High resolution Kirchhoff seismic migration via 1-norm regularized least-squares

For decades Kirchhoff migration has been one of the simplest migration algorithms and also the most frequently used method of migration in industry. This is due to its relatively low computational cost and its flexibility in handling acquisition and topography irregularities. The standard seismic migration operator can be regarded as the adjoint of a seismic forward modeling operator, which acts on a set of subsurface parameters to generate the observed data. Such adjoint operators are able to provide an approximate inverse of the forward modeling operator and only recover the time of the events (Claerbout, 1992). They cannot retrieve the amplitude of reflections, thus leading to a decrease in the resolution of the final migrated image. The standard seismic migration (adjoint) operators can be modified to better approximate the inverse operators. Least-squares migration (LSM) techniques have been developed to fully inverse the forward modeling procedures by minimizing the difference between observed and modeled data in a least-squares sense. An LSM is able to reduce the (Kirchhoff) migration artifacts, enhance the resolution and retrieve seismic amplitudes. Although implementing LSM instead of conventional migration, leads to resolution enhancement. It also brings some new numerical and computational challenges which need to be addressed properly. Due to the ill-conditioned nature of the inverse operator and also incompleteness of the data, the method generates unavoidable artifacts which severely degrade the resolution of the migrated image obtained by the non-regularized LSM method. The instability of LSM methods suggests developing a regularized algorithm capable of including reasonable physical constraints. Including the seismic wavelet into the migration operator, migration will generate the earth reflectivity image which can be considered as a sparse image, so applying the sparseness constraint, e.g., via the minimization of the 1-norm of reflectivity model, can help to regularize the model and prevent it from getting noisy artifacts (Gholami and Sacchi, 2013).
In this article, based on the Bregmanized operator splitting (BOS), we propose a high resolution migration algorithm by applying sparseness constraints to the solution of least-squares Kirchhoff migration (LSKM). The Bregmanized operator splitting is employed as a solver of the generated sparsity-promoting LSKM for its simplicity, efficiency, stability and fast convergence. Independence of matrix inversion and fast convergence rate are two main properties of the proposed algorithm. Numerical results from field and synthetic seismic data show that migrated sections generated by this 1-norm regularized Kirchhoff migration method are more focused than those generated by the conventional Kirchhoff/LS migration.
Regular spatial sampling of the data at Nyquist rate is another major challenge which may not be achieved in practice due to the coarse source-receiver distributions and presence of possible gaps in the recording lines. The proposed model-based migration algorithm is able to handle the incompleteness issues and is stable in the presence of noise in the data. In this article, we tested the performance of our proposed method on synthetic data in the presence of coarse sampling and also acquisition gaps. The results confirmed that the proposed sparsity-promoting migration is able to generate accurate migrated images from incomplete and inaccurate data.