True amplitude recovery in reverse time extrapolation of plane and spherical waves

A challenging outstanding problem in reverse time extrapolation is recovering accurate amplitudes at reflectors from the receiver wavefield. Various migrations have been developed to produce accurate image locations rather than correct amplitude information because of inadequate compensation of attenuation, dispersion, and transmission losses. We have evaluated the requirements, and determined the theoretical feasibility, of true amplitude recovery of 2D acoustic and elastic seismic data by using the analytic Zoeppritz equations for plane-wave reflection and transmission coefficients. Then, we used synthetic acoustic and elastic wavefield data generated by elastodynamic finite differences to verify the recovery, in the reverse time propagation, of spherical waves and illustrated the salient differences between the incident wavefields reconstructed from reflection data only and from the combination of reflection and transmission data. These examples quantitatively verify that recovering an incident plane or a spherical wave requires the reverse time propagation of all reflections and transmissions in a model with the correct velocity and density. Accurate reconstruction of an incident wave is not possible by backward propagation of only reflections. As an application, we removed downgoing internal multiple reflections generated by upgoing waves incident at reflectors shallower than a horizontal well, in which geophones are deployed. The subtraction of the downgoing reflection involves wavefield reconstruction at depths shallower than the horizontal well and separation of upgoing and downgoing wavefields. This approach assumes that the correct acoustic (or elastic) velocity and density models are available in, and shallower than, the layer where the horizontal well is located. Incident-wave reconstruction works equally well for smooth models, as for models with sharp boundaries. Uncertainties in the model used for reconstruction, and incompleteness of the data aperture are propagated into the equivalent uncertainties, and incompleteness of the reconstruction.