Theoretical evaluation of Huber and Smalian methods applied to tree stem classical geometries

Abstract

SUMMARYHuber, Smalian and Newton methods, to estimate tree stem and log volume by sections, were theoretically evaluated applying them to the geometries of paraboloid, cone and neiloid. The study follows approximation procedures from calculus for volume estimation of solids of revolution as function of the number of segments and error analysis methods from forest measurement research. The errors in using Huber and Smalian methods have been determined. Additionally, it was algebraically proved that the Huber's error is exactly one half of the Smalian's error and opposite in sign. The results predict that, for any tree stem modeled classically, Huber's and Smalian's average absolute percent errors should be: less than 13.5 % and 27 %, respectively, for five or more segments; less than 7 % and 14 %, for 10 or more segments; less than 3.6 % and 7.2 %, for 20 or more segments; less than 2.5 % and 5 %, for 30 or more segments. This work provides a quantization on the classical theory of tree stem and log volume estimation. It could help to unify that theory and make it a more compact reference for forest measurement teaching and research.Key words: sectional methods, tree stem volume. RESUMENLos métodos de Huber, Smalian y Newton para estimar el volumen de troncos y trozas de árbol por secciones, fueron evaluados teóricamente, aplicándolos a las geometrías de paraboloide, cono, y neiloide. El estudio sigue procedimientos de aproximación del cálculo para estimación de volumen de sólidos de revolución, como función del número de segmentos, y métodos de análisis de error de investigación en mediciones forestales. Se han determinado los errores en el uso de los métodos de Huber y Smalian. Adicionalmente, se probó algebraicamente que el error de Huber es exactamente la mitad del error de Smalian y de signo opuesto. Los resultados predicen que, para cualquier tronco modelado clásicamente, los errores porcentuales absolutos promedio de Huber y de Smalian deberían ser: menores que 13,5 % y 27 %, respectivamente, para cinco o más segmentos; menores que 7 % y 14 % para 10 o más segmentos; menores que 3,6 % y 7,2 %, para 20 o más segmentos; menores que 2,5 % y 5 %, para 30 o más segmentos. Este trabajo proporciona una cuantificación sobre la teoría clásica de estimación de volumen de troncos y trozas de árbol. Podría ser de utilidad para unificar dicha teoría y hacerla una referencia más compacta para enseñanza e investigación sobre mediciones forestales.Palabras clave: métodos seccionales, volumen de tronco de árbol.

Comparison of Different Methods of Measuring Tree Volumes in the Municipality of Anapu, Pará, Brazil

“…Therefore, the practice of cubic scaling is more reliable because it accounts for the real geometric shape of the stem with greater precision. This feature is very important for the adjustment of volumetric equations, especially those that allow planning of future management within a sustainable forest management unit (DE LEÓN;VALENCIA, 2013).…”

“…To ensure greater volumetric accuracy, the distance between the sections at the time of cubic scaling should be the smallest possible (DE LEÓN;URANGA-VALENCIA, 2013). Thus, the calculation of VR, which uses measurements of sections of the stem that are 2 meters apart, is more accurate than the geometrical method, which uses variable sections depending on the size of the stems to be bucked, with an average distance between sections that is greater than 2 meters (BUZA et al, 2006).…”

“…The profile of the stem of the species of hardwood trees in the Amazon displays an exponentially decreasing diameter as the length of the log sections increases, up to the largest section, when the diameter may or may not increase (DE LEÓN;URANGA-VALENCIA, 2013). Thus, the greater the length of sections of the trunk during the measuring of the trees, the greater the likelihood that the volume will be overestimated (RIBEIRO et al, 2014).…”

Error predictions for average end diameter sectional formula in tree stem volume measurement

“…Standard forest measurement literature agrees that the geometry of most tree stem segments lies around a cone and between paraboloid and neiloid (Cruz de León and Uranga-Valencia 2013). The main objective in this work is to calculate the errors for the AED method applied to tree stems modeled through the former three geometries as function of the number of segments.…”

“…The four remaining methods are possibly the most conventional worldwide. Errors in the use of Smalian and Huber methods as function of the number of segments have been calculated by Cruz de León and Uranga-Valencia (2013) and for the frustum of cone method by Cruz de León et al 1 The present work is aimed at determining errors in the remaining method out of the four revised methods.…”