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A99.9 F764U

Comprehensive Tree Volume
Equations for Major Species
of New Mexico and Arizona:

I. Results and Methodology

David W. Hann
B. Bruce Bare

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Merchantable
Volume

Stump
Volume

UDSA Forest Service Research Paper INT-209
INTERMOUNTAIN FOREST AND RANGE EXPERIMENT STATION
FOREST SERVICE, U.S. DEPARTMENT OF AGRICULTURE

ERRATA SHEET

Hann, David W., and B. Bruce Bare.
1978. Comprehensive tree volume equations for major species of
New Mexico and Arizona: I.. Results and Methodology. USDA
For. Serv. Res. Pap. INT=209,.45 p.) Intermt.. For..and Range
Exp. Stn..,. Ogden, Utah 84401:

Page 9
Table 5.--forked tree merchantable gross cubte foot volume data base
and regression results
Independent variables
= D.b.h. in inches D.. = Top diameter, inside bark, in inches
H = Total tree height in feet P.=-Height to,farst fork divided ‘by Ho
T = 1 + (number of forks)
F Relative
eiens :mean squared
Species, National Forest, and regression equations ] of A ec ie
> observations | PRMSOR)
White pine on Lincoln, Santa Fe, and Carson
R. = = Cee? Cao 101 0.9581
ig
Co == 0.61725167 + 0.042565845D_
Ci = 0.00598501 - 0.00073445D_
OTR, 2 = 1.04518089 - 7.88395198E-07D" + 8.39079873E-08D8
i a ut
Engelmann spruce-corkbark fir on Lincoln, Santa Fe, and
Carson F
Ru 2 8 1.15S721355 =-1.5106/496E-08D- 91
Ms: m
Ponderosa pine on Coconino, Tonto, Lincoln, Santa Fe,
and Carson i
Ro» =O, 997113216. = 0,0253001 25D. + CaT> + Cyr 937 59761
io ee) mas
Ga = 0.00564984 + 0.00079885D_
Cy = 0.002886816 + 0.025350015D_,
OFR_ 2 = 1.01577735. - 1.68401836E-06D>
mM,
Douglas-fir on Lincoln; Santa Fe, and carson
R_ 2 = 1.04417090 - 9.55965562E-09*D" 118
Aspen on Santa Fe and Carson ab
R. » = 1.07666448 - 1.18155616E-05D2-° 184
4: ;
White fir on Santa Fe, Carson, and Lincoln
a Z 7 ple
Re = Cy + C; (8/02) 74 86
Co = 0.673711298 + 0.055948591D_
Cy = 1.709640750 - 0.348154558D
or" m
R.. . = 01985295776 -- 2.40166266E-05D>

CONTENTS

Page

INDRODWCITONG ces eee A ee re We tA R es ion departs dey poss
SOURCE AND NATURE OFZDA TA. o. nctms aye, 06% drasieierysne fe) i oS
Re SOME RSs Gane souks Nst Se oa thwciwa het dios oi es caeadrs ov gy, @ a 4
DISCUSSION AND: CONGILUSIONS, ... oc. csunienteicaine) ci "Re.,,.04)o.9 6) pansercte ope 0, 20
PUBLICARIONSICULE DS cx., aioe couon © chaos oipet icy eure” fees sreatyc* age 27
APPENDIXES SU CO et ictais atopsleta ake ae viet oie eae aicis lel ies! hp air eober teh Wh is OL
Details Gt Methodology. «wc. ticveg ie: 81g werniseticr arses) “erigd re Biers OA

I. --Total Stem Gross Cubic Foot Volume--Unforked Trees .. 32

Il. --Total Stem Gross Cubic Foot Volume--Forked Trees ... 33
Ill. --Merchantable Gross Cubic Foot Volume--Unforked Trees . 33

IV.--Merchantable Gross Cubic Foot Volume--Forked Trees . . 35
V.--Gross Cubic Foot ee Volume--Unforked and Forked
Trees ... o (et den ato) OO
VI. --Gross ie ceational ie thch Board Foot vonesé
Unforked Trees... ; re ee
VII. --Gross International {/A-tneh Board Foot volumess
Forked Trees ... i ee ae OT
VIII. --Gross Scribner Board Foot voline=<Unforked Trees > 638
IX. --Gross Scribner Board Foot Volume--Forked Trees... . 39
X.--Probability of a Tree Being Unsound in Cubic Foot
Volume--Unforked and Forked Trees .. . a e < oo
XI. --Fraction Cull in Total Stem Cubic Foot voll Given the
Tree is Unsound--Unforked and Forked Trees ... 40
XII. --Fraction Cull in Merchantable Cubic Foot Volume Caen the
Tree is Unsound--Unforked and Forked Trees ... 2 AL
XIII. --Probability of Tree Being Unsound in Board Foot velune --
Unforked and Forked Trees ... - 42

XIV.--Fraction Cull in International /4Siach Bodtd B08e. volunie
Given the Tree Is Unsound--Unforked and Forked Trees. . 43
XV.--Fraction Cull in Scribner Board Foot Volume Given the
Tree Is Unsound--Unforked and Forked Trees ...... 438

RESEARCH SUMMARY

Comprehensive tree volume equations and the methodology used to
derive them are presented for the major tree species of New Mexico and
Arizona. With these equations, the following gross and net volume types
can be computed for both unforked and forked trees: total stem cubic
foot volume; mercuantable cubic foot volume to any top diameter between
3 and 8 inches, inside bark; International 1/4-inch board foot volume to
a 6-inch top, inside bark; and Scribner board foot volume to a 6-inch
top, inside bark. These equations were developed in such a manner as
to eliminate many of the problems of consistency, compatibility,
reliability, and flexibility associated with previous sets of volume
equations. This approach also allows the user to divide total stem
cubic foot volume into top volume, merchantable volume, and stums
volume. This last feature is useful for predicting the ievel of residues
after harvesting to various top diameters.

INTRODUCTION

Today, as in the past, the accurate estimation of tree volume is an essential pre-
requisite for foresters involved with timber management planning, forest surveys, damage
appraisal, timber sale preparation, trespass, and condemnation proceedings as well as
growth and yield studies. To be of immediate value, volume estimates must be expressed
in units of measure related directly to the products derived from the tree. The board
foot and the cubic foot are traditional units of measure, although the latter is
increasing in importance as utilization of the total tree becomes more common. However,
it is also recognized that cubic foot volume, when combined with square feet of circum-
ferential surface and linear feet, provides for more consistent and accurate estimates
of a tree's product potential (Grosenbaugh 1954; Davis and others 1962; Bruce 1970).
Thus, future needs of foresters may well require estimates in these units of measure as
well as the traditional board foot. This paper is concerned solely with tree volume
estimation in the Southwest. Surface area equations for selected species in the same
area were provided by Hann and McKinney (1975).

All of the early volume tables and equations published for exclusive use in
Arizona, New Mexico, or both were for the prediction of unforked, gross Scribner board
foot volume to an 8-inch top. When volume equations were developed, the logarithmic
method described by Schumacicr 274 Hall (1933) was followed (Hornibrook 1936; Lexen and
Thomson 1938a, 1938b; Peterson 1939b, 1939c, 1939d; Lexen and Peterson 1939b, 1939c,
1939d; and Krauch and Peterson 1943). When this method did not work, the alinement chart
method was used (Peterson 1939a; Lexen and Peterson 1939a; and Krauch and Peterson
1943). These early equations and tables were for "blackjack" or immature ponderosa pine
(Hornibrook 1936; Peterson 1939b), ''yellow'' or mature ponderosa pine (Hornibrook 1936;
Peterson 1939c), combined ponderosa pine (Peterson 1939d), white fir (Lexen and Thomson
1938a; Lexen and Peterson 1939a), southwestern white pines (Lexen and Thomson 1938b;
Peterson 1939a), Apache pine (Lexen and Peterson 1939b), Arizona pine (Lexen and Peter-
son 1939c), Chihuahua pine (Lexen and Peterson 1939d), and Douglas-fir (Krauch and
Peterson 1934).

In addition to these, Meyer (1938) published gross volume tables for unforked
ponderosa pine derived from data collected in several western States, including the
Southwest. Volumes were given in three units of measure--total stem cubic foot, Inter-
national 1/8-inch, and Scribner board foot.

More recently, the logarithmic method has been used to develop equations for
unforked, gross Scribner board foot volume to an 8-inch top in white fir (Peterson 1958)
and in Englemann spruce (Peterson 1961), and for unforked, gross cubic foot volume to
both a 4- and an 8-inch top in immature and mature ponderosa pine (Gaines and Peterson
1960). Minor (1961) used weighted, least squares regression to develop an equation for
unforked, gross cubic foot volume to a 4-inch top in ponderosa pine, whereas Myers
(1963) used ordinary, least squares regression with segmented data to develop unforked,
gross volume equations for (a) total stem cubic foot volume, (b) merchantable cubic foot
volume to a 4-inch and variable top limit, and (c) International 1/4-inch and Scribner
board foot volumes to a variable top limit. (Myers used the term "variable top limit"

to indicate that the merchantable top diameter was a fixed function of a tree's diameter
breast height and not to indicate that the top diameter was an independent variable in
the volume equation. The latter type of equation can be converted to the former by
defining the functional relationship between d.b.h. and top diameter.)

The purpose of this study was to develop gross and net tree volume equations for
forked and unforked trees of the major tree species found in Arizona and New Mexico.
Species-National Forest (sp-NF) specific equations were developed for the following
units of measure--total stem cubic foot volume, merchantable cubic foot volume for any
top diameter between 3 and 8 inches, and International 1/4-inch and Scribner board foot
volumes to a 6.0-inch top. The resulting equations and methodology used to derive these
equations are described in this paper. Volume tables for use in the field appear in a
companion piece, ''Comprehensive tree volume equations for major species of New Mexico
and Arizona: II. Tables for unforked trees."

The present study provides a complete set of standardized gross and net volume

equations for the major species, forked or unforked, in Arizona and New Mexico. In
addition, all equations are derived from a common data base. Further, modeling proce-
dures were adopted to ensure that (a) net volumes do not exceed gross volumes,
(b) reasonable board foot-cubic foot ratios prevail, and (c) Scribner board foot volumes
do not exceed International 1/4-inch board foot volumes. Merchantable cubic foot volume
equations for any top diameter between 3 and 8 inches also are provided, thus permitting
users of the equations to adapt to changing merchantability standards. :

SOURCE AND NATURE OF DATA

The data were collected from five National Forests of Arizona and New Mexico by
field crews of the Division of Timber Management, Southwestern Region, Forest Service,
Albuquerque, New Mexico (table 1).

The data for the Coconino, Tonto, and Lincoln Nationai Forests are the same as used
by Hann and McKinney (1975) in the development of surface area equations. On the Santa
Fe and Carson National Forests, the measurement trees were selected by line sampling
with a 40 basal area factor angle. The lines were located and trees designated by
inventory field crews traveling to randomly selected continuous forest inventory plots.
Later, a second crew felled and measured as many of the designated trees as could be
finished in one working day, using procedures similar to those described by Stage and
others (1968).

Tree volumes were computed using a modified version of the NETVSL computer program
(Stage and others 1968). Merchantable volumes were determined to various top diameters
and to a 1-foot stump on all National Forests except the Lincoln, where stump height
was 1.2 feet.

Table 1.--Data sources

National Forest

Specues : Coconino : Tonto : Lincoln : Santa Fe: Carson

Blackjack pine! (Pinus ponderosa

Laws.) xX xX xX X X
Yellow pine! (Pinus ponderosa

Laws.) X xX xX xX X
Douglas-fir (Pseudotsuga menztestt

var. glauca [Mirb.] Franco) Xx x xX x
White fir (Abtes concolor [Gord.

and Glend.]) X X X
Southwestern white pine (Pinus

flextlis var. reflexa Engeln.) 4 x X
Engelmann spruce (Picea

engelmannit Parry) x x x
Aspen (Populus tremulotdes Michx.) é x X
Corkbark fir (Abtes lLastocarpa

var. artzontca [Merriam] Lemn.) X x

ltn the Southwest, blackjack and yellow pines are distinguished by their
bark colors; blackjack pines have very dark bark and yellow pines heavily
Plated, orange bark. This distinction is based on age, growth, and vigor
differences rather than botanical differences, but foresters have found it
useful in forest management planning.

RESULTS

Final equations for predicting the various types of tree volume follow, along with
an explanation of how to apply them. Details concerning the methodology can be found
in the appendixes.

A statistic that will be presented whenever possible is the relative mean-squared
residuals (RMSQR) of the final equation. RMSQR is the quotient of mean-squared resid-
uals divided by the variance of the dependent variable and, as such, is an index of fit
similar to the coefficient of determination (R2). In the case of RMSQR, however,
perfect fit would result in a value of 0. A fit that would not reduce the squared
residuals below the value around the mean would result in a value of 1, and a value
greater than 1 would indicate that the model has increased variability. The advantage
of RMSQR over R* is that only the former will reflect the loss in degrees of freedom
that results from adding another independent variable; therefore, it serves as a better
measure for comparing equations of different numbers of independent variables to see if
the added variable(s) reduced variance about the regression model.

The standard error of the estimate is not reported here because the methodology
used precluded the calculation of a meaningful statistic.

Gross volwnes.--The appropriate equations for computing gross volume are chosen by
the user on a tree-by-tree basis. If the tree is unforked, then the unforked tree vol-
ume equation is chosen based on the sp-NF combination of the tree. If the tree is
forked, the unforked tree volume is first computed for the tree and then corrected for
forking by the appropriate forked tree equation for the sp-NF combination of the tree.
These seemingly trivial facts are brought out because the choice of which net volume
equation to use is not on a tree-by-tree basis, but instead is determined for an entire
data set.

A tree is considered forked if a fork of any severity occurs between breast height
and the tip of the tree. For most of the species, two equations are given. The first
equation utilizes tree size information only, while the second equation incorporates
two independent variables that relate to the severity of forking. In all cases, the
second equation has a lower RMSQR value. The position of the first fork variable, P,
is defined as the height to the first fork divided by totai tree height. The number of
tips variable, T, can be counted directly or computed by adding 1 to the number of
forks. All forks between breast height and the tip of the tree should be counted
regardless of their severity.

Total Stem Gross Cubte Foot Voluwme--Unforked Trees

The total stem gross cubic foot volume of unforked trees is predicted from the
model:
V, = a + a,D*H

where

Predicted total stem gross cubic foot volume of an unforked tree

Diameter at breast height (d.b.h.) in inches

Total tree height in feet.
The sp-NF specific, weighted least squares regression coefficients, ag and a,, are found

in table 2, along with the number of trees used in their development and the resulting
RMSQR. Methodological details can be found in appendix I.

Table 2.--Unforked tree total stem gross cubte foot volwne data base and regression results

: N ae : Relative
, : National : f eee :mean squared
Species of Regression coefficients ;
E Forest SS Grane residual
: (RMSOR)
a9 a)
White pine Lincoln, Santa Fe, Carson 43 0.160888987 0.00203250045 0.0177
Engelmann spruce-corkbark fir Lincoln, Santa Fe, Carson 155 - 225466084 - 00216969983 -0329
Yellow pine Coconino, Tonto, Lincoln, 201 - 237204154 -00221122919 -0292
Santa Fe, Carson
Blackjack pine Coconino, Tonto, Lincoln 680 - 0810724804 - 00198351037 -0266
Blackjack pine Santa Fe, Carson 220 - 0483082948 - 00204968419 -0435
Douglas-fir Lincoln, Tonto 108 - 438373815 - 00175642739 -0162
Douglas-fir Santa Fe, Carson 81 - 341133398 -00191796994 -0368
Aspen Santa Fe, Carson 109 0327 - 00231123522 -0199
White fir Lincoln 43 - 210903832 - 00183995833 20213
White fir Santa Fe, Carson 58 -157776859 - 00200912252 30277

Total Stem Gross Cubte Foot Volumne--Forked Trees

The model for total stem gross cubic foot volume of forked trees is

V =V,x .R
tf t tsf
where
Vy f = Predicted total stem gross cubic foot volume of a forked tree
3
VG = Predicted total stem gross cubic foot volume of an unforked tree

Ry f = Predicted ratio of actual total stem gross cubic foot volume in a forked
sg tree divided by predicted total stem gross cubic foot volume in an unforked
tree of same dimensions.

The sp-NF specific equations for Ry f? their RMSQR, and the number of trees used are
SJ

found in table 3. An examination of RMSQR values indicates that, in some cases, models
are only slightly better than the mean of ratios. Therefore, the mean values for Ry f
3

are also included in table 3 for those who do not feel comfortable using the equations.
Methodological details can be found in appendix II.

Table 3.--Forked tree total stem cubte foot volwne data base and regression results

Independent variables

D = D.b.h. in inches

H = Total tree height in feet
P = Height to first fork divided by H
T = 1 + (number of forks)
: lative
Number Rene
: ; F : : Mean : :mean squared
Species, National Forest, and regression equations of t
7 R, Z a pees residual
, .

: . : : (RMSQR)

White pine on Santa Fe, Carson, and Lincoln

‘ po = 8.72597123E-01 + 2.96964121E-03(H) 1.03680 7, 0.8803
ad
oes 7 0-83242764 + 2.81237312E-05(H2) + 1.67572359E-01 (P) 1.03680 17 .8137
SI
Engelmann spruce-corkbark fir on Santa Fe, Carson, and Lincoln
Ri f = 9.92085233E-01 + 1.87806305 (D/H) 2 1.12229 16 - 7702
sd
mR o = 0.49137954 + 3.73898958E-02(D) - 5.87687943E-05 (H2) + 5.08620456E-01 (P) 1.12229 16 ~5139
ad
Ponderosa pine on Lincoln, Coconino, Tonto, Santa Fe, and Carson
R. f = 9.86799996E-01 + 4.82642048E-01 (D/H) 2 1.02667 158 -9889
>
as 7 0790259661 + 1.24540324E-02(T2) + 9.74033859E-02 (P) 1.02667 158 .9374
a
Douglas-fir on Santa Fe, Carson, Lincoln, and Tonto
R.. p = 9.40292266E-01 + 3.38586950(1/H) 1.00723 23 -9790
ad
Aspen on Santa Fe and Carson
R. f = 9,49182022E-01 + 2.82537481E-05 (H2) 1.02097 37 ~9559
>’
OFR, > = 1.36944003 + 3.87833769E-05(H2) - 5.61174592E-02 (T)
a +3.69440030E-01(P2) - 7.38880060E-01 (F) 1.02097 37. ~8975

White fir on Santa Fe, Carson, and Lincoln
Rep = 9+10060854E-01 + 2.67134214E-01 (H/D2) -97591 13 8752
3

NHS UE EE EEE IEEE ne een nyeInInInEnI III IEI EIEN SSIES SSSEIEESNEENSEERS SNIDER

Merchantable Gross Cubic Foot Volwne--Unforked Trees

By definition, merchantable gross cubic foot volume in an unforked tree is equal
to total stem gross cubic foot volume minus the cubic foot volume in both the top and

the stump (Husch 1963).

where

<
iT]

unforked tree

<<
=
non

This relationship can be written as

Predicted total stem gross cubic foot volume in an unforked tree

Predicted merchantable gross cubic foot volume to a specified top in an

Predicted unmerchantable gross cubic foot volume in an unforked tree (that is,
the gross cubic foot volume in the top and stump of an unforked tree).

Weighted least squares regression was used to develop the following model for unmer-
chantable gross cubic foot volume in an unforked tree:

Dabok.

oO z= o <=
iH}

3
iT]

u

in inches

D3H

V2 bo + by + bab?

D

Total tree height in feet

An sp-NF specific power on d.b.h.

Any specified top diameter between 3 and 8 inches inside bark

Predicted unmerchantable gross cubic foot volume in an unforked tree

The appropriate regression coefficients, their RMSQR, and the number of observa-
tions used can be found in table 4.

of trees used because each tree has an unmerchantable volume for each of thc

diameters computed for the tree. Methodological details can be found in appendix III.

Table 4.--lmforked tree wmerchantable gross cubic foot volwne data base and regression results

The number of observations differs from the number

several top

N = Regression coefficients Relative
Species National of b : b : b :mean squared
Forest : : 0 1 2 residual
observations

: : (RMSOR)

White pine Santa Fe, Carson,
Lincoln 250 -0.213005397 0.00491211378 0.00606061971 0.0229

Engelmann spruce- Santa Fe, Carson,
corkbark fir Lincoln 810 - .266475206 - 00612895037 - 00743111599 0503
Yellow pine All 1,206 -0185465259 .000788175798 .00505513624 1004

Blackjack pine Lincoln, Coconino
Tonto 4,086 -.125349396 - 00360421889 - 00540634204 0591
Blackjack pine Santa Fe, Carson 1232 -.133967845 - 00650174839 - 00490223789 0657
Douglas-fir Lincoln, Tonto 648 - .083148657 -00121904459 - 00541744598 0547
Douglas-fir Santa Fe, Carson 424 - .187630869 - 00671872085 - 00536451038 0580
Aspen Santa Fe, Carson 528 -.236432433 .- 00580208490 - 00608042504 0435
White fir Lincoln 244 -.182699690 - 00124824607 - 00624477874 3681
White fir Santa Fe, Carson 298 -.187563245 - 00632648558 - 00604132385 0293

Merehantable Gross Cubte Foot Volwme--Forked Trees

The equation form for forked tree merchantable gross cubic foot volume is

where
ve f = Predicted merchantable gross cubic foot volvyme of a forked tree
F
= Predicted merchantable gross cubic foot volume of an unforked tree

a f = Predicted ratio of actual merchantable gross cubic foot volume in a forked
° tree divided by predicted merchantable cubic foot volume in an unforked tree
of same dimensions.
The sp-NF specific equations for Ra f? their RMSQR, and the number of observations used
e]

can be found in table 5. For those who do not feel comfortable using these equations,
equations that predict RS f as a function of top diameter alone can also be found in
3

table 5. Methodological details can be found in appendix IV.

Table 5.--Forked tree merchantable gross cubte foot volwmne data base
and regression results

Independent variables
D.b.h. in inches dD, = Top diameter, inside bark, in inches

Total tree height in feet P = Height to first fork divided by H
T = 1+ (number of forks)

: : Relative

8 8 oe :mean squared
Species, National Forest, and regression equations . of - ;

: ‘ : residual

observations

: (RMSOR)
White pine on Lincoln, Santa Fe, and Carson
R = Co + C,H 101 0.9581
m, f

Co 0.61725167 + 0.042365845D__
0.00598501 - 0.00073445D__

i]

Cy

°FR = 1.04518089 - 7.88395198E-07D’ + 8.39079873E-08D8
m,f m m

Engelmann spruce-corkbark fir on Lincoln, Santa Fe, and

Carson
Ra perc oe Cy (D/H) 2 91 .9719
Co = 0.88977465 + 0.01260886D_
C, = 2.02369991 + 0.00029092D_,
ooh ve = 1.09275293 + 1.66824359E-06D7 - 2.22920473E-07D8

Ponderosa pine on Coconino, Tonto, Lincoln, Santa Fe,
and Carson

Raf 7 50 + Cy (D/H) 2 937 9968
Co = 1.025717571 - 0.01509060D_,
C, = 0.13600973 + 0.09709709D_,

Rg 2 = 0.997113216 - 0.923300125D, + CoT? + C1P 937 .9761
Cy = 0.00564984 + 0.00079885D__
C, = 0.002886816 + 0.02330013D_

* Ree = 1.01577735 - 1.68401836E-06D°

Douglas-fir on Lincoln, Santa Fe, and Carson

Ra, = Co + CHO! + Conm!D, 118 .9064
Co = 0.552116849

Ci en@ 2225535301

C2. = 0.284962111

or, 2 = 0.951125601 - 7.62321367E-04D4 + 2.09712628E-03D3-5
Ms | 7 m

Aspen on Santa Fe and Carson

Ron macy & Crh" 184 .9165
altgyy
Co = 1.143728271 - 0.08851433D__
Cy = -0.000006413 + 0.000018827D
or m
R_ .-= 1.07666448 - 1.18133616E-03D2-°
m,; m
White fir on Santa Fe, Carson, and Lincoln
Row = Co # C) (H/D2) 74 . 7286
of et
Co = 0.673711298 + 0.055948391D_
(om) = 1.709640750 - 0.348134538D
or m

R_ » = 0.983293776 - 2.40166266E-05D*

Gross Cubte Foot Stump Volwne--Unforked and Forked Trees

Unmerchantable gross cubic foot volume is composed of both stump volume and top
volume. Information concerning the amount of top wood left after logging can be very
useful for residue and fire assessments. Weighted least squares regression was used
to fit the following gross cubic foot stump volume equation:

= 2
Leen Jo . g,D

where

Predicted gross cubic foot stump volume in an unforked or forked tree

ae |
iT]

B.D.h. In inches,

The appropriate regression coefficients, their RMSQR, and the number of trees used can

be found in table 6. Stump height was 1.2 feet on the Lincoln National Forest, and 1.0
foot on the other National Forests. Methodological details can be found in appendix V.

Equations for predicting d.b.h. from stump diameter have been reported previously by
Hann (1976).

Table 6.--Unforked and forked tree gross cubte foot stwmp volwne data base and regression results

* Number’ Regression Soe Ripoten oes
Species National Forest 3 he 90 3 91 : se
: : (RMSOR)
White pine Santa Fe, Carson 18 -2.20720716E-02 6.02924330E-03 050553
White pine Lincoln 42 -9.81087640E-03 6.92301900E-03 -0159
Engelmann spruce-corkbark fir Santa Fe, Carson 161 -1.26475621E-02 6.47928905E-03 .0658
Yellow pine Santa Fe, Carson, Coconino, Tonto 220 -7.19317815E-02 5.21911020E-03 -0417
Yellow pine Lincoln 11 -8.93559070E-02 6.21218970E-03 03521
Blackjack pine Santa Fe, Carson, Coconino, Tonto 963 -8.24872240E-03 4.87880288E-03 -0505
Blackjack pine Lincoln 57 -3.04689829E-02 6.05850250E-03 .0268
Douglas-fir Santa Fe, Carson, Tonto 115 7.56124690E-04 5.27297220E-03 .0648
Douglas-fir Lincoln 93 -2.21569699E-02 6.09390425E-03 -0311
Aspen Santa Fe, Carson 145 5.55975850E-03 5.30326265E-03 0797
White fir Santa Fe, Carson 66 -6.57525870E-03 5.48812315E-03 -O512
White fir Lincoln 48 -3.25893626E-02 6.52844110E-03 -0171

Gross International 1/4-Inch Board Foot Volume--Unforked Trees

Gross International 1/4-inch board foot volume to a 6-inch top in an unforked tree

is predicted by

<<
"

Mr Nei) 1/6

in an unforked tree

eo)
"

Predicted gross International 1/4-inch board foot volume to a 6-inch top

Predicted gross merchantable cubic foot volume to a 6-inch top in an
unforked tree

Predicted ratio of actual gross International 1/4-inch board foot volume
to a 6-inch top in an unforked tree divided by predicted gross merchant-

able cubic foot volume to a 6-inch top in an unforked tree.

This equation
The resulting

where

Dabeh.

The sp-NF specific, weighted regression coefficients for R

= TR = do = d,D_

I/C

in inches.

Ric?
1

sxaape> = aap

number of trees used can be found in table 7.
is also included to help bridge several gaps where comparable values of RMSQR were not

computed.

Species

White pine

Engelmann spruce-corkbark fir
Yellow pine

Blackjack pine

Blackjack pine

Douglas-fir

Douglas-fir

was fitted using weighted least squares

ratio equations, are of the form:

3

regression through the origin.

, their RMSQR, and the

The coefficient of determination, R%,

Methodological details can be found in appendix VI.

Table 7.--Unforked tree arose International 1/4-inch board foot volume
to a 6-ineh top data base and regression results

National Forest

Santa Fe, Carson,

Santa Fe, Carson,
All
Lincoln, Coconino,
Santa Fe, Carson
Lincoln, Tonto
Santa Fe, Carson
Santa Fe, Carson
Lincoln

Santa Fe, Carson

Lincoln

Tonto

: Number i

of

trees —

Lincoln 39

a

Ww

a

a

fon)

w

- 69196746

- 98736288

-10051404

- 84751736

- 58122078

- 58735344

-59717456

- 68808485

- 24687520

- 73644537

N

_

@

7

1.

ay

-52011366

-84791839

-97921881

- 69491322

-51941410

- 89271640

- 89404735

- 27685063

- 01994022

72093361

Regression coefficients

dz

+ 348366
- 8128080
-556497
~ 377226
097535
+514909
.877967
- 5048044
- 9587285

- 57378982

2,855.

1,423.

00

342464

985244

00

: mean squared :

Relative * Coefficient of

residual esas enc
(RMSOR)

0.0179 ee
.1018 9338
aes .9756
0753 9265
=F 9339
= 9711
0536 9543
0492 9649
0290 Poqae
0488 Koel

a —— TTT. ree

Gross Internattonal 1/4-Inch Board foot Voluwme--Forked Trees

The model for gross International 1/4-inch board foot volume to a 6-inch top for
forked trees is

Vy oe Predicted gross International 1/4-inch board foot volume to a 6-inch top in
' a forked tree

Vy = Predicted gross International 1/4-inch board foot volume to a 6-inch top in
an unforked tree ;

Ry f = Predicted ratio of actual gross International 1/4-inch board foot voiume to
: a 6-inch top in a forked tree divided by predicted gross International 1/4-
inch board foot volume to a 6-inch top in an unforked tree.

The appropriate equations for Ry f? their RMSQR, the number of trees used, and the mean,

ratio Ry ~? can be found in table 8. Methodological details can be found in appendix VII.

-~_

Table 8.--Forked tree gross International 1/4-inch board foot volwne data base and regression results

Independent variables

D = D.b.h. in inches P = Height to first fork divided by H
H = Total tree height in feet T = 1 + (number of forks)
: Mean N ex anes
Species, National Forest, and regression equations r 2 of ; seh
: OR residual

1.f : :
: a : Frees 7 (RMSOR)

White pine on Santa Fe, Carson, and Lincoln

R f = 2.21682508E-01 + 5.95430677E-05(H2) + 2.29900234 (H/D2) 1.16873 17 0.4462
Engelmann spruce-corkbark fir on Santa Fe, Carson, and Lincoln

Rf = 3.70906603E-01 + 7.71762038E+01 (1/D2) 1.23734 16 -4278
OTR, f = -.46864127 + 1.74260009(H/D2) + 1.46864127 (P) 1.23734 16 ~4154
Yellow pine on Lincoln, Coconino, Tonto, Santa Fe, and Carson

Ry f = 1.16393889 - 2.03944633 (D/H) * -95773 31 ~6744
Blackjack pine on Lincoln, Coconino, and Tonto

Ry f* 5.49502730E-01 + 1.09541429E-01 (H/D) - 99776 62 -8895
ae f = .61350474 + 9.60847876E-02(H/D) - 1.52902459E-01(T) -99776 62 .8238

" + 7.72990514E-O1(P) - 3.86495257E-01 (P2)

Blackjack pine on Santa Fe and Carson
R, ¢ = 4-15854140E-01 + 8.37338276E+01 (1/D*) 1.03576 63 +7438
od

Douglas-fir on Santa Fe, Carson, and Lincoln
-1.30844077

Ry ¢ = 5.38416406E+02 (H ) ; 2.12047 20 .7158
od
Aspen on Santa Fe and Carson

Ry ¢ = 5-13375186E-01 + 4.20725160E+01(1/D*) 1.08110 31 -9188

fe

White fir on Santa Fe, Carson, and Lincoln

Ry ge 6.81631222E-01 + 1.35049045E-02(D) -95691 12 .7677
oR, ¢ = 3.05788475E-01 + 2.34528327E-05(H2) + 6.94211525E-01(P) 95691 12 .7643

8

12

Gross Sertbner Board Foot Volume--Unforked Trees

Gross Scribner board foot volume to a 6-inch top in an unforked tree is predicted by

ea ea
where
Vo = Predicted gross Scribner board foot volume to a 6-inch top in an unforked
tree
Vy = Predicted gross International board foot volume to a 6-inch top in an
unforked tree
Rory = Predicted ratio of actual gross Scribner board foot volume to a 6-inch top

in an unforked tree divided by predicted gross International 1/4-inch board
foot volume to a 6-inch top in an unforked tree.

Like International 1/4-inch board foot volume, this equation was fitted using weighted
least squares regression through the origin. The resulting ratio equations, Rei? are
of the form:
-1 -1.177748 =
Rep = 20 - e1D - end Seep
where

D = D.b.h. in-inches. --

The sp-NF specific, weighted regression coefficients for R their RMSQR, and the

S/T’
number of trees used can be found in table 9. Methodological details can be found in
appendix VIII.

Table 9.--Unforked tree cress Scribner board foot volwne to a €-inch top data base and rearession results
: 5 Regression coefficients 4 Relative
a er z mean squared
: : : : : : : z
Species National Forest of eo 2) e2 23 ww
: H 7 Fi : : residual
trees

: : : : : : (RMSQR)

White pine Santa Fe, Carson, Lincoln 32 1.00608608 2.38465985 0.00 0.00 0.0204
Engelmann spruce-corkbark fir Santa Fe, Carson, Lincoln TL -878453705 00 00 15.9984577 - 1638
Yellow pine All 200 - 982101210 -926027395 +00 14.49443523 - 0486
Blackjack pine Lincoln, Coconino, Tonto 402 - 96579222 - 40579028 -00 16.93678414 -1035
Blackjack pine Santa Fe, Carson 168 - 993986685 1.463486622 -00 12.40584877 -1196
Douglas-fir Lincoln, Tonto 83 1.000897473 -00 4.100072359 0.0 +0324
Douglas-fir Santa Fe, Carson 48 -870259997 -00 -00 19.49594193 -0455
Aspen Santa Fe, Carson 38 -887891 00 -00 17.19374 -0831
White fir Lincoln 36 1.0 1.88814412 +00 8.85144911 +0233
White fir Santa Fe, Carson 31 1.01724769 1.87056853 -00 8.51445088 - 0376

13

Gross Sertbner Board Foot Volume--Forked Trees
The model for gross Scribner board foot volume to a 6-inch top for forked trees is

V =V.xR

Sar S ew
where
Sf = Predicted gross Scribner board foot volume to a 6-inch top in a forked tree
Ve = Predicted gross Scribner board foot volume to a 6-inch top in an unforked
tree, and
R = Predicted ratio of actual gross Scribner board foot volume to a 6-inch top

Sf in a forked tree divided by predicted gross Scribner board foot volume to
a 6-inch top in an unforked tree.

The appropriate equations for R their RMSQR, the number of trees used, and the mean

S,f?

ratio, Rg ~? can be found in table 10. Methodological details can be found in appendix
IX. :

Table 10.--Forked tree gross Sertbner board-foot volwne to a 6-ineh top data base and regression results

Independent variables

D = D.b.h. in inches P = Height to first fork divided by H
H = Total tree height in feet T = 1+ (number of forks)
Mean Nummer. 2 Rope
Species, National Forest, and regression equations . . of c aa
: Ro rs 4 trees : residual
: (RMSOR)
White pine on Santa Fe, Carson, and Lincoln
Re f = 6.69426337E-01 + 1.41693541E-02(H) - 3.10631659E-02 (D) 1.00494 14 0.5093
Engelmann spruce-corkbark fir on Santa Fe, Carson, and Lincoln
Re > = 3.27406045E-01 + 9.24980748E+01(1/D2) - 98096 aL -9191
wd
oR, f = -0.53128818 + 2.17203301E-02(D) - 1.53128818 (P*) + 3.06257636 (P) - 98096 TL .- 3868
Yellow pine on Lincoln, Coconino, Tonto, Santa Fe, and Carson
R, f = 1.16596696 - 2.01070651 (D/H) 2 - 96266 31 ~7025
|
Blackjack pine on Santa Fe and Carson
Re f = 4.24761670E-01 + 1.23016529E-01 (H/D) 86497 53 8029
TR. f 7 6+97723772E-01 = 1.79745211E-02(D) + 8.11255589E-05 (H) .86497 53 7426
, + 3.02276228E-01 (P)
Blackjack pine on Lincoln, Coconino, and Tonto
Re f = 1.50684356 - 1.94318670 (D/H) 1.00836 59 - 8484
,
oR. = 5.46618888E-01 + 1.15209503E-01(H/D) - 1.75749455E-01 (T) 1.00836 59 7476
td - 4.53381112E-01 (P*) + 9.06762224E-01 (P)
Douglas-fir on Santa Fe, Carson, and Lincoln
R, ro 6.39052346E-01 + 1.18994308 (H/D*) ; -99510 17 -6622
vr
Aspen on Santa Fe and Carson
Re r = 4.03747500E-01 + 1.17109771E-01 (H/D) 1.01472 pS -7935
J
White fir on Santa Fe, Carson, and Lincoln
Re f = 6.31636481E-01 + 1.50605156E-02(D) - 93862 12 -7142

eee

Net volwmes.--The general form of the net volume equations is the same for all volume
types:

NV = V[1.0 - (PrC) (FC)]

where

NV = Predicted net volume of an unforked or forked tree

V = Predicted gross volume of an unforked or forked tree

PrC = Predicted probability that a tree of given characteristics will have some
cull (that is, the probability that the tree will be unsound)

FC = Predicted fraction cull of a tree given that it does have some cull.

The product (PrC) (FC) estimates the average cull proportion for any tree of given char-
acteristics. The advantage of separating the average cull rate into its components is
the added flexibility it provides when sampling a stand or forest. This method not only
allows the user to predict average cull proportion, but it also allows him to separate
the sampled population into completely sound and unsound trees. For example, suppose a
sample tree, Xs represents n, trees of the same characteristics. The values PrC and

FC are computed for the sample tree's characteristics and then the following estimates
are made:
Number of sound trees = n,(1.0 - Prc)

Number of unsound trees = n-(Prc)

(1.0)
(2.0 =; FC)
n,(1.0 = Pre)

n,(PrC) (1.0 - FC)

x

Net volume in one sound tree
Net volume in one unsound tree
Total net volume in sound trees

ul ANN
<<<
x x

x

Total net volume in unsound trees

where

V = Gross volume of a tree of given characteristics.

It is obvious that this kind of data provides the forest manager with much more infor-
mation concerning the structure of a population's net volume than does average cull
proportion.

As mentioned earlier, for a given species, National Forest, and volume type, the
specific equations chosen for PrC and FC are not determined on a tree-by-tree basis,
but rather by the type of information collected during sampling. In addition to d.b.h.
and total tree height, there are two types of additional information that a user could
collect and use. One is the forking information previously discussed and the other is
information about tree damaging agents. The specific equations chosen will depend upon
whether the user collects all, one, or none of these additional pieces of information.
Once the choice has been made, the equations will then apply to all trees, whether they
be unforked, forked, undamaged, or damaged.

1S

Probability of a Tree Being Unsound in Cubie Foot Volume--Unforked and Forked Trees

The model form for the probability of a tree being unsound in cubic “sot volume to
be applied to both total stem and merchantable predictions is

Pre = 1,0/ (1.0 +e. -)

u~

8

io)
iT}

Predicted probability of a tree being unsound in cubic foot volume

o<
iT]

A function of the tree's measured attributes.

Weighted, nonlinear regression procedures (Hamilton 1974) were used to determine
the equations for Xo The resulting equations, along with the number of trees used, are

presented in table 11. An F-statistic was used to test the significance of the models.
The equation for blackjack pine on the Lincoln National Forest was significant at the
95 percent level, and all other equations were significant at the 99 percent level.
Methodological details can be found in appendix X.

16

D = D.b.h.

Table 11.--Probability of a tree being unsound in
ecubte foot volwne--regresston results

Independent variables

in inches H = total tree height in feet T = 1+ (number of forks)

Species, National Forest, and regression equations

White pine on Santa Fe, Carson, and Lincoln

aS

-2.955536 + 0.05898932H
-3.765521 + 0.05932042H + 0.6089821T

spruce on Santa Fe, Carson, and Lincoln
-1.655268 + 0.001678439DH

= -2.931385 + 0.1798041D + 0.3704742T

Corkbark fir on Santa Fe and Carson

xiai=

Xo =

-2.994364 + 0.3277421D
-5.310682 + 0.3746394D + 1.414157T

Yellow pine on Santa Fe and Carson

Xo =

0.9767974 + 0.03559618D

Yellow pine on Lincoln, Coconino, and Tonto

Xo =
Blackjack
Xo =
Xe =

Blackjack
Xe =

Blackjack
Xo =
Blackjack
Xo =

X —

Cc

0.3547862 + 0.003783644D

pine on Coconino
-5.640547 + 0.04713862H

-6.206244 + 0.04502878H + 0.6055332T

Pine on Lincoln
-2.672635 + 0.05792241H

Pine on Tonto
-4.768802 + 0.2093156D

pine on Santa Fe and Carson
-1.418322 + 0.00006223427D°H

-3.627838 + 0.1540565D + 0.6277967T

Douglas fir on Lincoln and Tonto

Xa =

-1.606318 + 0.0008139147DH

Douglas-fir on Santa Fe and Carson

Xe =

Xo =

=2. 591352 + 0.1359878D
-2.919862 + 0.001301024DH + 0.8573843T

Aspen on Santa Fe and Carson

Xo =

White fir
Xe =

1.202390 + 0.2220774D

on Lincoln
-1.730640 + 0.1221281D

on Santa Fe and Carson
-3.118569 + 0.04665243H

17

:Number of

trees

60
60

136
136

35
35

53.

N79

306
306

oy)

380

283
283

114

96
96

146

48

66

Fraction Cull tn Total Stem Cubie Foot Volume Given the Tree ts Unsound--Unforked and
Forked Trees

To predict the fraction of an unsound tree's total stem cubic foot volume that is
in cull, the following functional relationship is used:

a
FC, = (1.0 - e *)

where
FC, = Predicted fraction cull in total stem cubic foot volume given the tree is
unsound
h, = a constant
Y, =A function of the tree's measured attributes.

t

The sp-NF specific values and functions of h, and Y,, respectively, the number of trees

t?
used, and the resulting RMSQR values are found in table 12. An examination of the RMSQR
values indicate that many of the equations might not reduce the squared deviations more
than a mean value of FC,. This is due both to the general difficulty of modeling cull
rates and to the relatively small number of sample trees with cull. It is felt,
however, that the equations do represent cull trends in at least the sample trees and
that they also seem to behave reasonably well. It is recommended, therefore, that the
equations be used but, for those who choose not to do so, mean values of FC, have also

been included in table 12. Methodological details can be found in appendix XI.

Table 12.--Fraction cull in total stem cubte foot volwne, given the tree
ts unsound, data base and regression results

Independent variables

D = D.b.h. in inches H = Total tree height in feet
: ; Re v
: Mean 3M ear m ee d
Species, National Forest, and regression results a 7 of 2 een ava’
FC}, : residual

Hy t :
: frees; _(RMSOR)
White pine on Santa Fe, Carson, and Lincoln

h, = 0.573840391

Y. = (-7.66323806E-04) D2 0.14358 34 0.8358

Engelmann spruce on Santa Fe, Carson, and Lincoln
h, = 0.938462956

Y, = -3.01894250E-01 + 4.30560743E-03(H) -13981 44 -8359
-6.43668344E-04 (D2)

Corkbark fir on Santa Fe and Carson
h, = 0.961637598

Y. = 1.36557290E-0l - 5.67626653E-02 So +09807 21 -8472

Yellow pine on Lincoln, Coconino, Tonto, Santa Fe, and Carson

a, = 0.915201231

Y, = (-1.63671565E-01) + (3.25503506E-05) DH +09303 154 -9716

+
Blackjack pine on Lincoln, Coconino, and Tonto

h, = 0.970852066

Y, = (-9.71749091E-03) - (6.33508782E-07)D“H -01878 80 +9364
Blackjack pine on Santa Fe and Carson

h, = 0.959154988

Y, = (-2.67485517E-02) - (7.47306053E-02)H/D2 - 04676 90 -9727

Douglas-fir on Santa Fe, Carson, Lincoln, and Tonto

h, = 0.236595402

Y, = (-2.74212212E-01) - (6.75308810E-06)D7H 08489 66 -9818

Aspen on Santa Fe and Carson
h, = 0.885502084

Y. = (-2.1547562E-01) - (9.32233189E-03)D* + (1.31076200E-03)DH . 28482 139 7441

White fir on Santa Fe, Carson, and Lincoln
h, = 0.440697488

Y* (-1.21221834E-03) D2 .15895 51 .7790

ee

18

Fractton Cull tn Merchantable Cubic Foot Volume Given the Tree Is Unsound--Unforked and
Forked Trees

The following model form is used to predict the fraction of an unsound tree's mer-
chantable cubic foot volume that is cull:

Y
m

cam = h (1.0 -e )
where -
FC. = Predicted fraction cull in merchantable cubic foot volume to a specified top
diameter given the tree is unsound
h, = A constant
se = A function of the tree's attributes and a specified top diameter between 3

and 8 inches, inside bark.

The appropriate values of he and functions for 1 the number of observations used, and

the resulting RMSQR values are found in table 13. While these models are recommended
for use, equations predicting mean EC as a function of top diameter are also included

in table 13 for those who choose not to use the FC equations. Methodological details
can be found in appendix XII.

19

Table 13.--Fraction cull in merchantable cubie foot volwne, given
the tree ts unsound, data base, and regression results

Independent variables
D = D.b.h. in inches D. = Top diameter, inside bark, in inches

H = Total tree height in feet

Relati

: y en :mean s cue

Species, National Forest, and regression equations : of 5 fe
residual

“observations *

: (RMSQR)

White pine on Santa Fe, Carson, and Lincoln

h_ = 0.525456732
ae ae Ys
¥, = dy 204 0.8386
d, = -7.62194840E-04 - 1.72E-10D’
or m
FC, = 0.139967204 + 1.40417133E-07D>

Engelmann spruce on Santa Fe, Carson, and Lincoln
h, = 0.809722162

Si d,p* 246 .8881
Jog = 74.79663334E-01

J, = 5.99578762E-03 is

jy = -3-54716909E-04 ~ 2.30591722E-0SD_,

FC, = 1.10092261E-01 + 4.70167336E-04D*

-8.80762813E-05D° + 4.83607897E-07D/

or

Corkbark fir on Santa Fe and Carson

h 0.945856475

5 Fe = Jg + J,H/D 120 -8010

Jo = -2.04082904E-02 + 4.51477415E-02D_,
J, = -1.16845115E-02 - 1.29460840E-02D__

FC, = 7.39331940E-02 + 6. 95210968E-03D_,
Yellow pine on Lincoln, Coconino, Tonto, Santa Fe,
and Carson

h = 0.912935298

or

ad J + j ,DH 924 - 9696
jg = -1-67513064E-01 - 2.39E-07D>
J, = 3.368982527E-05 + 8.684E-11D°

or m

FC. = 9.41334189E-02 + 4.35158264E-06D?

Blackjack pine on Lincoln, Coconino, and Tonto
h 0.294142214

t= J. Feo 475 .9351

Jo = -3.39989415E-02 + 1.48516345E-04D*
= -2.19796372E-06 - 8.55224950E-09D2

m
FC. = 1.92501222E-02

(con. )

20

Table 13.--(con.)

Species, National Forest, and regression equations

Blackjack pine on Santa Fe and Carson
h = 0.943058463

Y = jo + gj H/D2
Jo = -1.80176132E-02 - 1.626286 43E-06D°
+ 2.00018357E-07D"

Jj, = -9.68124280E-02 + 6. 49858343E-06D®
-8.00723544E-07D7
or Hi
FC,, = 4.19699099E-02 + 1.20336014E-05D?

-3.54771919E-O6D> + 2.55488876E-07D’

Douglas-fir on Santa Fe, Lincoln, and Tonto
ie = 0;.2433921.21
= . @)
Je On tay OSH
Jo = —2.80555445E-01 - 1.2792366E-06D>

" 1.24888494E-07D’
= -5.87585480E-06 + 8.02594124E-11D*

6x!
FC = 7.88426507E-02 + 4.88150784E-05D?

Aspen on Santa Fe and Carson
0. 881568806

ay
i}

a o. 5 2: <
j, = 1.77859273 - 1.93330892D9.5
0 m

+ 4.42422923E-O1D,,

J, = 79-36182562E-03 + 2.23064483E-08Dé
-3.90157892E-09D,,

ord> = 1+38034680E-03 + 9.42523449E-11D/

FC, = 2.87277280E-01 + 5 .42205159E-04D2- >

White fir on Santa Fe, Carson, and Lincoln

hm = 0.438946854
Re 200
Ym = J 1D
Jy = -1.211898445E-03 - 9.066E-09D*
or
PC = 1.54156056E-Ol + 8. 17555326E-06D*

8 se Relative
:mean squared

Of ;
‘observations’ Bopeduel
o : (RMSOR)
532 0.9864
390 -9785
682 - 7639
301 - 8046

Probability of a Tree Betng Unsound tn Board Foot Volume--Unforked and Forked Trees

The probability of a tree being unsound in board foot volume, to be applied to both
International 1/4-inch and Scribner board foot predictions, is predicted by

-X,
Pec, = 1.0/0.0 #e.*)

vu

4

(a)
"

B Predicted probability of a-tree being unsound in board foot volume

os
iT]

B A function of d.b.h., total tree height, number of tree tips, and the
presence or absence of significant damage.

The equations for X, were also fitted using weighted, nonlinear regression procedures,
and the results are given in table 14, along with the number of trees used. Again, the
equation for blackjack pine on the Lincoln National Forest proved to be significant at
the 95 percent level, whereas all the other equations were significant at the 99 percent
level.

Two damaging agents, sweep (or crook) and porcupine, proved to be useful in predict-
ing whether a tree was sound or not. A damaging agent is recorded only if it is severe
enough to (1) prevent the tree from surviving; (2) preclude the production of a market-
able product; or (3) diminish the quality or quantity of that product. Damage is
entered into the equations through the usage of dummy variable(s). If the tree has the
particular damage, the dummy variable is set to 1.0, otherwise it is 0. Only one damag-
ing agent is recorded per tree, and, therefore, equations with two or more dummy variables
in them would have at most only one of them set to 1.0 and the rest to 0. Methodological
details can be found in appendix XIII.

22

D = D.b.h. in inches
S = 1.0 Tree with sweep or crook Pd = 1.0
0.0 Tree with no sweep or crook = 0.0

Table 14.--Probability of a tree betng unsound in
board foot volune--regresston results

Independent variables

Species, National Forest, and regression equations

White pine on Santa Fe, Carson, and Lincoln

x, =
Engelmann

Xe =
or

x, =

-4.421720 + 0.1001612H

spruce on Santa Fe, Carson, and Lincoln
-2.057921 + 0.1664132D

-2.412705 + 0.1689685D + 0.2973478T

Corkbark fir on Santa Fe and Carson

xX =
B

-2.724328 + 0.3295897D

Yellow pine on Santa Fe and Carson

xX =
B

1.416597 + 0.08953296D

Yellow pine on Lincoln, Coconino, and Tonto

x, =
Blackjack
».4 —
B
Blackjack
Xo =
Blackjack
x, =
Blackjack
or Xe =
or XB 7
or *B
x =
B
Douglas fi
x =
Douglas-fi
or XB 7
Xe =
Aspen on S
x =
B
White fir
X =
B
White fir
x =
B 2
x, s.

-0.3255527 + 0.09002528D

pine on Coconino
-3.790069 + 0.002540676DH

Pine on Lincoln
-1.195571 + 0.03548680H

Pine on Tonto
-4.866131 + 0.06649043H

Pine on Santa Fe and Carson
-3.023414 + 0.05290037H

=3-5551965 + 0.1806999D + 0.6451722T

= -3.650835 + 0.06016044H + 1.278169(Pd) + 1.7851585S

-3.625993 + 0.1844583D + 0.5517561T + 1.474701S

r on Lincoln and Tonto
-4.201783 + 0.3419626D

r on Santa Fe and Carson
-1.185559 + 0.0001857571D°H

-1.363273 + 0.0001709440D2H + 2.3055845

anta Fe and Carson
-3.889285 + 0.9000166D

on Lincoln
-2.260617 + 0.05019542H

on Santa Fe and Carson
-955732 + 0.2320741D

-3.023744 + 0.2225154D + 1.2603675

5)

H = total tree height in feet T = 1 + (number of forks)
Tree with porcupine damage
Tree with no porcupine damage

:Number of
trees

56

103

103

3

53

179

260

49

320

27.

271.

27k

271:

106

84

84

119

45

54

54

Fractton Cull tn Internattonal 1/4-Inch Board Foot Volwne Given the Tree Is Unsound--
Unforked and Forked Trees

To predict the fraction of an unsound tree's International 1/4-inch board foot vol-
ume that is in cull, the following model is used:

Mt
FC, = hy (e )
where
FC, = Predicted fraction cull in International 1/4-inch board foot volume to a
6-inch top given the tree is unsound
hy = A constant
Yr = A function of the tree's attributes.

The sp-NF specific values and functions of #, and Y_, respectively, the number of trees
used, and the resulting RMSQR values are found in table 15. The models for Engelmann
spruce and corkbark fir were too poor to be useful and, therefore, only their mean FC
values are included. Mean values of FC, for the other sp-NF combinations are also
found in table 15. Methodological details can be found in appendix XIV.

Table 15.--Fraction cull in Internattonal 1/4-ineh board foot volume,
given the tree ts unsound, data base, and regression results

Independent variables

D =D.b.h. in inches H = Total tree height in feet
: A Relative
Number
: F ; : Mean A : mean squared
Species, National Forest, and regression results of 5
7 FC, Be es residual

: : : (RMSQR)

White pine on Santa Fe, Carson, and Lincoln
hy = 0.181977197

ay = 9.53653005E-04 (D<) 0.30592 39 0.9776

Engelmann spruce on Santa Fe, Carson, and Lincoln

No model--Use mean FC, - 41094 45 So

Corkbark fir on Santa Fe and Carson
No model--Use mean FC, -55402 2 --

Yellow pine on Santa Fe, Carson, Lincoln, Coconino, and Tonto
hy = 1.05452734

Y_ = -1.89706097E-92(H) - 28490 202 -9199

Blackjack pine on Lincoln, Coconino, and Tonto
hy = 0.420800123

Y_ = -1.74436591E-02(H) - 15676 138 - 9430

Blackjack pine on Santa Fe and Carson

hy = 0.467888236

yy = -1.89407426E-01 (H/D) -22145 114 -9992

Douglas-fir on Santa Fe, Carson, Lincoln, and Tonto

ny = 0.839814911

Y, = -2.92010520E-01(H/D) - 26258 116 -9479
Aspen on Santa Fe and Carson

hy = 1.39801055

Ys = -8.79518535E-03(H) -89429 114 - 9364
White fir on Santa Fe, Carson, and Lincoln

A_ = 0.251235533

YS = 5.01411097E-06 (DH) - 34497 62 9972

Fraction Cull tn Sertbner Board Foot Volwne Gtven the Tree Is Unsound--Unforked and
Forked Trees

The following model form is used to predict the fraction of an unsound tree's
Scribner board foot volume that is cull:

Ys
FC, = he )
where
FC, = Predicted fraction cull in Scribner board foot volume to a 6-inch top given
the tree is unsound
he = A constant
Yo = A function of the tree's attributes.

The appropriate values of he and functions for Y,, respectively, the number of trees

SZ
used, and the resulting RMSQR values are found in table 16. Again, the models for

Engelmann spruce and corkbark fir were too poor to be useful and, therefore, only their
mean FC, values are included. Several other models proved to be no better than the mean

so the user may also want to use the mean FC. values in table 16 rather than the models.

Methodological details can be found in appendix XV.

Table 16.--Fractton cull tin Scribner board foot volwne, gtven the tree
ts unsound, data base, and regresston results

Independent variables

D = D.b.h. in inches H = Total tree height in feet
Mean N i ecetne
Species, National Forest, and regression results ¥ of i ae
: FC. caper prt residual
: (RMSQOR)
White pine on Santa Fe, Carson, and Lincoln
he = 0.245464500
Y = 9.55078847E-04 (D2) 0.41683 39 1.0027
Engelmann spruce on Santa Fe, Carson, and Lincoln
No model--Use mean FC. -42864 45 --
Corkbark fir on Santa Fe and Carson
No model--Use mean FC. -57057 21 --
Yellow pine on Santa Fe, Carson, Lincoln, Coconino, and Tonto
he = 1.38266375
Yo = -1.94422273E-02(H) - 36288 202 -9028
Blackjack pine on Lincoln, Coconino, and Tonto
he = 0.617372520
YX = -1.96681561E-02 (H) - 20529 138 ~9310
Blackjack pine on Santa Fe and Carson
he = 0.646540139
Y = -2.20260760E-01 (H/D) - 27296 114 1.0062
Douglas-fir on Santa Fe, Carson, Lincoln, and Tonto
he = 1.24325723
Yy = -3.45196887E-01 (H/D) -31653 116 -9126
Aspen on Santa Fe and Carson
he = 1.23056500
X, = -5.46784441E-03(H) -93059 114 -9904
White fir on Santa Fe, Carson, and Lincoln
he = 0.294076429
Yo = 4.60611418E-06 (DH) - 39036 62 1.0026

25

DISCUSSION AND CONCLUSIONS

A summary of the methodology for predicting the various volume types is given in
table 17. The advantage of this approach to developing volume equations is that it
avoids or eliminates many of the problems of consistency, compatibility, reliability,
and flexibility found with previous sets of volume equations. The disadvantage is that
the method does not lend itself well to hand computations of tree volumes. To minimize
this problem, we have prepared volume tables for unforked trees (''Comprehensive tree
volume equations for major species of New Mexico and Arizona: II. Tables for unforked
trees''). For those with access to a computer, a FORTRAN subroutine has been written
and can be obtained in versions for the CDC 6400 or the UNIVAC F108 by writing:

Southwest Volume Subroutine
Forest and Range Evaluation Unit
z Intermountain Forest and Range Experiment Station
507 = 25th Street
Ogden, Utah 84401

Table 17.--Swnmary of methodology for predicting various tree volwne types

Predicted value : Method for prediction

Total stem gross cubic foot volume, unforked trees Ve
Total stem gross cubic foot volume, forked trees Vv =V, xR, >

tsf t ts]
Merchantable gross cubic foot volume, unforked trees oe = v, - +
Merchantable gross cubic foot volume, forked trees Vv =v xR.

m, f mi Ms]
Gross International 1/4-inch board foot volume, unforked trees Vy = WG") x Rive
Gross International 1/4-inch board foot volume, forked trees v, f = Vy x Ry ¢

’ ad
Gross Scribner board foot volume, unforked trees Ve = vy x Rot

i Vv =v R

Gross Scribner board foot volume, forked trees s,f s x s,f
Total stem net cubic foot volume, unforked trees NV, = v, x [1 - (Prc, x FC,)]
Total stem net cubic foot volume, forked trees NV =V2 p X —veanre:. x FC)

ty tJ ec t
Merchantable net cubic foot volume, unforked trees NV, = Vm x ie= (Prc, x Fo ))
Merchantable net cubic foot volume, forked trees Win, f = Vn f x {1 - (Prc, x Fc]
Net International 1/4-inch board-foot volume, unforked trees NV = Vy x (1 - (Prc,, x FC,)]
Net International 1/4-inch board foot volume, forked trees NV, f = Vy f x. [i= (Prc,, x FC)]

, ,

Net Scribner board foot volume, unforked trees NV, = Vs x [l - (Pre, x FC.)]
Net Scribner board foot volume, forked trees NV =v x (lL .=2(Prei x: FOX)

s,f S,f B s

PUBLICATIONS CITED

Assman, Ernst.
1970. The principles of forest yield study. Translated by Sabine H. Gardiner.
506 p., Pergamon Press, New York.
Avery ,«-li., Eugene:
1967. Forest measurements. 290 p. McGraw-Hill Book Co., San Francisco.
Bruce, David.

1970. Predicting product recovery from logs and trees. USDA For. Serv. Res. Pap.

PNW-107, 15 p. Pac. Northwest For. and Range Exp. Stn., Portland, Oreg.
Cunia., 1.

1964. Weighted least squares method and construction of volume tables. For. Sci.
10:180-191.

Davis, Kenneth P., Philip A. Briegleb, John Fedkiw, and L. R. Grosenbaugh.

1962. Determination of allowable annual timber cut on forty-two western National
Forests.: *‘USDA For. Serv., Washington, D.C. Board Rev. Rep. M-1299, 38 p.

Gaines, Edward M., and Geraldine Peterson.

1960. Cubic-foot volume tables for southwestern ponderosa pine. USDA For. Serv.,

Res: “Pap? 50; 1S ps "Rocky "Mt. For. and’ ‘Range’ Exp.’ 'Stn.., Fort Collins, Colo.
Gray, H. R.

1956. The form and taper of forest-tree stems. Imperial For. Inst. Pap. 32, 79 p.

Univ. Oxford, Great Britain.
Grosenbaugh, L. R.

1954. New tree measurement concepts: height accumulation, giant tree, taper and
shape’. . -USDA For Serv., South. *For. ‘ard ‘Range: Exp. Stn. Oceas.. Pap. 134, 32 p.
New Orleans, La.

Grosenbaugh, L. R.

1967. REX-FORTRAN-4 system for combinatorial screening or conventional analysis of
multivariate regressions. USDA For. Serv. Res. Pap. PSW-44, 47 p. Pac. Southwest
For. and Range Exp. Stn., Berkeley, Calif.

Hamilton, David A., Jr.,

1974. Event probabilities estimated by regressions. USDA For. Serv. Res. Pap.

INT-152, 18 p. Intermt. For. and Range Exp. Stn., Ogden, Utah.
Hann, David W.

1976. Relationship of stump diameter to diameter at breast height for seven tree
species in Arizona and New Mexico. USDA For. Serv. Res. Note INT-212, 16 p.
Intermt. For. and Range Exp. Stn., Ogden, Utah.

Hann, David W., and Robert K. McKinney, Jr.
1975. Stem surface area equations for four tree species of New Mexico and Arizona.

USDA For. Serv. Res. Note INT-190, 7 p. Intermt. For. and Range Exp. Stn., Ogden,
Utah.

Hornibrook, Ezra M.

1936. Scribner volume tables for cut-over stands of ponderosa pine in Arizona. J.
Agric. Res. 52:961-974.
Husch, Bertram.

1963. Forest mensuration and statistics. 474 p. The Ronald Press Co., New York.

27

Jensen, Chester E.

1964. Algebraic description of forms in space. 57 p. USDA For. Serv., Cent. States
For. Exp. Stn., Columbus, Ohio.

Jensen, Chester E.

1973. MATCHACURVE-3: Multiple-component and multidimensional mathematical models for
natural resource studies. USDA For. Serv. Res. Pap. INT-146, 42 p. Intermt. For.
and Range Exp. Stn., Ogden, Utah.

Jensen, Chester E.

1976. MATCHACURVE-4: Segmented mathematical descriptors for asymmetric curve forms.
USDA For. Serv. Res. Pap. INT-182, 16 p. Intermt. For. and Range Exp. Stn., Ogden,
Utah.

Jensen, Chester E., and Jack W. Homeyer.

1970. MATCHACURVE-1 for algebraic transforms to describe sigmoid- or bell-shaped

curves. 22 p. USDA For. Serv., Intermt. For. and Range Exp. Stn., Ogden, Utah.
Jensen, Chester E., and Jack W. Homeyer.

1971. MATCHACURVE-2 for algebraic transforms to describe:curves of the class Xo
USDA For. Serv. Res. Pap. INT-106, 39 p. Intermt. For. and Range Exp. Stn., Ogden,
Utah.

Kmenta, Jan.
1971. Elements of econometrics. 655 p. The MacMillan Co., New York.
Krauch, Hermann, and Geraldine Peterson.

1943. Two new board-foot volume tables for Douglas-fir. USDA For. Serv., Res. Note

107,-7-p. Southwest. For. and Range Exp. Stn., Tucson, Ariz.
Lexen, Bert R., and Geraldine Peterson.

1939a. Total height volume table for white fir. USDA For. Serv., Res. Note 54, 2 p.

Southwest For. and Range Exp. Stn., Tucson, Ariz.
Lexen, Bert R., and Geraldine Peterson.

1939b. Merchantable height volume table for Apache pine. USDA For. Serv., Res. Note

55, 2p. Southwest. For. and Range Exp.. Stn, ,. Tucson,.Ariz.
Lexen, Bert R., and Geraldine Peterson.

1939c. Merchantable height volume table for Arizona pine. USDA For. Serv., Res. Note

56, 2 p. Southwest. For. and Range Exp. Stn., Tucson, Ariz.
Lexen, Bert R., and Geraldine Peterson.

1939d. Merchantable height volume table for Chihuahua pine. USDA For. Serv., Res.

Note 57, 2 p. Southwest. For and Range Exp. Stn., Tucson, Ariz.
Lexen, Bert R., and Walter G. Thomson.

1938a. White fir merchantable height volume table. USDA For. Serv., Res. Note 27,

2p. Southwest. For. and Range Exp. Stn., Tucson, Ariz.
Lexen, Bert R., and Walter G. Thomson.

1938b. Merchantable height volume table for the southwestern white pines (Pinus
strobtformis and Pinus flexilts). USDA For. Serv., Res. Note 28, 2 p. Southwest.
For. and Range Exp. Stn., Tucson Ariz.

Meyer, Walter H.

1938. Yield of even-aged stands of ponderosa pine. U.S. Dep. Agric. USDA Tech.

Bull. 630, 59 p. Washington, D.C.
Minor, Charles O.

1961. Pulpwood volume tables for ponderosa pine in Arizona. USDA For. Serv., Res.

Note 69, 6 p. Rocky Mt. For. and Range Exp. Stn., Fort Collins, Colo.
Myers, Clifford A. ,

1963. Volume, taper and related tables for southwestern ponderosa pine. Revised
1972. USDA For. Serv. Res. Pap. RM-2, 24 p. Rocky Mt. For. and Range Exp. Stn.,
Fort Collins, Colo.

Peterson, Geraldine.

1939a. Total height volume table for the southwestern white pines. USDA For. Serv.,

Res. Note 53, 2 p. Southwest. For. and Range Exp. Stn., Tucson, Ariz.
Peterson, Geraldine.

1939b. Merchantable height volume table for immature ponderosa pine (Pinus ponderosa
Laws.). USDA For. Serv., Res. Note 73, 2 p. Southwest. For. and Range Exp. Stn.,
Tucson, Ariz.

=

Peterson, Geraldine.
1939c. Merchantable height volume table for mature ponderosa pine (Pinus ponderosa

Laws.). USDA For. Serv., Res. Note 74, 2 p. Southwest. For. and Range Exp. Stn.,
Tucson, Ariz.
Peterson, Geraldine.
1939d. Merchantable height volume table for ponderosa pine (Pinus ponderosa Laws.).
USDA For. Serv. , Res: Note’ 75,02 p.' ¢Southwest.. For.. and Range Exp. Stn., Tucson,
Ariz.
Peterson, Geraldine.
1958. Board-foot volumes of white fir to an 8-inch top. USDA For. Serv., Res. Note
50; 2>p.. Rocky Mt. For. “and: Range!'Exp. Stn., Fort Collins, Colo.
Peterson, Geraldine.
1961. Board-foot volumes of Engelmann spruce to an 8-inch top. USDA For. Serv., Res.
Note 56, 2p. Rocky Mt. For. and Range Exp. Stn., Fort Collins, Colo.
Schumacher, Francis X., and Francisco dos Santos Hall.
1933. Logarithmic expression of timber-tree volume. J. Agric. Res. 47:719-734.
Stage, Albert R., Richard C. Dodge, and James E. Brickell.
1968. NETVSL--a computer program for calculation of tree volumes with interior
defect. USDA For. Serv. Res. Pap. INT-51, 30 p. Intermt. For. and Range Exp. Stn.,
Ogden, Utah.

29

APPENDIXES

St

Details of Methodology
I. Total Stem Gross Cubie Foot Volwne--Unforked Trees
The basic model of
Vor, Con? a,D*H (1)

where

<
"

ca Predicted total stem gross cubic foot volume in an unforked tree

Die behis, in inehes

= Total tree height in feet

has been used for years for predicting total stem gross cubic foot volume (Husch 1963).
Weighting this model by

Wee (D2H) (2)

was suggested by Cunia (1964) as a means for homogenizing the variance about regression.
This model and weighting procedure was applied to each sp-NF combination and a plot of
residuals indicated that the variance was homogenized by the weighting. However,
Pearson's beta statistics computed for each equation indicated that the residuals were
not normally distributed. This problem precluded the usage of this model for testing
the possibility of combining species across National Forests or of combining some of
the species with others, using analysis of covariance.

It was then hypothesized that the following mocel might eliminate the problem:
In(V,) = Ag + @,ln(H) + agin(D). (3)

The plots of residuals again confirmed that variance was homogenized and Pearson's beta
statistics this time indicated that the residuals were closer to being normally distri-
buted. Analysis of covariance was then applied to see if species could be combined
across forests. The results indicated that blackjack pine, Douglas-fir, and white fir
on the Santa Fe and Carson National Forests could not be combined with the same species
on the other National Forests. All other species were combined across Forests.

Only two "'species'’ combinations were tested. One was whether yellow pine and
blackjack pine could be combined. Results of the testing indicated that they could not
be combined, a conclusion supported by the findings of Hornibrook (1936) and Myers
(1963). The other combination tested was Engelmann spruce with corkbark fir. This
test was made because of the small data set for corkbark fir. This test resulted in
the combining of Engelmann spruce with corkbark fir.

These final data sets were fitted to equation (1) using weighted least squares
regression. For aspen, this resulted in a negative intercept. Therefore, an intercept
value of 0.0327 (the volume of a tree 2 inches in diameter at the root collar and 4.5
feet tall, assuming conical shape) was forced on the equation instead of forcing it
through the origin.

All final equations were visually checked by plotting the predicted and actual
volumes over diameter by height classes.

52

II. Total Stem Gross Cubic Foot Voluwmne--Forked Trees

The ratios of actual forked volume divided by predicted unforked volume were
plotted over d.b.h. (D), total tree height (H), position to fork (P), and number of tips
(T) and examined for trends. From this examination, a set of 12 potential transforma-
tions on D, H, P, and T were developed and all possible combinations of three or less
were screened through program REX (Grosenbaugh 1967). Selected models from this screen-
ing run were then tabled and the one with the lowest RMSQR that "behaved well'' was
picked as the final model. (An equation was judged to "behave well" if it did not
produce unrealistic results over the expected range of usage and if it did not exhibit
an undulating behavior for different combinations of tree characteristics.) Yellow pine
was combined with blackjack pine, and white fir and Douglas-fir were pooled across
National Forests because of the need to strengthen the data sets.

~

III. Merechantable Gross Cubie Foot Volume--Unforked Trees

Expressing merchantable gross cubic foot volume as the difference between total
stem gross cubic foot volume and the unmerchantable gross cubic foot volume in the top
and stump is a convenient technique for examining and then characterizing the effect of
changing top diameter upon merchantable gross cubic foot volume. To see this advantage,
consider the following:

Unmerchantable Volume = (Top Volume) + (Stump Volume)

Zi kx ) (Toe peed es tenet
4 x 144 in Inches in Feet

i eee (Stump bianetes|”(SHimp Honen (4)
4 x 144 in Inches in Feet

where

k = A form factor.
It has been shown that stump diameter and d.b.h. are highly correlated (Hann 1976) and
it is also true that stump height is basically a constant; therefore, the second term
of equation (4) can be expressed as

ag + aD", (S)

For top volume, the top diameter is specified by the user; therefore, the unknown quan-
tities are top length and the form factor, kK. To help determine the relationship of top
length to d.b.h. and total tree height, the following model was fitted for each top
diameter:

1In(TL) = ao + a ,1n(D) a az 1n(H) (6)

where

TL = Top length in feet.

33

By examining the behavior of the coefficients across the various top diameter, it was
found that the model

E: H
TL = ain (7)
D
where
Dd, = Top diameter inside bark, in inches
m= "AN “Sp=NF ‘specirti¢e value’ oF 1.0) IS “or 220

was applicable. This model assumes that the top can be approximated by a conic form
(Gray 1956; Husch 1963; and Assman 1970). The power on d.b.h., m, can be thought of as
determining the diameter of the top cone projected to ground level. This is because
total tree height is measured as the distance from the ground level to the tip of the
tree and consequently the relationship

n (8)

will hold for similar triangles only if D” is directly proportional to diameter at
ground level.

With this background as a basis, the model for unmerchantable volume is therefore:.

Li 3_H 2
Mi bo + b\D- x + boD (9)
where
vs = Gross cubic foot volume in the top and stump for a specified top diameter.

Analysis of residuals indicated that weighting was necessary to homogenize variance.
A procedure described by Hann and McKinney (1975) was therefore used to obtain the
weight:

D> H]-1 yp
W = = = (10)
D D3? H
m

Weighted least squares regression was used on each sp-NF combination to obtain the
regression coefficients for n equal to 1.0, 1.5, and 2.0. The final regression equation
was that which minimized RMSQR. Further refinement for a value of m was not necessary
because of the small differences in the RMSQR values of the three values tested. The
final merchantable gross cubic foot volume equations were visually checked by plotting
the data points versus representative curves from the equations over diameter by height
classes.

A word of caution concerning the interpretation of the terms in this model is
appropriate. While the logic behind the two components is clear, the final equations
cannot be divided into components because of multicollinearity between the two (Kmenta

'1971). This can be seen by comparing the second component of the unmerchantable model
to the model derived separately for stump volume. The coefficients are in the same
proximity but are not equal. However, the presence of multicollinearity, so long as it
is not perfect, does not bias the estimation of unmerchantable volume (Kmenta 1971).

34

IV. Merchantable Gross Cubte Foot Volwne--Forked Trees

To maintain consistency across top diameters, the same model used in predicting
total stem gross cubic foot volume was fitted to each top diameter data set. The
resulting regression coefficients were examined for trends over top diameter. From this
examination, the regression coefficients were modeled as linear functions of top diam-
eter. The basic model with the coefficients as functions of top diameter was then
fitted to the whole data set for a final overall slope correction on the model. Models
with forking variables that proved not to be better than models without forking vari-
ables were dropped.

Because the RMSQR values of the models were high, it was decided to provide an
optional model that would predict the mean ratio as a function of just top diameter.
To do this, the mean ratio values for the six top diameters were weighted by their
respective number of observations and then screened against 12 power transformations
on top diameter. Selected models were then examined for reasonableness of behavior and
the one that behaved both reasonably and minimized RMSQR was picked as the final model.

V. Gross Cubte Foot Stwnp Volwne--Unforked and Forked Trees
The basic model:

Lee = ag + a,D? (11)

where

ee = Gross volume of the stump in cubic feet
was fitted to the separate sp-NF combinations. The residuals were plotted and they
revealed that the basic model was appropriate but that the variance increased with the
square of D*. Therefore, the model was refitted using a weight of

W=D". (12)
The residuals were plotted and this time they were homogeneous.

Analysis of covariance was then used to determine if species could be combined
across all National Forests except the Lincoln, where stump height was 1.2 feet instead
of the standard of 1.0 foot. The results indicated that they could be so combined.
Analysis of covariance was also used to check to see if Engelmann spruce and corkbark
fir could still be legitimately combined, and the results indicated that they could.

VI. Gross International 1/4-Ineh Board Foot Volume--Unforked Trees

The approach of modeling the ratio of gross International 1/4-inch board foot
volume to a 6-inch top divided by the gross merchantable cubic foot volume to a 6-inch
top was used to provide the needed control on the behavior of the International 1/4-inch
model. It has long been recognized that this ratio starts with a zero value at a d.b.h.
near 6 inches and increases monotonically, as d.b.h. increases, to an asymptotic ratio
value between 6 and 8 (Husch 1963; Avery 1967). It has also been shown that this ratio
can be affected by tree height as well (Husch 1963).

35

With this in mind, two models were hypothesized for trial:

ie)
|

= a - a,H/D - apH/(D*) (13)

Sl -2
= bo - b,D - aoD (14)

Rric = Ratio of actual gross International 1/4-inch board foot volume to a 6-inch
top divided by predicted gross merchantable cubic foot volume to a 6-inch
top in an unforked tree.

The signs on the coefficients a,, a2, and b,, b> must be positive for the model to
behave as expected.

Before these models were tried, the ratio values were formed for all trees 6.1
inches and larger and plotted for the sp-NF combinations previously delineated. An
examination of these plots resulted in the decision to combine blackjack pine data sets
with the yellow pine data sets and to combine both Douglas-fir data sets. The plots
also showed a few outliers that were eliminated from the appropriate data sets.

The two models, (13) and (14), were then fitted to the data sets to determine which
of the two minimized RMSQR. The model with tree height (13) proved to be the worst and,
therefore, was eliminated. The fits-for model (14) were reasonable for only the ~
ponderosa pine and the Douglas-fir data sets. The other data sets either had a negative
sign on bd, or by or the plots of data sets versus the predicted curves indicated a poor
fit. The ponderosa pine model was then ''forced'' onto the white pine and the white fir
on the Lincoln National Forest data sets with good results. This was done by scaling
the ponderosa pine model through an overall slope correction using least squares regres-
sion. Techniques similar to those described by Jensen (1964, 1973, and 1976) and Jensen
and Homeyer (1970 and 1971) were used to develop better behaved (within a reasonable
usage range) ratio models for Engelmann spruce-corkbark fir, aspen, and white fir on
the Santa Fe and Carson National Forests data sets.

The resulting model form for white fir was the same as model (14) but, of course,
these parameter estimates are not least squares estimates in this case. The form of
the models for aspen and Engelmann spruce-corkbark fir was:

Rye = 20 - Biperwd 5H er 3.030: (15)
For both species groups, at least one of the parameters was negative which resulted in
an undesirable ''peak"’ and "valley'' in the ratio equation outside the available data
range. For aspen, the undesirable points also occurred outside the reasonable usage
range, but for Engelmann spruce-corkbark fir, the points did occur within the reasonable
usage range. Fortunately, the magnitudes of the "peak" and "'valley'' were small and it
was felt that they would not cause serious problems when predicting International 1/4-
inch board foot volume.

The appropriate ratio models were then multiplied by their predicted merchantable
cubic foot volume to a 6-inch top for each sp-NF combination and then fitted to actual
International 1/4-inch board foot volume to a 6-inch top, using least squares regression
through the origin to provide a slope correction. The resulting basic model form and
its error structure, therefore, is the following:

Mae = Go MRegd) Sa WV

(16)

1/C neyo:

36

where

Vy = Predicted gross International 1/4-inch board foot volume to a 6-inch top
Vin(6"") = Predicted gross merchantable cubic foot volume to a 6-inch top

dy = Least squares regression coefficient

€ = Residual about regression.

Next, the necessity for weighting this model was examined through the procedure
described by Hann and McKinney (1975). This resulted in weights being formed of the
type:

e e e
W = (D 1H Ne (17)

Unfortunately, no common set of values €,, @, and @3 could be found for all sp-NF
combinations. The final sp-NF weights were used in weighted least squares regression
through the origin to obtain the final slope correction. These slope corrections were
then incorporated into the ratio models by multiplication. As a final check, the data
points and regression equations were plotted over d.b.h. and total tree height to
determine adequacy of fit.

When considering this approach to modeling International 1/4-inch board foot
volume, the basic model form of equation (16) must be kept in mind. If a set of pro-
posed ratio equations had been available, then the fitting of equation (16) could have
been done by screening the various ratio equations times cubic foot volume to determine
which was appropriate for the sp-NF combination. Unfortunately, a proposed set of ratio
equations was not available; therefore, the ratio equations had to be developed using
the same data set.

There is a tendency to think that what is really being fitted is the model involv-
ing the 11 to 15 independent variables that would be formed by multiplying the ratio
and merchantable cubic foot volume models together. Using this approach, it was found
that some independent variables proved to be insignificant while the signs and
magnitude of other regression coefficients changed because of multicollinearity
problems. The result was an equation with all of the undesirable properties that
were to be avoided. The approach adopted here produces a model that behaves in a rea-
sonable and consistent fashion, but sacrifices some statistical niceties.

VII. Gross Internattonal 1/4-Inch Board Foot Volume--Forked Trees

The method used to determine the ratio correction for forking upon International
1/4-inch board foot volume was the same as that used to develop the equations for fork-
ing in total stem gross cubic foot volume. All selected equations were tabulated and
checked for reasonableness. From this it was discovered that the Douglas-fir equation
did not behave as expected. Therefore, the following power model was fitted to Douglas-
£iv:

ln (R = ag+ a,H (18)

1,f)

Ry f = Predicted ratio of actual gross International 1/4-inch board foot volume
: to a 6-inch top in a forked tree divided by predicted International 1/4-
inch board foot volume to a 6-inch top in an unforked tree.

37

This was transformed back by taking the antilog of both sides and a least squares slope
correction was made on the resulting model to correct it for possible log bias. Tabu-
lation of this model showed that it did behave reasonably.

VIII. Gross Sertbner Board Foot Volune--Unforked Trees

As described by Avery (1967), the ratio of Scribner board foot volume divided by
International board foot volume starts well below 1.0 at small diameters and then
increases monotonically toward an asymptotic value of 1 as diameter increases. This
relationship can be expressed as
2

R Sapa aie Saab.

S/T (19)

Sy alten Predicted ratio of actual gross Scribner board foot volume to a 6-inch top
divided by predicted gross International 1/4-inch board foot volume to a
6-inch top.

The signs on a, and a> must be positive and ag must be near 1 for the model to behave
reasonably.

Individual values of Re were formed and plotted across d.b.h. for all sp-NF
combinations used in gross ehbac foot equation development. All data below 9 inches ~
d.b.h. and certain outliers were eliminated based on the results of the plots. This
was necessary because the ratio values below 9 inches started to turn upward and exceed
1 as d.b.h. decreased. This problem was attributed to the way in which Scribner volume
is estimated in program NETVSL. The elimination of trees under 9 inches should not
cause problems because Scribner volume is seldom computed for trees under that limit.

Equation (19) was fitted to the corrected data set. Weights were developed using
the basic model:

where

<=
I

Sur Predicted gross Scribner board foot volume to a 6-inch top
bi

€

Least squares regression coefficient

Residual about regression

and the same process as described for International 1/4-inch board foot volume. An
examination of the final weighted least squares regression coefficients, after the slope
correction was multiplied through the ratio model, revealed that only the models for
yellow pine, both blackjack pines, and white fir on the Santa Fe and Carson National For-
ests were reasonable.

An abbreviated ratio model:

R [CONS e\Die

S/I (21)

was then tried using the same weighted scheme as previously described. This model

proved reasonable for Engelmann spruce-corkbark fir, aspen, and Douglas-fir on the Santa
Fe and Carson National Forests.

38

A model of the form:

In (1.0 = Re 71) = do + dD (22)
was then tried on the remaining data sets to determine the power in the model:
2 -dy
Roy = @9 - ée,D (23)

This was also weighted and proved reasonable for Douglas-fir on the Coconino, Tonto,
and Lincoln National Forests.

Both white pine and white fir on the Lincoln National Forest proved to be particu-
larly troublesome, with all previous attempts failing to provide a reasonable model.
In order to force a reasonable model on these two data sets, the following model was
fitted:

-1 -2
1.0: Rey =g 9D + 9;D (24)
This was then transformed to
Ring = 1,0 = ogb Sa (25)
Sfi.- = J0 91 .

“Model (25) could not be corrected with a weighted slope because doing so caused the
intercept to exceed l.

Again, all final models, with the final slope correction multiplied into them, were
checked by plotting. The caution given in the section on gross International 1/4-inch
board foot volume for unforked trees concerning the interpretation of this approach
applies even more strongly here. In this case, if all of the model components were
multiplied out, there would be from 23 to 63 independent variables to fit and the result
would be even more unreasonable than in the International 1/4-inch board foot volume
case.

IX. Gross Serthner Board Foot Volwne--Forked Trees
The techniques used were the same as for International 1/4-inch board foot volume
in forked trees. In this case, however, the basic model for Douglas-fir behaved reason-
ably so no special effort was necessary to model it.
X. Probability of a Tree Being Unsound in Cubte Foot Volume--Unforked and Forked Trees
By definition, the probability of a tree being unsound must take on a value between

0 and 1. A form that constrains itself between these two values is the logistic
function:

2y
Pre, = 1.0/(1.0 +e °) (26)
where
Pre = Predicted probability of a tree being unsound in total stem and in merchant-
able cubic foot volume
xX = A function of the tree's measured attributes.

Hamilton (1974) developed program RISK to fit this function to a dichotomous dependent
variable. The approach basically uses the first degree term of the Taylor Series expan-
sion of the function in weighted nonlinear regression. Output, in part, consists of
the regression coefficients, an F-statistic that tests the significance of the

39

model, t-statistics which test the significance of the regression coefficients from

a value of 0, "...and a chi-square table that evaluates goodness-of-fit over the range
of predictions" (Hamilton 1974). Of the three, Hamilton suggests use of the chi-square
statistic for screening alternative models, but he also states that the final choice

we aaets eke to the idiscretionso£ the<user ."'

The dichotomous dependent variable is defined to be 1 if the tree has any cull in
its total stem and 0 if not. The resulting equations for cubic foot volume, however,
are to be applied to both total stem and merchantable cubic foot net volume estimation.
It was felt that, in most cases, if the tree had cull in the total stem it would also
have cull in the merchantable portion. In those cases where removal of the top also
removed the cull portion, the fraction cull in the tree would then be 0, and this is a
legitimate value in the final fraction cull model for merchantable cubic foot volume.

For each sp-NF combination, 10 models were fitted with d.b.h., total tree height,
and number of tips as the tree attributes used for independent variables. Five models
had one independent variable (H,D,T,DH, or D*H) and the rest had two independent varia-
bitessi(H, DEHs Te Dal sDH,.: or D2H,T). Generally, the final runs picked were the ones that
minimized the chi-square statistic while maintaining the highest significance level
attained by the F-statistic for the one- or two-independent variable models (that is,
if some of the one- or two-variable models were significant at the 99 percent level,
then the final model was picked from those that minimized the chi-square statistic).

In some cases where the chi-square statistics were close, a model with a slightly higher
chi-square statistic was picked if the F- and t-statistics were considerably larger.

One- and two-variable models are presented in those instances where the second
variable improved the model. The added variable, number of tips in all cases, was not
always statistically significant, but the resulting model behaved as expected (that is,
as the number of tips increased, so did the probability of cull).

Some data sets were combined to gain data strength. Before combining, however,
the preliminary models and mean values of PrC. were 2xamined to assure compatibility.
Engelmann spruce and corkbark fir were separated because it was found that their cull
structures were very different.

Damage information was available on the data sets from the Santa Fe and Carson
National Forests. Therefore, in those cases where enough common damage was present to
justify the effort, the previously picked models were fitted with dichotomous damage
variables added. These damage variables did not prove to be significant.

All final equations were plotted and checked for reasonableness.

XI. Fraetton Cull in Total Stem Cubie Foot Volume Given the Tree Is Unsound--Unforked
and Forked Trees

Like the probability of a tree being unsound, fraction cull takes on a value that
ranges from 0 to 1. To provide some control over the value of fraction cull, the
following model was used:

FC h,(1.0 ‘e) 27)
| go Sete (LeOsrge 7) (
where

FC, = Predicted fraction cull in tutal stem cubic foot volume given the tree is
unsound

Y, = A function of the tree's attributes

h, = A final constant value.

40

“1 ATO = TE

With this model, fraction cull will have an asymptotic maximum value of h, and, there-
fore, h, should be less than or equal to 1. Unfortunately, this model doés not prevent
fraction cull from becoming less than 0.

The following process was used to determine the appropriate sp-NF parameters for
model (27). First, the model:

In(1.0 - ml ey (28)
whexe

FC = Actual fraction cull in total stem cubic foot volume given the tree is unsound

A function of the tree's attributes

= An intermediate constant value

was screened through linear regression using various functions of d.b.h., total tree
height, number of tips, position to first fork, and damage information as independent
variables in Y, and also using various dependent variables formed by varying the value
of h from the maximum FC in the data set to a value of 1. For each combination group
with a specified number of independent variables from the screening run, that combina-
tion of Y and hk which minimized RMSQR was picked, along with several other combinations,
that were almost as good. These models were next fitted, transformed back into model
(27), and then tabulated and checked for reasonableness. The model with the lowest
RMSQR that gave reasonable results was then fitted with least squares regression through
the origin for a slope correction to adjust the model for possible log bias. This slope
correction was then incorporated into the initial value of h to give the final value of
h, in model (27). Y, is the final function of Y. As a final check, the models were
again tabulated.

On the Douglas-fir data set, the final model proved to be worse than a mean value
of FC,. Therefore, the following model was fitted:

Y
FC, = bo = bye t, (29)

The resulting model was better than the mean, and tabulation revealed that it behaved
correctly. Model (29) was easily transformed back to model (27) by setting bg equal to
h, and adding 1n(b,/bo) to Yy-

Again, data sets were combined based on the results of the screening runs and upon

the mean FC, values of the data sets to be combined.

XII. Fraction Cull in Merchantable Cubie Foot Volwne Given the Tree Is Unsound--Unforked
and Forked Trees

For each sp-NF combination, the same independent variables used in the total stem
cubic foot fraction cull models were fitted to each top diameter data set with a vari-
able h in model (28). An examination of the various runs across top diameter was made
and a value of h was picked that did the best overall job of minimizing the RMSQR
values.

41

Using the value of h, the resulting regression coefficients of Y were modeled as
functions of top diameter to obtain the final equations of oa in the model:

Y
FC A(1.0 =e ™) (30)
where
FC = Predicted fraction cull in merchantable cubic foot volume given the tree is
unsound
sa = A function of the tree's attributes and top diameter
h = An intermediate constant value.

Model (30) was then fitted to the data set incorporating all top diameters with least
squares regression to obtain a slope correction to adjust the model for possible log
bias, and this slope correction was incorporated with k to give the final value, h_, in
the standard form: ne

Y
- m m
FC. = h (1.0 e ™) (31)
where

h_ = A final constant value.

The final models were tabulated and checked for reasonableness. This procedure
proved to work well on all data sets except Engelmann spruce and Douglas-fir. On these
two data sets, a model of the form:

¥
FC = bo - bie ™ (32)

was fitted. Tabulation of these runs indicated that they performed well. They were

then converted to the standard form of model (32) using the fact that bg equals h_ and
; m

by adding 1n(b,/bg) to Ye

Because of the high values of RMSQR, it was decided to provide an optional model
to predict mean FC_ as a function of top diameter. Plots of the means over top diameter
indicated that power functions of top diameter would be appropriate. The means were
then weighted by their number of observations and screened with numerous power trans-
forms on top diameter. The maximum number of independent variables in the screening
run was limited to the value of 1 plus the number of maxima or minima the plots had
shown. A final model was selected based on its behavior and the magnitude of its RMSQR.

XIII. Probability of Tree Being Unsound in Board Foot Volume--Unforked and Forked Trees

An examination of the data indicated that International 1/4-inch and Scribner board
foot volumes were always sound or unsound together. Therefore, one model was developed
that is applicable to both.

The techniques used were basically the same as those used for cubic foot volume.
The damage information did prove, however, to be statistically significant at the 95
percent level on the blackjack pine and Douglas-fir data sets. The model with damage
for white fir was almost significant and, because it behaved reasonably, it was also
included.

Again, all curves were plotted as a final check.

42

XIV. Fractton Cull in Internattonal 1/4-ineh Board Foot Volwne Given the Tree Is
Unsound--Unforked and Forked Trees

For fraction cull in International 1/4-inch board foot volume, the following model
was used:

FC, = hye ! (33)
where
FC, = Predicted fraction cull in International 1/4-inch board foot volume given the
tree i¢ unsound
Yy = A function of the tree's attributes
hy = A constant.

This model differs from models (27) and (32) because (33) is asymptotic to 0 instead of
1 and, therefore, it will allow predicted values of FC, to exceed 1, which is just the
opposite from models (27) and (32). This change in model form was by oversight rather
than design, even though it does not affect the legitimacy of the results.

To fit model (33), the following model was first fitted and screened using the same
_ basic procedures as in the cubic foot fraction cull equations:

In(FC) = Y _ 7 (34)
where
FC = Actual fraction cull in International 1/4-inch board foot volume given the
tree is unsound
Y =A function of the tree's attributes.

This model was then transformed back to the form of model (33) and hk, was found by least
squares regression through the origin to correct the model for possible log bias. The
models were tabulated and those that were reasonable and minimized RMSQR were picked.
All reasonable models examined for Engelmann spruce and for corkbark fir proved to be
much worse than the mean and, therefore, the models were dropped.

To gain data strength, yellow pine and Douglas-fir were combined across all
National Forests (but not with each other).

XV. Fractton Cull tn Sertbner Board Foot Volume Given The Tree Is Unsound--Unforked
and Forked Trees

The procedure and model forms used for fraction cull in Scribner board foot volume

were virtually the same as those used for fraction cull in International 1/4-inch board
foot volume.

43

Headquarters for the Intermountain Forest and
Range Experiment Station are in Ogden, Utah.
Field programs and research work units are
maintained in: ~_

Billings, Montana

Boise, Idaho

Bozeman, Montana (in cooperation with
Montana State University)

Logan, Utah (in cooperation with Utah State
University)

Missoula, Montana (in cooperation with
University of Montana)

Moscow, Idaho (in cooperation with the
University of Idaho)

Provo, Utah (in cooperation with Brigham
Young University)

Reno, Nevada (in cooperation with the
University of Nevada)

<y US. GOVERNMENT PRINTING OFFICE:1978—777-095 / 13

Hann, David W., and B. Bruce Bare.
1978. Comprehensive tree volume equations for major species of
New Mexico and Arizona: I. Results and Methodology. USDA
For. Serv. Res. Pap. INT-209, 43 p. Intermt. For. and Range

Exp. Stn., Ogden, Utah 84401.

Comprehensive tree volume equations andthe methods used to derive
them are presented for the major tree species of New Mexico and Ari-
zona. Selected tree volume tables for unforked trees can be found in the
companion publication, "Comprehensive tree volume equations for major
species of New Mexico and Arizona: II. Tables for unforked trees."'

KEYWORDS; volume equations, majortree species, New Mexico, Arizona,
gross volume, net volume, unforked tree, forked tree, total stem cubic
foot volume, merchantable cubic foot volume, Scribner board foot volume,
International 1/4-inch board foot volume.

Hann, David W., and B. Bruce Bare.
1978. Comprehensive tree volume equations for major species of
New Mexico and Arizona: I. Results and Methodology. USDA
For. Serv. Res. Pap. INT-209, 43 p. Intermt. For. and Range

Exp. Stn., Ogden, Utah 84401.

Comprehensive tree volume equations andthe methods used to derive
them are presented for the major tree species of New Mexico and Ari-
zona. Selected tree volume tables for unforked trees can be found in the
companion publication, ''Comprehensive tree volume equations for major
species of New Mexico and Arizona: II. Tables for unforked trees."'

KEYWORDS: volume equations, majortree species, New Mexico, Arizona,
gross volume, net volume, unforked tree, forked tree, total stem cubic
foot volume, merchantable cubic foot volume, Scribner board foot volume,
International 1/4-inch board foot volume.

BN

ERRATA SHEET

Hann, David W., and B. Bruce Bare.
1978. Comprehensive tree volume equations for major species of
New Mexico and Arizona: I. Results and Methodology. USDA
For. Serv. Res. Pap. INT-209,,.43 p. Intermt.. Ror. and Range
Exp. Stn., Ogden, Utah 84401.

Page 9
Table 5.--Forked tree merchantable gross cubte foot volwne data base
and regresston results
Independent variables
= D.bwh. in inches B. = Top diameter, inside bark, in inches
= Total tree height in feet P = Height to first fork divided by H
T ="1° +" (numberof forks)
Sales ee
Species, National Forest, and regression equations : of i me
: (obser Ole (RMSOR)
White pine on Lincoln, Santa Fe, and.’Carson
R, 2 = Co + CH 101 0.9581
Co = 0.61725167 + 0.042365845D_
C} = 0.00598501 - 0.00073445D_
OTR, 2 = 1-04518089 - 7.88395198E-07D7 + 8.39079873E-08D8
Engelmann spruce-corkbark fir on Lincoln, Santa Fe, and
Carson
R ¢ = 1.15372135 - 1.51067496E-08D" 91
Ponderosa pine on Coconino, Tonto, Lincoln, Santa Fe,
and Carson
Ro » = 0.997113216 - 0.023300125D_ + Cat- CaP 937, PES /Howk
MyJ /
Co = 0.00564984 + 0.00079885D_
Cy = 0.002886816 + 0.02550015D_
oR 2 = 1.01577735 - 1.68401836E-06D>
Douglas-fir on Lincoln, Santa Fe, and Carson
a 2 = 1.04417090 - 9, 53965362E-09*D" 118
Aspen on Santa Fe and Carson :
R. - = 1.07666448 - 1.18133616E-03D<-° 184
White fir on Santa Fe, Carson, and Lincoln
Re a C, (H/D2) 74 . 7286
Co = 0,.673711298 + 0.055948591D_
Cy = 1.709640750 - 0.348154558D
or m
R. » = 0.983293776 - 2. 40166266E-0SD*

ec erTrErenenTEssneenEnsnE en