Weighted Lagrange Multiplier Method for Robust Source-Independent Waveform Inversion

The Lagrange multiplier method has proven highly effective for mitigating the ill-conditioning of full waveform inversion (FWI), enabling robust and computationally efficient algorithms that converge to accurate velocity models even from poor initial estimates. Classical multiplier-based FWI methods optimize an augmented Lagrangian (AL) functional with a scalar penalty parameter that uniformly weights wave-equation constraint violations. While this balances data fit and wave-equation satisfaction, it applies uniform relaxation across the model, disregarding source locations and the natural decay of seismic energy. We propose a weighted proximal-point Lagrangian formulation that introduces spatially varying regularization, applying weaker enforcement near sources and progressively stronger enforcement with increasing distance. This compensates for the energy decay, promotes balanced wave-equation enforcement, and improves the convexity of the optimization landscape. The method also eliminates the need for explicit source signature estimation and relaxes the requirement for sources to lie on finite-difference grid points, increasing practical applicability. Enhanced computational efficiency is achieved through our dual-space ADMM implementation, which avoids repeated LU factorizations of the forward operator. Only a few LU factorizations are required, with all subsequent iterations solved via efficient forward-backward substitution, making the approach scalable to large-scale 2D and 3D problems. Numerical experiments on challenging synthetic benchmarks demonstrate that the proposed method broadens the basin of attraction of the AL objective, improves robustness to poor initial models and strong noise, and achieves faster, more stable convergence compared with standard multiplier-based methods.