Average Exterior and Average Interior Link Lengths and Statistical Laws of Topological Structure of Rill Networks in Sloping Fields

We determined the average exterior link length l_ex and the average interior link length l_in for each studied rill network in sloping fields using survey maps of rill networks. We used the results to investigate the relationships between l_ex, l_in and l_in/l_ex and the number of sources of each rill network, as well as the relationships between l_ex and l_in for each study rill network. Most of the average exterior and interior link lengths ranged from 2m to 5m for each rill network with n ≥29 sources, and l_ex was nearly equal to l_in for each rill network with n ≥50 sources, except for a rill network with 89 sources. Furthermore, by using a magnitude theory of channel networks, we suggested two equations: one to predict the length of the mainstream of the rill network and another to predict the average total length of rill links in sub-basins with Strahler's stream orders in a rill network. These equations express statistical laws of topological structure of rill networks and contain parameters l_ex and l_in . The validity of the equations was substantiated by the observed morphological data from the study rill networks.