Seismic wave propagated in viscoelastic media always suffers from amplitude attenuation and velocity dispersion. It not only reduces seismic resolution, but also affects the quality of traditional stationary normal moveout (NMO) correction by distorting wavelets. To address both issues of attenuation and the wavelet stretch simultaneously, we present a non-stationary NMO correction method along with prestack high-resolution processing. Instead of invariant wavelets, spatiotemporal variant wavelets are adopted via an inversion scheme for implementing NMO correction and removing the attenuation effect. In the inversion, two goal-oriented constraints including the temporal sparsity and horizontal coherence of prestack reflectivity are combined into a structural sparsity constraint to impose on the data misfit between the synthetic non-stationary common-midpoint (CMP) gather and the observed one. Both constraints approximately represented by a convex l2,1 norm limit the optimal solution to structural sparsest reflectivity, contributing to construct the flattened corrected gather with invariant wavelets. Synthetic and field data examples illustrate performances of the inversion-based non-stationary NMO correction. Moreover, we compared the corrected gathers obtained by the improved approach with those by the traditional and inversion-based stationary NMO correction approaches, and took the gabor spectrum as a quality control. The proposed method produces more flatten events without stretch and gives consistent frequency components, thus improving prestack seismic resolution.