Assessing reduction of cluster size to estimate wood volume in an Amazonian forest

While the Brazilian National Forest Inventory (NFI) is in progress, there is a growing demand to understand the effect of cluster size on the accuracy and precision of forest-attribute estimation. We aimed to find the minimum cluster size (in area) to estimate merchantable volume (MV) with the same accuracy and precision as the estimates derived from the original cluster of 8,000 m2. We used data from an inventory carried out in a forest unit (Bom Futuro National Forest) in the southwestern Brazilian Amazon, where 22 clusters were distributed as a two-stage sampling design. Three products were evaluated: (i) MV of trees with a diameter at breast height (DBH) ≥ 20 cm (P1); (ii) MV of trees with DBH ≥ 50 cm (P2); and (iii) MV of commercial species with DBH ≥ 50 cm and stem quality ‘level 1’ or ‘level 2’ (P3). We assessed ten scenarios in which the cluster size was reduced from 8,000 m2 to 800 m2. The accuracy of P1, P2 and P3 was highly significantly lower for reductions < 2,400 m2. The precision was more sensitive to variations in cluster size, especially for P2 and P3. Minimum cluster sizes were ≥ 2,400 m2 to estimate P1, ≥ 4,800 m2 to estimate P2, and ≥ 7,200 m2 to estimate P3. We concluded that it is possible to reduce the cluster size without losing the accuracy and precision given by the original NFI cluster. A cluster of 2,400 m2 provides estimates as accurate as the original cluster, regardless of the evaluated product.


INTRODUCTION
Forest inventories are the main tool to quantify and characterize forest resources and their composition, providing useful information to forest management, conservation, and policy . Based on the degree of coverage, the inventories can be classified either as complete, when the entire forest is measured (i.e., a census), or incomplete, when only a forest sample is evaluated (Loetsch et al. 1973). In the latter case, a representative sample is crucial, which depends on distinct factors, such as the dimension of sampling units and the sampling design, which define the precision of estimates.
Increasing forest inventory efficiency is a trade-off between attaining higher accuracy and precision, and seeking practices that allow cost reduction (Westfall et al. 2011;Westfall et al. 2016;Räty et al. 2018). In general, costs are directly related to data detail level and sampling intensity (Druszcz et al. 2012), as well as sampling unit area (Westfall et al. 2016). Plot area rules the precision and accuracy of estimates of forest attributes, while precision is also dependent on available resources, as field assessments generally are a demanding component of the forest inventory (Westfall et al. 2016;Räty et al. 2018;Westfall et al. 2019). Therefore, defining minimal sampling effort that ensures both high accuracy and precision in the estimation of variables of interest will increase inventory efficiency (Gregoire and Valentine, 2007;Westfall et al. 2016).
The challenge of finding the optimal trade-off between precision/accuracy and sampling effort is intensified in the Brazilian National Forest Inventories (NFIs). The NFI is one of the main surveys carried out by the federal government to produce information on Brazilian forest resources (SFB 2020). In the NFI, the ratio between sampled area and population area is notably smaller than in small-scale inventories. NFI SUs usually are field plots structured as clusters of smaller sub-units . These SUs are systematically distributed across the national territory (Westfall et al. 2016). The Brazilian NFI SUs usually are clusters of 4,000 or 8,000 m². The preference for clusters is, among other reasons, the lower variation among clusters, thus requiring the use of fewer SUs (Cochran 1977).
There is a growing demand to understand the relationship of cluster size with accuracy and precision of estimates, i.e., what is the effect of sampling unit size on the precision and accuracy of the estimates of merchantable volume? We aimed to estimate the minimum sampling unit size that provides merchantable volume estimates as accurate and precise as the standard NFI sampling unit of 8,000 m² used in the Amazon.

Study area
The study area covers about 80% (82,918.33 ha) of the Bom Futuro National Forest (9°27'39.0"S, 63°54'26.4"W), a protected area of native forest partially open for sustainable use, located between the municipalities of Porto Velho and Buritis in the state of Rondônia, Brazil (Wikiparques 2019) ( Figure 1). The region has a mean annual precipitation of 2,400 mm and a mean temperature of 25.2 °C . According to the Köppen climate classification, the local climate is Am humid (Alvares et al. 2013). The

Forest inventory and sampling design
Installation of sampling units (SUs) and data collection were performed following the Brazilian NFI guidance, defined by the Brazilian Forestry Service (SFB 2019). The NFI SUs are clusters composed of four crosswise sub-units (as a Maltese cross). Each cluster had 8,000 m 2 , divided into four sub-units of 20 m x 100 m (2,000 m 2 ), which is the standard for the Amazon biome. Further details can be found in the NFI manual (SFB 2014).
The sampling design was of two stages. The first was a systematic distribution of points 2.5 km apart from each other, forming a regular grid where each point represented a cell of 6.25 km² (2.5 x 2.5 km). These 6.25-km² cells correspond to the fourth level of density of the NFI [the first level is 400 km² (20 x 20 km), the second 100 km² (10 x 10 km), and the third 25 km² (5 x 5 km] (Vibrans et al. 2010). The Each standard sub-unit was subdivided into 20 sub-plots of 10 m x 10 m (100 m²) ( Figure 2). To assess how MV estimates change with decreasing SU area, we performed a stepwise reduction in the SU area by eliminating the last two sub-plots of each sub-unit. Because each SU consists of four sub-units, eight sub-plots (or 800 m²) of each SU were removed per reduction. After these successive reductions, the SU areas were: 8,000 m² (original area); 7,200 m² (first reduction); 6,400 m² (second); 5,600 m² (third); and so on, until the ninth reduction, i.e., 800 m², which was the smallest area. Since there were nine area reductions plus the original area, our sampling design comprised ten sampling-area sizes ( Figure 2, Table 1).

Wood products
MV was our variable of interest, defined as the wood volume until either the first stem bifurcation or the beginning of a tortuosity or of the tree crown. Equation 1 was used to obtain the individual MV of the trees. This equation comes from the Schumacher-Hall linear model adjusted to the sustainable forest management plan of the Jamari National Forest, located in northeastern Rondônia state and with a similar forest type to that in the Bom Futuro National Forest.
After estimating individual tree MV, three products (P1, P2 and P3) were assessed. P1 consists of MV of trees with DBH ≥ 20 cm; P2 is MV of trees with DBH ≥ 50 cm; and P3 is MV of commercial species with DBH ≥ 50 cm and stem quality 'level 1' (high stem straightness, cylindrical stem, without apparent defects) or 'level 2' (moderate stem straightness, cylindrical stem, and with acceptable defects) (see Table 2 for product statistics for 8,000-m 2 SUs).

Precision and accuracy
We calculated MV (m³ ha -1 ) of the products per scenario ( Figure 2; Table 1; Table 2). We assumed the 8,000-m 2 SU as the benchmark for the MV estimates. The accuracy of each other scenario (Table 1) was evaluated by comparing their MV means to the means of scenario A in the 22 replicates through Student's t-tests (α = 0.05). The normality of the distribution of MV estimates (α = 0.05) was assessed and confirmed by Kolmogorov-Smirnov tests. The 95% confidence interval (95% CI) of the mean was calculated for all scenarios and products.
The precision of the MV estimates for 800 to 7,200 m 2 relative to 8,000 m 2 was assessed through the coefficient of variation (CV) and the sampling error for each product in the 22 cluster replicates. The sampling error was expressed as the absolute error (E a ) (Equation 2) and relative error (E r ) (Equation 3).
where: s x the standard deviation of the MV; n is the number of SUs; and t is the Student's t-test value.
Scenarios exhibiting a decrease of up to five percentage points in precision in E r relative to 8,000-m 2 clusters were considered to display a precision in the MV estimates equivalent to that of the standard SU.

RESULTS
Mean MV varied from 158.5 to 128.7 m 3 ha -1 for P1, 69.6 to 45.6 m 3 ha -1 for P2, and 12.3 to 4.40 m 3 ha -1 for P3 ( Figure 3). SUs of 4,000 m² produced the highest means in all products, while SUs of 800 m² produced the lowest. The means for all products were very similar for SUs with areas ≥ 2,400 m².
The 95% CI remained relatively constant with increasing SU area, except for 800-m² SUs, in which the CI was wider, thus the uncertainty greater than for the other scenarios ( Figure 3). P3 in 800-m² SUs had a negative lower limit of the CI, which was owed to the low density of trees of commercial value with DBH ≥ 50 cm in forest stands, which made them vary scarce in the sampled areas of 800-m 2 SUs ( Figure 2; Table 1).
MV for P1 and P2 did not differ significantly from scenario A (original SU area of 8,000 for m 2 ) for areas ≥ 2,400 m 2 (Table 3). Accordingly, the 95% CI tended to be more stable around the mean from 2,400 m 2 onwards (Figure 3). For P3, the MV of SU areas of 1,600 m 2 or larger did not difer significantly from the original SU area (Table 3).
For P1 and P2, we observed a strong trend towards the stabilization of the Er of the MV in SUs ≥ 1,600 m² ( Figure  4b). The Er for P3 decreased constantly with the increase of SU area, while Ea did not vary noticeably, unlike it did for P1 and P2 between 800 and 1,600 m² ( Figure 4a). The product with the highest Ea presented the lowest Er and vice-versa (Figure 4a).
Adopting a SU of 2,400 m 2 to estimate P1 resulted in a precision loss of 4.4% relative to the 8,000-m 2 SU (Er increased from 13.9% to 18.3%, Figure 4b). For P2, using a SU of 4,800 m 2 resulted in an error increase from 19.9% to 24.5% (4.6% increase, Figure 4b). The precision was reduced in 3.4% when using a SU of 7,200 m 2 for P3 (E r increased from 50.6% to 54%, Figure 4b).
The relationship between CV and SU area followed a negative exponential trend for the three products ( Figure 5). The CV is linked to the abundance of each product in the sampled area, thus the largest variation range was observed for the least abundant product (P3) (120 to 260%), while P2 had the intermediate range (50 to 130%), and P1, being the most abundant product, exhibited the smallest range (34 to 64%). As P2 is less abundant than P1, its CV was 1.5 to 2.5  times higher than that of P1. Likewise, P3 exhibited a CV two to three times higher than that of P2.

Minimum sampling-unit size
Our objective was to determine the minimum SU size that estimates the MV as accurately and precisely as the 8,000-m² standard SU in an area of Amazon rainforest, which resulted to be 2,400 m 2 for P1, 4,800 m 2 for P2, and 7,200 m 2 for P3.
Considering accuracy as the only criterion to choose the minimum SU size, an area of 2,400 m² would be suitable for all products. However, given the observed variation in precision, additional caution is needed in the assessment of minimum SU size. Sampling errors directly affect the confidence interval of an estimate. The larger the error in an estimate, the wider the confidence interval, and viceversa (Loetsch et al. 1973;Péllico Netto and Brena 1997). Although E r and E a are complementary to each other, E r has greater applicability in practical terms (Cavalcanti et al. 2009;  Cavalcanti et al. 2011;Oliveira et al. 2014). Specifically, E a lacks information about the magnitude of its value relative to the mean. For example, an absolute error of 20 m³ ha -1 for a mean of 40 m³ ha -1 corresponds to a relative error of 50%, which is considered high in forest inventories. Conversely, for this same absolute error, but with a mean of 200 m³ ha -1 , the relative error would be only 10%, which is acceptable for most inventories.
In a similar study in an area of Amazon forest in Pará state, Flores et al. (2012) evaluated the effect of distance between clusters and among sub-plots within clusters on three products based on total wood volume of trees with DBH ≥ 30 cm (77 forestry species, ten most abundant species, and ten most commercialized species), and concluded that the 8,000-m 2 cluster is efficient in estimating the analyzed products. Likewise, Queiroz et al. (2011) also concluded that a 8,000-m 2 Maltese-cross cluster was efficient in estimating three wood-volume products (DBH ≥ 25 cm) in the Tapajós Natinal Forest, also in Pará state (all species, species with an international market, and only Manilkara huberi (Ducke) Standl). Our study showed that it is possible to reduce the SU area while attaining similar accuracy and precision as with an 8,000-m 2 cluster, allowing to reduce inventory costs while obtaining reliable MV estimates.
The recommendations of a larger SU by Queiroz et al. (2011) andFlores et al. (2012) is due to their use of the maximum curvature method to obtain the minimum SU size, which is a method based on the stabilization of the coefficient of variation. Our results also indicate that the coefficients of variation tend to stabilize with SUs ~8,000 m², however, we did not aim to determine the SU size that stabilizes the MV estimation by using the maximum curvature method, not to determine where the coefficient of variation stabilizes, since 8,000 m² was the largest SU area evaluated. Cavalcanti et al. (2009) assessed the effect of SU area on the precision and accuracy of inventories in the Amazon forest through the MV of trees with DBH ≥ 40 cm, and noted an exponential decrease in MV variation as SU area increased, which was also observed by Flores et al. (2012) in our study. Cavalcanti et al. (2009) tested SU areas ranging from 2,500 to 20,000 m² and found that the coefficient of variation tended to stabilize with SUs ≥ 7,500 m², which was also observed in our study, with a stabilization in the coefficient of variation of P1 at SUs ≥ 7,200 m². Flores et al. (2012) found a range of coefficients of variation comparable to our results, with ~100-150% for the most abundant product, ~130-200% for intermediate abundance, and ~140-240% for the less abundant product. Likewise, Queiroz et al. (2011) also observed coefficients of variation ranges of ~100%, ~100-300%, and ~150-450%, respectively for the most, intermediate, and less abundant products.

Effect of sampling-unit area on merchantable volume
Similar to Flores et al. (2012) and Queiroz et al. (2011), we also observed that scarcer products and smaller SUs resulted in larger coefficients of variation. The high coefficients of variation were caused by the high errors observed in P3 and, less intensely, by the errors in P2 and P1. As the abundance of each product varies with the successional stage and disturbance state of the forest, our recommendation applies to forests with products of similar abundance to those evaluated in here.

Sampling intensity and merchantable volume
The sampling intensity of 0.0212% adopted in our study for the 8,000-m 2 cluster sufficed to meet the minimum precision of 20% for P1 and P2 (see Figure 4b). If more precise estimates are required, a higher sampling intensity would likely be necessary. For example, in the study conducted by Cavalcanti et al. (2011) in an Amazon forest, a sampling intensity of 14% (~660x greater than 0.0212%) was needed to meet a precision of ~8% of the MV of commercial species of trees with DBH ≥ 40 cm. Thus, the sampling intensity adopted in our study may be too low to achieve an error smaller than 10%, especially for scarcer products such as P2 and P3. In a study on the effect of SU size on tree wood and carbon-stock estimates in an Amazonian forest, Oliveira et al. (2014) found best results with SUs ≥ 1,200 m 2 for DBH ≥ 20 cm, with mean uncertainty of 9.07%, and with SUs between 2,000 m 2 and 3,000 m 2 for DBH ≥ 25 cm, with mean uncertainty of 6.37%. These results also corroborate our findings that the increase of SU area increases precisions. Andrade et al. (2015) also evaluated tree-volume estimates with varying SU area [1,500 m² (30 m x 50 m) for DBH 10-25 cm, 3,000 m² (30 m x 100 m) for DBH 25-50 cm, and 7,500 m² (30 m x 250 m) for DBH ≥ 50 cm] and obtained an error smaller than 10% for the volume of trees with DBH ≥ 10 cm and trees with DBH ≥ 50 cm. This accuracy was considerably higher than that observed in our study, which may be due to the higher sampling intensity (~10 times higher) in the study by Andrade et al. (2015) and shows that sampling intensity is directly related to inventory accuracy. Since the intensity depends on the area and number of SUs, it is expected that more SUs will contribute to improve the accuracy.
We evaluated the effect of SU area while keeping SU number fixed. An alternative approach would be to assess SU area reduction and to vary the number of SUs (e.g., Kauai et al. 2019). Varying SU area usually has a different effect than varying number of SUs (Oliveira et al. 2014). For example, in forest remnants of an ombrophilous forest in southern Brazil, gains in precision on the basal area estimation were less evident when varying SU area than when varying number of Sus (Augustynczik et al. 2013). Future studies should address both number of SUs and SU area, as well as other variables of interest, to improve our ability to increase the efficiency of forest inventories.

CONCLUSIONS
Our results indicate that smaller clusters could replace the original standard 8,000-m² cluster for forest inventory estimates in Amazon rainforest without losing precision and accuracy of merchantable volume estimation. We found that the minimum sampling unit size to estimate merchantable volume is 2,400 m² for trees with DBH ≥ 20 cm, 4,800 m² for trees with DBH ≥ 50 cm, and 7,200 m² for commercial species with DBH ≥ 50 cm and stem quality of level 1 or 2. The precision of the estimates was affected more strongly by reduction of cluster size than accuracy. Regardless of the wood product abundance in the forest, a cluster size of 2,400 m² yielded estimates as accurate as those of the original standard cluster of 8,000 m². Precision, however, was dependent on cluster size and product abundance. We encourage other researchers to further assess the relationship between cluster size and accuracy and precision of other variables of interest of the Brazilian NFI, focusing on determining smaller yet reliable cluster sizes for wood product estimates.