SOME PROBLEMS ON THE DEVELOPMENTAL PROCESS OF CHANNEL NETWORKS

Abstract

The present report aims at seeking a possible way to simulate the developmental process of channel networks through the geologic time. Firstly, problems regarding application of the stochastic branching process theory to the networks were briefly reviewed. Kirkby's (1976) conversion or mapping of a network to a path of a random walk was appreciated, but the definition of mapping was expanded for the sake of its application to all possible paths. The total number of possible paths of 2 n steps is 22n. Each path is represented by a binary string of 1's and 0's. In our model, when the walk returns to zero after some steps, a mapped drainage system finishes with its development and another new walk starts at that point. The paths which lie in the negative half-plane are regarded as the same in the positive just like as the reflected ones with the X-axis. The very few walks only return to zero just after the 2 n steps. In this case, the area having n water sources (or n first-order channels) is divided to drainage basins, the number of which is equal to the times of return to zero in a path of 2 n steps. After the theorems of random walk it can be expected that the area will be divided to nearly _??_2n drainage basins with the most probability. A great majority of the walks does not finish at zero after 2 n steps. In this case, distance from the X-axis at x=2 n indicates the number of active channels that have possibility to develop to the higher-order ones. Such channels are regarded as inflow channels from the outside to the preliminarily bounded area. All paths of random walk of 2 n steps are thus mapped to a full set of networks. The theorem of unbiased random walk says that the probability that no return to zero will happen, or only once it will do, is highest, and that the probability becomes lower as the supposed times of return to zero increase. This means that with the most probability the preliminarily bounded area will be embraced by a bigger drainage system, or divided to one independent drainage basin and the remainder which forms a part of a bigger drainage system. This tendency was named as the law of oligopoly of drainage basin occupation. Entropy defined after the information theory was discussed for the network systems. The cyclic systems have comparatively low entropy because of its limitation which gives the cyclic property to the developmental process of systems. Shreve's (1966) random system has lower entropy than the system which allows the area to be divided to some drainage basins. The system which allows inflow channels from the outside to the area has the maximum entropy because it has not any limitation on generating a new branch channel at each step. Limitation which keeps low entropy from the autogenetic increasing tendency should not be thought to result from the cyclic property or the restraint of a drainage network strictly to an .area, but if there is, it results from geomorphological reasons. In conclusion, it will become possible to simulate the evolutional process of drainage systems as a branching process with a transition probability defined as a function of location (geologic control) and time (environmental change).