最も確からしい水路網およびそれによって構成される流域の諸パラメーターについて

水路網に関する集合論的考察

抄録

An investigation is made of some other geomorphic parameters as well as the bifurcation ratio of the most probable channel networks based on the set-theory.
The following equations are derived for the set S (nl, nl+1, …, nη, …, nλ) of all topologically distinct networks having nl, nl+1, …, nη, …nλ, streams of order l, (l-1), η, … λ respectively and the set Smax (nl, nl+1, …, nη…, nλ) having the largest number of topologically distinct networks among S (nl, nl+1, …, nη2, …, nλ) for the given value of nl and various values of nl+1, …, nη, …, nλ, where l is the lowest order.When nl is infinite,
S_max (nl, nl+1, …, nη, …, nλ)
=S (nl, 1/4 nl, … (1/4) η-l nl…, (1/4) λ- nl)
S_max (nl, nl+1, …, nη, …, nλ-1)
⊃S_max (nl, nl+1, …, nη, …, nλ)
Then the mean me of number of streams of order k entering into a stream of order mfrom sides in S_max (nl, nl+1, …, nη, nη, …, nλ) is given as follows.
mε_m=2^m-k-1
It is, therefore, proved that the bifurcation ratio of the most probable networks of infinite magnitude quite agrees with the mean bifurcation ratio of all infinite topologically distinct networks and me, shows the same value in both. When ni is finite, nl+1, …, nη, …, nλ take the closed values to (1/4) nl, … (1/4) λ-^lnl…, (1/4) λ-^lnl respectively. The same conclusion is deduced for the other geomorphic parameters, for example, the basin area ratio, the stream length ratio, etc, which are derived from mε_k.