Value chain · CLM-0955
In plain terms
The earlier formula for how much beam error is left after interpolating between sparse sensors turns out to be pessimistic by somewhere between 20 per cent and a factor of three, when the structure is modelled as a handful of bending modes rather than as random roughness. So the sensor spacings it permits are on the safe side, by 10 to 70 per cent.
The claim as recorded
CLM-0912's single-correlation-length closed form is a safe bound, not an estimate: for a deformation field composed of the first 5 to 10 free-free beam modes with amplitudes falling as n^-p for p from 0 to 2, it overstates the linear-interpolation residual by a factor of 1.2 to 3 at every detector spacing up to its own 2.35-correlation-length crossover. The ratio of modal residual to closed form runs from 0.33 (steep spectrum) to 0.84 (flat spectrum, ten modes) across twelve cases, and understates only beyond the crossover for flat spectra, where interpolation is already worse than no correction. The permitted detector spacing at the 0.1 dB budget is correspondingly 1.1 to 1.7 times larger than CLM-0912's design table gives. Length drops out entirely once spacing is expressed in correlation lengths.
Evidence behind this: T0-derivation. A numerical comparison of two models of the same quantity, with every input either derived in a held CALC or stated as an assumption and swept.
Scope and limits
THE DIRECTION IS ROBUST AND THE SIZE IS NOT. Conservative in every case swept; the factor depends on p and M, and more modes at a flat spectrum push the ratio toward 1. One-dimensional: a plate has two-dimensional modes and the CASSIOPeiA helix has coupled ones, and neither has stiffness data in the register (CLM-0907, CALC-109), so the modes here are a beam's, not the candidate geometry's. The Gaussian fit window moves the fitted correlation length by tens of per cent but not the modal residual.
What it assumes
A one-dimensional beam-mode basis on a length; the spectral exponent p, which nothing held determines, swept rather than known; random relative sign between modes at an instant.
Where it comes from
This is the project's own working, shown in full:
CALC-110: a field of M free-free beam modes with random relative signs, 64-draw ensemble, linear interpolation at eight grid offsets, residual at cell midpoints, against eq.(7) with a Gaussian correlation length fitted from the ensemble autocorrelation. Half the discrepancy is the Gaussian shape and half is non-stationarity - eq.(7) with the field's own exact autocorrelation sits at 0.5 to 0.97 of the modal answer. Why: a field of M modes has no content below about L/M, while a Gaussian-correlated field with the same fitted length has a smooth tail of short-scale content, and interpolation error is set by short-scale curvature. A second implementation, written separately, gives 0.61, 0.68 and 0.83 at D/Lc of 0.5, 1.0 and 2.0 for five modes, p = 1, 10 mm rms.
Basis: dimensionless (residual ratio; spacing in correlation lengths)
Related records
- Answers question
- QST-013 not published in this release
Cite this
energyaperture.com/c/clm-0955
CLM-0955, Energy Aperture, as of 2026-09-17.