There are other associated phenomena such as where certain plants grow, energy drains on wildlife and livestock, shading of ponds and rivers, and heating-cooling degree-days for buildings. South-facing slopes may be seeded earlier in the spring and later in the autumn, for example (McKee 1965)

One day there may be added here maps, tables, and graphs that show the relative slopes and slope and aspect statistics in widely accepted classes. They will be fundamental in Alpha unit analyses.

One initially recommended set of slope classes (with aspects) :

1. 0-20 percent - where rubber-tired and other forestry and conventional land-development equipment can be used
2. 21-40 percent - where only cleat-track vehicles can be used in forest operations and where roads are not advised
3. >40 - where slopes cannot be logged by any conventional terrestrial methods.

Suggested plot of " rose "of slope frequency classes (sample from previous work with all aspects from within a county). Aspect direction and frequency are shown. Slope direction classes are coded with different symbols. Compass coordinates from E-W and N-S as zero are shown as axes.

Suggested plot of bar graph of aspect-frequency classes (sample from previous work with all aspects from within a county)

A table showing total acres, and another showing areas in slopeclasses within each relevant (usually 8) aspect groups can provide information on ownership potentials - for example, possible operation on productive sites.

Aspect (The following is based on a a manuscript "Alternative Transformations for Land Surface Aspect" by J.W. McCombs, R.H. Giles, Jr., T.G. Gregoire, S.M. Martin, and S.D. Klopfer, 1998 given to Prof. James Campbell, College of Natural Resources, Virginia Tech in 2005 for completion.

Aspect has been an important measure in physical and biological sciences such as forestry, plant and animal ecology, geomorphology, and climatology (Hack and Goodlett 1960, Geiger 1965, Whittaker and Niering 1965, Howard and Mitchell 1985, Grafton and Dickerson 1969, Callaway 1983, Tajchman 1975, Gemborys 1979, Griffm 1971, Stage 1976, Beers et a1.1966, Osmond et al. 1990, Schallenberger 1966, Shoulders and Tiarks 1980, and Swanson et a1.1988). Orographic (mountain-effect) rainfall has been studied and reported by many authors (Swanson 1988, Howard and Mitchell 1985, and Osmond et al. 1990). Due to food, energy, or other relations, animals may be related to aspect. For example, Schallenberger (1966) observed bighorn sheep (Ovis canadensis) feeding and using southward-facing slopes 79% of the time. Rose (1984:287) observed that winter pellet densities of sitka black-tailed deer (Odocoileus hemionus sirkensis) were 2.5 to 15.3 times greater on southerly than on northerly slopes. Rose (1984:289) embedded "aspect" in a host of related factors: " Deer preference for southerly exposures in winter may be related to such factors as forage quality and quantity, or to snow and temperature conditions on southfacing slopes," as did Taber and Dasmann 1958, Julander 1966, Jones 1975, Taber and Hanley 1979. Southerly aspects receive a greater intensity and duration of sunlight than do northerly aspects in winter (Klopfer 1997). South-facing slopes may be seeded earlier in the spring and later in the autumn (McKee et al. 1965). Southern slopes are also exposed to warm southeasterly storms, resulting in lower snow depths and earlier snow melt, even at higher elevations.

Aspect with slope was used in the very parsimonious forest model Forclim by Bugmann (1996). He used it to adjust radiation as it influenced evapotranspiration. Aspect is rarely encountered in the agricultural literature since such lands are relatively flat and without conspicuous aspect. Aspect has been found to play an important role in tree survival, growth potential, growing season, and seedling survival (Shoulders and Tiarks 1980, Oemborys 1979, Rehteldt 1993, Griffin 1971, Muick and Bartolome 1987). Trimble and Weitzman (1956), Einspahr and McComb (1951) and Ike and Huppuch (1968) found site index of oaks and some other trees strongly related to aspect. See also Table 1. Scribner et al. (1991) found differing amounts of toxins in Populus tremuloides and Liriodendron tuhpiffra on a north-to-south gradient.

aspect image

Aspect is generally defined as the direction downhill, the slope orientation, the direction the slope faces, or the prevailing slope direction. We believe a precise definition is needed and suggest: Aspect is the difference in direction (in degrees from 0<>N or OOS in the southern hemisphere) of the vertical line of a right-angle triangle anywhere along the horizontal base (B) to the lower elevation apex (A) on a relatively flat land surface plane. (Most land surfaces have some irregularity. The scale of project work or factors observed will determine when a mere irregularity achieves a magnitude sufficient to require additional slope and aspect analyses. We suggest a tentative gross criterion for "relatively flat", it being that the ratio of the height of an irregularity to mean diagonal or diameter length should not exceed 1 in 100 units.) A flat plane with no slope has an indeterminate aspect. See Fig. 3.

Aspect is one of 9 major parameters of land surface shape, the others being elevation, slope, relief, length, area, roughness and texture, and pattern (Martin 1988). Aspect is a metric, like time, that can be readily discerned by people, but is probably only reflective of other forces and changes. It is probably no more than an index, an amalgamation of several factors readily expressed as one metric.

Ike and Huppuch (1968) developed a slope-aspect factor said to be useful in estimating forest site index. It was the product of aspect A, (e.g., 90) and slope S, (e.g., 10%; resulting in a factor of 900) but many combinations of A and S can produce the same index.

Aspect is now readily computed in many GIS packages using differences in elevations at raster centers from surrounding rasters (cells or pixels). Data are typically obtained from digital elevation models DEMS obtained, for example, from the U.S. Geological Survey, Reston, Virginia 22092).

Baker et aI. (1986) (studying the behavior of released insects) as did Cain (1989), analyzed the aspect angles as mean vectors and tested whether that vector r differs significantly from a standard (e.g., due West) or from an hypothesized " best" aspect, or when it is used as a factor (e.g., the direction of predominant winds; an expected angle; or a pre-treatment angle) using Rayleigh's test (Batchelet 1981). Rayleigh's test (Zar 1974:316) of the null hypothesis that non-transformed aspect values are not uniformly distributed around a circle may be computed in studies of sampling intensity, effectiveness of randomization efforts, and occurrences of items or phenomena observed. If rejected, the test results may then lead to Watson's V test (Zar 1974:324), the test that samples have the same direction (i.e., are not significantly different).

Transformations of the degree measures as shown above have been recommended to create a linear continuous hinction from the circular finction. In the field, 3550 is nearly the same as 5 degrees and 0 degrees is identical to 360 degrees. Angles " wrap on top of themselves" (Cain 1989). Conditions in most natural systems would rarely be hypothesized to be different given these 2 very different numerical measures. Statistical operations on raw measures, as if they were of a linear continuous variable, would generally be inappropriate (Cain 1989).

Stage (1976) discussed the transformations of measurements made in degrees. Burkhart and Wathen (1977) placed observed aspects into numerical classes related to moisture and other site factors. Ike and Hupuch (1968) said that " although most reports in the literature distinguish between eastand west-facing slopes, usually with northeast and southwest representing the extremes in the influence of aspect on [forest] site quality, we found no meaningfiil differences between east slopes and west slopes other than that of species composition." Their aspect was measured in degrees east or west from magnetic north (southwest was assigned equivalent to southeast; a maximum value was 180 degrees for a south. facing slope). De Pietri (1995) assigned values of 0 to N, NE, E, SE, and S aspects and 1.0 to SW, W, NW, and flat areas. These were used as one factor among 6 to define different areas for analyzing livestock degradation of land. Others have used 9 classes (i.e., 1 = " no aspect (flat)" and 8 other equal 45 degree separations). Smalley (1983) used 315degrees to 135 degrees as " North" ; other aspects as " South." This practice avoided the need for making transformations but assumes that each category is ecologically equivalent and generalizes otherwise precise data. Gaiser (1951) seemed to have first transformed aspect measures so that they related to tree growth. The transformation accommodated the expected best growth of a variety of tree species on northeastern slopes, the poorest on southwestern slopes. Trimble and Weitzman (1956) made similar transformations and assigned northeast 45 degrees a high value of 2. See Fig. 4. The shift of 45 degrees allowed the modified aspect value to relate well in regression analyses to highest forest site quality. Beers et al. (1966) recommended a general transformation of aspect as

Eq.l
K₁ = sin [A + (90 - A_max)] +1

Eq.2
K₂ = cos (A_max - A) +1

with A being the observed aspect in degrees and A_max being the assigned direction of maximum importance (e.g., site quality for plant growth). When A_max of 450 degrees is selected, the transformed values for select aspects are shown in Table 2.

Table 2. Select aspect values transformed as K, using Equation l (based on Trimble and Weitzman 1956; Beers et al. 1966; Stage 1976).
Aspect (Degrees)	K 1 - 4 0 degrees shift	K max 45 degrees shift
0	2.000	1.707
45	1.707	2.000
90	1.000	1.707
135	0.293	1.000
180	0.000	0.293
225	0.293	0.000
270	1.000	0.293
315	1.707	1.000
360	2.000	1.707

Baker et al. (1986) observed that studies of dispersal of irradiated sterile flies released for contral purposes had been flawed. The assumption of dispersal in an independent radially symmetrical manner may not be supportable. Analogously, tests of equal plant growth or animal behavior among all slope directions may be appropriate. Rejecting the null hypothesis tat all are equal may be a step to determining if'... significant directionality is present before performing further experiments or subjecting data to lumping and modeling" (Baker et al. 1986). They rejected chi-square and anova procedures and used a bivariate technique appropriate for grid-related analyses.

We have become aware that the transformations used by foresters, i.e., changing compass direction value A to K, do not express "aspect" but are surrogates of a tree growth index. Not merely a mathemati cal transformation, K is an expression of many natural relations strongly correlated with aspect, largely solar, but also moisture and others. K is an index to tree growth suitability, easily hypothesized to be correlated with a readily-measured factor. The 45 degree shift suggested is one that gives a low index to a site having a high solar radiation surface.

We suggest that the transformations of aspect suggested in the past may not serve well the very wide interests in and applications of knowledge about aspect. Sine or cosine transformations result in a northerlysoutherly continuous variable with no differences in numerical values for due East (1.0) and due West (1.0) (Table 2, column 2). When the 45 degree shift is made (Trimble and Weitnnan; Beers et al. 1966) (Table 2, column 2), equivalent numerical values are obtained equidistant, eastward and westward from the 45 degree position (Fig. 2) and other key positions (e.g., if a 33 degree shift was required, 303 degrees (90 degrees west) or 123 degrees (90 degrees east) would have equal values of 1.0. We believe a North-South index is needed, one ranging, for example, from 2.0 to 0 as derived from Eq. 1 and shown in Table 2. Other simple transformations are:

K₃=cos(-A)           Eq. 3

K₄= sin (A + 90) +1          Eq. 4

K₅= | A - 180 |           Eq. 5

K₆= integer (K₅/18)           Eq. 6

K₇= K_{3 / 0.2}          Eq. 7

where A is the observed compass direction of the slope, the aspect as usually defined. K₅, using absolute values, results in a northerly-southerly gradient of 180 units at the North to zero at the south. K₆results in a North-South "score" from 10 to O. As with the other transformation, no information is available to discriminate between easterly or westerly influences on the topic of interest, whatever they may be.

We believe that there are differences in many easterly-westerly phenomena and that these may have been masked by some transformations and shifts used historically. Thus, we recommend that aspect, as least for ecological studies (but probably for others), be perceived as 2 continuous functions (e.g., the two notably different effects of latitude or longitude on the occurrence of events). We recommend creating an easterly-westerly aspect measures, Alpha II. Similar to the N-S expressions (Alpha I), these equations are:

K₈= cos(90-A) + 1       Eq. 8

K₉= (sinA)+1      Eq.9

K₁₀= ||A-90| - 180|      Eq.10

K₁₁= integer(K₁₀ /18)      Eq. 11

K₁₂= K₈ / 0.2      Eq.12

See Table 3.

Northerly-Southerly					Easterly-Westerly
Degrees	K 1-4	K 5	K 6	K 7	K 8	K 9	K10	K 11	K 12
0	2.00	180	10	10.00	1.00	1.0	90	5	5.00
45	1.71	135	7	8.54	1.71	1.7	135	7	8.54
90	1.00	90	5	5.00	2.00	2.0	180	10	10.0
135	0.29	45	2	1.46	1.71	1.7	135	7	8.54
180	0.00	0	0	0.00	1.00	1.0	90	5	5.0
225	0.29	45	2	1.46	0.29	0.3	45	2	1.46
270	1.00	90	5	5.00	0.00	0.0	0	0	0.00
315	1.71	135	7	8.54	0.29	0.3	45	2	1.46
360	200	180	10	10.00	1.00	1.0	90	5	5.00

These values provide opportunities for studies of the regression Smodel of form:

Y = a + b log (K₇ + 1) + c log (K₁₂ + 1)

allowing the coefficients (b and c) to express the relative role of usually quite different latitudinal and longitudinal forces in estimating values of Y (where Y may be, for example, plant growth, basal area, wind damage to buildings). Rather than use a simple 45 degree shift to the north-east in the northern hemisphere resulting in an approximate solar maximum of 205degrees, Laichani and Davis (1982) suggested using 188 degrees based on their studies of Helianthemum. Computations such as those by Stage (1976) may be made that may aid in controlling variance at other specific sites and for other topics of interest. He found a desirable phase shift of 46 degrees for western pine sites; 201 degrees for basal area measures.

The differences in the transformed compass directions resulting from selecting the linear transformation as "aspect," K₆, or selecting the nonlinear form, K₇, are notable. Depending on the size of the area, the variety of slope directions, and the precision of the dependent variable to which aspect is being studied, the differences can be meaningful. We recommend using K₇, calling it "alpha I" and K₁₂, calling it "alpha II" where feasible. When these values are separated, then, as an example, influence of wind vectors, East-West in some areas, can be separated from insolation influences that are predominantly North-South in magnitude of effect. Other differences in East-West forces such as coastal or continental precipitation, temperature, and geological strata may be expressed by alpha II. When aspect of areas is considered, the influence of size and elliptical orientation of area may be further controlled. Areas with their longer axis East-West will be expressed by a single latitude, variation minimized, and contribution to an estimator of any y maximized. The opposite is true for areas with long axis North-South.

Table 3. Representative values of K₁ to K₁₂ given a range of degrees designating the aspect of a slope.

References

McKe, W.H. et al. 1965; Microclimate conditions found on highway slope facings as related to adaptation of species. Highway Res. Rec 93: 38-43.