Interactions of Elevation, Aspect, and Slope in Models of Forest Species Composition and Productivity

We present a linear model for the interacting effects of elevation, aspect, and slope for use in predicting forest productivity or species composition. The model formulation we propose integrates interactions of these three factors in a mathematical expression representing their combined effect in terms of a cosine function of aspect with a phase shift and amplitude that change with slope and elevation. This model allows the data to determine how the aspect effect changes with elevation and slope. Earlier articles concerning the interactions of slope, aspect, and elevation have been incomplete by either treating elevation as fixed or ignoring the possibility that aspect effect must also involve slope. The proposed set of variables is illustrated in four applications: (1) a hypothetical data set for probability of stocking by "species" having different adaptations to elevation, (2) in a discriminant function for forest/nonforest classification of data from Utah, (3) estimating mean annual increment of Utah forests, and (4) estimating the height asymptote in a mixed-model differential equation predicting Douglas-fir height growth. FOR .S CI. 53(4):486-492. SPECT, SLOPE, AND ELEVATION have been demon- strated to be useful surrogates for the spatial and temporal distribution of factors such as radiation, precipitation, and temperature that influence species com- position and productivity. Predictive models using slope, aspect, and elevation can be useful for extrapolating present or historical effects to areas where observations of species composition or productivity are not available. These predic- tive models can also be used to assess whether more com- plex models of direct effects of radiation, precipitation, and temperature adequately estimate historical integration of effects of these factors. Thus, they can serve as a reality check before predicting effects of changing climate using the more detailed models. Our references to the effect of aspect are phrased for the northern hemisphere; in the southern hemisphere, of course, the roles of north and south are reversed. However, to retain the appropriate roles of east and west, aspect is here defined as the azimuth measured clockwise from north by an observer facing downslope. Earlier proposals of how to represent interactions of aspect, slope, and elevation in models of species composi- tion and productivity have been incomplete. Beers et al. (1966) observed that southwest aspects are often the most severe sites for forest regeneration and growth. Accord- ingly, they recommended using cosine of aspect with a predetermined phase shift of 45° to create a variable (cos( 45°)) that would have its maximum of unity at northeast and its minimum of minus unity at southwest. Then the amplitude of the contrast between these extremes is estimated by the regression coefficient of cos( 45°). Stage (1976) demonstrated a transformation that permitted the phase shift of the cosine to an optimum aspect and an amplitude that is a function of slope to be estimated directly from the coefficients of a linear model. Roise and Betters (1981) extended the discussion to relations with elevation, but omitted slope interactions. Their observation that the optimum aspect at high elevations could be in the opposite quadrant to the optimum at the lower elevations was an important contribution. However, by representing the phase shift as arccos((a E)/b)) of elevation (E) scaled symmet- rically about the middle elevation (a) between the lowest elevation of a b and the highest of a b, their expression switches the optimum from south at high elevation to north at low elevation by passing the optimum aspect through east or west at a mid-elevation. We argue that, instead, it should be the amplitude of the cosine of aspect that passes through zero as the phase shift to optimality reverses quadrants at some mid-elevation. Effects of aspect, furthermore, should be greatest at the extremes of elevation for the species being represented. At the lower elevation limit for a species, the absolute value of the magnitude of the interaction with the trigonometric aspect factors should increase at an increasing rate as elevation decreases. Conversely, at the upper eleva- tion range for a species, the rate of change of the absolute magnitude should increase, either because of the physiolog- ical limits of the species or as a complement (through competitive exclusion) to the behavior of species better adapted to the higher elevation. Furthermore, the amplitude should be able to change with elevation and slope, particu- larly because aspect is undefined on flat ground. If the amplitude of cos( ) passes through zero at some mid-elevation, then this behavior explains insignifi- cant effects of slope/aspect variables in analyses of some data sets. If the range of elevations in a particular study is nearly centered on the elevation at which the switch occurs, then fitting a model having only a single transformation of aspect will have opposing aspect effects above and below the transition. Analysis would show no net effect of aspect.