Choice of regularization weight in basis pursuit reflectivity inversion

Seismic inverse problem of estimating P- and S-wave reflectivity from seismic traces has recently been revisited using a basis pursuit denoising inversion (BPI) approach. The BPI uses a wedge dictionary to define model constraints, which has been successful in resolving thin beds. Here we address two fundamental problems associated with BPI, namely, the uniqueness of the estimate and the choice of regularization weight λ to be used in the model norm. We investigated these using very fast simulated re-annealing (VFSR) and gradient projection sparse reconstruction (GPSR) approaches. For a synthetic model with two reflectors separated by one time sample, we are able to demonstrate convergence of VFSR to the true model with different random starting models. Two numerical approaches to estimating the regularization weight were investigated. One uses λ as a hyper-parameter and the other uses this as a temperature-like annealing parameter. In both cases, we were able to obtain λ fairly rapidly. Finally, an analytic formula for λ that is iteration adaptive was also implemented. Successful applications of our approach to synthetic and field data demonstrate validity and robustness.