ALBERT

Description: To enhance the feasibility of seismic full waveform inversion (FWI) for various types of geological structures, the model parameters should be updated along directions such that both long- and short-wavelength structures can be properly resolved. These long- and short-wavelength structures are primarily influenced by the low- and high-frequency components of the gradients, respectively. In some cases, however, the gradients are not flexible to reconstruct both the long- and the short-wavelength structures. This problem can be related to the scaling method using the Hessian matrix and the effect of the source spectrum. In this study, we analyse the problems of conventional scaling methods in frequency-domain FWI and propose a weighting method to compensate for these problems. The weighting method is applied to the conventional elastic FWI, where the gradient is scaled by the diagonal of the pseudo-Hessian matrix inside the frequency loop so that the effect of the source spectrum can be removed through cancellation. The weighting factors are designed using the backpropagated wavefields incited by the deconvolved residuals, which play a role in making the descent directions appropriately reflect the spectral differences between the observed data and the initial (or the inverted) modelling responses. We analyse the characteristics of the Jacobians and residuals and compare the descent directions of the two conventional waveform inversion methods with descent directions of the weighting method for thick rectangular-shaped and thin-layers models. The results indicate that the descent directions computed using the conventional inversion methods do not reflect the characteristics of deconvolved residuals and that particular frequency components are always emphasized regardless of geological models, while the spatial resolution of the descent direction calculated using the weighting method is flexibly determined depending on the differences between the true and the assumed models. Inversion results for the Marmousi-2 model show that the weighting method is not sensitive to the initial guess even though we do not apply the frequency marching strategy. Numerical examples for the SEG/EAGE salt model show that the weighting method can properly recover the high velocities of the salt body and the low velocities below the salt body. Our numerical examples are based on the assumption that low frequencies are available. Further study is needed to apply the weighting method to real field data that are noisy and do not have low-frequency components.