Stream Lengths and Basin Areas in Topologically Random Channel Networks

Abstract

In order to comprehend the geometry of drainage basins and channel networks, which is prerequisite to explaining their mechanics, it is necessary to understand the close connection between network topology and such planimetric elements as stream lengths and basin areas. Topologically random channel networks constitute an important theoretical case. In an infinite topologically random network, (1) the expected magnitude of a randomly drawn link of order $\omega$ is ( $2^{2{\omega}-1} + l$ )/3, (2) 2/3 of all links and 1/6 of the interior links head streams, (3) complete subnetworks of any given order have the same distribution of magnitudes as all networks of that order, which explains why it seems to make little difference whether or not basins chosen for investigation are complete, (4) the probability that a randomly drawn stream of order $\omega$ will consist of $\lambda$ links is $2^{-({\omega}-1)}(1 - 2^{-({\omega}-1)})^{{\lambda}-1}$ , and (5) the average stream of order $\Omega$ will have $2^{{\Omega}-1} - 1$ tributaries entering from the sides, of which $2^{{\Omega}-{\omega}-1}$ will on the average be of order $\omega$ . Link lengths measured on 1:24,000-scale maps of eastern Kentucky are approximated more closely by a gamma density with parameter v = 2 than by either a log normal or an exponential. The mean length of the exterior links is almost twice that of the interior links, in agreement with the findings of others. In an infinite topologically random channel network whose interior link lengths are gamma distributed with v = 2, (1) the densities of the logarithms of the stream lengths are slightly left-skewed and increase in dispersion with increasing order, and (2) the densities of the logarithms of the Schumm total length of all streams in a subbasin of given order are highly symmetrical, decrease slightly in dispersion with increasing order, and for orders 1 and 2 agree well with Schumm's observations at Perth Amboy, New Jersey. If the link lengths are exponentially distributed, then the stream lengths will also be exponentially distributed. In finite topologically random channel networks with specified stream numbers, the expected number of links per stream and per subnetwork of given order are given by closed but complicated formulas, from which the expected stream lengths and Schumm lengths can be calculated. As observed in natural networks, in a topologically random population, (1) the expected stream lengths do not satisfy Horton's geometric-series law as well as the expected stream numbers, whereas the expected Schumm lengths do, (2) on Horton diagrams the curves are straightest for the most probable networks, and are concave upward or downward for networks of order lower or higher, respectively, than the most probable, and (3) the geometric-mean stream-length and Schumm-length ratios are about 2 and 4.5, respectively. The corresponding results for basin areas are exactly the same as for Schumm lengths. In the Perth Amboy basin studied by Schumm, the average area draining directly overland into unit length of channel is not a constant independent of the particular position of the channel in the basin as he proposed, but is less for exterior links than for interior links. Substantial differences in the relative areas draining into sources and in other geomorphic characteristics are not reflected in Horton diagrams because the Horton variables are averages that are mainly determined by network topology. Of most fundamental significance, therefore, are not the Horton variables, but the more elementary quantities, such as link lengths and source areas.