Determining the parameters of the parabolic approximation of the sawlog based on Nilson’s sawlog volume formula

Although parabolic approximations of sawlogs have become widespread, in current specialized literature, there are no parameter values of such approximations or dependencies for their calculation that could be used to analyze the influence of log dimensions on the volume yield of edged boards. The developed technique for determining the parameters of a truncated paraboloid, which approximates a sawlog, is based on Nilson’s formula for estimating its volume. Based on this, the equation of the generatrix of the truncated paraboloid as a function of the top diameter and the length of the sawlog (considering the species) and the formula for calculating its taper (in particular, total and main) are derived. The error of approximating sawlogs with a paraboloid was determined by comparing the taper estimate with literature data on the taper of Scots pine (Pinus sylvestris L.), Norway spruce (Picea abies (L.) H. Karst.), and common oak (Quercus robur L.) sawlogs in Ukraine, Sweden, and Finland. This error ranges from -36% to -9%. Using the method of parametric identification of taper distributions based on literature data on their statistical characteristics (unbounded metalog distribution with four parameters and normal distribution were used), the probability that a sawlog has a taper smaller than its estimate was calculated. These probabilities range from 20% to 36%. The third quartiles of the sawlogs’ taper distributions are, respectively, between 200% and 150% of the taper estimate. Therefore, the obtained formulas for determining the parameters of the approximating paraboloid led to a rather conservative estimate of the taper (however, acceptable for use in log sawing planning), which confirms their suitability for analyzing the influence of log dimensions on the yield of edged boards.