Detection of ocean bathymetry from surface wave measurements

The detection of ocean bathymetry is one of the most important and difficult problems in oceanography. On the theoretical side it is a classical inverse problem which features severe ill-posedness found in similar problems from a wide array of applied sciences. From a practical standpoint, standard procedures based upon “Underwater Acoustics” are quite dangerous and expensive as the desired surface is separated from measuring devices by (at least) the entire ocean layer providing a very hostile and unpredictable sampling environment. In this research we take a rather different approach to this inverse problem as we rely upon nonlinear features of the governing fluid mechanical equations to detect information about the ocean bathymetry. This is also in contrast to similar methods in the literature which rely solely upon the variations in the dispersion relation. Using a formulation of the water wave problem due to Zakharov, and Craig and Sulem, and the analyticity of the “Dirichlet-Neumann operator” we find surprisingly convenient formulas involving the ocean bathymetry. Of course, these formulas are ill-conditioned and nonlinear, however, we have found that application of standard techniques from the theory of inverse problems allow us to predict the shape of bottom topography with excellent precision.