Wave-Propagation Geometry Emerges from Scalar Travel-Time Relations
Abstract
Scalar travel-time relations can identify propagation geometry without velocity inputs, ray labels or equation supervision during learning. Physical interpretation is applied after learning through the high-frequency isotropic eikonal relation (Aki and Richards, 2002). Here we show that local value differences constrain derivatives through sampling geometry and smoothness. Controlled neural fields recover propagation directions with 5.34±0.07∘ median angular error, and local P- and S-wave velocities in a prospectively locked, test-sealed realization. Classical representations also recover geometry. Scalar-matched perturbations separate value accuracy from derivative reliability; regularized inversion can recover structure when direct gradients fail. A field trained on 2.90 million Chinese arrivals retains continental crust--mantle boundary structure: correlation with a withheld reference is ρ=0.910 raw and 0.592 after quadratic detrending (spatial-shift p=0.020). Direct catalogue values yield ρ=0.909, and thickness contrasts are compressed. Manual-only California arrivals support broad velocity organization with limited lateral fidelity. Synthetic quantum fields containing phase information support probability-flow and trajectory readouts without supervising those quantities. These results establish conditional identification across representations and physical settings: informative scalar relations can constrain geometry, while known physical relations supply its interpretation. They do not imply that accurate scalar fitting alone guarantees reliable physical derivatives.
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BibTeX
@misc{yu2026wavepropagation,
title = {{Wave-Propagation Geometry Emerges from Scalar Travel-Time Relations}},
author = {Ziye Yu},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.31238},
eprint = {2609.31238},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.31238},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Wave-Propagation Geometry Emerges from Scalar Travel-Time Relations
AU - Ziye Yu
PY - 2026
DA - 2026/09/25
PB - arXiv
DO - 10.48550/arxiv.2609.31238
UR - https://arxiv.org/abs/2609.31238
AB - Scalar travel-time relations can identify propagation geometry without velocity inputs, ray labels or equation supervision during learning. Physical interpretation is applied after learning through the high-frequency isotropic eikonal relation (Aki and Richards, 2002). Here we show that local value differences constrain derivatives through sampling geometry and smoothness. Controlled neural fields recover propagation directions with 5.34±0.07^∘ median angular error, and local P- and S-wave velocities in a prospectively locked, test-sealed realization. Classical representations also recover geometry. Scalar-matched perturbations separate value accuracy from derivative reliability; regularized inversion can recover structure when direct gradients fail. A field trained on 2.90 million Chinese arrivals retains continental crust--mantle boundary structure: correlation with a withheld reference is ρ=0.910 raw and 0.592 after quadratic detrending (spatial-shift p=0.020). Direct catalogue values yield ρ=0.909, and thickness contrasts are compressed. Manual-only California arrivals support broad velocity organization with limited lateral fidelity. Synthetic quantum fields containing phase information support probability-flow and trajectory readouts without supervising those quantities. These results establish conditional identification across representations and physical settings: informative scalar relations can constrain geometry, while known physical relations supply its interpretation. They do not imply that accurate scalar fitting alone guarantees reliable physical derivatives.
ER -