分形学理论在城市排水管网中的应用研究
|
摘要
分形是近二、三十年才发展起来的一门新兴的学科,以非规则几何形态为研究对象,是研究和处理不规则图形强有力的工具,已经在各学科及工程技术中得到了广泛应用。
城市排水管道系统作为隐藏于城市道路或绿化带下庞大的网络工程,虽难以用传统几何形式表达其几何特征,但其具有一定的随机特性和自相似性,可以用分形几何学进行研究。而目前关于管道分形的研究很少,本文以西安市排水系统为例,采用《西安市1995～2020年排水管网规划图》进行分形研究。
首先,针对排水管道系统构成的空间网络,在总结前人研究成果与经验的基础上,研究其网络分形特性。由于管网分布的均匀程度在某种意义上具有重要的作用,因此,本论文应用基于分形理论的网络覆盖评价指标——分形维数,采用网格维数计算方法,编制网格维数计算程序,计算其分形维数。
对西安市东郊、南郊、西郊和北郊的污水及雨水管道系统的分维值进行了计算,并对其结果进行了分析。计算结果表明雨水管网分维值普遍大于污水管网分维;分维值越高,表示管网覆盖度高,其管线占据空间的能力越强,同时管线分布也越均匀。即西安市雨水管网布置的均匀程度要大于污水管网。雨水管网覆盖情况从大到小排列为:南郊＞北郊＞东郊＞西郊。污水管网覆盖情况为:南郊＞西郊＞东郊＞北郊。这主要与西安市各地的实际情况以及经济发展情况密切相关的。
其次,本文利用重力流枝状排水管道系统与河流水系的相似性,借鉴分形地貌学理论研究成果,提出了“等级分形”的概念,建立了重力流枝状排水管道等级分形模型,与等级划分方案,并给出了合适的分形算法和计算程序,能够方便的计算出重力流枝状排水管道等级分形的参数。利用西安市某些枝状排水干管进行了验证,并对其产生原因以及区别加以说明。计算结果表明:重力流枝状排水管道系统在等级空间上分枝比r_b、长度比r_l都符合自然界等级分形优化的结果,分枝比r_b为3～5,长度比r_l在1.5～3之间。等级维数D反映排水管道系统在等级空间的分形情况。
在管网分形理论中,对于同一排水管网构成的分形集可以有不同种类的维数,这些维数值反映这个集合不同方面的数学特征。并且是分形客体形态或结构的有效量度。
Fractal theory is a science newly developed in the past 20 to 30 years. The object of research of the fractal theory is irregular geometric shape. The fractal geometry is a powerful tool to study the irregular figures. It has been extensively used in different subject, and engineering and technology.
The urban drainage network is the huge network project under the urban road. Although the geometric characteristic of it can not be expressed by the traditional geometry forms, it presents a close structural similarity and stochastic characteristics, so they can be studied with fractal technology. And so on the fractal study of the drainage network was very rare. As a case study of Xi'an drainage network system, "The drainage network Plan drawing of Xi'an city from 1995 to 2020" was used in the fractal study.
Firstly, the fractal characteristic of the spatial network formed by the drainage network was studied by summarizing the formers' research results and research experience extensively. Because of the level of equality of the drainage network in some sense has vital function, the criteria for evaluating the network coverage on the basis of fractal theory——fractal dimension and the grid dimension was used, the grid dimension program was written to calculate fractal dimensions.
The fractal dimensions of the network system of the sewage and rainwater of the eastern suburbs, southern suburbs, western suburbs and the northern suburbs in Xi'an were calculated and the results of them were analyzed. The results indicate that the fractal dimension of rainwater network was larger than sewage network. The larger the fractal dimension is, the better the indices of network coverage formation is and the ability of the network occupied the spatial, and the more equality the network distributed. So, the distribution of rainwater network is more equality than sewage network in Xi'an. The indices of the rainwater networks covering formation in Xi'an are southern suburbs, western suburbs, eastern suburbs, and the northern suburbs in order. And this is related to the reality situation as well as the economical development situation of Xi'an.
Secondly, the similarity between the branching drainage channel system by gravity and the river system and the research results of Fractal Geomorphology for reference were used. The "rank fractal" was put forward. Then rank fractal model and the definition of rank of branched drainage network by gravity were built and proper algorithm was put forward too, so is the program. With them, we can calculate rank fractal parameters of the branching drainage network by gravity conveniently. Applying the model, we verified it by calculating several trunk tube of the branching drainage network in Xi'an. Further more, factors that may affect fractal dimension value were also commentated.
The research indicated that, the branching ratio r_b and length ratio r_t in rank space of thebranching drainage channel system by gravity were conform to the optimization of the rank fractal in the nature .The branching ratio is 3～5, the length compares is1.5～3. Rank dimension D reflected the fractal situation of the drainage network in rank space.
In brief, in the fractal theory of network, as a assemble composed by the same drainage network ,it can have different dimensions, and they reflect the different mathematics characteristic of it , Also they are the effective measurements of fractal assemble.
城市排水管道系统作为隐藏于城市道路或绿化带下庞大的网络工程,虽难以用传统几何形式表达其几何特征,但其具有一定的随机特性和自相似性,可以用分形几何学进行研究。而目前关于管道分形的研究很少,本文以西安市排水系统为例,采用《西安市1995～2020年排水管网规划图》进行分形研究。
首先,针对排水管道系统构成的空间网络,在总结前人研究成果与经验的基础上,研究其网络分形特性。由于管网分布的均匀程度在某种意义上具有重要的作用,因此,本论文应用基于分形理论的网络覆盖评价指标——分形维数,采用网格维数计算方法,编制网格维数计算程序,计算其分形维数。
对西安市东郊、南郊、西郊和北郊的污水及雨水管道系统的分维值进行了计算,并对其结果进行了分析。计算结果表明雨水管网分维值普遍大于污水管网分维;分维值越高,表示管网覆盖度高,其管线占据空间的能力越强,同时管线分布也越均匀。即西安市雨水管网布置的均匀程度要大于污水管网。雨水管网覆盖情况从大到小排列为:南郊＞北郊＞东郊＞西郊。污水管网覆盖情况为:南郊＞西郊＞东郊＞北郊。这主要与西安市各地的实际情况以及经济发展情况密切相关的。
其次,本文利用重力流枝状排水管道系统与河流水系的相似性,借鉴分形地貌学理论研究成果,提出了“等级分形”的概念,建立了重力流枝状排水管道等级分形模型,与等级划分方案,并给出了合适的分形算法和计算程序,能够方便的计算出重力流枝状排水管道等级分形的参数。利用西安市某些枝状排水干管进行了验证,并对其产生原因以及区别加以说明。计算结果表明:重力流枝状排水管道系统在等级空间上分枝比r_b、长度比r_l都符合自然界等级分形优化的结果,分枝比r_b为3～5,长度比r_l在1.5～3之间。等级维数D反映排水管道系统在等级空间的分形情况。
在管网分形理论中,对于同一排水管网构成的分形集可以有不同种类的维数,这些维数值反映这个集合不同方面的数学特征。并且是分形客体形态或结构的有效量度。
Fractal theory is a science newly developed in the past 20 to 30 years. The object of research of the fractal theory is irregular geometric shape. The fractal geometry is a powerful tool to study the irregular figures. It has been extensively used in different subject, and engineering and technology.
The urban drainage network is the huge network project under the urban road. Although the geometric characteristic of it can not be expressed by the traditional geometry forms, it presents a close structural similarity and stochastic characteristics, so they can be studied with fractal technology. And so on the fractal study of the drainage network was very rare. As a case study of Xi'an drainage network system, "The drainage network Plan drawing of Xi'an city from 1995 to 2020" was used in the fractal study.
Firstly, the fractal characteristic of the spatial network formed by the drainage network was studied by summarizing the formers' research results and research experience extensively. Because of the level of equality of the drainage network in some sense has vital function, the criteria for evaluating the network coverage on the basis of fractal theory——fractal dimension and the grid dimension was used, the grid dimension program was written to calculate fractal dimensions.
The fractal dimensions of the network system of the sewage and rainwater of the eastern suburbs, southern suburbs, western suburbs and the northern suburbs in Xi'an were calculated and the results of them were analyzed. The results indicate that the fractal dimension of rainwater network was larger than sewage network. The larger the fractal dimension is, the better the indices of network coverage formation is and the ability of the network occupied the spatial, and the more equality the network distributed. So, the distribution of rainwater network is more equality than sewage network in Xi'an. The indices of the rainwater networks covering formation in Xi'an are southern suburbs, western suburbs, eastern suburbs, and the northern suburbs in order. And this is related to the reality situation as well as the economical development situation of Xi'an.
Secondly, the similarity between the branching drainage channel system by gravity and the river system and the research results of Fractal Geomorphology for reference were used. The "rank fractal" was put forward. Then rank fractal model and the definition of rank of branched drainage network by gravity were built and proper algorithm was put forward too, so is the program. With them, we can calculate rank fractal parameters of the branching drainage network by gravity conveniently. Applying the model, we verified it by calculating several trunk tube of the branching drainage network in Xi'an. Further more, factors that may affect fractal dimension value were also commentated.
The research indicated that, the branching ratio r_b and length ratio r_t in rank space of thebranching drainage channel system by gravity were conform to the optimization of the rank fractal in the nature .The branching ratio is 3～5, the length compares is1.5～3. Rank dimension D reflected the fractal situation of the drainage network in rank space.
In brief, in the fractal theory of network, as a assemble composed by the same drainage network ,it can have different dimensions, and they reflect the different mathematics characteristic of it , Also they are the effective measurements of fractal assemble.
引文
[1]方丽达.地下空间 一眼望穿[J].市场观察,2002,6.88-89
[2]林向明.城市排水管网的规划设计[J].科技资讯,2005,25:42-43
[3]苗秀荣.城市给水管网的优化布置[J].科技情报开发与经济,2005,15(5):178-180
[4]陈胜兵,娄金生.排水管网平面布置的优化[J].工业用水与废水,2005,36(6):47-49
[5]贾树宝.污水管网的优化设计研究[J].灌溉排水,2000,19(1):63-65
[6]李树平,刘遂庆。城市排水管道系统设计计算的进展[J].给水排水,1999,25(10):9-12
[7]Batty M.Physical Phenomena[J].Geograph icalM agazine,1992,(7):35-361
[8]La Barbera P,Ro sso R.On the fractal dimension of stream networks[J].Water Resources Research,1989,25(4):735-7411
[9]李军锋.基于GIS的陕北黄土高原地貌分形特征研究[D].西安:西学大学硕士学位论文,2006
[10]Mandelbrot B B.The Fractal Geometry of Nature[M].San Francisco:Freeman.1983
[11]龚礼华,客观世界中普遍存在的分形与分维[J].达县师范高等专科学校学报(自然科学版)2004,14(5):22-23
[12]崔灵周,肖学年,李占斌,基于GIS的流域地貌形态分形盒维数测定方法研究[J].水土保持通报2004,24(2):37-38
[13]王薇.分形理论在表面粗糙度非接触测量中的应用[D].吉林:吉林大学硕士学位论文,2004
[14]刘湘南,许红梅,黄方.土地利用空间格局及其变化的图形信息特征分析[J].地理科学,2002,22(1):79-84.
[15]刘彦随,吴传钧,鲁奇.21世纪中国农业与农村可持续发展方向和策略[J].地理科学,2002,22(4):385-389.
[16]陈国阶.论地理学面临的挑战与发展[J].地理科学,2003,23(2):129-137
[17]张济忠.分形.北京:清华大学出版社.1995:7-110
[18]高安秀树.分数维[M].北京:地震出版社(中译本).1989:15-26
[19]Mandelbrot B B.Fractal:Form,Chance and Dimension.San Francisco:Freeman.1977:1-20
[20]龚礼华.客观世界中普遍存在的分形与分维[J].达县师范高等专科学校学报(自然科学版),14(5)2004:22-23
[21]刘秉正,彭建华编著.非线性动力学与混沌基础[M].长春:东北师范大学出版社.2004.
[22]韩向峰,基于分形理论的创新设计及其应用[D].山东师范大学硕士学位论文,2005
[23]于红志.分形图形生成研究[D].大连理工大学硕士论文,2003
[24]王丽.分形图形的研究与应用[D].昆明理工大学,2005
[25]束庆年.市政管网分形研究[D].北京工业大学硕士论文2003
[26]张宇星.城市与城市群形态空间的分形特性.新建筑,1995 3:42-46
[27]陈勇,艾南山.城市结构分形研究.地理学与国土研究.,1994,10(4):35-41
[28]R White,G Engelen.Cellular autumata and fractal urban form:a cellularmodeling approach to the evolution of urban land-use patterns.Enviornment and Planning A,1993,Volume 25,Pages1175-1199
[29]Batty M,Longley P,Fotheringham S.Fractal-based description of urbanform:scaling,fractal geometry,and diffusion-limited aggregation.Environment andPlanning A 21,1987,Volume 25,Pages1447-1472
[30]1 9 Batty M,Longley P.Fractal-based description of urban form.Environment and Planning B:Planning and design 14 123-134
[31]Susanna C.Manrubia,Transient dynamics and scaling phenomena in urban growth.Fractals C,1999Vol.7,No.1 Pages1-8
[32]Batty M,.Geneating urban forms from diffusive growth.Environment and Planning A,1991 23511-544
[33]Lam,Warren M.;Wornell,Gregory W.Multiscale Analysis and Control of Networks with Fractal Traffic.Applied and Computational Harmonic Analysis Volume:11,Issue:1,July,2001,pp.124-146
[34]Kim,Kwang Sik;Bengnigui,Lucien;Marinov,Maria.The fractal structure of Seoul's public transportation system.Cities Volume:20,Issue:1,February,2003,pp.31-39
[35]Ayedemir,M.;Bottomley,L.;Coffin,M.;Jeffries,C.;Kiessler,P.;Kumar,K.;Ligon,W.;Marin,J.;et.al.Two tools for network traffic analysis.Computer Networks Volume:36,Issue:2-3,July,2001,pp.169-179
[26]Liangxiu,Han;Zhiwei,Cen;Chunbo,Chu;Chuanshan,Gao.A new multifractal network traffic model.Chaos,Solitons and Fractals Volume:13,Issue:7,June,2002,pp.1507-1513
[37]陈彦光,刘继生.水系结构分形和分维---Horton水系定律的模型重建及其参数确定[J].地球科学进展,2001,16(2):118-123
[38]李华晔,黄志全,姜彤.河流水系分形的初步研究[J].华北水力水电学院学报,1998,14(4):28
[39]汪富泉,曹叔尤,丁晶.河流网络的分形与自组织及其物理机制[J].水科学进展,200213(3):368-376
[40]Veltri,M.Veltri,P.Maiolo,M.On the fractal description of natural channelnetworks.Journal of Hydrology Volume:187,Issue:1-2,December,1996:137-144
[41]陈彦光 刘继生.中心地体系与水系分形结构的相似性分析[J].地球科学进展,2000,20(1),81-88
[42]冯金良 张稳.海滦河流域水系分形.泥沙研究.Vol.1,1992
[43]沈晓华等.长江河道分形和流域构造特征的关系.浙江大学学报.Vol.28 No.1,2001:33
[44]罗文峰 李后强 丁晶 艾南山.Horton定律及分支网络结构的分形描述[J].水科学进展,19989(2):118-123
[45]Babadagli,Tayfun.Fractal analysis of 2-D fracture networks of geothermal reservoirs in south-western Turkey.Journal of Volcanology and GeothermalResearch Volume:112,Issue:1-4,December,2001,pp.83-103
[46]Schuller,D.J.;Rao,A.R.;Jeong,G.D.Fractal characteristics of dense stream networks.Journal of Hydrology Volume:243,Issue:1-2,March 1,2001,pp.1-16
[47]Puente,Carlos E.;Castillo,Pablo A.On the fractal structure of networks and dividers within a watershed.Journal of Hydrology Volume:187,Issue:1-2,December,1996,pp.173-181
[48]Tarboton,David G.Fractal river networks,Horton's laws and Tokunaga cyclicity.Journal of Hydrology Volume:187,Issue:1-2,December,1996,pp.74 105-117
[49]Moussa,Roger;Bocquillon,Claude.Fractal analyses of tree-like channelnetworks from digital elevation model data.Journal of Hydrology Volume:187,Issue:1-2,December,1996,pp.157-172
[50]Ayedemir,M.;Bottomley,L.;Coffin,M.;Jeffries,C.;Kiessler,P.;Kumar,K.;Ligon,W.;Marin,J.;et.al.Two tools for network traffic analysis.Computer Networks Volume:36,Issue:2-3,July,2001,pp.169-179
[51]Troutman,Brent M.;Over,Thomas M.River flow mass exponents with fractalchannel networks and rainfall.Advances in Water Resources Volume:24,Issue:9-10,November-December,2001,pp.967-989
[52]Niemann,JeffreyD.;Bras,RafaelL.;Veneziano,Daniele;Rinaldo,Andrea.Impacts of surface elevation on the growth and scaling properties of simulated river networks.Geomorphology Volume:40,Issue:1-2,September,2001,pp.37-55
[53]Crave,A.;Davy,P.Scaling relationships of channel networks at large scales:Examples from two large-magnitude watersheds in Brittany,France Tectonophysics Volume:269,Issue:1-2,January 30,1997,pp.91-111
[54]祝安,祝进,张朝晖.喀斯特流域水系分形、分维问题[J].贵州师范人学学报(自然科学版)18(4)2000:5-8
[55]Frankhouser P.A spects fractals des structures urbans[J].L'Esp ace Geog rap h ique,1990,19(1):45-69
[56]刘继生,陈彦光.交通网络空间结构的分形维数及其测算方法探讨[J].地理学报1999.54(5):471-478
[57]丁以中,楼勇.分形理论在交通网络评价中的应用[J].上海海运学院学报,1998.19(4):7-12
[58]刘立明.分形法在河北省高速公路规划中的应用[J].河北工业大学成人教育学院学报,2001,16(4):21-23
[59]李文华,杨兆升,王希伟.基于分形几何学的区域公路网布局评价指标的研究[J].交通运输系统工程与信息,2005,5(5):50-53
[60]许志海,翟京生.分形理论在空间网络分布特征研究中的应用[J].海洋测绘,2005,25(6):26-33
[61]李桀,朱金兆,朱清科.分形维数计算方法研究进展[J].北京林业大学学报,24(2)2002:71-78
[62]张济忠.分形[M].北京:清华大学出版社,1995
[63]沈莉芳.污水干管系统选线最优化决策方法研究[D].重庆大学工程硕士学位论文,2004
[64]牛文元.理论地理学EB
[65]李慧博,柴立和.给水管网的分形生长分析[R].北京:中国原子能科学研究院,2005
[66]Barbera P La,Rosso R.On the Fractal Dimension of Stream Networks[J].Water Resources Research,1989,25(4):735-741
[67]Tarboton D,Bras R & Rodriguez-Iturbe 1.The Fractal Nature of River Networks,Water Resources Research.1988,24(8):1317-1322
[68]洪时中,洪时明.地学领域中的分维研竞[J]:太自然探索,1988,24(8):33-40
[69]冻彦光,刘继生.水系结构分形和分维-Horton水系定律的模型重建及其参数确定[J].地球科学进展,2001,16(2):1118-123
[70]Tarhoton DG,Bras R L,Rodriguez-turbe I.The fraclal nature of river networks[J].Water Resources Research,1988,24(8):1317-1322
[71]朱嘉伟,赵云章,闫振鹏等.黄河下游河道地貌分形分维特征研究[J].测绘科学,2005;30(5):28-30.
[2]林向明.城市排水管网的规划设计[J].科技资讯,2005,25:42-43
[3]苗秀荣.城市给水管网的优化布置[J].科技情报开发与经济,2005,15(5):178-180
[4]陈胜兵,娄金生.排水管网平面布置的优化[J].工业用水与废水,2005,36(6):47-49
[5]贾树宝.污水管网的优化设计研究[J].灌溉排水,2000,19(1):63-65
[6]李树平,刘遂庆。城市排水管道系统设计计算的进展[J].给水排水,1999,25(10):9-12
[7]Batty M.Physical Phenomena[J].Geograph icalM agazine,1992,(7):35-361
[8]La Barbera P,Ro sso R.On the fractal dimension of stream networks[J].Water Resources Research,1989,25(4):735-7411
[9]李军锋.基于GIS的陕北黄土高原地貌分形特征研究[D].西安:西学大学硕士学位论文,2006
[10]Mandelbrot B B.The Fractal Geometry of Nature[M].San Francisco:Freeman.1983
[11]龚礼华,客观世界中普遍存在的分形与分维[J].达县师范高等专科学校学报(自然科学版)2004,14(5):22-23
[12]崔灵周,肖学年,李占斌,基于GIS的流域地貌形态分形盒维数测定方法研究[J].水土保持通报2004,24(2):37-38
[13]王薇.分形理论在表面粗糙度非接触测量中的应用[D].吉林:吉林大学硕士学位论文,2004
[14]刘湘南,许红梅,黄方.土地利用空间格局及其变化的图形信息特征分析[J].地理科学,2002,22(1):79-84.
[15]刘彦随,吴传钧,鲁奇.21世纪中国农业与农村可持续发展方向和策略[J].地理科学,2002,22(4):385-389.
[16]陈国阶.论地理学面临的挑战与发展[J].地理科学,2003,23(2):129-137
[17]张济忠.分形.北京:清华大学出版社.1995:7-110
[18]高安秀树.分数维[M].北京:地震出版社(中译本).1989:15-26
[19]Mandelbrot B B.Fractal:Form,Chance and Dimension.San Francisco:Freeman.1977:1-20
[20]龚礼华.客观世界中普遍存在的分形与分维[J].达县师范高等专科学校学报(自然科学版),14(5)2004:22-23
[21]刘秉正,彭建华编著.非线性动力学与混沌基础[M].长春:东北师范大学出版社.2004.
[22]韩向峰,基于分形理论的创新设计及其应用[D].山东师范大学硕士学位论文,2005
[23]于红志.分形图形生成研究[D].大连理工大学硕士论文,2003
[24]王丽.分形图形的研究与应用[D].昆明理工大学,2005
[25]束庆年.市政管网分形研究[D].北京工业大学硕士论文2003
[26]张宇星.城市与城市群形态空间的分形特性.新建筑,1995 3:42-46
[27]陈勇,艾南山.城市结构分形研究.地理学与国土研究.,1994,10(4):35-41
[28]R White,G Engelen.Cellular autumata and fractal urban form:a cellularmodeling approach to the evolution of urban land-use patterns.Enviornment and Planning A,1993,Volume 25,Pages1175-1199
[29]Batty M,Longley P,Fotheringham S.Fractal-based description of urbanform:scaling,fractal geometry,and diffusion-limited aggregation.Environment andPlanning A 21,1987,Volume 25,Pages1447-1472
[30]1 9 Batty M,Longley P.Fractal-based description of urban form.Environment and Planning B:Planning and design 14 123-134
[31]Susanna C.Manrubia,Transient dynamics and scaling phenomena in urban growth.Fractals C,1999Vol.7,No.1 Pages1-8
[32]Batty M,.Geneating urban forms from diffusive growth.Environment and Planning A,1991 23511-544
[33]Lam,Warren M.;Wornell,Gregory W.Multiscale Analysis and Control of Networks with Fractal Traffic.Applied and Computational Harmonic Analysis Volume:11,Issue:1,July,2001,pp.124-146
[34]Kim,Kwang Sik;Bengnigui,Lucien;Marinov,Maria.The fractal structure of Seoul's public transportation system.Cities Volume:20,Issue:1,February,2003,pp.31-39
[35]Ayedemir,M.;Bottomley,L.;Coffin,M.;Jeffries,C.;Kiessler,P.;Kumar,K.;Ligon,W.;Marin,J.;et.al.Two tools for network traffic analysis.Computer Networks Volume:36,Issue:2-3,July,2001,pp.169-179
[26]Liangxiu,Han;Zhiwei,Cen;Chunbo,Chu;Chuanshan,Gao.A new multifractal network traffic model.Chaos,Solitons and Fractals Volume:13,Issue:7,June,2002,pp.1507-1513
[37]陈彦光,刘继生.水系结构分形和分维---Horton水系定律的模型重建及其参数确定[J].地球科学进展,2001,16(2):118-123
[38]李华晔,黄志全,姜彤.河流水系分形的初步研究[J].华北水力水电学院学报,1998,14(4):28
[39]汪富泉,曹叔尤,丁晶.河流网络的分形与自组织及其物理机制[J].水科学进展,200213(3):368-376
[40]Veltri,M.Veltri,P.Maiolo,M.On the fractal description of natural channelnetworks.Journal of Hydrology Volume:187,Issue:1-2,December,1996:137-144
[41]陈彦光 刘继生.中心地体系与水系分形结构的相似性分析[J].地球科学进展,2000,20(1),81-88
[42]冯金良 张稳.海滦河流域水系分形.泥沙研究.Vol.1,1992
[43]沈晓华等.长江河道分形和流域构造特征的关系.浙江大学学报.Vol.28 No.1,2001:33
[44]罗文峰 李后强 丁晶 艾南山.Horton定律及分支网络结构的分形描述[J].水科学进展,19989(2):118-123
[45]Babadagli,Tayfun.Fractal analysis of 2-D fracture networks of geothermal reservoirs in south-western Turkey.Journal of Volcanology and GeothermalResearch Volume:112,Issue:1-4,December,2001,pp.83-103
[46]Schuller,D.J.;Rao,A.R.;Jeong,G.D.Fractal characteristics of dense stream networks.Journal of Hydrology Volume:243,Issue:1-2,March 1,2001,pp.1-16
[47]Puente,Carlos E.;Castillo,Pablo A.On the fractal structure of networks and dividers within a watershed.Journal of Hydrology Volume:187,Issue:1-2,December,1996,pp.173-181
[48]Tarboton,David G.Fractal river networks,Horton's laws and Tokunaga cyclicity.Journal of Hydrology Volume:187,Issue:1-2,December,1996,pp.74 105-117
[49]Moussa,Roger;Bocquillon,Claude.Fractal analyses of tree-like channelnetworks from digital elevation model data.Journal of Hydrology Volume:187,Issue:1-2,December,1996,pp.157-172
[50]Ayedemir,M.;Bottomley,L.;Coffin,M.;Jeffries,C.;Kiessler,P.;Kumar,K.;Ligon,W.;Marin,J.;et.al.Two tools for network traffic analysis.Computer Networks Volume:36,Issue:2-3,July,2001,pp.169-179
[51]Troutman,Brent M.;Over,Thomas M.River flow mass exponents with fractalchannel networks and rainfall.Advances in Water Resources Volume:24,Issue:9-10,November-December,2001,pp.967-989
[52]Niemann,JeffreyD.;Bras,RafaelL.;Veneziano,Daniele;Rinaldo,Andrea.Impacts of surface elevation on the growth and scaling properties of simulated river networks.Geomorphology Volume:40,Issue:1-2,September,2001,pp.37-55
[53]Crave,A.;Davy,P.Scaling relationships of channel networks at large scales:Examples from two large-magnitude watersheds in Brittany,France Tectonophysics Volume:269,Issue:1-2,January 30,1997,pp.91-111
[54]祝安,祝进,张朝晖.喀斯特流域水系分形、分维问题[J].贵州师范人学学报(自然科学版)18(4)2000:5-8
[55]Frankhouser P.A spects fractals des structures urbans[J].L'Esp ace Geog rap h ique,1990,19(1):45-69
[56]刘继生,陈彦光.交通网络空间结构的分形维数及其测算方法探讨[J].地理学报1999.54(5):471-478
[57]丁以中,楼勇.分形理论在交通网络评价中的应用[J].上海海运学院学报,1998.19(4):7-12
[58]刘立明.分形法在河北省高速公路规划中的应用[J].河北工业大学成人教育学院学报,2001,16(4):21-23
[59]李文华,杨兆升,王希伟.基于分形几何学的区域公路网布局评价指标的研究[J].交通运输系统工程与信息,2005,5(5):50-53
[60]许志海,翟京生.分形理论在空间网络分布特征研究中的应用[J].海洋测绘,2005,25(6):26-33
[61]李桀,朱金兆,朱清科.分形维数计算方法研究进展[J].北京林业大学学报,24(2)2002:71-78
[62]张济忠.分形[M].北京:清华大学出版社,1995
[63]沈莉芳.污水干管系统选线最优化决策方法研究[D].重庆大学工程硕士学位论文,2004
[64]牛文元.理论地理学EB
[65]李慧博,柴立和.给水管网的分形生长分析[R].北京:中国原子能科学研究院,2005
[66]Barbera P La,Rosso R.On the Fractal Dimension of Stream Networks[J].Water Resources Research,1989,25(4):735-741
[67]Tarboton D,Bras R & Rodriguez-Iturbe 1.The Fractal Nature of River Networks,Water Resources Research.1988,24(8):1317-1322
[68]洪时中,洪时明.地学领域中的分维研竞[J]:太自然探索,1988,24(8):33-40
[69]冻彦光,刘继生.水系结构分形和分维-Horton水系定律的模型重建及其参数确定[J].地球科学进展,2001,16(2):1118-123
[70]Tarhoton DG,Bras R L,Rodriguez-turbe I.The fraclal nature of river networks[J].Water Resources Research,1988,24(8):1317-1322
[71]朱嘉伟,赵云章,闫振鹏等.黄河下游河道地貌分形分维特征研究[J].测绘科学,2005;30(5):28-30.