The ‘gap’ between seismic ray theory and ‘full’ wavefield extrapolation
Bibliographic record
Abstract
A derivation is given for the factorization of the generally anisotropic, elastic wave equation into two operators which are first order with respect to a preferred direction of propagation. One factor gives the ‘one-way’ equation for forward-going body waves. A 3 × 3 matrix notation is used which emphasizes the role of the Christoffel equation (or what might be called its ‘preferred coordinate projection’) familiar from ray theory. To recognize this it helps to obtain first an exact factorization of the wave equation for homogeneous anisotropic media. When inhomogeneities exist it is necessary to use a Fourier representation for differential operators with respect to the transverse coordinates, but still the Christoffel equation is the key. The so-called ‘square root operator’ required in the factorization becomes more correctly the ‘root of a matrix operator ratic equation’ intimately connected to the algebraic Christoffel equation. The factorization does not assume narrow-angle propagation or involve an a priori reference phase. It includes the P and two S waves and the forward coupling between them. It remains valid at slowness-surface singularities such as conical points (also known as acoustic axes—a most stringent test) and for wave fronts which fold. In the frequency domain this wide-angle one-way equation involves Fourier synthesis with respect to transverse position/slowness and its time-domain analogue involves 2-D slant stacking (Radon transformation), multiplication by a 3 × 3 ‘propagator’ matrix and an inverse 2-D slant stack. The one-way equation can be solved ‘analytically’ by using ‘path integrals’. These in turn may be reduced by stationary-phase arguments to standard ray theory, Maslov and Kirchhoff representations. In this sense, the ‘gap’ between ray theory and the numerical solution of the full wave equation is bridged or filled in, at least for forward propagation. Alternatively, the one-way wave equation can be solved by numerical ‘forward-stepping’ algorithms and here again a number of choices are available. Narrow-angle Taylor or rational approximations to the matrix propagator in the Fourier/slant-stack representation lead to the anisotropic analogues of the classic 15° and 45° partial differential equations. The integral wide-angle form is also reminiscent of the ‘split-step’and ‘elastic phase screen’ approaches of Tappert, Wu and others, suggesting other implementations.
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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 | 0.002 | 0.003 |
| Meta-epidemiology (narrow) | 0.001 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.001 | 0.001 |
| Science and technology studies | 0.001 | 0.004 |
| Scholarly communication | 0.002 | 0.006 |
| Open science | 0.002 | 0.003 |
| Research integrity | 0.002 | 0.005 |
| Insufficient payload (model declined to judge) | 0.008 | 0.003 |
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".