Frequency-Domain Elastic Full Waveform Inversion Using the New Pseudo-Hessian Matrix: Experience of Elastic Marmousi-2 Synthetic Data

A proper scaling method allows us to find better solutions in waveform inversion, and it can also provide better images in true-amplitude migration methods based on a least-squares method. For scaling the gradient of a misfit function, we define a new pseudo-Hessian matrix by combining the conventional pseudo-Hessian matrix with amplitude fields. Because the conventional pseudo-Hessian matrix is assumed to neglect the zero-lag autocorrelation terms of impulse responses in the approximate Hessian matrix of the Gauss–Newton method, it has certain limitations in scaling the gradient of a misfit function relative to the approximate Hessian matrix. To overcome these limitations, we introduce amplitude fields to the conventional pseudo-Hessian matrix, and the new pseudo-Hessian matrix is applied to the frequency-domain elastic full waveform inversion. This waveform inversion algorithm follows the conventional procedures of waveform inversion using the backpropagation algorithm. A conjugate-gradient method is employed to derive an optimized search direction, and a backpropagation algorithm is used to calculate the gradient of the misfit function. The source wavelet is also estimated simultaneously with elastic parameters. The new pseudo-Hessian matrix can be calculated without the extra computational costs required by the conventional pseudo-Hessian matrix, because the amplitude fields can be readily extracted from forward modeling. Synthetic experiments show that the new pseudo-Hessian matrix provides better results than the conventional pseudo-Hessian matrix, and thus, we believe that the new pseudo-Hessian matrix is an alternative to the approximate Hessian matrix of the Gauss–Newton method in waveform inversion.