Pseudo-Hessian matrix for the logarithmic objective function in full waveform inversion

Ha, W.S., Chung, W.K. and Shin, C.S., 2012. Pseudo-Hessian matrix for the logarithmic objective function in full waveform inversion. Journal of Seismic Exploration, 21: 201-214. The pseudo-Hessian matrix of the J, objective function has been used in previous inversion studies because it was observed to yield satisfactory results in both the frequency and Laplace domains. However, there are fundamental differences between the pseudo-Hessian of the logarithmic objective function and that of the J, objective function. In this study, we first derive the pseudo-Hessian matrix for full waveform inversion using the logarithmic objective function. We then compare the two pseudo-Hessian matrices in both Laplace-and frequency-domain waveform inversions using the logarithmic objective function. A Laplace-domain inversion using the pseudo-Hessian of the logarithmic objective function yields a better result than that obtained using the pseudo-Hessian of the /, objective function. On the other hand, frequency domain inversion results using the two pseudo-Hessians yield similar results. The differences originate from the different behaviour of wavefields in the Laplace and frequency domains.