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

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 (10⁹) 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 loudspeakers 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.