Stable and efficient Q-compensated least-squares migration with compressive sensing, sparsity-promoting, and preconditioning

The anelastic effects of subsurface media decrease the amplitude and distort the phase of propagating wave. These effects, also referred to as the earth's Q filtering effects, diminish seismic resolution. Ignoring anelastic effects during seismic imaging process generates an image with reduced amplitude and incorrect position of reflectors, especially for highly absorptive media. The numerical instability and the expensive computational cost are major concerns when compensating for anelastic effects during migration. We propose a stable and efficient Q-compensated imaging methodology with compressive sensing, sparsity-promoting, and preconditioning. The stability is achieved by using the Born operator for forward modeling and the adjoint operator for back propagating the residual wavefields. Constructing the attenuation-compensated operators by reversing the sign of attenuation operator is avoided. The method proposed is always stable. To reduce the computational cost that is proportional to the number of wave-equation to be solved (thereby the number of frequencies, source experiments, and iterations), we first subsample over both frequencies and source experiments. We mitigate the artifacts caused by the dimensionality reduction via promoting sparsity of the imaging solutions. We further employ depth- and Q-preconditioning operators to accelerate the convergence rate of iterative migration. We adopt a relatively simple linearized Bregman method to solve the sparsity-promoting imaging problem. Singular value decomposition analysis of the forward operator reveals that attenuation increases the condition number of migration operator, making the imaging problem more ill-conditioned. The visco-acoustic imaging problem converges slower than the acoustic case. The stronger the attenuation, the slower the convergence rate. The preconditioning strategy evidently decreases the condition number of migration operator, which makes the imaging problem less ill-conditioned and significantly expedites the convergence rate. Numerical experiments verify the stability, benefits and robustness of the method proposed and support our analysis and findings.