The time domain approach

The sparse solver in the time domain employs a regularization term that enforces spikyness in the model space Nichols (1994). This approach is well-suited for time-domain processing and makes use of iterative solvers as opposed to direct inversion as described in equations (14) and (15). For instance, by choosing an approximation of the $\ell^1$ norm for the model space regularization, we can focus the energy of the model vector ${\bf m}$ into its main components. Ulrych et al. (2000) advocate a regularization by the Cauchy-Gauss model. In any case, the objective function to minimize becomes

$$f({\bf m}) = \Vert{\bf Lm-d} \Vert^2+\epsilon\vert{\bf m}\vert _{sparse},$$ (17)

where $\Vert.\Vert$ is the $\ell^2$ norm and where |.|_sparse induces a sparse model. Iteratively reweighted least-squares algorithms with the proper weighting function produce an artifact-free model space Bube and Langan (1997). A more ambitious Huber norm can be used as well Guitton and Symes (1999) for the $\ell^1$ case.