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Conventional amplitude inversion assumes that the migrated image preserves relative-amplitude information. However, illumination effects caused by complex geologic settings, undersampled acquisition geometry, and limited recording aperture pose a challenge to even the most advanced imaging algorithms. In addition, standard depth-migration images can suffer from lack of resolution caused by wavelet stretch, attenuation, and suboptimal deghosting. Least-squares migration (LSM) can mitigate many of these problems and produce better resolved migration images suitable for AVO inversion. However, whether formulated in the data domain or the image domain, LSM is an inversion algorithm and is sensitive to inaccuracies in the source wavelet, velocity model, data preprocessing, and the propagator used. Practical considerations to mitigate these problems under nonideal conditions and cost-reduction strategies differ between the data- and image-domain formulations. The relative merits of each approach are evaluated by using example inversions for complex synthetic models, including free-surface ghost and attenuation effects. When a data-domain implementation of LSM is considered necessary, the image-domain implementation should be considered at the same time, especially when targeting localized reservoir targets under complex overburdens. Application of image-domain least-squares migration on a Gulf of Mexico field data set produces significant improvements in resolution and event continuity in the subsalt target region.

Because of the conversion of elastic energy into heat, seismic waves are attenuated and dispersed as they propagate. The attenuation effects can reduce the resolution of velocity models obtained from waveform inversion or even cause the inversion to produce incorrect results. Using a viscoacoustic model consisting of a single standard linear solid, we discovered a theoretical framework of viscoacoustic waveform inversion in the time domain for velocity estimation. We derived and found the viscoacoustic wave equations for forward modeling and their adjoint to compensate for the attenuation effects in viscoacoustic waveform inversion. The wave equations were numerically solved by high-order finite-difference methods on centered grids to extrapolate seismic wavefields. The finite-difference methods were implemented satisfying stability conditions, which are also presented. Numerical examples proved that the forward viscoacoustic wave equation can simulate attenuative behaviors very well in amplitude attenuation and phase dispersion. We tested acoustic and viscoacoustic waveform inversions with a modified Marmousi model and a 3D field data set from the deep-water Gulf of Mexico for comparison. The tests with the modified Marmousi model illustrated that the seismic attenuation can have large effects on waveform inversion and that choosing the most suitable inversion method was important to obtain the best inversion results for a specific seismic data volume. The tests with the field data set indicated that the inverted velocity models determined from the acoustic and viscoacoustic inversions were helpful to improve images and offset gathers obtained from migration. Compared to the acoustic inversion, viscoacoustic inversion is a realistic approach for real earth materials because the attenuation effects are compensated.

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