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.