Decoupling of elastic parameters with iterative linearized inversion

Three model parameters as a function of position describe wave propagation in an isotropic elastic medium. Ideally, imaging of data for a point scatterer that consists of a perturbation in one of the elastic parameters should only provide a reconstruction of that perturbation, without cross-talk into the other parameters. This is not the case for seismic migration, where a perturbation of one elastic parameter contributes to the images of all three model parameters. For a reliable reconstruction of the true elastic reflectivity, one can apply iterative migration or linearized inversion, where the misfit cost function is minimized by the conjugate-gradient method. We investigated the decoupling of the three isotropic elastic medium parameters with the iterative linearized approach. Instead of iterating, the final result can be obtained directly by means of Newton's method, using the pseudo-inverse of the Hessian matrix. Although the calculation of the Hessian for a realistic model is an extremely resource-intensive problem, it is feasible for the simple case of a point scatterer in a homogeneous medium, for which we present numerical results. We consider the iterative approach with the conjugate-gradient method and Newton's method with the complete Hessian. Experiments show that in both cases the elastic parameters are decoupled much better when compared to migration. The iterative approach achieves acceptable inversion results but requires a large number of iterations. For faster convergence, preconditioning is required. An optimal preconditioner, if found, can be used in other iterative methods including L-BFGS. We considered two well known types of preconditioners, based on diagonal and on block-diagonal Hessian approximations. Somewhat to our surprise, both preconditioners fail to improve the convergence rate. Hence, a more sophisticated preconditioning is required.