Robust acoustic and elastic full waveform inversion by adaptive Tikhonov-TV regularization

TL;DR

This work tackles the ill-posed, non-convex nature of full waveform inversion by introducing adaptive Tikhonov-TV (TT) regularization that decomposes the model into a smooth component $oldsymbol{m}_2$ and a blocky component $oldsymbol{m}_1$ with $ oldsymbol{m}=oldsymbol{m}_1+oldsymbol{m}_2$. Implemented within an ADMM framework, TT leverages a robust, automated balancing strategy based on median absolute deviation to adaptively tune the relative influence of the two terms, and it extends naturally from acoustic to elastic FWI with parameter-specific handling. Numerical experiments on Gaussian void, SEAM, and 2004 BP salt models, as well as elastic examples (inclusion and overthrust), show that TT consistently improves convergence and reconstruction quality compared to standalone Tikhonov or TV, reduces cycle skipping, and remains robust to noise and sparse data with minimal computational overhead. The results suggest TT regularization offers a practical, scalable path to accurate high-resolution seismic imaging in complex geological settings, including challenging subsalt environments and multi-parameter elastic inversions.

Abstract

Full Waveform Inversion (FWI) is a powerful wave-based imaging technique, but its inherent ill-posedness and non-convexity lead to local minima and poor convergence. Regularization methods stabilize FWI by incorporating prior information and enforcing structural constraints like smooth variations or piecewise-constant behavior. Among them, Tikhonov regularization promotes smoothness, while total variation (TV) regularization preserves sharp boundaries. However, in the context of FWI, we highlight two key shortcomings of these regularization methods. First, subsurface model parameters (P- and S-wave velocities, density) often exhibit complex geological formations with sharp discontinuities separating distinct layers, while parameters within each layer vary smoothly. Neither Tikhonov nor TV regularization alone can effectively constrain such piecewise-smooth structures. Second, and more critically, when the initial model is far from the true model, these regularization assumptions can lead to a local minimum. To address these issues, we propose adaptive Tikhonov-TV (TT) regularization, which decomposes the model into smooth and blocky components, enabling robust recovery of piecewise-smooth structures. Implemented within the ADMM framework, TT regularization incorporates an automated balancing strategy based on robust statistical analysis. Numerical experiments on acoustic and elastic FWI using benchmark geological models demonstrate that TT regularization significantly improves convergence and reconstruction accuracy compared to Tikhonov and TV regularization when applied separately. We show that for complex models and remote initial models, both Tikhonov and TV regularization tend to converge to local minima, whereas TT regularization effectively mitigates cycle skipping through its adaptive combination of the two regularization strategies.

Figures (31)

• Figure 1: An example of a piecewise smooth model. (a) The desired Gaussian void model, $\bold{m}$, that can be decomposed into a piecewise constant component, $\bold{m}_1$, (b) and a smooth component, $\bold{m}_2$, (c); $\bold{m}=\bold{m}_1+\bold{m}_2$. (d,e,f) extracted horizontal profiles along the dashed lines in (a,b,c) for each model. (g) The gradient of (d) which contains non-Gaussian distributed jumps embedded in a smooth Gaussian distributed trend that is a linear summation of: (h) the gradient of (e) with sparse non-Gaussian distributed nature and (i), the gradient of (f) with smooth Gaussian nature; $\nabla \bold{m} = \nabla \bold{m}_1 + \nabla \bold{m}_2$. In (g) and (h), amplitudes are clipped to $\pm0.01$ for display.
• Figure 1: Gaussian void test. Recovered velocity models obtained using different approaches. From left to right: without regularization (No Reg), and with Tikhonov, TV, and TT regularizations. (a) 2D velocity images. (b) 1D velocity profiles along the horizontal dashed lines in the top row, comparing the recovered models (red) with the true model (blue).
• Figure 2: Gaussian void test. (a) Evolution of the RME (%) over iterations for the velocity models shown in \ref{['fig:inclusion_res']}. (b) The behavior of iteratively adjusted $\beta$ (blue curve) and $\phi(\beta)$ (red curve) for the case of TT regularization.
• Figure 3: SEAM test. (a) The true velocity model. (b) The initial model. Seismograms computed at the true model (c) and initial model (d). (e) An interleaved display of computed seismograms, showing alternating segments of traces from (c) and (d). A close-up of a segment of the seismograms is also provided for a detailed comparison. The source position is shown in panel (a) by a red star at $X=9.3$ km.
• Figure 4: SEAM test. (a) Comparison of velocity model reconstructions without regularization (No Reg, first row) and different regularization techniques, Tikhonov (second row) TV (third row) and TT (fourth row) for data frequency of 1.5 Hz after 1000 iterations. (b) The difference between true and estimated velocity model. (c) The evolution of RME (%) versus iteration for each method.