HORTON'S LAW OF STREAM NUMBERS FOR TOPOLOGICALLY RANDOM CHANNEL NETWORKS

Abstract

HORTON‘S LAW OF STREAM NUMBERS FOR TOPOLOGICALLY RANDOM CHANNEL NETWORKS CHRISTIAN WERNER University of California, Irvine ACCORDING to Horton’s law of stream numbers, the bifurcation ratio in a drainage network is fairly constant with an average value between 3.5 and 4, so that the stream numbers tend to form a geometric progression. This paper investigates drainage net- works whose structure is controlled by chance only. The mathematical analysis shows that the expected stream numbers also approach a geometric progression, and that the corresponding bifurcation ratios approximate the value 3.6 18. INTRODUCTION A scientific investigation of real world phenomena usually concentrates on selected properties and disregards all others. An example is the sequential pattern of merg- ing rivers in a drainage network. The apparent hierarchy of their mergers can be studied disregarding all other components (topography, hydrology, morphometry). What remains is the information regarding the number of tributaries the system contains, and how they are interconnected, i.e., the topological structure of the net- work. This structure is the subject of Horton’s famous law. To understand its content, a few network parameters have to be defined. Since the original concept of stream order introduced by Horton (1945, p. 281) still contains a geometrical element (angles), the refined version of Strahler (1952, p. 1120) will be used here. The stream segments starting from a source are called streams of first order, and the stream segments starting from the confluence of two streams of ith order are called streams of order i + 1 . The end of each stream is defined as the point where a higher-order stream starts. The bifurcation ratio of streams of order m is defined as the ratio between the number of streams (stream numbers) of order m and the number of order m + 1 . An example is shown in Figure 1. To avoid the problem of ambiguity, only rivers with a “mature” topography will be considered here; i.e., rivers with lakes and islands are excluded. We further assume that no more than two rivers merge in one point. FIGURE 1. Stream orders in a river net- work (after Strahler, 1952). XIV, 1, 1970 CANADIAN GEOGRAPHER,