Weighting method for the l₁-norm frequency-domain elastic full waveform inversion

To make the l₁-norm frequency-domain elastic full waveform inversion (FWI) less sensitive to initial guesses, we propose the weighting method that incorporates weighting functions to gradients at each frequency. The weighting function should be designed so that the final gradient can properly describe the differences between assumed (initial or inverted) and true models. In the l₂norm FWI, the backpropagated wavefields incited by source wavelet-deconvolved residuals, which can be easily obtained during the inversion process, can be used for the weighting function. However, in the l₁-norm FWI, since the residuals normalized by their magnitudes are backpropagated to compute gradients, the normalized residuals do not reflect the differences between assumed and true models. For this reason, we approximate the residuals of the l₂-norm objective function to define the weighting function for the l₁-norm FWI. We demonstrate the weighting function for synthetic data with outliers obtained for the 2D section of the SEG/EAGE salt model. Numerical examples show that while the conventional l₁-norm waveform inversion does not provide stable solutions for the salt model without good initial guesses, the weighting method gives inversion results comparable to true models even with the poorly estimated initial guesses.