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

TL;DR

The paper tackles penalty-parameter selection in full-waveform inversion by integrating Residual Whiteness Principle (RWP) and Robust Generalized Cross-Validation (RGCV) within a dual-space Augmented Lagrangian framework. This combination enables automatic, data-driven updates of the penalty parameter $\mu$ with negligible per-iteration cost, thanks to a fixed background operator and a single LU factorization per frequency. Across acoustic and elastic FWI experiments with white and colored noise, RWP demonstrates strong noise robustness and near-optimal parameter choices, while Dual-AL delivers substantial computational speedups over traditional reduced/penalty formulations. The work offers a practical, scalable solution for large-scale seismic inversion, reducing manual tuning and improving reconstruction quality under realistic noise conditions.

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.

Figures (27)

• Figure 1: Examples of signals in the time domain (left) and their magnitude spectra in the frequency domain (right). The figure demonstrates the inverse relationship: signals transitioning from a sharp spike to a flat shape in the time domain (decreasing kurtosis) correspond to spectra evolving from flat to a spike in the frequency domain (increasing kurtosis).
• Figure 2: Denoising Example: (a) Noisy Morlet wavelet. (b) Denoised wavelets using first-order Tikhonov regularization comparing DP, RGCV, and RWP methods. (c) RME, DP \ref{['mu_dp']}, RGCV \ref{['phi_rgcv']}, and RWP \ref{['phi_rwp']} curves versus regularization parameter $\mu$; vertical dashed lines indicate the minimum location of each curve. (d–f) Autocorrelation functions of the predicted noise using (d) DP, (e) RGCV and (f) RWP.
• Figure 3: Acoustic FWI setup. (a) True 2004 BP velocity model. (b) Initial 1D velocity model used to start the inversion.
• Figure 4: Noise-free acoustic inversion. Multiscale inversion results comparing the Reduced (a), Penalty (b), and Dual-AL (c) FWI methods after each frequency cycle, using the DP for penalty parameter selection.
• Figure 5: Same as figure \ref{['fig:BP_DP_noise_free']} but using the RGCV parameter selection.