A quasi‐Monte Carlo approach to 3-D migration: Theory

Abstract

A mathematical breakthrough was recently achieved in understanding the tractability of multidimensional integration using nearly optimal quasi‐Monte Carlo methods. Inspired by the new mathematical insights, we have studied the feasibility of applying quasi‐Monte Carlo methods to seismic imaging by 3-D prestack Kirchhoff migration. This earth imaging technique involves computing a large [Formula: see text] number of 3-D or 4-D integrals. Our numerical studies show that nearly optimal quasi‐Monte Carlo migration can produce the same or better quality earth images using only a small fraction (one fourth or less) of the data required by a conventional Kirchhoff migration. The explanation is that an image migrated from a coarse quasi‐random array of seismic data is less likely, on average, to be aliased than an image migrated from a regular array of data. In migrating these data, the geophones act as an incoherent arrangement of loud‐speakers that broadcast the reflected wavefield back into the earth; the broadcast will produce reinforcement or cancellation of seismic energy at the diffractor or grating lobe locations, respectively. Thus quasi‐Monte Carlo migration contains an inherent anti‐aliasing feature that tends to suppress migration artifacts without losing bandwidth. The penalty, however, is a decrease in the dynamic range of the migrated image compared to an image from a regular array of geophones. Our limited numerical results suggest that this loss in dynamic range is acceptable, and so justifies the anti‐aliasing benefits of migrating a random array of data.