Numerical analysis of a narrow-angle, one-way, elastic-wave equation and extension to curvilinear coordinates
Bibliographic record
Abstract
Abstract In this paper, we review the finite-difference implementation of a narrow-angle, one-way vector wave equation for elastic, 3D media. Extrapolation is performed in the frequency domain, where the second-order-accurate lateral spatial-difference operators are sufficiently accurate for narrow-angle propagation. We perform a numerical analysis of the finite-difference scheme to highlight the stability and dispersion characteristics. The von Neumann stability criterion indicates that extracting a reference phase during the extrapolation step noticeably improves the forward marching scheme, and dispersion analysis shows that numerical grid anisotropy is minimal for the propagation path lengths, source pulse spectral content, and angular range of forward propagation of interest. Although the algorithm is reasonable, its computational efficiency is limited by the second-order-accurate extrapolation step; therefore, the extrapolation scheme can be improved. We extend the Cartesian narrow-angle formulation to curvilinear coordinates, where the computational grid tracks the true wavefront in a reference medium and the wavefield derivative normal to the reference wavefront is evaluated locally using the Cartesian propagator. An example of curvilinear extrapolation for a simple model consisting of a high-velocity sphere within a homogeneous background velocity structure shows that the narrow-angle propagator is capable of modeling frequency-dependent geometric spreading and diffraction effects in curvilinear coordinates.
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How this classification was reachedexpand
Full frame machine prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. The Gemma side is a direct model label for every work in the frame, read from the title-only record. The Codex side is a classifier learned from the 10,348 direct Codex labels and calibrated to design-weighted sample rates; fields without enough sample support carry no Codex call. Candidate is the union of the two sides; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels.
Distilled classifier scores by category (both heads)
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.001 | 0.002 |
| 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 | 0.001 |
| Scholarly communication | 0.001 | 0.001 |
| Open science | 0.001 | 0.001 |
| Research integrity | 0.001 | 0.001 |
| Insufficient payload (model declined to judge) | 0.002 | 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 source (direct Gemma or distilled Codex), 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".