Semi-empirical estimation of log taper using stem profile equations

In January 2019 the forest industry in Ukraine adopted European standards for measuring and grading of round wood based on mid-point diameters, which caused major discrepancies from traditionally used estimates of timber volume using top diameters. To compare methods of merchantable wood volume estimation, we investigated the stem form inside bark for two dominant tree species in Ukraine, i.e. Scots pine (Pinus sylvestris L.) and common oak (Quercus robur L.). We used tree stem measurements to fit stem profile equations, whereas simulation was applied to derive log taper. We found that Newnham's (1992) variable-exponent taper equation performed well for predicting stem taper for both tree species. Then, we simulated the structure of harvested wood, so that it replicated annual distribution of logs by their length and diameters. As a result, the average log taper was estimated at 0.836 ÷ 0.855 cm·m^–1 and 1.180 ÷ 0.121 cm·m^–1 for pine and oak, respectively. The study also indicated that log taper varied along stems. The higher rates of diameter decrease were found for butt logs, for which the taper was 2.5–3.5 times higher than its average for the whole stem. The results of our study ensure the stacked round wood volume conversion between estimates obtained using top and mid-point diameters.

COMPATIBILITY BETWEEN THE SPURR FUNCTION VOLUME AND THE KOZAK’S TAPER FUNCTION AND QUINTIC POLYNOMIAL VOLUMES FOR BLACK WATTLE TREES

When modeling the taper and volume, it is desired that the volume estimates obtained by using these two methods are compatible, where the total stem volume estimates shall not differ when using a total volume equation and the volume calculated by integrating the taper equation. There are several of such systems proposed in the literature, in which modifications in the volume and taper models were made to obtain compatible systems. This paper introduces an idea to obtain compatibility in a simpler way, without the need to modify the volume and taper models. Thus, the overall objective of this study was to develop and present a procedure to obtain compatibility between the Spurr function volume and the Kozak’s taper function and quintic polynomial volumes for Acacia mearnsii De Wild trees and compare the results to the traditional method of the same system of equations. The procedures proposed were applied on data on the Acacia mearnsii De Wild (black wattle) species in the towns of Cristal, Piratini, and Encruzilhada in the south of the state of Rio Grande do Sul, Brazil. The data set included 343 trees ranging from 5 to 10.75 years of age. The quality of the fitting for the volume and taper equations fitted using procedures 1 and 2 is similar, and both are compatible. The system of equations presented in procedure 2 is simpler to be applied when compared to procedure 1.

Development of a Taper Equation for Teak (Tectona grandis L.f.) Growing in Western Thailand

Teak is an important and valuable tropical hardwood species. In this study, we developed and evaluated suitable taper equations for teak growing in Western Thailand using a formulation of Goodwin cubic polynomial model combined with a bark thickness model. The best taper model calibration was selected based on goodness-of-fit and leave-one-out cross validation statistical testing. In total, 12 different model calibrations were tested, with Thong Pha Phum (TPP) 2 being the most suitable for teak in Western Thailand. The mean prediction error of three validation statistics: (prediction of diameter under bark given height; prediction of height given diameter under bark; and prediction of under bark volume given log length) were within 10% and the overall validation index was 5.454, which was the lowest when compared to other calibrations. A comparison of TPP 2 with a teak taper equation developed for Northern Thailand, using a graphical analysis of the stem shape and bark thickness, indicated that the teak trees growing in the two regions have similar stem shapes, but the trees in Western Thailand tend to have a thicker bark. These results will also help in further work as they indicate that bark thickness equations are particularly important.

Variable-Exponent Taper Equation Based on Multilevel Nonlinear Mixed Effect for Chinese Fir in China

A variable-exponent taper equation was developed for Chinese fir (Cunninghamia lanceolate (Lamb.) Hook.) trees grown in southern China. Thirty taper equations from different groups of models (single, segmented, or variable-exponent taper equation) were compared to find the excellent basic model with S-plus software. The lowest Akaike information criteria (AIC), Bayesian information criteria (BIC), and -2loglikelihood (-2LL) was chosen to determine the best combination of random parameters. Single taper models were found having the lowest precision, and the variable-exponent taper equations had higher precision than the segmented taper equations. Four variable-exponent taper models that developed by Zeng and Liao, Bi, Kozak, Sharma, and Zhang respectively, were selected as basic model and had no difference in fit statistics between them. Compared with the model without seldom parameter, the nonlinear mixed-effects (NLME) model improves the fitting performance. The plot-level NLME model was found not to remove the residual autocorrelation. The tree-level and two-level NLME model had better simulation accuracy than the plot-level NLME model, and there were no significant differences between the tree-level and two-level NLME model. Variable-exponent taper model developed by Kozak showed the best performance while considering two-level or tree-level NLME model, and produced better predictions for medium stems compared to lower and upper stems.

A Taper Equation for Loblolly Pine Using Penalized Spline Regression

Stem profile needs to be modeled with an accurate taper equation to produce reliable tree volume assessments. We propose a semiparametric method where few a priori functional form assumptions or parametric specification are required. We compared the diameter and volume predictions of a penalized spline regression (P-spline), P-spline extended with an additive dbh-class variable, and six alternative parametric taper equations including single, segmented, and variable-exponent equation forms. We used taper data from 147 loblolly pine (Pinus taeda L.) trees to fit the models and make comparisons. Here we show that the extended P-spline outperforms the parametric taper equations when used to predict outside bark diameter in the lower portion of the stem, up to 40% of the tree height where the more valuable wood products (62% of the total outside bark volume) are located. For volume, both P-spline models perform equal or better than the best parametric model, with taper calibration, which could result in possible savings on inventory costs by not requiring an additional measurement. Our findings suggest that assuming a priori fixed form in taper models imposes restrictions that fail to explain the tree form adequately compared with the proposed P-spline.

What factors should be accounted for when developing a generalized taper function for black wattle trees?

Taper functions have been widely used for various purposes. Several functions were developed and successfully applied; however, most of these functions fail to account for the influence of stand-level and individual-tree effects of variation on the stem profile. Hence, we aimed in this study to assess how these factors influence the stem profile of black wattle (Acacia mearnsii De Wild.) trees in southern Brazil. There is a notable necessity for developing a domestic market for black wattle solid wood. The database was composed of 218 black wattle trees at age 10 years distributed across the state of Rio Grande do Sul, Brazil. A dimensionally compatible taper equation combined with the mixed-effect modeling approach was used. Additionally, auxiliary variables were included to build a generalized taper function that explains stem form variations. In general, all variables showed a significant influence on the stem profile, except the crown ratio. The inclusion of relative spacing and tree hierarchical position in the taper function resulted in higher accuracy when estimating stem diameters and total tree volume. This study indicates that accounting for attributes at the stand and individual-tree levels may improve stem profile predictions, as well as the biological soundness of the taper function.

A Comparative Evaluation of Three Stem Profile Equations for Three Precious Tree Species in Southern China

Accurately describing the stem curve of precious tree species and estimating the quantity of various types of wood and their volume in the tropics can provide technical support for reasonable bucking. This study utilized Erythrophleum fordii, Castanopsis hystrix and Tectona grandis as study objects. Forty replicates of each species were used for a total of 120 individual trees. Their tape equations were constructed using simple tape equations, segmented taper equations and variable form taper equations. Statistical indicators were utilized to determine the best taper equation for the three types of precious tree species. A number of methods were compared and analyzed, including the index of correlation, the residual sum of squares, the mean prediction error, the variance of prediction errors and the root mean square error. Finally, a preliminary quantitative analysis was conducted to determine the trends of these three types of tree species. The result shows that the precision of the three predictions developed for each species is high, and, in particular, the segmented taper equations with optimized algorithms is the best. The tendency of the three species to vary was shown to be the highest for T. grandis in the range of 0.0 to 0.8 for its relative height, followed by E. fordii, while the variation of C. hystrix was the smallest. However, in the range of 0.8 to 1.0 relative height, the variation of Castanopsis hystrix was the largest, and the variation of both E. fordii and T. grandis were almost the same. Therefore, the segmented taper equations with optimization algorithms was recommended to fit the three types of tree species in the tropics. These types of equations can be used to estimate the stumpage and timber quantity and as a guide reasonable bucking for these three species.

The conic-paraboloid formulae for coarse woody material volume and taper and their approximation

The conic-paraboloid volume equation is receiving increased use with downed coarse woody material (CWM), but the consequences for taper have not been identified mathematically. Requiring that subdivision of a conic-paraboloid yields two smaller conic-paraboloids leads to an exact taper equation intermediate between those of cones and second-order paraboloids. This exact taper equation does not have an explicit inverse, however. An alternative, naive approach does have an explicit inverse, but subdivision does not yield two conic-paraboloids. The exact conic-paraboloid is closely approximated by Fermat’s paraboloid with exponent 7/5. The exact and naive conic-paraboloids match in volume; differences in taper are ≤2.2% of large-end cross-sectional area and ≤5.9% of large-end diameter, while differences in inverse taper are ≤3.7% of total length. Fermat’s paraboloid is always within 1.2% of total volume; differences in taper are ≤0.8% of large-end cross-sectional area and ≤2.0% of large-end diameter, while differences in inverse taper are ≤1.1% of total length. Such differences are negligible given the variety of CWM shapes and practical measurement challenges. Either the exact conic-paraboloid or the corresponding Fermat’s paraboloid provides appropriate equations for estimating the volume and taper of CWM that is intermediate between conical and ordinary paraboloid frusta.