A utilitarian approach to modeling non‐Gaussian characteristics of a topographic field

This paper develops a general framework for the analysis of the moments greater than 2 of a topographic field. This framework uses “iterated” expectation to reduce a statistical moment function to component parts involving the vertical (disjoint) moment of the same order, lower moments, and two‐point conditional expectations. In this way it is possible to isolate the unique informational contribution of each moment. Use of this framework necessitates a “bootstrap” or perturbation method, where lower moments are modeled first and then are used as constraints in the modeling of higher moments. Functional modeling of any moment is thus reducible to characterization of the disjoint moment (e.g., skewness, kurtosis) and the two‐point conditional expectation. In this paper, I demonstrate how it is possible to “design” a statistical model most sensitive to specific non‐Gaussian topographic characteristics by solving for the two‐point conditional expectation under an invertable mapping between Gaussian and non‐Gaussian fields of interest. Mappings of this sort are useful both for the fact that they can be intuitive descriptions of non‐Gaussian characteristics and for their utility in generating non‐Gaussian synthetic topography. The primary intent of this methodology is to forge a link between physical topographic characteristics, the information we want to know, and statistical moments, our tool for quantitatively measuring topographic fields. In addition, mapping models can be used to calculate the skewness and kurtosis (or higher moments) of topographic slopes directly. The applicability of these methods is demonstrated for mapping models which create vertical and lateral asymmetry and peakiness in a topographic field.