On efficient frequency-domain full-waveform inversion with extended search space

Abstract

Efficient frequency-domain full-waveform inversion (FD-FWI) of wide-aperture data is designed by limiting inversion to a few frequencies and by solving the Helmholtz equation with a direct solver to process multiple sources efficiently. Some variants of FD-FWI, which process the wave equation as a weak constraint, have been proposed to increase the computational efficiency or extend the search space. Among them, the contrast-source reconstruction inversion (CSRI) reparameterizes FD-FWI in terms of contrast sources (CS) and contrasts and updates them in an alternating mode. This reparameterization allows for one lower-upper (LU) decomposition of the Helmholtz operator to be performed per frequency inversion hence further improving the computational efficiency of FD-FWI. However, iteratively refined wavefield reconstruction inversion (IR-WRI) relies on the alternating-direction method of multipliers to extend the search space by matching the data from the early iterations via an aggressive relaxation of the wave equation while satisfying it at the convergence point thanks to the defect correction performed by the Lagrange multipliers. In contrast to CSRI, IR-WRI requires redoing one LU decomposition when the subsurface model is updated. In both methods, the CSs or the wavefields are computed by solving in a least-squares sense an overdetermined linear system gathering an observation equation and a wave equation. A drawback of CSRI is that CSs are estimated approximately with one iteration of a conjugate gradient method, whereas the wavefields are reconstructed exactly by IR-WRI with a Gauss-Newton method. We have combined the benefits of CSRI and IR-WRI to decrease the number of LU decomposition during IR-WRI with a fixed-point (FP) algorithm while preserving its search space extension capability. Application on the 2D complex Marmousi and the BP salt models shows that our FP-based IR-WRI manages to reconstruct these models as accurately as the classic IR-WRI while reducing the number of LU factorizations considerably. A theoretical complexity analysis and a recent application of 3D FD-FWI based upon direct solver suggest that the FP algorithm should reduce the cost of IR-WRI by a factor of approximately 2 and 10 for 3D dense ocean bottom cable and sparse ocean bottom node acquisitions, respectively.