HV Metric For Time-Domain Full Waveform Inversion

TL;DR

This paper introduces the HV metric, a new transport-based distance for time-domain full waveform inversion that improves convergence and robustness over traditional L2 and Wasserstein metrics by naturally handling signed signals.

Contribution

The paper proposes the HV metric, a novel signed-signal transport distance with closed-form derivatives, enabling efficient FWI and tunable between L2 and transport norms.

Findings

HV metric improves convergence speed in FWI.

HV metric demonstrates robustness to poor initial models.

Synthetic tests show superior results over L2 and Wasserstein.

Abstract

Full-waveform inversion (FWI) is a powerful technique for reconstructing high-resolution material parameters from seismic or ultrasound data. The conventional least-squares (\(L^{2}\)) misfit suffers from pronounced non-convexity that leads to \emph{cycle skipping}. Optimal-transport misfits, such as the Wasserstein distance, alleviate this issue; however, their use requires artificially converting the wavefields into probability measures, a preprocessing step that can modify critical amplitude and phase information of time-dependent wave data. We propose the \emph{HV metric}, a transport-based distance that acts naturally on signed signals, as an alternative metric for the \(L^{2}\) and Wasserstein objectives in time-domain FWI. After reviewing the metric's definition and its relationship to optimal transport, we derive closed-form expressions for the Fr\'echet derivative and Hessian of the map \(f \mapsto d_{\text{HV}}^2(f,g)\), enabling efficient adjoint-state implementations. A spectral analysis of the Hessian shows that, by tuning the hyperparameters \((\kappa,\lambda,\epsilon)\), the HV misfit seamlessly interpolates between \(L^{2}\), \(H^{-1}\), and \(H^{-2}\) norms, offering a tunable trade-off between the local point-wise matching and the global transport-based matching. Synthetic experiments on the Marmousi and BP benchmark models demonstrate that the HV metric-based objective function yields faster convergence and superior tolerance to poor initial models compared to both \(L^{2}\) and Wasserstein misfits. These results demonstrate the HV metric as a robust, geometry-preserving alternative for large-scale waveform inversion.