Discrete Spherical Harmonic Transforms for Equiangular Grids of Spatial and Spectral Data
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Bibliographic record
Abstract
Discrete Spherical Harmonic Transforms for Equiangular Grids of Spatial and Spectral Data Spherical Harmonic Transforms (SHTs) which are non-commutative Fourier transforms on the sphere are critical in global geopotential and related applications. Among the best known global strategies for discrete SHTs of band-limited spherical functions are Chebychev quadratures and least squares for equiangular grids. With proper numerical preconditioning, independent of latitude, reliable analysis and synthesis results for degrees and orders over 3800 in double precision arithmetic have been achieved and explicitly demonstrated using white noise simulations. The SHT synthesis and analysis can easily be modified for the ordinary Fourier transform of the data matrix and the mathematical situation is illustrated in a new functional diagram. Numerical analysis has shown very little differences in the numerical conditioning and computational efforts required when working with the two-dimensional (2D) Fourier transform of the data matrix. This can be interpreted as the spectral form of the discrete SHT which can be useful in multiresolution and other applications. Numerical results corresponding to the latest Earth Geopotential Model EGM 2008 of maximum degree and order 2190 are included with some discussion of the implications when working with such spectral sequences of fast decreasing magnitude.
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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 | 0.001 | 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 | 0.000 |
| Research integrity | 0.000 | 0.000 |
| 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 it