HORTON LAWS AND THE FRACTAL NATURE OF STREAMS

Stream networks and catchment basins are characterized by numerous fractal dimensions, the traditional one being the mainstream length as a function of length scale which we denote by d. We consider an ideal Horton system and define fractal dimensions based on (1) the subcatchment length-area relationship, which gives D1 ; (2) the total length of streams as a function of scale, which gives a network similarity dimension D2; (3) the mainstream length-area relationship (D3), and (4) the total area of streams as a function of areal scale (D4). These fractal dimensions are related to the values of the bifurcation ratio, the stream length ratio, and the stream area ratio. Their value should be indirectly estimated from these Horton parameters. An improved understanding of the relationships between the fractal dimension and Horton's laws is obtained by defining an eigenarea of a stream of a particular Horton order. This is the area of the drainage basin which is not drained by streams of immediately lower order. The variation of eigenareas with scale controls the value of D1 , which equals D3 for hydrologically realistic systems. In such cases D4 = 2. This paper assumes that river systems are Hortonian in nature and that measured departures from Hortonianity are due to statistical variability. Estimates of the Horton parameters and of the fractal dimensions are therefore subject to uncertainty. Two sources of statistical variability are considered herein: The first source of statistical variability is due to the fluctuations in stream numbers, stream lengths, and stream areas. These fluctuations affect the values of the Horton parameters, which are traditionally estimated using a graphical technique. We show that an analytical technique (which corresponds to a constrained graphical technique) is preferable. The second source of variability that we consider arises from the fact that estimates of Horton's parameters and fractal dimensions are always made on truncated systems because the particular scales of measurement determine the order of the catchment. In a Horton system it is straightforward to estimate D1, which is defined exactly at all scales. Direct attempts to estimate D2 and D3 (as opposed to indirect attempts via the Horton parameters) are difficult because their true value is approached asymptotically in the limit of infinite-order catchments.