United States 

Department of 

Agriculture 

Forest Service 

Southern 

Research Station 

Research Paper 

SRS-? 0 

White Pine Site Index for the 

Southern Forest Survey 

Bernard R. Parresol and John S. Vissage 
White Pine Site Index for the Southern 

Forest Survey 

Bernard R. Parresol and John S. Vissage 

Abstract Table 14econd-growth white pine age-height data a 

A base-age invariant polymorphic site index equation was used to model 

the white pine (Pznus strobus L.) site-quality data provided by 

Frothingham (1914). These data are the accepted standard used by the 

Southern Forest Inventory and Analysis unit of the U.S. Department of 

Agriculture, Forest Service. An all possible growth intervals data structure 

was used, and autocorrelation parameters were incorporated into the site 

index model. It has recently been shown that these measures are necessary 

to obtain unbiased, efficient parameter estimates. The model is invertible, 

hence site index can be explicitly determined without the need for a 

numerical evaluation procedure. The site index model can be solved to 

provide an equation for any base age, hence it is applicable regardless of 

the choice of rotation age. Site index curves are graphed for base ages 25 

and 50 years, and example calculations are provided. 

Keywords: Autocorrelation, base-age invariant, Pinus strobus, 

polymorphic. 

Introduction 

Forest Inventory and Analysis units of the U.S. Department 

of Agriculture, Forest Service, have been estimating site 

productivity since 1962 (Josephson 1962). Within the 

southern forest survey, the standard for white pine (Pinus 

strobus L.) site productivity is the data published by 

Frothingham (1 9 14) for second-growth natural stands. 

While these data have been used previously to fit a white 

pine site index equation (Scott and Voorhis 1986), the 

equation used-a generalized form of the ChapmanRichards model (Carmean and Hahn 198 1 )--has proven 

unsatisfactory. The equation is not invertible, that is, one 

cannot directly solve for site index given height and age, 

and iterative procedures used to solve for site index often 

fail to converge to a solution. For purposes of the southern 

forest survey, what is desired is a well behaved white pine 

site index equation, capable of producing an estimate of 

dorninanticodominant average height given site index and 

age; and, conversely, an estimate of site index given 

dorninanticodominant height and age. 

Data 

Frothingham (1 9 14) published the results of measurements 

of 196 fully stocked second-growth white pine stands in 

New Hampshire. He divided the data into three site-quality 

Site quality I Site quality I1 Site quality I11 

Height6 Height Height 

Yrs 

10 

15 

20 

25 

30 

35 

40 

45 

50 

55 

60 

" Data from Frothingham (19 14), Tables 3,4, 5; p. 2 1-22. 

Average height of dominantlcodominant trees. 

classes, which he labeled quality I, 11, and I11 (S,, S,,, S,,,). 

These data are reproduced in table 1 and represent mean 

values for each age and site class. It should be pointed out 

that data developed in this manner do not represent the 

variability in the population. Hence, for any equation fitted 

to these data, parameter estimates cannot be tested for 

statistical reliability and prediction confidence intervals 

cannot be validly constructed. Nonetheless, unbiased 

predictions can still be made, and if the data represent true 

site-quality relationships in the white pine population, then a 

model that can capture the relationships among the data will 

give credible results in application. 
Figure l-Plot of the ratios of the site-quality data of table 1 showing 

changing ratios through time for each line. This indicates a polymorphic 

relationship. 

Methods 

Anamorphic vs. Polymorphic 

Today it is generally believed that anamorphic curves do not 

adequately represent site relationships (Carmean 1968, 

1975; Goelz and Burk 1992). An anamorphic site index 

curve is one in which the curve for a given site index is a 

constant proportion of the curve for another site index. A 

polymorphic curve, by contrast, is not a constant proportion 

of the other curves. To see if the implicit relationships in the 

data were anamorphic or polymorphic, we graphed the 

ratios (S,IS,,), (S,IS,,,), and (S,,/S,,,) over age (fig. 1). 

Because the ratios change over time (i.e., the lines are not 

horizontal), the definition of anamorphism is not met, which 

signifies that a polymorphic model should best capture the 

relationships in the data. We therefore ruled out fitting an 

anamorphic model to this data. 

Procedures for Fitting a Site Index Equation 

A height-by-age equation can be differentiated to provide an 

equation for height growth rather than accumulated height. 

An equation in this form is referred to as an algebraic 

difference equation (ADE) and is the preferred forrn for 

fitting site index models (Clutter and others 1983: 50-54; 

Goelz and Burk 1992, 1996; Ramirez-Maldonado and 

others 1988). 

A crucial fact pointed out by Goelz and Burk (1 992, 1996) 

is that modeling height in the context of site index involves 

two processes: (1) estimating site index at base age given 

height at some other age, and (2) estimating height at some 

desired age given height (site index) at base age. These two 

processes may be modeled by individual functions, such as 

was done by Carmean and others (1989). Height is assumed 

to be measured without error when on the right-hand side of 

the equation (dependent or Y variable), but possessing error 

when on the left-hand side (independent or X variable). 

This assumption causes a bias in the parameters of the 

curves; neither the height-prediction equation nor the site 

index prediction equation will possess a shape that 

represents the true functional relationship between height 

and age across levels of site index. 

To simultaneously optimize the regression of Yon X and X 

on Y and avoid parameter bias, Goelz and Burk (1 992, 

1996) recommend fitting a base-age invariant site index 

model (a single equation) in the form of an ADE to an all 

possible growth intervals data structure. The typical 

procedure in fitting an ADE is to use forward moving first 

differences. If plot height is measured at ages 15,20,25, 

and 30 years, you would set up three growth intervals. You 

would fit height Hz, at age A,, given the previous height HI, 

at age A,, as predictor variables; that is, you would fit Hz at 

A, = 20 given HI at Al = 15, then H, at A, = 25 given HI at 

Al = 20, and finally H, at A, = 30 given H, at A, = 25. In the 

all possible growth intervals data structure, you use all 

possible differences in both directions. In this example, 

with 4 age measurements, 12 age intervals are possible: 15- 

20, 15-25, 15-30,20-25,20-30,25-30; and 30-25, 30-20, 

30-15,25-20,25-15,20-15. 

With the all possible growth intervals data structure, you 

automatically introduce a lack of independence among 

observations. With stem analysis data, you would have 

temporal or autocorrelation within each tree, with 

Frothinghm7s plot-averaged data you should have 

autocorrelation within a site-quality class. A nonlinear 

model can be written 
where Model 

the error terms are assumed to be independent and 

identically distributed. A number of researchers have pointed to desirable attributes 

for site index equations (e.g., Bailey and Clutter 1974, 

Under nonindependence, the nomal procedure is to expand Goelz and Burk 1992, Scott and Voorhis 1986). The most 

frequently listed criteria are: (1) polymorphism, (2) sigmoid the error term to allow f~st-order autocorrelation: (S-shaped) growth pattern, (3) asymptote is a function of 

where 

the ei are now independent and identically distributed. 

Goelz and Burk (1992) point out that when using all 

possible growth intervals, the model is more complex: 

where 

Y,j depicts prediction of height i by using Y, (height j), X, 

(age i), and4 (age j .c- i) as predictor variables. 

Consequently, the error term must be further expanded: 

e., = + E,, j + ye,, j-1 II IJ 

site index (increases with increasing site index), and 

(2) (4) logical behavior (height should be zero at age zero and 

equal to site index at base age). With these criteria in mind, 

we examined a variety of models, selected several to fit to 

the data, and finally settled on one described by Clutter and 

Jones (1 980). The function, in ADE format, is 

Equation (4) represents the autocorrelation structure 

inherent in fitting site index models to an all possible 

growth intervals data structure. The parameter p accounts 

for the autocorrelation between the current residual and the 

residual from estimating Y,-, using as a predictor. The y 

parameter accounts for the autoconelation between the 

current residual and the residual from estimating Y, using 

Y,-! as a predictor variable. Goelz and Burk (1 992, 1996), 

as well as many other modelers (e.g., Parresol 1993), also 

recommend correcting for heteroscedasticity. 

A final point to be noted is that when using the all possible 

growh intervals data structure, standard errors for the 

parameter estimates will be too small, because the number 

of observations are artificially inflated. Goelz and Burk 

(1996) give a simple correction factor. Because we will not 

be constructing confidence intervals, we are not concerned 

with this point (for this application). In summary, to fit site 

index models one should (1) use an ADE form, (2) use all 

possible growth intervals, (3) account for autocorrelation 

and heteroscedasticity, and (4) inflate standard errors. 

1 1.4 = ..P[P~(; - ;)I (lnq - P2- A, + P,) 

(5) 

where 

ln stands for natural logarithms, H, and A, represent the 

predictor height and age, 4 is the predicted height at age 

A,, the p's are model parameters, and e is stochastic error. 

This function meets most of the above criteria and, in 

addition, has the benefit of being base-age invariant. Baseage invariant curves are more general than base-age-specific 

equations, because they can predict height at any age given 

height at any other age. This function behaves very well for 

ages 10 and greater, giving small values then increasing in a 

sigmoid pattern to the asymptote value, which increases 

with increasing site index. 

To use model (5) to estimate average stand height (H) for 

some desired age (A), given site index (S) and its associated 

base age (A,), simply substitute S for I-i, and A, for A ,: 

Similarly, to estimate site index at some chosen base age, 

given stand height and age, simply substitute S for H2 and A, 

for A, in model (5): 
Application 

Generally, the correlation parmeters can be ignored when 

using equations (6) and (7) for predicting height and site 

index. The quantities e ,-,,, and e ,, will probably not be 

known unless one is working repeatedly on the same plot, in 

which case they could be determined and utilized to predict 

a future value. Our main purpose in using the 

autocorrelation error structure was to obtain unbiased, 

efficient estimates of the P vector. 

Results 

Base Age 25 System 

The data listed in table 1 were expanded into the all possible 

growth intervals data structure. Because each site-quality 

class has 19 age measurements, there are 19x 18 or 342 age 

intervals per site-quality class, or 1,026 total observations 

for fitting model (5). We first fit model (5) without the 

autocorrelation parameters using ordinary nonlinear least 

squares. Weighted least squares was not used because the 

log transformation effectively homogenized the variance. 

We tested the residuals for autocorrelation using Durbin's ttests (Durbin 1970). Durbin's t-tests consist of regressing 

the ordinary least squares residuals on the hypothesized 

autoregressive error structure (in our case equation 4) and 

testing the significance of the correlation parameter 

estimates. The tests show the residuals to be highly 

correlated, as was expected (lj= 0.33, t = 20.68, P < 0.0001; 

f= 0.63, t = 36.19, P < 0.0001; R2 = 0.83). We refit model 

(5), appending the autocorrelation parameters, with the 

SASIETS@ MODEL procedure (SAS Institute Inc. 1993), 

which allows for dynamic updating of residuals. The final 

fitted parameter estimates are listed in table 2. It should be 

noted that model (5) is parsimonious, having only three 

parameters (excluding the correlation parameters), and fits 

the data as well as, or better than, other models we tried. 

The R2 was 0.9994. 

Table 2-Parameter estimates for model (5) 

If we choose to work with a base age of 25 years (i.e., A, = 

25), then inserting the values from table 2 into equations (6) 

and (7), we obtain the following: 

lnhH = 0.7084 exp (8.6188iA) (1nS + 0.9022) 

(8a) 

lnAs = 1.41 16 exp (-8.6188IA) (In H + 74.70991A 

(8b) 

Figure 2 displays a graph of the site index curves generated 

from equation (8a). Some example calculations follow. 

Suppose we want to estimate average dominantlcodominant 

height at age 35 on an area where the site index (base age 

25) is reported to be 30 ft. Inserting 35 for A and 30 for S 

into (8a) results in 

therefore, 

Parameter Estimate I? = exp(3.851) = 47 ft 

-8.6188 Now suppose we want to estimate site index in a 15-year- -74.7099 old stand where the average dominantlcodominant height is 

-2.0862 determined to be 20.7 ft. Inserting 15 for A and 20.7 for H 

0.3599 into (8b) gives: 

0.6348 
Site 

Index 

Figure 2-White pine polymorphic site index curves based on Frothingham's data, base age 25. 
therefore, 

Now suppose we want to estimate site index in a 35-yearold stand where the average do~nantfcodominant height is 

determined to be 52.6 ft. Inserting 35 for A and 52.6 for H 

into (9b) gives: 

lh = 1 .I881 exp (-8.6188135)(1n52.6 

+ 74.7099135 2.0862) + 0.5920 = 4.317 

S^ = exp (3.806) = 45 ft 

therefore, 

Base Age 50 System S^ = exp (4.317) = 75 ft 

If we choose to work with a base age of 50 years (is., A, = 

50), then inserting the values from table 2 into equations (6) 

and (7), we obtain the following: Discussion 

Developing and fitting site index equations for unbiased, 1;H = 0.8417 exp (8.6188/A) (In S - 0.5920) 

(9,) efficient estimates is an involved process, even with 

- 74.70991A + 2.0862 relatively simple data as in this application. Goelz and 

Burk (1 992, 1996) have devised principles and an ad hoc 

method which can be used as a guide in developing or 

selecting an appropriate site index model, and for fitting 

that model to obtain unbiased, efficient parameter estimates. 

ln?' = 1.1881 exp (-8.6188/A) (1nH + 74.7099/A With the ready availability of data management and 

- 2.0862) + 0.5920 

(9b) statistical software, it is now relatively easy to restructure 

data for all possible growth intervals and fit nonlinear 

autoregressive site index models. 

Figure 3 displays a graph of the site index curves generated The southern forest survey has chosen to use the data of 

from equation (9a). Some example calculations follow. Frothingham (1914) as the standard for white pine site 

Suppose we want to estimate average dominant/codominant evaluation. We recognize that, because they are from a 

height at age 70 on an area where the site index (base age northern latitude, these data are not ideal for representing 50) is reported to be 80 ft. Inserting 70 for A and 80 for S southern site relationships. For good or bad, the southern 

into (9a) results in survey considers Frothingham's study to be the best source 

for white pine site relationships in the Appalachians. 

I& = 0.84 17 exp (8.61 88/70) (1n 80 - 0.5920) 

- '74.7099170 + 2.0862 = 4.627 

therefore, 

I? = exp (4.627) = 102.2 ft 

Previously, a base-age-specific site index function was used 

to model these data; however, the current selection of a 

base-age invariant site index function gives great flexibility 

to accommodate changing management paradigms which 

might necessitate a change in the desired base age. 

Equations (6) and (7) can be solved for any base age, as 

demonstrated in the Application section of this paper. 

Estimates of white pine stand height and site index are 

easily obtained through direct evaluation of the functions, 

there is no need for iterative numerical evaluation methods. 
n 

loo 

10 20 30 40 50 60 70 80 90 100 

Age (Y rs) 

Figure 3-White pine polymorphic site index curves based on Frothingham's data, base age 50. 

Site 

Index 
Acknowledgments 

The authors thmk Dr. Robert M. Farrar, Jr., retired project 

leader from the Southem Research Station now living in 

Mississippi, and Dr. Jeffery G. Goelz with the Southern 

Hardwoods Laboratory of the Southern Research Station, 

Stoneville, MS, for their reviews and advice. The authors 

would also like to thank the three anonymous reviewers for 

their helpfbl comments and suggestions. 

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fir and white spruce in the Lake States. Res. Note NC-269. St. Paul, 

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forest tree species in the eastern United States. Gen. Tech. Rep. NC-128. 

St. Paul, MN: U.S. Department of Agriculture, Forest Service, North 

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old-field slash pine plantations. Res. Pap. SE-217. Asheville, NC: U.S. 

Department of Agriculture, Forest Service, southeastern Forest 

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1983. Timber management: a quantitative approach. New York: John 

Wiley & Sons. 333 p. 

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when some of the regressors are lagged dependent variables. 

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Washington, DC: U.S. Department of Agriculture. 70 p. 

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index equation: jack pine in north central Ontario. Canadian Journal of 

Forest Research. 22: 776-784. 

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index equations. Canadian Journal of Forest Research. 26: 1585-1 593. 

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of Agriculture. Forest Service, Interagency Memorandum, November 16, 

1962. 53 p. 

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conference on forest growth modeling and prediction, 1987 August 23- 

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forest survey. Res. Pap. SRS-10. Asheville, NC: U.S. Department of Agriculture, 

Forest Service, Southern Research Station. 8 p. 

A base-age invariant polymorphic site index equation was used to model the white 

pine (Pinus strobus L.) site-quality data provided by Frothingham (1914). These 

data are the accepted standard used by the Southern Forest Inventory and Analysis 

unit of the U.S. Department of Agriculture, Forest Service. An all possible growth 

intervals data structure was used, and autocorrelation parameters were incorporated 

into the site index model. It has recently been shown that these measures are 

necessary to obtain unbiased, efficient parameter estimates. The model is invertible, 

hence site index can be explicitly determined without the need for a numerical 

evaluation procedure. The site index model can be solved to provide an equation for 

any base age, hence it is applicable regardless of the choice of rotation age. Site 

index curves are graphed for base ages 25 and 50 years, and example calculations are 

provided. 

Keywords: Autocorrelation, base-age invariant, Pinus strohus, polymorphic.