River network fractal geometry and its computer simulation

The hierarchical ordinal and statistical models of river networks are proposed. Their investigation has been carried out on the basis of river networks computer simulation as well as on empirical data analysis. The simulated river networks display self‐similar behavior on small scales (the fractal dimension D ≈ 1.52 and Hurst's exponent H = 1.0) and self‐affine behavior on large scales (the lacunary dimension DG ≈ 1.71, H ≈ 0.58). Similar behavior is also qualitatively characteristic for natural river networks (for catchment areas from 142 to 63,700 km2 we obtained DG ≈ 1.87 and H ≈ 0.73). Thus in both cases one finds a region of scales with self‐affine behavior (H < 1) and with DG < 2. Proceeding from fractal properties of the river networks, the theoretical basis of scaling relationships L ∼ Aβ and ℒ ∼ Aε, widely used in hydrology, are given (L, ℒ and A denote the main river length, the total length of the river network, and catchment area, respectively); β = 1/(1+H) and ε=DG/2.