Fast Single-Step Least-Squares Reverse-Time Imaging via Adaptive Matching Filters in Beams

Least-squares reverse time migration (LSRTM) is a powerful tool in seeking broadband-wavenumber reflectivity images. It produces better images over reverse-time migration (RTM) at the expense of computational cost. The Hessian effect can be measured in the image domain with the point-spread function (PSF). Here, we try to measure the Hessian effect in the data domain with the so-called trace-spread function (TSF). The difference between PSF and TSF is that the former originates from <inline-formula> <tex-math notation="LaTeX">${{\mathbf{L}}^{T}}{\mathbf{L}}$ </tex-math></inline-formula> in the image domain while the latter from <inline-formula> <tex-math notation="LaTeX">${\mathbf{L}}{{\mathbf{L}}^{T}}$ </tex-math></inline-formula> in the data domain. By comparing the TSFs with their original corresponding traces (or beams), we can design adaptive matching filters for preconditioning to alleviate the Hessian effect. However, the full TSF matrix is expensive. In this article, we propose a multiscale solution, which first has a diagonal approximation to <inline-formula> <tex-math notation="LaTeX">${\mathbf{L}}{{\mathbf{L}}^{T}}$ </tex-math></inline-formula> in beams, and then handle the full submatrix composed of the one-beam traces using the Sherman–Morrison formula. The preconditioned beams are superimposed into a “deblurred” data for remigration. Through synthetic and real data examples, we see that: 1) single-step data-domain LSRTM can yield deblurred RTM images via adaptive matching filters and 2) the beam-by-beam consideration outperforms the trace-by-trace one.