Statistical moments of the hypsometric curve and its density function

A method, applicable to multivariate designs, describing the form of the percentage hypsometric curve is developed in this research. Emphasis is placed on the quantitative aspects of curve form, rather than on average slopes, inflection points, or hypsometric integrals. A question of uniqueness arises when values, like the integral, are used as landform surrogates in process-response models involving drainage basins. It is demonstrated that the hypsometric curve has a much greater potential for quantitative landform analysis than can be realized through employment of the integral value alone. Unlike the integral, the functional form of the curve is unique to a particular area, depicting, among other things, evolutionary changes in the form of drainage basins. The technique involves treating the decumulation of the hypsometric curve in its mirror image as a cumulative distribution function. Statistical moments of the curve, and expectations of (x)for the curve's density function are derived, projecting a vector of curve-form attributes.