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A comment on Horton's law of stream numbers
Author(s) -
Smart J. Samuel
Publication year - 1967
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/wr003i003p00773
Subject(s) - order (exchange) , mathematics , law of large numbers , law , combinatorics , statistics , random variable , finance , political science , economics
Horton's Law of Stream Numbers is shown to be internally inconsistent in the following sense: If there exists a large channel network of order S , with stream numbers that satisfy Horton's Law exactly, then the mean stream numbers for lower order networks contained in the large one will show definite deviations from Horton's Law. These mean stream numbers can be expressed by a recursive formula, which involves the probabilities P ij that a stream of order i terminates in a stream of order j > i . Reasonable assumptions about the nature of the P ij suggest that Horton curves (log N k versus k ) for the small order basins should be concave upwards, a result which is in general agreement with observation.