Abstract
Geographical AnalysisVolume 4, Issue 2 p. 119-133 Free Access Patterns of Drainage Areas with Random Topology* Christian Werner, Christian Werner Christian Werner is associate professor of geography, University of California, IrvineSearch for more papers by this author Christian Werner, Christian Werner Christian Werner is associate professor of geography, University of California, IrvineSearch for more papers by this author First published: April 1972 https://doi.org/10.1111/j.1538-4632.1972.tb00464.xCitations: 6 * The support of the National Science Foundation, Grant GS-2989, is gratefully acknowledged. AboutPDF ToolsRequest permissionExport citationAdd to favoritesTrack citation ShareShare Give accessShare full text accessShare full-text accessPlease review our Terms and Conditions of Use and check box below to share full-text version of article.I have read and accept the Wiley Online Library Terms and Conditions of UseShareable LinkUse the link below to share a full-text version of this article with your friends and colleagues. Learn more.Copy URL Share a linkShare onFacebookTwitterLinkedInRedditWechat LITERATURE CITED 1 Dacey, M. F. Probability distribution of number of networks in topologically random drainage patterns,” Water Resources Res., submitted, 1971. 2 Hall, Marshall, Jr. Combinatorial Theory. Waltham, Mass. (1967). 3 James, W. R. and W. C. Krumbein, “Frequency Distribution of Stream Link Lengths Journal of Geology, 77, (1969), pp. 544– 565. 4 Leopold, L. B., M. G. Wolman, and J. P. Miller. Fluvial Process in Geomorphology. San Francisco: W. H. Freeman and Co., (1964). 5 Riordan, J. An Introduction to Combinatorial Mathematics. New York: J. Wiley, (1958). 6 Riordan, J. Combinatorial Identities. New York: J. Wiley, (1968). 7 Shreve, R. Statistical Law of Stream Numbers,” Journal of Geology, 74 (1966) 17– 37. 8 Shreve, R., “Infinite Topologically Random Channel Networks,” Journal of Geology. 75, (1967) 178– 186. 9 Werner, C. Topological Randomness in Line Patterns,” Proc. Assoc, Amer. Geographers, 1 (1969) 157– 62. 10 Werner, C. Expected Number and Magnitude of Stream Networks in Random Drainage Patterns,” Proc. Assoc. Amer. Geographers, 3, (1971) 181– 185. 11 Woldenberg, M. J. Horton's Law Justified in Terms of Allometric Growth and Steady State in Open Systems,” Bull. Geol. Soc. Amer., 77 (1966) 431– 434. 12 Woldenberg, M. J. Spatial Order in Fluvial Systems: Horton's Laws Derived from Mixed Hexagonal Hierarchies of Drainage Basin Areas,” Bull. Geol. Soc. Amer., 80 (1969) 97– 112. Citing Literature Volume4, Issue2April 1972Pages 119-133 ReferencesRelatedInformation
Highlights
After several years of rather successful investigation of the geometry and topology of channel networks by theoretical analysis and simulation models, one is tempted to pose the question of whether other morphological patterns of the physical landscape can be approached with a similar methodology
What will be the expected distribution of drainage polygons when grouped according to the number of their sides? Do empirical d a b support the hypothetical assumption above through close correspondence to the theoretical distribution deduced from this assumption?
Rather than defining the subdivision on the basis of the total drainage, we will consider drainage by surface runoffonly. This definition is equivalent with an alternative definition based on the concept of channel links and the notion of steepest slope; namely, that all points of the area under consideration belong to the same subarea if and only if the lines of steepest decline passing through these points enter the same channel link
Summary
After several years of rather successful investigation of the geometry and topology of channel networks by theoretical analysis and simulation models, one is tempted to pose the question of whether other morphological patterns of the physical landscape can be approached with a similar methodology. Rather than defining the subdivision on the basis of the total drainage, we will consider drainage by surface runoffonly This definition is equivalent with an alternative definition based on the concept of channel links and the notion of steepest slope; namely, that all points of the area under consideration belong to the same subarea if and only if the lines of steepest decline passing through these points enter the same channel link. In any real case, the location of a drainage divide can never be identified with absolute (mathematical) precision, this observation will be adopted as an axiom, i.e., we will assume, that every node in a drainage divide pattern joins exactly three drainage divide links The following table shows maximum and minimum values of X for the range 2 5 n 5 00
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