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Agrociencia
versão On-line ISSN 2521-9766versão impressa ISSN 1405-3195
Agrociencia vol.50 no.1 Texcoco Jan./Fev. 2016
Natutal Renewable Resources
Comparison of wood volume estimators in medium height tropical forests of Mexico
1Campus Montecillo. Colegio de Postgraduados. 56230. Montecillo, Estado de México. México. (velasco.efrain@colpos.mx) (hectorm.delossantos@gmail.com).
2División de Ciencias Forestales-Universidad Autónoma Chapingo. 56230. Chapingo, Estado de México. México.
In Mexico, the project National Forest and Soil Inventory (Inventario Nacional Forestal y de Suelos-NFyS) uses ratio estimators that are easily applicable for statistical analysis of dasometric data. However, under diverse circumstances, it may be impossible to obtain data that fully comply with their theoretical assumptions. In this study, the results of the INFyS ratio estimator are compared with results using alternative estimators. With 2010 data from the sampling grid obtained in the medium height sub-evergreen and subdeciduous forests of Quintana Roo and Campeche, Mexico, the ratio of means estimator, mean of ratios estimator and the Forest Inventory and Analysis (FIA-USDA) were compared statistically. The relative bias of the ratio estimator (bias, relative to the standard error) in all cases was below 10 % and, therefore, considered insignificant. Under the conditions of this study, the results indicate that the three estimators are not statistically different. However, the mean of ratios estimator has the advantage of directly providing estimations per hectare. Moreover, its theoretical approach permits its analysis from a perspective of the Horvitz-Thompson estimator.
Key words: Ratio of means; mean of ratios; bootstrap; spatially disjointed inclusion zones
En México, el proyecto Inventario Nacional Forestal y de Suelos (INFyS) utiliza para el análisis estadístico de los datos dasométricos estimadores de razón cuya aplicación es fácil. Sin embargo, en circunstancias diversas podría ser imposible obtener datos que cumplan por completo con sus supuestos teóricos. En este estudio se compararon los resultados del estimador de razón del INFyS con estimadores alternativos. Con datos del 2010 provenientes de la malla de muestreo obtenida en selva mediana subperennifolia y subcaducifolia en Quintana Roo y Campeche, México, se compararon estadísticamente los estimadores: razón de medias, medias de razones e Inventario Forestal y Análisis (FIA-USDA). El sesgo relativo del estimador de razón (sesgo respecto al error estándar) en todos los casos fue menor a 10 %; por tanto se consideró insignificante. En las condiciones estudiadas, los resultados indican que los tres estimadores no son estadísticamente diferentes. Sin embargo, el estimador medias de razón presenta la ventaja de proporcionar estimaciones por hectárea de manera directa. Además, su planteamiento teórico permite analizarlo desde una perspectiva del estimador Horvitz-Thompson.
Palabras clave: Razón de medias; media de razones; bootstrap; zonas de inclusión espacialmente disjuntas
Introduction
The National Forest and Soils Inventory (Inventario Nacional Forestal y de Suelos-INFyS) uses the systematic stratified cluster sampling design. The clusters are located 5 km equidistance from each other in temperate forests, as well as in high and medium high tropical forests, 10 km in low tropical forests and semiarid communities, and 20 km in arid communities (CONAFOR, 2012).
The cluster, or primary sampling unit (PSU), from a conceptual point of view, is a 1 ha circular plot (56.42 m radius), in which four secondary sampling units (SSU), or sites, are evaluated. SSU are 400 m2 each, placed geometrically in an inverted Y-shape with respect to the north. They are circular in the case of temperate forests and arid zone vegetation, and rectangular in tropical forests. SSU number 1 is the center of the PSU and is the georeferencing point of the cluster; SSU 2, 3 and 4 are peripheral (satellite). The distance from the center of SSU 1 (site 1) to the center of the other SSU (sites 2, 3 and 4) is 45.14 m when the sites are circular and 36.42 m when they are rectangular. The azimuth for locating SSU 2, 3 and 4 from the center of SSU 1 is 0°, 120° and 240°, respectively. The study units (SU) are the trees or other biological elements in the SSU (Velasco et al., 2005; CONAFOR, 2012). In each SSU, diameter at breast height (DBH) and total height of trees with DBH larger than 7.5 cm are measured. The cluster design is similar to the Forest Inventory and Analysis (FIA) program for evaluation of forest resources in the USA (Bechtold and Scott, 2005; McRoberts et al., 2005).
Location of the clusters in the field is subject to conditions and vegetation of the terrain. In a cluster, it might be impossible to install one or several SSU since they might be placed in canyons, lakes, farmland, roads or mines (non-existent SSU). In other circumstances, by scale in the cartographic base (Land use and Vegetation SERIES IV, INEGI, 1:250 000), it is possible that in the field part of the cluster is located in a stratum different from that defined in the cartography. In other cases, because of borders or topographic barriers, only a part of the cluster can be installed, and thus, the SSU will be inaccessible. This situation of clusters of different sizes has led to the use of ratio estimators in the INFyS.
Even so, when a cluster comprises two or more disjointed subplots, it is treated as a single plot because the cluster is tied to a single sampling point. Indeed, the cluster of plots serves to identify those elements whose spatially disjointed inclusion zones include the sample point (Gregoire and Valentine, 2008). Thus, there are different ways to estimate the forest parameters of interest.
The contribution of medium height rainforest is poor to the reality and potential of lumber production, although it is useful to obtain some non-timber forest products. Besides biodiversity, the sub-evergreen medium height tropical forest is negligible in terms of its capacity for carbon storage. Eaton and Lawrence (2009) report that in the medium height sub-evergreen tropical forest of Campeche and Quintana Roo live biomass on the soil oscillates between 4.8 Mg ha-1 in a three-year-old secondary forest and 73.5 Mg ha-1 in a mature forest, whereas, aboveground carbon combined with soil carbon varies from 192 Mg ha-1 in a 12-year-old secondary forest to 469 Mg ha-1 in a mature rain forest. These qualities diminish the importance of studying the medium height sub-evergreen tropical forest from a quantitative perspective.
Although the linear relationship between the variable of interest (volume, basal area, number of trees) and the auxiliary (area) is weak and the line does not always pass through the intercept, the ratio estimator is used in the National Forest and Soils Inventory (INFyS) project of Mexico. Therefore, and to evaluate the behavior of alternative estimators in complex forest ecosystems, the objective this study was to compare the estimators means of ratios, ratio of means and FIA, and timber volume (m3 ha-1) was the parameter of interest in sub-evergreen and sub-deciduous tropical forests of Quintana Roo and Campeche, Mexico. The hypothesis was that the three estimators are not statistically different.
Materials and Methods
Cluster design and inclusion zone
The INFyS cluster design is a cluster of subplots (circular or rectangular) that comprises a central georeferenced subplot at a sampling point (x s , z s ) and n-1 satellite non-overlapping subplots in a fixed arrangement. This type of arrangement of subplots defines an inclusion zone, spatially disjointed around the central point of any tree (Valentine et al., 2006). A cluster of subplots includes one tree if the sampling point falls within any part of the inclusion zone of the tree (Valentine et al., 2006; Gregoire and Valentine, 2008).
In order to present the possible estimators and their respective variances, in the following section presents the notation in detail.
Notation
A: forested area of the region of interest (ha); U
k
: k
th element in the population; m: sampling points; P
s
: s
th cluster, s=1,2,..., m; (x
s
, z
s
): coordinates of the cluster P
s
; (x
k
, z
k
): geographic location of an element of the population, U
k
; a
sj
: area of each subplot (400 m2) in PS; ns: number of subplots effectively evaluated in the field in plot s, n
s
=1,2,3,4; a
k
: area of the sub-zone of inclusion of the element U
k
; n
s
a
k
: area of the zone of inclusion of the element U
k
; πk=n
s
a
k
/A: probability of inclusion of element U
k
; y
k
: value of an attribute of interest associated with U
k
(basal area, volume, biomass, carbon);
Estimators
Under the principle that clusters can be of different sizes, Velasco et al. (2005) proposed ratio of means estimators to analyze data of the National Forest Inventory of Mexico. In the numerator, they considered the attribute of interest (volume) and in the denominator the area of the cluster as the auxiliary variable. In this sense, the estimator is expressed as follows:
In (1), y s is the total quantity of the attribute in sampling plot s and a s is the area (ha) of the sampling plot s effectively evaluated in the field. The variance estimated in (1), according to Cochran (1993) is:
where
The ratio estimator is a biased estimator; however, there may be certain circumstances in which it can be unbiased. Ratio estimator (1) is more effective when 1) the relationship between y s and a s is a straight line that passes through the origin, and 2) the variance of y s around this line is proportional to a s . In practice, the use of (1) is justified if 1) the coefficient of correlation between y s and a s is above 0.5, and 2) the sample size is above 30 (Scheaffer et al., 1987; Cochran, 1993).
Scott et al. (2005), within the framework of the Forest Inventory and Analysis (FIA) program of the United States, proposed an estimator for forest attributes similar to a mean of ratios, but for the estimation in each sample plot s, an adjusted area is considered:
where a
0 is the total area used to observe the attribute of interest in a plot (four times the area of the subplot) and p
0 is the mean proportion of plot areas that fall within the population of interest, obtained with
The values y ha (s) of (3) are used to estimate the mean per hectare:
The variance of (4) is estimated in the following manner:
The estimator of Scott et al. (2005) is a mean of ratios estimator that considers, in each sample plot s, the complete area of the cluster adjusted by the mean proportion of the areas of the plots observed that fall within the population of interest.
The probability of including a U k element in each sample point allows deriving another estimator from the viewpoint of repeated sampling. Thus, T y can estimate with an unbiased Horvitz-Thompson (HT) estimator, that is:
In Equation (6), ρ k = y k / n s a k is the value of y k prorated per unit of area and ps is the sum of all the prorated values for the P S cluster. Installation of multiple independent P S clusters in A, followed by the HT estimation of T y with data from each P S (Equation 6) is replicated sampling (Barabesi and Fattorini, 1998; Barabesi and Pisani, 2004). According to Schreuder et al. (1993) and Gregoire and Valentine (2008), the T y estimator based on replicated sampling of m P S is:
The variance of
The total quantity of an attribute of interest divided by the area of the population results in the average quantity per unit of area; that is, λ
yπ,rep
. This quantity can be estimated unbiasedly with data from the sample with
where
Equation (9) does not require explicit determination of the inclusion probabilities of each tree, but it does assume that regardless of the number of satellite subplots that finally integrate the cluster in the field, the sub-zones of inclusion are complete (not truncated). Thus, (9) implies that for each element, the attribute of interest is divided by the area of the cluster effectively evaluated in the field (n s a k = n s a sj ) which is common for all the trees belonging to the same cluster.
Assuming that the sub-zones of tree inclusion are complete, estimator (9) can be considered a mean of ratios (Smelko and Merganic, 2008). The method mean of ratios was recommended for systematic sampling with sample plots of different sizes (Smelko and Saborowski, 1999). Therefore, in each sample plot s, the sample data ys need to be calculated to an equal area (1 ha) using the following formula (Smelko and Merganic, 2008):
where y s is the total quantity of the attribute in the sample plot s and a s is the area (ha) of the sample plot s effectively evaluated in the field.
These values per hectare y ha (s) are used to estimate the mean per hectare:
There are no estimators for the variance of (12) when systematic sampling is used. It is therefore estimated in the following way:
this estimation was shown as a conservative variance in applications of systematic sampling (Smelko and Saborowski, 1999).
Estimation of the variance for the estimators (1), (4) and (12) considers random sampling even when the sampling design of the National Forest Inventory is systematic. However, in forest inventories, it is reasonable to assume that systematic sampling is approximately equivalent to simple random sampling (Zarnoch and Bechtold, 2000). Velasco et al. (2005) refer to the use of random sampling in forest inventories when the data are collected in a systematic grid. Cochran (1993) states that if the population is random, the formula for the variance in systematic sampling is the same as for simple random sampling.
Another way to estimate the variance of (1), (4) and (12) is with the non-parametric bootstrap method; with this, the variance is calculated on the basis of a large number of estimations corresponding to bootstrap samples obtained from the original sample (Efron, 1979; Särndal et al., 1992; Pérez, 2000). In any case, assuming for
Database
In this study, data from medium height sub-evergreen and sub-deciduous tropical forests of Quintana Roo and Campeche taken in the field in 2010 were used within the framework of the project National Forest and Soils Inventory. For Quintana Roo, m=206 clusters of medium height sub-evergreen tropical forest and m=22 medium height sub-deciduous tropical forest were used. For Campeche, the data included m=127 clusters of medium height sub-evergreen tropical forest and m=58 clusters of medium height sub-deciduous tropical forest. The equations used were those of trunk volume reported by SAG (1976) and SARH (1985), which were applicable at the state level and by botanical group. Timber volume was selected as the parameter of interest because at the regional level there are functions of volume in the Mexican Southeast. The variables volume and carbon are highly correlated, so that similar results are expected when the variable carbon is of interest. For analysis of the information, a program was created with the software Statistical Analysis System Version 9.2 (SAS, 2009). In this program, a matrix (mxp) was constructed. Columns correspond to longitude, latitude, basal area, tree density, trunk timber volume and area of the cluster effectively sampled; rows identified the clusters.
The medium height sub-evergreen and sub-deciduous tropical forest of the Mexican southeast is of little importance from the quantitative perspective per se, but we decided to conduct this study in four of these tropical forest populations in Quintana Roo and Campeche to evaluate the statistical behavior of the estimators with different sample sizes. In these conditions, we hope the results will be generalizable for other forest populations of interest.
The timber volume per hectare was estimated with the following methods: 1) Mean of ratios (Smelko and Merganic, 2008), 2) ratio of means (Velasco et al., 2005) and 3) mean of ratios with adjusted area (Scott et al., 2005). The variances were obtained according to the conventional expressions reported for each estimator. Besides the point estimations, estimations by interval were also obtained. Additionally, for the ratio estimator, bias was estimated with
The overlap of the confidence intervals and the similarity in the sampling error were used as criteria to compare the estimators studied.
Results and Discussion
The correlation between timber volume and cluster area effectively sampled in the field was around 0.43. Under the conditions studied, the correlations were significant at a 5 %.
The dispersion diagrams indicated that, in all cases, the variance of volume was proportional to the area of the cluster. In the medium height sub-evergreen tropical forest (Quintana Roo and Campeche) the small clusters (0.04, 0.08 and 0.12 ha) were well represented. The opposite occurred in the medium sub-deciduous tropical forest. Thus in Quintana Roo (m=22) there were no clusters with an area of 0.12 ha, and in Campeche (m=58) there was only one with an area of 0.04 ha and another with an area of 0.08 ha. The latter situation may have been due to the relatively small sample size.
The regressions of volume over area of the cluster, y=β0+β1 a indicated that, for the medium height sub-evergreen tropical forest of Quintana Roo and Campeche, the intercept was significant at 5 % (value of p of the t test below 0.05); that is, H o : β0 =0 was rejected. In the medium height sub-deciduous tropical forest of the two states, H o : β0 =0 was not rejected. Given the dispersion of the observations, in no case did the coefficient of determination was higher than 0.5.
The above situation, puts the ratio of means estimator at a disadvantage. To be effective it requires a coefficient of correlation between the variable of interest and the auxiliary variable above 0.5, and the relationship must be a straight line that passes through the origin. However, it must be kept in mind that in three of the four populations studied, the number of observations greatly surpasses the suggested sample size, which should be at least 30; thus, use of the ratio estimator is justified.
The point values per hectare calculated with the three estimators in each population were also similar. Consistently, the mean of ratios estimator provides slightly lower values (conservative) compared with the other two estimators. The relative biases estimated by the ratio estimator (bias relative to the standard error) were 0.18, 0.73, 0.43 and 0.22 %, for SMSUPQROO, SMSUCQROO, SMSUPCAMP and SMSUCCAMP, respectively. The relative bias of 0.73 % corresponded to the sampled population with only 22 clusters. All of these values are below 10 %, and so the bias of the ratio estimator is practically insignificant (Table 1).
Table 1 Point estimation and by interval of timber volume (m3 ha-1) in medium height sub-evergreen and sub-deciduous tropical forests of Quintana Roo and Campeche, Mexico, obtained with the estimators Mean of ratios (MR), Ratio of means (RM) and Forest Inventory and Analysis (FIA).
| Población de interés | Estimador | Estimación puntual | Límite inferior | Límite superior | Error de muestreo (%) |
|---|---|---|---|---|---|
| SMSUPQROO (m=206) | MR | 105.90 | 96.54 | 115.25 | 8.83 |
| RM | 109.23 | 99.95 | 118.52 | 8.50 | |
| FIA | 109.23 | 99.26 | 119.21 | 9.13 | |
| SMSUCQROO (m=22) | MR | 112.35 | 77.30 | 147.40 | 31.20 |
| RM | 118.75 | 83.46 | 154.05 | 29.72 | |
| FIA | 118.75 | 80.00 | 157.50 | 32.63 | |
| SMSUPCAMP (m=127) | MR | 106.10 | 94.32 | 117.87 | 11.10 |
| RM | 111.50 | 99.85 | 123.14 | 10.44 | |
| FIA | 111.50 | 98.19 | 124.81 | 11.94 | |
| SMSUCCAMP (m=58) | MR | 98.32 | 86.10 | 110.55 | 12.43 |
| RM | 99.92 | 87.68 | 112.17 | 12.25 | |
| FIA | 99.92 | 86.78 | 113.07 | 13.15 |
SMSUPQROO: medium height sub-evergreen tropical forest of Quintana Roo, SMSUCQROO: Medium height sub-deciduous tropical forest of Quintana Roo, SMSUPCAMP: medium height sub-evergreen tropical forest of Campeche, SMSUCCAMP: Medium height sub-deciduous tropical forest of Campeche.
In all cases, the ratio of means estimator, even though there is generally a weak correlation between the variable of interest and the auxiliary, it is slightly more precise (lower sampling error), followed by the mean of ratios estimator and FIA. This may explain why the sample size was larger than 100 in the medium height sub-evergreen tropical forest. In the case of the medium height sub-deciduous tropical, although the sample size was smaller than 100, there is evidence of a linear relationship to the origin of volume and cluster area. Because there is an overlap of the 95 % confidence intervals, it can be considered that the three estimators are not statistically different. In addition, in each of the studied populations the sampling errors of the evaluated estimators had a high degree of similarity.
In the two populations of each state, the FIA estimator was slightly less precise than the other two. Consistently, it is observed that it has a larger sampling error, even when the sample size is larger than 100. The reason for this may be that the denominator of the expression that permits obtaining the values per hectare considers a common adjusted area; the area of the cluster originally desirable (0.16 ha) is adjusted by a mean proportion of all the areas effectively sampled in the field.
The above results are congruent with other studies that compared mean of ratios and ratio of means estimators in the field of forestry. In this respect, Smelko and Saborowski (1999), with data from the forested region of northern Slovakia, 32 sample plots of different sizes (from 100 to 1000 m2 in ranges of 100) collected in 1991 and 1995, studied the mean of ratios (method A) and ratio of means estimators (method B) in the estimation of number of trees and timber volume per hectare. The correlations between number of trees and the area of the plots were low (0.323 and 0.278 for 1991 and 1995, respectively), whereas the correlations between volume and area of the plots were small (0.323 and 0.278 for 1991 and 1995, respectively). In the second case, no relationship of a straight line passing through the origin was observed. In both years, method A gave a lower volume per hectare than B (1991: A 423.3 and B 453.9 m3 ha-1; 1995: A 443.5 and B 470 m3 ha-1) and standard errors were also lower than those of B. The opposite was true for the number of trees per hectare. On the two occasions and for the two parameters, the differences in standard errors in both methods were not above 2.7 %.
Smelko and Merganic (2008) evaluated the ratio of means and the mean of ratios estimators with data of the National Forest Inventory and Monitoring of Slovakia. For commercial timber volume, they report the following results per hectare:
With the ratio estimator, the values of the sampling units are treated globally; that is, the total of the variable of interest and the total of the auxiliary variable are obtained, and finally, the quotient of both is obtained. With the mean of ratios estimator, the first level of inference is at the hectare (plot) level, then at the population level. One powerful statistical method for analyzing inventory data obtained by panelized sampling is Generalized Least Squares, which considers the covariance between re-measured plots. Thus, it is preferable to maintain the inference level at the hectare level.
Sample size has a notable effect on the sample error. Thus, for example, for the volume estimated by FIA when m=206, the maximum sampling error is 9.15 %; when m=127, the maximum sampling error is 12.35 %; when m=58, the maximum sampling error is 13.24 %; and when m=22, the maximum sampling error shoots up to 32.85 %. With 22 observations, in the ratio and mean of ratio estimators there was also a sampling error of around 30 %. This situation is not a reason for concern considering that INFyS is designed for the study of forest populations of great magnitude. Low sample sizes of these populations will require greater sampling intensity to achieve reliable estimations.
When considering that
This situation leads us to assume that in other forest populations different from tropical rainforests, the MR, RM and FIA estimators would behave as they did in this study.
Conclusions
In each of the situations tested, the overlap of the confidence intervals at 95 % leads us to conclude that the evaluated estimators are equivalent, the ratio estimator being slightly more precise.
Sample size has a strong effect on the precision of the estimators. Thus, in the population where fewer than 30 clusters were studied, the sampling errors of the three estimators analyzed ascend to approximately 30 % in the estimation of timber volume.
All of the relative biases of the ratio estimator were less than 10 %, and so considered practically insignificant.
Literatura Citada
Barabesi, L., and L. Fattorini 1998. The use of replicated plot, line and point sampling for estimating species abundance and ecological diversity. Environ. Ecol. Stat. 5: 353-370. [ Links ]
Barabesi, L., and C. Pisani 2004. Steady-state ranked set sampling for replicated environmental designs. Environmetrics 15: 45-46. [ Links ]
Bechtold, W. A., and C. T. Scott. 2005. The forest inventory and analysis plot design. In: Bechtold W. A., and P. L. Patterson (eds). The enhanced Forest Inventory and Analysis program- national sampling design and estimation procedures. General Technical Report SRS-80. Forest Service. United States Department of Agriculture. Southern Research Station, Asheville, North Carolina. USA. pp: 27-42. [ Links ]
Cochran, W. G. 1993. Técnicas de Muestreo. Compañía Editorial Continental. México. 513 p. [ Links ]
CONAFOR. 2012. Manual y procedimientos para el muestreo de campo: Re-muestreo 2012. Secretaría de Medio Ambiente y Recursos Naturales-Comisión Nacional Forestal (CONAFOR). México. 136 p. [ Links ]
Eaton, J. M., and D. Lawrence. 2009. Loss of carbon sequestration potential after several decades of shifting cultivation in the Southern Yucatán. For. Ecol. Manage. 258: 949-958. [ Links ]
Efron, B. 1979. Bootstrap methods: another look at the Jacknife. Annals Stat. 7: 1-26. [ Links ]
Gregoire, T. G., and H. T. Valentine. 2008. Sampling Strategies for Natural Resources and the Environment. Chapman & Hall/CRC. New York, USA. 474 p. [ Links ]
McRoberts, R. E., W. A. Bechtold, P. L. Patterson, C. T. Scott, and G. A. Reams. 2005. The enhanced Forest Inventory and Analysis program of the USDA Forest Service: Historical perspective and announcement of statistical documentation. J. For. 103: 304-308. [ Links ]
Pérez L., C. 2000. Técnicas de Muestreo Estadístico. Teoría, Práctica y Aplicaciones Informáticas. Alfaomega Grupo Editor. México. 603 p. [ Links ]
Särndal, C. E., B. Swensson, and J. Wretman. 1992. Model Assisted Survey Sampling. Springer-Verlag. New York, USA. 694 p. [ Links ]
SAS Institute Inc. 2009. SAS/STAT 9.2. User’s Guide, Second Edition. SAS Institute Inc. North Carolina. United States America. 78-86 p. [ Links ]
Scheaffer, R. L., W. Mendenhall, y L. Ott. 1987. Elementos de Muestreo. Rendón S., G., y J. R. Gómez A. (trad). Grupo Editorial Iberoamérica. México. 321 p. [ Links ]
SAG. 1976. Inventario forestal del estado de Quintana Roo. Publicación Número 41. Secretaría de Agricultura y Ganadería (SAG). México. 44 p. [ Links ]
SARH. 1985. Inventario forestal del estado de Campeche. Publicación Especial Número 56. Secretaría de Agricultura y Recursos Hidráulicos (SARH). México. 119 p. [ Links ]
Smelko, S., and J. Saborowski. 1999. Evaluation of variable size sampling plots for monitoring of forest condition. J. For. Sci. 45: 341-347. [ Links ]
Smelko, S. , and J. Merganic. 2008. Some methodological aspects of the National Forest Inventory and Monitoring in Slovakia. J. For. Sci. 54: 476-483. [ Links ]
Scott, C.T., W. A. Bechtold, G. A. Reams, W. D. Smith, J. A. Westfall, M. H. Hansen, and G. G. Moisen. 2005. Samplebased estimators used by the Forest Inventory and Analysis national information management system. In: Bechtold, W. A. , and P. L. Patterson (eds). The enhanced Forest Inventory and Analysis program-national sampling design and estimation procedures. General Technical Report SRS80. Forest Service. United States Department of Agriculture. Southern Research Station, Asheville, North Carolina. USA. pp: 43-67. [ Links ]
Schreuder, H.T., T. G. Gregoire, and G. B. Wood. 1993. Sampling Methods for Multiresource Forest Inventory. John Wiley & Sons, Inc. New York, USA. 446 p. [ Links ]
Valentine, H. T., M. J. Ducey, J. H. Gove, A. Lanz, and D. L. R. Affleck. 2006. Corrections for cluster-plot slop. For. Sci. 52: 55-66. [ Links ]
Velasco B., E., H. Ramírez M., F. Moreno S., y A. de la Rosa V. 2005. Estimadores de razón para el inventario nacional forestal de México. Rev. Ciencia For. Méx. 28: 23-43. [ Links ]
Zarnoch, S. J., and W. A. Bechtold. 2000. Estimating mappedplot forest attributes with ratios of means. Can. J. For. Res. 30: 688-697. [ Links ]
Received: March 01, 2015; Accepted: September 01, 2015
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