Applications of the random model of drainage basin composition

EARTH SURFACE PROCESSES, VOL. 1, 219-233 (1976) APPLICATIONS OF THE RANDOM MODEL OF DRAINAGE BASIN COMPOSITION J. S. SMART IBM Thomas .I. Watson Research Center, Yorktown Heights, New York, U.S.A. AND C . WERNER School of Social Sciences, University of California, Irvine, California, U.S.A. SUMMARY The random model of drainage basin composition is founded on the assumptions that (a) natural channels are topologi- cally random in the absence of geological controls and (b) for channel networks developed in similar environments, the exterior and interior link lengths are independent random variables with a common distribution for each type. The effectiveness of this model in estimating the values of geomorphic variables and in explaining and predicting geomorphic relationships is illustrated by several examples. The data required for these examples were obtained from map studies of 30 channel networks, comprising a total of about 8700 links, in eastern Kentucky. A common factor in the success of all three applications of the model is the way in which the planimetric features of drainage basins are determined by their underlying topologic structure. KEY WORDS Channel networks Drainage basins Stochastic models INTRODUCTION The development and maintenance of drainage networks involve many complex processes, such as weath- ering, soil creep, fluid flow, and sediment transport. Although the physical bases of these processes are reasonably well understood, the overall complexity is such that an exact deterministic explanation of individual stream network properties appears impossible (Krumbein and Shreve (1970) p. 40, Shreve (1975)). It should however be possible to provide a good statistical description of the properties of drainage basins developed in a given environment. The observed values of some particular variable, such as drainage density, will vary randomly from basin to basin but will belong to a well-defined probability distribution determined by the local climate and geology. In addition, correlations between pairs of geomorphic variables are frequently relatively insensitive to variations in the environment and may persist in samples of data obtained from a rather wide range of climates and lithologies. As Shreve (1975) has noted, these points are clearly illustrated by the various empirical relationships between geomorphic variables that have been reported in the literature. Examples of such regularities are the linear relation between drainage density and the square of stream frequency (Melton (1958)) and the 0.6 power law dependence of mainstream length on area (Hack (1957)). Although such relationships are often stated in deterministic form, they are almost always obtained from regression analyses, and the results of individual measurements are seen to scatter randomly about the regression line. The deviations are consistently larger than is to be expected from measurement errors alone and can be attributed almost entirely to geomorphic randomness. Such regularities provide useful information about drainage basin composition but they remain for the most part a series of isolated results, giving little hint of the complex interrelations of geomorphic Received 1 March 1976 @ 1976 by John Wiley & Sons, Ltd.