Profile Integral, a robust and unified metric for measuring profile concavity

Abstract— To measure the shape of topographic profiles in two dimensions (height v. distance), we propose a new metric (and develop its related toolbox) based on integrating the area under the profile. It is applicable to any cross-valley profile, slope profile, long profile or arbitrary profile. By analogy with the Hypsometric Integral, we term the metric the Profile Integral. It is a normalized value between 0 and 1 with the value of 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. Correlations with the V-index, the VWDR, the K-curve, the power curve and the quadratic polynomial are analyzed. The advantages of the Profile Integral are (1) its 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). Our toolbox generates smoothed streamlines (thalwegs) to provide starting points for a series of cross-profiles. Applications to glaciated valleys in the Tian Shan (Daxi) and northern Iceland (Eyjafjarđardalur) are illustrated.