A generalized form of Snell's law in anisotropic media
Bibliographic record
Abstract
Abstract We have reformulated the law governing the refraction of rays at a planar interface separating two anisotropic media in terms of slowness surfaces. Equations connecting ray directions and phase-slowness angles are derived using geometrical properties of the gradient operator in slowness space. A numerical example shows that, even in weakly anisotropic media, the ray trajectory governed by the anisotropic Snell's law is significantly different from that obtained using the isotropic form. This could have important implications for such considerations as imaging (e.g., migration) and lithology analysis (e.g., amplitude variation with offset). Expressions are shown specifically for compressional (qP) waves but they can easily be extended to SH waves by equating the anisotropic parameters (i.e., ε = δ ⇒ γ) and to qSV and converted waves by similar means. The analytic expressions presented are more complicated than the standard form of Snell's law. To facilitate practical application, we include our Mathematica code.
Fetched live from OpenAlex and de-inverted. Abstracts are not stored in this database: the inverted indexes are 8.6 GB of the frame’s 9.3 GB of text, and the host has 13 GB free.
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 | 0.004 |
| Meta-epidemiology (narrow) | 0.001 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.002 | 0.001 |
| Science and technology studies | 0.001 | 0.002 |
| Scholarly communication | 0.002 | 0.004 |
| Open science | 0.001 | 0.001 |
| Research integrity | 0.001 | 0.001 |
| Insufficient payload (model declined to judge) | 0.005 | 0.001 |
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".