Many existing migration schemes cannot simultaneously handle the two most important problems of migration: imaging of steep dips and imaging in media with arbitrary velocity variations in all directions. For example, phase-shift (ω, k) migration is accurate for nearly all dips but it is limited to very simple velocity functions. On the other hand, finite-difference schemes based on one-way wave equations consider arbitrary velocity functions but they attenuate steeply dipping events.

We propose a new hybrid migration method, named “Fourier finite-difference migration,” wherein the downward-continuation operator is split into two downward-continuation operators: one operator is a phase-shift operator for a chosen constant background velocity, and the other operator is an optimized finite-difference operator for the varying component of the velocity function. If there is no variation of velocity, then only a phase-shift operator will be applied auto matically. On the other hand, if there is a strong variation of velocity, then the phase-shift component is suppressed and the optimized finite-difference operator will be fully applied.

The cascaded application of phase-shift and finite-difference operators shows a better maximum dip-angle behavior than the split-step Fourier migration operator. Depending on the macro velocity model, the Fourier finite-difference migration even shows an improved performance compared to conventional finite-difference migration with one downward-continuation step. Finite-difference migration with two downward-continuation steps is required to reach the same migration performance, but this is achieved with about 20 percent higher computation costs. The new cascaded operator of the Fourier finite-difference migration can be applied to arbitrary velocity functions and allows an accurate migration of steeply dipping reflectors in a complex macro velocity model. The dip limitation of the cascaded operator depends on the variation of the velocity field and, hence, is velocity-adaptive.

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