Inversion in Higher Dimensions

In Chapter 2, formulas were created to determine the wavespeed profile of a medium with one dimension of parameter variability only, via high-frequency inversion of plane-wave data. The original plan of Chapter 2 was to invert the data for the actual wavespeed profile or, rather, the perturbation α(y) from a known background wavespeed profile. However, results influenced by the bandlimited nature of the data, represented by a symmetric filter F(w) motivated a change to the new goal of imaging the discontinuities of the wavespeed profile—the reflectors. This yields a new output, the “reflectivity function” β(y), which was found to consist of bandlimited delta functions having peak amplitudes occurring at reflector locations, with size scaled by the normal-incidence, plane-wave reflection coefficient. The reflectivity function is analogous to a similar reflection coefficient series that may be obtained in the process of creating a synthetic well log from seismic data. Equivalent results in higher dimensions will be the goal of all subsequent inversion formulations found in this text.