Discussion

In this work, the GMRES method is used to invert the non-Hermitian matrices. Similar to the CG method, the full GMRES method (with no restarts) converges in no more than $n$ steps where $n$ is the total size of the model. However, the GMRES method requires additional memory to store the previous stepping directions. In the numerical examples presented above, because the model space is small enough, no restarts are needed. However, for large 3D models, restarts might be required for a large number of iterations, which could compromise global optimality. Fortunately, the proposed method, as well as other types of preconditioners, is designed to achieve a satisfying result within only a small number of iterations. In practical applications where each iteration consumes large computing resources, only a small number of iterations is usually afforded.
The goal of preconditioning LSRTM in viscoacoustic media using $Q$-RTM is to alleviate the computational burden on iterative inversion by compensating for attenuation explicitly in wave propagation. Therefore, the iterations can be spent on removing migration artifacts and compensating irregularities in subsurface illumination caused by other reasons, such as acquisition footprint. Due to attenuation, the events of the reflectors beneath the attenuating zone yield a smaller amplitude compared with un-attenuated events, and approximately correspond to smaller eigenvalues of the forward operator (Blanch and Symes, 1994). Inversion routines based on Krylov subspace methods, such as CG and GMRES, will favor large eigenvalues, which approximately correspond to shallower and un-attenuated reflectors. This leads to the observed behavior of LSRTM without Q-compensation, which first focused on improving shallow reflectors above the attenuation zone. Blanch and Symes (1994) suggested a simple way of assigning more weights to deeper reflectors, by post-conditioning the seismic record with an increasing function of time. The proposed method is similar in spirit but more accurate, in that $Q$-compensation removes the true effect of attenuation in the gradient by accurately compensating for attenuation along the entire wave path.