Magnitude-frequency relations and the morphology of debris-flow fans

Debris-flow fans are constructed by repeated debris flows that both transport sediment and help shape the associated catchment. As well as being important geomorphic agents, these flows represent a significant hazard to communities and infrastructure on debris-flow fans. Because of the brevity of the historical record, we generally lack solid constraints on debris-flow magnitude-frequency relations over time scales that are long enough for accurate hazard assessment. Here, we investigate the use of fan form and surface morphology as a long-term, integrated record of debris-flow occurrence. Fan construction depends on (1) the pattern of deposition along each debris-flow path and (2) the distribution of debris-flow volumes that exit the catchment. Because both of these parameters are difficult (or impossible) to estimate over geological time scales, we use field observations and simple scaling relations to inform a two-dimensional forward model of fan building. Debris flows are chosen at random from a specified volume distribution at the fan apex and routed down the path of steepest descent into a subsiding basin. Deposition is controlled by the downstream rate of volume loss, termed the lag rate, which critically depends on both initial flow volume and local slope. Flows continue downslope until their volume is completely depleted. We find that the overall form of the fan, in terms of its surface slope, radial length, and cross-sectional profile, is sensitive to the formulation of the lag rate but is remarkably insensitive to the details of the volume distribution, provided the mean flow volume is held approximately constant. The volume distribution does, however, play a large role in determining debris-flow runout paths and the spatial pattern of debris-flow snouts. These results raise the possibility that mapping of snout positions, and thus flow runout lengths, may yield important constraints on the debris-flow volume distribution over the time scales preserved by fan surfaces, which are typically 10s to 100s of kyr.