Multiparameter elastic imaging improved by preconditioning with an incomplete inverse hessian approximation

The result of multiparameter isotropic elastic linearized inversion carries information about the reflectivity of each of the three elastic parameters, but these reflectivities are cross-coupled, i.e., a perturbation in one of the elastic parameters appear as a cross talk in the reflectivity image of the other. We propose an efficient way to decouple the reflectivity images of the three elastic parameters reliably within a multiparameter frequency-domain linearized approach. This is achieved by computing selected elements of the Hessian matrix and subsequent construction of its approximate sparse inverse rather than the inverse of its sparse approximation. The resulting preconditioning matrix can be treated either as an improved imaging condition for multiparameter elastic reverse-time migration or as an efficient preconditioner for the conjugate gradient method. The numerical study shows the potential of the proposed preconditioning method and provides an insight into the best attainable resolution and quality of elastic reflectivity images in the context of linearized inversion.