An Efficient Dual-Space Formulation for Robust Crustal-Scale Full-Waveform Inversion

Full-waveform inversion (FWI) has emerged as a powerful tool for high-resolution seismic imaging. However, its application to crustal-scale and long-offset problems remains severely challenged by strong nonlinearity arising from long propagation paths, pronounced velocity contrasts, and the lack of sufficiently low-frequency data. These factors exacerbate cycle skipping and often cause gradient-based optimization methods to stagnate. In practice, crustal-scale FWI is predominantly performed within the standard reduced-space time-domain formulation, largely due to its favorable memory requirements and the efficiency of time-stepping schemes for solving large-scale wave equations. Nevertheless, this memory efficiency comes at the cost of increased ill-conditioning of the inverse problem, which is difficult to address adequately within the reduced-space framework.

Extended-space formulations based on Lagrange multiplier methods have proven effective in alleviating ill-conditioning and mitigating cycle skipping in FWI. However, time-domain implementations of these multiplier-based approaches for large-scale crustal imaging can be computationally demanding, primarily due to the cost associated with constructing and inverting the data-space Hessian. Recent developments employing Fourier-domain block-diagonal approximations and direct inversion strategies have improved the tractability of time-domain extended FWI. Despite these advances, the approach remains computationally intensive for realistic crustal-scale applications, as the Hessian must typically be recomputed at each iteration.

In this work, we introduce a dual-space formulation that recasts the inversion in the data space to estimation data-side Lagrange multipliers, or dual variables. These variables encode the multiple-scattering components of the data that are neglected in the conventional first-order Born approximation. Unlike standard FWI approaches, which iteratively update the velocity model to reduce data misfit, the proposed method focuses on estimating the dual variables responsible for the mismatch while keeping the background model fixed. Once these dual variables are estimated, the data are matched and the inverse problem is effectively solved. A key advantage of this formulation is that the wave-equation operators and the associated Hessian remain fixed throughout the inversion and therefore need to be constructed only once prior to the iterations. As a result, each iteration requires only two wave-equation solves. Moreover, the use of the exact Hessian eliminates the need for step-length tuning and leads to more stable and accurate updates.

Numerical experiments on large-scale acoustic models demonstrate that the proposed method achieves rapid convergence, enhanced robustness against cycle skipping, and computational efficiency suitable for crustal-scale time-domain seismic imaging.