Abstract
SYMBOLS AND DEFINITIONS
m, l Order of basin or stream channel.
mεl Mean value of numbers of the l-th order streams joining directly to one m-th order tream, but, when (l+1)=m, the two streams that join to form the m-th order are excluded.
mεl (n; _??_) Weighted mean value of mεl of topologically distinct stream networks of order _??_ with n first order streams. (Denominator is the total number of the m-th order streams.)
mμl Mean value of numbers of the l-th order streams in one m-th order stream network.
_??_am Mean area of the m-th order basins in a large _??_-th order basin. _??_Lm Mean length of the m-th order stream segments in an _??_-th order stream network. k' Constant.
Horton's law of stream numbers is a valid statistical law to describe approximate relationships beween stream orders and numbers in both actual and randomly-generated stream networks. In a more exact sense, however, many networks exhibit certain systematic deviations from this law in Strahler systems. Many students have called attention to the fact that, in many cases, the set of points number versus order plotted on the semilogalithmic graph shows distinct upconcavity.
In a previous paper (Tokunaga, 1966), an equation was proposed to modify Horton's law of stream numbers in order to satisfy this tendency. The following assumptions were then needed to construct the equation. Assumptions:
1. mεm-1=m-lεm-2=………=m-λεm-λ-1=………=2ε1=ε1
mεm-2=m-1εm-3=………=m-λεm-λ-2=………=3ε1=ε2 _??_ mεl=m-lεl-1=………=εm-λεl-λ=………εm-l
2. _??_
The equation is given by the following expression:
_??_ (1)
_??_ (2)
_??_(2)'
The above assumptions were checked by populations of topologically distinct networks with the first order streams to be the most probable for given orders. Calculated values of mεl (n; 5) are tabulated in Table 3.
Further, these assumptions lead to the following equation for the relationship between basin order and area:
_??_ (3)
Assuming the next relationship between basin area and stream length.
_??_ (4)
a law of stream length is written
_??_ (5)
Now, let m→∞ in equations (1), (3), we get log Q, as gradients of symptotes of these equations on the semilogalithmic graph.
Consequently, Q means the bifurcation ratio and the basin area ratio and _??_ the length ratio of infinite networks.
Because the fifth order network is the most probable, when n=171, putting ε1=2ε1 (171; 5) and K=3ε1(171; 5) /2ε1(171; 5), (2) becomes
Q=4.03
consequently
_??_=2.01
These values are in good agreement with the bifurcation ratio or the basin area ratio and the length ratio in infinite topologically random stream network by R. E. Shreve (1967).
