A Stochastic Differential Equation Approach to Studying Landslide Failure and Size Distributions

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A major unsolved problem in geomorphology is this: when a slope begins to fail, how big will the ensuing landslide be? The span of possible landslide sizes is enormous, with potential failures ranging in scale from meters to kilometers - so it's unfortunate, particularly for hazard assessment, that we remain unable to predict such sizes. This inability persists despite the fact that we now have rather good constraints on the size-frequency distribution of landslides at a regional scale. This project takes a new theoretical tack, one that involves the use of stochastic differential equations, to address in concert the issues of individual landslide propagation and ensemble landslide sizedistribution. We have found that a simple, stochastic calculus model for slope failure can explain the full size-frequency distribution of landslides, including the mean landslide size and both the power-law scaling and non-scaling components of the distribution. We are developing this theory in order to address the following key questions: (1) how does the mean landslide size in an ensemble distribution relate to reality? - if it is not an artifact of mapping resolution, does it relate to physical properties such as soil depth, cohesion, and lithology, or is it simply a function of mean hillslope scale? (2) are the physical assumptions of the stochastic theory borne out by field observations? (3) can we be more precise in our data analysis and modeling of different slope failure mechanisms? (4) do rockfalls occur by an entirely different stochastic process with an altogether different size-frequency distribution? The outcome of our efforts will be a deeper understanding of the stochastic behavior of hillslope failure and landslide hazard.