We propose a Kirchhoff-style algorithm that migrates coefficients obtained by wavelet decomposition of seismic traces over time. Wavelet-based prestack multiscale Kirchhoff migration involves four steps: wavelet decomposition of the seismic data, thresholding of the resulting wavelet coefficients, multiscale Kirchhoff migration, and image reconstruction from the multiscale images. The migration procedure applied to each wavelet scale is the same as conventional Kirchhoff migration but operates on wavelet coefficients. Since only the wavelet coefficients are migrated, the cost of wavelet-based migration is reduced compared to that of conventional Kirchhoff migration. Kirchhoff migration of wavelet-decomposed data, followed by wavelet reconstruction, is kinematically equivalent to and yields similar migrated signal shapes and amplitudes as conventional Kirchhoff migration when data at all wavelet scales are included.
The decimation in the conventional discrete pyramid wavelet decomposition introduces a translation-variant phase distortion in the wavelet domain. This phase distortion is overcome by using a stationary wavelet transform rather than the conventional discrete wavelet transform of the data to be migrated.
A wavelet reconstruction operator produces a single composite broadband migrated space-domain image from multiscale images. Multiscale images correspond to responses in different frequency windows, and migrating the data at each scale has a different cost. Migrating some, or only one, of the individual scale data sets considerably reduces the computational cost of the migration. Successful 2D tests are shown for migrations of synthetic data for a point-diffractor model, a multilayer model, and the Marmousi model.