Profile Integral: A robust and flexible metric to measure the concavity/convexity of a topographic profile

Abstract

In geomorphology, topographic profile shapes are used to analyze landform characteristics, interpret erosional and depositional processes, and understand the evolution of landscapes over time. To more consistently measure the shape of topographic profiles in two dimensions (height vs. distance), we provide a new metric based on integrating the area under the profile. The metric is applicable to any cross-valley profile, slope profile, long profile, or arbitrary profile. By analogy with the Hypsometric Integral, we name this metric as Profile Integral (PI). It is a standardized value between 0 and 1, with 0.5 representing either a straight-line long profile or a V-shaped cross-profile, and values of >0.5 for convex profiles and < 0.5 for concave profiles. We develop a toolbox to assist in determining the PI and for comparison with other measures of profile shape, including the V-index, the VDWR, the k-curve, the power curve, and the quadratic polynomial. Our toolbox provides tools to delineate smoothed streamlines (thalwegs), generate a series of cross-profiles along streamlines, and derive the PI and a set of other metrics for both cross-valley and long profiles. We illustrate the toolbox with examples for an area in the Tian Shan, China, and another in Iceland, including situations where some cross profiles should be manually deleted. In both areas, PI provides a good measure of the degree of glacial modification of valleys, with expected down-valley variations. PI has strong correlations with most metrics that measure profile concavity based on a single parameter, while its correlations with metrics derived from those curve-fitting models that define profile shape with multiple parameters are relatively low. The advantages of the PI are (1) its low-skewed distribution and flexibility in providing a metric with similar interpretations for long profiles, slope profiles and valley cross-profiles and (2) its applicability to asymmetric cross-profiles in full, including those close to the confluence of tributary valleys (asymmetric cross-profiles, reaching different heights on each side, are common yet are excluded from most analyses of valleys and troughs). We, therefore, propose PI as the preferred and unified index for measuring the concavity/convexity of any topographic profile.