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. This equation was integrated to obtain cubic
volume estimates. Regression analysis was not used, but "actual" volume was compared 4 to estimated volume and resulted in an average difference of -1% with a standard deviation of 9%.
MacLean and Berger (1976) developed volume equations for major conifer species of California. They used data gathered before the 1950's and lacked information for trees less than eleven inches (28 cm) in diameter. The data were extrapolated down to diameters of five inches (13 cm). Weighted least squares regression was used, with homogeneity of variance obtained by multiplying by a weight of 1/ 0.005454154D2H
The new equations, although biased for lodgepole pine (Pinus contorta Dougl.) and incense-cedar (Calocedrus decurrens Toff.), were said to be less biased than older tables.
Volume equations were also developed using Girard's form class. It was found that these equations only gave a better prediction of volume if the individual tree's form class was known; therefore, MacLean and Berger did not recommend use of these equations if the form class was unknown.
Wensel (1977) constructed volume equations and tables for several young-growth conifer species from northern California. He used the logarithmic regression model
log V = loga +blogD+clogH where
V= total stem volume
D = diameter at breast height
H = total height of tree
a,b, c = regression coefficients. 5 Girard's form class was estimated and was not found to significantly improve the volume estimate. The smallest trees in these tables were ten inches (25 cm) in diameter.
Hann and Bare (1978) presented tree volume equations for the major tree species of Arizona and New Mexico including ponderosa pine (Pinus ponderosa Dougl. ex
Laws.) and white fir (Abies concolor (Gord. & Glend.) Lindl. ex Hildebr.), for both forked and unforked trees using the equation
V = a + bD2 H where
V= total stem volume
D = diameter at breast height
H = total height of tree
a,b = regression coefficients.
The equations were weighted by The development of these equations
eliminated many of the historical problems concerning sets of volume equations,
addressing compatibility, reliability, and flexibility. In addition, the equations allowed
the user to separate top and stump volumes from merchantable volume.
Walters et al. (1985) presented volume equations and tables for several species in
second growth, mixed-conifer stands of southwestern Oregon. Height and crown ratio
were measured prior to felling, to mimic standard inventory procedures. Trees were
felled and cut at into 8.4-foot (2.56 m) sections. At each section, inside and outside bark
diameters were measured at their longest and shortest axes, and geometric means of those
two measurements were calculated. Volume above breast height and below breast height 6 were calculated with separate equations, then added to obtain total volume. Weighted least squares regression was used for volume above breast height, using the following equation weighted by
where
V = volume above breast height
H = total height minus 4.5 feet (1.4 m)
D = diameter at breast height
a, b = regression coefficients.
This approach was found to be a better predictor of total volume than the "one-equation" methods. It was found that the crown ratio variable was a considerable improvement over the equation without it. The crown ratio form of the equation was significant for
Douglas-fir (Pseudotsuga menziesii (Mirb.) Franco), grand fir (Abies grandis Lindl.), and white fir. The data showed that for the same height and diameter, trees having large crown ratios had less volume than trees with small crown ratios. It was suggested to use the crown ratio equations when possible.
Another approach to estimating tree volume is to utilize taper equations. Taper equations are functional relationships that predict upper stem diameters given easily measured variables. These predicted diameters are then used to calculate volumes of stem segments, which are then summed to obtain stem volume. A major advantage of this approach to volume estimation is that volume can be estimated to various
merchantable upper stem sizes (Wensel and Krumland 1983). Several factors influence 7 the rate of taper such as tree height, diameter, age and species (Avery and Burkhart
2002). Taper of a single tree can be determined by obtaining diameter measurements at recorded intervals along the tree stem.
Behre (1923) studied tree form using "western yellow pines" (Pinus ponderosa), from which he obtained diameter measurements at every tenth of the height from breast height to tip. He presented an equation, the "Behre Hyperbola":
where
d= diameter at distance L from the tip of tree stem
D = diameter at breast height
L = relative distance from tip of tree stem
a, b = constants varying with form quotient, a + b =1.
He compared this formula to an equation developed earlier by Hojer (Behre 1923):
where C and c are constants.
Behre felt that his new equation conformed to nature more consistently than that of
Hojer. Behre's hyperbola has been used in many studies to approximate tree taper.
Kozak et al. (1969) began studying taper of trees in British Columbia. They
developed the following parabolic taper function:
where
d = diameter inside bark in inches at any given h in feet 8 D = diameter breast high, outside bark, in inches
h= height above the ground in feet
H= total height of the tree in feet
b1= regression coefficient.
They confirmed prior observations that no practical advantage in estimating total volume was gained by obtaining measurements in addition to DBH and total height. This equation carried significant bias when predicting diameters near the ground and in other parts of the stem.
Later Kozak (1988) published an improved version, called a variable-exponent taper function, which represented a new approach to developing taper equations. A single function containing a continuously changing exponent described the shape of the tree stem:
where 9 This equation produced very small biases from the ground to the top of the tree, resulting
in more accurate estimations of stem shape and reduced computing time. Kozak also examined the usefulness of obtaining an additional upper stem outside bark diameter
measurement to improve the accuracy of the taper equation. He found that although
upper stem measurements improved the equation slightly, the additional cost and lack of
accuracy in obtaining these measurements on standing trees was not justified (Kozak
1998).
Kozak's variable form taper equation was studied in Alberta for white spruce
(Huang et al. 2000). This equation performed well for many tree species. It was found
that several of the seven independent variables were insignificant. Huang recommended,
however, that Kozak's original taper equation be used.
Although there have been many studies and volume tables prepared for conifers
with diameters over ten inches (Czaplewski et al. 1989, MacLean and Berger 1976,
Pillsbury and Kirkley 1984, Wensel 1977, Wensel and Krumland 1983), I found very
little quantitative information about smaller trees. Environmental requirements of the
Sierra Nevada Ecosystem Project (SNEP) report (University of California, 1996)
preclude harvesting of larger trees, while smaller trees are becoming increasingly
important in the timber market. Therefore, it appears that future management of forest
stands will be concentrated on smaller trees. Recent catastrophic fires in the western
United States have shown the need for more aggressive fuels management, which often
dictates removal of smaller understory trees. In addition, the Pacific Southwest Research 10 Station of the U.S. Forest Service desires volume data for northern California conifers to incorporate into CONIFERS, a young stand simulator.
The primary objective of this study was to develop cubic foot volume tables for small diameter ponderosa pine (Pinus ponderosa Dougl. ex Laws.) and white fir (Abies concolor (Gord. & Glend.) Lindl. ex Hildebr.) trees in the Shasta-Trinity National Forest.
Individual tree volumes were developed from felled tree data. Statistically derived
volume equations were based on measured tree diameters and heights. In addition, the
tree form variable, measured as crown ratio, was examined to determine the extent to
which tree form affects the predictive ability of volume equations.
The following hypotheses were tested:
The model does not account for a significant amount of
variation in the data.
The model does not account for a significant
amount of variation in the data,
where:
V = Cubic volume of tree stem
D = Diameter at breast height
H = Total height of tree
CR = Crown ratio
bo , b1 = Regression coefficients
Ɛ = residual error 11 A secondary objective of this study was to fit taper equations to the same trees to estimate inside bark diameter at any desired height (Kozak 1988). Volume of any length
log can then be calculated using these estimates. Taper functions can also be used to
estimate height from ground for any top utilization standard.
The following hypothesis was tested:
The model
does not account for a significant amount of variation in the data,
where: 12 The inflection point p of 0.225 was used by Kozak (1988). For more information on inflection points see Demaerschalk and Kozak (1977). METHODS
Trees were sampled from the Shasta-McCloud Management Unit of the Shasta-
Trinity National Forest in Siskiyou County, northern California (Figure 1). A total of 79 ponderosa pine and 111 white fir trees were measured. Trees were purposely selected to cover the full range of diameters between 1.0 and 12.9 inches (2.5 - 32.8 cm) (Table 1).
Trees with defects or stem malformations such as forked or broken tops were not selected. Three independent variables were measured: diameter at breast height (DBH), total height (H), and crown ratio (CR). These variables were measured in the standing tree to mimic actual forest inventory work. Diameter was measured with a steel diameter tape to the nearest tenth of an inch. Heights and crown lengths were measured with a laser or a clinometer to the nearest whole foot (30.5 cm), or half foot (15.2 cm) for the smallest trees.
After the standing tree measurements were taken, trees were felled and cut into 8- foot (2.4 m) lengths. Diameters inside and outside bark were measured with a transparent plastic ruler to the nearest 0.05 inch (1.3mm) at each cut. In each case the diameter of the longest axis was measured as well as the axis perpendicular to that. The quadratic mean diameter of these two measurements was calculated. For a large percentage of the trees, the diameter outside bark of the middle of the butt log was measured with steel calipers to the nearest 0.05 inch (1.3mm).
The length of each log, including the top section, was measured with a steel tape to the nearest 0.5 inch (1.3 cm). For maximum accuracy in determining volume, height measured on the ground was used as the independent height variable. To evaluate the
13 14
Figure 1. Map of sampling area in Sisk you County, California. Latitude is approximately 41.5 degrees and longitude is approximately 121.5 degrees. 15 Table 1. Summary of tree data collected in Sisk you County, California.
Ponderosa pine White fir
Sample Size 79 111
Mean Range Mean Range
DBH 6.8 in. 1.2 - 12.6 in. 6.7 in. 1.1 - 12.7 in.
17.3 cm 3.0 - 32.0 cm 17.0 cm 2.8 - 32.3 cm
Total Height 34.5 ft. 7.5 - 79.8 ft. 38.2 ft. 8.5 - 81.1 ft.
10.5m 2.3 - 24.3 m 11.6m 2.6 - 24.7 m 16 accuracy of the standing height measurements, they were compared to the total felled stem length.
The crown ratio measurement was not as straightforward as diameter and height.
The base of the live crown is commonly defined as the lowest contiguous whorl of three live branches (Maguire and Hann 1987). However, it is not always easy to determine the lowest contiguous whorl of three live branches in the field. In plantations, the live crown is easy to see and is typically uniform. Young trees in naturally regenerated stands, especially tolerant species such as firs, are often growing underneath larger trees causing
a lack of uniformity in the crown and poor visibility of the entire crown. The crown
length was measured as precisely as possible given these factors. To calculate crown
ratio, the crown length was divided by the total tree length.
Appendix A contains all tree measurements. Brief definitions of the items in the
Appendix A are given here. Trees were numbered in order of selection and were listed in
columns by tree number. Three digit tree numbers beginning with "1" or "3" were
ponderosa pines, while numbers starting with "2" or "4" were white firs.
Stump height was measured from the ground on the uphill side of the tree to the
first cut, to the nearest 0.5 inch (1.3 cm). Diameter was measured to the nearest 0.05 inch
(1.3 mm) at the top of the stump and at the top of each log. A ruler was placed with zero
at the left edge of the longest axis and three measurements taken. Three additional
measurements were made at the perpendicular axis (Figure 2).
The length of each log, including the top length, was measured to the nearest 0.5
inch (1.3 cm). Top length was the remaining stem length after all eight-foot (2.4 m) logs 17
Figure 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. 18 were cut. "Middle Log 1 DOB" represented two perpendicular caliper measurements taken at the midpoint of the first log. These measurements were not taken on all trees.
"Top last log" measurements were repeats of the highest diameter measurements taken on each tree. The remaining information (except DBH and Crown Ratio) in Appendix A consists of calculations based on the above measurements. The quadratic mean diameter
(QMD) inside bark was calculated using the two perpendicular measurements taken at each cut. The QMD outside bark was determined using the same method. Bark thickness (BT) was calculated by subtracting the QMD inside bark from the QMD outside bark. Basal area (BA) was based on the quadratic mean diameter inside bark.
The last three rows in Appendix A (Height, DBH, and Crown Ratio) are the three independent variables used in the regression analysis. Height in feet was calculated by
adding the length of each section in inches (stump, logs, and top) and dividing by twelve.
DBH and Crown Ratio were calculated from measurements made in the field to the
nearest 0.05 inch (1.3mm) and 1 foot (30.5 cm) respectively.
The dependent variable volume, was calculated in cubic inches and converted to
cubic feet and cubic meters. Three volume formulas were used corresponding to three
parts of the tree in order to most accurately estimate true volume given the measurements
taken. The top section was treated as a cone and the volume calculated appropriately:
V = 1/3BL
where
V = cubic volume
B = basal area at the base of the top section 19 L = length of section.
The stump volume was calculated as a cylinder but this volume was not included in the total cubic foot volume figure. The remaining sections were calculated as cylinders as well. Two commonly used formulas for calculating volume of logs are (Avery and
Burkhart 2002):
where
V= cubic volume
B = basal area at large end of log
b = basal area at small end of log
B1/2 = basal area of the log midpoint
L = length of log.
For those trees with caliper measurements taken for the middle of Log 1,
Newton's formula was used to calculate the volume of the butt log. This formula is accepted as the most accurate estimator of cubic log volume when B112 is known, although its use is limited due to the expense of measuring logs at the midpoint (Avery and Burkhart 2002). Since the butt log of a tree contains a large percent of its entire volume (Bell and Dilworth 1997), the extra time involved in obtaining the additional caliper measurement was determined to be worthwhile for this study. Smalian's formula was used for all other sections. 20 All analyses were performed using the NCSS 2000 statistical software package
(Hintze 2000). To begin the volume prediction analysis, multiple regression was run on three different models for comparison:
Additionally, the NCSS variable selection program was used to examine several independent variables: D2, H, D2 H, and CR (crown ratio) in order to assess the correlation of each variable with volume.
A preliminary analysis of Model 1 demonstrated that one of the major assumptions of regression, homogeneity of variance, was violated. This problem can be resolved through weighted least squares, which gives greater weight to observations with smaller residual variance (Neter et al. 1989). The consequences of heteroscedasticity in least squares regression are inefficiency and biased standard errors. Biased standard errors result in biased test statistics and confidence intervals (Allison 1999). Weighted regression has been used in past small tree volume studies (Allen et al. 1976). An appropriate weighting variable was selected through an iterative procedure, discussed later. Volume tables were constructed using these equations and coefficients.
Next, taper equations were produced with the tree measurements collected for the volume study (111 firs, 79 pines) using Kozak's variable-exponent function:
(4) 21 Multiple regression was used with di, the inside bark diameter at each section, as the dependent variable. The seven independent variables were calculated using height and diameter data at each section. After examining the regression output, the variable selection program in NCSS was used to determine if a simpler model would predict di accurately. RESULTS
The model V = bo + b1 D2 + b2 H was discarded due to lower R-squared value and higher mean square error (MSE), collinearity between the variables; and a desire for a parsimonious equation (Table 2). The model V = bo + b1D 2 + b2H + b3 D2H had MSE and R-squared values similar to Model 1 but with high multicollinearity. Additionally, there was no justification for selecting this more complicated model rather than Model 1.
Variable selection results for ponderosa pine and white fir are given in Tables 3 and 4. These results indicate that the independent variable D2H was the most significant
and other variables tested did not add significantly to the variation explained.
Some researchers include a measure of tree form in volume equations. Crown
ratio was a surrogate for stem form that was examined in this study. Trees with large
crown ratios have smaller volumes for a given diameter and height than trees with small
crown ratios. The addition of crown ratio was not found to increase the accuracy of
volume prediction in this study.
Due to the above results the model chosen for this study was
V = bo + b1 D2 H
where:
V= volume in cubic feet for the English equation and cubic meters for the metric,
D = diameter in inches (English) and centimeters (metric), and
H = height in feet (English) and meters (metric).
22 23 Table 2. Regression statistics comparing the three models tested. Number of R-Squared R-Squared Change Variables Variables
1 0.980717 N/A D2H
2 0.980832 0.000115 D2H, H
3 0.981070 0.000239 D2H, H, D2
4 0.981293 0.000223 D2H, H, D2, CR 25 Table 4. Comparison of R-squared values with an increasing number of variables added to the model for volume of white fir.
Model Size R-Squared R-Squared Change Variables
1 0.991203 N/A D2H
2 0.991718 0.000515 D2H, D2
3 0.992075 0.000357 D2H, D2, H
4 0.992079 0.000003 D2H, D2, H, CR 26 Least squares regression was then used to predict volume from D2H. When the regression residuals were plotted against predicted volume, the residual variance increased in magnitude as tree diameter and height increased, resulting in a "megaphone" pattern (Figures 3, 4). Therefore, the assumption of homoscedasticity was not met and it became necessary to weight the variances of the residuals in order to make them approximately equal (Draper and Smith 1998). Since the residual variance increased as
D2H increased, it was logical to weight each residual with the inverse of a function of its variance. Several weights were tried with the goal of choosing the weight that gave approximately equal variances to all the residuals. Since the variance was unknown but was positively correlated with D2H (referred to as x), the inverse of various functions of x were examined including An iterative procedure was used to determine that a weight of resulted in approximately equal residuals of
variance for all values of x (Figures 5, 6). Figures 7 and 8 plot the relationship between
calculated volume and D2H and the corresponding regression line.
Weighted regression coefficients in English measurements and t-test results for
Equation 1 are given in Table 5. Metric equivalent coefficients for Equation 1 are found
in Table 6. These coefficients were algebraically derived from the English equations and
therefore have the same R-squared and t-test results as the English equations. English
and metric volume tables for these species are given in Tables 7-10.
Similar volume results were obtained for the pines and the firs in this study;
regression coefficients varied slightly between the two species. A pine had less volume 27
Figure 3. Plot of unweighted residuals versus D2H for volume equation 1 for ponderosa pine. 28
Figure 4. Plot of unweighted residuals versus D2H for volume equation 1 for white fir. 29
Figure 5. Plot of weighted residuals versus D2H for volume equation 1 for ponderosa pine. 30
Figure 6. Plot of weighted residuals versus D2H for volume equation 1 for white fir.
32
Figure 8. Plot of volume versus D2H with regression line, for white fir. 33 Table 5. Weighted regression coefficients and coefficients of determination (R2) for Equation 1. D is diameter in inches and H is height in feet.
Ponderosa pine White fir bo (intercept) 0.05249978 0.04784340 t-value 0.9747 1.2490
P-value 0.3327 0.2143
b1 (D2H) 0.001942181 0.002167353 t-value 75.7528 130.1452
P-value <0.0000009 <0.0000009
R2 0.9868 0.9936
MSE 0.009944 0.006080 n 79 111 34
Table 6. Weighted regression coefficients for Equation 1. D is diameter in centimeters and H is height in meters.
Ponderosa pine White fir bo (intercept) 0.000757889 0.000674134 bi b1 2 H) 0.000028030 0.000031284 Table 7. Cubic-foot volume table for ponderosa pine.
Ponderosa Pine Cubic-Foot Volume Height (feet) sample DBH (in.) 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 size 1.0 0.1 0.1 0.1 1 1.5 0.1 0.1 0.1 0.1 3 2.0 0.1 0.1 0.2 0.2 2 2.5 0.1 0.2 0.2 0.3 4 3.0 0.2 0.3 0.4 7 3.5 0.3 0.4 0.5 0.6 5 4.0 0.4 0.5 0.7 0.8 1.0 1.1 2 4.5 0.4 0.6 0.8 1.0 1.2 1.4 7 5.0 0.8 1.0 1.3 1.5 1.8 1 5.5 0.9 1.2 1.5 1.8 2.1 1 6.0 1.1 1.5 1.8 2.2 2.5 2.8 3 6.5 1.3 1.7 2.1 2.5 2.9 3.3 3.7 3 7.0 2.0 2.4 2.9 3.4 3.9 4.3 3 7.5 2.2 2.8 3.3 3.9 4.4 5.0 3 8.0 3.2 3.8 4.4 5.0 5.6 6.3 6 8.5 3.6 4.3 5.0 5.7 6.4 7.1 7.8 4 9.0 4.8 5.6 6.3 7.1 7.9 8.7 9.5 10.3 1 9.5 5.3 6.2 7.1 7.9 8.8 9.7 10.6 11.4 12.3 4 10.0 6.8 7.8 8.8 9.8 10.7 11.7 12.7 13.6 1 10.5 8.6 9.7 10.8 11.8 12.9 14.0 15.0 16.1 2 11.0 10.6 11.8 13.0 14.2 15.3 16.5 17.7 18.9 4 11.5 11.6 12.9 14.2 15.5 16.7 18.0 19.3 20.6 6 12.0 14.0 15.4 16.8 18.2 19.6 21.0 22.4 4 12.5 15.2 16.7 18.3 19.8 21.3 22.8 24.3 2 Sample size 1 8 12 7 6 7 10 5 3 1 3 5 8 2 0 1 79 Bold indicates extent of data. 3 5 Table 8. Cubic-foot volume table for white fir.
White Fir Cubic-Foot Volume Height (feet) sample DBH (in.) 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 size 1.0 0.1 0.1 0.1 1 1.5 0.1 0.1 0.1 0.1 1 2.0 0.1 0.1 0.2 0.2 0.3 7 2.5 0.1 0.2 0.2 0.3 0.4 0.5 4 3.0 0.2 0.3 0.4 0.5 0.6 0.7 6 3.5 0.3 0.4 0.6 0.7 0.8 1.0 1.1 10 4.0 0.4 0.6 0.7 0.9 1.1 1.3 1.4 6 4.5 0.5 0.7 0.9 1.1 1.4 1.6 1.8 2.0 2 5.0 0.9 1.1 1.4 1.7 1.9 2.2 2.5 6 • 5.5 1.0 1.4 1.7 2.0 2.3 2.7 3.0 3.3 5 6.0 1.2 1.6 2.0 2.4 2.8 3.2 3.6 3.9 7 6.5 1.4 1.9 2.3 2.8 3.3 3.7 4.2 4.6 5.1 3 7.0 2.2 2.7 3.2 3.8 4.3 4.8 5.4 5.9 2 7.5 2.5 3.1 3.7 4.3 4.9 5.5 6.1 6.8 7.4 6 8.0 3.5 4.2 4.9 5.6 6.3 7.0 7.7 8.4 7 8.5 4.0 4.7 5.5 6.3 7.1 7.9 8.7 9.4 10.2 4 9.0 5.3 6.2 7.1 7.9 8.8 9.7 10.6 11.5 12.3 2 9.5 5.9 6.9 7.9 8.8 9.8 10.8 11.8 12.8 13.7 14.7 2 10.0 7.6 8.7 9.8 10.9 12.0 13.1 14.1 15.2 16.3 17.4 3 10.5 8.4 9.6 10.8 12.0 13.2 14.4 15.6 16.8 18.0 19.2 9 11.0 10.5 11.8 13.2 14.5 15.8 17.1 18.4 19.7 21.0 5 11.5 12.9 14.4 15.8 17.2 18.7 20.1 21.5 23.0 7 12.0 15.7 17.2 18.8 20.3 21.9 23.5 25.0 4 12.5 18.7 20.4 22.1 23.8 25.4 27.1 2 Sample size 0 10 11 13 10 9 7 9 6 5 5 4 5 13 3 1 111 36
Bold indicates extent of data. Table 9. Cubic-meter volume table for ponderosa pine.
Ponderosa Pine Cubic-Meter Volume Height (meters) DBH (cm.) 2 4 6 8 10 12 14 16 18 20 22 24 2.0 0.001 0.001 4.0 0.002 0.003 0.003 6.0 0.003 0.005 0.007 8.0 0.008 0.012 0.015 10.0 0.012 0.018 0.023 0.029 12.0 0.017 0.025 0.033 0.041 14.0 0.023 0.034 0.045 0.056 16.0 0.044 0.058 0.073 0.087 0.101 18.0 0.055 0.073 0.092 0.110 0.128 20.0 0.090 0.113 0.135 0.158 0.180 22.0 0.109 0.136 0.164 0.191 0.218 0.245 0.272 24.0 0.162 0.195 0.227 0.259 0.291 0.324 0.356 26.0 0.190 0.228 0.266 0.304 0.342 0.380 0.418 28.0 0.264 0.308 0.352 0.396 0.440 0.484 0.528 30.0 0.404 0.455 0.505 0.556 0.606 32.0 0.460 0.517 0.575 0.632 0.690 3 7 Table 10. Cubic-meter volume table for white fir.
White Fir Cubic-Meter Volume Height (meters) DBH (cm.) 2 4 6 8 10 12 14 16 18 20 22 24 2.0 0.001 0.001 4.0 0.002 0.003 0.004 6.0 0.003 0.005 0.007 0.010 8.0 0.009 0.013 0.017 0.021 10.0 0.013 0.019 0.026 0.032 0.038 12.0 0.019 0.028 0.037 0.046 0.055 0.064 14.0 0.025 0.037 0.050 0.062 0.074 0.087 16.0 0.033 0.049 0.065 0.081 0.097 0.113 0.129 18.0 0.061 0.082 0.102 0.122 0.143 0.163 0.183 20.0 0.101 0.126 0.151 0.176 0.201 0.226 22.0 0.122 0.152 0.182 0.213 0.243 0.273 0.304 24.0 0.181 0.217 0.253 0.289 0.325 0.361 0.397 26.0 0.212 0.254 0.297 0.339 0.381 0.424 0.466 0.508 28.0 0.295 0.344 0.393 0.442 0.491 0.540 0.589 30.0 0.395 0.451 0.507 0.564 0.620 0.676 32.0 0.513 0.577 0.641 0.705 0.770 38 39 than a fir of the same DBH and height. Since this study predicts only inside bark volume yet the volume tables are based on DBH an outside-bark measurement, the species difference in volume is probably due to thicker bark in pines.
Taper functions were fitted using Kozak's variable-exponent taper function:
where
d, D = diameters in inches for the English equation and centimeters for the metric,
H = height in feet (English) and meters (metric).
Regression coefficients in English measurements and summary statistics for Equation 4 are given in Table 11. Metric equivalent coefficients for Equation 4 are found in Table
12. These coefficients were algebraically derived from the English equations and therefore have the same R-squared and MSE values as the English equations.
The equation fit the data very well, with high R-squared values and low mean square error (MSE) values. Taper curves were generated by inserting data for three different sized trees into the taper equation. These curves describe the shape of the bole and are shown in Figures 9 and 10.
Many of the variables in Kozak's original seven-factor taper equation had extremely high collinearity. Although multicollinearity does not affect the predictive
ability of a regression equation, it suggests that a simpler model may suffice (Allison
1999). In order to choose a more parsimonious model variable selection was run with the
seven factors of Kozak's original taper equation. Results showed that a model using only
the variables In D and In Xez : 40 Table 11. Regression coefficients and coefficients of determination (R2) for Kozak's taper function (Equation 4) using English units.
Ponderosa pine White fir
bo 1.1139200 1.0873640
b1 0.6440522 0.7566335
b2 1.0438190 1.0259540
b3 0.2569733 0.7770160
4 b -0.1208996 -0.1315949
b5 0.5009987 0.7417985
b6 -0.0500276 -0.3144156
7 b 0.2367626 0.6457469
p 0.225 0.225
R2 0.9713000 0.9770000
MSE 0.0159800 0.0142300
n 371 575 41 Table 12. Regression coefficients for Kozak's taper function (Equation 4) using metric units.
Ponderosa pine White fir
b0 1.5522208 1.3642627
b1 0.6440522 0.7566335
b2 1.0170276 1.0101388
b3 0.2569733 0.7770160
b4 -0.1208996 -0.1315949
b5 0.5009987 0.7417985
b6 -0.0500276 -0.3144156
b7 0.0284115 0.0774896 42
Figure 9. Generalized taper curves for ponderosa pine. 43
Figure 10. Generalized taper curves for white fir. 44
described 96.0 percent of the variation in di for ponderosa pine and 97.4 percent for white fir, and had mean square error values of 0.02244 and 0.01625 respectively. These
figures show that the original seven factor equation is not significantly more precise than
the simplified version. Furthermore, the shorter equation is easier to interpret and serves
the purposes of this study adequately. DISCUSSION
The desire for simplicity in the volume equation makes the function D2H logical since the volume of a cylinder equals the area of its circular base multiplied by its length.
In this study, the model V = bo + b1 D2H fit well. It is unknown if larger trees or other conifer species would fit this model as closely. The negligible effect of adding crown ratio to the model is probably due to the small size of the trees studied. Walters et al.
(1985) found that adding crown ratio significantly improved their equation. However, that equation was produced for trees from 10 to 200 inches DBH.
The volume equations can be used to calculate volumes of individual trees or groups of trees, taking caution not to extend the model beyond the range of data presented here. As an alternative to using the equations to calculate volume, the tables presented on pages 35-38 can be used to look up the volume of a tree of given DBH and height.
Since some products are sold on a weight basis instead of volume, an equation is provided that will convert cubic volume to weight. The basic equation follows (Briggs
1994):
Wt .(lb.) = Vol .(ft3) * SG * Den.(lb / ft3) * (1+ MC(%)/100)
or
Wt.(kg)=Vol.(m3)* SG * Den.(kg / m3)* (1+ MC(%)/100) where
SG = average green wood specific gravity of the species (unitless)
45 46 Den. = density of water (lb/ft3 or kg/m3)
MC = green wood moisture content of the species on an oven-dry basis (percent).
Inserting species-specific values into the above equations resulted in the following equations to convert cubic volume to weight, in English or metric units.
Ponderosa pine:
White fir:
The appropriate volume and weight equations can be combined so that one can use the diameter and height measurement of a tree to obtain the weight of that tree. For example the weight of a 10-inch DBH, 60-foot tall ponderosa pine may be calculated using equations (1) and (3) with coefficients from Table 5 as follows:
A common use of the taper equation is to predict the inside-bark diameter of a tree at a given height. Suppose for example one measured a 10-inch DBH, 60-foot tall ponderosa pine and wished to know its inside bark diameter at 50 feet. Using Equation 2 and coefficients from Table 7, the calculations are as follows: 47 One may also estimate the height of a tree at a given merchantable top diameter by manipulating Equation 2 to solve for hi.
The objective of this study was to find the model that best predicted volume of small conifers in northern California. The combined variable model, V = bo + b1 D2 H , was found to be the most parsimonious while maintaining a high level of precision.
Contrary to the findings of other researchers, the addition of a form variable did not significantly increase predictive power.
Kozak's seven-factor taper function modeled stem taper well. A high level of predictability was maintained using a much simpler form. LITERATURE CITED
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Avery, T.E. and H.E. Burkhart. 2002. Forest Measurements, 5th ed. McGraw Hill Book Company, Boston, Massachusetts
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Bell, J.E. and J.R. Dilworth. 1997. Log scaling and timber cruising. Cascade Printing Company, Corvallis, Oregon.
Briggs, D. 1994. Forest products measurements and conversion factors: with special emphasis on the U.S. Pacific Northwest. College of Forest Resources, University of Washington, Seattle, Washington.
Curtis, R.O., D. Bruce, and C. VanCoevering. 1968. Volume and taper tables for red alder. United States Department of Agriculture, Forest Service, Pacific Northwest Forest and Range Experiment Station, Research Paper PNW-56. Portland, Oregon.
Czaplewski, R.L., A.S. Brown, and D.G. Guenther. 1989. Estimating merchantable tree volume in Oregon and Washington using stem profile models. United States Department of Agriculture, Forest Service, Rocky Mountain Forest and Range Experiment Station, Research Paper RM-286. Fort Collins, Colorado.
Demaerschalk, J.P. and A. Kozak. 1977. The whole-bole system: a conditioned dual- equation system for precise prediction of tree profiles. Canadian Journal of Forest Research. 7: 488-497.
Draper, N.R. and H. Smith. 1998. Applied regression analysis, 3d. ed. John Wiley & Sons, Inc., New York, New York.
Gerrard, D.J. 1966. The construction of standard tree volume tables by weighted multiple regression. University of Toronto, Faculty of Forestry Technical Report No. 6. Toronto, Ontario.
48 49 Hann, D.W. and B.B. Bare. 1978. Comprehensive tree volume equations for major species of New Mexico and Arizona: I. results and methodology. United States Department of Agriculture, Forest Service, Research Paper INT-209. Ogden, Utah.
Hintze, J.L. 1999. Number cruncher statistical system (NCSS 2000). Released April 15, 1999. Kaysville, Utah.
Huang, S., D. Price, D. Morgan, and K. Peck. 2000. Kozak's variable-exponent taper equation regionalized for white spruce in Alberta. Western Journal of Applied Forestry 15: 75-85.
Kozak, A. 1988. A variable-exponent taper equation. Canadian Journal of Forest Resources 18: 1363-1368.
Kozak, A. 1998. Effects of upper stem measurements on the predictive ability of a variable-exponent taper equation. Canadian Journal of Forest Resources 28: 1078-1083.
Kozak, A., D.D. Munro, and J.H.G. Smith. 1969. Taper functions and their application in forest inventory. The Forestry Chronicle 45: 278-283.
MacLean, C.D. and J.M. Berger. 1976. Softwood tree volume equations for major California species. Portland, Oregon. United States Department of Agriculture, Forest Service, Research Paper PNW-266. Portland, Oregon.
Maguire, D.A. and D.W. Hann. 1987. Equations for predicting sapwood area at crown base in southwestern Oregon Douglas-fir. Canadian Journal of Forest Research 17: 236-241.
Neter, J., W. Wasserman, and M.H. Kutner. 1989. Applied linear regression models, 2d ed. Irwin Publishers, Inc., Homewood, Illinois.
Pillsbury, N.H. and M.L. Kirkley. 1984. Equations for total wood and saw-log volume for thirteen California hardwoods. United States Department of Agriculture, Forest Service, Pacific Northwest Forest and Range Experiment Station, Research Note PNW-414. Portland, Oregon.
Spurr, S.H. 1952. Forest inventory. The Ronald Press Company, New York, New York.
University of California. 1996. Summary of the Sierra Nevada ecosystem project report. University of California and SNEP Science Team and Special Consultants. Berkeley, California. 50
Walters, David K., David W. Hann, & Merlise A. Clyde. 1985. Equations and tables predicting gross total stem volumes in cubic feet for six major conifers of southwest Oregon. Oregon State University, Forest Research Laboratory, Research Bulletin 50. Corvallis, Oregon.
Wensel, L.C. 1977. Volume tables for young-growth conifers in the northern regions of California. University of California, Division of Agricultural Sciences, Bulletin 1883. Berkeley, California.
Wensel, L.C. and B. Krumland. 1983. Volume and taper relationships for redwood, Douglas fir, and other conifers in California's north coast. University of California, Division of Agricultural Sciences, Bulletin 1907. Berkeley, California.
Wiant, H.V. and W.S. Berry. 1965. Cubic-foot volume and tarif access tables for tanoak in Humboldt County, California. Humboldt State College, Department of Forestry, Research Report No. 2. Arcata, California. Appendix A. Tree measurements in inches.
51 Appendix A. Tree measurements in inches (continued). 52 Appendix A. Tree measurements in inches (continued). 53 Appendix A. Tree measurements in inches (continued). 54 Appendix A. Tree measurements in inches (continued). 55 Appendix A. Tree measurements in inches (continued). 56 Appendix A. Tree measurements in inches (continued). 57 Appendix A. Tree measurements in inches (continued). 58 Appendix A. Tree measurements in inches (continued). 59 Appendix A. Tree measurements in inches (continued). 60 Appendix A. Tree measurements in inches (continued). 61 Appendix A. Tree measurements in inches (continued). 62 Appendix A. Tree measurements in inches (continued). 63 Appendix A. Tree measurements in inches (continued). 64 Appendix A. Tree measurements in inches (continued). 65 Appendix A. Tree measurements in inches (continued). 66 Appendix A. Tree measurements in inches (continued). 67 Appendix A. Tree measurements in inches (continued). 68 Appendix A. Tree measurements in inches (continued). 69 Appendix A. Tree measurements in inches (continued). 70 Appendix A. Tree measurements in inches (continued). 71 Appendix A. Tree measurements in inches (continued). 72 Appendix A. Tree measurements in inches (continued). 73 Appendix A. Tree measurements in inches (continued). 74 Appendix A. Tree measurements in inches (continued). 75 Appendix A. Tree measurements in inches (continued). 76 Appendix A. Tree measurements in inches (continued). 77 Appendix A. Tree measurements in inches (continued). 78 Appendix A. Tree measurements in inches (continued). 79 Appendix A. Tree measurements in inches (continued). 80 Appendix A. Tree measurements in inches (continued). 81 Appendix A. Tree measurements in inches (continued). 82 Appendix A. Tree measurements in inches (continued). 83 Appendix A. Tree measurements in inches (continued). 84 Appendix A. Tree measurements in inches (continued). 85 Appendix A. Tree measurements in inches (continued). 86 Appendix A. Tree measurements in inches (continued). 87 Appendix A. Tree measurements in inches (continued). 88 Appendix A. Tree measurements in inches (continued). 89 Appendix A. Tree measurements in inches (continued). 90 Appendix A. Tree measurements in inches (continued). 91 Appendix A. Tree measurements in inches (continued). 92 Appendix A. Tree measurements in inches (continued). 93 Appendix A. Tree measurements in inches (continued). 94