收稿日期: 2000-05-23
修回日期: 2000-08-12
网络出版日期: 2001-04-01
基金资助
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:40071035)中关于“城市环境支持系统”的部分研究内容资助。
FRACTALS AND FRACTAL DIMENSIONS OF STRUCTURE OF RIVER SYSTEMS:MODELS RECONSTRUCTION AND PARAMETERS INTERPRETATION OF HORTON’S LAWS OF NETWORK COMPOSITION
Received date: 2000-05-23
Revised date: 2000-08-12
Online published: 2001-04-01
陈彦光, 刘继生 . 水系结构的分形和分维——Horton水系定律的模型重建及其参数分析[J]. 地球科学进展, 2001 , 16(2) : 178 -183 . DOI: 10.11867/j.issn.1001-8166.2001.02.0178
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 Rb≈ Dr/ Dn, b=Dl/ Da=ln Rl/ln Ra≈ Dr/ Ds, and σ=Db/ Da=ln Rb/ln Ra≈ Dn/ Ds. These equations show the physical distinction and mathematical relationships between varied dimensions of a system of rivers.
Key words： River network; Fractals; Symmetry; Horton' s laws; Hack' s law.
[1] Batty M.Physical phenomena[J]. Geographical Magazine,1992, (7):35~36.
[2] La Barbera P, Rosso R.On the fractal dimension of stream networks[J]. Water Resources Research, 1989, 25(4):735~741.
[3] Li Houqinag, Ai Nanshan. Fractal geomorphology and fractal models of landform evolution[J].Journal of Nature, 1992,15(7):516~519.[李后强,艾南山.分形地貌学及地貌发育的分形模型[J].自然杂志,1992,15(7):516~519.]
[4] Turcotte D L.Fractals and Chaos in Geology and Geophysics[M]. Cambridge: Cambridge University Press,1992.
[5] Takayasu H. Fractals[M]. Translated by Shen Buming,Chang Ziwen. Beijing: Earthquake Press, 1989.[高安秀树著.分数维[M].沈步明,常子文译.北京:地震出版社,1989.]
[6] Feng Ping, Feng Yan. Calculation on fractal dimension of river morphology [J]. Acta Geographica Sinica, 1997, 52(4):324~330.[冯平,冯焱.河流形态特征的分维计算方法[J].地理学报,1997,52(4):324~330.]
[7] Tarboton D G, Bras R L, Rodriguez-Iturbe I. The fractal nature of river networks[J]. Water Resources Research, 1988,24(8):1 317~1 322.
[8] Feder J. Fractals[M]. New York:Plenum Press,1988.208~211.
[9] Rosso R, Bacchi B, La Barbera P. Fractal relation of mainstream length to catchment area in river networks[J]. Water Resources Research, 1991,27(3):381~387.
[10] Marani A, Rigon R,Rinaldo A. A note on fractal channel network [J]. Water Resources Research, 1991, 27 (12):3 041~3 049
[11] Robert A,Roy A G. On the fractal interpretation of the mainstream length-drainage area relationship[J]. Water Resources Research,1990,26(5):839~842.
[12] Kaye B H. A Random Walk Through Fractal Dimensions[M]. New York: VCH Rublishers,1989.
[13] Ai Nanshan, Chen Rong, Li Houqiang. A way to fractal geomorphology[J]. Geography and Territorial Research, 1999,15(1):92~96.[艾南山,陈嵘,李后强.走向分形地貌学[J].地理学与国土研究,1999,15(1):92~96.]
[14] He Longhua, Zhao Hong. The fractal dimension of river networks and its interpretation[J]. Scientia Geographica Sinica,1996, 16(2): 124~128.[何隆华,赵宏.水系的分形维数及其含义[J].地理科学,1996,16(2):124~128.]
/
| 〈 |
|
〉 |
甘公网安备62010202000687
