Seismic Wavelet Estimation: A Fast Learning Algorithm Using Cumulant Matching and Natural-Gradient
Bibliographic record
Abstract
Abstract In this paper, a simple, fast and local learning algorithm for estimating mixed-phase seismic wavelets is developed. This learning algorithm minimizes the nonlinear cumulant matching criterion using the natural gradient. It is shown that the nonlinear cumulant matching criterion has a Riemannianstructure. Therefore its steepest-descent is given by its natural gradient descent. The Riemannian metric tensor of the pth-order cumulant matching criterion, p?3, is determined. Then thenatural-gradient learning algorithm minimizing the pth-order cumulant matching criterion is derived for recovery of mixedphase seismic wavelets. Computer simulations, for the extraction of a broadband mixed-phase seismic wavelet from synthetic noisy seismic data, illustrate the powerfulness of the developed natural-gradient learning algorithm with respect to the standard-gradient learning algorithm. Introduction Seismic wavelet estimation is required for solving many seismic signal processing problems. Such problems may includetime-lapse seismic inversion, linearized seismic AVO amplitude-varying-with-offset) inversion, mono-/multi-channelseismic deconvolution, phase correction of stacked seismic sections, seismic forward modeling, etc. To estimate mixedphase seismic wavelets, the cumulant matching criterion iscommonly used(2)(3)(5). In this paper, a simple, fast and local learning algorithm for recovery of mixed-phase seismic wavelets is developed. This learning algorithm minimizes the nonlinear cumulant matching criterion using the natural gradient. It is shown that the nonlinear cumulant matching criterion has a Riemannian structure. Therefore the steepest-descent of the nonlinear cumulant matching criterion isnot defined by its standard-gradient descent, as suggested in Lazear (1993) (2), but rather by its natural-gradient descent. The Riemannian metric tensor of the pth-order cumulant matching criterion, p?3, which is a positive-definite matrix, is determined. Then the natural-gradient learning algorithmminimizing the pth-order cumulant matching criterion is derived for the extraction of mixed-phase seismic wavelets. The standard-gradient learning algorithm is derived as a special case of the natural-gradient learning algorithm by substituting the Riemannian metric tensor by the identity matrix. This paper is organized as follows. In section 2, a general deterministic cost function is proposed, and then its natural-gradient descent isdetermined and analysed. In section 3, the nonlinear cumulant matching criterion for the extraction of mixed-phase seismic wavelets is treated as a special case of the deterministic cost function proposed in section 2. Then, the natural-gradient descent of this nonlinear cumulant matching criterion is derived from the natural-gradient descent of the deterministic costfunction proposed in section 2. In section 4, a couple of computer simulations for estimating a broadband mixed-phase seismic wavelet from synthetic noisy seismic data are carried out. In the first computer simulation, the seismic wavelet is recovered by minimizing the fourth-order cumulant matching criterion using both natural- and standard-gradients. In the second computer simulation, the seismic wavelet is extracted by minimizing the joint third- and fourth-order cumulant matching criterion using both natural- and standard-gradients. Both computer simulations illustrate that the natural-gradient learning algorithm requires much less iterations to converge than thestandard-gradient learning algorithm. Section 5 concludes the paper.
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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 | 0.000 | 0.000 |
| Research integrity | 0.000 | 0.000 |
| Insufficient payload (model declined to judge) | 0.001 | 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".