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On the fractal interpretation of the mainstream length‐drainage area relationship
Author(s) -
Robert André,
Roy André G.
Publication year - 1990
Publication title -
water resources research
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.863
H-Index - 217
eISSN - 1944-7973
pISSN - 0043-1397
DOI - 10.1029/wr026i005p00839
Subject(s) - fractal , exponent , allometry , fractal dimension , scale (ratio) , interpretation (philosophy) , mathematics , fractal analysis , geometry , streams , cartographic generalization , generalization , drainage basin , geology , hydrology (agriculture) , geography , mathematical analysis , cartography , computer science , geotechnical engineering , paleontology , philosophy , linguistics , programming language , computer network
The exponent of the mainstream Length‐Drainage area relationship is interpreted as a fractal quantity, superseding the previous allometric interpretation. The fractal interpretation is based on the assumption that cartographic generalization is applied evenly to all map scales. Twenty‐three drainage basins of the Eaton River (Quebec, Canada) were delineated from topographic maps at three different scales (1:20,000, 1:50,000, and 1:125,000). The exponent of the length‐area relation is much lower (0.546) at the largest scale than at the smallest scales (0.65), and its value corresponds to that obtained from a Richardson analysis of 10 interior stream segments. At the 1:20,000 map scale, the exponent is entirely fractal. The larger exponent values obtained at the smallest scales exceed the fractal value and incorporate an allometric component. This component is not functional, however, and it merely reflects the generalization process of cartographic abstraction of stream heads as scale is reduced. The fractal dimension of streams should not be inferred from the exponent of the length‐area relation because its value is likely to be scale‐dependent.