Solving the eikonal equation for compressional and shear waves in anisotropic media using peridynamic differential operator
Bibliographic record
Abstract
SUMMARY Traveltimes of compressional (P) and shear (S) waves have proven essential in many earthquake and exploration seismology applications. An accurate and efficient traveltime computation for P and S waves is crucial for the success of these applications. However, solving the eikonal equation with a complex phase velocity field in anisotropic media is challenging. The eikonal equation is a first-order nonlinear hyperbolic partial differential equation. It represents the high-frequency asymptotic approximation of the wave equation. The fast marching and sweeping methods are commonly used due to their efficiency in numerically solving the eikonal equation. However, these methods suffer from numerical inaccuracy in anisotropic media with sharp heterogeneity, irregular surface topography and complex phase velocity fields. This study presents a new method for the solution of the eikonal equation by employing the peridynamic differential operator (PDDO). The PDDO provides the non-local form of the eikonal equation by introducing an internal length parameter (horizon) and a weight function with directional non-locality. The operator is immune to discontinuities in the form of sharp changes in field or model variables and invokes the direction of traveltime in a consistent manner. The weight function controls the strength of association among points within the horizon. Solutions are constructed in a consistent manner without upwind assumptions through simple discretization. The robustness of this approach is established by considering different types of eikonal equations on complex velocity models in anisotropic media. The examples demonstrate its unconditional numerical stability and results compare well with the reference solutions.
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How this classification was reachedexpand
Full frame distilled prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. Learned from the 10,348 direct Codex labels and 10,348 direct Gemma labels. Candidate is the union of thresholded teacher heads; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels or direct frontier model labels.
Codex and Gemma teacher scores by category
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.000 | 0.000 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.000 | 0.000 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.001 | 0.001 |
| Research integrity | 0.000 | 0.002 |
| Insufficient payload (model declined to judge) | 0.000 | 0.000 |
Machine scores (provisional)
The two teacher heads of the student model, read on this work. A score orders the frame for review; it never asserts a category, and the validation status ships verbatim with every row.
Baseline scores from an immature model (maturity gate not passed, 7 training rounds). Scores rank; they never assert a category.
score_only:v0-immature-baseline · verbatim from the scoring run: score_only means the number may rank works, and no category label ships from itClassification
machine, unvalidatedMachine predicted; a candidate call from one teacher head, not a consensus.
How this classification was reached, model by model and score by score, is at the end of the page under "How this classification was reached".