Cascaded f-k migration; removing the restrictions on depth-varying velocity

Stolt's frequency-wavenumber (f-k) method is computationally efficient and has unlimited dip accuracy for constant-velocity media. Although the f-k method can handle moderate vertical velocity variations, errors become unacceptable for steep dips when such variations are large. This paper describes an extension to the f-k method that removes its restrictions on vertical velocity variation, yielding accuracy comparable to phase-shift migration at only a fraction of the computational time.This extension of the f-k method is based on partitioning the velocity field, just as in cascaded finite-difference migration, and performing a number of stages of f-k migration. In each stage, the migration-velocity field is closer to a constant--the ideal situation for the f-k migration method--than when the migration is done conventionally (i.e., in just one stage). Empirical results and error analyses show that, at most, four stages of the cascaded f-k algorithm are sufficient to migrate steep events as accurately as by the phase-shift method for virtually any vertically inhomogeneous velocity field. Given its accuracy and efficiency, cascaded f-k migration can become the method of choice for 2-D, two-pass 3-D, and single-pass 3-D time migrations.