A Note on Fractal Channel Networks

This paper studies the relation between the structure of river networks and the features of their geomorphologic hydrologic response. A mathematical formulation of connectivity of a drainage network is proposed to relate contributing areas and the network geometry. In view of the connectivity conjecture, Horton's bifurcation ratio R(B) tends, for high values of Strahler's order OMEGA of the basin, to the area ratio R(A), and Horton's length ratio R(L) equals, in the limit, the single-order contributing area ratio R(a). The relevance of these arguments is examined by reference to data from real basins. Well-known empirical results from the geomorphological literature (Melton's law, Hack's relation, Moon's conjecture) are viewed as a consequence of connectivity. It is found that in Hortonian networks the time evolution of contributing areas exhibits a multifractal behavior generated by a multiplicative process of parameter 1/R(B). The application of the method of the most probable distribution in view of connectivity contributes new inroads toward a general formulation of the geomorphologic unit hydrograph, in particular generalizing its width function formulation. A quantitative example of multifractal hydrologic response of idealized networks based on Peano's construct (for which R(B) = R(A) = 4, R(L) = 2) closes the paper.