--&jlq -

U.S. Department of Agriculture S.E.F.ES. Libra’Y

Forest Service

Research Pa.per SO-154

Growth and Yield Estimation for Loblolly Pine

in the West Gulf

PAULA.MURPHYANDHERBERTS.STERNITZKE
SUMMARY

An equation system is developed to estimate current yield, projected

basal area, and projected volume for merchantable natural stands on a

per-acre basis. These estimates indicate yields that can be expected from

woods-run conditions.

1979
Growth and Yield Estimation for Loblolly Pine in the West Gulf

PA C’I, A. MURPHY

A,VLj

HERBERT S. STERiVITZKb;

INTRODUCTION

Loblolly (E%zus taedu L.) is the keystone of

the southern pine forest products industry.

Except in Florida, where slash pine (P. c>lliottii

Engelm.) prevails, it dominates in the Atlantic

and Gulf coastal States south of New Jersey,

accounting for nearly half the total southern pine

inventory. About a third of the loblolly resource is

west of the Mississippi River (Sternitzke and

Nelson 1970).

According to the most recent State-by-State

field inventories conducted by the Forest Service

as part of.the nationwide Forest Survey, standing

volume of southern pine on commercial forest

land’ west of the Mississippi is more than 22

billion cubic feet (table 1); nearly 13 billion is

loblolly pine (table 2).

Table’ l.- Volume by state of southern pine growing stock on

commercial forest land west of the Mississippi

Kil-er

Date

State of Growing Stock

Survey

--------~~~~~o~_-_-_-_-

cubic feet

Arkansas 1969 6,192.2

Louisiana’ 1974 6,779.4

Missouri 1972 305.0

Oklahoma 1976 1,002.4

Texas 1975 8375.2

All States 22354.2

lExcludes southeast Louisiana parishes.

‘Forest land either producingor capableof producinycropsof

industrial wood and not withdrawn from timber utilization.

West of the Mississippi Valley, loblolly pine is

restricted to the Coastal Plain in Arkansas,

Louisiana, southeast Oklahoma, and east Texas.

The main factor limiting the northern extension

of its natural range probably is temperature.

Also, damage by snow and sleet may be a factor.

The western extension of loblolly probably is

limited by lack of precipitation (USDA Forest

Service 1965).

Information on growth and yield of natural

stands of loblolly pine in the West Gulf Coastal

Plain is deficient. Older studies do not provide

information on stands not fully or normally

stocked. More recent studies were conducted in

other regions of the South and sampled pure,

uniformly stocked stands, but stands encountered

in practice do not approach the pure, ideal

conditions sampled in these studies. Many are not

uniformly stocked, most have a significant

hardwood component, and their yields are less.

The growth and yield relationships

represented here have been observed on inventory plots throughout the West Gulf. Inventory

data have more unexplained variation than do

data from controlled studies, and variables tend

to be clustered rather than dispersed throughout

their range of possible values. This type of

information, however, has the advantage of

depicting actual conditions.

Data

The data are from remeasured plots of

statewide surveys made by the Forest Service.

The interval between surveys is about 10 years.

Sample plots are at the intersections of a s-milesquare grid. Information was collected according

Paul A. Murphy is Principal Mensurationist. Southern Forest Experiment Station, Monticello, Arkansas, and Herbert S.

Sternitzke is Principal Resource Analyst, Southern Forest Experiment Station, Forest Service-USDA, New Orleans,

Louisiana.
to standardized inventory procedures.” Plots have

lo-point samples spaced 66 feet apart over about

1 acre.

Plots within the Coastal Plain of the West

Gulf region were screened by use of these criteria:

(1) at least half the basal area in trees 5 inches and

larger in the initial survey had to be loblolly pine;

(2) all 10 points at each location had to be

recovered in the remeasurement tally; (3) site

quality information had to be collected from

loblolly pine sample trees; (4) plots had to be free

from catastrophic mortality, (5) sampled stands

had to be even-aged and natural in origin and (6)

plots had to be undisturbed by logging or other

activity between measurements. I fates of surveys

are Louisiana, 1964 and 19i3; east Texas, 1965

and 1975; and Arkansas,:’ 1969 and 1978.

The geographical distribution of 145 sample

plots that met these criteria is shown in figure 1.

Numbers of plots by age, site, and density are

tabulated in table 3. Average stand age is 38;

average basal area, 70 square feet per acre; and

average site index, 90 feet. The range of data is

from 17 to 137 square feet of basal area at

remeasurement; site index varies from 68 to 127

feet; and stand age ranges from 18 to 85 years at

remeasurement.

These stand and tree statistics were

calculated: (1) site index from equations

developed by Farrar (1973); (2) stand age at first

measurement and at remeasurement; (3) basal

area in square feet per acre of loblolly pine trees 5

inches and larger at the first measurement and of

‘Renewable Resources Evaluation Staff. Forest resources

inventory work plan. Yuppl.. For. Serv. Handb. 4809.11,83 p.

U.S. Dep. Agric. For. Serv., South. For. Exp. Stn.

‘Includes only counties in the Coastal Plain.

the same trees at remeasurement; and (4) solid

wood volume from a l-foot stump to a 4-inch top,

outside bark, of trees included in the basal area

calculations. The data were then summarized

into plot statistics according to Beers and Miller

(1964).

Estimation in Equation Systems

Much recent research in the growth and yield

of even-aged natural timber stands has used data

from remeasured permanent plots. This information has prompted interest in estimating not only

current volumes but also future yields. Some

variables used to predict stand volume, such as

site quality, are assumed to be constant. Others,

such as stand age, change by a known amount.

Others, such as basal area, change by an unknown

amount, and their future values must be estimated. The result is an interdependent set of

prediction equations.

Clutter (1963) fitted a yield model, a basal

area growth model, and a volume growth model

by least squares. He derived basal area projection

and volume projection equations by integrating

the growth functions to produce a five-equation

system for estimating growth and yield of loblolly

pine. Buckman (1962) Curtis (1967) and Moser

and Hall (1969) also have developed equations

based on the concept that the yield function is the

integral form of the growth function.

Remeasurements on the same plot introduce

the problem of correlated errors, which violates

one of the basic assumptions of the standard

linear model. Sullivan and Clutter (1971)

modified Clutter’s cubic yield equation and

projected basal area equation by combining them

into a single function. Parameters were esTable 2.- Volume by St& und diumetrr class of loblolly pine growing stock on commercial forest land west of the Mississippi

Ricer

Diameter Class

(inches at breast height)

Date 29.0-

State of All 5.w 7.0. 9.0. ll.O- 13.0- 15.0- 17.0- lY.O- 21.0. and

Survey classes 6.9 8.11 10.9 12.9 14.9 16.9 18.9 20.9 28.9 larger

___---_____________--___-------_---_-__........_-------,~~~~~~~~~ cubic ]~et_____________------------__------------_____..~..~_______.

Arkansas 1969 2603.8 250.2 283.5 357.9 398.5 374.1 313.9 246.8 173.5 196.7 8.7

Louisiana] 1974 4,804.b: :3w.3 614.0 650.4 690.4 668.9 523.7 47U.fi 319.8 449.1 34.4

Oklahoma 1976 63.7 :1.7 6.7 10.1 6.2 10.3 2.8 5.0 10.2 8.7 ______

‘Texas 1975 5.1868 339.9 545.2 721.8 808.0 779.9 655.1 513.3 367.2 429.9 26.5

All States 12658.9 977.1 1,449.4 1,740.2 1.903.1 1,8X3.2 1.495.5 1.235.7 870.7 1,084.4 69.6

‘Excludes southeast Louisiana parishes.

2
LOUISIANA

Figure l.-Gcoy,np/ticul tiistri6ution of loblolly piw suwplr plots irr Arka~as. Louisiana. and Texas.
Table 3.- Plot distribution by age, sitr index, and loblolly basal area at wmPaaurwnent

Age Site index Basal area per acre

years &et -----------------___-____---sguarr fQQt_____________________________________------

< 20 21-40 41-60 61-80 x1-100 100+ Total

1 l-20 70 ____ _._. 0

80 ____ ____ ____ ____ 0

90 _._. _._. 0

100 1 1 ____ ____ 2

110+ ..-- ---- ____ ____ ____ ____ 0

Total 0 1 1 0 0 0 2

21-30 70 1 1

80 ___. 3 1 1 __._ 5

90 1 3 4 2 _ _ _ _ _ _ _ _ 10

100 ____ 1 2 3 1 3 10

110+ ---- 2 2 3 2 ____ 9

Total 2 9 8 9 4 3 35

31-40 70 ____ 4 3 2 1 ____ 10

80 1 2 9 2 3 1 18

90 ____ 4 6 10 5 1 26

100 __._ 3 4 2 9

110+ ___. 1 1

Total 1 10 18 17 13 5 64

41-50 70 1 1 2

80 ____ 1 1 1 2 2 7

90 __._ 2 3 ____ 7 12

100 2 1 2 5

110+ ---- ---- ____ ____ ____ ____ 0

Total 0 2 3 6 4 11 26

51-60 70 ____ 1 ____ 1 ____ ____ 2

x0 ____ ____ 1 ____ 1 2

90 __._ ___. ____ 1 2 ____ 3

100 ____ 3 ____ 1 4

110+ ____ 0

Total 0 1 0 6 2 2 11

61+ 70 _._. 1 _._. 1

80 _._. ____ 2 1 ____ 3

90 ____ ____ ____ 1 1 1 3

100

::::

____ ____ ____ ____ ____ 0

110+ ---- ._._ ____ ___. ____ 0

Total 0 0 1 3 2 1 7

All ages 70 1 6 4 3 2 0 16

80 1 6 10 7 7 4 35

90 1 7 12 17 8 9 54

100 0 2 3 11 6 8 30

110+ 0 2 2 3 2 1 10

Total 3 23 31 41 25 22 145

timated by maximum likelihood. The fact that

errors between two measurements of the same

plot were correlated was taken into account in the

maximum likelihood procedure. Seegrist and

Arner (1978) extended this maximum likelihood

procedure to more than one remeasurement.

Furnival and Wilson (1971) dealt with the

problems of estimating parameters in a system of

equations to predict growth and yield of forest

4

stands. They pointed out that using ordinary least

squares may not be an appropriate means of

estimation: often it is necessary to treat avariable

as both independent and dependent; coefficients

in one equation may be related to coefficients in

other equations; and residuals from equations

may be correlated. They described several

simultaneous equation techniques widely used in

econometrics that might be applied togrowth and
yield equation systems. These techniques explicitly treat the problems of equation systems.

One such technique, three-stage least squares

(3SLS), is applied here to an equation system that

estimates cubic yield and basal area of even-aged

stands of loblolly pine.

Model Development

The equations are those of Clutter (1963) and

Sullivan and Clutter (1971). In modified form,

they are:

(1) E(lnISL) = In(HI)A,/AL+rrl(l-Al/AZ)+cl:! (I-AI/A2)ln(Bd+

tr.,(I-AI/AP)I~(BI)~

(2) k;(lnL’,) = P,, + P ,S + P,Al -I + P:, (InB,)

where

A. =

Bf =

s =

V. =

1

Ui, Pi =

stand age at the ith occasion

basal area in square feet per acre of

loblolly pine trees 5 inches in diameter

and greater at breast height

site index in feet at base age 50

cubic volume (inside bark) per acre of

loblolly pine trees 5 inches in diameter

and greater at breast height from a lfoot stump height to a 4-inch top

diameter, outside bark

parameters to be estimated.

The symbol E( . ..) denotes expected value of the

quantity within the parantheses. The form of the

basal area equation (1) was suggested by

Seegrist.

Variables in simultaneous equation models

may be designated by standard econometric

terminology (Furnival and Wilson 1971, Theil

1971, Pindyck and Rubinfeld 1976) as exogenous

(independent) or endogenous (dependent). The

variables stand age Ai and site S in equation (1)

are uncontrolled covariates and exogenous to the

equation system. The variables ln(Bi) and ln(Vi)

are endogenous variables. The original set of

relationships, equations (1) and (Z), are the

structural equations.

The estimation procedure depends upon

whet,her unique estimates of parameters can be

obtained. If so, the system is said to be exactly

4Personal communication, March 5, 1979.

identified. If more than one estimate is possible,

the system is overidentified. When none is

possible, it is unidentified. The degree of identification can be determined by changing the

system of equations to its reduced form, where the

exogenous variables occur on the right side, and

the endogenous variables occur on the left. The

reduced form of equation (2) is:

(is) E(lnV,)= h,,+X,S+ AzAZ-‘+ h.,(A,/A,)ln(B,)+ h,

(I- A,/A,) + h, (I-AJA,)ln(B,) + h,(l-A,/A,)ln(B,)Z

where L=P,,. X,=P,. h,=P2. h,=& h4=Pfo,, X5=

/3 gL and h,, =&I:,.

Equation (1) is in reduced form. Estimates of

the model parameters can be determined from

those of the reduced form equations. Examination of the reduced form equations shows that

there are two sets of estimates for the parameters

a17 a,, and ay. One set comes directly from

equation (l), and the other is a, = A J h 3, a2 =

A J A 3, a3 = A SJ h 3.

Since the system is overidentified and the

endogenous variables probably are correlated,

using ordinary least squares is inappropriate.

Two-stage least squares (ZSLS) could be used to

estimate the parameters of an overidentified

system. The basal area equation (1) is fitted using

ordinary least squares, and expected values are

used as independent variables in fitting equation

(2).’ But using 3SLS is more efficient because it

takes account of possible correlation in errors

between equations in overidentified systems.

The 3SLS is an extension of BSLS. Estimates

of the second stage are used to obtain estimates of

residuals. Residuals are used to obtain estimates

of cross-equation correlations. Zellner and Theil

(1962) originated this technique, and estimates

obtained are consistent. The efficiency of 3SLS

ususally exceeds that of the two-stage procedure.

Its efficiency is matched by 2SLS only if there are

no correlated errors between the system

equations (Madansky 1964).

But 3SLS would not estimate the likely serial

correlation between original and remeasured

observations in equation (2). However, equation

(2) can be decomposed into two components - a

function that predicts current yield and another

that forecasts future volume -so that these three

equations can be solved:

C-1) E(lnH,)= In(B,)A,/A,+tr,(l-A,/A,)+tr,(l-A,/AL)ln(BI)+

tr,(l-A,/A,)I~I(B,)~

(5) E(lnV,) = /3,, + fllS + PLAl -I + &ln(B,)

5
(6) E(lnV,) = 0” + ~7,s + beA, -I + P31n(B,).

Data were evaluated with the Statistical Analysis

System (SAS) 3SLS procedure (Barr, Goodnight,

Sall, Helwig 1976) that handles side conditions on

the parameters.

Results

The results of analyses are shown in table 4.

Cross-equation correlations of residuals from the

second stage of the analysis are:

In(B,) WV,) In(V,)

In(B) 1.0 -.34457 -.37965

In(V,)

[

-.34457 1.0 .60611

In(V,) -37965 .60611 1.01

A negative correlation results between the

basal area projection (4) and the two yields (5,6).

Also, a stronger correlation results between

projected basal area and projected volume than

between projected basal area and current yield.

Residuals between current and projected yields

are positively correlated. The same pattern of

correlated errors has been reported by Seegrist

and ArneP. Sullivan and Clutter (1971) observed

a positive correlation between residuals of

current versus projected yields.

5Personal communication, March 5, 1979.

Although refined estimates of parameters

might be obtained by calculating the residuals

and obtaining new values for cross-equation

correlations, Madansky (1964) found that

iterated 3SLS offers no practical gain in efficienCY.

Whether any advantage results from using

simultaneous equation procedures, such as SSLS,

cannot be settled on the basis of the simple

equation system we presented here. But as the

prediction of growth and yield evolves from

reliance upon a single equation to a set of

interdependent ones, the applicability of these

techniques becomes increasingly relevant. Furnival and Wilson (1971) and Sullivan and

Reynolds (1976) have outlined the hazards of

applying ordinary least squares to equation

systems. Three-stage least squares has an advantage over other simultaneous equation

procedures such as maximum likelihood.

Software packages are readily available, and

extension to more than one remeasurement is

straightforward.

Table 4.- Estimates of model coefficients and their standard

errors

Parameter

al

a2

a3

PO

Estimate Standard Error

1.55265 0.81079

1.55846 0.48408

-0.15086 0.07115

3.02439 0.12324

0.00515 0.00114

-20.17542 1.04924

1.06693 0.02804

Table 5.- Evaluation oj model predictions

Criterion

Equation

Current volume

V, Projected volume

V, Basal area

Mean Value D RMSE2

____________________Cu& jeet________________----

897 82 193

1,754 -24 235

____________________SQuare fee&__ ______--_-_ -----

69.8 -1.5 8.2

D%3 RMSE%*

__________Percent ____ _ ____

12 30

1 15

___-_-_Peroe& _____---_

-1 12

‘Mean difference 5 =

c

-

(4i _ Y j)

n

2Root mean square error RMSE = J&x (?i - Yi)2

3Percentage of mean difference E% = k

4Percentage of root mean square error RMSE% = 1; 2 (qi - Yi)2

Yi z

x 1oo

where n = number of observations, Yi = actual value, and 9 i = predicted value.

6
Our yield equations accounted for 90.7 and

92.9 percent of the variation about mean cubicfoot volume for current V, and projected V2 stand

volumes. The basal area equation explained 91.3

percent of the variation about mean projected

basal are B,.

Table 5 provides information on the prediction performance of the equation system. The

mean difference indicates that the current

volume equation overestimates volume, and the

projected volume and basal area equations give

slight underestimates on the average. The root

mean square errors penalize large deviations

from the observed values more than average

differences. Percentage of mean difference and

percentage of root mean square error depict error

as a proportion of observed values.

Application of the equations is straightforward. Projected basal area can be predicted

with the equation

(7) In(E,) = In(B,)A,/A, + 1.55265(1-AI/AL) + 1.55846(1-A,/A,)

In(E,)- .lsOXG(I-A,/A,)ln(U,)’

Current and projected yields can be estimated by

(8) In(\‘) = :k024:<9 + 0.005155 - 20,17542/A + l.O66931n(B).

For example, suppose estimates of current yield

and projected volume and basal area in 10 years

are wanted for a loblolly pine stand at age 40 with

a site index of 85 feet and a basal area of 70 square

feet per acre. Current yield estimate is:

V, = exp [LO2439 + 0.00515(85) - 20.17542/40 + l.O6693ln(70)]

= 1791 cubic feet.

Projected basal area is:

B, = exp [ln(70)40/50 + 1.55265(1-40/50) + 1.55846

(I-40/50)ln(70)- .150s6(1-4o/5o)ln(70)Z].

= 89 square feet.

Projected volume is:

Vr = exp (X02439 + 0.00515(85) 20.17542/50 + 1.0669Xln(89)].

= 2560 cubic feet.

Periodic annual increment for merchantable

volume would be 77 cubic feet. Figure 2 illustrates cubic yields expected for different

densities for site index 90; figure 3 shows

projected basal area yields for various initial

densities; and figure 4 depicts projected volumes

for different initial densities for site index 90. The

estimates indicate yields that can be expected

from extensive areas of woods-run conditions.

0330

STAND AGE (YEARS)

Figure 2.- Relationship of current yield to basal area and

age for natural even-aged loblolly pine stands in

the West Gulf Coastal Plain, site index 90.

Figure 3.-

7

40 50

STAND AGE (YEARS)

Relationship of projected basal area to age and

initial basal area at age 3Ofor natural even-aged

loblolly pine stands in the West Gulf Coastal

Plain.
STAND AGE (YEARS)

LITERATURE CITED

Barr, A.J., J. H. Goodnight, J. P. Sall, and J. T.

Helwig. ,

1976. A user’s guide to SAS 76. SAS Institute,

Inc., Raleigh, N. C., 329 p.

Beers, T. W., and C. I. Miller.

1964. Point sampling: research results, theory,

and applications. Purdue Univ. Agric. Exp.

Stn. Res. Bull. 786, 56 p.

Buckman, R. E.

1962. Growth and yield of red pine in

Minnesota. U.S. Dep. Agric. Tech. Bull.

1272, 50 p.

Clutter, J. L.

1963. Compatible growth and yield models for

loblolly pine. For. Sci. 9:354-371.

Curtis, R. 0.

1967. A method of estimation of gross yield of

Douglas-fir. For. Sci. Monog. 13, 24 p.

Farrar, R. M.

1973. Southern pine site index equations. J.

For. 71:696-697.

Furnival, G. M., and R. W. Wilson.

1971. Systems of equations for predicting forest

growth and yield. In Statistical Ecology 3:43-

55. G. P. Patil, E. C. Pielow, W. E. Waters,

eds. Penn. State University Press, University

Park.

Madansky, A.

1964. On the efficiency of three-stage least

squares estimation. Econometrica 32:51-56.

Moser, J. W., and 0. F. Hall.

1969. Deriving growth and yield functions for

uneven-aged forest stands. For. Sci. 15:183-

188.

Pindyck, R. S., and D. L. Rubenfield.

1976. Econometric models and economic forecasts. 580 p. McGraw-Hill, Inc., New York.

Seegrist, D. W., and S. L. Arner.

1978. Statistical analysis of linear growth and

yield models with correlated observations

from permanent plots remeasured at fixed

intervals. Ikr Growth models for long-term

forecasting of timber yields. p. 203-223, Pub.

FWS-1-78. J. Fries, H. E. Burkhart, T. A.

Mar, eds., Va. Polytech. Inst. State Univ.,

Blacksburg.

Sternitzke, H. S., and T. C. Nelson.

1970. The southern pines of the United States.

Econ. Bot. 243142-150.

Sullivan, A. D., and J. L. Clutter.

1971. A simultaneous growth and yield model

for loblolly pine. For. Sci. 18:76-86.

Sullivan, A. D., and M. R. Reynolds, Jr.

1976. Regression problems from repeated

measurements. For. Sci. 22:382-385.

Theil, H.

1971. Principles of econometrics. 736 p. John

Wiley and Sons, Inc., New York.

U.S. Dep. Agric., Forest Service.

1965. Silvics of forest trees of the United States.

U.S. Dep. Agric. For. Serv. Agric. Handb.

271, 762 p.

Zellner, A., and H. Theil.

1962. Three-stage least squares: simultaneous

estimation of simultaneous equations.

Econometrica 30:54-78.
MURPHY, PAUL~A., AND HERBERT S. STERNITZKE,

1979. Growth and yield estimation for loblolly pine in the

West Gulf. U.S. Dep. Agric. For. Serv. Res. Pap. SO-154,

8 p. South. For. Exp. Stn., New Orleans, La.

Presents an equation system developed to estimate current

yield, projected basal area, and projected volume for

merchantable natural stands on a per-acre basis. These

estimates indicate yields that can be expected from woods-run

conditions.

Additional keywords: Pinus taeda, volume prediction, basal

area projection, stand volume, natural stand yields.