Robust Frequency Domain Full-Waveform Inversion via HV-Geometry

TL;DR

This work proposes the HV metric $d_{HV}$ as a robust, transport-inspired misfit for frequency-domain full-waveform inversion that naturally handles complex-valued signals and signed data, addressing cycle-skipping and noise sensitivity of $L^2$ and limitations of $W_2$. It develops a two-block alternating-direction algorithm to compute $d_{HV}$ and its Fréchet derivative, enabling gradient-based inversions, and demonstrates superior performance on Marmousi seismic tests and USCT breast-imaging, including real clinical data. The key contributions are (i) a complex-valued HV framework with analytical representations for subproblems, (ii) a practical, gradient-friendly derivative theory, and (iii) comprehensive validation showing improved robustness and detail recovery over standard misfits. Limitations include the lack of a complete theoretical justification of noise robustness and the increased computational cost for large-scale 3D problems, motivating future work toward 3D extensions and perturbation analyses.

Abstract

Conventional frequency-domain full-waveform inversion (FWI) is typically implemented with an $L^2$ misfit function, which suffers from challenges such as cycle skipping and sensitivity to noise. While the Wasserstein metric has proven effective in addressing these issues in time-domain FWI, its applicability in frequency-domain FWI is limited due to the complex-valued nature of the data and reduced transport-like dependency on wave speed. To mitigate these challenges, we introduce the HV metric ($d_{\text{HV}}$), inspired by optimal transport theory, which compares signals based on horizontal and vertical changes without requiring the normalization of data. We implement $d_{\text{HV}}$ as the misfit function in frequency-domain FWI and evaluate its performance on synthetic and real-world datasets from seismic imaging and ultrasound computed tomography (USCT). Numerical experiments demonstrate that $d_{\text{HV}}$ outperforms the $L^2$ and Wasserstein metrics in scenarios with limited prior model information and high noise while robustly improving inversion results on clinical USCT data.

Figures (11)

• Figure 1: (a) Simulated frequency-domain data is generated for the wave speed $c(x)$, ranging from 1300 to 1700$\,\mathrm{m/s}$. (b) Misfit functions are computed by comparing reference data, based on the true wave speed $c^* = 1500 \,\mathrm{m/s}$, with simulated data corresponding to different wave speeds $c(x) = c \in (1300, 1700) \,\mathrm{m/s}$. Three misfit functions are evaluated: the squared $L^2$ norm, the squared 2-Wasserstein metric, and the squared HV metric. (c) HV metric between the reference data, based on the true wave speed $c^* = 1500 \,\mathrm{m/s}$, and the simulated data corresponding to different wave speeds $c(x) = c \in (1350, 1650) \,\mathrm{m/s}$, are computed. Three HV metric with different $\varepsilon\in\{10^{-5}, 10^{-6}, 10^{-7}\}$ are evaluated.
• Figure 2: Signal $f$ and the shifted signal $g\left(t;\frac{1}{3}\right)=f\left(t-\frac{1}{3}\right)$.
• Figure 3: The normalized loss between $f(t)$ and $g(t) = f(t-s)$ using the: (a) $L^2$ norm; (b) HV metric with $\kappa=\lambda=\varepsilon=10$; (c) HV metric with $\kappa=\varepsilon=10^{-7}$ and $\lambda=1$; (d) HV metric with $\kappa=\lambda=10^{-5}$ and $\varepsilon=10^{-3}$.
• Figure 4: (a) True velocity of the Marmousi model; (b) The initial velocity in all inversions; (c)-(e) Noise-free inversion results using the $L^2$ norm, the HV metric and the $W_2$ metric, respectively; (f)-(h) The 1D data residual of sources 1, 5, 18 (the reference data subtracted by the simulated traces from velocity models in (c)-(e)) for the $L^2$ norm, the HV metric and the $W_2$ metric, respectively.
• Figure 5: Computational time of 3Hz seismic inversion of the Marmousi model using three different metrics as objectives.

Theorems & Definitions (6)

• Theorem 1
• Proposition 1
• Lemma 1
• Proposition 2
• Proposition 3
• Proposition 4