The Optical Spectroscopy Pipeline

BOSS/eBOSS has two spectrographs with 500 fibers each, grouped in 25 bundles of 20 fibers each. The light is split into a blue channel and a red channel, for a total of 4 CCD images per exposure. The CCD y-coordinate is the spectral dispersion direction (larger y is larger λ) and larger x is larger fiber number, though the spectral "traces" in y vs. x are curved and do not exactly align with CCD columns.

Raw electrons are extracted from the CCD images using row-by-row extractions similar to Horne 1986 by fitting Gaussians, plus a polynomial background, to each CCD row for each bundle of 20 fibers.

Fiber flats correct for fiber-to-fiber variations by comparing the differences between fibers equally illuminated by a smooth flat lamp spectrum.

Sky model is derived from flat-fielded electrons of sky fibers and then interpolated to the locations of every science fiber and subtracted. Flux calibration vectors model the instrument and atmospheric throughput per exposure by comparing standard star spectra to a set of models of known flux.

Flux correction vectors adjust for flux mis-calibrations with low-order polynomials per-fiber per-exposure to make different exposures of the same object consistent with each other.

Flux distortion vectors model variations in the throughput across the focal plane.

Putting these together, the "flat-fielded sky-subtracted electrons" in spFrame:

$F_{e} = electrons / \big(superflat \cdot fiberflat\big) - skymodel$

become the "calibrated flux" in spCFrame:

$F = \big(F_{e}/calib\big) \cdot fluxcorr \cdot fluxdistort$

The following sections describe each of these steps in more detail.

Flux correction

Individual exposures are initially flux-calibrated with no constraint that the same object has the same flux across different exposures. Empirical "fluxcorr" vectors are broadband corrections to bring the different exposures into alignment for each object prior to coaddition. In DR13 and prior, these were implemented for each spectrum by minimizing

$\chi_{i}^{2} = \sum_{\lambda}\frac{\big(f_{i\lambda}-f_{ref,\lambda}/a_{i\lambda}\big)^{2}}{\big(\sigma^{2}_{i\lambda}-\sigma^{2}_{ref,\lambda}/a^{2}_{i\lambda}\big)^{2}}$

where f_iλ is the flux of exposure i at wavelength λ; f_ref,λ is the flux of the selected reference exposure; and a_iλ are low-order Legendre polynomials. The number of polynomial terms is dynamic, up to a maximum of 5 terms. Higher order terms are added only if they improve the χ² by 5 compared to one less term. This approach is biased toward small a_iλ, since that inflates the denominator to reduce the χ².

Since DR14, we solve the fluxcorr vectors relative to a common weighted coadd F_λ which is treated as noiseless compared to the individual exposures.

$\chi_{i}^{2} = \sum_{\lambda}{\frac{\big(f_{i\lambda}-F_{\lambda}/a_{i\lambda}\big)^{2}}{\sigma^{2}_{i\lambda}}}$

We additionally include an empirically-tuned prior that a_iλ ~ 1 to avoid large excursions in the solution for very low signal-to-noise data.

The fluxcorr terms are Chebyshev polynomials instead of Legendre polynomials. We actually solve for ((f - Fa)/σ)² and then return 1/a. The prior is weighted by the data weights such that the relative strength of the prior vs. the data is approximately independent of S/N.

Flux distortion

The flux distortion vectors are parameterized in terms of magnitude (i.e. log-flux) that are achromatic with x, y, x², y², xy, where those are linear coordinates XFOCAL, YFOCAL from the plugmap.

There are also chromatic terms that scale as $\widetilde{\lambda} = 1- \big(5070/\lambda\big)^{2}$, since that function gives an equal effect between 3900 and 5070 Å as between 5070 and 9000 Å. There are also magnitude offsets as a function of spectrograph ID, and a chromatic offset as a function of spectrograph ID. The 13 parameters are:

$F_{new} = F_{orig}(1+a_{0}s_{1}+a_{1}s_{2})exp(\\ \ \ \ \ \ a_{2}x + a_{3}y + a_{4}xy +a_{5}x^{2} + a_{6}y^{2} + \\ \ \ \ \ \ a_{7}\widetilde{\lambda}x + a_{8}\widetilde{\lambda}y + \\ \ \ \ \ \ a_{9}\widetilde{\lambda}s_{1} + a_{10}\widetilde{\lambda}s_{2} + a_{11}\widetilde{\lambda}^{2}s_{1} + a_{12}\widetilde{\lambda}^{2}s_{2})\\),$

where s₁ = 1 if specid = 1 else 0 and s₂ = 2 if specid = 1 else 0.

Only SPECTROPHOTO_STD or REDDEN_STD objects are used to compute the flux distortion. The procedure minimizes differences between spectro-flux and photometry (CALIBFLUX). Only g, r and i-bands are used.