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 qua si-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 imagin g technique involves computing a large (10(9)) number of 3-D or 4-D in tegrals. Our numerical studies show that nearly optimal quasi-Monte Ca rlo migration can produce the same or better quality earth images usin g only a small fraction (one fourth or less) of the data required by a conventional Kirchhoff migration. The explanation is that an image mi grated from a coarse quasi-random array of seismic data is less likely , on average, to be aliased than an image migrated from a regular arra y of data, In migrating these data, the geophones act as an incoherent arrangement of loudspeakers that broadcast the reflected wavefield ba ck into the earth: the broadcast will produce reinforcement or cancell ation of seismic energy at the diffractor or grating lobe locations, r espectively, Thus quasi-Monte Carlo migration contains an inherent ant i-aliasing feature that tends to suppress migration artifacts without losing bandwidth. The penalty, however, is a decrease in the dynamic r ange of the migrated image compared to an image from a regular array o f geophones. Our limited numerical results suggest that this loss in d ynamic range is acceptable. and so justifies the anti-aliasing benefit s of migrating a random array of data.