Equations for Merchantable

Volume for Subtropical Moist and

Wet Forests of Puerto Rico

Thomas J. Brandeis, Olaf Kuegler, and 

Steven A. Knowe

United States

Department of

Agriculture

Forest Service

Southern

Research Station

Research Paper

SRS-39
Thomas J. Brandeis, Research Forester, USDA Forest Service, 

Southern Research Station, Knoxville, TN 37919; Olaf Kuegler, 

Biometrician, USDA Forest Service, Portland Forestry Sciences 

Laboratory, Portland, OR 97205; and Steven A. Knowe, Biometrician, 

American Forest Management, Charlotte, NC 28273.

The Authors

November 2005

Southern Research Station

P.O. Box 2680

Asheville, NC 28802

The use of trade or firm names in this publication is for reader information and does not imply endorsement 

by the U.S. Department of Agriculture of any product or service.
i

Equations for Merchantable Volume 

for Subtropical Moist and Wet Forests

of Puerto Rico

Thomas J. Brandeis, Olaf Kuegler, and Steven A. Knowe
ii
iii

Contents

Page

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

 Study Area and Datasets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

 Measurements and Calculations for Estimating Stem Volume . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

 Fitting of Regression Equations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

 Volume Estimates and Comparisons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

 Model Building . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

 Equation Testing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

Literature Cited . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

Species List . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
iv
1

Equations for Merchantable Volume for

Subtropical Moist and Wet Forests of

Puerto Rico

Thomas J. Brandeis, Olaf Kuegler, and Steven A. Knowe

Abstract

In Puerto Rico, where locally grown woods are primarily used for furniture 

and crafts production, estimation of wood volume makes it possible to 

estimate the monetary value of one of the many commodities and services 

forests provide to society. In the Puerto Rican forest inventories of 1980 

and 1990, workers calculated stem volume directly by applying geometric 

formulae to bole sections of merchantable trees. Field crews recorded 

several diameter and height measurements along the bole of each tree. If 

tree volume estimates were based on fewer tree measurements, this would 

significantly increase field crew productivity. For this reason, tree volume 

equations have been derived from Puerto Rican forest inventory data by 

directly calculating stem volume, then creating regression equations that 

estimate inside and outside bark merchantable stem volume from tree 

diameter at breast height and total height. 

Keywords: Forest Inventory and Analysis, merchantable stem volume, 

Puerto Rico, tropical forest, volume equations.

Introduction

Estimates of wood volume are derived from forest inventory 

data. This is done by means of volume tables, geometric 

formulae, or regression equations that convert tree measurements such as diameter at breast height (d.b.h.) and total tree 

height to some measure of wood volume in the tree. Even in 

the absence of large-scale harvesting and forest-based industries, which is the case for Puerto Rico where locally grown 

woods are primarily used for furniture and crafts production 

(Kicliter 1997), estimation of wood volume makes it possible

to estimate the monetary value of one of the many commodities and services that forests provide to society. Additionally, 

studies of forest resource sustainability and international 

assessments of forest resources, such as the Food and Agriculture Organization of the United Nations Forest Resource 

Assessments, require the reporting of growing-stock volumes

(FAO 2001).

Volume equations have been developed for some Puerto 

Rican forests and species. Wadsworth (1949) developed a 

linear regression model that estimates total cubic foot volume

from d.b.h. and total tree height for the species found in the 

subtropical wet forest life zone’s tabonuco forest type (Ewel 

and Whitmore 1973). Francis (1988, 1989) produced linear 

and nonlinear equations that estimate tree volumes from 

d.b.h. and merchantable bole length for plantation-grown 

oxhorn bucida (Bucida buceras) and mahoe (Hibiscus elatus).

Bauer and Gillespie (1990) produced linear volume equations

using d.b.h. and total tree height for Honduras mahogany x 

West Indian mahogany (Swietenia macrophylla x S. 

mahagoni). Equations for noncommercial subtropical hardwoods are available from the U.S. Department of Agriculture

Forest Service’s National Information Management Systems 

(NIMS); these will be used to estimate tropical tree volumes 

for southern Florida, an area that has some tree species and 

genera in common with Puerto Rico. However, in Puerto 

Rico where an island-wide forest inventory has the potential 

to encounter over 700 tree species (Francis and Liogier 

1991, Little and others 1974), additional resources must be 

used to produce tree volume estimates.

In Puerto Rican forest inventories conducted in 1980 (Birdsey

and Weaver 1982) and 1990 (Franco and others 1997), workers

calculated merchantable volumes directly by applying a geometric formula to bole sections of trees classified as growing 

stock. For a tree to be considered growing stock, one-third 

or more of the gross volume in its saw-log section must meet

grade, soundness, and size requirements for commercial logs,

or the tree must have the potential to meet these requirements

if it is poletimber size with 12.5 cm ≤ d.b.h. < 27.5 cm. 

Inventory field crews took multiple diameter and height 

measurements along the bole of each tree so that these calculations could be made—a time-consuming process. If tree 

volume estimates could be based on fewer tree measurements,

this would significantly increase the productivity of field 

crews in future forest inventories.

For this reason, we have derived tree volume equations from 

Puerto Rican forest inventory data by first directly calculating

stem volumes using geometric formulae, and then fitting 

linear and nonlinear regression equations for estimating 

merchantable stem volume inside and outside bark from 

d.b.h. and total tree heightmeasurements. Equations were 

fitted to data for all species combined, and then for species 

and genera represented by 25 or more growing-stock trees. 
2

To determine which equations would produce the best estimates of merchantable volume from future Caribbean forest 

inventory data, we compared directly calculated volumes 

from a test dataset to estimates made using our models, 

estimates made using the Wadsworth (1949) method, and 

estimates made using the NIMS equations. 

Methods

Study Area and Datasets

The forests surveyed in 1980 and 1990 by the U.S. Department

of Agriculture Forest Service’s Forest Inventory and Analysis

Program in Puerto Rico were those considered to have the 

potential for commercial wood production (i.e., timberland). 

These surveys excluded forested lands unsuitable for commercial timber production, such as watersheds in which it is 

necessary to maintain forest cover, forests in areas with 

excessive rainfall or inoperably steep slopes, dry forests, and 

all public forests (Birdsey and Weaver 1982, 1987). The 

surveyed areas were in the subtropical rain, wet, and moist 

forest life zones (Ewel and Whitmore 1973). The 1990 forest

inventory dataset we used in constructing volume equations 

in this study consisted of data for 1,247 trees of 102 species 

with d.b.h. ≥ 12.5 cm after the removal of data for palms, 

tree ferns, and poorly formed cull trees that did not meet the 

standards for growing stock. The 1980 inventory dataset 

used to test the equations consisted of data for 591 trees of 

76 species with d.b.h. ≥ 12.5 cm, again with data for cull 

trees, palms, and tree ferns removed. Figure 1 presents the 

diameter distribution of trees from the 1990 and 1980 forest 

inventory datasets by 10-cm d.b.h. classes. Table 1 shows 

maximum and minimum d.b.h. and heights for species or 

genera in the 1980 and 1990 datasets that were represented 

by more than 25 trees in both datasets combined.

The resource bulletins based on the 1980 and 1990 Puerto 

Rican forest inventories (Birdsey and Weaver 1982, Franco 

and others 1997) gave volume estimates (in m3

) that were 

calculated directly by applying the formula for the volume 

of a conic frustum to bole sections and summing these 

section volumes to obtain total merchantable volume. Total 

merchantable volume was defined as the inside bark portion 

of the tree’s stem between a stump 30 cm tall and a 10-cm 

upper-stem diameter outside bark (d.o.b.), excluding wood 

volume in branches. The details of these direct volume 

calculations are given below.

0 10 20 30 40 50 60 70 80 90 100

1990 1980

Diameter class (cm)

Number of trees

0

100

200

300

400

500

Figure 1—Distribution of trees by 10 cm diameter classes (d.b.h.) and year.

Figure 1—Distribution of trees by 10-cm diameter classes (d.b.h.) and year.
3

Table 1—Number of trees, mean, standard error, maximum, and minimum measurements for d.b.h. and height by 

species and year, Puerto Rico

D.b.h. Height

 Standard Max- Min- Standard Max- MinSpecies Trees Mean error imum imum Mean error imum imum

number - - - - - - - - - - - - cm - - - - - - - - - - - - - - - - - - - - - - - - - m - - - - - - - - - - - -

Andira inermis

 1980 33 22.1 1.3 41.6 12.5 12.9 0.6 21.0 7.0

 1990 56 24.3 1.4 55.4 12.8 15.0 0.6 30.0 8.0

Cecropia schreberiana

 1980 28 30.3 1.9 53.8 17.5 16.1 0.7 23.0 6.0

 1990 70 32.8 1.4 57.5 12.7 17.8 0.6 29.0 7.0

Cordia spp.

 1980 18 24.4 1.7 35.8 14.3 14.3 0.8 24.0 9.0

 1990 30 23.6 1.3 42.3 13.5 14.7 0.9 25.0 3.0

Erythrina poeppigiana

 1980 15 29.7 2.1 49.9 20.2 13.1 1.0 18.0 5.0

 1990 26 35.8 2.5 68.7 18.9 16.5 1.3 31.0 6.0

Guarea guidonia

 1980 94 31.9 1.3 62.8 12.7 14.7 0.3 21.0 8.0

 1990 187 34.2 1.2 76.5 12.7 16.1 0.3 29.0 5.0

Inga laurina

 1980 25 27.0 1.5 48.5 15.5 16.2 0.8 23.0 9.0

 1990 50 31.5 1.6 65.3 14.7 17.6 0.9 34.0 5.0

I. vera 

 1980 70 26.9 1.1 51.3 12.7 14.1 0.4 22.0 8.0

 1990 110 29.4 1.0 61.8 13.0 16.8 0.4 30.0 7.0

Mangifera indica

 1980 22 57.0 4.2 85.1 27.1 14.7 0.7 22.0 8.0

 1990 35 57.1 3.4 87.9 17.1 17.5 0.7 27.0 10.0

Ocotea spp.

 1980 16 22.4 1.6 36.5 13.4 12.5 0.8 22.0 9.0

 1990 33 24.0 1.4 44.5 13.1 12.4 0.6 19.0 4.0

Schefflera morototonii

 1980 22 27.8 3.4 65.3 13.6 15.9 0.9 23.0 9.0

 1990 38 28.5 2.4 72.5 12.7 15.6 1.0 31.0 6.0

Spathodea campanulata

 1980 8 30.8 3.6 46.8 15.5 17.0 1.7 23.0 11.0

 1990 121 25.8 1.0 68.5 12.8 14.2 0.4 28.0 3.0

Syzygium jambos

 1980 15 24.0 1.9 40.0 14.5 11.3 0.9 18.0 6.0

 1990 39 22.1 1.2 43.4 13.5 11.6 0.6 22.0 6.0

Tabebuia heterophylla

1980 15 21.1 2.1 37.1 13.1 13.8 0.7 18.0 8.0

 1990 53 21.1 1.0 42.8 12.5 14.4 0.4 20.0 8.0

Othera 

 1980 209 26.8 0.8 67.4 12.7 13.8 0.7 26.0 5.0

 1990 399 28.2 0.6 76.5 12.6 15.4 0.3 37.0 5.0

All species

 1980 590 28.4 0.5 85.1 12.5 14.2 0.2 26.0 5.0

 1990 1,247 29.4 0.4 87.9 12.5 15.5 0.1 37.0 3.0

D.b.h. = diameter at breast height.

a Species for which a species- or genus-specific equation was not fitted.
4

Measurements and Calculations for

Estimating Stem Volume

During the Puerto Rican forest inventories, the following 

measurements were taken on all trees with a d.b.h. ≥ 12.5 

cm (fig. 2A):

• D.b.h. outside bark 

• Double bark thickness (twice the bark thickness measured 

with a bark gauge, at breast height [1.37 m])

• Stump d.o.b. taken at 30 cm height

• Total bole length (length from a stump 30 cm tall to a

10-cm top d.o.b.)

• Pole top d.o.b. (if not 10 cm)

• Total height (HT) from ground

For trees with d.b.h. ≥ 27.5 cm (sawtimber size), additional 

measurements were taken (fig. 2B):

• Saw-log length to d.o.b. of 22.5 cm

• Saw-log d.o.b. (if not 22.5 cm)

Stem volume was calculated by applying the formula for a 

conic frustum to sections of the bole. 

For poletimber-size trees (fig. 2A), there were two bole 

sections:

1. Stump top to breast height

2. Breast height to pole top

For sawtimber-size trees (fig. 2B), the sections were: 

1. Stump top to breast height

2. Breast height to saw-log top

3. Saw-log top to pole top

Estimates of outside bark merchantable stem volume were 

based on outside bark diameters, and no deductions were 

made for bark thickness. To estimate inside bark merchantable stem volume, the diameters for all sections were 

converted from outside bark to inside bark. To deduct for 

bark thickness, bark ratio was first calculated using the 

following formula for hardwoods:

(1)

Figure 2—Measurements taken on (A) poletimber-size trees (minimum d.b.h. = 12.5 cm, minimum top d.o.b. = 10.0 cm) and (B) sawtimber-size trees 

(minimum d.b.h. = 27.5 cm, minimum saw-log top d.o.b. = 22.5 cm) in the 1980 and 1990 forest inventories of Puerto Rico.
5

where

BR = bark ratio

DBH = d.b.h. outside bark

DBT = double bark thickness

Then, diameters inside bark at stump, breast, saw-log top, 

and pole-top heights were calculated with the following 

formula for hardwoods:

(2)

where

DIB = diameter inside bark

DOB = diameter outside bark

Section lengths and inside and outside bark diameters were 

used in the formula for a conic frustum to calculate the wood

volume of individual sections (example for inside bark 

section volume):

(3)

where

VSEC = section volume in m3

HSEC = section length in m

DIB1 = diameter in cm inside bark at one end of section

DIB2 = diameter in cm inside bark at other end of section

0.00007854 is a constant derived from the expression

 (Husch and others 1993) (4)

where

Di = section diameter in cm 

The section volumes are then added together to obtain a total

stem volume. Total stem volume is defined here as the inside 

or outside bark portion of the tree’s stem between a 30-cmtall stump and a 10-cm upper stem diameter. This volume 

estimate does not include branchwood volume. Stem volumes

were calculated using this method for all trees in the 1990 

forest inventory dataset that met minimum diameter requirements and were not abnormally formed.

Fitting of Regression Equations

Figure 3 presents directly calculated mean volumes for the 

1990 and 1980 Puerto Rico forest survey datasets by 10-cm 

d.b.h. classes (there was only one tree in the 90-cm class so 

no standard error is presented for that class). First, equations 

were fitted to the 1990 inventory data for all species combined.

These equations used d.b.h. and total tree height in different 

combinations, including the often used model DBH 

2

HT . 

Mallow’s Cp, PRESS, Variance Inflation Factor, and R2 statistics (Stafford and Sabin 1997) were examined to determine 

which model fit best. Separate equations for poletimber and 

sawtimber-size stems were also tried, but this did not improve

the fit appreciably. Once equations were fitted to the entire 

dataset, separate equations were fitted to individual species 

or genera represented by more than 25 individual trees.

The equations that fit the data best were those in which d.b.h.

and total tree height were the independent variables and logarithmic transformations were applied to both the dependent 

and independent variables to stabilize variance. Two other 

models were also fitted to the data. An equation with only 

d.b.h. as an independent variable was fitted so that volume 

can be estimated when forest inventories do not include data 

on tree height, which is often hard to measure accurately in 

forest settings. An equation using only d.b.h. could also be 

used where tree crowns have been damaged by hurricanes, 

which is common in the Caribbean. 

We also fit a nonlinear equation that incorporates a taper 

function based on the models mathematically described by 

Clutter (1980) and applied by Zarnoch and others (2003) to 

the same 1990 dataset (by species and genera) used for linear

regression model fitting. This was done despite concerns 

that fewer upper stem measurements (only two—saw and 

pole-log top diameters) were available in the inventory datasets than are usually taken on each tree when developing a 

taper function (Cost 1978, Thomas and Parresol 1991, Van 

Deusen and others 1982). This results in more extrapolation 

from breast height to pole and saw-log heights, and a less 

than ideal representation of tree taper. The advantage of an 

equation with an incorporated taper function is the flexibility 

to change the upper stem diameter merchantability limit, 

currently set for this study at 10 cm d.o.b., if future needs 

change. We used the SAS NLIN (nonlinear regression) procedure (Cody and Smith 1997) with the Marquardt method 

as the fitting algorithm, and provided a starting value grid to 

avoid local minima (Freund and Littel 2000). 

Bias introduced into the volume estimate by the logarithmic 

transformation was corrected for as suggested by Baskerville

(1972) by multiplying estimated individual tree volume by 

e0.5MSE. As the main focus of this analysis is on volume and 
6

not the logarithm of the volume, the mean square error and 

R2 were calculated using estimated volume and measured 

volume as follows (Kvålseth 1985):

(5)

where

Vmeas = measured volume

Vstem = merchantable stem volume

The R2 obtained when volume is used is slightly lower than 

that obtained when the logarithm of volume is used. 

Volume Estimates and Comparisons

The 1980 Puerto Rico forest survey data were used as a test 

dataset. Stem volume was calculated directly using the forest

inventory methods described above for all trees with a d.b.h. 

≥ 12.5 cm in the 1980 forest survey (fig. 3). Stem volume 

was also estimated by inputting each tree’s d.b.h. and total 

tree height values into the regression equations. For species 

without a species-specific volume equation, the general equations were used. For the nonlinear equation with incorporated

taper function, the upper stem diameter merchantability limit

was set at 10 cm.

For the sake of comparison, volume was also estimated for 

each tree in the 1980 dataset using the Wadsworth (1949) 

volume equation for trees in the tabonuco forest type in the 

Caribbean National Forest for trees ranging in size from a 

d.b.h. of 5 to 125 cm and total tree height from 3.0 to 36.5 m.

[Note that Wadsworth (1949) used English units, where stem 

volume is in cubic feet, d.b.h. is in inches, and total tree height

is in feet. The volume estimates produced by Wadsworth’s 

equation (equation 6) for this comparison were converted to 

metric units.]

(6)

where

YS = stem volume

Volume was also estimated using the NIMS noncommercial 

subtropical hardwood equations that will be used to estimate 

subtropical tree volumes for southern Florida. NIMS has an 

equation for poletimber-size (d.b.h. < 27.5 cm) trees,

1990 1980

5

4

3

2

1

0

0 10 20 30 40 50 60 70 80 90 100

Mean stem volume (m3

)

Diameter class (cm)

Figure 3—Mean stem volumes with standard errors directly calculated from multiple stem measurements, by 10 cm diameter 

classes (d.b.h.) and year. (Stem volume measurement is between a 30 cm tall stump and a 10 cm outside bark upper stem 

diameter, excluding wood volume in branches.)

Figure 3—Mean stem volumes with standard errors directly calculated from multiple stem measurements, by 

10-cm diameter classes (d.b.h.) and year. (Stem volume measurement is between a 30-cm tall stump and a 10-cm 

outside bark upper stem diameter, excluding wood volume in branches).
7

where

Dupper = upper stem diameter merchantability limit in cm

K = 0.00007854 · (1-2 / b3)

The taper function implied by this merchantable volume 

equation (equation 11) can be used to estimate upper stem 

diameter at any merchantable height, and height at any upper

stem diameter as follows:

(12)

and

(13)

where

h = merchantable height in m

When volume was estimated with the nonlinear equation, the 

upper stem diameter merchantability limit was set at 10 cm. 

Coefficients of equations for merchantable stem volume 

inside and outside bark for all species combined and 13 individual species or genus groupings are given in tables 2, 3, 

and 4 (table 2 for d.b.h. and total tree height equations, table 

3 for d.b.h. only equations, and table 4 for nonlinear d.b.h. 

and total tree height equations). If one is using the speciesspecific equations to estimate a forest’s volume, one should 

use the “other species” equation coefficients in tables 2, 3, 

and 4 to estimate volumes for species that don’t have their 

own equations.

Equation Testing

Volume comparisons were made for all trees in the poletimber

size class (12.5 cm ≤ d.b.h. < 27.5 cm), sawtimber size class 

(d.b.h. ≥ 27.5 cm), and by 10-cm d.b.h. classes up to 70 cm. 

The results with standard errors and 95-percent confidence 

limits are presented in table 5. Note that the 10-cm d.b.h. class

consists only of trees with d.b.h. ranging from 12.5 to 14.9 cm.

Only comparisons up to the 70-cm d.b.h. class are presented 

in table 5 because the 80- and 90-cm classes had only two 

trees each.

The log-transformed linear equation with d.b.h., the nonlinear

taper equation with d.b.h., and the log-transformed linear 

equation with d.b.h. and total tree height produced merchantable stem volume estimates that were very similar overall 

and were acceptable in accuracy when compared to the 

directly calculated volumes. Assessing model performance 

(7)

where

YPOLE = poletimber volume

and another equation for sawtimber-size trees (d.b.h. ≥ 27.5 

cm),

(8)

where

YSAW = sawtimber volume

Again, the volume estimates produced by these equations 

have been converted to metric units.

Pair-wise t-tests were used to determine whether the mean 

differences between the four volume estimates (log transformed

linear d.b.h. and total tree height equation, log transformed 

d.b.h. alone equation, Wadsworth equation, and NIMS equations) and the directly calculated merchantable volume were 

greater than zero by chance alone (Cody and Smith 1997). 

Comparisons were made on all trees in the poletimber size 

class (12.5 ≤ d.b.h. < 27.5 cm), sawtimber size class (d.b.h. 

≥ 27.5 cm), and by 10-cm d.b.h. classes up to 70 cm. There 

were too few trees in the 80- and 90-cm d.b.h. classes to 

permit accurate testing.

Results

Model Building

For all species combined, the best fitting equation that used 

d.b.h. and total tree height was:

(9)

where 

Vstem = merchantable stem volume in m3

A second regression was fitted with only DBH as the independent variable,

(10)

The nonlinear taper equation used had the form:

(11)
8

Table 2—Model coefficients for estimating inside and outside bark merchantable stem volume from d.b.h. and total 

height (log transformed) by species, Puerto Rico, 1990a 

Species Trees b0 b1 b2 b3 R2 MSE

number

Andira inermis 56

Inside bark -10.73531 2.33133 0.77468 -0.00748 0.89225 0.02102

 Outside bark -10.28954 2.28989 0.71767 -0.00729 0.89010 0.02697

Cecropia schreberiana 70

 Inside bark -13.56336 3.55071 0.78797 -0.04695 0.87448 0.04774

 Outside bark -13.14265 3.44680 0.78694 -0.04487 0.87223 0.06064

Cordia spp. 30

 Inside bark -11.95066 3.06860 0.57666 -0.02958 0.92034 0.00572

 Outside bark -11.74761 3.07771 0.58133 -0.03390 0.92032 0.00681

Erythrina poeppigiana 26

Inside bark -14.27432 3.63718 0.73037 -0.03270 0.94862 0.06127

 Outside bark -14.50486 3.83840 0.69531 -0.03801 0.95913 0.06134

Guarea guidonia 187

Inside bark -12.56150 3.05691 0.74885 -0.02617 0.89648 0.06431

 Outside bark -12.27119 3.01970 0.75533 -0.02674 0.89667 0.07929

Inga laurina 50

 Inside bark -12.70218 2.83593 1.01689 -0.02070 0.92292 0.01989

 Outside bark -13.00902 3.00540 1.02641 -0.02627 0.93027 0.02186

I. vera 110

 Inside bark -10.83036 2.50263 0.73109 -0.01560 0.91807 0.01670

 Outside bark -10.84585 2.55560 0.73265 -0.01738 0.91968 0.01999

Mangifera indica 35

 Inside bark -10.16260 1.92058 1.08297 -0.00294 0.73210 0.41411

 Outside bark -9.41269 1.82991 0.97767 -0.00155 0.73835 0.48507

Ocotea spp. 33

 Inside bark -11.10688 2.19819 1.05004 -0.00256 0.97535 0.00148

 Outside bark -10.95222 2.20138 1.04221 -0.00322 0.97829 0.00163

Schefflera morototonii 38

 Inside bark -10.70757 2.16136 0.97939 -0.00323 0.99144 0.01111

 Outside bark -10.12920 2.06841 0.89646 -0.00022 0.99523 0.00808

Spathodea campanulata 121

 Inside bark -11.83160 2.70894 0.86993 -0.01669 0.98328 0.00497

 Outside bark -11.14226 2.55388 0.83876 -0.01385 0.98075 0.00759

Syzygium jambos 39

 Inside bark -10.54334 2.08250 0.89368 0.00008 0.83125 0.01450

 Outside bark -10.33813 2.02344 0.91053 0.00132 0.83866 0.01607

Tabebuia heterophylla 53

 Inside bark -11.57430 2.49205 0.89277 -0.00703 0.95610 0.00222

 Outside bark -11.36222 2.48892 0.90955 -0.01059 0.95513 0.00286

Otherb 399

 Inside bark -12.13380 2.70518 0.95270 -0.01870 0.92474 0.02722

 Outside bark -11.82546 2.64837 0.94063 -0.01708 0.92436 0.03570

All species 1,247

 Inside bark -12.02348 2.75276 0.88961 -0.02072 0.90827 0.04140

 Outside bark -11.68173 2.69449 0.87782 -0.01975 0.90924 0.05188

d.b.h. = diameter at breast height.

MSE = mean square error.

a , where Vstem = merchantable stem volume in m3

, DBH = diameter at 1.37 m, and HT = total tree height in m.

b Species for which a species- or genus-specific equation was not fitted.

(ln ) (ln ) ( ) o 1 BH 2 T 3 DBH b b D b H b

stem V = e + + +
9

Table 3—Model coefficients for estimating inside and outside bark merchantable stem volume from 

d.b.h. (log transformed) by species, Puerto Rico, 1990a

Species Trees b0 b1 b2 R2 MSE

number

Andira inermis 56

 Inside bark -9.89365 2.74883 -0.01023 0.78206 0.04251

 Outside bark -9.50640 2.67666 -0.00983 0.78106 0.05373

Cecropia schreberiana 70

 Inside bark -13.66618 4.36740 -0.06043 0.78402 0.08234

 Outside bark -13.24115 4.26242 -0.05833 0.78183 0.10355

Cordia spp. 30

 Inside bark -13.26963 4.24623 -0.06536 0.81290 0.01342

 Outside bark -13.07582 4.26489 -0.06997 0.79597 0.01744

Erythrina poeppigiana 26

 Inside bark -15.87797 4.82072 -0.04694 0.84485 0.18501

 Outside bark -16.02184 4.96512 -0.05157 0.86773 0.19853

Guarea guidonia 187

 Inside bark -11.37162 3.29002 -0.02417 0.86666 0.08283

 Outside bark -11.06816 3.25483 -0.02471 0.86490 0.10367

Inga laurina 50

Inside bark -14.76853 4.70259 -0.06525 0.75718 0.06266

Outside bark -15.09003 4.88954 -0.07125 0.76266 0.07440

I. vera 110

 Inside bark -9.95936 2.89215 -0.01964 0.81276 0.03817

 Outside bark -9.97137 2.94596 -0.02143 0.82018 0.04475

Mangifera indica 35

 Inside bark -10.49736 3.07171 -0.02132 0.58428 0.64259

 Outside bark -9.69613 2.86912 -0.01815 0.60684 0.72887

Ocotea spp. 33

 Inside bark -9.53696 2.54212 -0.00418 0.78678 0.01282

 Outside bark -9.39237 2.54274 -0.00482 0.78423 0.01618

Schefflera morototonii 38

 Inside bark -10.75646 3.08376 -0.01346 0.93960 0.07838

 Outside bark -10.15879 2.91271 -0.00958 0.94095 0.09998

Spathodea campanulata 121

 Inside bark -11.08799 3.19974 -0.01805 0.91919 0.02400

 Outside bark -10.42197 3.02710 -0.01516 0.91732 0.03259

Syzygium jambos 39

 Inside bark -9.65948 2.53683 -0.00496 0.60120 0.03428

 Outside bark -9.43608 2.48633 -0.00381 0.60566 0.03927

Tabebuia heterophylla 53

 Inside bark -11.75282 3.58078 -0.04111 0.93698 0.00318

Outside bark -11.54393 3.59811 -0.04531 0.93484 0.00415

Otherb 399

 Inside bark -11.25341 3.23528 -0.02031 0.82853 0.06202

 Outside bark -10.95213 3.17175 -0.01866 0.83424 0.07823

All species 1,247

 Inside bark -11.41282 3.35947 -0.02733 0.81503 0.08347

Outside bark -11.07369 3.29316 -0.02627 0.81746 0.10434

d.b.h. = diameter at breast height.

MSE = mean square error.

a , where Vstem = merchantable stem volume in m3

 and DBH = diameter at 1.37 m.

b Species for which a species- or genus-specific equation was not fitted.

(ln ) ( ) b0 b1 D BH b2 DBH

stem V e

(ln ) ( ) b0 b1 DBH b2 DBH

stem V e ? ? ?

= + +
10

Table 4—Model coefficients for estimating inside and outside bark merchantable stem volume from

d.b.h. and total height in a nonlinear regression taper equation by species, Puerto Rico, 1990a

Species Trees b0 b1 b2 b3 SSE

number

Andira inermis 56

 Inside bark 0.00004 1.81263 1.11977 3.35135 2.52271

Outside bark 0.00002 1.90521 1.21305 3.24424 4.17579

Cecropia schreberiana 70

 Inside bark 0.00010 1.81935 0.83918 3.29290 7.92211

 Outside bark 0.00007 1.95023 0.83456 3.26138 11.48275

Cordia spp. 30

 Inside bark 0.00006 2.23743 0.50421 3.46169 0.29389

Outside bark 0.00003 2.52160 0.44382 3.19032 1.13561

Erythrina poeppigiana 26

 Inside bark 0.00002 1.98406 1.12400 3.51253 1.91242

 Outside bark 0.00005 1.97705 0.93928 3.44792 2.28506

Guarea guidonia 187

 Inside bark 0.00008 1.92664 0.70748 2.96476 25.98568

 Outside bark 0.00010 1.88734 0.73166 2.96413 33.09052

Inga laurina 50

 Inside bark 0.00002 2.02317 1.15868 3.29652 3.05359

 Outside bark 0.00001 2.04250 1.22339 3.25192 4.16100

I. vera 110

 Inside bark 0.00008 1.85341 0.87317 3.27694 6.42406

 Outside bark 0.00007 1.92919 0.84339 3.22199 9.87309

Mangifera indica 35

 Inside bark 0.00009 1.56418 1.19588 2.93596 25.90205

 Outside bark 0.00015 1.54875 1.09195 2.91972 28.84107

Ocotea spp. 33

 Inside bark 0.00003 1.93042 1.15826 3.57446 0.19879

 Outside bark 0.00001 2.13112 1.26462 3.37728 0.71280

Schefflera morototonii 38

 Inside bark 0.00008 1.94121 0.78641 3.71168 1.63683

 Outside bark 0.00006 2.03835 0.80043 3.54624 2.77364

Spathodea campanulata 121

 Inside bark 0.00006 1.86027 0.96010 3.69325 3.43735

 Outside bark 0.00005 1.94002 1.00073 3.54513 6.77271

Syzygium jambos 39

 Inside bark 0.00002 2.09996 1.14408 3.42865 0.67420

 Outside bark 0.00001 2.43478 1.05685 3.22420 1.29064

Tabebuia heterophylla 53

 Inside bark 0.00004 2.08728 0.77360 3.52155 0.31078

 Outside bark 0.00002 2.35746 0.82028 3.19140 1.60281

Otherb 399

 Inside bark 0.00011 1.79592 0.80149 3.13759 35.14657

 Outside bark 0.00008 1.92245 0.78102 3.13227 47.89941

All species 1,247

 Inside bark 0.00009 1.76072 0.95231 3.18682 4.29 3 1020

 Outside bark 0.00011 1.75178 0.93862 3.20789 5.18 3 1020

d.b.h. = diameter at breast height.

SSE = sum of square error.

a , where Vstem = merchantable stem volume in m3

, DBH = diameter at 1.37 m (in cm),

HT = total tree height in m, Dupper = upper stem diameter merchantability limit in cm, and K = 0.00007854 · [1-(2 / b3)].

b Species for which a species- or genus-specific equation was not fitted.

( ) ( 1.3 7) 0 ( 2) 3

3

1 2

b T

B H

b

b u p p e r

T

b

ste m B H H

D

D

V b D H K

( ) ( 1.37) 0 ( 2) 3

3

1 2 ? ? ? ? ? ? b ? T

BH

b

b upper

T

b

stem BH H

D

D

V b D H K

= • – – • • –
11

Table 5—Mean volume difference between volume equation estimate and directly calculated merchantable stem volume, 

standard error of the mean, confidence limits, and pair-wise t-test result by size class, diameter class, and equation type, 

Puerto Rico, 1990

Confidence limits

 Mean Standard Lower Upper

Class Trees difference error 95 percent 95 percent Pr > |t|

number - - - - - - - - - - - - - - - - - - m3 - - - - - - - - - - - - - - - - -

Size class

Sawtimber (d.b.h. ≥ 27.5 cm) 330

 D.b.h. / HT equationa 0.0081 0.0027 0.0028 0.0134 0.0027

 D.b.h. equationb 0.0159 0.0033 0.0095 0.0224 < 0.0001

 Nonlinear taper equationc 0.0034 0.0029 -0.0022 0.0091 0.2333

 NIMS equationd -0.0164 0.0029 -0.0222 -0.0106 < 0.0001

 Wadsworth equatione 0.0606 0.0028 0.0550 0.0661 < 0.0001

Poletimber (12.5 cm ≤ d.b.h. < 27.5 cm) 261

 D.b.h./ HT equationa 0.0450 0.0141 0.0172 0.0727 0.0016

 D.b.h. equationb 0.1143 0.0192 0.0765 0.1520 < 0.0001

 Nonlinear taper equationc -0.0685 0.0124 -0.0929 -0.0442 < 0.0001

 NIMS equationd -0.1112 0.0146 -0.1400 -0.0824 < 0.0001

 Wadsworth equatione 0.3287 0.0326 0.2646 0.3928 < 0.0001

All stems 591

 D.b.h./ HT equationa 0.0244 0.0064 0.0117 0.0370 0.0002

 D.b.h. equationb 0.0594 0.0089 0.0419 0.0768 < 0.0001

 Nonlinear taper equationc -0.0284 0.0059 -0.0399 -0.0168 < 0.0001

 NIMS equationd -0.0583 0.0069 -0.0719 -0.0447 < 0.0001

 Wadsworth equatione 0.1790 0.0155 0.1486 0.2093 < 0.0001

Diameter class

10 cm (5.0 cm ≤ d.b.h. ≤ 14.9 cm)f 53

 D.b.h./ HT equationa 0.0002 0.0020 -0.0037 0.0042 0.9020

 D.b.h. equationb 0.0045 0.0024 -0.0003 0.0093 0.0658

 Nonlinear taper equationc 0.0031 0.0030 -0.0029 0.0092 0.3015

 NIMS equationd -0.0102 0.0018 -0.0139 -0.0065 < 0.0001

 Wadsworth equatione 0.0292 0.0018 0.0256 0.0329 < 0.0001

20 cm (15.0 cm ≤ d.b.h. ≤ 24.9 cm) 230

 D.b.h./ HT equationa 0.0066 0.0033 0.0001 0.0131 0.0451

 D.b.h. equationb 0.0135 0.0037 0.0061 0.0208 0.0004

 Nonlinear taper equationc 0.0031 0.0035 -0.0038 0.0099 0.3801

 NIMS equationd -0.0148 0.0037 -0.0221 -0.0074 < 0.0001

 Wadsworth equatione 0.0592 0.0031 0.0530 0.0653 < 0.0001

30 cm (25.0 cm ≤ d.b.h. ≤ 34.9 cm) 165

 D.b.h./ HT equationa 0.0119 0.0074 -0.0027 0.0265 0.1100

 D.b.h. equationb 0.0423 0.0097 0.0231 0.0614 < 0.0001

 Nonlinear taper equationc -0.0268 0.0081 -0.0427 -0.0109 0.0011

 NIMS equationd -0.0744 0.0081 -0.0904 -0.0584 < 0.0001

 Wadsworth equatione 0.1143 0.0084 0.0978 0.1308 < 0.0001

continued
12

Table 5—Mean volume difference between volume equation estimate and directly calculated merchantable stem volume, 

standard error of the mean, confidence limits, and pair-wise t-test result by size class, diameter class, and equation type, 

Puerto Rico, 1990 (continued)

Confidence limits

 Mean Standard Lower Upper

Class Trees difference error 95 percent 95 percent Pr > |t|

number - - - - - - - - - - - - - - - - - - m3 - - - - - - - - - - - - - - - - -

Diameter class (continued)

40 cm (35.0 cm ≤ d.b.h. ≤ 44.9 cm) 86

 D.b.h./ HT equationa 0.0154 0.0182 -0.0208 0.0516 0.3994

 D.b.h. equationb 0.0588 0.0242 0.0106 0.1069 0.0173

 Nonlinear taper equationc -0.0913 0.0193 -0.1297 -0.0529 < 0.0001

 NIMS equationd -0.1654 0.0182 -0.2016 -0.1292 < 0.0001

 Wadsworth equatione 0.2031 0.0194 0.1645 0.2417 < 0.0001

50 cm (45.0 cm ≤ d.b.h. ≤ 54.9 cm) 31

 D.b.h./ HT equationa 0.0786 0.0465 -0.0163 0.1735 0.1012

 D.b.h. equationb 0.1588 0.0589 0.0384 0.2792 0.0114

 Nonlinear taper equationc -0.0736 0.0400 -0.1552 0.0081 0.0756

 NIMS equationd -0.1853 0.0428 -0.2726 -0.0980 0.0002

 Wadsworth equatione 0.4722 0.0410 0.3885 0.5560 < 0.0001

60 cm (55.0 cm ≤ d.b.h. ≤ 64.9 cm) 12

 D.b.h./ HT equationa 0.2318 0.1297 -0.0536 0.5172 0.1013

 D.b.h. equationb 0.5007 0.1395 0.1937 0.8077 0.0042

 Nonlinear taper equationc -0.0425 0.1117 -0.2884 0.2034 0.7108

 NIMS equationd 0.0360 0.1322 -0.2549 0.3270 0.7904

 Wadsworth equatione 1.1151 0.1288 0.8315 1.3986 < 0.0001

70 cm (65.0 cm ≤ d.b.h. ≤ 74.9 cm) 10

 D.b.h./ HT equationa 0.1383 0.1508 -0.2028 0.4794 0.3830

 D.b.h. equationb 0.4316 0.2238 -0.0747 0.9378 0.0859

 Nonlinear taper equationc -0.2042 0.1552 -0.5552 0.1468 0.2207

 NIMS equationd -0.1715 0.1521 -0.5155 0.1726 0.2888

 Wadsworth equatione 1.3288 0.2165 0.8391 1.8185 0.0002

 Total 587

 Pr > |t| = the probability that the mean difference between estimated and calculated volumes is greater than zero by chance alone.

a Vstem = e , where Vstem = merchantable stem volume in m3

, DBH = diameter at 1.37 m, and HT = total tree height in m.

b Vstem = e .

c , where Vstem = merchantable stem volume in m3

, DBH = diameter at 1.37 m (in cm),

HT = total tree height in m, Dupper = upper stem diameter merchantability limit in cm, and K = 0.00007854 · [1-(2 / b3)].

d Poletimber size trees: , where YPOLE = poletimber volume.

 Sawtimber size trees: , where YSAW = sawtimber volume.

e , where YS

 = stem volume.

f The 10-cm d.b.h. class consists only of trees with d.b.h. ranging from 12.5 to 14.9 cm.

YPOLE DBH HT = − + ⋅ ⋅ 2 0.378830 0.001855 ( )

YSAW DBH HT = + ⋅ ⋅ 2 0.82350 0.001630 ( )

( ) = 2.2020 ⋅ ( ) + 0.4434 ⋅ ( ) −1.8374 Log YS Log DBH Log HT

, (ln ) (ln ) ( ) bo b1 D BH b2 H T b3 D BH

stem V e

, (ln ) (ln ) ( ) 

1 BH 2 T 3 D BH 

b D b H b 

? ? 

+ + +

. (ln ) ( ) b0 b1 DBH b2 DBH

stem V e

. 

n ) ( ) 

DBH b2 DBH

? 

+ +

( ) ( 1.3 7) 0 ( 2) 3

3

1 2

b T

B H

b

b u p p e r

T

b

ste m B H H

D

D

V b D H K

( ) ( 1.37) 0 ( 2) 3

3

1 2 ? ? ? ? ? ? b ? T

BH

b

b upper

T

b

stem BH H

D

D

V b D H K

= • – – • • –
13

for trees with a d.b.h. > 60 cm is difficult due to their 

scarcity in the test dataset; however, it appears that NIMS 

equations might produce estimates that are comparable in 

accuracy to our equations for these trees.

Conclusions

Regression equations that satisfactorily estimate stem volume

from d.b.h. and total tree height, and from d.b.h. alone, will 

facilitate future forest inventory field work and analysis in 

Puerto Rico. Not surprisingly, equations developed from the 

local inventory data generally outperformed the equations 

used in NIMS, which were developed from data collected in 

south Florida.

Applying the regression equations developed from the 1990 

forest inventory dataset produces stem volume estimates 

acceptably close to those obtained by direct calculation, the 

method used in previous resource bulletins for Puerto Rico. 

However, to estimate volume growth and change over the 

past 10 years, 1990 tree volumes should be re-estimated using

these new regression equations and the directly calculated 

1990 values should not be employed in comparisons. 

These volume equations for Puerto Rico may have application on other Caribbean islands that share the same species 

and forest types, and the nonlinear taper equations have the 

flexibility to accommodate different upper stem diameter 

merchantability limits. However, some very important 

restrictions will have to be adhered to if these equations are 

applied to other forest inventory datasets in the Caribbean:

1. Use of these volume equations requires the assumption 

that stand d.b.h. and height distributions, tree forms, and 

log lengths are similar to those in the 1990 Puerto Rico 

forest inventory dataset.

2. Unsound cull trees (for example, those too rotten or 

poorly formed to be usable) and portions of trees would 

have to be accounted for separately and subtracted from 

the gross total volume for a more accurate estimate of net 

timber and sound cull volume.

3. Palms and tree ferns were excluded from the modelbuilding and testing datasets, so other equations or 

methods would have to be used to estimate their volume.

4. These equations estimate only stem volume and do not 

take into account smaller material such as branchwood, 

which can be used as fuel, or in craft items, small furniture components, or the like, and could have substantial 

value in the local markets.

Acknowledgments

The authors are grateful to Joseph Donnegan of the USDA 

Forest Service’s Pacific Northwest Research Station for his 

review of the manuscript. We also thank the following USDA

Forest Service, Southern Research Station personnel for their

assistance: Mark Brown, Sonja Oswalt, and Stanley Zarnoch 

for their reviews of the manuscript; Raymond Sheffield, 

James Rosson, and Larry Royer for their guidance and 

suggestions; and Andy Hartsell for his help in replicating 

previous volume estimation methods.

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15

Species List

Common namea Scientific name

Cabbagebark tree Andira inermis

Oxhorn bucida Bucida burceras

Pumpwood Cecropia schreberiana

Cordia Cordia spp. 

Mountain immortelle Erythrina poeppigiana

American muskwood Guarea guidonia

Mahoe Hibiscus elatus

Sacky sac bean Inga laurina

River koko I. vera

Mango Mangifera indica

Sweetwood Ocotea spp.

Matchwood Schefflera morototonii

African tuliptree Spathodea campanulata

Honduras mahogany Swietenia macrophylla

West Indian mahogany S. mahagoni

Malabar plum Syzygium jambos

White cedar Tabebuia heterophylla

a Nomenclature retrieved (January 25, 2005) from the Integrated Taxonomic 

Information System online database (http://www.itis.usda.gov).
16
Brandeis, T.J.; Kuegler, O.; Knowe, S.A. 2005. Equations for merchantable volume for subtropical 

moist and wet forests of Puerto Rico. Res. Pap. SRS–39. Asheville, NC: U.S. Department of 

Agriculture Forest Service, Southern Research Station. 15 p.

In Puerto Rico, where locally grown woods are primarily used for furniture and crafts production, 

estimation of wood volume makes it possible to estimate the monetary value of one of the many 

commodities and services forests provide to society. In the Puerto Rican forest inventories of 1980 

and 1990, workers calculated stem volume directly by applying geometric formulae to bole sections 

of merchantable trees. Field crews recorded several diameter and height measurements along the bole 

of each tree. If tree volume estimates were based on fewer tree measurements, this would significantly increase field crew productivity. For this reason, tree volume equations have been derived 

from Puerto Rican forest inventory data by directly calculating stem volume, then creating regression 

equations that estimate inside and outside bark merchantable stem volume from tree diameter at 

breast height and total height. 

Keywords: Forest Inventory and Analysis, merchantable stem volume, Puerto Rico, tropical forest, 

volume equations.
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Agriculture (USDA), is dedicated to the principle 

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