Full-waveform inversion (FWI) is a powerful seismic imaging technique used to estimate high-resolution physical properties of subsurface structures by minimizing the misfit between observed and modeled seismic data. FWI is inherently a highly non-linear and ill-posed inverse problem. Extended-source approaches, such as the augmented Lagrangian (AL) method, are employed to improve solution convexity and robustness. A key component of this formulation is the penalty parameter, which controls the trade-off between data fitting and satisfaction of the wave-equation constraint, strongly influencing convergence in the presence of noise. The main challenge lies in selecting the penalty parameter. Traditional strategies such as the Discrepancy Principle (DP) require an accurate estimate of the noise level, which is often unknown or poorly characterized. Moreover, trial-and-error tuning requires repeatedly solving the inverse problem, making it computationally expensive. To overcome these limitations and develop a parameter-free, computationally efficient extended-source FWI algorithm, we integrate two data-driven parameter-selection strategies--the Residual Whiteness Principle (RWP) and a stable variant of Generalized Cross-Validation (RGCV)--within a multiplier-oriented AL framework. Specifically, we adopt a dual-space AL formulation, which allows the background wave-equation operator to remain fixed and requires only a single LU factorization per frequency, significantly improving efficiency. This design enables dynamic adjustment of the parameter at negligible cost during iterations, making the algorithm scalable for large-scale applications. Numerical experiments on acoustic and elastic FWI with white and colored noise show that, combined with the dual-space formulation, RWP provides strong noise robustness, resulting in a reliable automated solution for large-scale seismic inversion.
Full-waveform inversion (FWI) is a powerful seismic imaging technique used to reconstruct subsurface physical properties (Tarantola 1984;Pratt et al. 1998;Operto et al. 2023;Métivier et al. 2025), with broad applications in wave-based imaging (e.g., Thrastarson et al. 2022;Fachtony et al. 2025).
FWI operates by iteratively minimizing the discrepancy (misfit) between simulated and observed seismic data. However, FWI is generally viewed as a highly non-linear and ill-posed inverse problem (e.g., Mulder & Plessix 2008;Kirsch & Rieder 2014), necessitating the use of regularization techniques to stabilize the inversion and obtain meaningful solutions. Extended-source approaches, such as the augmented Lagrangian (AL) method (Bertsekas 1982), offer a robust means of improving the problem’s convexity (Gholami & Aghazade 2024). A key component of this formulation is the penalty parameter (µ), which governs the critical trade-off between minimizing the data misfit (residual) and satisfying the wave equation constraint. The specific choice of this parameter significantly influences conver-gence and practical performance, particularly in the presence of noise (van Leeuwen & Herrmann 2013;Huang et al. 2018;Aghamiry et al. 2019;Gholami et al. 2022;Lin et al. 2023).
The challenge lies in efficiently and reliably selecting the optimal µ. Traditionally, µ is determined empirically, often set as a fraction of the maximum eigenvalue of the augmented wave-equation operator (e.g., van Leeuwen & Herrmann 2016;Aghamiry et al. 2019;Operto et al. 2023). Alternatively, the Discrepancy Principle (DP) is used, which selects µ such that the norm of the data residual matches the estimated noise level (Fu & Symes 2017;Gholami & Aghazade 2024;Symes et al. 2025). The DP approach is severely limited because it relies on an accurate estimate of the noise standard deviation (σ), which is often unknown or poorly characterized in practical applications. Furthermore, the DP considers only the zero-lag value of the residual autocorrelation, thereby neglecting potentially informative correlations at nonzero lags. Consequently, the conventional trial-and-error tuning or methods reliant on accurate σ estimates become computationally prohibitive. Therefore, developing efficient, data-driven strategies for penalty parameter estimation is essential for computationally demanding FWI problems.
To address these limitations, we propose employing the Residual Whiteness Principle (RWP) (Almeida & Figueiredo 2013;Lanza et al. 2013) and the Robust Generalized Cross-Validation (RGCV) (Lukas 2006). The primary contribution of this work is the integration of these two data-driven parameter selection strategies within the highly efficient multiplier-oriented formulation of the AL method (Gholami & Aghazade 2024), specifically utilizing the dual-space AL (Dual-AL) method (Aghazade & Gholami 2025a). This choice of framework is essential due to its superior computational efficiency.
The Dual-AL method is rooted in the perspective that the Lagrange multipliers are the fundamental unknowns, allowing the background wave equation operator, A(m 0 ), to remain fixed for each frequency inversion. This design only requires a single LU matrix factorization per frequency inversion.
In contrast, standard primal algorithms necessitate repeated factorizations at every iteration. This crucial efficiency gain enables the dynamic and efficient update of the penalty parameter µ within each iteration at negligible computational cost, thereby making the resulting algorithm scalable and practical for large-scale applications.
RWP is an automatic, parameter-free strategy that overcomes DP’s limitations by evaluating the full autocorrelation function (Pragliola et al. 2023). It selects the optimal µ by minimizing a measure of non-whiteness of the residual data. This measure is based on kurtosis minimization, forcing the residual to resemble white noise, which has a maximally sparse autocorrelation function. We also adopt RGCV, a stable extension of GCV, which addresses the instability of standard GCV for smaller datasets that often leads to under-regularization. Extensive numerical experiments on both acoustic and elastic FWI demonstrate that the combination of RWP/RGCV with the Dual-AL formulation exhibits exceptional noise robustness and computational efficiency. Specifically, RWP consistently achieves the closest approximation to the optimal parameter choice among the tested methods, making it a robust, automated solution for large-scale seismic inversion.
This section establishes the mathematical context for solving ill-posed inverse problems and reviews the specific FWI formulations and parameter selection principles employed in this work.
We consider the problem of estimating an unknown model m from noisy observations d, typically formulated as a discrete ill-posed inverse problem. FWI is a nonlinear and ill-posed problem that requires regularization to stabilize the solutio
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