Forward Global Photometric Calibration of the Dark Energy Survey

The response of the combined Blanco and DECam instrumental system can be factored into parts that are characterized and determined in different ways,

$$\begin{eqnarray}{S}_{b}^{\mathrm{inst}}(x,y,t,\lambda ) & = & {S}_{b}^{\mathrm{flat}}(\mathrm{pixel},\mathrm{epoch})\times {S}_{b}^{\mathrm{starflat}}(\mathrm{pixel},\mathrm{epoch})\\ & & \times {S}_{b}^{\mathrm{superstar}}(\mathrm{ccd},\mathrm{epoch})\times {S}^{\mathrm{optics}}(\mathrm{MJD})\\ & & \times {S}_{b}^{\mathrm{DECal}}(\mathrm{ccd},\lambda ),\end{eqnarray} \tag{ 18 }$$

where the independent variables (described in greater detail below) have been replaced with units that appropriately match the granularity and stability of the system: pixels in each CCD and MJD dates during an epoch of stable instrumental performance. New epochs are defined at the start of yearly operations and whenever the instrumental complement or performance of sensors is known to change. There were five such epochs defined over the course of Y3A1—two each in Y1 and Y2, while Y3 was an epoch unto itself. The epochs in Y1 and Y2 are each three to four months in duration and include ∼100 nights (or half nights) of DES observing.

None of the factors in Equation (18) include variations that might occur on hourly timescales such as those that could be caused by instability in the temperatures of the sensors or electronics. The average temperature of the DECam focal plane is maintained by an active thermal system and is found to vary by no more than 0.1°C over periods of weeks. The response of the sensors over this range is expected to vary less than 0.1% (Estrada et al. 2010), and any such instability is included in the FGCM performance metrics discussed below. We note also that only ${S}_{b}^{\mathrm{DECal}}$ has a specific wavelength dependence, and that this quantity includes nearly all of the loss of light through the system (Figure 1 below).

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3.1.1. Flat Fields and Star Flats: ${S}_{b}^{{flat}}$ and ${S}_{b}^{{starflat}}$

3.1.2. Superstar Flats: ${S}_{b}^{{superstar}}$

3.1.3. Opacity of Optical System: ${S}^{{optics}}$

3.1.4. Wavelength Dependence from DECal: ${S}_{b}^{{DECal}}$

The in situ "DECal" system provides nearly monochromatic illumination of the DECam focal plane through the Blanco input pupil. This system was used to measure the detailed wavelength dependencies ${S}_{b}^{\mathrm{DECal}}(\mathrm{ccd},\lambda )$ of the grizY instrumental passbands at the beginning of DES operations and once per year during the campaign. These measurements are made with 2 nm FWHM spectral bins stepped in 2 nm increments across the nominal passband of each filter and in 10 nm increments at wavelengths that are nominally "out-of-band" (defined as wavelengths approximately 10 nm or more outside the main passband of the filter). These data account for the wavelength dependence of the reflectivity of the primary mirror, the filter passbands, and the sensor efficiency. The passbands are measured individually for each CCD in the DECam focal plane as shown in Figure 1. The normalization is arbitrarily chosen to be the average of the CCD responses over the central $\pm 400\,{\rm{\mathring{\rm A} }}$ of the i band. The light gray lines show the per-CCD variation, which is especially pronounced for the g band due to variations in the quantum efficiencies of the sensors at the blue side. In addition, Figure 2 shows the variation in the blue edge of the i-band passband as a function of radius from the center of the field of view; this is caused by the variation of the transmittance of the filter with incidence angle. The shapes of the passbands are measured with better than 0.1% precision and are found to be stable over the Y3A1 campaign to the accuracy with which they are measured.

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3.1.5. Instrumental Fit Parameters

The vector of the parameters of the instrumental system used to fit the observed DES data,

$$\begin{eqnarray}&&{{\boldsymbol{P}}}^{{\rm{i}}{\rm{n}}{\rm{s}}{\rm{t}}}\equiv ({\rm{o}}{\rm{p}}{\rm{t}}{\rm{i}}{\rm{c}}{\rm{s}}({\rm{w}}{\rm{a}}{\rm{s}}{\rm{h}}{\rm{\_}}{\rm{M}}{\rm{J}}{\rm{D}}),{\rm{r}}{\rm{a}}{\rm{t}}{\rm{e}}({\rm{w}}{\rm{a}}{\rm{s}}{\rm{h}}{\rm{\_}}{\rm{M}}{\rm{J}}{\rm{D}})),\end{eqnarray} \tag{ 19 }$$

includes the opacity ${S}^{\mathrm{optics}}(\mathrm{MJD})$ of the optics after the primary mirror is washed on $\mathrm{wash}\_\mathrm{MJD}$ and the linear rate of change in the throughput of the optics during the period of time following each washing. All other characteristics of the instrumental system are measured quantities.

Processes that attenuate light as it propagates through the atmosphere include absorption and scattering (Rayleigh) by molecular constituents (${{\rm{O}}}_{2}$, ${{\rm{O}}}_{3}$, and trace elements), absorption by precipitable water vapor (PWV), scattering (Mie) by airborne macroscopic particulate aerosols with physical dimensions comparable to the wavelength of visible light, and shadowing by larger ice crystals and water droplets in clouds that is independent of wavelength (gray).

The FGCM fitting model for atmospheric transmittance is written as

$$\begin{eqnarray}{S}^{\mathrm{atm}}(\mathrm{alt},\mathrm{az},t,\lambda ) & = & {S}^{\mathrm{molecular}}(\mathrm{bp},\mathrm{zd},{\rm{t}},\lambda )\\ & & \times {S}^{\mathrm{pwv}}(\mathrm{zd},{\rm{t}},\lambda )\times {e}^{-({\rm{X}}(\mathrm{zd})\times \tau (t,\lambda ))},\end{eqnarray} \tag{ 20 }$$

where ${S}^{\mathrm{molecular}}$ accounts for absorption and scattering by dry gases, ${S}^{\mathrm{pwv}}$ accounts for absorption by water vapor, τ is the aerosol optical depth, and ${\rm{X}}$ is the airmass. The barometric pressure $\mathrm{bp}(\mathrm{mm})$, the zenith distance of the observation $\mathrm{zd}(\deg )$, and the time of the observation t (MJD) are acquired for each exposure. The airmass $X(\mathrm{zd})$ is computed separately for each exposure at the center of each CCD on the focal plane and includes corrections for the curvature of the Earth (Kasten & Young 1989), which become important at larger zenith distances. As discussed below, the model does not explicitly include possible extinction by cloud cover. We now describe in greater detail each of these terms and their corresponding parameterizations.

3.2.1. Molecular Absorption: ${S}^{{molecular}}$

3.2.2. Water Vapor Absorption: ${S}^{{pwv}}$

The FGCM parameterization of atmospheric transmission accounts for the time variations in PWV during observing nights. These are taken from measurements made with auxiliary information when they are available. Auxiliary data are assigned to a DES survey exposure if their MJD acquisition dates are most closely matched to, and are within $\pm 2.4\,\mathrm{hr}$ of, that survey exposure. The procedure adapts to missing auxiliary data by inserting a model that is linear in time through the night.

If a calibration exposure is successfully matched by auxiliary data, then the PWV is parameterized as

$$\begin{eqnarray}\mathrm{PWV}(\mathrm{exposure}) & = & {\mathrm{pwv}}_{0}(\mathrm{night})+{\mathrm{pwv}}_{1}\\ & & \times {\mathrm{pwv}}_{\mathrm{AUX}}(\mathrm{exposure}),\end{eqnarray} \tag{ 21 }$$

where ${\mathrm{pwv}}_{\mathrm{AUX}}$ is the value from the auxiliary instrumentation matched to the exposure. The nightly ${\mathrm{pwv}}_{0}$ term accounts for possible instrumental calibration offsets in the auxiliary data, and the constant ${\mathrm{pwv}}_{1}$ (a single value for the entire run) accommodates possible theoretical or computational scale differences between the FGCM and auxiliary data reductions. If the auxiliary instrument does not provide data for an exposure, then the PWV is parameterized with the less accurate approximation

$$\begin{eqnarray}\mathrm{PWV}(\mathrm{exposure}) & = & \mathrm{pwv}(\mathrm{night})+{\mathrm{pwv}}_{s}(\mathrm{night})\\ & & \times \mathrm{UT}(\mathrm{exposure}),\end{eqnarray} \tag{ 22 }$$

where the value at $\mathrm{UT}=0$ ($\mathrm{pwv}(\mathrm{night})$) and the time derivative (${\mathrm{pwv}}_{s}(\mathrm{night})$) are FGCM fit parameters. The code allows for both cases within each night, and so requires three parameters per calibratable night plus the one overall scale parameter ${\mathrm{pwv}}_{1}$.

3.2.3. Aerosol Absorption: ${e}^{-(X\tau )}$

Scattering by aerosols can be more complex, but the corresponding optical depth for a single particulate species is well-described with two parameters as

$$\begin{eqnarray}&&\tau (\lambda )={\tau }_{7750}\times {(\lambda /7750{\rm{\mathring{\rm A} }})}^{-\alpha }.\end{eqnarray} \tag{ 23 }$$

The normalization ${\tau }_{7750}$ and optical index α depend on the density, size, and shape of the aerosol particulate. The FGCM does not use any of the available MODTRAN aerosol models, as these are specific to types of sites.

Aerosol optical depth, like water vapor, can vary by several percent over hours, so the calibration measurements and process must account for variations of this magnitude on these timescales. The aerosol normalization ${\tau }_{7750}$ is parameterized in a manner similar to the PWV. When auxiliary data are available,

$$\begin{eqnarray}&&{\tau }_{7750}(\mathrm{exposure})={\tau }_{0}(\mathrm{night})+{\tau }_{1}\times {\tau }_{\mathrm{AUX}}(\mathrm{exposure}),\end{eqnarray} \tag{ 24 }$$

or if no auxiliary data are available,

$$\begin{eqnarray}&&{\tau }_{7750}(\mathrm{exposure})=\tau (\mathrm{night})+{\tau }_{s}(\mathrm{night})\times \mathrm{UT}(\mathrm{exposure}).\end{eqnarray} \tag{ 25 }$$

Again, the code allows for both cases within each night, so it requires three parameters per calibratable night plus the one overall scale parameter ${\tau }_{1}$.

For our present modeling, we assume that the aerosols on any given night are dominated by a single species. Therefore, we require one value for the aerosol optical index α(night) for each calibratable night.

3.2.4. Atmospheric Fit Parameters

The vector of the atmospheric parameters used to fit the observed DES data,

$$\begin{eqnarray}&&{{\boldsymbol{P}}}^{\mathrm{atm}}\equiv ({{\rm{O}}}_{3},{\mathrm{pwv}}_{0},{\mathrm{pwv}}_{1},\mathrm{pwv},{\mathrm{pwv}}_{s},{\tau }_{0},{\tau }_{1},\tau ,{\tau }_{s},\alpha ),\end{eqnarray} \tag{ 26 }$$

includes the vertical column height of ozone (Dobson), the vertical column height of PWV (mm), the vertical optical depth of aerosol (dimensionless), and the aerosol optical index (dimensionless). Section 4.5.3 below summarizes the input auxiliary data available during Y3A1 and the results of the FGCM fit.

Observing operations for the DES are generally carried out only when the sky is relatively free of cloud cover. Even so, condensation of water droplets and ice can produce thin clouds that are invisible to the naked eye and have intricate spatial structure (e.g., Burke et al. 2014). This condensation process occurs along sharp boundaries in temperature and pressure determined by the volume density of PWV; this leads to the common characterization of observing conditions as either "photometric" or not.

The FGCM fitting model does not include a specific component for extinction by clouds, but a rigorous procedure is followed to identify photometric, or nearly photometric, exposures for use in the calibration fit. Estimates of the standard magnitudes $\overline{{m}_{b}^{\mathrm{std}}}$ of the calibration stars obtained in each cycle of the FGCM fitting process are used to estimate the extinction of each exposure that is not accounted for by the fitted parameter vectors. (See Section 4.2 below for a discussion of this process.) This estimate is used to select the sample of calibration exposures to be used in the next cycle of the fitting process. During the DES observing season, the conditions at CTIO are such that cloud formation does not occur for large periods of time on many nights. The FGCM finds that nearly 80% of the exposures taken in the Y3A1 campaign were acquired under photometric conditions and are used in the final fit cycle.

In the final step of the FGCM, an estimate of a "gray" correction is made from the observed ${m}_{b}^{\mathrm{std}}$ on each exposure that accounts for cloud extinction. This step is discussed in detail in Section 6.3 below, but we note here that this "gray" correction is an estimate of the cumulative effect from a number of sources that are not explicit in the fitting model. This includes possible instrumental effects (e.g., dome occultations and shutter timing errors) and residual errors in assignments of ADU counts to celestial sources (e.g., aperture corrections and subtraction of sky backgrounds). These may depend on the band of the exposure, but are assumed to have no explicit wavelength dependence across each band and so are labeled "gray."

Standard passbands were defined for the Y3A1 campaign, and look-up tables (LUTs) were pre-computed to allow rapid evaluation of the passband integrals ${{\mathbb{I}}}_{0}^{\mathrm{obs}}$ and ${{\mathbb{I}}}_{1}^{\mathrm{obs}}$ over a wide range of model parameter vectors.

The standard instrumental system responses were chosen to be the average responses of the CCDs in the focal plane shown in Figure 1. These were synthesized from DECal scans taken during the first two years of DECam operations. The standard atmospheric transmittance was computed with the MODTRAN IV code (Berk et al. 1999) with the parameters given in Table 1 chosen as typical of those encountered during the Y3A1 campaign. The transmissions of the various components of the standard atmosphere are shown in Figure 3, and the combined set of standard passbands is shown in Figure 4. Subsequent observations of the SDSS standard BD+17°4708 (Fukugita et al. 1996) indicate that these passbands should be multiplied by a factor $\approx 0.55$ if approximate normalization is desired. A code that contains these passbands, as well as tools to use them, is available for download.³⁷ The photon-weighted average wavelength and the ${{\mathbb{I}}}_{0}^{\mathrm{std}}$ and ${{\mathbb{I}}}_{1}^{\mathrm{std}}$ integrals, and their ratio ${{\mathbb{I}}}_{10}^{\mathrm{std}}$ for these passbands, are given in Table 2.

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Table 2.  Standard Photometric Passband Parameters

Band	${\lambda }_{b}$	${{\mathbb{I}}}_{0}^{\mathrm{std}}$	${{\mathbb{I}}}_{1}^{\mathrm{std}}$	${{\mathbb{I}}}_{10}^{\mathrm{std}}$
	(Å)		(Å)	(Å)
g	4766.0	0.163	4.333	26.58
r	6406.1	0.187	1.850	9.89
i	7794.9	0.174	1.344	7.72
z	9174.4	0.136	−2.163	−15.90
Y	9874.5	0.052	0.911	17.52

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Look-up tables of the ${{\mathbb{I}}}_{0}^{\mathrm{obs}}$ and ${{\mathbb{I}}}_{1}^{\mathrm{obs}}$ integrals were computed at discrete points over a broad range of the atmospheric parameter vector ${{\boldsymbol{P}}}^{\mathrm{atm}}$ using the focal-plane averaged instrumental passbands in Figure 1. Variations across the focal plane in the ${{\mathbb{I}}}_{0}^{\mathrm{obs}}$ values are corrected by application of the superstar flats. The ${{\mathbb{I}}}_{1}^{\mathrm{obs}}$ integrals are corrected for the variation of the wavelength profile of the instrumental passband across the focal plane by using data acquired with DECal for individual CCDs and assuming standard atmospheric parameters. By definition, all of these corrections average to zero across the focal plane, so the standard passbands remain the reference. Interpolation of these discrete LUTs to continuous parameter space is done during the FGCM fitting and analysis procedures.

The FGCM includes several steps that are done once at the outset (cf., Appendix D). This includes SQL queries of the "FINALCUT" catalogs produced by DESDM to obtain both a "demand" list of exposures that are to be calibrated and a catalog of data from all observations of objects that are candidates to be used as calibration stars. These observations are then cross-matched by location on the celestial sky to assign them to unique objects. Selection is then made of candidate calibration stars, and a catalog of all observations of each candidate, which serves as the basis for the calibration fit and subsequent computation of calibration data products, is created.

The FGCM does not use any prior knowledge of properties of potential calibration stars. It begins with a bootstrap that uses the parameters of the standard passbands (Tables 1 and 2) as initial guesses for the model to be used to fit the observed data. Initial estimates of the ${m}_{b}^{\mathrm{obs}}$ magnitudes of the candidate calibration stars are made using a "bright observation" algorithm that identifies groups of observations near the brightest observation found. These are assumed to be approximately photometric, and the algorithm yields estimates of the magnitudes of stars and rough estimates of residual "gray" errors on each exposure.

From this start, the process becomes cyclical with the steps illustrated in Figure 5 (see also Appendix E):

• 1.  
  Select calibration stars from those in the candidate pool with at least two observations in each of griz (Figure 5(a)).
• 2.  
  Select calibration exposures with at least 600 calibration stars from those in the campaign exposure demand list (Figure 5(b)).
• 3.  
  Select "calibratable" nights with at least 10 calibration exposures from those in the campaign exposure demand list (Figure 5(c)).
• 4.  
  Iteratively fit all griz observations of all calibration stars on all calibration exposures taken on calibratable nights to obtain the best parameter vectors ${{\boldsymbol{P}}}^{\mathrm{atm}}$ and ${{\boldsymbol{P}}}^{\mathrm{inst}}$ (Figures 5(d), (e), and (f)).
• 5.  
  When the fit converges (or reaches a maximum number of iterations): compute best estimates for the magnitudes of calibration stars and dispersions of their repeated measurements, update estimates of residual "gray" extinction on individual exposures, and update the superstar flats (Figures 5(e), (g), and (h)).
• 6.  
  If the sample of calibration exposures shows sign of residual "gray" loss of flux (see Section 4.3), then remove occulted exposures and start a new cycle with updated parameter vectors and analysis data products.

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The Y-band observations are not used in the fit because those data are preferentially taken when observing is not optimal. We "dead reckon" the Y-band magnitudes for calibration stars using the parameter vectors obtained from the fit to the griz magnitudes:

• 1.  
  Select Y-band calibration stars and exposures (Figures 5(a) and (b)).
• 2.  
  Compute Y-band magnitudes (${m}_{Y}^{\mathrm{std}}$) for the subset of griz calibration stars that were also observed on Y-band calibration exposures (Figure 5(e)).
• 3.  
  Compute Y-band dispersions, estimate "gray" extinction on individual exposures, and update Y-band superstar flats (Figures 5(g) and (h)).
• 4.  
  If the sample of Y-band exposures shows sign of significant loss of flux, then reselect Y-band calibration exposures and repeat.

We note that the precision of the final Y-band magnitudes is a useful internal "blind" diagnostic for the accuracy of the FGCM process (see Section 5.1).

As a final step following the fitting cycles, the FGCM process uses the final calibration star magnitudes ${m}_{b}^{\mathrm{std}}$ and parameter vectors ${{\boldsymbol{P}}}^{{inst}}$ and ${{\boldsymbol{P}}}^{\mathrm{atm}}$ to compute output data products for CCD images on science exposures in the campaign. At this point, the procedure is:

• 1.  
  Compute ${{\mathbb{I}}}_{0}^{\mathrm{obs}}$ and ${{\mathbb{I}}}_{10}^{\mathrm{obs}}$ values from the FGCM fit parameter vectors.
• 2.  
  Compute zeropoint values and chromatic corrections from the FGCM fit parameter vectors (cf., Equation (16)).
• 3.  
  Use the standard magnitudes ($\overline{{m}^{\mathrm{std}}}$) of calibration stars to estimate residual "gray" corrections.
• 4.  
  Assign quality flags and estimate errors of data products.

These computations are described in more detail in Section 6.

The FGCM fitting step minimizes the weighted dispersion of repeated measurements of the ${m}_{b}^{\mathrm{std}}$ magnitudes (cf., Equation (16)) of calibration stars,

$$\begin{eqnarray}&&{\chi }^{2}=\displaystyle \sum _{(i,j)}\displaystyle \frac{{({m}_{b}^{\mathrm{std}}(i,j)-\overline{{m}_{b}^{\mathrm{std}}(j)})}^{2}}{{\sigma }^{\mathrm{phot}}{(i,j)}^{2}},\end{eqnarray} \tag{ 27 }$$

where the summation is over all calibration objects j found on all griz calibration exposures i. The photometric error is defined as

$$\begin{eqnarray}&&{\sigma }^{\mathrm{phot}}{(i,j)}^{2}\equiv {\sigma }^{\mathrm{inst}}{(i,j)}^{2}+{({\sigma }_{0}^{\mathrm{phot}})}^{2},\end{eqnarray} \tag{ 28 }$$

where the instrumental error ${\sigma }^{\mathrm{inst}}(i,j)$ is computed by DESDM from source and background ADU counts, and the parameter ${\sigma }_{0}^{\mathrm{phot}}=0.003$ is introduced to control possible underestimates of the errors assigned to the brightest objects. This value is estimated from the residuals of measurements of magnitudes of stars in the exposures used to construct the DESDM star flats (cf., Section 3.1.1). The error-weighted means of the calibrated magnitudes ${m}_{b}^{\mathrm{std}}(i,j)$ of each calibration star j,

$$\begin{eqnarray}&&\overline{{m}_{b}^{\mathrm{std}}(j)}=\displaystyle \frac{{\displaystyle \sum }_{i}{m}_{b}^{\mathrm{std}}(i,j){\sigma }^{\mathrm{phot}}{(i,j)}^{-2}}{{\displaystyle \sum }_{i}{\sigma }^{\mathrm{phot}}{(i,j)}^{-2}},\end{eqnarray} \tag{ 29 }$$

are taken as the best estimates of the true standard magnitudes; here, the summation is over exposures i in band b. The SciPy bounded fitting routine FMIN_L_BFGS_B³⁸ (Zhu et al. 1997) is used to minimize the ${\chi }^{2}$ with the function value and derivatives with respect to all fit parameters explicitly computed. We note that it would be preferable here to use model-based estimates of the photometric errors rather than those taken from the data. However, the errors would themselves then depend on the fit parameters. The stability of this approach will be investigated for possible use in future versions and applications of the FGCM code.

The ${\chi }^{2}$ fitting statistic uses the standard magnitudes of stars with SEDs that span much of the stellar locus. This provides sensitivity to both the amplitude and shape of the observing passband. If the fit parameters are wrong for a given exposure, so too will be the chromatic corrections included in the computation of ${m}_{b}^{\mathrm{std}}$. As discussed in Appendix B, even within a single exposure, there is typically a sufficient range of stellar spectra to constrain the FGCM fit parameters with reasonable accuracy. It is an important feature of the FGCM that it extracts as much information as possible from each star that samples the observational passband of each exposure.

As introduced in Section 3.3, there are a number of factors that affect the photometry that are not included in the FGCM fitting model. A key to the success of the FGCM fitting process is the ability to isolate a set of "photometric" exposures free of clouds and significant instrumental errors. To do this, the residual of each measurement i of the magnitude of each calibration star j is computed using the parameter vectors from the most recent fit cycle,

$$\begin{eqnarray}&&{E}^{\mathrm{gray}}(i,j)\equiv \overline{{m}_{b}^{\mathrm{std}}(j)}-{m}_{b}^{\mathrm{std}}(i,j).\end{eqnarray} \tag{ 30 }$$

The average value of this residue is then computed for the calibration stars j that are observed on each CCD image of each candidate calibration exposure i,

$$\begin{eqnarray}&&{\mathrm{CCD}}^{\mathrm{gray}}(i,\mathrm{ccd})=\displaystyle \frac{{\displaystyle \sum }_{j}{E}^{\mathrm{gray}}(i,j){\sigma }^{\mathrm{phot}}{(i,j)}^{-2}}{{\displaystyle \sum }_{j}{\sigma }^{\mathrm{phot}}{(i,j)}^{-2}},\end{eqnarray} \tag{ 31 }$$

with statistical error

$$\begin{eqnarray}&&{\sigma }^{\mathrm{phot}}{(i,\mathrm{CCD})}^{2}=\displaystyle \frac{1}{{\displaystyle \sum }_{j}{\sigma }^{\mathrm{phot}}{(i,j)}^{-2}}.\end{eqnarray} \tag{ 32 }$$

The statistical error on ${\mathrm{CCD}}^{\mathrm{gray}}$ is typically $\sim 1\mbox{--}2\,\mathrm{mmag}$, so structure on physical scales larger than the $\sim 0.2^\circ$ size of a DECam sensor can be resolved. To take advantage of this, the average and variance of the residual extinctions of the CCD images on each exposure are computed as

$$\begin{eqnarray}&&{\mathrm{EXP}}^{\mathrm{gray}}(i)=\displaystyle \frac{{\displaystyle \sum }_{\mathrm{ccd}}{\mathrm{CCD}}^{\mathrm{gray}}(i,\mathrm{ccd}){\sigma }^{\mathrm{phot}}{(i,\mathrm{ccd})}^{-2}}{{\displaystyle \sum }_{\mathrm{ccd}}{\sigma }^{\mathrm{phot}}{(i,\mathrm{ccd})}^{-2}}\end{eqnarray} \tag{ 33 }$$

and

$$\begin{eqnarray}{\mathrm{VAR}}^{\mathrm{gray}}(i) & = & \displaystyle \frac{{\displaystyle \sum }_{\mathrm{ccd}}{\mathrm{CCD}}^{\mathrm{gray}}{(i,\mathrm{ccd})}^{2}{\sigma }^{\mathrm{phot}}{(i,\mathrm{ccd})}^{-2}}{{\displaystyle \sum }_{\mathrm{ccd}}{\sigma }^{\mathrm{phot}}{(i,\mathrm{ccd})}^{-2}}\\ & & -{\mathrm{EXP}}^{\mathrm{gray}}{(i)}^{2}.\end{eqnarray} \tag{ 34 }$$

Both ${\mathrm{EXP}}^{\mathrm{gray}}$ and ${\mathrm{VAR}}^{\mathrm{gray}}$ are used in the selection of calibration exposures to use in the FGCM fit (cf., Appendix E). A night will be "calibratable" if a sufficient number of such calibration exposures were taken anytime during that night. The ${\mathrm{CCD}}^{\mathrm{gray}}$ and ${\mathrm{EXP}}^{\mathrm{gray}}$ quantities are also used in the second step of the FGCM process to estimate the "gray" corrections for residual errors that are not included in the FGCM model (see Section 6.3).

The FGCM yields detailed passbands for nearly all exposures taken on calibratable nights; this includes exposures that were not used in the fit as well as those that were. These passbands can be used to compute both ${{\mathbb{I}}}_{0}$ zeropoints for these exposures as well as chromatic corrections either with known or hypothetical SEDs of sources (Equation (7)), or linearized approximations based on the measured magnitudes of objects (Equation (16)).

The Y3A1 fit was completed in four cycles. The distributions of ${\mathrm{EXP}}^{\mathrm{gray}}$ obtained in the i band at the conclusion of these cycles are shown in Figure 6. Similar distributions are obtained for all other bands; we show these for illustrative purposes. There is clearly an asymmetric outlier population with significant loss of flux (${\mathrm{EXP}}^{\mathrm{gray}}\lt 0)$ seen after the initial cycle (top-left panel in the figure). These bias the computed magnitudes of the calibration stars and prevent the fitting process from finding the optimal solution for the passband parameter vectors. Selection of calibration exposures at the end of each cycle of the fit is done by removing those found to be occulted with a cut selected by examining the non-occulted side of the distribution. The precision of the subsequent cycle improves and allows the cut value to be tightened. The FGCM fitting process is deemed to have converged when the distributions shown in the figure become nearly symmetrical, and the bias of the distribution is reduced to an acceptable level. We find that 2%–3% of candidate calibration exposures are removed on each cycle of the fitting process and that this is sufficient to reduce the bias by typically a factor of two. After completion of the final fit Cycle 3, the bias of the fitted Gaussian peak has been reduced to $\mu \approx 0.5\,\mathrm{mmag}$ and the width to $\sigma \approx 5\,\mathrm{mmag}$.

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It can be seen in Figure 6 that the final sample includes a residual excess of exposures with some unaccounted loss of flux. Cutting too tightly on ${\mathrm{EXP}}^{\mathrm{gray}}$ will bias the calibration star magnitudes in the sense opposite to that caused by exposures taken through very thin cloud layers that remain in the calibration sample. The FGCM minimizes this ambiguity by explicitly including in the final step of the process the "gray" correction introduced above.

It is important to note that the magnitudes of the calibration stars ($\overline{{m}_{b}^{\mathrm{std}}(j)}$) are not explicitly free parameters of the fit; they are functions of the observed ADU counts and the fitted parameter vectors ${{\boldsymbol{P}}}^{\mathrm{inst}}$ and ${{\boldsymbol{P}}}^{\mathrm{atm}}$. So, the ${\chi }^{2}$ function does not rigorously have the statistical properties of a chi-square, but it is what we seek to minimize. It is also very efficient and highly constrained. The FGCM calibration of the Y3A1 campaign used 2552 parameters to fit 133,265,234 degrees of freedom (DOF), and converged after four cycles (Figure 5) of 25, 50, 75, and 125 iterations.

We provide here a summary of the statistics of the Y3A1 campaign and the parameters obtained from the FGCM fitting step. A summary of statistics for the Y3A1 campaign is given in Table 3. A query of the DESDM FINALCUT tables found demand for calibration of 41,562 griz and 9770 Y-band wide-field and supernova field exposures that were taken during the campaign.³⁹ There were 11,710,194 candidate calibration stars found on exposures for which a calibration was requested. At least one griz calibration exposure was taken on 351 scheduled observing nights or half nights, while at least 10 were taken on 335 nights. The Y3A1 FGCM calibration fit used 32,368 griz calibration exposures (78% of the total) taken on 317 calibratable nights. It produced standard magnitudes for 8,702,925 griz calibration stars spaced nearly uniformly across the DES footprint (≈0.5 calibration stars per square arcminute). Of these, 6,225,680 were also Y-band calibration sources.

Table 3.  DES Y3A1 Release Statistics

Statistic	Value	Comment
Total exposures	51,332	DESDM demand file
griz exposures	41,562
griz cal exposures	32,368
Y exposures	9770
Nights with >1 cal exposure	351	Dome at least opened
Nights with >10 cal exposures	335	Minimum to attempt calibration
Calibratable nights	317	Some photometric time
Number of griz cal stars	8,702,925	Require ≥2 cal observations in each band
Number of Y cal stars	6,225,680	A griz cal star with ≥2 Y cal observations
Number of ${\mathrm{ZPT}}^{\mathrm{FGCM}}$	3,182,584	All CCD images (Table 5)

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4.5.1. Superstar Flats

Five epochs of camera operations were identified and captured in the superstar flats ${S}_{b}^{\mathrm{superstar}}$. These are not parameters of the fit, but are computed and updated after each calibration cycle from the ${\mathrm{CCD}}^{\mathrm{gray}}$ values (Equation (31)) averaged over the DES wide-field calibration exposures taken in each band. A typical g-band superstar flat, shown in Figure 7, is dominated by differences in the shorter wavelength sensitivity of the CCDs and their AR coatings, while the i-band superstar flat in Figure 8 exhibits smooth gradients of a percent or so across the focal plane. These gradients are consistent with known ambiguities in the fitting technique used to create the initial DESDM star flats.

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4.5.2. Opacity Fit Parameters

The primary mirror was washed on seven dates during the three-year Y3A1 campaign, so the linear model used for the accumulation of dust ${S}^{\mathrm{optics}}$ requires 14 free parameters. The resulting history, shown in Figure 9, is consistent with laboratory engineering measurements taken on the wash dates. It exhibits overall worsening of optimal transmission over time as expected for the aluminum mirror surface exposed to air and possible buildup of dust on the downward-facing DECam external window that was not cleaned during this period of time.

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4.5.3. Atmospheric Fit Parameters

A summary of the atmospheric parameters obtained by the FGCM fit is given in Figure 10. The auxiliary aTmCAM instrument was unavailable for the first year of the Y3A1 campaign, and analysis of the data obtained in the latter two years was not available for inclusion in the Y3A1 calibration. So these data are not used in the Y3A1 calibration. The SUOMINET GPS network provided measurements of atmospheric water vapor on 90% of the nights of the DES Y3A1 campaign, and these data were used to compute (Equation (21)) most of the PWV values shown in the figure. The less precise linear form of Equation (22) was used for the remainder of the campaign observations. The aerosol depth values were fit in all cases with the linear form (Equation (25)), and as discussed in Section 3.2.3, the aerosol optical index α is assumed to be constant during each night. Note that for clarity the figure shows only nightly averages of the computed PWV and ${\tau }_{7750}$ values.

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A detailed analysis of the patterns and correlations in the meteorological parameters has not been done, but the aerosol and water vapor distributions are consistent with historical data from the CTIO site. There is evidence for seasonal variation in the fitted aerosol optical index consistent with the smaller sized particulates (larger optical indices) being prevalent during the early spring start of the DES observing periods and particulates of larger cross-sections dominating in the later summer periods. Even with these trends removed, the index remains noisy. This may be due to the presence of multiple peaks in the likelihood function as would be the case if there were more than one component of aerosol particulate in the atmosphere (not an unreasonable expectation). Future incorporation of data from the auxiliary aTmCAM instrument may allow the inclusion of two components of aerosol particulates in the atmospheric model. The residual error in these parameters remains reflected in the overall performance of the Y3A1 calibration, discussed next.

To evaluate the reproducibility (precision) of the FGCM calibration, we consider the distributions of the residuals ${E}^{\mathrm{gray}}(i,j)$ (Equation (30)) in the b = griz bands on exposures used in the final Cycle 3 of the Y3A1 FGCM fit. Those measurements made with ${\sigma }^{\mathrm{phot}}\lt 0.010\,\mathrm{mag}$ are shown in Figure 11. By design, the fit simultaneously minimizes the photon-statistics weighted variance of these in all bands. For diagnostic purposes, we analyze these band by band:

$$\begin{eqnarray}&&{\delta }^{2}{E}^{\mathrm{gray}}(b)\equiv \overline{{({E}^{\mathrm{gray}}(b))}^{2}}-{(\overline{{E}^{\mathrm{gray}}(b)})}^{2}.\end{eqnarray} \tag{ 35 }$$

We interpret these data in terms of the combined random errors in the FGCM fit without attribution to particular sources of these errors.

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The DES survey is carried out in multiple "tilings" of the footprint and produces repeated observations of each calibration star that are generally well-separated in time; a star is seldom observed in the same band more than once on a given night. Exceptions are the supernova fields that are often observed with successive exposures when they are targeted. With this exception, the errors in the FGCM fit parameters evaluated on different tilings are approximately independent of each other. So, we approximate the variances of the residuals as

$$\begin{eqnarray}&&\displaystyle \frac{\overline{{N}_{{\rm{b}}}(j)}}{\overline{{N}_{{\rm{b}}}(j)}-1}\times {\delta }^{2}{E}^{\mathrm{gray}}(b)\approx {\delta }^{2}\mathrm{FGCM}(b)+\displaystyle \frac{{\displaystyle \sum }_{(i,j)}{\sigma }^{\mathrm{phot}}{(i,j)}^{2}}{{N}_{b}},\end{eqnarray} \tag{ 36 }$$

where N_b(j) is the number of observations i of the calibration star j in band b, and N_b is the total observations of all calibration stars in band b. In this approximation, ${\delta }^{2}\mathrm{FGCM}(b)$ is the variance of the parent distribution of the measurements of the magnitude of a star introduced by random errors in the FGCM fit parameters. We assume it is constant over the three-year survey and over the survey footprint. The magnitudes of each calibration star are computed as the average of the measured values, so the observed dispersion of ${E}^{\mathrm{gray}}(j)$ underestimates the true dispersion. We estimate a correction from the average number of observations of a calibration star in each band (typically ∼4 in the Y3A1 release).

After subtraction of the photon statistical uncertainties, values of 6–7 mmag are obtained for the precision $\delta \mathrm{FGCM}(b)$ of the Y3A1 fit in the griz bands. Table 4 summarizes the means and variances of Gaussian fits to the distributions and includes the fractions of observations found outside the $2\sigma$ of the mean. The Gaussian fits are reasonably good, but the outlier populations are seen to be approximately twice that expected for purely random error. These results are robust to variations in the cut on photon statistics over the range $0.005\lt {\sigma }^{\mathrm{phot}}\,\lt 0.020$. We note that these results are consistent with the precision implied by the residual exposure-averaged gray term shown in Figure 6, where the photon statistical errors are negligible. Analysis of the observations made of the DES supernova fields discussed in Section 5.2 below support the hypothesis that the values determined from the entire data sample can be used to represent the error in any single measurement.

Table 4.  Summary of FGCM Calibration Fit Results

Band	Mean Offset (mag)	Gaussian σ (mag)	δFGCM (mag)	Fraction $\lt -2\sigma$	Fraction $\gt 2\sigma$
g	−0.00000	0.0087	0.0073	0.036	0.034
r	−0.00000	0.0080	0.0061	0.049	0.051
i	−0.00012	0.0080	0.0059	0.045	0.049
z	0.00000	0.0087	0.0073	0.034	0.033
Y	−0.00023	0.0097	0.0078	0.020	0.042

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5.1.1. The Y-band Calibration

As described in Section 4.1, the Y-band magnitudes are "dead reckoned" from the atmospheric parameters derived from the griz exposures. Therefore, the Y-band data offer a useful internal check on the calibration precision of the FGCM fit. The subset of griz calibration stars that are also observed on at least two Y-band exposures are taken as candidate Y-band standard stars. Final Y-band calibration stars and exposures are selected with a cyclical process to remove non-photometric exposures. This process is identical to that used for the griz bands except that no additional fits are made to the calibration parameter vectors. The ${m}_{Y}^{\mathrm{std}}$ magnitudes are then computed from the ${{\boldsymbol{P}}}^{\mathrm{atm}}$ parameters obtained from the griz fit, and the distribution of the residuals ${E}^{\mathrm{gray}}(i,j)$ is computed in the same manner as the griz samples. The result is shown in Figure 12 and included in Table 4. We find that the Y-band calibration precision $\delta \mathrm{FGCM}(Y)=7.8\,\mathrm{mmag}$ is comparable to that of the griz bands. This provides assurance that the FGCM models and fitted parameters are sufficiently accurate to account for subpercent variations in the photometry in the reddest bands.

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5.1.2. Precision of Calibration Star Magnitudes

We examine in this section the internal precision with which the magnitudes of calibration stars are determined; we note that, as in all cases, these magnitudes are only approximately normalized to an external scale. With the assumption that the repeated measurements of the magnitudes of the calibration stars are independent of each other, we estimate the random error in their mean magnitudes,

$$\begin{eqnarray}&&{\delta }^{2}\overline{{m}_{b}^{\mathrm{std}}(j)}\approx \displaystyle \frac{{\delta }^{2}\mathrm{FGCM}(b)}{({N}_{{\rm{b}}}(j)-1)}+\displaystyle \frac{1}{{\displaystyle \sum }_{i}{\sigma }^{\mathrm{phot}}{(i,j)}^{-2}}+{({\sigma }_{0}^{\mathrm{std}})}^{2},\end{eqnarray} \tag{ 37 }$$

where, as before, N_b(j) is the number of observations i of calibration star j in band b. The random error from the fit reduces to $\lt 4\,\mathrm{mmag}$ in all cases, the statistical photometric error becomes $\lt 0.050\,\mathrm{mag}$ for nearly all calibration stars (initially chosen with ${\rm{S}}/{\rm{N}}\gt 10$ per exposure), and for the brighter objects, the overall errors approach the somewhat arbitrary control value ${\sigma }_{0}^{\mathrm{std}}=3\,\mathrm{mmag}$. The precision with which the calibration star magnitudes are known plays an important role in the final assignment of calibration data products (Section 6.3) and in particular the ability to provide accurate calibrations of non-photometric exposures.

The DES survey targets the supernova fields repeatedly every few days, so these are used as a quality check on the stability of the FGCM calibration. Shown in Figure 13 are the ${\mathrm{EXP}}^{\mathrm{gray}}$ values (Equation (33)) for all SN calibration exposures taken during the Y3A1 campaign. Exposures in all bands are plotted in the figure. The deviation of the mean of the residuals over the three-year survey is well below $5\,\mathrm{mmag}$.

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The FGCM calibration will introduce correlations in the errors of measurements made closely spaced in time, and these can be imprinted on the uniformity of the calibration error across the celestial sky by the survey tiling strategy. The DECam focal plane is approximately 2° in diameter, so it is expected that structure on this scale will appear particularly in regions of the footprint that were not observed a large number of times in Y3A1.

To look for possible spatial structure in the calibration, we compare with the Gaia DR1 results (Lindegren et al. 2016) that were taken above the atmosphere. The Gaia G band is a broad passband that is centered approximately on the DES r band and spans most of the DES g, r, and i filters. We fit a color transformation that combines weighted combinations of the DES g, r, and i instrumental passbands with the Gaia G published response. Stars with DES color $0.5\lt (g-i)\lt 1.5$ are spatially matched to stars in the Gaia catalog, and the Gaia G-band magnitudes are compared with the transformed DES gri magnitudes. The mean differences of these magnitudes are binned in HEALPIX pixels (NSIDE = 256) and shown in Figure 14. The differences are statistically well-described by a Gaussian with $\sigma =6.6\,\mathrm{mmag}$. Spatial structure at small and large scales, which is caused by calibration and depth issues from both surveys, can be seen at this level. We note that comparisons to the broad G band are likely to be contaminated by Milky Way dust, so that this may be interpreted as an upper limit on the uniformity.

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The FGCM uses linearized chromatic correction (Equation (16)) to compute the standard magnitudes of calibration stars during the fitting stage. Every evaluation of the ${\chi }^{2}$ requires these to be recomputed, and the fully integrated correction is computationally too slow to use for this purpose. We discuss in this section both the size of the fully integrated correction and the accuracy of the linear approximation for the stellar SEDs used in the fit. Note that this does not address the accuracy of the calibration fit parameter vectors, only the robustness of the linear computations.

To examine the accuracy of the linear approximation, we use the stellar spectral library of Kelly et al. (2014), which combines SDSS spectra (Aihara et al. 2011) and the Pickles spectral library (Pickles 1998). For each template star, we synthesize colors by integrating the SED with the standard FGCM passbands. We then randomly sample 50,000 exposures/CCD pairs from Y3A1 observing, and compute the chromatic correction for each in two ways. These are shown in Figures 15–17 for a sample blue star, middle-color star, and red star, respectively.

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We first integrate the stellar spectrum with the passband for each exposure and CCD combination. These are plotted with red solid histograms in the figures and can be taken as estimates of the systematic chromatic offset that needs to be included in the computation of the magnitude. These offsets vary from as little as $1\mbox{--}2\,\mathrm{mmag}$ (e.g., i- or z-band observations of blue stars) to as much as $40\mbox{--}50\,\mathrm{mmag}$ (e.g., g-band observations of red stars). As discussed further in Appendix C, the chromatic corrections are dominated by instrumental effects in the r and i bands, by water vapor variation in the z band, and by a mix of instrumental and atmosphere effects in the g band.

We next use the synthetic colors to estimate the linearized correction as done during the FGCM calibration (Equation (16)). The residual of the linearized correction is shown with the blue dashed histograms in the figures. The median offset and rms estimated via median absolute deviation is also shown. The linearized corrections reduce the residual error by factors of 2 to 10. In most cases, the linearized correction reduces the systematic chromatic error to an rms of $\sim 2\,\mathrm{mmag}$, though in some cases (particularly in the g band for the reddest stars with $g-i\sim 3.0$) the residuals have an rms of $\sim 5\,\mathrm{mmag}$. These systematic uncertainties are reasonably well matched to the overall precision of the FGCM calibration.

While the previous analysis applies to single observations (as is the case with transients such as supernovae), the impact of the linearized residuals on co-add (average) magnitudes should be smaller if the corrections are uncorrelated. To test this, we start by computing the mean color of each calibration star with no corrections. After the g − i color is computed, we match this to the closest match in the Kelly et al. (2014) spectral library. Chromatic corrections are then computed using the matched spectrum. We then compute the offset between the co-add average magnitudes using the linearized corrections and the integrated corrections for blue, middle, and red stars as above. Figure 18 shows a map of these offsets over the footprint (left panel) and a histogram of residuals (right panel) for red stars in the g band. The linearized residuals for these stars are fit well by a Gaussian with $\sigma =2\,\mathrm{mmag}$. As can be seen in Figure 17, this case has the largest co-add residuals. The residuals from the linearized corrections are $\lt 0.7\,\mathrm{mmag}$ for every other band/color combination; this validates our assumption that use of the linearized correction in the fit does not produce significant loss of precision.

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The assignment of values to FGCM data products follows one of several paths identified by the parameter ${\mathrm{FLAG}}^{\mathrm{FGCM}}$ summarized in Table 5 and detailed in this section. Data acquired from griz calibration exposures are given ${\mathrm{FLAG}}^{\mathrm{FGCM}}=1$, while data from Y-band calibration exposures are given ${\mathrm{FLAG}}^{\mathrm{FGCM}}=2$. Data from exposures that are not deemed to have been taken in photometric conditions, but that were taken on calibratable nights, are in the ${\mathrm{FLAG}}^{\mathrm{FGCM}}=4$ category. These first three categories comprise the exposures taken on calibratable nights that are themselves calibratable; 90.5% of the exposures on the Y3A1 demand list are in one of these categories. Objects observed on these exposures will have valid ${m}^{\mathrm{std}}$ values with accurate chromatic correction.

Table 5.  FGCM Calibration Quality Flags

${\mathrm{FLAG}}^{\mathrm{FGCM}}$	Y3A1	Description
	Exposures (%)
1	77.7	CCD image on a griz calibration exposure (% of griz exposures)
2	76.0	CCD image on a Y calibration exposure (% of Y exposures)
4	13.2	CCD image taken on a calibratable night, but not calibration exposure
8	0.4	CCD image recovered on a night with no calibration fit
16	2.9	Unable to assign data products
33–36	6.1	CCD image on a calibratable night; unable to estimate gray correction
Any	100.0	All CCD images on all exposures

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There are a few individual CCD images that have a large number of calibration stars, but these were taken on nights with no calibration fit. These are ${\mathrm{FLAG}}^{\mathrm{FGCM}}=8$ entries and might be useful for some science cases, although the chromatic corrections are purely instrumental. A few percent of Y3A1 exposures that are in the initial demand list are found to be of too poor photometric quality to complete a calibration; these are the ${\mathrm{FLAG}}^{\mathrm{FGCM}}=16$ entries. Finally, there are also a number of CCD images that were acquired on a calibratable night, but contain too few calibration stars to determine reliable "gray" corrections. These individual CCDs are retained as ${\mathrm{FLAG}}^{\mathrm{FGCM}}={\mathrm{FLAG}}^{\mathrm{FGCM}}+32$ for completeness, but are not deemed science quality.

With sufficient knowledge of the SED of the target source, it is possible to compute the full chromatic correction for observations taken on calibrated nights (${\mathrm{FLAG}}^{\mathrm{FGCM}}=1,2\,\mathrm{or}\,4$) using the fit parameter vectors. However, the linear approximation to the chromatic correction can be easily implemented and is sufficient for many applications. The FGCM provides calibration data products for both the fully integrated and linearized corrections that are discussed in this section.

A calibrated measurement (on exposure number $\mathrm{EXP}$ and sensor $\mathrm{CCD}$) of the ${m}_{b}^{\mathrm{std}}$ magnitude of a celestial object can be computed as (cf., Equation (16))

$$\begin{eqnarray}{m}_{b}^{\mathrm{std}}(\mathrm{EXP},\mathrm{CCD}) & = & {m}_{b}^{\mathrm{inst}}+{\mathrm{ZPT}}^{\mathrm{FGCM}}(\mathrm{EXP},\mathrm{CCD})+2.5{\mathrm{log}}_{10}\\ & & \times \left(\displaystyle \frac{1+{{ \mathcal F }}_{\nu }^{{\prime} }({\lambda }_{b})\times {{\mathbb{I}}}_{10}^{\mathrm{FGCM}}(\mathrm{EXP},\mathrm{CCD})}{1+{{ \mathcal F }}_{\nu }^{{\prime} }({\lambda }_{b})\times {{\mathbb{I}}}_{10}^{\mathrm{std}}(b)}\right),\end{eqnarray} \tag{ 38 }$$

where ${m}_{b}^{\mathrm{inst}}$ is the instrumental magnitude computed by DESDM from source and background ADU counts, nightly flats, and star-flat corrections. The zeropoint ${\mathrm{ZPT}}^{\mathrm{FGCM}}$ is computed from the integral of the observing passband ${{\mathbb{I}}}_{0}^{\mathrm{obs}}$ as in Equation (4), and for reasons of flexibility includes the exposure time normalization

$$\begin{eqnarray}{\mathrm{ZPT}}^{\mathrm{FGCM}} & = & 2.5{\mathrm{log}}_{10}({\rm{\Delta }}T)+2.5{\mathrm{log}}_{10}({{\mathbb{I}}}_{0}^{\mathrm{obs}}(\mathrm{EXP},\mathrm{CCD}))\\ & & +{\mathrm{ZPT}}^{\mathrm{gray}}+{\mathrm{ZPT}}^{\mathrm{AB}}.\end{eqnarray} \tag{ 39 }$$

The passband integral ${{\mathbb{I}}}_{0}^{\mathrm{obs}}(\mathrm{EXP},\mathrm{CCD})$ is computed as in Equations (17) and (18):

$$\begin{eqnarray}{{\mathbb{I}}}_{0}^{\mathrm{obs}}(\mathrm{EXP},\mathrm{CCD}) & = & {S}_{b}^{\mathrm{superstar}}(\mathrm{CCD},\mathrm{epoch})\\ & & \times {S}^{\mathrm{optics}}({{\boldsymbol{P}}}^{\mathrm{inst}}(\mathrm{EXP}),\mathrm{MJD})\\ & & \times {\displaystyle \int }_{0}^{\infty }{S}^{\mathrm{atm}}({{\boldsymbol{P}}}^{\mathrm{atm}}(\mathrm{EXP}),\lambda )\\ & & \times {S}_{b}^{\mathrm{DECal}}(\mathrm{CCD},\lambda )\times {\lambda }^{-1}d\lambda ,\end{eqnarray} \tag{ 40 }$$

and the "gray" correction ${\mathrm{ZPT}}^{\mathrm{gray}}$ is described in the following section.

The normalized chromatic integral ${{\mathbb{I}}}_{10}^{\mathrm{FGCM}}(\mathrm{EXP},\mathrm{CCD})$ is similarly defined:

$$\begin{eqnarray}&&{{\mathbb{I}}}_{10}^{\mathrm{FGCM}}(\mathrm{EXP},\mathrm{CCD})\,\equiv \,\displaystyle \frac{{\displaystyle \int }_{0}^{\infty }{S}^{\mathrm{atm}}({{\boldsymbol{P}}}^{\mathrm{atm}}(\mathrm{EXP}),\lambda )\times {S}_{b}^{\mathrm{DECal}}(\mathrm{CCD},\lambda )\times (\lambda -{\lambda }_{b})\times {\lambda }^{-1}d\lambda }{{\displaystyle \int }_{0}^{\infty }{S}^{\mathrm{atm}}({{\boldsymbol{P}}}^{\mathrm{atm}}(\mathrm{EXP}),\lambda )\times {S}_{b}^{\mathrm{DECal}}(\mathrm{CCD},\lambda )\times {\lambda }^{-1}d\lambda },\end{eqnarray} \tag{ 41 }$$

with the cancellation of the superstar and optics terms that do not depend on wavelength. This integral is approximated via the LUTs described in Section 3.4.

The FGCM takes advantage of the extensive network of calibration stars with well-determined magnitudes to correct for residual errors due to effects not included in the fit model. With no guidance on the possible wavelength dependence of such failures, the algorithm uses the "gray" parameters computed after the last cycle of the fitting process (Section 4.2). The FGCM gray zeropoint correction, ${\mathrm{ZPT}}^{\mathrm{gray}}$, is determined from the stars on the CCD whenever possible, an estimate from other CCDs on the same exposure will be used as a second choice, and as a last resort ${\mathrm{FLAG}}^{\mathrm{FGCM}}$ will be set to ${\mathrm{FLAG}}^{\mathrm{FGCM}}+32$.

For each CCD image, we compute ${\mathrm{CCD}}^{\mathrm{gray}}$ and ${\sigma }^{\mathrm{phot}}(\mathrm{CCD})$ as described in Equations (31) and (32). On CCD images with at least five calibration stars and ${\sigma }^{\mathrm{phot}}(\mathrm{CCD})\lt 5\,\mathrm{mmag}$, we assign the CCD estimate to the zeropoint gray value:

$$\begin{eqnarray}&&{\mathrm{ZPT}}^{\mathrm{gray}}={\mathrm{CCD}}^{\mathrm{gray}}(\mathrm{EXP},\mathrm{CCD}).\end{eqnarray} \tag{ 42 }$$

It is possible for a CCD image to contain few, if any, calibration stars even though the entire exposure may contain CCD images with many stars (e.g., CCD images on exposures taken at the edges of the DES footprint). In this case, we compute ${\mathrm{EXP}}^{\mathrm{gray}}(\mathrm{EXP})$ and ${\mathrm{VAR}}^{\mathrm{gray}}(\mathrm{EXP})$ as in Equations (33) and (34). If there are at least five CCDs with valid ${\mathrm{CCD}}^{\mathrm{gray}}$ values and variance ${\mathrm{VAR}}^{\mathrm{gray}}\,\lt {0.005}^{2}$, then we assign the exposure estimate to the zeropoint gray value:

$$\begin{eqnarray}&&{\mathrm{ZPT}}^{\mathrm{gray}}={\mathrm{EXP}}^{\mathrm{gray}}(\mathrm{EXP}).\end{eqnarray} \tag{ 43 }$$

No explicit limit is placed on the size or sign of the ${\mathrm{ZPT}}^{\mathrm{gray}}$ correction. If both attempts fail, then the CCD image will be uncorrected for gray extinction, and the value of ${\mathrm{FLAG}}^{\mathrm{FGCM}}$ will be increased by 32.

We note that application of the ${\mathrm{ZPT}}^{\mathrm{gray}}$ correction reduces to the traditional use of a catalog of "standard stars" to estimate residual errors in the calibration fit; in this case, the standard catalog is created by the FGCM fitting step itself. It relies on the presence of a sufficient number of "photometric" observations of these stars in the fitted data sample to provide the needed reference magnitudes. No chromatic correction can be attached to a gray correction, but the retrieved chromatic integrals discussed in Appendix B might usefully flag cases that have significant residual chromatic effects; we have not yet studied this possibility in detail.

The FGCM assigns an error to the zeropoint ${\mathrm{ZPT}}^{\mathrm{FGCM}}$ (Equation (39)) that includes contributions from the error in ${{\mathbb{I}}}_{0}$ from the global calibration fit and from the error in the gray correction ${\mathrm{VAR}}^{\mathrm{ZPT}}$ computed for each CCD image. If ${\mathrm{ZPT}}^{\mathrm{gray}}$ is derived directly from ${\mathrm{CCD}}^{\mathrm{gray}}$, then the error in the gray correction is computed solely from photon statistics ${\mathrm{VAR}}^{\mathrm{ZPT}}={\sigma }^{\mathrm{phot}}{(\mathrm{EXP},\mathrm{CCD})}^{2}$ (Equation (32)). If it is derived from the average of other CCD images on the exposure, then ${\mathrm{VAR}}^{\mathrm{ZPT}}={\mathrm{VAR}}^{\mathrm{gray}}(\mathrm{EXP})$ (Equation (34)). Note that spatial structure in the residual errors on scales below the size of a CCD will not be included in this estimate.

With the assumption previously discussed that the random errors in the FGCM fit parameters from each tiling of the footprint are independent, the random error in ${\mathrm{ZPT}}^{\mathrm{FGCM}}$ can be estimated,

$$\begin{eqnarray}{\sigma }^{\mathrm{ZPT}}(\mathrm{EXP},\mathrm{CCD}) & = & \left(\displaystyle \frac{{\delta }^{2}\mathrm{FGCM}(b)}{{N}_{\mathrm{tile}}-1}\right.\\ & & +{\mathrm{VAR}}^{\mathrm{ZPT}}(\mathrm{EXP},\mathrm{CCD})\\ & & +{{({\sigma }_{0}^{\mathrm{ZPT}})}^{2})}^{1/2},\end{eqnarray} \tag{ 44 }$$

where the global fit error is $\delta \mathrm{FGCM}(b)$ (Equation (36)) and ${N}_{\mathrm{tile}}(\mathrm{EXP},\mathrm{CCD})$ is the average number of observations in band b per calibration star found on the CCD image (Equation (37)). The ${\sigma }_{0}^{\mathrm{ZPT}}$ systematic control term is again set to $3\,\mathrm{mmag}$. As discussed in Section 4.2, exposures used in the calibration fit (i.e., ${\mathrm{FLAG}}^{\mathrm{FGCM}}\lt =2$) were selected to have little, if any, gray extinction, so the ${\sigma }^{\mathrm{ZPT}}$ values for these exposures are typically $5\,\mathrm{mmag}$ or less. While the ${\mathrm{ZPT}}^{\mathrm{gray}}$ corrections averaged over the focal plane can be equally accurate, errors in magnitudes measured on non-photometric exposures (i.e., ${\mathrm{FLAG}}^{\mathrm{FGCM}}\gt =4$) can be significantly larger.

Shown in Figures 19 through 21 are the calibration errors ${\sigma }^{\mathrm{ZPT}}$ averaged over exposures with ${\mathrm{FLAG}}^{\mathrm{FGCM}}\lt =\,4$ in HEALPIX pixels across the full DES footprint that would be typical of the Y3A1 co-add catalog. We show only the g, i, and Y bands as examples, as the r and z bands look very similar. The structure seen in these plots is primarily due to correlations in the varying number of tilings of regions on the sky. For example, the supernova fields can be readily identified as individual DECam focal-plane footprints with estimated errors near $3\,\mathrm{mmag}$ (the error floor set by ${\sigma }_{0}^{\mathrm{ZPT}}$) in the griz bands.

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The DESDM software package accepts raw DECam exposures and produces "FINALCUT" catalogs of observed quantities for each object detected on science exposures. A set of quality cuts is applied to eliminate exposures on which one or more of a number of recognizable hardware failures occurred or that were taken through easily detectable cloud cover. The exposures selected for Y3A1 include wide-field survey and supernova fields, but no standard-star observations. A control log that drives the calibration process is then created.

A query (SQL) of the DESDM FINALCUT catalogs is done to make an initial selection of observations that are candidates to be used in the FGCM calibration. This query requires (Source Extractor data products are indicated in capital letters):

• 1.  
  SExtractor flag = 0 (objects that were not deblended, saturated, or had other processing problems).
• 2.  
  The object image is not within 100 pixels of any CCD sensor edge.
• 3.  
  A successful MAG_PSF fit with 0.001 < MAGERR_PSF < 0.100 mag (no explicit cut is made on the instrumental magnitude).
• 4.  
  Selection of stellar sources with CLASS_STAR > 0.75 and −0.003 < SPREAD_MODEL < 0.003.

The following process is used to identify candidate calibration stars:

• 1.  
  Remove observations that were "blacklisted" by DESDM due to known instrumental or imaging problems.
• 2.  
  Select i-band observations and identify star candidates from detections that are within 1 arcsec of each other.
• 3.  
  Remove candidates that have another candidate within 2 arcsec separation.
• 4.  
  Randomly remove candidates to limit the density of stars to approximately one per square arcmin as found at the south Galactic Cap; HEALPIX (NSIDE = 128) is used in this step.
• 5.  
  Seek observations in the remaining grzY bands that match an i-band candidate.
• 6.  
  Identify candidate griz calibration stars as those with at least two observations in each of the four bands.
• 7.  
  Identify the subset of griz calibration stars that also have two observations in the Y band.

A catalog of grizY FINALCUT observations of candidate calibration stars is created for use during the FGCM calibration process that follows.