Wavelength Fitting Algorithm

Overview

Wavelength calibration establishes the mapping between CCD pixel positions and wavelengths using arc lamp emission lines. pyKOSMOS++ uses Chebyshev polynomial fitting with Bayesian Information Criterion (BIC) order selection.

Method

  1. Detect arc lines in 1D collapsed arc spectrum

  2. Match to catalog wavelengths (He-Ne-Ar, Ar, Kr, Ne)

  3. Fit Chebyshev polynomial with iterative sigma-clipping

  4. Select optimal order using BIC to balance fit quality vs overfitting

  5. Validate RMS residual < threshold (default 0.1Å)

Chebyshev Polynomials

Chebyshev polynomials are preferred over standard polynomials for numerical stability:

\[\begin{split}\\lambda(x) = \\sum_{i=0}^{n} c_i T_i(x_{norm})\end{split}\]

where \(T_i\) are Chebyshev polynomials and \(x_{norm}\) is pixel position normalized to [-1, 1].

Advantages:

  • Numerically stable for high orders (up to n=10)

  • Orthogonal basis reduces coefficient correlations

  • Better conditioned than power series

BIC Order Selection

Bayesian Information Criterion balances fit quality with model complexity:

\[\begin{split}BIC = n \\ln(\\sigma^2) + k \\ln(n)\end{split}\]

where:

  • n = number of arc lines

  • σ² = variance of residuals

  • k = number of parameters (polynomial order + 1)

Algorithm:

  1. Fit polynomials for orders 3-7 (configurable)

  2. Compute BIC for each order

  3. Select order with minimum BIC

  4. Validate RMS < 0.1Å threshold

Iterative Sigma-Clipping

Removes outlier lines from fit:

  1. Fit polynomial to all matched lines

  2. Compute residuals: \(r_i = \\lambda_i - f(x_i)\)

  3. Calculate RMS: \(\\sigma = \\sqrt{\\sum r_i^2 / n}\)

  4. Reject lines with \(|r_i| > 3\\sigma\)

  5. Repeat until convergence or max iterations (default 5)

Typical results:

  • 80-95% of lines retained

  • RMS improves by 20-40% after clipping

Parameters

order_range: (3, 7)

Range of polynomial orders to test

sigma_clip: 3.0

Threshold for outlier rejection (sigma)

max_iterations: 5

Maximum sigma-clipping iterations

rms_threshold: 0.1

Maximum acceptable RMS in Angstroms

Performance

Typical Metrics:

  • Lines detected: 40-60 for He-Ne-Ar

  • Lines used in fit: 35-55 (after clipping)

  • RMS residual: 0.05-0.15Å

  • Optimal order: 5-7 for KOSMOS

References

  • Chebyshev polynomials: Press et al. “Numerical Recipes” Ch. 5

  • BIC: Schwarz, G. 1978, Annals of Statistics, 6, 461

  • pyKOSMOS: Davenport et al. 2023, DOI:10.5281/zenodo.10152905

See Also