Fractal structure and properties of stream networks

This paper studies the fractal structure and properties of stream networks through a comparison between stream networks and loopless random aggregate trees. Five fractal dimensions are recognized for stream networks, namely, topological dimension (dt), minimum path dimension (dm), fractal dimension of stream networks as a whole (df), diffusion dimension (dw), and spectral dimension (ds). Mathematical definitions and derivations of the fractal dimensions are proposed, which provide a theoretical basis for some recognized empirical relationships in drainage basin geomorphology. Relationships among the fractal dimensions are examined, and the average values of the fractal dimensions for natural stream networks are estimated. The hydrological implications of the fractal dimensions of stream networks are also discussed.