Since the introduction of the geomorphological instantaneous unit hydrograph (GIUH) by Rodriguez-Iturbe and Valdes (1979) there have been a number of different approaches to expressing the geomorphology of a catchment for determining the hydrological response of the catchment. We examine three approaches by which geomorphology can be introduced through the probabilities and lengths of the pathways available within a network: (1) using the Horton order ratios to derive analytical expressions for these pathway parameters, (2) extracting these probabilities and lengths directly from a Strahler ordered network without using the Horton order ratios, and (3) using a contributing area-flow distance function extracted directly from the digital elevation model (DEM) of a catchment without the assumptions of Strahler stream ordering. We use these techniques to derive the GIUH and the geomorphological dispersion coefficient as defined by Rinaldo et al. (1991) for two catchments in the southwestern region of Australia. The geomorphological dispersion coefficient derived from the area-distance function expresses the natural dispersion within the catchment in a more fundamental manner than the other methods, and the GIUHs derived from this function exhibit an underlying dispersion of an order of magnitude greater than either of the approaches based on Strahler ordering. We look at the concept of the “completeness” of a network from which we derive catchment parameters and how this concept is related to the threshold area supporting a network derived from a catchment DEM.