Automatic Penalty Parameter Selection by Residual Whiteness Principle (RWP) and GCV for Full Waveform Inversion

Abstract

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.