Flow paths for geologic mass movements driven by gravity, such as lahars, floods, mudflows, debris flows, and lava flows, are determined by the local topography. In this work, we consider the problems of predicting the path of a geologic mass movement on a discrete topographic surface that is described by a Digital Elevation Model (DEM). The emphasis of this work is to develop a path finding algorithm that addresses the problems arising from the discrete nature of DEM topography, rather than the physics of mass movements, and improves upon existing approaches employed by commercial and public domain algorithms. We impose three criteria for the algorithm: 1) it must have a rigorous basis in analytic calculus, 2) the decision criteria used within the path finding algorithm must be clearly identified and tested for real flow conditions, and 3) it must be validated by comparing predictions with actual flows. The fundamental assumption of the path finding algorithm is that the definition of ‘downhill’ can be found from the local 3-dimensional gradient to the surface at each point. This definition differs fundamentally from the ‘lowest nearest neighbor’ approach employed by some commercially available software. While conceptually similar to a contour normal approach, we show that integration between contours is necessary to eliminate systematic errors in path orientation. We discuss several specific decision criteria that affect the practical implementation of the algorithm to discrete DEM surfaces: how integration is approximated through a small step size and averaging, our approach to addressing regions of low slope, and how we have accounted for residuals along the path. Finally, we show two convincing examples where the algorithm predicts the paths for two lava flows in Hawaii that traveled several tens of kilometers. The agreement between the predicted and actual flow path is remarkable and validates the algorithm and decision criteria for this type of mass flow.

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