and thus H may still be measured from the degraded profile (Figure 12.2b). The midsection of the degraded profile is defined as the straight inflection segment separating the basal concavity from the crestal convexity. The degraded excess midsection slope angle, β, is defined as the angle by which the inclination of the degraded midsection exceeds θ.
For hillslopes with horizontal crests and bases (i.e., θ=0, a definite relationship exists among H, c, α, β, and t, the elapsed time since the initiation of the second stage of scarp degradation. A single value of tan α/tan β is the unique result of a single value of (tc/H2) tan2α: (Nash, 1984). This relationship (Figure 12.14) may be used as the basis for morphologic dating. Although the relationship breaks down for hillslopes with steeply inlined crests and bases (θ>20°), negligible errors (less than 2 percent in the calculated value of t) result from using the following dating procedure on scarps on which θ<10° and where α+θ<35°. According to Figure 12.14, one-quarter as much time is required for a scarp to degrade from α to β if c is quadrupled or if H is halved. To date a scarp, H, β+θ, and θ are measured from a profile of the scarp, and α+θ is found indirectly by measuring the angle of repose, Φ, of the underlying debris, tan α/tan β is calculated and the corresponding value of (tc/H2) tan2α is taken from Figure 12.14. This value of (tc/H2) tan2α is multiplied by H2 and divided by tan2α to yield tc. If t is known then c may be calculated. If c has been calculated for a nearby scarp of known age or has been derived by some other means, t may be calculated. Because the time required for completion of the first, loosening-limited stage of degradation is relatively short (rarely more than a few centuries), t is generally assumed to be equal to the total age of the scarp.
The accuracy of t is dependent on the validity of assuming that a scarp had an initial morphology similar to that shown in Figure 12.2a and on the accuracy of c. Calculated values of c vary widely: 12×10−3 m2/yr for wave-cut bluffs underlain by sandy morainal material in Michigan (Nash, 1980b); 2×10−3 m2/yr for fluvial terrace scarps underlain by cohesionless obsidian sand and gravel near West Yellowstone, Montana; to a minimum of (0.9–1.0)×10−3 m2/yr calculated for the Lake Bonneville scarps by Colman and Watson (1983) and Hanks et al. (1984). [Hanks et al. (1984) also proposed that 1 m2/1000 yr be termed a G.K.G. in recognition of G.K.Gilbert’s contribution to the study of hillslopes.] Pierce and Colman (in press) observe a correlation between scarp aspect and c and also find a disturbing relationship between c and scarp height. It is likely that c is a function of underlying material, climate, and scarp aspect and thus is highly site specific. The identical values for c determined by Hanks and Wallace (in press) for the
Lake Bonneville and Lake Lahontan shoreline scarps, however, suggest that a value of c=(1.0–1.1)×10−3 m2/yr may be appropriate for many scarps underlain by alluvium in the Basin and Range province. Ideally, the value used for c should be derived from a nearby scarp of known age, underlain by the same material, and having the same aspect as the scarp to be dated. It is assumed that c does not change with time—questionable given the extent to which Holocene climatic conditions differed from those of the Pleistocene. Pre-Holocene dates are thus suspect, and it is possible that the moderation of climate during the xerotherm 4000 yr BP may have been sufficient to have changed c significantly.