Statistical Properties of Stream Lengths

Abstract

Two basic assumptions are employed in this treatment of the statistics of stream lengths: (1) All topologically distinct networks with a given number of sources are equally likely (after Shreve). (2) Lengths of interior links in a given network are independent random variables drawn from the same population. The mathematical development leads to an approximate expression for L• that contains no adjustable parameters and that depends only on the stream numbers and the mean link length. This expression gives somewhat better agreement with data on actual stream systems than does Horton's law of stream lengths; other advantages over Horton's law are cited. Quantitatively, our procedure appears to account for about 65% of the variance in mean stream length data for third-and fourth-order basins. If exact values of the link numbers are introduced into the calculation, the unexplained variance is reduced to about 15%. s. s•AgTThe Horton analysis m•y be regarded as an attempt to give a quantitative description of drainage basin composition in terms of five parameters' Rs, RL, RA, •, and •, or some equivalent set [Horton, 1945, p. 295]. Once these five numbers are known, approximate values for other geomorphic parameters, such as total stream length, total basin area, drainage density, and stream frequency, can easily be obtained. Leopold and Miller [1956] have shown how the analysis can be extended to predict width, depth, discharge, sediment load, and other hydraulic variables. Although the Horton and Strahler ordering techniques have been much used by hydrologists and geomorphologists, they have also attracted a certain amount of criticism. In

The River Systems in Small Catchments in the Context of the Horton’s and Schumm’s Laws – Implication for Hydrological Modelling. The Case Study of the Polish Carpathians

“…Shreve 1966, 1969, Smart 1968, Kirchner 1993) have discussed Horton's and Schumm's laws on the basis of "statistical models", and they have compared the results to natural river systems. In this way, Kirchner (1993) showed that, statistically speaking, roughly 95% of river systems obey Horton's and Schumm's laws, regardless of whether the systems are topologically random or not.…”

“…For natural catchments, the R L ratio usually ranges from 1.5 to 3.5, with the mean equal to 2.0 (Smart 1968, 1972, Bras, Rodriguez-Iturbe 1989). An R L ratio ranging from 1.5 to 3.5 was found to be typical for the large Carpathian catchments (Bajkiewicz-Grabowska 1987).…”

Sensitivity of channel network planform laws and the question of topologic randomness

“…[property 1] the distributions and numerical values of RB (Shreve, 1966), Rz• (Smart, 1968;Shreve, 1967Shreve, , 1969, and ( Shreve, 1969) [property 2] the correlations between RB and R L and between stream numbers and stream lengths (Smart, 1968) [property 3] the statistical distributions of second-order stream lengths, Schumm lengths, and areas (Shreve, 1969) [property 4] Meltoh's (1958) proportional relationship between stream frequency and the square of the drainage density (Shreve, 1967) [property 5] Hack's (1957) variation of mainstream length with basin area (e.g., Shreve, 1974) [property 6] Langbein et al 's (1947) relationship between distance-weighted area and basin area (Werner and Smart, 1973) [property 7] Gray's (1961) relationship between distance from outlet to "centroid" and mainstream length (Werner and Smart, 1973).…”

Applications of the random model of drainage basin composition

“…For drainage networks developed under similar environmental conditions, the exterior and interior link lengths are independent random variables with a single common distribution for each type (Smart (1968) p. 1005, Shreve (1967) p. 184, (1969) p. 402)).…”