VOLUME TABLES AND TAPER EQUATIONS FOR SMALL DIAMETER CONIFERS by Sally Joann Schultz A Thesis Presented to The Faculty of Humboldt State University In Partial Fulfillment Of the Requirements for the Degree Masters of Science In Natural Resources: Forestry May, 2003 VOLUME TABLES AND TAPER EQUATIONS FOR SMALL DIAMETER CONIFERS By Sally J. Schultz Approved by the Master's Thesis Committee Gerald M. Allen, Major Professor Date William L. Bigg, Committee Member Date Lawrence Fox, III, Committee Member Date Coordinator, Natural Resources Graduate Program Date 02-F-473-05/03 Natural Resources Graduate Program Number Approved by the Dean of Graduate Studies Donna E. Schafer Date ABSTRACT Volume Tables and Taper Equations for Small Diameter Conifers Sally Joann Schultz Although volume tables have been in existence for years and there are many available, there is very little empirically-based volume information for small diameter trees. For the current study, a total of 79 young ponderosa pine and 111 young white fir trees were selected from naturally regenerated stands and plantations in the McCloud Ranger District of the Shasta-Trinity National Forest. Tree diameters ranged from 1 to 13 inches (2.5 to 33 cm) at breast height and tree heights ranged from 7 to 81 feet (2.1 to 24.7 m). From field measurements, total cubic foot volume inside bark was calculated for each tree. Equations that estimate tree volume given height and diameter at breast height were constructed using weighted least squares regression techniques. In addition, crown ratio was examined for significance in predicting volume. Diameter and height were found to significantly influence volume while crown ratio was found to be insignificant. The model V = ß0 + ß1 D2H was chosen and the results of weighted least squares regression showed that the data fit this model well. Statistical analysis showed that the simplest model was the most appropriate and was very precise, with R-squared values of 0.981 for the pines and 0.991 for the firs. From these equations, cubic foot volume tables were constructed. iii Kozak's variable-exponent taper function was analyzed with the same tree data using multiple regression and was found to fit the data closely. Upon analysis it was determined that this seven-factor equation could be simplified to a two-factor model with very little compromise in precision. Additionally, this shorter taper equation eliminated the problem of multicollinearity that was present in the seven-factor model, and was easier to interpret. iv ACKNOWLEDGEMENTS Funding and sample data for this study were provided by the McCloud Ranger District of the Shasta-Trinity National Forest. Additional funding was provided by a grant from the McIntire-Stennis Forest Research Program. I would like to thank the following professors of the Forestry Department at Humboldt State University: Dr. Jerry Allen, an exceptional advisor, teacher, and friend; and Drs. Bill Bigg and Larry Fox, committee members. Dr. Butch Weckerly, Biometrics professor, also assisted. Dr. Martin Ritchie of the USDA Forest Service, Pacific Southwest Research Station provided advice for data collection and visited the study sites. I wish to thank the following employees of the Shasta-Trinity National Forest: Bill Branham, Operations Forester, for giving me the opportunity to begin forestry work; Bob Hammond, District Ranger; Ron Howard, Timber Sale Administrator, for ideas; Joyce Tausch, Silviculturist, for encouraging me toward a forestry degree and for providing sample trees; the fire suppression department for tree falling; the timber sale administration department for sample trees and assistance; and co-workers Mark Lorraine for tree falling and David Hawkins for data collection. The following individuals provided assistance beyond the call of duty in field data collection: my father, Bill Schultz, and friends Tom Barnett and Kristin Carter. Beverly Bulaon gave moral support during the writing process. I would especially like to thank my parents Sandra and Bill Davidson, and Bill and Hanne Schultz, for their love, encouragement, and support. TABLE OF CONTENTS Page ABSTRACT ..iii ACKNOWLEDGEMENTS ..v LIST OF TABLES vii LIST OF FIGURES .viii INTRODUCTION 1 METHODS 13 RESULTS 22 DISCUSSION .45 LITERATURE CITED .48 APPENDIX A 51 vi LIST OF TABLES Table Page 1. Summary of tree data collected in Siskiyou County, California 15 2. Regression statistics comparing the three models tested 23 3. Comparison of R-squared values with an increasing number of variables added to the model for volume of ponderosa pine 24 4. Comparison of R-squared values with an increasing number of variables added to the model for volume of white fir 25 5. Weighted regression coefficients and coefficients of determination (R2) for Equation 1. D is diameter in inches and H is height in feet 33 6. Weighted regression coefficients for Equation 1. D is diameter in centimeters and H is height in meters 34 7. Cubic-foot volume table for ponderosa pine 35 8. Cubic-foot volume table for white fir 36 9. Cubic-meter volume table for ponderosa pine 37 10. Cubic-meter volume table for white fir 38 11. Regression coefficients and coefficients of determination (R2) for Kozak's taper function (Equation 4) using English units 40 12. Regression coefficients for Kozak's taper function (Equation 4) using metric units...41 vii LIST OF FIGURES Figure Page 1. Map of sampling area in Siskiyou County, California. Latitude is approximately 41.5 degrees and longitude is approximately 121.5 degrees 14 2. Illustration of diameter measurements at each stem section. A ruler was placed with zero at the left edge of the section at its longest axis and three measurements were taken. This procedure was repeated at the axis perpendicular to the longest ....17 3. Plot of unweighted residuals versus D2H for volume equation 1 for ponderosa pine 27 4. Plot of unweighted residuals versus D2H for volume equation 1 for white fir 28 5. Plot of weighted residuals versus D2H for volume equation 1 for ponderosa pine 29 6. Plot of weighted residuals versus D2H for volume equation 1 for white fir 30 7. Plot of volume versus D2H with regression line, for ponderosa pine 31 8. Plot of volume versus D2H with regression line, for white fir 32 9. Generalized taper curves for ponderosa pine 42 10. Generalized taper curves for white fir 43 viii INTRODUCTION Tree volume tables have been used in forest management applications for over two hundred years, their primary purpose being to estimate volume of standing trees (Spurr 1952). Then as now, it was thought that tree volume was best calculated by the relationship v = bhf where v = volume of the tree stem b= cross-sectional area h= height f= form factor. Form factor, and form in general, is an abstract variable since it cannot be directly measured in a standing tree (Spurr 1952). For example, Girard form class is defined as the ratio between stem diameter inside bark at the top of the first sixteen-foot (4.9 m) og and the diameter at breast height outside bark (Avery and Burkhart 2002). • Efforts to predict tree volume from easily measured variables, primarily diameter and height, have continued for years. Much difference of opinion still exists with regard to the equation which best predicts total volume. Board-foot volumes and merchantable lengths further complicate the problem, since they add commercial factors to the biological expression of volume in cubic feet (Gerrard 1966). More variables can be measured, such as crown ratio and factors can be added to volume equations, such as site class or other environmental variables. Generally the added time and difficulty involved 1 2 in obtaining these measurements overshadows the gain in precision of the volume equation and often fails to improve accuracy of the volume estimate (Gerrard 1966). Heinrich Cotta published the first modem volume table ih 1804 for beech trees (Spurr 1952). Cotta based his tables on volume calculation of theoretical geometric shapes. In 1846, Bavarians published the first volume tables based on actual measurements of thousands of tree stems rather than basing volume calculations on a theoretical solid (Spurr 1952). Wiant and Berry (1965) produced tanoak (Lithocarpus densiflorus (Hook. & Am.) Rehd.) volume tables using the logarithmic combined variable equation log V = log a + b log (D2H) where V= total stem cubic-foot volume D = diameter at breast height H= total height of tree a, b = regression coefficients The combined variable equation was transformed to a logarithmic model for fitting purposes. The inherent underestimation in volume when using logarithms was corrected by multiplying resulting values by 101.506 percent. Their volume tables represented diameters as small as five inches (13 cm). Gerrard (1966) developed a method of constructing standard tree volume tables using weighted multiple regression with jack pine (Pinus banksiana Lamb.) data from Ontario. Forest mensurationists were beginning to consider mathematical expressions 3 more complicated than logarithmic volume equations due to advancing electronic computer technology. Gerrard used the equations and where V= total stem volume D= diameter at breast height H= total height of tree bo, b1, etc. = regression coefficients. Gerrard used an original method for determining weights: where = variance of ith data point from the regression line. He found that the terms D, D2, and D2H described the majority of the variation in the pines he used for the study, but recommended use of all six terms. Curtis et al. (1968) published volume and taper tables for red alder (Alnus rubra Bong.) from data collected in Oregon, Washington, and British Columbia. Their volume tables covered tree diameters as small as two inches (5 cm). The data were summarized in order to express diameter inside bark as a function of its relative height from the ground, the tree's DBH, and total height.