Optimal Space-Variant Anisotropic Tikhonov Regularization for Full Waveform Inversion of Sparse Data

TL;DR

The paper tackles the ill-posed nature of full waveform inversion under sparse data by introducing an optimal space-variant anisotropic Tikhonov regularization, where local tilt angles $[\theta]_i$ and anisotropy weights $[\sigma]_i$ are treated as inverse variables and updated jointly with the model $m$. The regularizer $\mathcal{R}(m,\theta,\sigma)=\frac{1}{2}\sum_{i=1}^n \|\Sigma_i \mathbf{R}([\theta]_i)[\nabla m]_i\|_2^2$ enables adaptive, directionally focused smoothing, and the ADMM-based algorithm performs updates for $\theta$, $\sigma$, $u_s$, and $m$, including a Gauss-Newton step for $\theta$ and a rotated-gradient-based update for $\sigma$. Numerical tests on a denoising task and a sparse Marmousi II FWI scenario show significant gains over isotropic regularization, with improved preservation of geological features and robust reconstructions under sparse sampling. The approach offers practical impact for crustal-scale imaging and OBS-type surveys where dense data coverage is impractical.

Abstract

Full waveform inversion (FWI) is a challenging, ill-posed nonlinear inverse problem that requires robust regularization techniques to stabilize the solution and yield geologically meaningful results, especially when dealing with sparse data. Standard Tikhonov regularization, though commonly employed in FWI, applies uniform smoothing that often leads to oversmoothing of key geological features, as it fails to account for the underlying structural complexity of the subsurface. To overcome this limitation, we propose an FWI algorithm enhanced by a novel Tikhonov regularization technique involving a parametric regularizer, which is automatically optimized to apply directional space-variant smoothing. Specifically, the parameters defining the regularizer (orientation and anisotropy) are treated as additional unknowns in the objective function, allowing the algorithm to estimate them simultaneously with the model. We introduce an efficient numerical implementation for FWI with the proposed space-variant regularization. Numerical tests on sparse data demonstrate the proposed method's effectiveness and robustness in reconstructing models with complex structures, significantly improving the inversion results compared to the standard Tikhonov regularization.

Figures (6)

• Figure 1: Top row: Geometric illustration of the effect of a single term in the anisotropic regularizer equation \ref{['Reg']} for different values of $\sigma$ and $\theta$. The horizontal and vertical components of the model gradient vary along the respective axes. Each ball represents the set of points where the regularization function equals a given value. Bottom row: Illustration of the behavior of the solution of a linearized version of equation \ref{['FWI_const']} for $\theta=0$ and various values of $\sigma$. Finding a solution is equivalent to finding a point constrained to a given line while minimizing the anisotropic regularization term; this is done by expanding the regularization ball until it becomes tangent to the line (see Tarantola_2005_IPT).
• Figure 2: (a) True signal. (b) Noisy signal. Estimated signal by (c) the isotropic L2-norm, (d) the anisotropic L2-norm with fixed $\sigma$, (e) the adaptive anisotropic L2 with the estimated space-variying (f) $\theta$, (g) $\sigma_x$, and (h) $\sigma_z$.
• Figure 3: (a) The true Marmousi II velocity model; (b) The 1D starting velocity model.
• Figure 4: Comparison of FWI results using anisotropic and isotropic regularizations for different seismometer spacing ($\Delta S$). (a-d) anisotropic regularization for varying receiver spacings, (e-h) the corresponding isotropic results.
• Figure 5: Comparison of velocity models after inverting the first frequency (3 Hz) for the sparsest acquisition setup ($\Delta S=1500$ m). (a)-(c) Models obtained using anisotropic regularization for the first three inversion cycles. (d)-(f) Corresponding models obtained with standard isotropic regularization.