FRACTALS AND FRACTAL DIMENSIONS OF STRUCTURE OF RIVER SYSTEMS:MODELS RECONSTRUCTION AND PARAMETERS INTERPRETATION OF HORTON’S LAWS OF NETWORK COMPOSITION
Yan Chen · 2001
Based on standard fractal stream system model and mirror image symmetry of series of channel classes, the first three models of Horton's laws of network composition can be ‘reconstructed’ by mirror writing the ordinal numbers of channels,i.e., writing ordinals from the highest level to the grass roots. ① From the first and the second laws, we deduce out a three parameter Zipf's model, L(r)=C(r-a) -dz ,where r is the rank of a river in a network which is marked in order of size, L(r) is the length of the r th river, as for parameters C=L 1[R b /(R b -1)] dz , a=1/(1-R b),and dz=ln R l/ln R b=1/ D. In the parameter expressions, R b and R l are the bifurcation ratio and length ratio respectively, and D is the fractal dimension of river hierarchies. ② From the second and the third laws, a generalized Hack's model is derived out as L m=μA b m, where L m is the length of the mth order river, A m is the corresponding catchment area, μ=L 1A -b 1,b= ln R l/ln R a, and in the parameters, R a is basin area ratio, L 1 is the main stream length, and A 1 is the drainage area of the mainstream. It is evident that L 1=μA b 1 is the classical Hack model. ③ From the first and the third laws, an allometric relationship is deduced as N m= ηA -σ m,where N m is the number of mth order rivers, A m is corresponding catchment area, η=N 1A σ 1,σ= ln R b/ln R a. As an attempt, the geographical space is divided into three: Space 1, existence space real space; Space 2, evolution space phase space; Space 3, correlation space order space. Defining D r, D n, and D s as the fractal dimension of rivel, network, and catchment area in real space, and D l, D b, and D a as the generalized dimension corresponding to D r, D n, and D s, we can construct a set of fractal dimension equations as follows, d z = D l/ D b=ln R l/ln R b≈ D r/ D n, b=D l/ D a=ln R l/ln R a≈ D r/ D s, and σ=D b/ D a=ln R b/ln R a≈ D n/ D s. These equations show the physical distinction and mathematical relationships between varied dimensions of a system of rivers.