Weighted pseudo-Hessian for frequency-domain elastic full waveform inversion

In full waveform inversion, an appropriate preconditioner for scaling the gradient improves the inversion results and accelerates the convergence speed. The Newton-based method uses the Hessian to scale the gradient and ensures a correctly inverted image with good convergence. However, calculating the full Hessian or approximated Hessian and obtaining their inverse matrices require extensive computation, which has remained an obstacle to the application of Newton-based methods to full waveform inversion. Many attempts have been made to reduce the computational cost of obtaining the Hessian and its inverse; among these alternatives, the pseudo-Hessian method has been widely used. The use of the pseudo-Hessian reduces the computational cost by regarding the zero-lag correlation of the impulse response as a unit matrix, but the pseudo-Hessian cannot properly scale the deep portion of a model. Therefore, we proposed a weighted pseudo-Hessian which can overcome the limitations of the conventional pseudo-Hessian. The weighted pseudo-Hessian was generated by combining a weighting matrix with the conventional pseudo-Hessian. The weighting matrix is an amplitude field and helps the pseudo-Hessian to scale the gradient properly. Therefore, the weighted pseudo-Hessian can effectively scale the gradient from the shallow part to the deep part of the model with great balance. Calculating the weighting function required minimal computation, such that the computational cost of generating the weighted pseudo-Hessian was nearly the same as the computational cost needed to calculate the conventional pseudo-Hessian. To verify the proposed algorithm, Marmousi-2 data were used for the synthetic test. The results indicate that the weighted pseudo-Hessian can effectively scale the gradient from the shallow to deep portions of a model.