分形理论在水文水资源研究中的应用
摘要
本文着眼于水文尺度与水文现象的自相似性，运用分形理论研究了水文变量的时空分布特性，不同时空尺度下水文变量的变化规律以及水文变量与尺度之间的相互关系。
首先用盒子数法来计算日降雨过程和日径流过程的分形维数，研究了无标度区间的范围，并计算了日降雨过程和日径流过程的多重分形谱。
在洪水区域分析中一般采用洪水指标法，但该法的基本假定与实际情况存在矛盾，因此本文采用一种新的分析方法——标度分析法(或称为分形分析法)来研究洪峰的区域变化，将标度不变性引入年最大洪峰流量——汇流面积关系中，并将其用于嘉陵江流域的洪水，另外，本文在Smith提出的具有标度性质的二参数对数正态分布模型基础上创造性地提出了三参数对数正态分布模型来表征年最大洪峰流量分布中汇流面积的尺度影响。
暴雨公式是基于暴雨资料而建立起来的经验性公式，尚无合理的理论解释。本文将标度不变性引入暴雨强度——历时关系中。由年最大平均暴雨强度随历时变化的标度性质推导出暴雨公式的形式，找到了暴雨公式的理论根基——暴雨在时间上分配具有自相似性的结果。另外，本文提出了暴雨强度的对数正态分布模型来表征年最大暴雨强度分布中历时的尺度影响。
最后，本文将标度不变性引入洪水洪量——历时关系中，对大流域年最大洪量随历时变化的标度性质行了尝试性的研究。依照暴雨公式的思路，根据洪
量随历时变化的标度性质建立了洪水强度历时公式。并提出了洪水洪量的对数
正态分布模型来表征年最大洪量分布中历时的尺度影响。
总之，本文针对水文现象中广泛存在的自相似性，将分形理论引入水文水
资源的研究中，研究了时空尺度对于水文变量变化规律的影响，加深了对水文
运动规律的认识，并为水文水资源的研究提供了新的思路。
In this paper, the self-similarity and scaling problem in hydrology are investigated by using the fractal theory. Emphases are given on the variation of hydrology variables on different scale and relationship of hydrology variables and scale.
Firstly, the fractal dimensions of daily rainfall and daily runoff are calculated by using the box-counting method. The result demonstrated that daily rainfall and daily runoff has multifractal property in a range of temporal scale. So the multifractal spectra are calculated.
Index flood method used in flood regional analysis has a shortcoming that one of its base assumption is not in accord with empirical observation. So a new method?scale analysis method(or called fractal analysis method) is applied to study the flood of Jialing River basin. The scaling hypotheses is applied to the relationship of annual maximum flood and drainage area. And basing on the scaling lognormal model with two parameters introduced by Smith, a lognormal model with three parameters of flood is introduced to represent the scale effect of drainage area in annual flood peak distributions.
The rainfall Intensity-Duration-Frequency form is an experimental form based on a large number of rainfall data, but the theory base is unclear. The scaling hypotheses
is applied to the relationship of annual maximum rainfall intensity and duration. The rainfall Intensity-Duration-Frequency form is proved based on the temporal scaling property of rainfall. And a scaling lognomial model of rainfall intensity is introduced to represent the affection of temporal scale of duration in annual maximum rainfall intensity distributions.
Lastly, the scaling hypotheses is applied to the relationship of flood volume and duration in this paper. The flood Intensity-Duration-Frequency form is proved based on the temporal scaling property of flood. And a scaling lognormal model of flood volume is introduced to represent the affection of temporal scale of duration in annual maximum flood volume distributions.
Above all, base on the self-similarity in hydrology, the fractal theory is applied to the hydrology and water resource research to study the affection of scale in the distribution of hydrology variables. It makes the hydrology law clearer and suggests a new idea for hydrology and water resource research.
首先用盒子数法来计算日降雨过程和日径流过程的分形维数，研究了无标度区间的范围，并计算了日降雨过程和日径流过程的多重分形谱。
在洪水区域分析中一般采用洪水指标法，但该法的基本假定与实际情况存在矛盾，因此本文采用一种新的分析方法——标度分析法(或称为分形分析法)来研究洪峰的区域变化，将标度不变性引入年最大洪峰流量——汇流面积关系中，并将其用于嘉陵江流域的洪水，另外，本文在Smith提出的具有标度性质的二参数对数正态分布模型基础上创造性地提出了三参数对数正态分布模型来表征年最大洪峰流量分布中汇流面积的尺度影响。
暴雨公式是基于暴雨资料而建立起来的经验性公式，尚无合理的理论解释。本文将标度不变性引入暴雨强度——历时关系中。由年最大平均暴雨强度随历时变化的标度性质推导出暴雨公式的形式，找到了暴雨公式的理论根基——暴雨在时间上分配具有自相似性的结果。另外，本文提出了暴雨强度的对数正态分布模型来表征年最大暴雨强度分布中历时的尺度影响。
最后，本文将标度不变性引入洪水洪量——历时关系中，对大流域年最大洪量随历时变化的标度性质行了尝试性的研究。依照暴雨公式的思路，根据洪
量随历时变化的标度性质建立了洪水强度历时公式。并提出了洪水洪量的对数
正态分布模型来表征年最大洪量分布中历时的尺度影响。
总之，本文针对水文现象中广泛存在的自相似性，将分形理论引入水文水
资源的研究中，研究了时空尺度对于水文变量变化规律的影响，加深了对水文
运动规律的认识，并为水文水资源的研究提供了新的思路。
In this paper, the self-similarity and scaling problem in hydrology are investigated by using the fractal theory. Emphases are given on the variation of hydrology variables on different scale and relationship of hydrology variables and scale.
Firstly, the fractal dimensions of daily rainfall and daily runoff are calculated by using the box-counting method. The result demonstrated that daily rainfall and daily runoff has multifractal property in a range of temporal scale. So the multifractal spectra are calculated.
Index flood method used in flood regional analysis has a shortcoming that one of its base assumption is not in accord with empirical observation. So a new method?scale analysis method(or called fractal analysis method) is applied to study the flood of Jialing River basin. The scaling hypotheses is applied to the relationship of annual maximum flood and drainage area. And basing on the scaling lognormal model with two parameters introduced by Smith, a lognormal model with three parameters of flood is introduced to represent the scale effect of drainage area in annual flood peak distributions.
The rainfall Intensity-Duration-Frequency form is an experimental form based on a large number of rainfall data, but the theory base is unclear. The scaling hypotheses
is applied to the relationship of annual maximum rainfall intensity and duration. The rainfall Intensity-Duration-Frequency form is proved based on the temporal scaling property of rainfall. And a scaling lognomial model of rainfall intensity is introduced to represent the affection of temporal scale of duration in annual maximum rainfall intensity distributions.
Lastly, the scaling hypotheses is applied to the relationship of flood volume and duration in this paper. The flood Intensity-Duration-Frequency form is proved based on the temporal scaling property of flood. And a scaling lognormal model of flood volume is introduced to represent the affection of temporal scale of duration in annual maximum flood volume distributions.
Above all, base on the self-similarity in hydrology, the fractal theory is applied to the hydrology and water resource research to study the affection of scale in the distribution of hydrology variables. It makes the hydrology law clearer and suggests a new idea for hydrology and water resource research.
引文
1 夏军，水文尺度问题，水利学报．1993，(5)．
2 李长兴，论流域水文尺度化和相似性，水利学报．1995，(1)．
3 Preface to the special section on scale problem on hydrology. Water Resour. Res., Vol. 33, No. 12, pp. 2881, 1997.
4 D. Schertzer, and S. Lovejoy, Physical modeling and analysis of rain and clouds by anisotropic scaling multiplicative processes, J. Geophys. Res., Vol.92, No. D8, pp.9693-9714, 1987.
5 S. Lovejoy, and D. Schertzer, Multifractals, university classes and satellite and radar measurements of cloud and rain fields, J. Geophys. Res., Vol.95, No.D3, pp.2021-2034, 1990.
6 Y. Tessier, S. Lovejoy, and D. Schertzer, Universal multifractals: theory and observations for rain and clouds, J. Appl. Meteorol., Vol.32, No.2, pp.223-250,1993.
7 李后强，张国祺，汪富泉．分形理论的哲学发轫．四川大学出版社，1993年。
8 汪富泉，李后强．分形几何与动力系统．黑龙江教育出版社，1993年。
9 李后强．关于多分形的Kohmoto熵函数．熵、信息与交叉学科．云南大学出版社，1994年。
10 赵凯华、朱照宣、黄昀．非线性物理导论．北京大学非线性科学中心，1992年
11 Gupta, V. K., and E. Waymire, Multiscaling properties of spatial rainfall and river flow distributions. J. Geophys. Res., Vol. 95, No. D3, pp. 1999-2009, 1990.
12 I. Rodriguez-Iturbe, A. Rinaldo, Fractal river basins—Change and self-organization. Cambridge University Press, 1997.
13 Book review: Scale dependence and scale invariance in hydrology. Hydrological Sciences-Journal-des Sciences Hydrologiques, vol. 45, No. 1, pp. 164, 2000.
14 D. Dietler and Y. Zhang, Fractal aspects of the swiss landscape, Physica A, 191, pp.213-219, 1992.
15 L. barbera and R,Rosso, On fractal geometry of river networks, Water Resour. Res., Vol. 25. No. 4, pp. 735-741, 1989.
16 Tarboton, D.G., Bras, R. L., and I. Rodriguez-Iturbe, The fractal nature of river networks, Water Resour. Res., Vol. 24, No. 8, pp.1317-1322, 1989.
17 Mckerhar, A.I., Ibbitt, R.P., Brown, S.L.R., and Duncan, M.J., Data for Ashley river to test channel network and river basin heterogeneity concepts, Water Resour. Res., Vol. 34, No. 1, pp. 139-142, 1998.
18 I. Rodriguez-Iturbe, M. Marani, R. Rigon and A. Rinaldo, Self-organized river basin landscapes: fractal and multifractal characteristics, Water Resour. Res., Vol. 30, No. 12, pp. 3531-3539, 1994.
19 汪富泉，泥沙运动及河床演变的分形特征与自组织规律研究，四川大学博士论文，1999。
20 Gupta, V. K., Statistical self-similarity in river networks parameterized by elevation, Water Resources Bulletin, Vol. 25, No. 3, 1989.
21 罗文锋，李后强，丁晶，艾南山．Horton定律及分枝网络结构的分形描述．水科学进展．Vol．9，No．2，P118-123，1998．
22 Robert, A., and Roy, A. G., On the fractal interpretation of the main-stream length-drainage area relationship, Water Resour. Res., Vol. 26, No. 9, pp. 839-842, 1990.
23 洪时中，洪时明．地学领域中的分维研究：水系、地震及其他．大自然探索．Vol．7，No．2，1988．
24 R. Rosso, B. Bacchi and Barbera, R L., Fractal relation of mainstream length to catchment area in river networks, Water Resour. Res., Vol. 27, No. 3, pp. 381-387, 1991.
25 V. Nikora, Fractal structures of river plan forms, Water Resour. Res., Vol. 27, No. 6, pp. 1327-1333, 1991.
26 李后强，艾南山．分形地貌学及地貌发育的分形模型．自然杂志．No．7，P516-519，1992．
27 冯平，冯炎．河流形态特征的分维计算方法，地理学报．Vol．52，No．4，P．324-330，1993．
28 傅军，丁晶，邓育仁．嘉陵江流域形态及流量过程分维研究．成都科技大学学报．Vol．82，No．1，P74-79，1995．
29 A. Marnani, R. Rigon, and A. Rinaldo, A note on fractal channel networks, Water Resour. Res., Vol. 27, No. 12, pp. 3041-3049, 1991.
30 R. Rigon, A. Rinaldo, I. Rodriguez-Iturbe, E. Ijjasz-Vasquez and Bras, R. L., Optimal channel networks: a fremework for the study of river basin morphology, Water Resour. Res., Vol. 39, No. 6, pp. 1635-1646, 1993.
31 Howard, A. D., Theoretical model of optimal drainage networks. Water Resour. Res., Vol. 24, No. 9, pp. 2107-2117, 1990.
32 R. Rigon, A. Rinaldo, I. Rodriguez-Iturbe, On landscape self-organization, J. Geophys. Res., Vol. 99, No. 11, pp. 993, 1994.
33 A. Rinaldo, I. Rodriguez-Iturbe, R. Rigon, E. Ijjasz-Vasquez and Bras, R. L., Self-organized fractal river network, Phys. Rev. Lett., Vol, 70, pp. 822-826, 1992.
34 T Sun, P. Meaken and T, Jossang, A minimum energy dissipation model for river basin geometry, Phys. Rev. E, Vol. 49, No. 6, pp. 4865-4872, 1994.
35 Tarboton, D.G., Bras, R. L., and I. Rodriguez-Iturbe, Scaling and elevation in river networks, Water Resour. Res., Vol. 25, No. 9, pp. 3037-2051, 1989.
36 H. de Vries, T. Becher, and B. Echhardt, Power law distribution of discharge in ideal networks, Water Resour. Res., Vol. 30, No. 12, pp. 3541-3543, 1994.
37 I. Rodriguez-Iturbe, E. Ijjasz-Vasquez and Bras, Tarboton, D. G., Power law distribution of discharge and energy in river basins, Water Resour. Res., Vol. 28, No. 4, pp. 1089-1093, 1992.
38 G. Wharton and Tomlinson, Flood discharge estimation from river channel dimensions: results of application in Java, Burunda,Ghana and Tanzania, Hydrological Sciences:Journal-des Sciences Hydrologiques, vol. 44, No. 1, pp.97-111, 1999.
39 金德生，陈浩，郭庆伍．河道纵剖面分形—非线性形态特征．地理学报．Vol．52，No．2，P．154-161，1993．
40 王协康，方铎，姚令侃．非均匀沙床面粗糙度的分形特征．水利学报．No．7，P70-74，1999．
41 Nikora, V. I., Fractal structures of river plan forms, Water Resour. Res., Vol. 27, No. 6, pp. 1327-1333, 1991.
42 Nikora, V. I., Sapozhnikov, V. B., Noever, D. A., Fractal geometry of individual river channels and its computer simulation, Water Resour. Res., Vol. 29, No.10, pp. 3561-3568, 1993.
43 Nikora, V. I., Sapozhnikov, V. B., River network fractal geometry and its computer simulation, Water Resour. Res., Vol. 29, No.10, pp. 3569-3575, 1993.
44 Sapozhnikov, V. B., and E. Foufpula-Georgou, Self-affinity in braided rivers, Water Resour. Res., Vol. 32, No.5, pp. 1429-1439, 1996.
45 E. Foufpula-Georgou, and Sapozhnikov, V. B., Anisotropic scaling in braided rivers: an integrated theoretical framework and results from application to an experimental river, Water Resour. Res., Vol. 34, No.4, pp. 863-867, 1998.
46 Nykanen, D. K., E. Foufpula-Georgou, and Sapozhnikov, V. B., Study of spatial scaling in braided river patterns using synthetic aperture radar images, Water Resour. Res., Vol. 34, No.7, pp. 1795-1807, 1998.
47 E. Foufpula-Georgou, and Sapozhnikov, V. B., Do the current landscape elevation models show self-organizaed criticality? Water Resour. Res., Vol. 32, No.4, pp. 1109-1112, 1996.
48 姚令侃，方铎．非均匀自组织临界性及其应用研究．水利学报．No．3，P26-32，1997．
49 S. Lovejoy, D. Schertzer and Tsonis, A. A., Functional box-counting and multiple elliptical dimensions in rain, Science, Vol. 235, pp. 1036-1038, 1987．
50 J. Olsson, J. Niemczynowics and R. Bemdtsson, Fractal analysis of high-resolution rainfall time serials, J. Geophys. Res., Vol.98, No.D12, pp.23265-23274, 1993．
51 B. Sivakumar, A preliminary investigation on the scaling behaviour of rainfall observed in two different climates, Hydrological Sciences-Journal-des Sciences Hydrologiques, vol. 45, No. 2, pp.203-219, 2000．
52 Shu-chen Lin, Chang-ling Liu and Tzong-yeang Lee, Fractal of rainfall: identification of temporal scaling law, Fractals, Vol. 7, No. 2, pp. 123-131, 1997．
53 C. Svensson, J. Olsson and R. Berndtsson, Multifractal properties of daily rainfall in two different climates, Water Resour. Res., Vol. 32, No.8, pp.2463-2472, 1996．
54 M. I. P. de Lima, J grasman, Multifractal analysis of 15-min and daily rainfall from a semi-arid region in Portugal, J. Hydrol., pp. 1-1 1,1999．
55 E. Waymire, Scaling limits and self-similarity in precipitation fields, Water Resour. Res., Vol. 21, No.8, pp.1272-1281, 1985．
56 E. Foufpula-Georgou, V. Gupta, V. K., and E. Waymire, Scaling considerations in the modeling of temporal rainfall, Water Resour. Res., Vol. 20, No. 1 1, pp.161 1-1619, 1984．
57 S. Lovejoy, and D. Schertzer, Generalized scale invariance in the atmosphere and fractal models of rain, Water Resour. Res., Vol. 21, No.8, pp. 1233-1250, 1985．
58 B.Keden and Chiu, L. S., Are rain rate processes self-similar? Water Resour. Res., Vol. 23, No.10,pp.1816-1818, 1987．
59 I. Zawadzki, Fractal structure and exponential decorrelation in rain, J. Geophys. Res., Vol.92, No.D8, pp.9586-9590, 1987．
60 E. Waymire, V. Gupta and I. Rodriguez-Iturbe, A spectral theory of rainfall intensity at the meso-β scale, Water Resour. Res., Vol. 20, No.10, pp. 1453-1465, 1984．
61 I. Rodriguez-Iturbe, Scale of fluctuation of rainfall models, Water Resour. Res, Vol. 22, No.9, pp. 15S-37S, 1986．
62 V. Gupta, E. Waymire, A statistical analysis of mesoscale rainfall as a random cascade, J. Appl. Meteor., Vol. 32, pp. 251-267, 1993．
63 Over, T. M., V. Gupta, Statistical analysis of mesoscale rainfall: dependence of a random cascade generator on large-scale forcing. J. Appl. Meteor., Vol. 33, pp. 1526-1542, 1994．
64 A. Davis, A. Marshak, W. Wiscombe and R. Cahalan, Multifractal characterizations of nonstationarity and intermittency in geophysical fields: observed, or simulated, J. Geophys. Res., Vol.99, No.D4, pp.8055-8072, 1994．
65 M. Menabde, A. Seed, D. Harris and G. Austin, Multiaffine random field model of rainfall, Water Resour. Res., Vol. 35, No. 2, pp. 509-514, 1999．
66 C. Jothityangkoon, M. Sivapalan, Viney, N. R., Test of a space-time model of daily rainfall in southwestern Australia based on nonhomogeneous random cascade, Water Resour. Res., Vol. 36, No.1, pp. 267-284, 2000．
67 I. Rodriguez-Iturbe, M. Marani, P. D'Odorico, A. Rimaldo, On space-time scaling of cumulated rainfall fields, Water Resour. Res., Vol. 34, No.11, pp. 3461-3469, 1998．
68 R. Deidda, R. Benzi, F. Siccardi, Multifractal modeling of anomalous scaling laws in rainfall, Water Resour. Res., Vol. 35, No.6, pp. 1853-1867, 1999．
69 M. Menabde, D. Harris, A. Seed, G. Austin, D. Stow, Multiscaling properties of rainfall and bounded random cascades, Water Resour. Res., Vol. 33, No. 12, pp. 2823-2830, 1997．
70 R Kumar, and E. Foufpula-Georgou, A Multicomponent decomposition of spatial rainfall fields 1． Segration of large-and small-scale features using wavelet transforms, Water Resour. Res., Vol. 29, No.8, pp. 2515-2532, 1993．
71 P. Kumar, and E. Foufpula-Georgou, A Multicomponent decomposition of spatial rainfall fields 2． Self-similarity in fluctuations, Water Resour. Res., Vol. 29, No.8, pp. 2533-2544, 1993．
72 D.Koutsoyiannis, and E. Foufoula-Georgiou, A scaling model of a storm hyetograph, Water Resour. Res., Vol. 29, No.7, pp. 2345-2361, 1997．
73 M.Menabbde, A. Seed, G.Pegram, A simple scaling model for extreme rainfall, Water Resource.Res., Vol.35, No.1, pp.335-339, 1999．
74 Smith, J. A., Representation of basin scale in flood peak distributions. Water Resour. Res., Vol. 28, No. 11, pp. 2993-2999, 1992．
75 Gupta, V. K., Mesa, O. J., and Dawdy, D. R., Multiscaling theory of flood peaks: Regional quantile analysis. Water Resour. Res., Vol. 30, No. 12, pp. 3405-3421, 1994．
76 Dawdy, D. R, and Gupta, V. K, Multiscaling and skew separation in regional floods. Water Resour. Res., Vol. 31, No. 11, pp. 2761-2767, 1995.
77 Ouarada, T. B. M. J., Bobée, B., and Rasmussen, P. R., Bernier, J., Comment on "Multiscaling and skew separation in regional floods" by David R. Dawdy. Water Resour. Res., Vol. 33, No. 1, pp. 271-272, 1997.
78 Dawdy, D. R, and Gupta, V. K, Reply, Water Resour. Res., Vol. 33, No. 1, pp. 273-275, 1997.
79 Bates, B. C., A. Rahman, Mein, R. G., Wienmann, P. E., Climatec and physical factors that influence the homogeneity of regional floods in southeastern Australia, Water Resour. Res., Vol. 34, No. 12, pp. 3369-3381, 1998.
80 Robison, J. S., M. Sivapalan, An investigation into the physical causes of scaling and heterogeneity of regional flood frequency, Water Resour. Res., Vol. 33, No. 5, pp. 1045-1059, 1997.
81 丁晶，刘国东．日径流过程分维估计．四川水力发电．Vol．18，No．4，P74-76，1999．
82 刘德平．分形理论在水文过程形态特征分析中的应用．水利学报．No．2，P20-25，1998．
83 傅军，丁晶，邓育仁．嘉陵江流域形态及流量过程分维研究．成都科技大学学报．No．1，P74-79，1995．
84 李贤彬，子波分析及其在水文水资源研究中的应用，四川大学博士论文，1999。Mandelbrot, B. B, The fractal geometry of nature, Freeman, New York, 1983.
85 Mesa, O. J., G. Poveda, The Hurst effect: the scale of fluctuation approach, Water Resour. Res., Vol. 29, No. 12, pp. 3995-4002, 1993.
86 Anderson, P. L., and Meerschaert, M. M., Modeling river flows with heavy tails, Water Resour. Res., Vol. 34, No. 9, pp. 2271-2280, 1998.
87 Hosking, J. R. M., Modeling persistence in hydrological time series using fractional differencing, Water Resour. Res., Vol. 20, No. 12 pp. 1898-1908, 1984.
88 A. Montanari, R. Rosso and Taqqu, M. S., Fractionally differenced ARIMA models applied to hydrologic time series: identification, estimation, and simulation, Water Resour. Res., Vol. 33, No. 5, pp. 1035-1044, 1997.
89 Rasmussen, P. F., Salas, J. D., L. Fagherazzi, Rassam, J. C., and B. Bobee, Estimation and validation of contemporaneous PARMA models for streamflow simulation, Water Resour. Res., Vol. 32, No. 10, pp. 3151-3160, 1996.
90 S. Tokinaga, H. Moriyasu, A. Miyazaki and N. Shimazu, Forecasting of time series with fractal geometry by using scale transformations and parameter estimations obtained by the wavelet transform, Electronics and Communications in Japan, Part 3, Vol. 80, No. 8, pp. 2054-2062, 1997.
91 Jayawardena, A. W., and Feizhou Lai, Analysis and prediction of chaos in rainfall and steam flow time series, J. Hydrol., 153, pp.23-52, 1994.
92 A. Porporato and L. Ridolfi, Nonlinear analysis of river flow time sequences, Water Resour. Res., Vol. 33, No. 6, pp. 1353-1367, 1997.
93 I. Tchiguirinskaia, S. Lu, Molz, F. J., Williams, T. M., D. Lavallee, Multfractal versus monofrctal analysis of wetland topography, Stochastic Environmental Research and Risk Assessment, Vol. 14, No. 1, pp. 8-32, 2000.
94 Liu. H. H., and Molz, F. J., Multifractal analyses of hydraulic conductivity distributions, Water Resour. Res., Vol. 33, No. 11, pp. 2483-2488, 1997.
95 S. Lovejoy, D. Schertzer and P. Silas, Diffusion in one-dimensional multifractal porous media, Water Resour. Res., Vol. 34, No. 12, pp. 3283-3291, 1998.
96 徐永福，田美存．土的分形微结构．水利水电科技进展．Vol．16，No．1，P25-29，1996．
97 徐永福，孙婉莹，吴正根．我国膨胀土的分形结构的研究．河海大学学报．Vol．25，No．1，P18-23，1997．
98 晁晓波，赵文谦，邱大洪．表面分形原理在研究泥沙吸附乳化油特征中的应用．水利学报．No．9，P1-5，1997．
99 K. Loague and Kyriakidis, Spatial and temporal variability in the R-5 infiltration data set: Déjà vu and rainfall-runoff simulations, Water Resour. Res., Vol. 33, No. 12, pp. 2883-2895, 1997.
100 Grayson, R. B., Western, A. W., and Chiew, F. H. S., Preferred states in spatial soil moisture patterns: local and nonlocal controls, Water Resour. Res., Vol. 33, No. 12, pp. 2897-2908, 1997.
101 B. Merz, Plate, E. J., An analysis of the effects of spatial variability of soil moisture on runoff, Water Resour. Res., Vol. 33, No. 12, pp. 2909-2922, 1997.
102 Dooge, J. C. I., M. Bruen, Scale effects on moisture fluxes at unvegetated land surfaces, Water Resour. Res., Vol. 33, No. 12, pp. 2923-2927, 1997.
103 Franks, S. W., and Beven, K. J., Estimation of evapotranspiration at the langscape scale: a fuzzy disaggregation approach, Water Resour. Res., Vol. 33, No. 12, pp. 2929-2938, 1997.
104 Woods, R. A., and M. Sivapalan, A connection between topographically driven runoff generation and channel network structure, Water Resour. Res., Vol. 33, No. 12, pp. 2939-2950, 1997.
105 Goodrich, D. C., Lane, L. J., Shillito, R. M., Miller, S. N., Syed, K. H., and Woolhiser, D. A., Linearity of basin response as a function of scale in a semiarid watershed, Water Resour. Res., Vol. 33, No. 12, pp. 2951-2965, 1997.
106 Bl schl, G., ang M. Sivapalan, Process controls on regional flood frequency: Coefficient of variatin and basin scale. Water Resour. Res, Vol. 33, No. 12, pp. 2967-2980, 1997.
107 Robinson, J. S., and M. Sivapalan, Temporal scale and hydrological regimes: implications for flood frequency scaling, Water Resour. Res, Vol. 33, No. 12, pp. 2981-2999, 1997.
108 Wood, E. F., and Hebson, C. S., On hydrologic similarity: 1. Derivation of the Dimensionless flood frequency curve. Water Resour. Res, Vol. 22, No. 11, pp. 1549-1554, 1986.
109 M. Sivapalan, K. Beven, and Wood, E. F., On hydrologic similarity: 2. A scaled model of storm runoff production. Water Resour. Res, Vol. 23, No. 12, pp. 2266-2278, 1987.
110 M. Sivapalan, Wood, E. F., and Beven, E. J., On hydrologic similarity: 3. A dimensionless flood frequency model using a generalized geomorphologic unit hydrograph and partial area runoff generation. Water Resour. Res, Vol. 26, No. 1, pp. 43-58, 1990.
111 A. Becker, and P. Braun, Disaggregation, and aggregation spatial scaling in hydrological modeling. J. Hydrol, 217, pp. 239-252, 1999.
112 Kurothe, R. S., Goel, N. K., and Mathur, B. S., Derived flood frequency distribution for negatively correlated rainfall intensity and duration, Water Resour. Res, Vol. 33, No. 9, pp. 2103-2107-1554, 1986.
113 彭成彬，陈藕．地震中的分形结构．中国地震．Vol．5，No．2，P19-26，1989．
114 王碧泉，杨锦英，王玉秀．自相似地震活动特征的提取．地震研究．Vol．11，No．3，P241-248，1988．
115 李凡华，刘慈群，宋伏权．分形在油气田开发中的应用．力学进展．Vol．28，No．1，P101-110，1998．
116 同登科，陈钦雷．分形油藏渗流问题的精确解及动态特征．水动力学研究与进展，A．Vol．14，No．2，P201-209，1999．
117 李后强，艾南山．分形地貌学及地貌发育的分形模型．自然杂志．No．7，P516-519，1992．
118 高旭，徐永安，张牧．分形曲线曲面在三维地貌中的应用．河海大学学报．Vol．24，No．6，P83-87，1996．
119 冯平，王仁超．水文干旱的时间分形特征探讨．水利水电技术．Vol．28，No．11，P48-51，1997．
120 冯利华，许晓路．金忂盆地旱涝统计特征初步分析．水科学进展．Vol．10，No．1，P84-88，1999．
121 朱晓华，王建．长期降水的分形性质研究．水文水资源．Vol．70，No．3，P39-41，1999．
122 候玉，吴伯贤，邓国权．分形理论用于洪水分期的初步探讨．水科学进展．Vol．10，No．2，P140-143，1999．
123 付立华．分形方法分析和预测月平均海面水温．海洋预报．Vol．12，No．1，P49-54，1995．
124 丁晶，赵永龙，邓育仁．年最大洪峰序列统计混沌性的初步研究．水利学报．No．10，P66-71，1997．
125 傅军，丁晶，邓育仁．洪水混沌特性初步研究．水科学进展．Vol．7，No．3，P226-229，1996．
126 丁晶，邓育仁，傅军．探索水文现象变化的新途径——混沌分析．水利学报．增刊，P242-246，1997．
127 赵永龙，丁晶，邓育仁．混沌分析在水文预测中的应用和展望．水科学进展．Vol．9，No．2，P181-186，1998．
2 李长兴，论流域水文尺度化和相似性，水利学报．1995，(1)．
3 Preface to the special section on scale problem on hydrology. Water Resour. Res., Vol. 33, No. 12, pp. 2881, 1997.
4 D. Schertzer, and S. Lovejoy, Physical modeling and analysis of rain and clouds by anisotropic scaling multiplicative processes, J. Geophys. Res., Vol.92, No. D8, pp.9693-9714, 1987.
5 S. Lovejoy, and D. Schertzer, Multifractals, university classes and satellite and radar measurements of cloud and rain fields, J. Geophys. Res., Vol.95, No.D3, pp.2021-2034, 1990.
6 Y. Tessier, S. Lovejoy, and D. Schertzer, Universal multifractals: theory and observations for rain and clouds, J. Appl. Meteorol., Vol.32, No.2, pp.223-250,1993.
7 李后强，张国祺，汪富泉．分形理论的哲学发轫．四川大学出版社，1993年。
8 汪富泉，李后强．分形几何与动力系统．黑龙江教育出版社，1993年。
9 李后强．关于多分形的Kohmoto熵函数．熵、信息与交叉学科．云南大学出版社，1994年。
10 赵凯华、朱照宣、黄昀．非线性物理导论．北京大学非线性科学中心，1992年
11 Gupta, V. K., and E. Waymire, Multiscaling properties of spatial rainfall and river flow distributions. J. Geophys. Res., Vol. 95, No. D3, pp. 1999-2009, 1990.
12 I. Rodriguez-Iturbe, A. Rinaldo, Fractal river basins—Change and self-organization. Cambridge University Press, 1997.
13 Book review: Scale dependence and scale invariance in hydrology. Hydrological Sciences-Journal-des Sciences Hydrologiques, vol. 45, No. 1, pp. 164, 2000.
14 D. Dietler and Y. Zhang, Fractal aspects of the swiss landscape, Physica A, 191, pp.213-219, 1992.
15 L. barbera and R,Rosso, On fractal geometry of river networks, Water Resour. Res., Vol. 25. No. 4, pp. 735-741, 1989.
16 Tarboton, D.G., Bras, R. L., and I. Rodriguez-Iturbe, The fractal nature of river networks, Water Resour. Res., Vol. 24, No. 8, pp.1317-1322, 1989.
17 Mckerhar, A.I., Ibbitt, R.P., Brown, S.L.R., and Duncan, M.J., Data for Ashley river to test channel network and river basin heterogeneity concepts, Water Resour. Res., Vol. 34, No. 1, pp. 139-142, 1998.
18 I. Rodriguez-Iturbe, M. Marani, R. Rigon and A. Rinaldo, Self-organized river basin landscapes: fractal and multifractal characteristics, Water Resour. Res., Vol. 30, No. 12, pp. 3531-3539, 1994.
19 汪富泉，泥沙运动及河床演变的分形特征与自组织规律研究，四川大学博士论文，1999。
20 Gupta, V. K., Statistical self-similarity in river networks parameterized by elevation, Water Resources Bulletin, Vol. 25, No. 3, 1989.
21 罗文锋，李后强，丁晶，艾南山．Horton定律及分枝网络结构的分形描述．水科学进展．Vol．9，No．2，P118-123，1998．
22 Robert, A., and Roy, A. G., On the fractal interpretation of the main-stream length-drainage area relationship, Water Resour. Res., Vol. 26, No. 9, pp. 839-842, 1990.
23 洪时中，洪时明．地学领域中的分维研究：水系、地震及其他．大自然探索．Vol．7，No．2，1988．
24 R. Rosso, B. Bacchi and Barbera, R L., Fractal relation of mainstream length to catchment area in river networks, Water Resour. Res., Vol. 27, No. 3, pp. 381-387, 1991.
25 V. Nikora, Fractal structures of river plan forms, Water Resour. Res., Vol. 27, No. 6, pp. 1327-1333, 1991.
26 李后强，艾南山．分形地貌学及地貌发育的分形模型．自然杂志．No．7，P516-519，1992．
27 冯平，冯炎．河流形态特征的分维计算方法，地理学报．Vol．52，No．4，P．324-330，1993．
28 傅军，丁晶，邓育仁．嘉陵江流域形态及流量过程分维研究．成都科技大学学报．Vol．82，No．1，P74-79，1995．
29 A. Marnani, R. Rigon, and A. Rinaldo, A note on fractal channel networks, Water Resour. Res., Vol. 27, No. 12, pp. 3041-3049, 1991.
30 R. Rigon, A. Rinaldo, I. Rodriguez-Iturbe, E. Ijjasz-Vasquez and Bras, R. L., Optimal channel networks: a fremework for the study of river basin morphology, Water Resour. Res., Vol. 39, No. 6, pp. 1635-1646, 1993.
31 Howard, A. D., Theoretical model of optimal drainage networks. Water Resour. Res., Vol. 24, No. 9, pp. 2107-2117, 1990.
32 R. Rigon, A. Rinaldo, I. Rodriguez-Iturbe, On landscape self-organization, J. Geophys. Res., Vol. 99, No. 11, pp. 993, 1994.
33 A. Rinaldo, I. Rodriguez-Iturbe, R. Rigon, E. Ijjasz-Vasquez and Bras, R. L., Self-organized fractal river network, Phys. Rev. Lett., Vol, 70, pp. 822-826, 1992.
34 T Sun, P. Meaken and T, Jossang, A minimum energy dissipation model for river basin geometry, Phys. Rev. E, Vol. 49, No. 6, pp. 4865-4872, 1994.
35 Tarboton, D.G., Bras, R. L., and I. Rodriguez-Iturbe, Scaling and elevation in river networks, Water Resour. Res., Vol. 25, No. 9, pp. 3037-2051, 1989.
36 H. de Vries, T. Becher, and B. Echhardt, Power law distribution of discharge in ideal networks, Water Resour. Res., Vol. 30, No. 12, pp. 3541-3543, 1994.
37 I. Rodriguez-Iturbe, E. Ijjasz-Vasquez and Bras, Tarboton, D. G., Power law distribution of discharge and energy in river basins, Water Resour. Res., Vol. 28, No. 4, pp. 1089-1093, 1992.
38 G. Wharton and Tomlinson, Flood discharge estimation from river channel dimensions: results of application in Java, Burunda,Ghana and Tanzania, Hydrological Sciences:Journal-des Sciences Hydrologiques, vol. 44, No. 1, pp.97-111, 1999.
39 金德生，陈浩，郭庆伍．河道纵剖面分形—非线性形态特征．地理学报．Vol．52，No．2，P．154-161，1993．
40 王协康，方铎，姚令侃．非均匀沙床面粗糙度的分形特征．水利学报．No．7，P70-74，1999．
41 Nikora, V. I., Fractal structures of river plan forms, Water Resour. Res., Vol. 27, No. 6, pp. 1327-1333, 1991.
42 Nikora, V. I., Sapozhnikov, V. B., Noever, D. A., Fractal geometry of individual river channels and its computer simulation, Water Resour. Res., Vol. 29, No.10, pp. 3561-3568, 1993.
43 Nikora, V. I., Sapozhnikov, V. B., River network fractal geometry and its computer simulation, Water Resour. Res., Vol. 29, No.10, pp. 3569-3575, 1993.
44 Sapozhnikov, V. B., and E. Foufpula-Georgou, Self-affinity in braided rivers, Water Resour. Res., Vol. 32, No.5, pp. 1429-1439, 1996.
45 E. Foufpula-Georgou, and Sapozhnikov, V. B., Anisotropic scaling in braided rivers: an integrated theoretical framework and results from application to an experimental river, Water Resour. Res., Vol. 34, No.4, pp. 863-867, 1998.
46 Nykanen, D. K., E. Foufpula-Georgou, and Sapozhnikov, V. B., Study of spatial scaling in braided river patterns using synthetic aperture radar images, Water Resour. Res., Vol. 34, No.7, pp. 1795-1807, 1998.
47 E. Foufpula-Georgou, and Sapozhnikov, V. B., Do the current landscape elevation models show self-organizaed criticality? Water Resour. Res., Vol. 32, No.4, pp. 1109-1112, 1996.
48 姚令侃，方铎．非均匀自组织临界性及其应用研究．水利学报．No．3，P26-32，1997．
49 S. Lovejoy, D. Schertzer and Tsonis, A. A., Functional box-counting and multiple elliptical dimensions in rain, Science, Vol. 235, pp. 1036-1038, 1987．
50 J. Olsson, J. Niemczynowics and R. Bemdtsson, Fractal analysis of high-resolution rainfall time serials, J. Geophys. Res., Vol.98, No.D12, pp.23265-23274, 1993．
51 B. Sivakumar, A preliminary investigation on the scaling behaviour of rainfall observed in two different climates, Hydrological Sciences-Journal-des Sciences Hydrologiques, vol. 45, No. 2, pp.203-219, 2000．
52 Shu-chen Lin, Chang-ling Liu and Tzong-yeang Lee, Fractal of rainfall: identification of temporal scaling law, Fractals, Vol. 7, No. 2, pp. 123-131, 1997．
53 C. Svensson, J. Olsson and R. Berndtsson, Multifractal properties of daily rainfall in two different climates, Water Resour. Res., Vol. 32, No.8, pp.2463-2472, 1996．
54 M. I. P. de Lima, J grasman, Multifractal analysis of 15-min and daily rainfall from a semi-arid region in Portugal, J. Hydrol., pp. 1-1 1,1999．
55 E. Waymire, Scaling limits and self-similarity in precipitation fields, Water Resour. Res., Vol. 21, No.8, pp.1272-1281, 1985．
56 E. Foufpula-Georgou, V. Gupta, V. K., and E. Waymire, Scaling considerations in the modeling of temporal rainfall, Water Resour. Res., Vol. 20, No. 1 1, pp.161 1-1619, 1984．
57 S. Lovejoy, and D. Schertzer, Generalized scale invariance in the atmosphere and fractal models of rain, Water Resour. Res., Vol. 21, No.8, pp. 1233-1250, 1985．
58 B.Keden and Chiu, L. S., Are rain rate processes self-similar? Water Resour. Res., Vol. 23, No.10,pp.1816-1818, 1987．
59 I. Zawadzki, Fractal structure and exponential decorrelation in rain, J. Geophys. Res., Vol.92, No.D8, pp.9586-9590, 1987．
60 E. Waymire, V. Gupta and I. Rodriguez-Iturbe, A spectral theory of rainfall intensity at the meso-β scale, Water Resour. Res., Vol. 20, No.10, pp. 1453-1465, 1984．
61 I. Rodriguez-Iturbe, Scale of fluctuation of rainfall models, Water Resour. Res, Vol. 22, No.9, pp. 15S-37S, 1986．
62 V. Gupta, E. Waymire, A statistical analysis of mesoscale rainfall as a random cascade, J. Appl. Meteor., Vol. 32, pp. 251-267, 1993．
63 Over, T. M., V. Gupta, Statistical analysis of mesoscale rainfall: dependence of a random cascade generator on large-scale forcing. J. Appl. Meteor., Vol. 33, pp. 1526-1542, 1994．
64 A. Davis, A. Marshak, W. Wiscombe and R. Cahalan, Multifractal characterizations of nonstationarity and intermittency in geophysical fields: observed, or simulated, J. Geophys. Res., Vol.99, No.D4, pp.8055-8072, 1994．
65 M. Menabde, A. Seed, D. Harris and G. Austin, Multiaffine random field model of rainfall, Water Resour. Res., Vol. 35, No. 2, pp. 509-514, 1999．
66 C. Jothityangkoon, M. Sivapalan, Viney, N. R., Test of a space-time model of daily rainfall in southwestern Australia based on nonhomogeneous random cascade, Water Resour. Res., Vol. 36, No.1, pp. 267-284, 2000．
67 I. Rodriguez-Iturbe, M. Marani, P. D'Odorico, A. Rimaldo, On space-time scaling of cumulated rainfall fields, Water Resour. Res., Vol. 34, No.11, pp. 3461-3469, 1998．
68 R. Deidda, R. Benzi, F. Siccardi, Multifractal modeling of anomalous scaling laws in rainfall, Water Resour. Res., Vol. 35, No.6, pp. 1853-1867, 1999．
69 M. Menabde, D. Harris, A. Seed, G. Austin, D. Stow, Multiscaling properties of rainfall and bounded random cascades, Water Resour. Res., Vol. 33, No. 12, pp. 2823-2830, 1997．
70 R Kumar, and E. Foufpula-Georgou, A Multicomponent decomposition of spatial rainfall fields 1． Segration of large-and small-scale features using wavelet transforms, Water Resour. Res., Vol. 29, No.8, pp. 2515-2532, 1993．
71 P. Kumar, and E. Foufpula-Georgou, A Multicomponent decomposition of spatial rainfall fields 2． Self-similarity in fluctuations, Water Resour. Res., Vol. 29, No.8, pp. 2533-2544, 1993．
72 D.Koutsoyiannis, and E. Foufoula-Georgiou, A scaling model of a storm hyetograph, Water Resour. Res., Vol. 29, No.7, pp. 2345-2361, 1997．
73 M.Menabbde, A. Seed, G.Pegram, A simple scaling model for extreme rainfall, Water Resource.Res., Vol.35, No.1, pp.335-339, 1999．
74 Smith, J. A., Representation of basin scale in flood peak distributions. Water Resour. Res., Vol. 28, No. 11, pp. 2993-2999, 1992．
75 Gupta, V. K., Mesa, O. J., and Dawdy, D. R., Multiscaling theory of flood peaks: Regional quantile analysis. Water Resour. Res., Vol. 30, No. 12, pp. 3405-3421, 1994．
76 Dawdy, D. R, and Gupta, V. K, Multiscaling and skew separation in regional floods. Water Resour. Res., Vol. 31, No. 11, pp. 2761-2767, 1995.
77 Ouarada, T. B. M. J., Bobée, B., and Rasmussen, P. R., Bernier, J., Comment on "Multiscaling and skew separation in regional floods" by David R. Dawdy. Water Resour. Res., Vol. 33, No. 1, pp. 271-272, 1997.
78 Dawdy, D. R, and Gupta, V. K, Reply, Water Resour. Res., Vol. 33, No. 1, pp. 273-275, 1997.
79 Bates, B. C., A. Rahman, Mein, R. G., Wienmann, P. E., Climatec and physical factors that influence the homogeneity of regional floods in southeastern Australia, Water Resour. Res., Vol. 34, No. 12, pp. 3369-3381, 1998.
80 Robison, J. S., M. Sivapalan, An investigation into the physical causes of scaling and heterogeneity of regional flood frequency, Water Resour. Res., Vol. 33, No. 5, pp. 1045-1059, 1997.
81 丁晶，刘国东．日径流过程分维估计．四川水力发电．Vol．18，No．4，P74-76，1999．
82 刘德平．分形理论在水文过程形态特征分析中的应用．水利学报．No．2，P20-25，1998．
83 傅军，丁晶，邓育仁．嘉陵江流域形态及流量过程分维研究．成都科技大学学报．No．1，P74-79，1995．
84 李贤彬，子波分析及其在水文水资源研究中的应用，四川大学博士论文，1999。Mandelbrot, B. B, The fractal geometry of nature, Freeman, New York, 1983.
85 Mesa, O. J., G. Poveda, The Hurst effect: the scale of fluctuation approach, Water Resour. Res., Vol. 29, No. 12, pp. 3995-4002, 1993.
86 Anderson, P. L., and Meerschaert, M. M., Modeling river flows with heavy tails, Water Resour. Res., Vol. 34, No. 9, pp. 2271-2280, 1998.
87 Hosking, J. R. M., Modeling persistence in hydrological time series using fractional differencing, Water Resour. Res., Vol. 20, No. 12 pp. 1898-1908, 1984.
88 A. Montanari, R. Rosso and Taqqu, M. S., Fractionally differenced ARIMA models applied to hydrologic time series: identification, estimation, and simulation, Water Resour. Res., Vol. 33, No. 5, pp. 1035-1044, 1997.
89 Rasmussen, P. F., Salas, J. D., L. Fagherazzi, Rassam, J. C., and B. Bobee, Estimation and validation of contemporaneous PARMA models for streamflow simulation, Water Resour. Res., Vol. 32, No. 10, pp. 3151-3160, 1996.
90 S. Tokinaga, H. Moriyasu, A. Miyazaki and N. Shimazu, Forecasting of time series with fractal geometry by using scale transformations and parameter estimations obtained by the wavelet transform, Electronics and Communications in Japan, Part 3, Vol. 80, No. 8, pp. 2054-2062, 1997.
91 Jayawardena, A. W., and Feizhou Lai, Analysis and prediction of chaos in rainfall and steam flow time series, J. Hydrol., 153, pp.23-52, 1994.
92 A. Porporato and L. Ridolfi, Nonlinear analysis of river flow time sequences, Water Resour. Res., Vol. 33, No. 6, pp. 1353-1367, 1997.
93 I. Tchiguirinskaia, S. Lu, Molz, F. J., Williams, T. M., D. Lavallee, Multfractal versus monofrctal analysis of wetland topography, Stochastic Environmental Research and Risk Assessment, Vol. 14, No. 1, pp. 8-32, 2000.
94 Liu. H. H., and Molz, F. J., Multifractal analyses of hydraulic conductivity distributions, Water Resour. Res., Vol. 33, No. 11, pp. 2483-2488, 1997.
95 S. Lovejoy, D. Schertzer and P. Silas, Diffusion in one-dimensional multifractal porous media, Water Resour. Res., Vol. 34, No. 12, pp. 3283-3291, 1998.
96 徐永福，田美存．土的分形微结构．水利水电科技进展．Vol．16，No．1，P25-29，1996．
97 徐永福，孙婉莹，吴正根．我国膨胀土的分形结构的研究．河海大学学报．Vol．25，No．1，P18-23，1997．
98 晁晓波，赵文谦，邱大洪．表面分形原理在研究泥沙吸附乳化油特征中的应用．水利学报．No．9，P1-5，1997．
99 K. Loague and Kyriakidis, Spatial and temporal variability in the R-5 infiltration data set: Déjà vu and rainfall-runoff simulations, Water Resour. Res., Vol. 33, No. 12, pp. 2883-2895, 1997.
100 Grayson, R. B., Western, A. W., and Chiew, F. H. S., Preferred states in spatial soil moisture patterns: local and nonlocal controls, Water Resour. Res., Vol. 33, No. 12, pp. 2897-2908, 1997.
101 B. Merz, Plate, E. J., An analysis of the effects of spatial variability of soil moisture on runoff, Water Resour. Res., Vol. 33, No. 12, pp. 2909-2922, 1997.
102 Dooge, J. C. I., M. Bruen, Scale effects on moisture fluxes at unvegetated land surfaces, Water Resour. Res., Vol. 33, No. 12, pp. 2923-2927, 1997.
103 Franks, S. W., and Beven, K. J., Estimation of evapotranspiration at the langscape scale: a fuzzy disaggregation approach, Water Resour. Res., Vol. 33, No. 12, pp. 2929-2938, 1997.
104 Woods, R. A., and M. Sivapalan, A connection between topographically driven runoff generation and channel network structure, Water Resour. Res., Vol. 33, No. 12, pp. 2939-2950, 1997.
105 Goodrich, D. C., Lane, L. J., Shillito, R. M., Miller, S. N., Syed, K. H., and Woolhiser, D. A., Linearity of basin response as a function of scale in a semiarid watershed, Water Resour. Res., Vol. 33, No. 12, pp. 2951-2965, 1997.
106 Bl schl, G., ang M. Sivapalan, Process controls on regional flood frequency: Coefficient of variatin and basin scale. Water Resour. Res, Vol. 33, No. 12, pp. 2967-2980, 1997.
107 Robinson, J. S., and M. Sivapalan, Temporal scale and hydrological regimes: implications for flood frequency scaling, Water Resour. Res, Vol. 33, No. 12, pp. 2981-2999, 1997.
108 Wood, E. F., and Hebson, C. S., On hydrologic similarity: 1. Derivation of the Dimensionless flood frequency curve. Water Resour. Res, Vol. 22, No. 11, pp. 1549-1554, 1986.
109 M. Sivapalan, K. Beven, and Wood, E. F., On hydrologic similarity: 2. A scaled model of storm runoff production. Water Resour. Res, Vol. 23, No. 12, pp. 2266-2278, 1987.
110 M. Sivapalan, Wood, E. F., and Beven, E. J., On hydrologic similarity: 3. A dimensionless flood frequency model using a generalized geomorphologic unit hydrograph and partial area runoff generation. Water Resour. Res, Vol. 26, No. 1, pp. 43-58, 1990.
111 A. Becker, and P. Braun, Disaggregation, and aggregation spatial scaling in hydrological modeling. J. Hydrol, 217, pp. 239-252, 1999.
112 Kurothe, R. S., Goel, N. K., and Mathur, B. S., Derived flood frequency distribution for negatively correlated rainfall intensity and duration, Water Resour. Res, Vol. 33, No. 9, pp. 2103-2107-1554, 1986.
113 彭成彬，陈藕．地震中的分形结构．中国地震．Vol．5，No．2，P19-26，1989．
114 王碧泉，杨锦英，王玉秀．自相似地震活动特征的提取．地震研究．Vol．11，No．3，P241-248，1988．
115 李凡华，刘慈群，宋伏权．分形在油气田开发中的应用．力学进展．Vol．28，No．1，P101-110，1998．
116 同登科，陈钦雷．分形油藏渗流问题的精确解及动态特征．水动力学研究与进展，A．Vol．14，No．2，P201-209，1999．
117 李后强，艾南山．分形地貌学及地貌发育的分形模型．自然杂志．No．7，P516-519，1992．
118 高旭，徐永安，张牧．分形曲线曲面在三维地貌中的应用．河海大学学报．Vol．24，No．6，P83-87，1996．
119 冯平，王仁超．水文干旱的时间分形特征探讨．水利水电技术．Vol．28，No．11，P48-51，1997．
120 冯利华，许晓路．金忂盆地旱涝统计特征初步分析．水科学进展．Vol．10，No．1，P84-88，1999．
121 朱晓华，王建．长期降水的分形性质研究．水文水资源．Vol．70，No．3，P39-41，1999．
122 候玉，吴伯贤，邓国权．分形理论用于洪水分期的初步探讨．水科学进展．Vol．10，No．2，P140-143，1999．
123 付立华．分形方法分析和预测月平均海面水温．海洋预报．Vol．12，No．1，P49-54，1995．
124 丁晶，赵永龙，邓育仁．年最大洪峰序列统计混沌性的初步研究．水利学报．No．10，P66-71，1997．
125 傅军，丁晶，邓育仁．洪水混沌特性初步研究．水科学进展．Vol．7，No．3，P226-229，1996．
126 丁晶，邓育仁，傅军．探索水文现象变化的新途径——混沌分析．水利学报．增刊，P242-246，1997．
127 赵永龙，丁晶，邓育仁．混沌分析在水文预测中的应用和展望．水科学进展．Vol．9，No．2，P181-186，1998．