C1-approximation of seafloor surfaces with large variations

In many problems of geophysical interest, when trying to describe surfaces, one has to deal with data that exhibit rapid variations. This occurs for instance when describing the topography of mountain ranges, volcanoes, seafloor surfaces (bathymetry maps), islands, or the shape of geological entities, that can present large and rapid variations due for instance to the presence of faults in the structure. The correct description of such geological surfaces, by a fitting process from a given set of points, is therefore of great importance. Usual methods give good results in the case of curve fitting, but less accurate results in the case of surface fitting. The new method the authors propose uses scale transformations (spline under tension), and is applied without any particular a priori knowledge of the data. They first propose a construction of these scale transformations families, and then show the efficiency of this innovative approach by applying it to seafloor surfaces around the Big Island in Hawaii in order to get a regular surface with at least continuity of the first derivatives