HV Metric For Time-Domain Full Waveform Inversion

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.