Frequency Distributions of Stream Link Lengths
W. Ryan James, W. C. Krumbein · The Journal of Geology · 1969
Shreve's concept of infinite topologically random channel networks focuses attention on exterior and interior channel links, which are the basic elements of the network. The population distribution of link lengths has importance in its relation to the statistical properties of entire networks, and it plays a basic role in computer simulation models for generating stream patterns. Published data on link lengths are sparse, and earlier workers have inclined toward a lognormal model for the length distribution of exterior links (Strahler first-order streams). More recently, Smart has proposed an exponential model for interior links, on the basis of a postulated link-generating mechanism. The model agrees with observed interior link lengths except for a deficiency of short links as required by his theory. Analysis of interior links in a magnitude 1311 basin shows that the probability of successive tributaries entering main channels from the same side is only about two-thirds the probability of opposite-side entry. These higher-order structures in the data (i.e., the spatial arrangement of successive tributaries) display a first-order Markovian property in sequences of link type. These structures are examined in their substantive context, and a composite gammalike population density for interior link lengths is developed theoretically. This density contains the higher-order structures and agrees, at least approximately, with observed relations in main channels of magnitude 10 or greater.