Power law between the apparent drainage density and the pruning area

Self-similar structures of river networks have been quantified as having diverse scaling laws. Among these, we investigated a power function relationship between the apparent drainage density ρa and the pruning area Ap, with an exponent ≠. We analytically derived the relationship between ≠ and other known scaling exponents of fractal river networks. The analysis of 14 real river networks covering a diverse range of climate conditions and free-flow connectivity levels supports our derivation. We further linked ≠ with non-integer fractal dimensions found for river networks. Synthesis of our findings through the lens of fractal dimensions provides an insight that the exponent ≠ has fundamental roots in the fractal dimension of the whole river network organization.