摘要
基于标准分形水系等级序列的镜象对称性 ,重建水系构成定律 :从 Horton第一、第二定律出发 ,导出关于河流长度与位序关系的三参数 Zipf模型 ;从 Horton第二、第三定律出发 ,导出广义的 Hack定律 ;从 Horton第一、第三定律出发 ,导出关于河流长度—流域面积关系的异速生长方程。根据上述数学变换结果建立了水系分维方程式 ,澄清了不同空间、不同类别的各种维数之间的数理关系。
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.
出处
《地球科学进展》
CAS
CSCD
2001年第2期178-183,共6页
Advances in Earth Science
基金
国家自然科学基金!资助项目"城市体系空间网络的分形结构及其演化机制"(编号 :40 0 710 35 )中关于"城市环境支持系统"的部分研究内容
