discussion and conclusion

I have presented a regularized inversion scheme in the SODCIGs to deal with the artifacts caused by insufficient offset coverage. The inversion scheme concentrates the migrated energy at zero-offset locations and removes incoherent and weak noise by constraining the solution with the DSO and sparseness operator. Though I tested my approach only on 2-D data sets, it would be quite easy to extend it to 3-D, since I approximate the Hessian with a diagonal matrix, which reduces the computational expense.

Compared to regularizing in the ADCIGs with a roughening operator acting along the angle axis, regularization in the SODCIGs has the advantage of being computationally cheaper. More importantly, with proper selection of the trade-off parameter $\epsilon$, it can preserve the velocity information correctly when a wrong migration velocity is used. Therefore it may have the potential to update the velocity more accurately, since it can produce much cleaner angle gathers.

The proposed inversion scheme may also be dangerous if we choose inappropriate hyperparameters $\epsilon$ and $\sigma$. Since by adding the sparseness constraint in the image cube, we run the risk of penalizing true reflections that have very weak energy, over-regularization may lead to too-sparse solutions, forfeiting the ability to image weak reflections. For the example of the Marmousi model, we can clearly see that some weak reflections are greatly attenuated.

The deconvolution effects, i.e. the wavelet-squeezing effects, in the above examples are not obvious; this is because of the approximation of the Hessian with a diagonal matrix. The approximated diagonal deconvolution filter is not sufficient to deconvolve the image accurately, especially for the complex Marmousi model. A more accurate but also more expensive way is to compute the full Hessian with wave equations instead of approximating it with a diagonal matrix.