The Inverse Problem of Seismic Velocities
To reasonable approximation, the construction of structural images of the earth's interior, using seismic reflection data as input, boils down to an inverse problem for the wave equation. This inverse problem requires that the spatially varying wave velocity (a coefficient in the wave equation) be determined from samples of solutions near the boundary of the space-time domain of propagation. The data-fitting techniques that have proven effective in other science and engineering inverse problems encounter fundamental mathematical obstacles in application to this one. In response, the seismic prospecting industry has devised a collection of methods to extract earth structure from data, that appear at first glance to have little to do with data-fitting. I will describe the seismic inverse problem and the qualities that make it resistant to data-fitting, as well as an underlying mathematical structure that encompasses both the data-fitting and industrial approaches. This structure supports variational principles, more general than the data-fitting or least squares principle, which in some cases have proven effective tools for velocity estimation.
William Symes, Rice University