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An Expression for the Effect of Aspect, Slope, and Habitat Type on Tree Growth Note by A. R. Stage Abstract. An expression for describing the effect of aspect and slope on tree growth is illustrated by the relation of western white pine site index to aspect, slope, and habitat type. Forest Sci. 22: 457-460. Additional key words. Site, Pinus monticola, western white pine. STUDIES of the effect of environmental factors on the growth of trees generally assume that the effect of aspect can be coded as a cosine function with the minimum shifted to the southwest quadrant and the maximum to the northeast quadrant. This assumption, first proposed by Gaiser (1951) with respect to site index, has been found to be an acceptable approximation in numerous later site index studies. However, for other measures of tree growth such as periodic diameter increment, the optimum aspect can be quite different. Indeed, in regression models, the aspect vari- able may have different effects depending on the choice of other factors to be included in the model. Beers and others (1966) have shown how the phase of the cosine function can be shifted to place the optimum at an arbitrary aspect. Searching techniques to locate the optimum for a given set of data as used by Hartung and Lloyd (1969) can be quite inefficient and depend for their success on a good initial guess of the location. Finally, the use of higher order sine or cosine functions to represent asymmetries was introduced by Carmean (1967). The purpose of this note is to show how the foregoing developments can be combined into a single technique whereby the a priori as- sumption of a maximum in the northeast quadrant (45 ø azimuth) and a minimum in the southwest quadrant can be replaced by an empirically determined location of the optimum without repeated calculations of the regression fit. In addition, it is argued that expressions for the effect of aspect should always be considered as terms involving an interaction with slope. The simple symmet- rmal cosine curve assumption can also be modified to accommodate more asymmetrical forms of the response to aspect. The effect of aspect proposed by Beers and others is stated mathematically as B cosine (a-O) where B is the amplitude, a is the azi- muth measured clockwise from north, and 0 is the phase shift that generally has been assumed to be 45 ø (Trimble and Weitzman 1956). Suppose now that instead of assuming the optimum aspect a priori we wish to esti- mate the phase shift angle from the data at hand. This can be accomplished quite readily by introducing both the cosine of a and the sine of a as two independent variables in the multiple regression for predicting the response to aspect (e.g., equation (1) in Figure 1). The regression coefficients estimated by least squares for these two independent variables can then be used to determine the phase shift (0) and the amplitude (B) from the relations given in Figure 1. The expressions for 0 in degrees depend on the signs of b• and b2 and the arc tan of the absolute value of the ratio: [ b2/b•l. These equations are commonly found among the "formulae for reference" in ap- pendices to many mathematical texts, but their application to growth analyses apparently is not widely recognized. The simple variables, sine and cosine of azimuth, would suffice in the regression if all data were obtained from plots having about the same slope. However, plots on level ground supply no information on the effect of aspect on tree growth. Furthermore, it is reasonable to assume that the effect of aspect will increase on an adverse aspect up to the angle of slope that is perpendicular to the sun's rays. For greater slopes, the slight advantage of a decrease in incoming radiation is likely to be offset by decreasing soil depth. For these reasons, I recommend that the variables to represent the combined effect of slope and aspect be defined as the tangent of slope (slope percent) times the sine and cosine, respectively, of the azimuth. In this way, plots on flat ground will have a zero value for these two variables, but plots on steep ground will have high weights for the sine and cosine of aspect. In addition, the tangent of slope Principal Mensurationist, USDA Forest Ser- vice, Intermountain Forest and Range Experiment Station, Ogden, Utah 84401, stationed in Moscow, Idaho, at the Forestry Sciences Laboratory, main- tained in cooperation with the University of Idaho. Manuscript received April 15, 1976. volume 22, number 4, 1976 / 457
should also be included as a separate indepen- dent variable to overcome the symmetry of the sine and cosine functions. Otherwise, the steeper slopes on favorable aspects would be predicted to be more favorable than the flatter slopes. The additive variable for slope percent permits the model to describe adverse or negligible slope effects on the more favorable aspects and increasingly severe conditions with •ncreasing slopes on the more unfavorable aspects. Example of the application of this technique •s taken from the study of the effect of en- vironmental factors on the site index (S.I.) of western white pine (Pinus monticola Dougl.). The regression of site index on the factors of aspect, slope, and habitat type (Daubenmire and Daubenmire 1968) was calculated to be: ln(S.l.): 0.08070s cos(a) + 0.08423s sin(a) - 0.12634s + hab where ln(S.l.) = logarithm of site index for western white pine a = azimuth from north s = slope percent + 100: tan (slope angle) 4.16974 for Abies grandis/ Pachistima habitat type hab : 4.15770 for Tsuga/Pachistima habitat type 4.41374 for Thuja/Pachistima habitat type. Substituting these regression coefficients in the expressions given in Figure 1, we find that the data indicate a phase shift of arc tan (0.08423/0.08070) = 46 ø. The amplitude of the cosine function with the phase shift •s ¾0.08070"+ 0.08423": 0.11665. Conse- quently, the simplified prediction equation is. ln(S.l.) • 0.11665s cos(a-46 ø) - 0.12634s + hab. Note that the regression coefficient for tan (slope angle) is -0.12634. It is opposite in sign and of almost the same magnitude as the amplitude of cos (a-O) in the northeast quad- rant. However, on the southwest facing slopes, the slope effect accentuates the adverse aspect effect. Figure 2 illustrates the solution of the final regression equation for two habitat types. The plotting of the A bies grandis/ Pachistima habitat type is omitted because •t would nearly coincide with the Tsuga/Pachis- tima habitat type. In some cases, the symmetrical cosine func- tion may not truly represent the effect of aspect on tree growth. For example, the function represented by the above coefficients for site index of western white pine is illus- trated by the solid line in Figure 3. Suppose now that instead of this relationship an asym- metrical form, defined by the points repre- sented by the filled circles, represents the true relationship. That is, the northeast and north- west aspects are equally favorable, and the Relations for calculating amplitude (B) and phase shift (0) to express the effect of aspect and slope on growth: If: G = be + b•scos(a) + ba sin(a) + b•s (1) where: G: growth response be: constant term or sum of other predictor effects in the regression a = azimuth in degrees from north s: slope in percent + 100 then: G = be q- Bs cos(a-O) q- b• where: B = ¾ b• '• + b• • and 0 is given by: b2•• positive negative positive arctan Ib2/bxl xs0 ø - arctan negative -arctan Ib2/bxl xs0 ø + arctan FIGURE 1. Equations for calculating amplitude (B) and phase shiit (o) to express the effect of aspect on growth. (2) 458 / Forest Science