Anumerical method for seismic waveform inversion is developed, using the methods of quantum scattering theory. The method, which is based on a convenient matrix reprentation of the relevant integral operators, is completely general, but we focus on novel applications to 4D seismics. It is shown that the unknown scattering potential matrix (diagonal) is related to the partially observable T- matrix (non-diagonal) in a nonlinear manner, associated with the well-known Lippmann-Schwinger integral equation. We develop an iterative numerical scheme for inversion of 4D seismic waveform data with respect to the scattering potential, where the T-matrix is updated after each iteration, either by inverting a relatively large matrix or by using the forward scattering series. The connection between this simple numerical realization of the inverse scattering series and the ISS employed by Weglein et al. (which appears to contain redundant terms) will also be discussed. In a numerical experiment, where we generated synthetic 4D seismic waveform data using a full numerical (T-matrix) solution of the Schwinger-Dyson integral equation, we obtained a dramatic improvement on the Born inversion result after just 3 iterations. This suggest that internal multiples should be regarded as an additonal source of information, and not just noise.
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