A frequency-approximated approach to Kirchhoff migration

Abstract

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, this process is commonly formulated as a time-domain operation 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 frequency domain, enabling robust antialias filtering through appropriate dip limiting of each frequency and implicit accommodation of true dispersion. Full frequency decomposition of the input seismogram can be approximated by band-pass filtering (or correlation with band-limited source sweeps for Chirp/Vibroseisdata) 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 improves the preservation of higher frequencies and cancellation of steeply dipping aliased energy over traditional time-domain antialiasing methods. If, however, two bands per octave cease to be a robust approach, our frequency-approximated approach provides the processor with ultimate control over the frequency decimation, 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-dependent antialias filtering parameters.