Wavefield migration plus Monte Carlo imaging of 3D prestack seismic data

Prestack wave-equation migration has proved to be a very accurate shot-by-shot imaging tool. However, 3D imaging with this technique of a large field acquisition, especially one with hundreds of thousands of shots, is prohibitively costly. Simply adapting the technique to migrate many superposed shot-gathers simultaneously would render 3D wavefield prestack migration cost-effective but it introduces uncontrolled non-physical interference among the shot-gathers, making the final image useless. However, it has been observed that multishot signal interference can be kept under some control by averaging over many such images, if each multishot migration is modified by a random phase encoding of the frequency spectra of the seismic traces. In this article, we analyse this technique, giving a theoretical basis for its observed behaviour: that the error of the image produced by averaging over M phase encoded migrations decreases as M-1. Furthermore, we expand the technique and define a general class of Monte-Carlo encoding methods for which the noise variance of the average imaging condition decreases as M-1; these methods thus all converge asymptotically to the correct reflectivity map, without generating prohibitive costs. The theoretical asymptotic behaviour is illustrated for three such methods on a 2D test case. Numerical verification in 3D is then presented for one such method implemented with a 3D PSPI extrapolation kernel for two test cases: the SEG-EAGE salt model and a real test constructed from field data.