HV Metric For Time-Domain Full Waveform Inversion

2025-08-23

AI Summary

To address cycle-skipping and nonconvexity issues in full-waveform inversion (FWI) arising from L² misfit, this paper introduces the HV metric—grounded in optimal transport theory—as the objective function for time-domain FWI. Unlike Wasserstein-based approaches, HV operates directly on signed wavefields without requiring probability measure conversion, thereby preserving both amplitude and phase information. It continuously interpolates between L², H⁻¹, and H⁻² norms via a tunable hyperparameter, enabling flexible trade-offs between local and global data fidelity. We derive closed-form expressions for the Fréchet derivative and Hessian of the HV metric and integrate them into an efficient gradient computation framework using the adjoint-state method. Numerical experiments on the Marmousi and BP benchmark models demonstrate that HV significantly accelerates convergence compared to both L² and Wasserstein misfits and exhibits superior robustness to initial model errors.

Technology Category

• Search and Optimization: Heuristic Search
• Computer Vision: Learning & Optimization for CV
• Planning, Routing, and Scheduling: Optimization of Spatio-temporal Systems

Application Category

• Search and Retrieval-Augmented AI: Web evaluation methodologies and metrics
• Systems and Infrastructure for Web, Mobile and WoT: Web performance, measurement, and characterization
• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs

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 Fréchet derivative and Hessian of the map (f mapsto d_{ ext{HV}}^2(f,g)), enabling efficient adjoint-state implementations. A spectral analysis of the Hessian shows that, by tuning the hyperparameters ((κ,λ,ε)), 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.

Problem

• Addressing cycle skipping in full-waveform inversion misfit functions
• Eliminating artificial signal conversion required by optimal transport metrics
• Providing tunable trade-off between local and global waveform matching

Innovation

• HV metric for signed signals in FWI
• Tunable interpolation between L2 and transport norms
• Closed-form derivatives enabling efficient adjoint implementation