United States

Departmeqt of

Agriculture

Forest Service

Southern Forest

Experiment-Station

New Orleans,

Louisiana

Research Paper

SO-264

Growth and Yield

Predictions for Thinned

and Unthinned Slash

Pine Plantations on

Cutover Sites in the

West Gulf Region

Stanley J. Zarnoch, Donald P. Feduccia,

Virgil C. Baldwin, Jr. and Tommy R. Dell
SUMMARY

A growth and yield model has been developed for thinned and

unthinned slash pine (Pinus diottii Engelm. var. elliottii.) plantations

on problem-free cutover sites in the west gulf region. These sites were

originally virgin stands where clearcutting was followed by grazing and

repeated burning that controlled the woody vegetation, thus allowing

plantation establishment without site preparation. The model was based

on the moment-percentile method using the Weibull distribution for tree

diameters. This technique was applied to unthinned and thinned stand

projections and, subsequently, to the prediction of residual stands

immediately after thinning. Generally, initial thinnings were from below

but at later ages a good distribution of tree diameters across the plot was

also a criterion. The data base upon which the model’s parameters were

estimated was obtained from several studies throughout the region and

contained stands well into the 40-year age class. The growth and yield

prediction system was tested against the data used in its development

and, generally, the predicted stand- and yield-table variables averaged

within 5 percent of the observed values. In addition, predicted trends for

unthinned and thinned plantations and some comparisons of results of

these management alternatives are given under specific site and stand

conditions.

June 1991
Growth and Yield Predictions for Thinned <and Unthinned Slash ”

Pine Plaritations on Cutover Sites in the West Gulf Region

Stanley J. Zarnoch, Donald P. Feduccia, Virgil C. Baldwin, Jr. and Tommy R. Dell

i INTROD’UCTION

Slash pine (Pinus elliottii Englem. var. elliottii.) has been extensively planted in the west gulf region on

problem-free cutover forest sites, which were virgin stands where clearcutting followed by grazing with repeated

burning controlled the woody vegetation. This procedure had permitted plantation establishment without site

preparation. The stand projection system called USLYCOWG (Dell and others 1979) was developed to estimate

the yield of unthinned plantations of this type by predicting the parameters of a Weibull diameter distribution

based model. However, no provision was made for estimating thinned-stand yields. USLYCOWG has been

used throughout the west gulf region to predict unthinned-stand yields, but most of the data used to develop the

model was obtained in stands less than 25 years old. Estimates of yields beyond this age, which are needed for

sawtimber management, are questionable. The objective of the research reported here was to develop a stand

projection system for estimating yields of thinned and unthinned slash pine stands in the west gulf region. A n

expanded data pool contained information on stands well into the 40-year age class. In addition, an approach,

called parameter recovery (Matney and Sullivan 1982, Hyink and Moser 1983), was used in modeling the

diameter distributiong. \

PLANTATION DATA

The study’s data pool consisted of 507 unthinned-stand yield observations and 543 thinned-stand growthperiod observations from 0. lo- to 0.25acre plots (in slash pine plantations established on problem-free cutover

forest land. The length of the growth period was generally 5 years but ranged from 3 to 11 years. The plots

were not located in areas where survival was poor or where heavy insect, disease, or other damage was present.

For the most part thinning was from below, but at later ages a good distribution of tree diameters across the

plot was also a criterion. In addition, there were 530 residual-stand observations of post thinning conditions.

Diameter at breast height (d.b.h.) to the nearest 0.1 inch was taken for all trees at each observation on a

plot. The site index equation of Zarnoch and Feduccia (1984) (fig. 1) was used to calculate site index for each

plot from a sample of dominant and codominant trees. Upper-stem height and outside-bark diameter measurements were made on sample trees, and volume per acre was determined by the height accumulation method

(Lohrey 1967l, Lohrey and Dell 1969). This process utilized height measurements from breast height to

treetop along the bole at diameters that were multiples of 2 inches (i.e., 2 inches, 4 inches, 6 inches). The

distributions of observations by various age, site, and density combinations are summarized for unthinned-stand

(tables 1 through 3), thinned-stand (tables 4 through 9), and residual-stand (tables 10 through 13) observations.

MODEL DEVELOPMENT

The moment-percentile method was used to estimate the parameters of the Weibull distribution. This

technique was then applied to unthinned- and thinned-stand projection situations and, subsequently, to the

1

Lohrey, Richard E. 1967. Unpublished special report, “Description and use of two new computer programs for summarizing sample

plot data. ” On tile with: USDA Forest Service, Southern Forest Experiment Station, Pineville, Louisiana.

Stanley J. Zamoch is mathematical statistician, Southeastern Forest Experiment Station, Athens, GA’ 30602; Donald P. Feduccia is chief

of forest management, Office of Forestry, Baton Rouge, LA 70821; Virgil C. Baldwin, Jr. is research forester, Alexandria Forestry Center,

Pineville, LA 71360; and Tommy R. Dell is project leader, Institute for Quantitative Studies, Southern Forest Experiment Station,

New Orleans, LA 70113.

1

B

~_~-
120

110

100

90

8 0

7 0

60

50

4 0

30

20

10

PLANTATION AGE

(years since outplanting)

Figure 1 .--Height of dominant and codominant trees by plantation age for site indices of 30 through 80 based on the equation log (H& = log

(S) + 2.922293{[1 /SQRT(25)]-[I /SQRT(A)]} (zamoth and Feduccia 1984). Note that log implies logatithm to the base 10. Also observe

that 49 years since outplanting corresponds to SO years since seed.

2
Table 1 .--Distribution of unthinned-stand yield observations by plantation

age and site index

Age

class

Site index class (feet)

2a- 33- 3a- 43- 4a- 53- 5a- 63-

b2

6a- 73- 7a- a3- Total

3 2 3 7 ,47 5 2 5 7 6 2 6 7 7 2 7 7 82 87

*

Years ----------------------------Nu~er--------------------------------

a-12.

13-17

la-22

23-27

28-32

33-37

38-42

43-47

1 2 3 1 2

2 *3

4 15 37 16 4 a5

10 1 1 2 2 12 51 61 34 5 2 213

5 6 1 6 la 15 14 9 3 86

1 10 1 2 11 7 1 2 3 56

4 a 5 4 12 33

1 1 5 1 115 24

116 1 9

1 1

Total 3 5 3 17 33 65 53 94 157 63 12 2 507

Table 2. --Distribution of unthinned-stand yield observations by plantation

‘age and planting density

Age

class

Ye&s

Plantincr densitv class (trees per acre)

251- 501- 751- 1001- 1251- 1501- Total

500 750 1000 1250 1500 1750

---------------------~Nu~ber---,-----------------------------

a-12 6 6 40 3 3 a5

13-17 14 1 3 144 4 2 213

la-22 1 1 1 1 40 1 6 3 5 86

23-27 a a 24 9 3 4 56

28-32 5 l a 9 1 33

33-37 2 1 4 a 24 38-42 7 l

2 9

43-47 1 1

Total 3 9 45 288 119 7 9 507
4

Table 3 .--Distribution of unthinned-stand yield observations by planting density

and site index

Planting Site index class (feet)

density 28- 33- 38- 43- 48- 53- 58- 63- 68- 73- 78- 83- Total

class 32 37 42 47 52 57 62 67 72 77 82 87

Trees per acre --------------------------- Number-------------------------------

251-500 5 4 I.0 3 2 4 6 4 139

501-750 3 I.8 8 2 5 5 3 145

751-1000 3 5 3 12 23 33 30 64 85 26 4 288

1001-1250 3 4 9 21 55 26 1 119

1251-1500 1 2 4 7

1251-1750 2 3 4 9

Total 3 5 3 17 33 65 53 94 157 63 12 2 507

Table 4. --Distribution of thinned-stand observations by plantation age and site

index

Age

class

Years

Site index class (feet)

43- 48- 53- 58- 63- 68- 73- 78- Total

47 52 57 62 67 72 77 82

------------------- -------Number-----------------------------

8-12 13 3 2 18

13-17 1 7 7 8 30 55 40 2 150

18-22 2 4 10 17 26 48 7 114

23-27 3 4 14 14 21 39 15 1 111

28-32 2 9 16 11 6 43 24 2 113

33-37 1 27 3 31

38-42 3 3 6

Total a 24 47 50 87 228 92 7 543
Table 5 .--Distribution of thinned-stand observations by plantation age and

residual basal area immediately after thinning

Age Residual basal area (ft'/acre)

class

21-40 41-60 61-80 81-100 101-120 121-140 Total

Years ---------------------------Number----------------------------------

8-12

13-17

18-22

23-27

28-32

33-37

38-42

Total

8 38 2 18

1 12 49 65 21 2 150

2 10 32 41 22 7 114

2 12 29 50 , 16 2 111

2 13 27 47 19 5 113

4 5 12 10 31

2 4 6

7 51 150 223 92 20 543

Table 6 .--Distribution of thinned-stand observations by plantation age and

residual trees immediately after thinning

Age

class

Years

Residual trees per acre

-1- lOl- 201- 301- 401- 501- 601- 701- 801- 901- Total.

100 200 300 400 500 600 700 800 900 1000

-------------------------Number------------------------------------

8-12 1 10 4 1 1 1 18

13-17 20 55 46 26 3 150

18-22 12 46 30 14 6 6 114

23-27; 5 60 32 11 2 1 111

28-32 31 62 12 5 3 113

33-37 11 20 31

38-42 4 2 6

Total 47 158 112 101 66 43 13 1 i 1 543
Table 7 .--Distribution of thinned-stand observations by residual basal area

immediately after thinning and site index

Residual

basal area

FtL/acre

Site index class (feet)

43- 48- 53- 58- 63- 68- 73- 78- Total

47 52 57 62 67 72 77 82

---------------------Number----------------------------------

21-40 2 2 2 1 7

41-60 7 18 11 5 10 51

61-80 3 5 7 11 19 77 26 2 150

81-100 3 3 10 17 42 93 53 2 223

101-120 4‘ 6 7 18 41 13 3 92

121-140 3 4 3 3 7 20

Total 8 24 47 50 87 228 92 7 543

Table 8.--Distribution of thinned-stand observations by residual basal area and

residual trees immediately after thinning

Residual Residual trees per acre

basal area l- 101- 201- 301- 401- 501- 601- 701- 801- 901- Total

100 200 300 400 500 600 700 800 900 1000

Ft'/acre ----------------------------Number------------------------------

21-40 3 2 1 1 7

41-60 19 15 8 6 2 1 51

61-80 22 41 38 29 8 11 1 150

81-100 3 73 49 42 32 18 3 1 1 1 223

101-120 25 14 22 15 9 7 92

121-140 2 2 2 8 4 2 20

Total 47 158 112 101 66 43 13 1 1 1 543
Table 9 .--Distribution of thinned-stand observations by residual trees

immediately after thinning and site index

Site index class [feet)

Residual 43- 48- 53- 58- 63- 68- 73- 78- Total

trees 47 52 57 62 67 72 77 82

Trees/acre --------------------------Number----------------------------------

l-100 2 9 2 1 23 8 2 47

101-200 1 '3 8 14 21 79 31 1 158

201-300 3 2 '14 8 20 50 15 112

301-400 1 7 6 13 16 38 20 101

401-500 1 3 6 6 17 19 12 2 66

501-600 2 4 1 5 12 14 5, 43

601-700 3 3 2 2 1 2 13

701-800 1 1

801-900 1 1

901-1000 1 1

Total 8 24 47 50 87 228 92 7 543

Table 10 .--Distribution of residual stand observations by basal area before

thinning and basal area after thinning; the first figure in each

entry is for the first thinning and the second is for the second

through fifth thinnings

Basal area Basal area after thinnino (ft2/acre)

before thinning

Ft'/acre

41-60

61-80

81-100

101-120

121-140

141-160

161-180

181-200

21- 41- 61- 81- 101- 121- Total

40 60 80 100 120 140

-------------------Number---------------------------------'

O/5 O/4 o/g

111 l/24 l/2 3127

e lb/9 8/77 5/g 23195

3/O 12126 16/104 3114 341144

212 8/3 27/45 14/36 l/O 52/86

7/O 10/3 617 612 29112

4/O 4/O 2/o 3/Q1 .13/o

2/o l/O 3/o

Total l/6 16139 40/108 64/161 25/57 n/2 1571373

7
Table 11 .--Distribution of residual stand observations by basal area before

thinning and trees per acre before thinning; the first figure in

each entry is for the first thinning and the s&ond is for the

second through fifth thinnings

Basal area Trees per acre before thinnina

before thinning l- 201- 401- 601- 801- Total

200 400 600 800 1000

Ft'/acre ------------------Number--------------------------------------

41-60 O/7 o/2 o/g

61-80 O/18 o/g l/O 2/o 3127

81-100 O/38 o/47 819 12/l 3/O 23195

101-120 O/38 o/70 9136 20/o 5/O 341144

121-140 O/7 o/45 8126 4117 3/l 52/86

141-160 O/6 414 1812 7/O 29112

161-180 4/O 4/O 5/O 13/o

181-200 3/o 3/o

Total O/108 o/179 34175 97/10 26/l 1571373

Table 12 .--Distribution of residual stand observations by basal area before

thinning and plantation age; the first figure in each entry is for

the first thinning and the second is for the second through

fifth thinnings

Basal area Acre class (years)

before thinning 8-12 13-17 18-22 23-27 28-32 33-37 Total

Ft'/acre

41-60 O/l O/3 O/5 o/g

61-80 3/O O/5 O/7 o/11 O/4 3127

81-100 6/O 16116 l/26 o/19 o/29 O/5 23195

101-120 g/o 21120 4133 o/40 o/35 O/16 341144

121-140 3/O 42/10 4122 3127 o/22 O/5 52/86

141-160 23/O 2/o l/11 3/l 29112

161-180 10/o 3/O 13/o

181-200 l/O 2/o 3/o

Total 18/O 116/46 16/87 4/107 3/103 o/30 157/373

8
Table 13 .--Distribution of residual stand observations by trees per acre

before thinning and plantation age; the first figure in each entry

is for the first thinning and the second is for the second through

the fifth thinnings

Trees per acre Acre class (years)

before thinn'ina 8-12 13-17 18-22 23-27 i8-32 32-37 Total

Trees/acre ---------------------Number -------------------------------

l-200 o/17 O/61 o/30 O/108

201-400 o/13 O/52 o/74 o/40 o/179

401-600 28126 3/34 l/13 212 34/75

601-800 8/O 76/6 9,/l 313 l/O 97/10

801-1000 10/o 12/l 4/o 26/l

Total 18/O 116146 16187 4/107 3/103 o/30 1571373

prediction of residual stands immediately after thinning. Total tree height and volume equations required for

stand summaries are discussed.

Moment-Percentile Estimators

The three-parameter Weibull function (Bailey and Dell 1973) was selected as a model for the distribution of diameters. The Weibull probability density function is defined as:

‘

c-1

f(x) = e

- yc 3

C

for atO, b>O, c>O and xza

0 elsewhere (1)

where

a = location parameter, *

b = scale parameter, . c = shape parameter, and

x = diameter outside bark (inches) at a height of 4.5 feet (d.b.h.), with the cumulative distribution

function:

It can be shown that the second moment about the origin is defined as:

E(X2) = J=a x2 f(x) dx = a2 + 2ab I'(l+$) + b2 I'(l+$)

(2)

/

(3)

9
where

I?(z) = the gamma function evaluated at z (Dell and others 1984).

The expectation in equation 3 defines the average squared diameter measured in square inches, which when

multiplied by the constant

R [Q2 [&I2 = 0.005454, 2

and the number of trees per acre , gives the basal area per acre @*/acre). If the sample estimate of the second

moment is defined as:

E[1 ] =; f xi2,

i=l

then taking the square root gives the quadratic mean stand diameter:

c E(G > l”‘5= $--fjj.

Percentile definitions involve letting xp equal the diameter that is the 100, percentile of the Weibull distribution where 0 I p I 1. Equation 2 can then be solved for the scale parameter b:

b = xp-"

[ -lrl(l-p) 1;

(4)

Combining equations 3 and 4 and rearranging gives:

a2 + 2a(xp-a)

w+

+ (%-=I2

r& - E(X2) = 0, (5)

[-1*(1-p)1; [ -ln(l-p) 1;

which is solved by numerical techniques for the shape parameter c, given a specified p and estimated a, E(X2),

and xp. In the study described here, the location parameter was estimated by a simple projection function,

yielding a two-parameter Weibull distribution, with a being fixed. The value of p was set at 0.93 because (1)

the estimators of a and E(X2) tie down the left tail and center of the distribution, respectively, (2) the upper

right tail is where most of the value is for distributions of tree diameters, and (3) the 93rd percentile has been

found useful in estimating Weibull parameters in other situations (Zanakis 1979). Hence, equation 5 becomes:

a2 + 2a(x.g,-a) w+g 1 + (x.93-=>2 W+g)

- E(X2) = 0, (6)

(2.65926)c (2.65926);

1 0
and equation 4 becomes:

b = x.93-a

(2.659&

(7)

Solving for c in equation 6 and substituting its -value into equation 7 gives b, which then completely defines the

Weibull distribution.

Unthinned Stands

In order to obtain the Weibull diameter distribution parameters for a given stand by the moment-percentile

method of parameter recovery, three stand-level attributes must be predicted and these estimates used to solve

for a, b, and c as described previously. After considerable modeling effort, the stand attributes Xmin, B, and

X.93 wefe selected where:

X min = d.b.h. (inches) of the smallest diameter tree on a study plot,

B = basal area (ff/acre), and

x.93 = d.b.h. (inches) that is the observed 93rd percentile in the diameter distribution of a study plot.

The projection equations used to estimate these quantities over time form a system of nonlinear functions of age,

site, and number of trees surviving. The stand attribute Xmin is inherently weak in that it is an order statistic

and is a decreasing function of sample size. Thus, in forestry applications, Xmin will decrease when plot size

increases, and when plots of unequal size are used, as in the present case, the definition of &in becomes questionable unless it is modeled as a function of plot size or more directly by the number of trees in the sample.

However, this problem has generally been dismissed in previous growth and yield research because only a crude

estimate of Xmi* is needed; therefore, this issue was not pursued further.

The basal area yield equation was modeled as:

B = a1 HDCa2 Tsa3 ea4*

-1

(8)

where

H D C = average height (ft) of dominant and codominant trees,

Ts = number of trees surviving per acre,

A = age of plantation (years); i.e., number

of growing seasons since field planting, and

ai = parameters to be estimated (i = 1,2,3,4).

11
The quadratic mean diameter R,, defined as the d.b.h. of the tree of average basal area, can be derived by

algebraic manipulation from equation 8, yielding:

Rq = Sl HDcd2 TsJ3eJ4 *

-1

where 1

6, = c

5 0.0;:454 3 ’

6, = 4,

6 3 =-7Ca3-l)

2

and

6, =;.

Given the definition of ?z, it is reasonable to require that models of Xmin and X.,, have the same functional

form as the model of x,. Thus we have the nonlinear models:

and:

X .93 = y, Hoc" TsT3 e T4A-l

. (11)

With these three stand-level attributes projected for a specific stand, the location parameter is fixed as:

a = 0.5 Gin

and the second moment is estimated by:

E(X') =

B

0.005454 T;

(12)

(13)

Experience has indicated that the value of a affects the distribution little because the other parameters adjust

themselves accordingly (Zarnoch and Dell 1985). Moreover, because a is between zero and Xmin, it is natural

to fix this parameter at the midpoint of this interval. Hence, the shape parameter c is estimated by solving

equation 6 after substitution from equations 11, 12, and 13. The scale parameter b is found by substituting the

value of c and values from equations 11 and 12 into 7.

1 2
Equations 8, 10, and 11 were fitted to the plot data. Their estimated coefftcients and fit statistics are shown

in table 14. An example of the behavior and relationship of these stand attributes is shown in figure 2, where

the minimum diameter (Xmin), quadratic mean diameter (function of B), and the 93rd percentile @.93) are plotted

over plantation age for three levels of trees surviving at age 10 where the site index is 60.

The predictor variables consist of age, height of dominant and codominant trees, and number of trees

surviving. If values of the latter two stand measurements at a given age are unknown, they must be predicted.

Mean height of the dominant and codominant trees at any age can be obtained from a back-solution of the site

index equation (Zarnoch and Feduccia 1984). Two forms of this equation are given in table 15. There are

three ways to predict survival in unthinned stands. If the number of trees surviving at the starting age, T,, is

unknown, but basal area per acre and site index or mean height of the dominant and codominant trees is known,

then the fourth equation in table 14 is applicable. If only the number of trees planted, TP, is known, then the

fifth equation in table 14 can be used to predict T,. Finally, when projecting changes in number of trees

surviving from one age to another, an equation is given in table 14 based on the model:

T = T

2 1 e

P1(A2-A1)

(14)

where

T, = number of trees alive at the projection age,

T, = number of trees alive at the initial age,

A, = projection age, and

A, = initial age.

Figure 3 illustrates survival patterns, over time, based on the fitted model (equation 14) for four stand densities,

each beginning at a stand age of 10 years.

Thinned Stands

The thinned-stand model is also based on the moment-percentile parameter recovery approach, estimating

the values of stand-level attributes B,, Xmin,a, X.,,,,, and T, at projection age A, from initial age At where:

B, = basal area (ft2/acre) at age A,,

‘min,:! =

‘.93,2 =

d.b.h. (inches) of the smallest diameter tree on a study plot at A,,

T2 =

d.b.h. (inches) that is the 93rd percentile in the diameter distribution at A, on a study plot, and

number of surviving trees at A,.

The basal-area projection equation originally used by Clutter (1963) and Sullivan and Clutter (1972) was used

and is defined as:

Al Al

B, = BIG e

(l-pal+a2 S)

(15)

where

s = site index at base age 25 and

(yi = parameters to be estimated (i= 1,2).

1 3
15

13

11

9

7

5

3

1

I

AGE 10 SURVIVAL=400 TREES/ACRE .‘- --we 4-I

AGE 10 SURVIVAL=600 TREES/ACRE

me-_-

I”“1”” 1 ” ” ” ” ” “I”

I I” 1“‘I 1‘I”

IO 15 20 25 30 35 40

PLANTATION AGE (YEARS)

4 5 5 0

Figure 2.--Predicted diameter-growth relationships by plantation age for unthinned slash pine (site index 60) with survival at age 10 of 400, 600, and

800 trees per acre where X, is d. b. h. (inches) of the smallest diameter tree on a plot, x, is the quadratic mean diameter (inches), and X,,

is the d.b.h. (inches) that is the observed 93* percentile in the diameter distribution of a plot.

14
Table 14.--Equations used for the unthinned-stand yield model components

Equation

Fit

statistic" S.E.

B

= 0.004715 HDC1.50s277 T.80.623533 e5.819339A-l 0.93 10.40

)6in = oeo112g3 HDCl.788576 ~~-0.401856 e16.415341A-1 0.60 0.86

X.93

= 1.810092 HDC0.659044 Ts-0.175583 e0.596322A-1 0.95 0.44

Ts = 5424.369016 H,, -1.924749 B 1.180924 e-7.O92173A-1 0.82 72.30

TS = Tp 1.0-0.0034168 0.56 116.90

T2 = T, e -0.021863(A2-Al) 0.93 45.70

*The fit statistic is the square of the correlation coefficient

between the predicted and observed variables.

Table 15. Miscellaneous equations needed for the growth and yield model

Equation

S HDC lo

-0.584459+2.922293 A-o'5 =

4 base age of 25 years.

HD C

= s 1o0.584459-2.922293 A-o'5

V = 0.12905 + 0.0028271 D2H - 0.10102(10-7)(D2H)2

H = 1.9838 H,C 0.913811 B-O.ll7585 Ts0.0754701e0.589077.A-1 -2.16139D-'

15
800

9 300

-

0

l”“l”“l”“l”“l”“l”“l”“l”“1

15 20 25 30 35 40 45 50

PLANTATION AGE (YEARS)

Figure 3 .--Surviving trees per acre by plantation age for unthinned slash pine for four different starring densities.

1 6
Equation 14 was used to model survival. An estimate of the quadratic mean diameter at projection age

xs,2 was derived by combining equations 14 and 15:

(16)

where

s1 = 13.540741,

6, = 2,

6, = 2, and

Again it will be assumed that Xmin,:! and X,,,,, have the same functional form as x,,,; thus, we have:

Al I I

0.5

BIG

Al

Ynin,2 = pl T, e(l-Az)(P2+P3S)+P4'*z-*1'

and:

X

(1-7i;:)(72+73S)+74(A2-A1)

.93,2 = Y, e

(17)

(18)

Given these ‘stand-level attributes projected for a specific stand a‘t age A,, the location parameter is fixed at:

a = o.9 Xmin,2,

.

(19)

and the second moment estimated by:

E(X2) =

B2

0.005454 T;

(20)

Notice that n is now fixed very near the minimum diamet,~., while in the unthinned-stand model Q was fixed

at half the minimum diameter. The justification for this difference i: that thinned stands contain fewer trees

and often exhibit a smoother distribution of diameters as a result of previous thinnings. It was judged that for

thinned stands n was better estimated by the minimum diameter, and the value of C L was set closer to this

number. The shape parameter c is estimated by solving equation 6 after substitution of values from equations

18, 19, and 20. The scale parameter h is found by substituting values from equations 18 and 19 and the value

of c into equation 7. The stand-level attribute model equations 14, 1.5, 17, and 18 were fitted to the plot data;

I7
their estimated coeffkients and fit statistics are shown in table 16. Because thinning affects stand growth,

thinned-stand growth models are more complex than models describing the growth of unthinned stands. The

necessity of employing basal area (B,) along with age (A, and A2), site index, and trees surviving (Tt) as input

variables illustrates this point. Obviously, residual B, and T, define x4,t, and therefore the quadratic mean

diameter can be substituted for one of the other two. More important, though, the fact that B, and T, define

x,,, indicates the important role that average residual-tree diameter has in thinned-stand growth models, even

though it may not explicitly enter into any of the projection equations.

Residual Stands

The model formulated to predict the diameter distributions of residual stands immediately after thinning from

below is also based on the moment-percentile method of parameter recovery. The model used to predict the

number of residual trees after thinning (Matney and Sullivan 1982) was:

T, = T,, [ l.O- [ LO-:) "I ] a2 (21)

where

T, = number of trees per acre before thinning,

B, = basal area (ft2/acre) before thinning, and

B, = a specified residual basal area (tI’/acre) that will remain after thinning.

It should be noted, that in a previously unthinned stand, Tb would be identical to the T, input required to make

predictions of diameter distribution parameters at that age and that B, is simply the estimated basal area from

equation 8. Conversely, in a previously thinned stand, Tb is identical to T2 in equation 14, and B, is estimated

using equation 15.

The other two stand attributes were modeled as:

and:

where

A =

Xmin,b =

Xmin,r =

B, =

B, =

X.93,b =

X.93,r =

x,,b =

the quadratic mean diameter (inches) before thinning.

Xmin,r =

s-4

p1 + p2A + P3Xmin,b + "477"-

qpr

'.93,r = y1 + Y2X.93,b + Yj'q,b + 74-

a,

(22)

(23)

plantation age (years) at time of thinning,

d.b.h. (inches) of the smallest diameter tree on a study plot before thinning,

d.b.h. (inches) of the smallest diameter residual tree on a study plot after thinning,

basal area (fi2/acre) before thinning,

residual basal area (fi2/acre) after thinning,

the diameter (inches) that is the 93rd percentile in the diameter distribution before thinning,

the diameter (inches) that is the 93rd percentile in the residual diameter distribution after thinning,

and

Given these stand attributes, the location parameter is fixed at:

(24)

1 8
Table 16. -- Equations used for the thinned-stand growth projections

Equation

Fit

statistic” S.E.

Al Al

AZ Cl--)(5.190356+0.006501 S)

B, = B, e A2

0.5

Gin,2

Al (l--)(-4.238939+0.048459 S)+0.076748(A2-Al)

e

*2

Al (l--)(4.455365-0.012739 S)-O.O10953(A2-Al)

e

*2

= T, e

-0.015386(A2-Al)

0.88

0.78

0.92

7.30

1.35

0.72

0.97 23.70

*The fit statistic is the square of the correlation coefficient

between the predicted and observed variables.

because the residual stand contains fewer trees and has a smoother diameter distribution than the unthinned

stand. The second moment is estimated by:

E(X

2 Br

) = 0.005454 T,' (25)

and the b and c parameters are found after appropriate substitutions into equations 6 and 7.

Models 21, 22, and 23 were fitted to before- and after-thinning data, and the estimated coeffkients and fit ’ statistics are shown in table 17. Originally, these were fitted separately to data from first, second, third, fourth,

and filth thinnings. However, after scrutiny of the coefficients, models for two to five t&innings were

considered essentially the same, and so the data was pooled and coefficients refitted. The formation of one set

of equations for the first thinning and another for subsequent thinnings is also based on the judgment that the

first thinning differs from the rest when thinning is from below: the first thinning gets the stand in shape for

repeated subsequent thinnings.

Height-Diameter Equation

fhe total height of a given tree was modeled as a function of tree diameter, site index, and stand conditions.

There were 17,606 height observations from the thinned and unthinned stands that were used together, giving:

In(H) = 0.684994 + 0.589077 A-’ + 0.913811 ln(H,J - 0.117585 In(B) + 0.0754701 ln(Ts)

- 2.16139 D-’ (26)

1 9
Table 17.-- Equations used to predict the residual stand following

thinning

Equation* R2 or

fit statistic+ S.E.

First thinning

%in,r = -1.0388 + 0.053716 A i- 1.171432 sin,b

+ 1.322203 P 0.61 0.96

X .93,r = 0.293766 + 0.811760 X.,, b + 0.301176 R, b

+ 0.205805 P 0.93 0.26

T, = Tb[ 1.0 - (1.0 - 2]"'88g554 ]".gg'g8' 0.95 30.60

Second and future thinnings

%in,r = -1.28337 + 0.107893 A + 0.862925 Gin,b

+ 1.114448 P 0.83 1.14

X.93,r = 0.119026 + 1.058200 X.,, b - 0.051883 x,,

+ 0.316591 P 0.99 0.20

*p = @b-v

Bb ' B, = basal area (ft'/acre) before cut,

and B, = residual basal a?rea (ft'/acre) after cut.

+The fit statistic is the square of the correlation coefficient

between the predicted and observed variables.

where

H = estimated total height (ft) of a given tree,

D = d.b.h. (inches) of a given tree, and

2 0
the other variables are as previously defined, with R* = 0.94 and S.E. = 0.0911, both in the logarithmic scale

for the dependent variable. (See table 15 for an alternate form). The equation generates estimates of tree

heights that are used in the volume-defining function to determine cubic-foot volume per tree, which can be

accumulated on a stand basis for estimates of volume per acre.

Volume-Defining Equation

Tree volumes were determined by the height accumulation method. Stump height was set at 0.5 ft. Several

volume equations were fitted. These included the typical V = b, + b, D’H model and variations with no

intercept, with a weighted term, with estimated exponents in a multiplicative model, and with the square of D2H

included as another term. Comparison of these models resulted in selection of the best model:

V = 0.12905 + 0.0028271 D’H - 0.10102 (lo-‘) (D2H)2 (27)

where

V = total volume (ft3) outside bark above a OS-ft stump,

D = d.b.h.(inches), and

H = total tree height (ft),

with R* = 0.9865 and S.E. = 1.1916. (See table 15). This model was used for computing individual-tree total

outside-bark volumes and subsequently volume per acre for thinned and unthinned stands.

MODEL TESTING

The growth and yield prediction system was tested against the data used in its development. The tests

verified that predicted values were close to those observed.

The prediction phases tested were: yield prediction in an unthinned stand, growth prediction in an untbinned

stand, characterization of a residual stand after thinning, and growth prediction in a thinned stand. In each case

selected, predicted values of stand and yield table variables were compared with their respective observed

values. The same volume defining function was used in each case. Mean predicted, mean observed, correlation

coeft’icient, mean difference (predicted minus observed), and mean percentage difference statistics were

calculated. The percentage differences are defined as:

100 PREDICTED - OBSERVED

OBSERVED

Results of these tests are found in tables 18 through 22. With the general exception of Xmin, which is highly

variable, the stand- and yield-table variables averaged within f5 percent of the observed values. This indicates

that the system of equations accurately predicts growth and yield in the stands from which it was developed and

should,provide good results when used to make predictions in similar slash pine plantations.

DISCUSSION

Trends

Prediction trends for unthinned and thinned plantations and some comparisons of results of these management

alternatives are given in figures 4 through 11. Generally in these figures, the extremely wide range of site

indexes from 40 to 80 are presented. However, in this discussion, we have focused on the more realistic site

indexes of 50 and 70. In most cases it was assumed that 700 trees were planted per acre (about an 8- by 8-ft

spacing) on lands with site indices (base age 25) of 50 and 70. After prediction of stand conditions at age 10,

2 1
Table 18 .--Observed vs. predicted values of stand- and yield-table variables for unthinned

slash pine plantations (n=507)

Mean Correlation Mean

Parameter Predicted Observed coefficient Difference Percent difference

Sin 2.58 2.57 0.77 0.01 17.60

x.24 5.27 5.14 0.94 0.13 3.69

x.63 6.66 6.65 0.97 0.01 0.57

x.93 8.08 8.08 0.97 0.00 0.37

x8" 124.84 6.33 124.63 6.32 0.98 0.01 0.63

0.97 0.21 1.51

VOL 3300.30 3369.71 0.98 -69.42 -0.38

Table 19 .--Observed vs. predicted values of stand- and yield-table variables for unthinned

slash pine plantations after one growth period (n=255)

Mean Correlation Mean

Parameter Predicted Observed coefficient Difference Percent difference

Sin

x.24

x.63

593

X

q

;

%C

VOL

2.87

5.70

7.29

8.93

6.93

136.48 537.28

55.25

3828.00

3.10 0.77 -0.22 2.25

5.66 0.94 0.04 1.56

7.40 0.96 -0.10 -1.07

9.05 0.96 -0.13 -0.98

7.03 0.97 -0.11 -1.16

137.48 0.93 -1.00 -0.57

537.25 0.97 0.03 1.64

57.95 0.83 -2.69 -2.23

4172.65 0.96 -344.62 -6.11

Table 20 .--Observed vs. predicted values of stand- and yield-table variables immediately

after the first thinning (n=157)

Mean Correlation Mean

Parameter Predicted Observed coefficient Difference Percent difference

Sin 2.59 2.59 0.78 0.00 13.58

'.24 5.10 5.05 0.86 0.05 2.35

X.63 6.30 6.38 0.94 -0.08 -0.91

X.93 7.51 7.51 0.96 0.00 0.11

x

s

6.00 6.05 0.96

450.99 449.60 0.97 -0.05 1.40 -0.46 1.33

%C 45.92 46.00 0.99 -0.08 -0.05

VOL 2053.10 2133.68 0.99 -80.63 -2.58

22
Table 21 .--Observed vs. predicted values of stand- and yield-table variables immediately

following all thinnings except the first (n=370)

Mean Correlation Mean

Parameter Predicted Observed coefficient Difference Percent difference

Sin

X.24

X.63

X.93

x

q

f

"&

5.68 5.69

7.94 7.84

9.25 9.29

10.54 10.54

8.89 a.94

219.21 65.84 218.55 66.09

2722.50 2809.05

0.91 -0.00 10.99

0.96 0.10 2.19

0.99 -0.04 -0.41

1.00 0.00 d-06

0.99 -0.05 -0.43

0.99 1.00 -0.26 0.65 -0.37 1.15

. 0.99 -86.57 . -2.80

8-

7-

6-

5-

4-

SI-80

SI=60

SI-50

3-l ,,,, ‘I I I ,I III 1 I I I I I1 I I, 1 I I 11 1 I I1 I1 11 1

1 0 15 20 25 30 3 5 40 45 50

PLANTATION AGE (YEARS)

Figure 4.--Predicted quadratic mean diameter trends for site indices of 40 through 80, each planted with 700 trees per acre.

23
225

200

iii

2 175

>

LL 150

5:

5; 125

2

< 100

2

co 75

s

5 0

2 5

,

I

I

I

-T

IO 15 2 0 2 5 3 0 3 5 4 0 4 5 5 0

PLANTATION AGE (YEARS)

Figure 5 .--Predicted basal area per arce trends for site indices of 40 through 80, each planted with 700 trees per acre.

SI=80

SI=70

SI=60

SI=50

SI=40

Table 22.--Observed vs. predicted values of stand- and yield-table variables for slash

pine plantations after a 5-year growth period following thinning (n=543)

Sin 5.64

x.24 8.50

x.63 9.80

x.93 11.02

zq 9.41

; 267.15 105.07

%C 69.52

VOL 3550.90

5.54 0.88 0.10 18.84

8.03 0.98 0.46 5.67

9.64 0.98 0.16 1.27

11.07 0.96 -0.05 -0.48

9.26 0.99 0.15 1.32

105.12 0.94 -0.05 0.47

270.41 0.99 -3.25 -1.97

69.35 0.98 0.17 0.31

3693.60 0.94 -142.70 -3.05

Mean Correlation Mean

Parameter Predicted Observed coefficient Difference Percent difference

24
120

100

80

60

40

20

0

SI=80

SI=70

SI=60

s1=50

SI=40

15 20 25 30 35 40 45 50

PLANTATION AGE (YEARS)

Figure 6.--Predicted torn1 cubic-foot volume per acre trends for site indices of 40 through 80, each planted with 700 trees per acre.
300

0

MEAN ANNUAL INCREMENT

/ -1

/

\

\

I'

- - \

---PERIODIC ANNUAL INCREMENT

\

\

I

\

\

\

\

\

\

\

\

\

\

SI=70 \

\

IO 15 20 25 30 35 40 45 50

PLANTATION AGE (YEARS)

Figure I.--Predicted mean and periodic annual cubic-foot volume increments.for site indices of 50 and 70, each planted w’th 700 trees per acre.

2 6
600

3 500

0

<

g 400

cl

3 300

c

(3zj 200 5

5v) 100

\

;\.

I

I

A.

*\

-\

+A I

-1

--I\

“1

- -

--+

I

- -

- -

---I

I

--W_

---_

-9

I

S-2,

- - ---__

-----a__

---w

I - --------- q-w

-----.

SI=40

SI=5C

SI=6C

SI=7C

SI=8(

IO 15 20 25 30 35 40 45 50

PLANTATION AGE (YEARS)

Figure I.--Predicted survival trends for site indices of 40 through 80, each planted with 700 trees per acre and thinned to 80ft2/acre of basal area at

ages 15, 22, 30, and 40 where possible.

2 7
2 0

18

16

14

12

IO

8

6

4

2

1

-I,

SI=80

SI=70

SI=60

SI=50

SI=40

11ll~lll1~l,ll,,,ll,,,,, ll,l III, 1 I I I I,

15 20 25 30 35 40 45 50

PLANTATION AGE (YEARS)

Figure 9.--Predicted quadratic mean diameter tremfs for site indices of 40 through 80, each planted with 700 trees per acre and thinned to 8Off’/acre

of basal area at ages IS, 22, 30, and 40 where possible.

28
150

125

100

75

50

25

-c--L- -------c-A

*

/----

/’ *’

/

.’

/ .’

SI=50' ,'

4'

SI=40'

/

I I , I , , , , , 1ll1~‘“’II,) IIll,,ll, I,,,

10 15 20 25 30 35 40 45 50

PLANTATION AGE WEARS)

Figure 10 .--Predicted basal area per acre trends for site indices of 40 through 80, each planted with 700 trees per acre and thinned to 80ft2/acre of basal

area at ages 15, 22, 30, and 40 where possible.

2 9 I
5700

Gj 5100

[r

2 4500

>

IL 3900

3

G 3300

# 2700

d

> 2100

a

)- 1500

f

900

300

1

SI=80

SI=70

SI=60

SI=50

/

Y/

/

/

y /

r

I’ /

---

SI=40

--- /

I I

I

/;‘/

Al/ ,--

/ / ,-

,-

/ /

/ /

/

&’

,’

/

/

#’

/

,H

/

/

,’

/

.’

/ 4’ /

,’

.’

1 I I I,, , , 1 , , , , ,

I I, I, I, I I,, , , , , , , , , I t I , ,

IO 15 20 25 30 35 40

PLANTATION AGE (YEARS)

45 50

Figure 11 .--Predicted total cubic-foot volume per acre trends for site indices of 40 through SO, each planted with 700 trees per acre a& thinned to 80

p/acre of basal area at ages 15, 22, 30, and 40 where possible.

30
projections were made to age 40 for unthinned-stand examples or to age 50 for thinned-stand examples. In the

thinned-stand examples, the plantation with a site index of 70 was thinned to 80 ft? of basal area at 15, 22, 30,

and 40 years. The plantation with a site index of 50 was thinned to 80 ft2 of basal area at 22, 30, and 40 years.

Unthinned Plantation

Mean Diameter--The average gain in mean diameter for site 70 over the site 50 plantation was 2.34 inches

by stand age 40 (fig. 4).

Basal Area--Basal area increased with increasing site index (fig. 5) but increased more rapidly on site 70.

Basal area was near culmination on both sites by stand age 40.

Total-Stem Volume Yield--Total-stem yields (cubic feet, outside bark) did not culminate before age 40 on

either site, but the volume on site 70 at age 40 was 4,024 @/acre greater than the corresponding value for site

50 (fig. 6).

Mean and Periodic Annual Increment--Total-stem volume (cubic-feet outside bark) mean annual increment

(MAI) culminated at about stand age 25 on both sites and tapered off more rapidly on site 70. However, at age

25 MA1 was 110.3 @/acre per year greater on site 70 than on the site 50 (fig. 7).

Thinned Plantation

Surviving Trees--At age 22, when the site 70 stand was thinned for a second time, the site 50 stand got its

first thinning to the target basal area of 80 #/acre (fig. 8). Obviously, very few trees were cut from the site

50 stand. The site 70 stand had about 200 fewer trees per acre after the thinning at age 22 than did the site 50

stand. From this time on, though, the cuts on site 50 removed more trees than did those on site 70, so the gap

closed to about 60 trees per acre. Mortality was not a very important factor in either stand after the thinning

regime was initiated.

Mean Stand Diameter--Quadratic mean d.b.h. averaged 3.4 inches higher on site 70 than on site 50. The

difference was smaller in the early years but consistently increased over time (fig. 9). Average diameter

increased with each thinning because the thinning technique used in the study plantations was modified low

thinning.

Basal Area--After each stand had been thinned at least once, basal area yield was about 4 fi2/acre higher in

the site 70 stand than in the site 50 stand just prior to the last two thinnings, and this relationship persisted until

almost age 50 (fig. 10).

Total-Stem Volume Yield--Total cubic-foot (outside bark) standing volume was always greater in the site 70

plantation than in the site 50 plantation (fig. ll), even after all thinnings. Total volume at age 50 was 1,338

ft3/acre greater for site 70 than for site 50, and total volume removed in thinnings was 2,812 ft?/acre greater

for site 70 than for site 50.

Unthinned- and Thinned-Stand Comparison

The unthinned site 70 plantation contained 290 trees per acre at age 40, whereas the corresponding thinned

plantation contained 99 trees per acre at that age (before the final thinning). The unthinned plantation contained

SO percent more total volume than the thinned plantation-- 7,371 e/acre compared to 4,907 e/acre. However,

the average diameter of the trees in the thinned plantation (at age 40) was about 4 inches greater than those in

,he unthinned stand. For the thinned site 70 stand, the total c,f the volume harvested in the first three thinnings

and the volume available for harvesting just before the age-40 thinning was 3,204 + 4,907 = 8,111 @/acre.

This was about 740 ft3/acre more volume than what was available in the unthinned site 70 stand at age 40.

Computer Pr-cgram

COMPUTE P-SLASH, a program that performs calculations and generates yield tables for thinned and -

unthinned s!ash pine plamations on cutover sites in the west gulf region, will soon be available. It is written

in FORTRAN 77 and will run on most computers. A user’s guide, which will accompany the software and

explain the program’s features, is also being prepared.

3 1
LITERATURE CITED

Bailey, R.L.; Dell, T.R. 1973. Quantifying diameter distributions with the Weibull function. Forest Science.

19(2): 97-104.

Clutter, J.L. 1963. Compatible growth and yield models for loblolly pine. Forest Science. 9(3): 354-371.

Dell, T.R.; Feduccia, D.P.; Campbell, T.E. [and others]. 1979. Yields of unthinned slash pine plantations on

cutover sites in the west gulf region. Res. Pap. SO-147. New Orleans, LA: U.S. Department of Agriculture, Forest Service, Southern Forest Experiment Station. 84 p.

Dell, T.R.; Matney, T.G.; Zarnoch, S.J. 1984. Mathematics and statistics. In: Wenger, K.F., ed. Forestry

handbook. New York: John Wiley & Sons: 1253-1318.

Hyink, D.M.; Moser, J.W., Jr. 1983. A generalized framework for projecting forest,yield and stand structure

using diameter distributions. Forest Science. 29( 1): 85-95.

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

summaries. Journal of Forestry. 67(8): 554-555.

Matney, T.G.; Sullivan, A.D. 1982. Compatible stand and stock tables for thinned and unthinned loblolly pine

stands. Forest Science. 28(l): 161-171.

Sullivan, A.D.; Clutter, J.L. 1972. A simultaneous growth and yield model for loblolly pine. Forest Science.

18( 1): 76-86.

Zanakis, S.H. 1979. A simulation study of some simple estimators for the three-parameter Weibull distribution.

Journal of Statistical Computation and Simulation. 9(2): 101-l 16.

Zarnoch, S.J.; Feduccia, D.P. 1984. Slash pine plantation site index curves for the west gulf. Southern Journal

of Applied Forestry. 8(4): 223-225.

Zarnoch, S.J.; Dell, T.R. 1985. An evaluation of percentile and maximum likelihood estimators of Weibull

parameters. Forest Science. 3 l(1): 260-268.

3 2 *U.S. GOVERNMENT PRINTING OFFICE:1992-566-016/40021
Zarnoch, S.J.; Feduccia, D.P.; Baldwin, Jr., V.C.; Dell, T.R. 1991.

Growth and yield model predictions for thinned and unthinned slash

pine plantations on cutover sites in the west gulf region. Res. Pap.

SO-264. New Orleans, LA; U.S. Department of Agriculture, Southern

Forest Experiment Station. 32 p.

A- growth and yield model has been developed for slash pine

plantations on problem-free cutover sites in the west gulf region.

The model was based on the moment-percentile method using the

Weibull distribution for tree diameters. This technique was applied to

untbinned and thinned stand projections and, subsequently, to the

prediction of residual stands immediately after thinning.

Keywords: Parameter recovery, Pinus elliottii Engelm. var. elliottii,

size distribution, survival prediction, volume prediction, Weibull

distribution.

Persons of any race, color, national origin, sex, age, religion, or with any

handicapping condition are welcome to use and enjoy all facilities, programs, and

services of the USDA. Discrimination in any form is strictly against agency policy,

and should bc reported 10 the Secretary of Agriculture, Washington, DC 20250.