Spline de Bézier e modelos segmentados e não segmentados para descrever o perfil do fuste e estimar sortimento de madeira

Abstract

This research aimed to develop a heuristic using the Bézier Spline Function and compare its performance in relation to the taper functions to describe the stem profile of Pinus taeda L. and estimate the wood assortments. For this purpose, a censo was carried out in the stand with na área of 1.21 hectare, in which there were 433 trees aged 17 years old. Sample data from 52 trees were used in proportional to the diameter distribution, collected in the municipality of Pinhão, in the state of Paraná, Brazil, from which 70% were randomly selected to adjust the taper functions and the Bézier Splines functions and 30% were used for validation. The taper functions 5th degree, Max and Burkhart and Bi and the Bézier Splines were adjusted. To adjust the taper models, the nonlinear regression technique (nls) was used, available in the R software, and to adjust the Bézier Splines, a search heuristic for the minimization of the sum of squares of residuals was developed and implemented in the R software, both for the estimation of diameters along the stem, diameters with stem stratified into three portions (basal, median and apical) and partial volume estimates for the same stratifications. The models were evaluated by statistical criteria: Pearson's correlation coefficient (𝑟𝑦𝑦^), square root mean error in percentage (RMSE(%)) and graphical residual analysis. The average profile Spline by diameter class showed better results compared to the taper functions and the average profile Spline, which use only DBH and h to estimate diameters along the stem, diameters in stratified stem and stratified volumes. The individual Spline profile, measuring: DBH, 15%h and h, for the adjustment data in the diameter estimates (base portion), the RMSE(%) was reduced by 31.37%, equivalent to 1.45 points percentage and for the validation data, the RMSE(%) was reduced by 26.23% equivalent to 1.39 percentage points, compared to the average of the three taper functions. The residual graphs of the Individual Spline for adjustment and validation showed great superiority, consequently improving the diameters between 10% and 35% of the total height. For the adjustment data in the volume estimates (basal portion), Individual Spline reduced the RMSE(%) by 58.36% equivalent to 4.58 percentage points and for the validation data, it reduced the RMSE(%) by 50.17% equivalent to 3.86 percentage points. The residuals graphs of volumes of this Individual Spline for the adjustment and validation also showed great superiority, with residuals very close to zero. Individual spline profile also improved the median portion mainly the diameters between 35% and 50% of the total height. Several scenarios were simulated with the inclusion of diameters along the stem above 15% of the total height, obtaining the lowest RMSE(%) for each portion, for 3 points (DBH, 25%h and h), 4 points (DBH, 25%h, 50%h and h) and 5 points (DBH, 25%h, 50%h, 75%h and h). The assortment processed by harvester tends to underestimate the volumes in all assortment classes (Lamination, Sawmill 1, Sawmill 2, Pulp and Energy) compared to the volume cubed by Smalian in the same processed lengths, with a difference in total volume of 9.95 m³ in the analyzed sample (52 trees). On the other hand, the volumetric loss per hectare was 68.49 m³ (loss of 18.69%) and in financial terms a loss of R$ 9,853.40 (loss of 19.24%). The downgrade coefficients were calculated for each assortment class in relation to the harvester the taper functions and the Bézier Splines functions and also in relation to the cubed by Smalian and the respective functions. Bézier Splines are an excelente option to describe the stem profile, estimate the assortments and are applicable to standing trees, using the heuristics proposed in this research.