Base. Age Invariant Polymorphic Site Curves

Transcription

1 Base. Age Invariant Polymorphic Site Curves ROBERT JEROME L. BAILEY L. CLUTTER Abstract. Polymorphic curves fitted by techniques in which site index is a predictor variable are specific to a preselected base age. A new approach eliminates this problem and the need to quantify site quality with the data before estimating parameters. Height-over-age curves for Pinus radiata D. Don in New Zealand are produced with linear least-squares estimation. Forest Sci. 20: Additional key words. Anamorphic curves, tree height, Pinus radiata. CURRENT METHODS for obtaining polymorphic site index curves from remeasurement or sectioned stem data require a data-based measure of site quality as a predictor variable in the model. This paper presents an approach which only requires that height-age data be grouped into unique, but not necessarily mutually exclusive, site classes. On the assumption that a linear relationship exists between logarithm of height and age to a negative power, we have derived an estimation technique which does not require iterative nonlinear least squares. Most approaches to polymorphic curves assume that data from either stem analysis or remeasurement plots are available. Data from sectioned stems are adjusted to correct for a bias in observed heights (Carmean 1971, Lenhart 1972). Data are sometimes combined by plots so that the basic observation becomes a plot's average height-age curve, which is assumed to represent height growth on one homogeneous site. One approach then calls for selection of a base age and identifies the plot curves by site index class at this age. A model is then adopted: where and H = f(,4,s; A = stand age, $ = site index (observed height at base age), H = average dominant height, Pk = a vector of k parameters. Depending on the form of f, the parameters are estimated either by linear or nonlinear least squares (Carmean 1972). Since S enters the model prior to estimation, the estimates of general parameters are unique to the preselected base age. Curves must be used for that base age only. Algebraic adjustment of the original parameter estimates to obtain curves for a different base age will not give curves equivalent to those obtained by refitting with the new base age. The shape of the curves as estimated from the data depends on an essentially arbitrary initial choice rather than, as is deskable, being invariant under choices of base age. Heger (1973) has found that curves obtained with different such choices can differ markedly in shape even though algebraically adjusted after fitting to pass through the same points at an index age. In a somewhat different approach, Stage (1963) determines growth rate when total height is 55 ft (16.8 m). Again, estimates of general parameters are unique to this choice; they would not necessarily be the same if 45 ft (13.7 m) or 60 ft (18.3 m) were the datum height. This approach, The authors are, respectively, Mathematical Statistician, Southern Forest Experiment Station, USDA Forest Service, New Orleans, La ; and Union Camp Professor of Forest Resources and Statistics, Univ. of Georgia, Athens Data furnished by N. Z. Forest Products, Ltd, Auckland, New Zealand. This paper is based on a part of the senior author's Ph.D. research (Bailey 1972). Manuscript received May 10, volume 20, number 2, 1974 / 155

2 however, does provide a better index of a site's uniqueness-of-curve-shape. To obtain a value for height increment or rate of growth, more of the data for a site are necessary--at least two points rather than only one as in the other approach. In the method discussed here, all the data for a plot are used to determine the site-specific parameter for that plot without any prehminary specification of an arbitrary datum. Developing the Model Background. One popular approach to anamorphic (proportional) site index curves is based on the assumption that tog (H) is a linear function of (l/a). -Further, if height-age data are available for rn sites (plots), the anamorphic system results from fitting log (H) = at + b(1/a)% (2) (i = 1, 2..., m); where at = a parameter specific to the i'th site, b = a common regression slope parameter, and c > 0 is a linearization parameter (often taken to be + 1). In (1) the values for S are analagous to the at's in (2) and are the site-specific parameters in the model. Thus, the general approach outlined for using (1) obtains estimates for a site-specific parameter based on one height-age pair in the data for a particular plot. This has the undesirable feature that measurement errors associated with that data point are weighted too heavily in the fitting process. Estimates for the at's are not required in the final form of the model. To see this, let the base age for which curves are desired be A and the corresponding height values be &. From (2) we have log (&) -= at + b(1/a ) c, at -- log (SO - b(1/ab). (3) Substitution of (3) for ai in (2) gives log (H) = log (SO + b{a -ø- A - } or H = & 10 't- -'tc q, (4) 1og (x) =base 10 logarithm of x and In (x) = base e logarithm of x. an equation for H in terms of A, A,, and &. With estimates for b and c, a height-age curve can be determined for any base age and site index combination with equation (4). If c in (2) is fixed, say c = 1, then an estimate for b using least squares is easily obtained without reference to a base age. If b < 0, height over age curves produced by (4) will all start at the origin and tend to different asymptotes as A--->. The asymptote for a curve through the point (A, S) will be S 10- b -ø. These are not unreasonable limits. Another desirable feature of this technique of parameter estimation is that when,4 = A,, H = &. This feature will not necessarily exist in models developed with the general approach using (1). Polymorphic Model. A model based on (2) would generate a system of proportional site index curves with a constant relative rate of growth, (dh/da)/h, across all sites at a given age. To obtain a polymorphic (nonproportional) system we can rewrite (2) as log (H) = a + bt (l/a)% (5) (i = 1, 2..., m). In (5), the b 's are the site-specific parameters and (dh/da)/h contains them so that relative rate of growth is a function of both age and site. If b, is expressed as a function of site index, as was done with a, in (2), then or log (/-/): a + [log(s0- a] [A /A] (6) H = l0 s {&/lo }r /. If S < l0 s and c > 0, limits are: limit H = 0,,4-->0 limit H: 10% The first limit is desirable. The second indicates that limiting height is the same for all sites. Thus, curves from (6) with selected values for St at a base age, A,, will start at the origin, have rates of increase dependent on &'s, and tend to an upper 156 / Forest Science

3 asymptote independent of &'s. Whether or not the latter property has biological significance is not important. The effective limit will depend on the life span of the species under study and therefore will not be constant across values of &. If AL is the limiting age, the effective limiting height is Parameter HL = Estimation with either (2) or (5) is easy if c is known. In the case of (2) with c known to be c* and nt > 1 height-age pairs on each site, an estimator for b is the common slope estimator in covariance analysis (Graybill 1961), t-1 j=l t=l J=l where Y j = log(hq) X o = (1/A j)** H o = the fth height on site i Ao = the fth age on site i. If nt--- 1 for all i (that is, the data consist of independent height-age points), this estimator will always give = 0. For this situation, (2) is fitted by minimizing the sums of squares about the model with the average value for at replacing at in (2). Let a=(1/m) a -- (l/m) [log(h ) - b(1/a ) *] t= = = (I/m) ' log(ht) - (b./m) 4=1 Replacing at with in (2), Yt = Y + b(xt- X), where the ] is dropped since it is one in every case. This is the simple linear regression model. If c is known to be c* in (5) and n > 1 height-age pairs are available on the?th site, the least squares estimator for a is d = (7) When nt = 1 for all i, (5) cannot be used as a model. When c is not known for either (2) or (5), the estimation problem is slightly more difficult. The discussion here will be confined to the more interesting case of (5). For (2), the steps which follow would lead to a similar estimation technique, but the system of curves would still be anamorphic. For an unknown c, parameters could be estimated by nonlinear least squares. However, m + 2 parameters would have to be estimated explicitly--a very time-consuming process. An easier approach can be effected through derivatives. With logarithmic bases converted, (5) is In(H) = a + bt(1/a)% (8) except for the constant ln(10). Differentiation of (8) with respect to A gives (l/h) (dh/da) = -b c(1/a) +, In [(l/h) (d///da)] = ln(-b,c) (9) + (c + 1) In(l/A). Equation (9) is linear in the new parameters and we have the model 4 t = ln(-btc) 4 = c + 1, so that Y = X (10) where Y i = In [(1/HO(dll j/dat )] X o = ln(1/ao) ß With stem analysis or remeasurement data, the average relative growth rates (Stage 1963) and interval midpoint ages can be obtained and (10) can be fitted by least squares to give a pooled estimate for (beas in a one-way analysis of covariance (Graybill 1961). This estimator is the same as for b in the case of (2). To esti- volume 20, number 2, 1974 / 157

4 TABLE 1. Number of plots by pre- PREDOMINANT SITE INOEX MEA..E G.T SASE AGE 2S} dominant mean height at age 25 and year Fm(,E*,S E,(, *,s Of planting. too,.o,40., Predominant mean height at age 25 in ft (m) Year 95 I I (29.0) (30.5) (32.0) (33.5) (35.1) (36.6) 130 (39.6) 175 (55.3) 120 (36.6) I10 (33.5) 50( ( (27.4) 80 ( I I I I I I I I I IOO ( H5.2) 25 {7.6) mate c, we can subtract one from the estimator of 4,2. That is, (x -œ,.)r, _1 = 1. (11) (Xs-X.) s i=1 We can en substitute d into (5) for c and use (7) to estimate e a. Wi (5) a hei t gro model and parameters estimated by (11) and (7), a polymorphic system of cu es can be produced th (6). The fitting process d s not require a preselected b e age, soaing, or iterative nonlne le t squares. In addkion, whatever the b e age, hei t will equ site index when age equ s base age. Application to Monterey pine Extensive data on height growth in Monterey pine (Pinus radiata D. Don) plantations over a period of 41 years were made available by N. Z. Forest Products, Ltd., of Auckland, New Zealand. Permanent plots were established in plantations of Monterey pine and measurements taken at 1- to 10-year intervals. Earliest measurements were at age 6. Most plots were remeasured at ages 7, 16, 19, and 23, then annually through age 39 or 40. The 54 plots were distributed among 5 management blocks on the North Island at elevations ranging from 505 to 1900 ft (153.9 to m). Slopes ranged from zero to 30 ø. Predominant mean height, defined AGE (YEARS} FIGURE I. Site index curves from a common asymptote model for Monterey pine plantations in New Zealand. by Beekhuis (1966) as the average height of the 40 tallest trees per acre, was recorded for the plots at each measurement age. Average height of dominants and codominants, the most common variable in the United States, was not easily available. The two variables very likely are closely correlated, and methods developed here could be directly applied in studies with average height of dominants and codominants. Numbers of plots by predominant mean height at age 25 and planting date (Table 1 ) indicate a high proportion of plots on the better sites. Data from each plot (with heights in feet) were transformed as required for (10)--derivatives approximated by first differences. From (11), the estimate for c was = When this estimate was substituted and (5) fitted to the data for each plot, the maximum unexplained variation in log(//) for any plot was 0.6 percent. In 48 of 54 cases r s was or higher. The pooled estimate for the a using (7) was = , giving H = 10 s.00øs (&/lo.ooo }r't /'tlø' (ft) (12) Forest Science

5 as a height growth model for P. radiata in New Zealand. Curves generated with (12) for base age 25 (Fig. 1) agreed well with plotted data when compared visually on a transparent overlay. The upper asymptote for these curves is ft ( m). However, the species rarely exceeds a lifespan of 150 years (Scott 1960). If we set 200 as a limiting age, then the upper limit on height is Hn = &0.aas (ft) for the base age 25 curves. For the site indices of Figure 1, we have: ft m ft m Polymorphism in the curves is apparent but more pronounced for younger ages. Extension of the Approach The method used to obtain nonproportional curves with a log(h) - (l/a) c model can also be applied to other equation forms. It essentially consists of identifying a parameter in the equation responsible for curve shape and allowing this parameter to be site-specific. An estimation technique not requiring identification of site in the data before fitting must then be derived. The final step is the solution for H as a function of A, Ab, and S. Literature Cited BAILEY, ROBERT L Development of unthinned stands of Pinus radiata in New Zealand. PhD Thesis, Univ of Ga, Athens. 73 p Diss Abstr Internat 33(9) :4061-B, BEEICUmS, I Prediction of yield and increment in Pinus radiata stands in New Zealand. N Z Forest Serv, Tech Pap No p. C MEAN, WILLAm> H Site index curves for black, white, scarlet, and chestnut oaks in the Central States. USDA Forest Serv Res Pap NC-62, 8 p. North Cent Forest Exp Stn, St Paul, Minn Site index curves for upland oaks in the Central States. Forest Sci GRAYBmL, FRANr, Zn A An introduction to linear statistical models, Vol I. Mc- Graw-Hill, NY, 463 p. HEGER, L Effect of index age on the precision of site index. Can J Forest Res 3' 1-6. LEN n T, J. DAYre An alternative procedure for improving height/age data from stem analysis. Forest Sci 18:332. ScoTr, C. W Pinus radiata. FAO Forest and Forest Prod Stud, No 14. Rome, Italy, 32 p. STAGE, ALBERT R A mathematical approach to polymorphic site index curves for grand fir. Forest Sci 9: volume 20, number 2, 1974 / 159