Improved Pseudo-hessian for Frequency-domain Elastic Waveform Inversion

In seismic waveform inversion using the adjoint operator, using the pseudo-Hessian matrix as a pre-conditioner can allow us to achieve computation efficiency. However, the former pseudo-Hessian matrices (i.e., the original and new pseudo-Hessian matrices) have a limitation to simulate the features of the approximate Hessian matrix for the deeper parts of given models, in particular in elastic waveform inversion. To compensate for this limitation, we propose the improved pseudo-Hessian matrix, which is obtained introducing an auxiliary matrix in the original pseudo-Hessian matrix. Comparing the values of the improved pseudo-Hessian matrix with those of the two former pseudo-Hessian matrices indicates that the improved pseudo-Hessian matrix is much closer to the approximate Hessian matrix. Inversion results for the elastic Marmousi-2 model show that the improved pseudo-Hessian matrix increases the convergence rate and yields better results, in particular for the deeper part.