The GPSPI algorithm uses a phase shift that is a high-frequency approximation to the exact solution: the square-root Helmholtz operator symbol. The square-root Helmholtz operator symbol dramatically changes character between high- and low-frequency limits, while the GPSPI phase shift does not. A frequency-dependent smoothing of the velocity model (and therefore the phase shift) is used to approximate the character change of the square-root Helmholtz symbol, and is implemented via the spatial resampling within the FOCI algorithm. This results in heavy smoothing of the phase shift at low frequencies, and virtually no smoothing at higher frequencies. The frequency-dependent smoothing results in an image of much higher quality than the image generated using the usual GPSPI phase shift.
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