1. INTRODUCTION
Forest fuel material consists in every living or dead biomass susceptible to ignition and burn in forest fires (^{Fuller, 1991}; ^{Agee & Skinner, 2005}) and is the main driver of nutrient cycling in forest formations. However, despite the ecological importance of biomass deposited on the soil, sampling of fuel material has rarely been addressed in forest inventories (^{Ribeiro et al., 2012}). Yet, surveying the amount of fuels in forest stands is important for adoption of preventive silvicultural practices (^{Soares, 2002}).
Normally, thinner materials with diameters up to 2.5 cm are the first to burn in a forest fire, and for that reason, they are considered dangerous or semidangerous (^{Beutling, 2009}). Thus, according to ^{Bartlett et al. (2001)}, the main objective of studies employing statistical inference is to collect data that may be representative of the whole population, considering a maximum acceptable error. However, the optimization problem in forest inventories is important to choose an optimal sample structure that maximizes the value of the necessary data, taking into account a limited available budget (^{Brassel & Lischke, 2001}).
Appropriate methodologies applied in forest fuel material inventories should aim to optimize the sampling procedure, to provide accurate assessments (^{Péllico & Brena, 1997}). Overestimation can result in economic adversities for forestry companies due to the high costs of inventories (^{Fink, 2003}). On the other hand, underestimation can negatively affect the planning of silvicultural activities and the allocation of resources for preventing and fighting fires. Other factors are related to greenhouse gas release from biomass combusted in forest fires (^{Smith et al., 2004}) and the production of clean energy (^{Nogueira et al., 2000}; ^{Souza, 2010}).
Thus, considering the hypothesis that production of forest residues has spatial variability and that systematization of sampling units allows obtaining statistically appropriate estimates in forest inventories, the objective of this study was to simulate and evaluate sampling procedures for estimation of forest fuel material with different sample sizes, and to model the spatially dependent variation of this material on the ground of a pine stand.
2. MATERIAL AND METHODS
2.1. Study area
The study was developed in the experimental station of the Federal University of Paraná located in the municipality of Rio Negro, Paraná State, Brazil (26° 06’ S and 49° 47’ W), with average altitude of 780 m. According to ^{Batista (1995)} and ^{Maack (2002)}, the climate of the area is temperate humid (Cfb), based on Koppen classification, with average temperatures of 22 °C in the hottest month and more than five frosts per year.
The experimental station covers an area of approximately 120 ha, with a significant part consisting of forest plantations. A Pinus elliottii Engelm stand established on flat terrain in 1984, with initial spacing of 2 × 2.5 m and without silvicultural treatment since its deployment, was selected for the present study. The last inventory in the area was conducted in 2005 and estimated a volume of 560 m^{3} ha^{–1} and basal area of 47.16 m^{2} ha^{–1} (^{Pereira, 2009}).
2.2. Fuel material collection
The experiment was carried out in the total area of four tracks positioned between planting rows on the NorthSouth direction. Each track was segmented in 15 rectangular sampling units of 1.2 m × 0.6 m (Figure 1), in which fuel material was collected and separated according to diametric classes following the methodology proposed by ^{Brown et al. (1982)}. For this, two classes of material were adopted: A) branches with diameters of up to 0.7 cm; and B) branches with diameters between 0.71 cm and 2.5 cm. Material with diameter greater than 2.5 cm was not sampled, since they represent a small fraction of fuel material, high moisture content variation, as well as slow deterioration speed and, thus, do not present high risk for fire occurrence (^{Beutling, 2009}).
2.3. Geostatistical modeling
Average production of fuel material and the coefficient of variation (cv) were calculated for classes A and B. Posteriorly, geostatistical analysis was employed to model spatial patterns of fuel material in the pine stand for the following sampling intensities: 100% (Figure 1), 50% (Figure 2a), 33% (Figure 2b) and 23% (Figure 2c) of the potential sample size, through calculation of experimental semivariance (1), taking into account the central geographical positioning of sampling units in the field, the subsequent calculation of distances (h) and numerical differences of the variable in the grid.
In order to estimate semivariances, Spherical (2), Exponential (3) and Gaussian (4) models were adjusted. The structure of the semivariogram included the nugget effect (C_{0}), corresponding to the semivariance for a distance value equal to zero; the sill (C_{0} + C), representing the stabilization of semivariogram values near to the variance of the data; the variance a priori (C), given by the difference between the sill (C_{0} + C) and the nugget effect (C_{0}); and the range (a), defined as the distance at which the semivariogram reaches a sill, indicating the boundary where sampling units correlate among themselves (^{Vieira 2000}).
Where:
The GS+ version demonstration program 5.1 (^{Robertson, 2008}) was used to adjust the semivariogram. Evaluation and selection of the best adjustments were based on the least sum of squared deviations (SQD) and the largest determination coefficient (R^{2}). Finally, with the parameters of adjusted semivariogram and with the observed values of neighboring sampling units, production of fuel material was spatialized along the forested area by punctual ordinary kriging (^{Andriotti, 2003}; ^{Yamamoto & Landim, 2013}).
2.4. Sampling procedures
Deposition of fuel material, expressed in megagrams per hectare (Mg ha^{–1}), was quantified in all sampling units, representing the maximum value of deposition of the two classes of sampled materials. Subsequently, estimates were calculated for 50% (Figure 2a), 33% (Figure 2b) and 23% (Figure 2c) of the potential sample size, in respectively 30, 20 and 14 sampling units, whereas tracks were the primary units (k_{1}) and plots within these tracks were the secondary units (k_{2}) systematically sampled.
Additionally, the statistical formulation presented by ^{Péllico & Brena (1997)} for systematic sampling was used to estimate the mean (5), variance of the mean (6), standard error (7), absolute sampling error (8) and sampling error in percentage (9), and confidence interval for the mean (10), which were calculated for the different simulated sample sizes as follows:
Sample mean
Variance of the mean
Standard Error
Absolute sampling error
Sampling error in percentage
Confidence interval for the mean
Where: m = number of tracks sampled; n_{j} = number of units sampled within tracks;
Moreover, each track was considered as a linear cluster (Figure 3a) and, for the application of this sampling procedure, the following estimators were calculated for the simulated sample intensities (^{Péllico & Brena, 1997}): sample mean (11), intracluster correlation coefficient (12), variance of the mean (13), standard error (14), absolute sampling error (15) and sampling error in percentage (16) and confidence interval for the mean (17).
Sample mean
Intracluster correlation coefficient
Variance of the mean
Standard Error
Absolute sampling error
Sampling error in percentage
Confidence interval for the mean
Where: M = number of subunits of the cluster; n = number of cluster sampled;
4. RESULTS AND DISCUSSION
The average production of fuel material was equal to 5.66 Mg ha^{–1} (cv = 16%) and 0.61 Mg ha^{–1} (cv = 28%) for classes A and B, respectively, where the spatial dependence of fuel material deposited in pine stand was confirmed by the possibility of adjustment of semivariogram models (Table 1). Lower values were obtained to a unit of nugget effect (C_{0}), varying from 0.060 to 0.143 for class A and from 0.010 to 0.013 for class B of fuel material. This showed that the variance caused by nonsampling errors or unidentified variations was low (^{Vieira, 2000}).
Fuel material  Geostatistical model  C_{0}  C  a (m)  R^{2}  SQD 

Class A  Spherical  0.060  0.593  3.542  0.991  2.2 × 10^{3} 
Exponential  0.086  1.083  4.575  0.968  7.5 × 10^{3}  
Gaussian  0.143  0.608  1.797  0.993  1.8 × 10^{3}  
Class B  Spherical  0.010  0.025  1.120  0.622  2.6 × 10^{5} 
Exponential  0.010  0.025  1.500  0.614  2.6 × 10^{5}  
Gaussian  0.013  0.025  1.100  0.660  2.3 × 10^{5} 
Where: C_{0} = nugget effect; C = variance a priori; a = range; R^{2} = coefficient of determination; and SQD = sum of squared deviations.
Range (a) expresses the maximum distance at which two sampling points were spatially correlated, corresponding to the radius of the areas within which spatial variability of neighboring samples was more similar. Thus, the ranges between 1.797 m and 4.575 m for class A, and 1.100 m and 1.500 m for class B (Table 1) represented the distances within which the analyses led to the estimates with greater precision (^{Vieira, 2000}; ^{Chig et al., 2008}), while evaluations in intervals beyond the range were independent from each other.
In addition, coefficients of determination (R^{2}) were above 0.6 for class A (Table 1), what shows that the sampling grid was efficient to detect spatial characteristics of fuel material deposition, particularly with the Gaussian model for classes A and B, with which the lowest values of the sum of squared deviations (SQD) were obtained.
Thus, the semivariograms selected for estimating the spatial distribution of fuel material deposition in the pine stand (Figures 3a, e) demonstrated the effect of increasing semivariances related to increasing distances until reaching a regular value (^{Pereira et al., 2011}). However, this was not identified for 50%, 33% and 23% sampling intensities, where semivariances showed irregular distribution in scaled semivariograms. This indicates that the spatial component for smaller distances was not detected.
Table 2 shows the parametric mean (
Fuel material 

n 







Mg ha^{–1}  Mg ha^{–1}  Mg ha^{–1}  
Systematic sampling procedure  
Class A  5.66  50%  5.69    0.08  0.16  2.7%  5.53 Mg ha^{–1} ≤

33%  5.64    0.11  0.23  4.1%  5.41 Mg ha^{–1} ≤


23%  5.68    0.18  0.39  6.9%  5.28 Mg ha^{–1} ≤


Class B  0.61  50%  0.62    0.02  0.05  7.7%  0.57 Mg ha^{–1} ≤

33%  0.62    0.04  0.08  12.8%  0.54 Mg ha^{–1} ≤


23%  0.63    0.04  0.10  15.2%  0.53 Mg ha^{–1} ≤


Sampling procedure in linear clusters  
Class A  5.66  50%  5.69  0.65  0.28  0.90  15.9%  4.79 Mg ha^{–1} ≤

33%  5.64  0.48  0.31  1.00  17.6%  4.64 Mg ha^{–1} ≤


23%  5.68  0.60  0.39  1.24  21.8%  4.44 Mg ha^{–1} ≤


Class B  0.61  50%  0.62  0.58  0.08  0.17  27.7%  0.45 Mg ha^{–1} ≤

33%  0.62  0.43  0.07  0.23  37.2%  0.39 Mg ha^{–1} ≤


23%  0.63  0.57  0.08  0.25  40.2%  0.37 Mg ha^{–1} ≤

Where:
Considering the selected semivariograms, thematic maps of fuel material deposition were built through punctual ordinary kriging (Figure 4). The apparent homogeneity of fuel material on the ground was contrasted with the real spatial heterogeneity (Figure 4a, b) in the pine stand detected through geostatistical evaluation of the fuel material deposition. This heterogeneity is mainly a result of the variability of growth and size of individuals measured in adjacent areas where the fuel material was collected. Therefore, sampling procedures that capture such spatial variability is essential for quantitative accurate statistical estimates of fuel material deposition.
Estimated fuel material deposition means
The intracluster correlation coefficient (r) showed greater heterogeneity of fuel material deposition between clusters than between sampling units within linear clusters. This indicates that this sampling procedure is not suitable for this type of evaluation, as it is recommended that, for the effective application of this sampling process, r values should not exceed the interval between 0 and 0.4 for intracluster correlation (^{Péllico & Brena, 1997}). Thus, r values between 0.43 and 0.65 observed in classes A and B of fuel material are related to greater spatial variability of deposition in the forest stand, indicating that the variance between clusters is superior to the variance within clusters and corroborating with the spatial variability observed in the geostatistical modeling (Figure 4).
Furthermore, the absolute sampling error
The confidence interval for the mean
Authors such as ^{Brown (1974)} and ^{Ribeiro et al. (2012)} have considered that sampling errors of up to 20% are acceptable for estimating forest fuel material, because of the high variability of dimensions of the objects measured. On the other hand, when there is an interest in the potential use of this product, errors of up to 10% are most appropriate. Consequently, the values found in the present study are within acceptable limits for the sample sizes of 23% for the class A and 50% for the class B when evaluating fuel material using the systematic sampling process.
5. CONCLUSION
Fuel material deposition of branches with diameters up to 0.7 cm, and branches with diameters from 0.71 cm to 2.5 cm is spatially dependent and presents a pattern that can be assessed through geostatistical analysis. Variability comes from the growth and size of individuals measured in adjacent areas where the fuel material was collected. Spatial behavior of fuel material confirms the need of using appropriate sampling methods to accurately detect this variable in forest inventories.
Systematic sampling is appropriate and recommended for estimating fuel material deposition in pine stands when representative spatial coverage of the population and accurate statistical estimators are employed. On the other hand, the sampling structure in linear clusters was considered inadequate for estimating this variable in the study area due to the heterogeneity among the clusters.
The reduction in the number of sampled units affect the estimators of fuel material deposition in the pine stand, but the respective sampling errors do not exceed the maximum limit of 10% for sample sizes of 23% in class A and of 50% in class B using the systematic sampling process, consequently resulting in consistent estimates.