# HV Metric For Time-Domain Full Waveform Inversion

arXiv: https://arxiv.org/abs/2508.17122 · HTML: https://mlsanity.com/paper/2508.17122/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 4 of its 30 references that are in the map.
- First posted on arXiv 23 Aug 2025.

## Builds on (most cited first, 4 of 4)

- [Optimal Transport Based Seismic Inversion: Beyond Cycle Skipping](https://mlsanity.com/paper/2002.00031/) (2020)
- [The quadratic Wasserstein metric for inverse data matching](https://mlsanity.com/paper/1911.06911/) (2019)
- [HV Geometry for Signal Comparison](https://mlsanity.com/paper/2304.11538/) (2023)
- [Robust Frequency Domain Full-Waveform Inversion via HV-Geometry](https://mlsanity.com/paper/2505.01817/) (2025)

## Abstract (arXiv)

Full-waveform inversion (FWI) is a powerful technique for reconstructing high-resolution material parameters from seismic or ultrasound data. The conventional least-squares ((L2)) misfit suffers from pronounced non-convexity that leads to 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 HV metric, a transport-based distance that acts naturally on signed signals, as an alternative metric for the (L2) 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 dHV2(f,g)), enabling efficient adjoint-state implementations. A spectral analysis of the Hessian shows that, by tuning the hyperparameters ((kappa,λ,ε)), the HV misfit seamlessly interpolates between (L2), (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 (L2) and Wasserstein misfits. These results demonstrate the HV metric as a robust, geometry-preserving alternative for large-scale waveform inversion.