Blocky regularization schemes for Full-Waveform Inversion

With ill-posed inverse problems such as Full-Waveform Inversion, regularization schemes are needed to constrain the solution. Whereas many regularization schemes end up smoothing the model, an undesirable effect with FWI where high-resolution maps are sought, blocky regularization does not: it identifies and preserves strong velocity contrasts leading to step-like functions. These models might be needed for imaging with wave-equation based techniques such as Reverse Time Migration or for reservoir characterization. Enforcing blockiness in the model space amounts to enforcing a sparse representation of discontinuities in the model. Sparseness can be obtained using the l1 norm or Cauchy function which are related to long-tailed probability density functions. Detecting these discontinuities with vertical and horizontal gradient operators helps constraining the model in both directions. Blocky regularization can also help recovering higher wavenumbers that the data used for inversion would allow, thus helping controlling the cost of FWI. While the Cauchy function yields blockier models, both l1 and Cauchy attenuate illumination and inversion artifacts.