With correct migration velocity and infinite survey length, wave-equation migration will focus the energy perfectly at the zero-offset locations. In reality, however, since the survey length can never be infinite, the offset domain suffers from the truncation effect. The situation gets even worse when the data sets are incomplete or irregular; severe amplitude smearing and aliasing artifacts may appear in the offset gathers and migrated images. This problem is more pronounced in 3-D, because of the irregular nature of 3-D seismic data.
One way to deal with this problem is to interpolate before migration, as is done in Radon-based interpolation schemes. Though Radon-based methods are acknowledged to be effective for data interpolation, they have severe theoretical limitations that require the events in CMP gathers to have the shape of hyperbolas or parabolas. These limitations prevent them from accurately interpolating in situations with very complex velocity structures.
Another approach is to pose the migration problem as a regularized inversion process. A reasonable regularization term, assuming lateral continuity along the reflection-angle axis, would be to smooth across offset-ray parameters in the Angle-Domain Common Image Gathers (ADCIGs). As shown in Prucha et al. (2000) and Kuehl and Sacchi (2001), by smoothing along the angle gathers, the illumination gaps can be successfully filled in and the migration artifacts caused by the insufficient survey length and lack of illumination can be attenuated to some extent.
In this paper, I describe another method based on least-squares migration with regularization in the Subsurface-Offset-Domain Common Image Gathers (SODCIGs). Regularizing in the SODCIGs instead of in the ADCIGs has the advantage of being computationally cheaper, since it saves the computational cost of transforming from SODCIGs into ADCIGs. The basic idea is to add a Differential Semblence Operator (DSO) in the SODCIGs along the offset dimension to penalize energy far from zero-offset locations Shen et al. (2003); Valenciano (2006), followed by applying a sparseness constraint which minimizes the model residuals in the L1 norm or Cauchy norm to enhance the resolution of the image cube.
I start with reviewing the theory of Bayes inversion and derive the weighting function for the sparseness constraint, then derive the objective function with DSO regularization and sparseness constraints based on the propagation of wavefields. I demonstrate that regularizing with DSO in the SODCIGs is equivalent to regularizing with a roughener along the offset-ray axes in the ADCIGs. To further reduce the computational cost, I approximate the Hessian with a diagonal matrix, which eliminates the need to propagate wavefields upward and downward within each iteration; however, the trade-off is a loss of accuracy. My approximated inversion scheme is tested on a simple two-layer model as well as the complex Marmousi model.