Surface fitting of rapidly varying data using rank coding: Application to geophysical surfaces

Addressing geophysical problems often implies the correct description of su rfaces with large local variations. This problem is of interest in many are as of geophysics-for instance, for the description of topography when study ing site effects in seismic wave propagation, or the propagation of lava or pyroclastic flows along the slopes of a volcano, or in the presence of geo logical structures with faults. However, surface fitting of rapidly varying data using classical functions like splines is known to be difficult. With out information about the location of the large variations in the data set, the usual approximation methods lead to instability phenomena or undesirab le oscillations. We propose a new approach that uses scale transformations and whose originality consists in a preprocessing and a postprocessing of t he data. Variations of the unknown function are reduced using a scale trans formation in the preprocessing phase. The transformed data ado not exhibit large variations, and therefore we can use a usual approximant that will no t create oscillations. An inverse scale transformation is subsequently appl ied. We discuss the convergence of the method when the number of data point s tends to infinity. We show the efficiency of this technique by applying i t to a Digital Elevation Model of the topography of the Piton de la Fournai se volcano (Reunion Island, France).