Optimal Extraction Algorithm

Overview

Optimal extraction (Horne 1986) maximizes signal-to-noise ratio by weighting spatial pixels according to the spatial profile and inverse variance. This outperforms simple aperture summation, especially for faint sources.

Method: Horne 1986 Algorithm

The optimal extraction algorithm computes extracted flux as:

\[\begin{split}f_\\lambda = \\frac{\\sum_y P(y) \\cdot D(y, \\lambda) / V(y, \\lambda)}{\\sum_y P(y)^2 / V(y, \\lambda)}\end{split}\]

where:

  • \(D(y, \\lambda)\) = 2D sky-subtracted data

  • \(P(y)\) = normalized spatial profile

  • \(V(y, \\lambda)\) = variance (read noise + Poisson)

  • \(y\) = spatial coordinate

  • \(\\lambda\) = wavelength coordinate

Algorithm Steps

  1. Fit Spatial Profile

    Fit Gaussian (or Moffat) to spatial cross-section:

    \[\begin{split}P(y) = A \\exp\\left(-\\frac{(y - y_0)^2}{2\\sigma^2}\\right)\end{split}\]

  2. Compute Variance

    Propagate noise from read noise and Poisson statistics:

    \[\begin{split}V(y, \\lambda) = (\\text{readnoise})^2 + \\text{gain} \\cdot D(y, \\lambda)\end{split}\]

  3. Weighted Extraction

    Apply profile weights normalized by variance

  4. Cosmic Ray Masking

    Pixels flagged as cosmic rays get zero weight

Advantages Over Aperture Extraction

Signal-to-Noise Improvement:

  • Typical gain: 10-30% for point sources

  • Greater improvement for faint sources

  • Down-weights noisy edge pixels

Optimal Use of Data:

  • All pixels in aperture contribute

  • Weights match actual spatial profile

  • Robust to seeing variations

Comparison: Optimal vs Boxcar

Aspect

Optimal Extraction

Boxcar (Aperture Sum)

SNR

Higher (weighted by profile)

Lower (equal weights)

Implementation

Complex (profile fitting)

Simple (sum pixels)

Robustness

Sensitive to profile errors

Robust, less optimal

Speed

Slower (~2×)

Faster

Best for

Faint sources, precision

Bright sources, quick-look

When to Use Boxcar Instead

Boxcar extraction preferred when:

  • Very bright sources (SNR > 50)

  • Poor spatial profile fit (extended/irregular)

  • Quick-look / exploratory analysis

  • Profile varies significantly along spectrum

Configure in YAML:

extraction:
  method: boxcar  # or 'optimal'
  aperture_width: 10

Spatial Profile Fitting

Gaussian Profile:

\[\begin{split}P(y) = A \\exp\\left(-\\frac{(y - y_0)^2}{2\\sigma^2}\\right)\end{split}\]

  • Best for: Point sources, good seeing

  • Parameters: amplitude, center, width

Moffat Profile:

\[\begin{split}P(y) = A \\left[1 + \\left(\\frac{y - y_0}{\\alpha}\\right)^2\\right]^{-\\beta}\end{split}\]

  • Best for: Extended wings, poor seeing

  • Parameters: amplitude, center, width, power

Empirical Profile:

  • Use actual data profile without parametric fit

  • Robust when profile shape unknown

  • Fallback if Gaussian/Moffat fail

Variance Propagation

Total variance combines read noise and Poisson noise:

\[\begin{split}V(y, \\lambda) = \\sigma_{read}^2 + \\frac{D(y, \\lambda)}{g}\end{split}\]

where:

  • \(\\sigma_{read}\) = read noise (electrons)

  • \(g\) = gain (e⁻/ADU)

  • \(D\) = data (ADU)

Uncertainty in extracted flux:

\[\begin{split}\\sigma_f = \\sqrt{\\sum_y \\frac{P(y)^2}{V(y, \\lambda)}}\end{split}\]

Parameters

aperture_width: 10 pixels

Width of extraction region around trace center

profile_type: ‘Gaussian’

Spatial profile model (‘Gaussian’, ‘Moffat’, ‘empirical’)

method: ‘optimal’

Extraction method (‘optimal’, ‘boxcar’)

Performance

Typical Metrics:

  • SNR improvement: 15-25% over boxcar

  • Extraction time: 1-3 seconds per trace

  • Profile fit χ²: <2.0 for good data

Failure Modes:

  1. Poor profile fit (χ² > 10):

    • Use empirical profile

    • Or fall back to boxcar

  2. Negative flux values:

    • Sky over-subtracted

    • Increase sky_buffer parameter

References

  • Horne, K. 1986, PASP, 98, 609 - “An Optimal Extraction Algorithm for CCD Spectroscopy”

  • Marsh, T. R. 1989, PASP, 101, 1032 - “The Extraction of Highly Distorted Spectra”

See Also