Advances in Earth Science ›› 2001, Vol. 16 ›› Issue (2): 178-183.

• Articles •

### FRACTALS AND FRACTAL DIMENSIONS OF STRUCTURE OF RIVER SYSTEMS:MODELS RECONSTRUCTION AND PARAMETERS INTERPRETATION OF HORTON’S LAWS OF NETWORK COMPOSITION

CHEN Yan-guang 1,  LIU Ji-sheng 2

1. 1.Department of Urban and Environmental Sciences,Peking University,Beijing  100871,China;
2.Department of Geography,Northeast Normal University,Changchun  130024,China
• Received:2000-05-23 Revised:2000-08-12 Online:2001-04-01 Published:2001-04-01

CHEN Yan-guang, LIU Ji-sheng. FRACTALS AND FRACTAL DIMENSIONS OF STRUCTURE OF RIVER SYSTEMS:MODELS RECONSTRUCTION AND PARAMETERS INTERPRETATION OF HORTON’S LAWS OF NETWORK COMPOSITION[J]. Advances in Earth Science, 2001, 16(2): 178-183.

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=L1[Rb /(Rb -1)] dz , a=1/(1-Rb),and dz=ln Rl/ln Rb=1/ D. In the parameter expressions, Rb and Rl 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 Lm=μAbm, where L m is the length of the mth order river, A m is the corresponding catchment area, μ=L1A-b1,b= ln Rl/lnRa, and in the parameters, Ra is basin area ratio, L1 is the main stream length, and A1 is the drainage area of the mainstream. It is evident that L1=μAb1 is the classical Hack model. ③ From the first and the third laws, an allometric relationship is deduced as Nm= ηA-σm,where N m is the number of mth order rivers, Am is corresponding catchment area, η=N1Aσ1,σ= lnRb/lnRa. 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 Dr, Dn, and Ds as the fractal dimension of rivel, network, and catchment area in real space, and Dl, Db, and Da as the generalized dimension corresponding to Dr, Dn, and Ds, we can construct a set of fractal dimension equations as follows, dz = Dl/ Db=ln Rl/ln RbDr/ Dn, b=Dl/ Da=ln Rl/ln RaDr/ Ds, and σ=Db/ Da=ln Rb/ln RaDn/ Ds. These equations show the physical distinction and mathematical relationships between varied dimensions of a system of rivers.

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