63F\ United States

Department of

Agriculture

f-R

v

6 s Forest Service

Southern Forest

Research Note

SO-378

Experiment Station September 1995

A Comparison of Height-Accumulation and Volume-Equation

Methods for Estimating Tree and Stand Volumes

R. B. Ferguson and V. C. Baldwin, Jr.

SUMMARY

Estimating tree and stand volume in mature plantations is time consuming, involving much manpower and

equipment; however, several sampling and volume-prediction techniques are available. This study showed that

a well-constructed, volume-equation method yields estimates comparable to those of the often more time-consuming, height-accumulation method, even though the

latter should be more accurate for any individual tree.

Plot volumes were estimated by both methods in a

remeasurement of trees in a 40-plot, planted slash pine

thinning study. The mean percent age difference in total

volume, inside bark, between the two methods ranged

from 1 to 2.5 percent across all the plots; differences

outside bark ranged from 7 to 10 percent. The results

were similar when the effects of site, plot mean values,

or tree-by-tree comparisons were incorporated.

Keywords: Growth and yield, finus elliotfii, plantation management

INTRODUCTION

Accurately estimating the volume of stands of trees

has jong been a goal of foresters. One method used extensively in the South is the height-accumulation (HA)

method, reported by Grosenbaugh (1954) and programmed for computer use by Lohrey and Dell (1969).

This method is accurate because it accounts for taper in

each tree that is measured; but, it has certain disadvantages. The required measurements on individual trees

are time consuming, and for maximum accuracy, upperstem diameters and heights should be measured directly-an almost impossible task when many large, old

stands are involved. This disadvantage can be overcome

by using reasonably accurate instruments such as

relaskops or tele-dendrometers. However, the accuracy

of these instruments is affected by the skill of the user,

the height and visibility of the tree bole, weather conditions, and the condition of the instrument itself. Careful

training of the personnel using the instruments, coupled

.with periodic calibration and maintenance, improves the

accuracy of the measurements but does not overcome

the problem of tree size nor does it greatly increase

speed. Even with improved instruments and training, a

typical field crew might be able to measure only 80 to

100 trees per day (Enghardt and Derr 1963).

The reduction in the number of workers, the aging of

research plantations, and the desirability of measuring

total height on all plot trees forced this research work

unit to seek an alternative to the HA method that had

been previously used in our research. The selected alternative was volume prediction equations developed for

each species of pine from sample-tree measurements

collected on plots with widely different site conditions,

thinning intensities, age classes, and locations. Several

of these equations had already been reported for each

of the species being studied (Baldwin and Feduccia 1987;

Baldwin and Saucier 1981; Clark and others 1991;

Zarnoch and others 1991).

Some felt that changing the method of estimating such

an important variable as volume would reduce the value

of the data and invalidate year-to-year comparisons.

Would the results be comparable, or would there be a

significant difference between the two methods? To answer these questions, measurement results of the two

methods were compared after conducting a specific case

study.

Results of this research should be generally applicable

to others because of the similarities between tree measurements on these research plots and those on standard inventory plots used by forest managers.

Southern Forest Experlment Station/T-10210 U.S. Postal Services Bldg., 701 Loyola Avenue, New Orleans, La. 70113

Forest Service, U.S. Department of Agriculture.

Serving Alabama, Arkansas, Louisiana, Mississippi, E. Oklahoma, Tennessee, E. Texas, Puerto Rico, U.S. Virgin Islands.

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METHODS

Volumes estimated by the HA method were compared

with volumes estimated by the volume-equation (VE)

method in a field study scheduled for periodic remeasurement in the winter of 1989-90. The study was a slash

pine (Pinus elhotfii var. elliottii Engelm.) test installed in

two plantations near Peason, Louisiana. Plantation A was

39 years old with an average site index of 81 ft (base

age 50), and plantation B was 41 years old with an average site index of 78 ft (base age 50). The study was

begun in the winter of 1964-65 with eight treatments

consisting of seven thinning intensities and an unthinned

check. Forty 0.15-acre plots were measured.

Field crews first measured all plot trees for total height

in feet using a relaskop and for diameter at breast height

(d.b.h.) in tenths of an inch. Then they returned to each

plot and selected sample trees following an established

sampling procedure. The relaskop was then used to

measure the height of these sample trees at each 2-inch

change in bole diameter to determine taper-the information needed for the HA method of estimating volume.

Individual tree cubic-foot volumes and finally total area

volumes, expressed in cubic feet per plot outside and

inside bark, were then computed using the height-accumulation and plot-summary computer programs developed by Lohrey and Dell (1969). These programs

calculate individual sample tree volumes by height accumulation and then, given the basal area of each tree,

estimate the total volume of each plot using the sampletree mean-volume to mean-basal-area ratio estimator.

Inside-bark measurements were estimated by applying

a constant mean-inside-bark-diameter to outside-barkdiameter ratio at each measured taper point. The mean

ratio selected, 0.87, had been previously determined from

earlier measurements of bark thickness of plantationgrown slash pines from this and other slash pine growthand-yield studies.’

The taper/volume functions of Thomas and Parresol

(1991) were applied separately to estimate the total volume of individual trees. The Thomas and Parresol volume equations were derived using linear and nonlinear

regression methods on felled-tree data from 199 sample

trees representing a range of ages, diameters, stand

‘Ferguson, R.B., Unpublished reports: “Thinning alternatives for

planted slash pines,” “The effect of age and residual basal area on

growth and yield of planted slash pine on a medium to poor site.” and

“Timing of first thinning of planted slash pine;” (on file: USDA Forest

Service, Southern Forest Experiment Station, RWU-4101, 2500

Shreveport Highway, Pineville, LA 71360).

densities, site indices, thinning treatments, and geographic locations.2 The Thomas and Parresol equations

were chosen for this comparison partly because some

of the data used to provide parameters for the VE models came from trees felled from plots within this thinning

study.The taper and corresponding volume functions are:

i2/D2 =b,(x-l)+b,sin(crrx)+b,cot(lrx/2) (1)

X”

C=kD2H[b,(x-l)+b2sin(c~x)+b,cot(~x/2)dx (2)

J

XI

Where:

d=

D=

diameter at a given height

diameter-outside-bark at breast height

(d.b.h.)

total height

height of observation/total height above

ground

lower relative height (h/H) and upper

relative height (h/H), respectively, where

h, is the lower height limit and h, is the

upper height limit

volume

3.14159

H=

x=

x, X” =

v=

Ic=

k =

b,,b2,b3,c =

ld40,OOO if diameters are measured in

centimeters and heights in meters or rd

576 if diameters are measured in inches

and heights in feet.

coefficients estimated from the data.

The estimated coefficient values for the slash pine data

are presented in table 1. Note that: (1) because these

equations involve predictions using relative heights and

diameters, the coefficients are valid for measurements

made in either Standard International (metric) or English

units when the appropriate k is selected, and (2) there

are separate sets of coefficients for trees from unthinned

stands and for trees from stands thinned at least once.

In the alternative application, individual tree volumes

were predicted using equation (2) and the corresponding coefficients from table 1, depending on whether the

tree was in a thinned or an unthinned plot. These volqhe Thomas and Parresol paper provides only coefficients for inside-bark taper or volume of planted slash pine. At our request, Mr.

Parresol refit this model to the original data to also provide coefficients for outside-bark taper and volume. We gratefully acknowledge

his contribution.

R. B. Ferguson is a forester, and V. C. Baldwin, Jr., is the principal forest biometrician at the U.S. Department of Agriculture, Forest Service,

Southern Forest Experiment Station, Silviculture of Artiiicially Regenerated Southern Pines Research Work Unit, 2500 Shreveport Highway,

Pineville. LA 71360.

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Table 1 .-Coeffioents for solving s/ash pine taper and volume equations by Thomas and Parresol

(1991)

Bole

measurement

Unthinned plantations

Inside bark

Outside bark

Thinned plantations

Inside bark

Outside bark

b,

-0.66200

-1 .01750

-0.74100

-1.02570

Coefficient values

b, b,

0.02780 0.00456

.01789 .01009

.03450 .00391

.01160 .00750

C

1.50

1.50

1.50

1.75

umes were then summed for each plot to obtain the predicted plot volumes.

The volume estimates from the two methods were then

statistically compared on both an individual-tree and a

plot basis. Differences were compared among all 40 plots

and between plots on the 2 different plantation sites.

Expected differences in volume due to the effect of thinning on taper (Baldwin and Feduccia 1991) were assumed to be accounted for by the use of the different

coefficients for equations applied to trees from thinned

or unthinned plots. The correlation coefficient (0 of the

observed and predicted values was computed for each

combination of volume estimates. In addition, the mean

percent difference, PDlfF =(100/n) Z (VE-HA)IHA; the

HA mean,m; the VE mean, E; and the mean difference, DlFF = (l/n) C ( E-HA) statistics were computed.

RESULTS AND DISCUSSION

Generally, the VE method predicted higher individual

tree and plot volumes than the HA method (table 2). The

total mean predicted plot volume, inside bark, for all 40

plots differed by only 1.9 percent between the 2 methods. The mean percent age difference for the outsidebark volumes was higher, differing by 9.0 percent.These

values remained about the same when age (site) and

tree-to-tree differences were considered. Figures 1 and

2 show an extremely close correlation between the two

volume-prediction procedures. The correlation would

have been even closer if the five highest volume plots,

which were unthinned, had been eliminated (fig. 1). However, all plots were included to generalize the comparisons. This does indicate, even though the taper-volume

equations were developed with different coefficients for

thinned and unthinned stands, that density still had a

strong influence on the volume predictions. Better predictions might be obtained by accounting for stand density either as a predictor variable in volume-taper

equations or by developing separate equations for different density classes. The latter method has been used

with live crown ratio as the density surrogate (Dell 1979).

The differences between the methods are similar for trees

and plots (table 2), so it does not appear that the sampling procedure and the procedure for estimating the

mean-volume to mean-basal-area ratio, when used with

height accumulation to estimate plot volumes, contributed to the differences.

The 9.0-percent difference for outside-bark volumes

was considered to be acceptable but large, which warranted additional study. A subset of 25 sample trees was

selected that differed in outside bark volume by 6 ft3 or

more between the HA and the VE methods. Using the

measured 2-inch bole diameter change points, their corresponding heights for each of these sample trees, and

the compatible taper equations based on equation (1)

the inside- and outside-bark diameters at each height

were predicted.The height-versus-diameter points for the

two methods were then plotted in an overlay fashion. A

consistent trend was noted in each of these graphs. The

bole taper of the lower portion of each sample tree was

similar for the two methods, even though the diameters

predicted by the taper equations were consistently larger.

The upper portion of each bole, roughly corresponding

to the live crown, tapered more sharply using the HA

method.This difference indicates lower total outside-bark

volume predictions by the HA method for the crown por-.

tion of the bole (fig. 3).

Although the study reveals differences between the

methods, it does not indicate which of the two methods

is more accurate. Possibly, outside-bark measurements

made on the population of trees in question should give

more accurate results than those from an equation developed from measuring some other population. Moreover, given that the field measurement error is low, it is

reasonable to assume that the HA method will be more

accurate for any given tree, particularly if the height for

each l-inch change in diameter rather than for each

2-inch change is measured. However, the value of that

precision has to be weighed against the time needed to

take those additional measurements.

The closeness of fit of these two methods may not

hold for all plantations. Tests should be conducted on

plantations with different site qualities, ages, densities,

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Table 2.-Closeness of fit statistics based on estimating total volume, inside and outside bark, by

height accumulation (HA) procedures versus the Thomas and Parresol(1991) volume

equations (VE)

Treatment category

for comparing methods

of estimating total

bole volume

Statistic’

n r PDlFF HA VE DIFF

Full plot predictions

All plots 40 0.991 1.92

Plantation A 16 ,968 2.49

Plantation B 24 ,993 1.57

All plots 40 0.995 9.03

Plantation A 16 ,996 9.51

Plantation B 24 ,995 8.72

Sum of sample trees only

All sample trees 40 0.966 1.70

Plantation A 16 ,954 2.90

Plantation B 24 ,972 .90

All sample trees 40 0.978 8.64

Plantation A 16 ,962 9.88

Plantation B 24 ,982 7.82

Tree-by-tree comparisons

All sample trees

Plantation A

Plantation B

206

314

0.990 1.27

,987 2.55

,992 .42

All sample trees 520 0.990 9.11

Plantation A 206 ,988 10.35

Plantation B - 314 ,992 8.30

______________ Fp/p/ot (i.b,) ___~~_~~~_~~~~

450.75 455.61 4.86

451.21 457.43 6.22

450.45 454.40 3.95

____ _____ _____ Fp/p/ot (0.b.) __--__________

603.24 658.00 54.76

604.00 659.84 55.84

602.74 656.78 54.04

______________ Sample (i.6,) _---_____-_--_

216.26 219.94 3.68

227.74 234.54 6.80

208.60 210.20 1.60

_______ _______ Sample (0.b.) ~~~~~~~~~_~~__

289.75 315.09 25.34

305.25 335.27 30.02

279.42 301.63 22.21

__ ____________ Fr/rree (i,b.) ~~~~~~~~_~~~~~~

16.67 16.92 0.25

17.71 18.22 .51

16.00 16.07 .07

______________ Ft3/1ree (0.b.) _______----_--

22.32 24.24 1.92

23.71 26.04 2.33

21.41 23.05 1.64

l n = sample size of the comparison;r= correlation coefficient; PO/Fe mean percent age

difference = (100/n) X (VE-HA)/‘HA = height accumulation mean; VE = Thomas and Parresol

(1991) volume equation mean: D/FF= mean difference = (l/n) I( VIZ-HA).

4
Inside Bark l.S@J

i Outslde Bark ,/i

i

100 3w 500 700 900 1,100 1,300 100 300 500 700 800 1,100 1,300

Total \FDhKM) (cubic feet) by HeQht Accumulatbn ComputaaOne

Figure 1 .-Comparison of the height-accumulation (HA) method and the Thomas and Parresol(l99 1) volume-equation (VE) method

for computing plot volumes, Inside and outside bark.

Inside Bark

_- 1. /

I 10 20 30 $0 50 80 70

Figure 2.-Comparison of the height-accumulation (HA) method and the Thomas and Parresol(1991) volume-equation (VE) method

(VE) computing sample-tree volumes, inside and outside bark.

5
\I

\:

1

87654321012345678 87654321012345678

Bole Radius Outside bk (ii) Bole Radius Inside Bark (inches)

8c

70

8u

!d

40

30

xl

10

0

I \ A/ \

/ \

I \

I \

Figure 3.-Comparison of bole taper, outside and inside bark, for a typical sample tree.
and thinning treatments as well as different species to

confirm these results. It would be better, but also more

time consuming, to develop volume equations from trees

sampled from each of the studies in question. Nevertheless, the predictions were close enough to justify the

change to the VE method in this case, and it appears

that this procedure is acceptable if time and field personnel are limited.

CONCLUSION

There was no significant loss of precision or year-toyear comparability when estimating stand volumes with

volume equations instead of height accumulation. This

result can largely be attributed to the quality of the database used to derive the equations. Thus, volume equations can be a viable alternative to the height-accumulation

method for estimating stand volume.

LITERATURE CITED

Baldwin, V.C., Jr.; Feduccia, D.P. 1987. Loblolly pine

growth and yield prediction for managed west gulf plantations. Res. Pap. SO-236. New Orleans, LA: U.S. Department of Agriculture, Forest Service, Southern

Forest Experiment Station. 27 p.

Baldwin, V.C., Jr.; Feduccia, D.P 1991. Compatible treevolume and upper-stem diameter equations for plantation loblolly pines in the west gulf region. Southern

Journal of Applied Forestry. 15(2): 92-97.

Baldwin, V.C., Jr.; Saucier, J.R. 1981. Weight and volume prediction of unthinned, planted longleaf pine. In:

Gresham, Charles A., ed. Proceedings, 1981 Southern forest biomass workshop; 1981 June 11-12;

Georgetown, SC: Clemson University Press: 27-38.

Clark, Alexander, III; Souter, Ray A.; Schlaegel, Bryce

E. 1991. Stem profile equations for southern tree species. Res. Pap. SE-282. Asheville, NC: U.S. Department of Agriculture, Forest Service, Southeastern

Forest Experiment Station. 113 p.

Dell, Tommy R. 1979. Potential of using crown ratio in

predicting product yield. In: Frayer, W.E., ed. Forest

resources inventories: workshop proceedings; 1979

July 23-26; Fort Collins, CO: Colorado State University Press: 843-851. Vol. 2.

Enghardt, Hans; Derr, H.J. 1963. Height accumulation

for rapid estimates of cubic volume. Journal of Forestry. 61: 134-l 37.

Grosenbaugh, L.R. 1954. New tree measurement concepts: height accumulation, giant tree, taper and shape.

Occas. Pap. SO-134. New Orleans, LA: U.S. Department of Agriculture, Forest Service, Southern Forest

Experiment Station. 32 p.

Lohrey, Richard E.; Dell, Tommy R. 1969. Computer programs using height accumulation for tree volumes and

plot summaries. Journal of Forestry. 67: 554-555.

Thomas, Charles E.; Parresol, Bernard R. 1991. Simple,

flexible, trigonometric taper equations. Canadian Journal of Forest Research. 21: 1132-l 137.

Zarnoch, Stanley J.; Feduccia, Donald P.; Baldwin, Virgil

C., Jr.; Dell, Tommy Ft. 1991. Growth and yield predictions for thinned and unthinned slash pine plantations

on cutover sites in the west gulf region. Res. Pap. SO264. New Orleans, LA: U.S. Department of Agriculture, Forest Service, Southern Forest Experiment

Station. 32 p.

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