A Slope-Area Relation From Global Hypsometry

Many properties of geomorphic systems such as order, length, and area have been assumed to exhibit power-law scaling. We have examined slope-area relations derived from global hypsometric data in order to determine magnitudes of slopes developed on subaerial crust and corresponding areal extents of upslope drainages. Cumulative areas were tabulated from crustal elevation data. Square roots of areas above any elevation are closely approximated by the inverse of an exponential distribution of elevation, and global hypsometry suggests that topography is similarly approximated by a conical form with exponential slopes. From that approximation, slope is calculated as the probability density function associated with the cumulative distribution of elevation. This provides an explicit solution for the relation of slope to upslope area. The underlying equation: E(A) = E₀ * e^-(kA){0.5} yields an excellent approximation of global hypsometry where E is elevation, A is upslope area, E₀ is the highest elvation and k is the exponential constant determined by the sum of areas and the highest elevation. Because this model is based on global area-elevation data, the slope-area relation only a first-order solution. The equation is not a strict power-law and therefore does not necessitate a break in the scaling of slope with area that has been previously suggested.