A frequency-approximated approach to Kirchhoff migration

The integral solution of the wave equation has long been one of the most popular methods for imaging Kirchhoff migration and inverting Kirchhoff inversion seismic data. For efficiency, thisprocessiscommonlyformulatedasatime-domainoperation on each trace, applying antialiasing through high-cut filtering of the operator or pre-/postmigration dip filtering. Migration in the time domain, however, does not allow for velocity dispersion; standard antialiasing methods assume a flat reflector and tend to overfilter the data. We have recast the Kirchhoff integral in the frequencydomain,enablingrobustantialiasfilteringthroughappropriate dip limiting of each frequency and implicit accommodationoftruedispersion.Fullfrequencydecompositionoftheinput seismogram can be approximated by band-pass filtering or correlationwithband-limitedsourcesweepsforChirp/Vibroseis data into a few narrow-band traces that cumulatively retain the full source bandwidth. From prior knowledge of the source waveform, we have defined suitable bandwidths to describe broadband 3.0 octaves data using just six frequency bands. Kirchhoff migration of these narrow-band traces using coefficients determined at their central frequencies significantly improvesthepreservationofhigherfrequenciesandcancellationof steeplydippingaliasedenergyovertraditionaltime-domainantialiasingmethods.If,however,twobandsperoctaveceasetobea robust approach, our frequency-approximated approach providestheprocessorwithultimatecontroloverthefrequencydecimation, balancing increased resolution afforded by more bands against computing cost, whereas the number of frequency bands is few enough to permit detailed control over frequency-dependentantialiasfilteringparameters.

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