Seismic absolute acoustic impedance inversion with L1 norm reflectivity constraint and combined first- and second-order total variation regularizations

Absolute acoustic impedance (AI) is generally divided into background AI and relative AI for linear inversion. In practice, the intermediate frequency components of the AI model are generally poorly reconstructed, so the estimated AI will suffer from an error caused by the frequency gap. To remedy this error, a priori information should be incorporated to narrow down the gap. With the knowledge that underground reflectivity was sparse, we solved an L1 norm constrained problem to extend the bandwidth of the reflectivity section, and an absolute AI model was then estimated with broadband reflectivity section and given background AI. Conventionally, the AI model is regularized with the total variation (TV) norm because of its blocky feature. However, the first-order TV norm that leads to piecewise-constant solutions will cause staircase errors in slanted and smooth regions in the inverted AI model. To better restore the smooth variation while preserving the sharp geological structure of the AI model, we introduced a second-order extension of the first-order TV norm and inverted the absolute AI model with combined first- and second-order TV regularizations. The algorithm used to solve the optimization problem with the combined TV constraints was derived based on split-Bregman iterations. Numerical experiments that were tested on the Marmousi AI model and 2D stacked field data illustrated the effectiveness of the sparse constraint with respect to shrinking the frequency gaps and proved that the proposed combined TV norms had better performances than those with conventional first-order TV norms.