Multiresolution modeling and seismic wavefield reconstruction in attenuating media

Abstract The propagation of the seismic wavefield through a viscoelastic medium is a multiresolution process, in which depth of propagation and scale are closely associated with one another. We propose the multiresolution wavefield reconstruction (MRWR) as a means to directly integrate such concepts of scale into the backpropagation component of an imaging method. MRWR produces a reconstruction of the wavefield (at some fixed depth d), in which each scale term reconstitutes the resolution that was lost as the wavefield propagated some step-length towards the measurement surface, away from d. Concurrently, MRWR provides a stable platform for this removal of the effects of absorption in propagation. In a multiresolution model of propagation, the viscoelastic propagation kernel is seen, mathematically, to fill the role of the scale function in multiresolution theory, as it operates on a wavefield to propagate it through some distance. The suppression of high-resolution components of the wavefield via this scaling/propagation kernel function is readily illustrated with a simple numerical example. We use this scale-based view of propagation and the differential inversion method to provide a physical and mathematical rationale for MRWR. Two cases of the MRWR formula are derived, and applications in one dimension and two dimensions, for synthetic data, and one field data example, are presented to demonstrate its use.