thinning will be applied.
One additional factor that must be considered in determining the sample size is what the current forest legislation establishes in terms of precision and reliability. In this regard, the Official Mexican Standard NOM-152-SEMARNAT-2006 demands that in the preparation of timber inventories through sampling, a minimum reliability of 95 % (( = 0.05) at the farm level and a maximum sampling error of 10 % must be obtained (Semarnat, 2008).
The efficient quantification of timber stocks gives certainty to face forest pledge processes, thereby contributing to the capitalization of the CFP as a business and forest company, because it is possible to place the value of the plantation as collateral for the banking transaction or good to improve securitization and market conditions.
Conclusions
The optimal sampling strategy in terms of precision to carry out the operational inventory and estimate the mean volume of PFC of Teak in Campeche, Mexico, was formed by associating the sampling design under simple random sampling with the estimator of specific ratio within sampling stratified, with basal area as an auxiliary variable and when stratifying by age class.
In the stratified sampling, the initial sample size could be reduced to 68.3 % to maintain the precision of 2.5 % (reliability of ( = 0.05), which implies less time and effort, with the consequent reduction of costs in the timber inventory.
The age of the CFP was more efficient when it was used for stratification than when it was used as an auxiliary variable in the ratio and regression estimators. To obtain the highest precision in the population parameters, it is recommended to stratify by age classes with intervals of one year.
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Received: January 20, 2021; Accepted: September 30, 2021