Height. Growth Patterns and Site Index of White Pine in the Southern Appalachians

Transcription

1 Height. Growth Patterns and Site Index of White Pine in the Southern Appalachians DONALD E. BECK Abstract. Recently devised methods of developing site-index curves are used to investigate patterns of height growth in eastern white pine (Pinus strobus L.) and the impact of these patterns on estimation of site index. An exponential growth model is fitted to sets of height-growth data obtained by internode measurement. Estimated parameters of the growth function are then used to analyze the pattern of growth and some factors which affect it. Pattern of height growth of dominant and codominant trees varies significantly within even-aged stands but is not related to present density of the surrounding stand. Highly significant differences in the growth curve and its parameters are associated with the quality of the site, per se. Pattern of growth also varies to some extent at a given level of site index and is related to some topographic features of the site. Forest Sci. 17: Additional key words. Pinus strobus, stand growth. POSSIBLE causes of the discrepancies between site-index curves and observed height-growth trends have been discussed at length by Spurt (1952), Smith et al. (1960), Stage (1963), and Curtis (1964). The major discrepancies apparently result from the mechanics of curve construction and the assumptions about height growth used in developing site-index curves. Harmonized site-index curves for most American species were developed with the techniques described by Bruce (1926); these involved measuring present height and age of many stands, fitting an average regression of height on age, and drawing a series of curves proportional to the average guiding curve. The principal reasons why site-index curves derived in this manner often do not accurately depict height growth may be summarized as (1) distortion of the mean curve caused by correlation of age with site quality in the sample stands, (2) failure to determine or describe the true shape of the growth curve, and (3) the assumption that the shape of the curve is the same for all sites (Bull 1931, Curtis 1964, Spurr 1952, 1955). Distortion of the mean curve as a result of sampling bias can be prevented by stem analysis, internode measurement, and remeasurement of permanent plots. Data obtained in this manner will also allow determination of true curve form by reducing the scatter of points about mean height for a given age. The question remains as to whether the shape of the height-over-age curve varies from site to site. Spurt (1955, p. 80) said, "... actually, the site-index curve is simply a growth curve for a given genetic entity under a given set of environmental conditions." Thus, we could expect to find as many different curve shapes as there are combinations of trees and environmental conditions--an almost infinite number. From the practical viewpoint of deriving and applying site-index curves for site evaluation, the main questions are the extent of the variation and whether curve shape is related to quanti- The author is Principal Silviculturist, Southeast. Forest Exp. Sta., USDA Forest Serv., Bent Creek Experimental Forest, Asheville, N. C. This paper is based on part of a Ph.D. thesis submitted to the School of Forest Resources, North Carolina State University at Raleigh. Manuscript received Sept. 18, / Forest Science

2 fiable characteristics of the tree, stand, or site. This paper presents the results of an investigation of height-growth patterns in eastern white pine (Pinus strobus L.) based on internode measurements. The study objectives were: (1) to determine the extent of variation in curve pattern; (2) to relate growth pattern to site index, per se, and to specific attributes of the tree, stand, and site; and (3) to assess the impact of variation in growth pattern on evaluation of relative site productivity by the site-index method. Field Procedure The data were obtained from 42 pure, even-aged stands of white pine of natural origin. These stands were in the mountains of southwestern Virginia, eastern Tennessee, western North Carolina, and northern Georgia, the southernmost range of the species. Stands were distributed over elevations from 1,350 to 3,050 ft, slopes from 0 to 57 percent, many aspects, and slope positions ranging from streambottoms to ridgetops. Site index (at age 50) ranged from 71 to 122. Most of the stands were between 45 and 70 years of total age and thus allowed direct measurement of height at index age 50 or required only short extrapolations. In each stand, three dominant or codominant trees were selected from a small portion of the stand apparently uniform in topography and soils. The maximum size plot needed to contain the three sample trees was approximately A acre; the majority of plots were % acre. Basal area in trees 4.6 inches dbh and greater, as measured on fixed plots surrounding the sample trees, ranged from 96 to 177 sq ft per acre. The trees selected for measurement of the height-over-age relationship were free from injury and disease and showed no evidence of past suppression of height growth. After the trees were felled, height at successive ages was determined by internode measurement. Nodes on the lower bole of the larger trees were frequently obscured by natural pruning and subse- quent diameter growth. When the nodes were obscured, the lower 15 to 20 ft of the bole was split to the pith to avoid missing nodes. As a check on internode measurements, rings were also counted. Hence, separate sets of height-over-age data were derived for each of the 126 sample trees. Each set consisted of the succession of annual heights of a given tree from age 1 to total age at time of measurement. For each sample tree, there were additional measurements of stand density, site index, and topographic features of the site. The Growth Model Attempts to fit mathematical formulae to height growth of trees date back to the 19th century and possibly beyond. The choice of the most appropriate model is, however, still highly problematical. Graphs of height over age showed that height growth for all trees followed the sigmoid trend typical of the size-time relationship. Graphic methods were used to investigate sigmoid functions discussed by Grosenbaugh (1965), including the Gauss, Gompertz, Bertalanffy, and logistic. None of these functions proved sufficiently flexible to describe the variety of shapes represented in the data. Investigation of possible modifications led to Richards' (1959) modification of the Bertalanffy equation (Bertalanffy 1938, 1957). Richards demonstrated that this function is capable of generating a wide range of shapes if certain restrictions based on theoretical considerations of animal growth are removed. The possible application of the model to tree growth was suggested by Cooper (1961), and it was applied to height growth of Engelmann spruce and inland Douglas-fir by Brickell (1966, 1968). The form of the equation tested in the present study was: H = A [1- exp (-kt) ] /o-, ) (1) where H is height at age t, and A, k, and m are parameters to be estimated. Equation 1, with suitable parameter values, represents an asymmetrical, sigmoid curve that passes through the origin and ap- volume 17, number 2, 1971 / 253

3 TABLE 1. Analysis of variance indicating the contributions of several sources to total variation in the set Of height and age observations. Source of variation DF MS MS is an estimate of-- F Site class ro q r? q r? q r 117'* Stands within site class l ro q r? q rs 2** Trees within stand l ro q r? 31'* Observations within tree l ro Total 1,348 ** Indicates significance at the.01 level of probability. a The variance component for: site class S, : , estimates r, ; stands within site class 291, estimates r?; trees within stand S? , estimates r?; and observations within tree 1.36, estimates ro 2. proaches some maximum height, A, as age approaches infinity. When equation 1 was fitted to the data for each of the 126 sample trees by an iterative least-squares procedure, it proved highly efficient in describing the height-over-age relationship of individual trees. 2 The root-mean-square residuals ranged from 0.33 to 3.00 ft; more than 95 percent were less than 2 ft. Graphs of estimated versus observed heights showed no evidence of bias. Furthermore, extrapolations beyond the limits of data were in reasonable agreement with observations from white pine of advanced age made by Spaulding (1899). It seemed reasonable to assume that the height-growth pattern of individual trees was adequately described by the three parameters of the model. Sources of Variation in Growth Pattern Sources of variation in the curves of height over age were examined by partitioning the sum of squared residuals about the mean height-over-age curve. This was ac- Marquardt, Donald W. Least-squares estimation of non-linear parameters. IBM Share Program 1428 NLIN 2, 32 p Only a part of the data for each tree was utilized. Starting with age one year, every fifth height-over-age pair was used, i.e., ages 1, 6, n. This procedure cut computational ttme significantly. The differences were inconsequential in trial curves fitted with all the data and those fitted with only a part of the data. complished by fitting equation 1 to (1) the total sample of 1351 paired observations of height and age to obtain one mean curve, (2) the data segregated into 5 site classes, s (3) the data for individual stands, and (4) each of the trees individually. If the 126 tree curves, 42 stand curves, 5 site-class curves, and 1 mean curve with 3 unknowns fitted per curve--are treated as repeated subsamples, the analysis of variance is as shown in Table 1 (Anderson and Bancroft 1952). Degrees of freedom for testing variation within tree are probably inflated because of serial (or auto-) correlation of error residuals. However, degrees of freedom for the other sources of variation, which were the primary concern of the analysis, are not affected by autocorrelation. The analysis of variance shows that the growth curves vary significantly within stands and among stands of similar site index, as well as among stands of different site indices. In Table 1, the relative contribution of each source is indicated by the variance components. As might be expected, the variance component for site class (S 2 = ) is large relative to the other sources because by definition the curves for site classes vary in level. And, as shown in previous studies, curve shape is likely to vary with site class as well. The relative contribution of stands within site class and of Site classes were derived by arbitrarily dividing the data into 10-ft intervals of site index. 254 / Forest Science

4 trees within stand, on the other hand, was not expected. If growth pattern varied greatly among stands of a given site-index class, we would expect the variance component for stands within site class to be large. However, the component for stands within site class (S? = 2.91) is only about one-fourth as large as the component for trees within stand (St ). This comparison suggests that the patterns of growth for stands of a given site-index class (in some cases separated by three degrees of latitude) are more nearly alike than those for individual trees located together in a small, apparently homogeneous portion of the same stand. The small component for observations within tree demonstrates the good fit of the model to growth curves of individual trees. The sections that follow explore each of the significant sources of variation in the growth curves. Patterns of Growth Within Stands. Although not the primary object of the study, the relatively large contribution of variation among trees within stand to total variation demanded more than cursory examination. In the restricted area of a stand selected for uniformity in site conditions, the most obvious source of variation among trees was the degree of competition or environmental space available to each. Point density in square feet of basal area (Spurr 1962) was calculated for each tree and was shown to vary over a wide range among trees in a given stand. However, graphic and regression analysis of the growth-curve parameters and measured periodic growth rates failed to show any relationship to point density. From inspection of the data, growth pattern appeared to be totally unrelated to point density; stands with a high degree of variability in overall growth pattern among the three trees often had relatively uniform distribution of basal area. Conversely, in some stands with practically identical growth trends for the three trees, point density varied greatly among the trees. For the remaining analyses, the best measure of curve form for a given stand was assumed to be the set of curve parameters estimated by fitting a common stand curve to the data from the three trees in the stand. Root-mean-square residuals for the 42 stand curves ranged from 0.90 to 5.19 ft and were less than 4.0 ft for 82 percent of the stands. Thus, even with the relatively large amount of variation among trees, the growth model described the average growth curve for a stand with reasonable accuracy. Pattern of Growth by Level of Site Index. In order to determine if pattern of height growth varied in a definable way with the level of site index, the parameters A, k, and m, which defined the growth curves of the stands, were examined in relation to the measured site indices of the stands. Graphic and bivariate regression analysis showed that estimates of the parameters A and k for individual stands were correlated with site indices of the stands The expressions best relating the coefficients to site index were: A -- b0 + b (S) (2) k = Co + c (S), (3) where S is site index of the stands and b0, b, Co, and c are coefficients to be estimated. The parameter rn showed no tendency to vary with site index. In order to show the total effect of growth pattern, site index was incorporated into the height-growth equation by substituting expressions 2 and 3 in equation 1 for.4 and k, respectively. The parameter rn was allowed to assume a constant value over all sites. Fitting this function to the combined data for all stands resulted in the equation expressing height (H) as a function of age (t) and site index (S): H: ( S) x (4) { 1 - exp [-( S)t] } /( -ø.47ø27) The root-mean-square residual for the equation is 3.7 ft. A series of curves for height and annual increment were calculated from equation 4 and its first derivative (Fig. 1). volume 17, number 2, 1971 / 255

5 1' INDEX AGE SITE INDEX loo o 9O 80 6o 7o 6O 5O 4O 3O 2O AGE (YEARS) FIGURE 1. Curves of annual increment (A) and cumulative height (B) for selected site indices. These curves show the changing pattern of growth with level of site index. The most striking features of the curves are the way in which age at inflection changes with site index and the convergence of relative growth rates after the maximum rate is obtained (Fig. 1 ). The age of maximum growth rate occurs at younger ages on better sites. Once the inflection is passed, the growth rate de- 256 / Forest Science

6 clines relatively more rapidly on the. better sites. By about age 55, the rate of annual growth is about equal on all sites. For example, in stands of site index 120, the rate of growth increases rapidly from an early age and reaches a maximum of 3.4 t per year at approximately 14 years of age. On the other hand, stands of site index 60 do not attain their maximum of 1.5 ft per year until age 23. After inflection, however, the decline in growth for site index 60 is relatively slow in contrast to that of site index 120. By age 55, the rate of growth on both site indices 120 and 60 is about 1 ft per year. This convergence of growth rates occurred well within the range of ages sampled and is not simply an artifact of the model. For example, analysis of variance of measured rate of height growth at 50 years of age showed no significant difference among sites. Extrapolation of the curves indicates a slightly higher rate of growth beyond about 50 years of age on the lower quality sites than on the higher quality sites. The data for advanced ages, however, are not sufficient to show this relationship conclusively. Somewhat similar systematic changes in curve shape with the level of site index have been demonstrated by Bull (1931) for red pine, Spurr (1955) and King (1966) for Douglas-fir, McGee and Clutter (1967) for slash pine, Stage (1963) for grand fir, and Brickell (1966, 1968) for Engelmann spruce and inland Douglas fir. Such changes in curve shape with the quality of the site can be accounted for in curve construction and, hence, do not necessarily cause serious difficulty in site evaluation. If this is the only sort of variation in curve shape, age and site index can be used to define height within rea- sonable bounds. Consequently, site index could be determined from measurement of stand height at any age. Pattern of Growth for a Given Site-Index Class. The variation in growth pattern among stands of the same site-index class, which is suggested by the analysis of variance in Table 1, is illustrated in Figure 2. Comparison of observed height-growth trends with the trends calculated from equation 4 shows that age and site index alone do not fully define growth patterns. For example, the observed height in stands 20 years old differs from estimated height by as much as 7 ft. Consequently, for these three stands, estimates of site index at age 20 from equation 4 could err by as much as 12 ft. In order to determine if the estimates of height and growth pattern could be further improved, curve parameters,4, k, and m for fixed levels of site index were examined in relation to latitude, elevation, slope, position on slope, and aspect. In the regression analyses, percentage of slope, position on slope, and aspect were significantly related to the three growth-curve parameters. However, incorporation of the best expressions of these relationships into equation 4 did not materially influence the precision of height estimates. A nine-parameter model expressing height as a function of age, site index, percentage of slope, position on slope, and aspect had a standard error of estimate of 3.7 ft, which was the same as that for the five-parameter model (equation 4). Curves of height over age calculated from the nine-parameter model for extremes of the topographic variables at a fixed level of site index differed most imperceptibly from the relationship predicted by the five-parameter model. Thus, there appeared to be no gain in ability to predict growth pattern by inclusion of the topographic variables. Discussion This investigation demonstrates that Richards' (1959) growth model offers a satisfactory fit to the height growth of white pine trees over an extensive range. It further shows that this growth equation can be used to analyze the pattern of the height-growth curve and the factors that affect it. Variation in growth pattern and periodic growth rates among trees within stands could not be explained by surrounding stand density. Thus, estimation of site quality can be made without reference to al- volume 17, number 2, 1971 / 257

7 11o loo OBSERVED GROWTH PREDICTED GROWTH ' 70 k. o 50 40,., '"1/ / ,,,,?,'.;/,,, m ' I I I i o AGE (YEARS) FIGURE 2. Observed height growth for three stands o! the same site index compared with the predicted growth trend from equation 4. level of stocking--at least within the range of stocking observed in this study. However, the relatively large degree of apparently random variation among trees re-emphasizes the need to follow welldefined rules, such as suggested by Stage (1963), when applying site-index curves based on stem analysis. The largest part of the variation in growth pattern among stands was due to differences in site quality. From regression of growth-curve parameters on site index, it was shown that there are progressive changes in the shape as well as the level of the height-over-age curve as s te index changes. Estimates of site index can be considerably improved by allowing for these differences in growth pattern. In Figure 3, deviations of estimated site index from observed site index are compared for proportional and polymorphic curves. In (A), the estimates are based on a series of curves harmonized to. have the same shape, i.e., proportional to the mean curve derived from the data of this study; in (B), the estimates are based on the polymorphic curves derived from equation 4. Errors of estimate based on the proportional curves show a highly significant tendency to vary with age and with observed site index in Figure 3A. Site index measured in stands less than index age (50 years) is underestimated on less-thanaverage sites and overestimated on betterthan-average sites. A similar pattern of bias exists at all ages less than index age (50 years), but the magnitude of errors decreases as index age is approached. In stands older than index age, the bias is reversed. Allowing for change in growth pattern (as in equation 4) with the level of site index eliminates the bias as well as sub- 258 / Forest Science

8 +30 (A) (s) -2' -30 i i i i i i i i I i i i I SiTE INDEX (OBSERYED/ SITE INDEX (OBSERYED/ FIGURE 3. (A) Biased estimates of site index result when proportional curves are used in stands 20 years o[ age. (B) Estimates o! site index at age 20 based on the polymorphic curves o! Figure 2 are unbiased. stantially reduces the magnitude of the errors (Fig. 3B). The average difference between observed and estimated site index for the 42 stands is reduced from 4.4 (with a standard deviation of the differences of 10.6 for the proportional curves) to 1.0 (with a standard deviation of differences of 6.1 for the polymorphic curves). Inclusion of the topographic variables in the growth equation did not mateddally improve the accuracy with which growth pattern could be predicted. Consequently, site-index curves incorporating topographic features did not improve estimates of site index beyond those estimates possible with curves that allow for changes in growth pattern with the level of site index (Fig. 1). The possibility remains, of course, that prediction of growth pattern and, thus, of site index could be further refined by considering soil factors or site attributes other than the topographic features included in this study. Literature Cited ANDERSON, R. L., and T. A. BANCROFT Statistical theory in research. II. Analyses of experimental models by least squares. McGraw-Hill Book Co, N.Y. BERTALANFFY, LUDWIG VON A quanti- tative theory of organic growth. II. Inquiries on growth laws. Human Biol 10: Quantitative laws in metabolism and growth. Quart Rev Biol 32: BRICKELL, JAMES E Site-index curves for Engelmann spruce in the northern and central Rocky Mountains. USDA Forest Serv Res Note INT-42, 8 p. Intermountain Forest Range Exp Sta, Ogden, Utah A method for constructing site index curves from measurements of tree age and height--its application to inland Douglas-fir. USDA Forest Serv Res Pap INT-47, 23 p. Intermountain Forest Range Exp Sta, Ogden, Utah. BRUCE, DONALD A method of preparing timber-yield tables. J Agr Res 32: BULL, HENRY The use of polymorphic curves in determining site quality in young red pine plantations. J Agr Res 43:1-28. COOPER, CHILES F Equations for the description of past growth in even-aged stands of ponderosa pine. Forest Sci 7:72-80 CURTIS, R.O A stem-analysis approach to site-index curves. Forest Sci 10: GROSENBAUGIt, L. R Generalization and reparameterization of some sigmoid and other non-linear functions. Biometrics 21' KING, JAMES E Site-index curves for Douglas-fir in the Pacific Northwest. Weyerhaeuser Forest Pap 8, 49 p. Centralia, Wash volume 17, number 2, 1971 / 259

9 MCGEE, C. E., and J. L. CLUTTER A study of site index for planted slash pine. J Forest 65: R CHARDS, F.J A flexible growth function for empirical use. J Exp Bot 10: SMITH, J. H. G., J. W. KER, and L. MEGER Natural and conventional height-age curves for Douglas-fir and some limits to their refinement. Fifth World Forest Congr Proc, p SPAULDING, V. M. (revised by B. E. Fernow) The white pine. USDA Div Forest Bull 22, 185 p. S?URR, STE?HEN H Forest inventory. Ronald Press Co, N.Y. 476 p Soils in relation to site-index curves. Soc Amer Forest Proc 1955: A measure of point density. Forest Sci 8: STAGE, ALBERT R A mathematical ap- proach to polymorphic site index curves for grand fir. Forest Sci 9: Manuscript Reviewers, 1970 THE MERIT of a scientific journal depends on the quality of its authors and reviewers alike, but the latter are usually unknown to the readers. Forest Science acknowledges with gratitude the services of the people listed below who reviewed or refereed manuscripts during the year Any omissions from this list should be made known to the Editor. Abbreviations of governmental agencies in the addresses are as follows: CFS--Canadian Forestry Service. Int.--Intermountain Forest and Range Experiment Station NC--North Central Forest Experiment Station NE- Northeastern Forest Experiment Station PNW--Pacific Northwest Forest and Range Experiment Station PSW- Pacific Southwest Forest and Range Experiment Station RM Rocky Mountain Forest and Range Experiment Station SE--Southeastern Forest Experiment Station SO--Southern Forest Experiment Station USFS- U.S. Forest Service Herschel G. Abbott, Univ. of Massachusetts Paul E. Aho, USFS PNW Martin Alexander, Cornell Univ. L. Hartwell Allen, Cornell Univ. Robert Allen, Clemson Univ. Hal E. Anderson, USFS RM Loukas Arvanitis, Univ. of Wisconsin Robert Benson, USFS Int. William Bentley, Univ. of Michigan John W. Benzie, USFS NC J. E. Bethune, Univ. of Georgia C. A. Bickford, State Univ. N. Y. College of Forestry Don Boelter, USFS NC J. M. Bonga, CFS Forest Research Lab, Fredericton David R. Bouldin, Cornell Univ. T. W. Bowersox, Pennsylvania State Univ. Robert B. Brander, USFS NC James E. Brickell, Univ. of Idaho Holger Brix, CFS Forest Research Lab, Victoria A. Broido, USFS PSW Claud Brown, Univ. of Georgia Gregory Brown, Univ. of Missouri James K. Brown, USFS Int. George Byram, USFS SE Martyn Caldwell, Utah State Univ. W. H. Carmean, USFS NC Craig Chandler, USFS RM Dan Chappelle, Michigan State Univ. T. W. Childs, USFS PNW 260 / Forest Science