The FOCI algorithm is a new approach to the design and implementation of explicit wavefield extrapolation for depth migration in the space-frequency domain. Operator instability is addressed by splitting the operator into two parts, one that controls phase accuracy and another that improves stability. The first operator is simply a windowed version of the exact operator for a half-step. The second operator is designed as a bandlimited inverse for the first. The final FOCI operator is formed as the convolution of the first operator with the conjugate of the second. By controlling the degree of evanescent filtering, the resulting operator can easily be stabilized for several thousand steps. During the wavefield extrapolation process, the data are divided into frequency chunks that are optimally resampled in the spatial coordinates to enhance the performance of the extrapolation. Lower frequencies are resampled to a larger sample size. Testing of the algorithm shows that it scales over three orders of magnitude as O(N) with run times comparable to the phase shift method of time migration. Images from trial depth migrations of the Marmousi model show very high resolution.
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