The sparseness promoted by the total variation norm is utilized to achieve superresolution amplitude-variation-with-offset (AVO) inversion. The total variation norm promotes solutions that have constant values within unspecified regions and thus are well suited for an earth model consisting of layers bounded by faults and erosion surfaces. Algorithmic developments from digital image and video restoration are utilized to solve the geophysical problem. A spatial point spread function is used to model the resulting effect of wave propagation, migration, and processing. The methodology is compared to current alternatives and discussed in the context of AVO inversion. Good results are obtained in a Barents Sea test case.