Refinements to the linear velocity inversion theory
The linear inversion method presented by Cohen and Bleistein in 1979 gives seriously degraded results when large reflectors are encountered. Obviously there is an irrecoverable loss of information when such a linear algorithm is applied to a nonlinear world. However, in many cases, excellent results can be achieved by suitable postprocessing of the output of the basic linear inversion algorithm. Although a certain degree of helpful postprocessing can be and has been performed by straightforward consideration of the linearization process, we present here a substantially improved postprocessing algorithm. The basis for these improvements is a more accurate scattering model due to Lahlou et al where, among other things, a WKB analysis of the wave equation led to a much more accurate accounting of the geometric spreading of the scattered wave. These notions plus an effective use of traveltime are used in the new algorithm to improve both the estimate of the reflector locations and the estimate of amplitude (velocity or acoustic impedance) change across the reflectors. The basic idea is to insert this idealized scattering data into the original linear algorithm, and then use the result of this computation as a guide in the interpretation of the numerical output of the algorithm. We demonstrate the result of computer implementation of this algorithm on synthetic data, with and without noise, and verify that the postprocessing algorithm produces dramatically improved reflector locations and speed estimates. Moreover, the new algorithm adds only very modest cost to the basic processing, which is, in turn, very competitive in cost to other multidimensional algorithms.