WATER RESOURCES RESEARCH, VOL. 30, NO. 7, PAGES 2311-2323, JULY 1994 On geomorphological dispersion in natural catchments and the geomorphological unit hydrograph John D. Snell and Murugesu Sivapalan Centre for Water Research, Department of EnvironmentalEngineering University of Western Australia, Nedlands Abstract. Since the introductionof 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 catchmentfor determining the hydrological responseof the catchment. We examine three approaches by which geomorphology can be introducedthroughthe probabilities and lengthsof 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 catchmentwithout the assumptions of Strahler stream ordering. We use these techniquesto derive the GIUH and the geomorphological dispersion coefficientas defined by Rinaldo et al. (1991) for two catchments in the southwestern region of Australia. The geomorphological dispersion coefficientderived 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 underlyingdispersion of an order of magnitudegreater 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 parametersand how this concept is related to the thresholdarea supporting a network derived from a catchment DEM. Introduction The concept of the geomorphological instantaneous unit hydrograph (GIUH) was introduced by Rodriguez-Iturbe and Valdes [1979] and later generalized by Gupta et al. [1980]. The underlying natural order in the morphology of the catchment, determining to some extent the distribution of arrival times at an outlet of a unit instantaneous pulse injected throughouta channel network, is the basic concept of the GIUH. The remaining part of the travel time distribu- tion is a product of the hydraulics of the channels them- selves. The original research by Rodriguez-Iturbe and Valdes expressedthe morphologicalorder in terms of the Horton order ratios, which are in turn based on Strahler ordering of the streams. They derived a set of generic pathways which are composed of "states" consisting of the Strahler orders experiencedby a droplet of water between its source on the catchmentand the outlet. They considered the progressionof the droplet through the set of states of a pathway as a progression through an embedded Markov process, so that the holding time within each state is inde- pendent of the destination state, as indeed it must be for the Markovian assumption to hold. Furthermore, they made the arbitrary decision that the holding time within each state is distributed exponentially. The probabilities associated with (1) the area of the land surface from which the droplet originated and (2) the transitions between states were for- Copyright 1994by the American Geophysical Union. Paper number 94WR00537. 0043-1397/94/94 WR-00537 $05.00 mulated in terms of the Horton order ratios of the catch- ment. Gupta et al. [ 1980], in generalizingthe GIUH concept, showed that the Markovian assumption and consequently the assumption of exponential holding times within each state were unnecessary in determining the distribution of arrival times at an outlet. Gupta et al. still expressed the catchmentmorphologythrough Horton order ratios derived from a network which had been Strahler ordered. They did not deduce the true distribution of the holding times in the channels but applied their analysis to two assumeddistribu- tions of holding times: the exponential distribution, as used by Rodriguez-Iturbe and Valdes, and a uniform distribution. Gupta and Waymire [1983] criticized the GIUH approach taken by Rodriguez-Iturbe and Valdes on the basis that the travel time distributions in each state are assumptions within the theory without any underlying basis for the actual physical response of each channel. This is compounded in the caseof a Strahler ordered systemin that the geomorphic states are averages across all the realizations of that order within the network. Kirshen and Bras [ 1983] have performed some work trying to elucidate the physical nature of the channel travel time which could be incorporated into the GIUH concept, but this as yet remains inconclusive. Gupta and Waymire [1983], in deriving their analytical formulation of the network response, suggestedthat the concept of Strahler ordering and ratios derived from such ordering play little if any part in the determination of the network re- sponse. $urkan [1969], as quoted by Beven [1986], has shown that topologically similar networks can have quite different channel network responses, indicating that the topology of the network on its own cannot fully capture the 2311