THE SLOAN DIGITAL SKY SURVEY COADD: 275 deg² OF DEEP SLOAN DIGITAL SKY SURVEY IMAGING ON STRIPE 82

The data selection criteria are listed in Table 1. We chose all runs on Stripe 82 with 125 ⩽ run ⩽5924, i.e., all data obtained in or before 2005 December 1, demanding that the runs were either on the N or S strip and rejecting ∼10 runs that were not offset (strip labeled "O"), plus one run that is a crossing scan at ∼45° inclination. We then select the fields in the r band, requiring seeing better than 2'', sky brightness less than 19.5 mag arcsec⁻², and less than 0.2 mag of extinction. The sky brightness cut corresponds to 2.5 times the median sky of 150 DN, allowing at most 0.5 mag increase in sky noise. The majority of the data is uncalibrated so we also require that the field has enough stars for our relative calibration method (discussed in Section 3.2) to work. We cut on the r-band parameters, but rejected all the corresponding data in ugiz. This choice maximizes homogeneity across filters, though not optimization for a given filter.

Table 1. Data Selection Criteria

Scope	Criterion	Description	Acceptance Rate
Run	125 ⩽ run ⩽5924	Data taken on or before 2005 Jan 12 on Stripe 82	...
Field	r $0.265\sqrt{\rm {neff\_psf}} \le 2.0$	Seeing <2''	91%
	r $\rm {sky\_frames} \le 375$ DN	Sky brightness less than 19.5 mag arcsec−2	84%
		2.5 × the median sky of 150 DN
		allowing at most 0.5 mag increase in sky noise
	r transparency >1/1.2	Less than 0.2 mag of extinction	95%
	Ncalibration ⩾ 1	Enough stars for relative calibration	95%

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The fraction of fields passing the seeing and sky noise cuts are 91% and 84%, respectively, and 77.5% pass both cuts jointly. All of the fields in the standard SDSS runs and 95% of supernova fields pass the transparency cut and meet the requirement to have enough stars to calculate the photometric scaling. This indicates that the SDSS had a high threshold for classifying a night as photometric. Overall, 1,124,075 frames were included in various fields of the coadd.

Table 2 summarizes the 123 runs included in the coadd from both main survey and supernova runs. Sixty-nine of these were calibrated (phot = 1) in the sense above. The remaining 54 runs were uncalibrated (phot = 0). These were on average twice as long as the calibrated ones, as they were taken by the supernova survey, which was unconcerned with photometricity or sky brightness. Not all of the images in the available overlapping runs were used in a given coadded frame. In order to prevent sharp discontinuities in the PSF in the output image at input run edges, we imposed the constraint that if a field overlaps the output image, other good fields from the same run must cover the output image R.A. completely.

There are a considerable number of additional runs that could be added to the coadd if data after our date cutoff were included, which would gain roughly 0.75 mag in depth. A coadd including all the available data is reported by Jiang et al. 2014. LSST has also produced coadds from the Stripe 82 data as part of their pipeline development process (LSST Collaboration 2013, Public Note). Neither of these used PHOTO for the object photometry. The coadd reported here is likely the last to be run through the standard SDSS processing.

The standard SDSS processing calibrates the single pass data using data obtained by the 20'' PT telescope. These include a set of star fields in the Stripe 82 area and extinction values measured on the night the SDSS telescope data are obtained. The PT pipeline (Tucker et al. 2006) calibration results in runs calibrated to an rms of 1% in gri, 2% in u and 3% in z (Abazajian et al. 2003; Ivezić et al. 2004). The DR8 data (Aihara et al. 2011) has been calibrated using the overlap between runs (Padmanabhan et al. 2008), but these calibrations were not available at the time this work was performed.

Much of the Stripe 82 data was nonphotometric, so we developed a method to calibrate them. In the process we also recalibrated those runs taken under photometric conditions. The runs were calibrated following the prescription of Bramich et al. (2008), which builds a catalog-level coadd of bright stars to match stars in the frames and compute a zero-point shift. The resulting photometric calibration of a given run is good to 0.02 mag in up to 1 mag of atmospheric extinction (see also Ivezić et al. (2007) for a discussion of calibration through clouds).

The PT telescope observed the calibration patches in Stripe 82 many times while measuring the extinction for each standard run independently. Averaging stars calibrated using these independent calibrations will increase the photometric accuracy. The increase is unlikely to scale as $\sqrt{N}$ as there are systematics floors from residual flat field variations and uncorrected atmospheric transmission variations.

We used 62 of the photometric runs for which normal SDSS PT calibrations were available to construct a standard star catalog. We start with a set of bright, isolated, unsaturated stars, with 14 < r < 18, taken from a set of high-quality photometric runs covering both strips of the whole stripe acquired over an interval of less than 12 months (2659, 2662, 2738, 2583, 3325, 3388). We then match the individual detections of these stars in each of the 62 runs, using a matching radius of 1 arcsec. On average, there are 10 independent measurements of each star among the 62 runs, and we only include in the reference catalog those with 5 or more measurements. We then compute the mean of the independent calibrated flux measurements of each star and adopt that mean flux, defining it separately in each band. We use the fluxes measured in the SDSS "aperture 7," which has a radius of 7.43 arcsec; this aperture is typically adopted in the SDSS as a reference aperture appropriate for isolated bright star photometry.

Using this standard star catalog, we computed the relative zero-point offset of all fields in all runs used in the coadd, regardless of whether they were photometric or nonphotometric runs initially. The relative zero-point offset was defined as the median fractional flux difference of the standard stars in each field in the run. These field-by-field offsets are the atmospheric transmission T, which we need for weighting in the coadd as well as to place the fields onto the same calibration. The quantity T is determined for each frame (it is determined separately in each filter), and can vary substantially with time on nonphotometric nights.

The u-band images have S/N significantly poorer than the other bands and require special treatment. All u runs have a provisional calibration applied, but in case of nonphotometric runs these calibrations are purely the average instrumental zero-point. We use this approximate calibration to eliminate lower S/N stars by rejecting those with u > 18. The remaining stars are used to match against the standard star catalog to find the relative zero-point.

The flux calibrations are relative magnitude offsets from a zero-point, defined to be −23.90. We interpret the relative magnitude offsets as variations in T with respect to the mean transparency 〈T〉. Later we use T to scale the individual images prior to coaddition.

The sky brightness varies both spatially and temporally. As the data we used for the coadd came from DR7 and earlier, the improved sky subtraction of the DR8 PHOTO was not implemented (Aihara et al. 2011; see also Blanton et al. 2011 for continued work on this subtle problem). The DR7 PHOTO algorithm, used in this work, produced a sky image by calculating the median of the 256 × 256 pixel boxes in the data image on a grid of 128 pixels in each dimension. The sky was then modeled as bilinear interpolation between the medians. This algorithm oversubtracts the extended parts of galaxies on the scale of the 128 × 128 pixel grid used in the sky calculation, leaving artifacts due to astrophysical objects in the data. Given that, the PHOTO sky subtraction engine was therefore deemed not suitable for our purposes, and we developed our own method.

The sky is time dependent, so subtracting a global sky for each frame worked poorly, leaving behind a row-dependent sky gradient. Instead we adopted a sky value that depended linearly on row number. For the central frame and its two flanking frames from the same run, the median of the 2048 pixels along each row was calculated. The resulting 3 × 1489 pixel sky vector is a time series estimate of sky. Bright stars were eliminated via a 3σ 5 iteration sigma clipping, and the rms was used to reject pixels more than 2σ from the mean along the vector. A linear least squares fit to the remaining vector was used to model the sky and was subtracted from the central image.

The coadd relies on the existing astrometry produced using the astrom pipeline (Pier et al. 2003). All of the images used in the coadd had astrometric calibrations. Pier et al. (2003) were able to achieve positions accurate to ∼45 mas rms per coordinate by calibrating to the U.S. Naval Observatory CCD Astrograph Catalogue (UCAC; Zacharias et al. 2000). The accuracy is limited primarily by the accuracy of the UCAC positions (∼70 mas rms at the UCAC survey limit of R ≈ 16) and the density of UCAC sources. This accuracy can be represented in the affine transformations that are standard in the WCS convention. Pier et al. (2003) also noted that there are systematic optical distortions due to the camera present in the data. We will use the astrom measurements to remove these distortions.

The process of forward mapping requires a transformation from R.A., decl. (α, δ) to the pixel location in the input image. The SDSS runs were taken along great circles. Thus, astrom worked in a coordinate system in which each run's great circle is the equator of the coordinate system. In this great circle coordinate system, the latitude of an observed star never exceeds about 13; thus, the small angle approximation may be used and lines of constant longitude are, to an excellent approximation, perpendicular to lines of constant latitude. Longitude and latitude in great circle coordinates are referred to as μ and ν, respectively. ν is equal to 0 along the great circle, μ increases in the scan direction, and the origin of μ is chosen so that μ = α₂₀₀₀ at the ascending node (where the great circle crosses the J2000 celestial equator). The conversion from great circle coordinates to J2000 celestial coordinates is then

$$\begin{eqnarray} \tan {(\alpha -\mu _0}) &=& \frac{\sin (\mu - \mu _0) \cos \nu \cos i - \sin \nu \sin i }{\cos (\mu - \mu _0) \cos \nu } \end{eqnarray} \tag{ 1 }$$

$$\begin{eqnarray} \hspace{26pt}\sin {(\delta)} &=& \sin (\mu - \mu _0) \cos \nu \sin i + \sin \nu \cos i, \end{eqnarray} \tag{ 2 }$$

where i and μ₀ are the inclination and J2000 right ascension of the great circle ascending node, respectively. μ₀ = 95° for all survey stripes, and for Stripe 82 i ≈ 0.

Given the great circle coordinates (μ, ν), we can transform to distortion corrected frame coordinates (x',y') using the affine transformation

$$\begin{eqnarray} \mu _{CMP} & = & a + b x^{\prime } + c y^{\prime } \end{eqnarray} \tag{ 3 }$$

$$\begin{eqnarray} \nu _{CMP} & = & d + e x^{\prime } + f y^{\prime }. \end{eqnarray} \tag{ 4 }$$

The transformation from (x', y') to (x, y) accounts for optical distortions that, in drift-scan mode, are a function of column only:

$$\begin{eqnarray} & & x^{\prime } = x + g_0 + g_1 y + g_2 y^2 + g_3 y^3 \end{eqnarray} \tag{ 5 }$$

$$\begin{eqnarray} & & y^{\prime } = y + h_0 + h_1 y + h_2 y^2 + h_3 y^3. \end{eqnarray} \tag{ 6 }$$

Equations (1)–(6) provide the pixel coordinates on the input image that corresponds to a given (R.A., decl.) position.

Bramich et al. (2008) performed a recalibration of the Stripe 82 astrometry using a run from the mid-time of the Stripe 82 observations as a reference. This removed an erroneous but measurable galaxy mean proper motion of ∼10 mas yr⁻¹ in both R.A. and decl. due to the proper motion of reference stars (see Bramich et al. (2008), Figure 4). The optics distortions removed in this section have a maximum peak to peak shift of 80 mas (see Pier et al. 2003, Figure 2) and are larger than the astrometry shifts induced by stellar motion over the five-year range of the data used in the coadd. Therefore, although ideally we should have removed the reference star proper motion drift, in practice these have little effect.

We geometrically map the input images onto the output image. We defined output frames aligned along the J2000 equator with rows aligned perpendicular to R.A., in a standard SDSS image format, from −50° ⩽ α ⩽ 60° and −125 ⩽ δ ⩽ 125.

Since the output image is simply a locally flat tangent projection of the sky, the mapping must remove optical distortions and provide a surface brightness estimate at the aligned pixel location. To perform the mapping we used a version of Swarp (Bertin et al. 2002) modified to perform the astrometric conversions described in Section 4.2.

Each pixel in the mapped image is estimated from the input image pixels using a Lanczos interpolation kernel, which is a truncated sinc interpolation. Given bandwidth limited signal of infinite extent, sinc function interpolations reproduce exactly the data after resampling. Our images are not undersampled (they have 04 pixels and seeing of ∼13), but they are not of infinite bandwidth either and this motivates the use of a truncated kernel.

We used a two-dimensional Lanczos-3 kernel, retaining 3 maxima on each side of the center in each dimension. The one-dimensional Lanczos-3 kernel is L(x) = sinc(x)sinc(x/3), for −3 < x < 3. Then the two-dimensional interpolation formula is

$$\begin{equation} \hat{I}(r,c) = \sum _{i,j} I(i,j) L(r-i) L(c-j), \end{equation} \tag{ 7 }$$

where r, c are the output image pixel coordinate and i, j are the input image pixel coordinate. The Lanczos-3 window is well behaved in the sense of having a minimum of aliasing and smoothing and minimal ringing, but a Lanczos-3 interpolation does use (2 × 3 + 1)² = 49 pixels to estimate the value of one output pixel. This is too large to use for bad pixels (e.g., saturated pixels), so we use the nearest neighbor interpolation and reduce the weight of bad pixels during Lanczos-3 interpolation by using a mask (see Section 4.8).

To keep track of the variance of data images, pixel by pixel inverse variance images are a natural choice, but they produce biases in the resulting mean. At low S/N, the upward fluctuations in signal are given more weight than downward fluctuations as a result of the one-sided nature of the Poisson distribution. This bias is deterministic and one could correct for it; however, for example, for u-band data with its 120 e⁻ of sky noise, pixels at 1σ above sky would be biased by 0.5% and this would be a fair fraction of our photometric error budget. Another problem is that per-pixel inverse variance weighting systematically changes the shape of the PSF as a function of the magnitude of the object. This would cause serious complications to our PSF-based photometry using PHOTO.

Because in the end we want the inverse variance map as a component in the accounting of the sky noise in the final coadd, we chose a method suitable for this. The variance of the sky was measured on each frame from the width of the sky histogram. As we used a linear gradient sky subtraction, each image is assigned a variance image that is the variance of the subtracted linear sky gradient. This assumes Poisson statistics while the data are in ADU. We did not include the effective gain, g_eff, in calculating this variance, but the gain varies by less than 30%.

The remapping stage will introduce sky pixel correlations in the data image at the scale of the interpolation kernel. Our approach was, as described in Section 4.8, to remap the inverse variance images in the same manner as the data.

The geometry mask keeps track of which pixels in the input image actually contribute to the coadd. In this mask definition we account for image defects found by PHOTO, in particular, cosmic rays, saturated pixels, and bad columns. We refer the reader to the PHOTO papers (Lupton et al. 2001; Stoughton et al. 2002; Lupton et al. 2012) for details on how PHOTO detects and characterizes these defects.

PHOTO produces a 16 bit mask image, the fpM file. For the geometry mask we are interested in the INTERP bit, which is set for any pixel for which PHOTO used interpolation to fill its value. This happens for cosmic rays, saturated pixels, and bad columns. These pixels are poor but not useless estimates of the true value of the pixel. We set the geometric weight of such pixels to a small value, 0.0000001. This ensures that if there is no input image that contributes a good estimate for a given output frame pixel, such as the center of a saturated star, the interpolated value is used.

The geometry of the SDSS images is also encoded into the geometry mask. In SDSS images, the first 124 rows of each frame are duplicates of the last 124 rows of the preceding frame. This replication of pixels was done so that objects could be well measured despite being on edge of a frame. To account for this, the first 124 rows of the mask are set to 0. There is also a roughly 1 arcmin overlap between adjacent camcols in the north and south strips, allowing the properties of objects near the edges to be measurable. We did not coadd the overlap between the camcols, but simply kept the north and south strip data separate.

PHOTO's bit mask also contains information about saturated pixels, encoded in the SATUR bit. The SATUR flag is set for any pixel that PHOTO determines to be saturated. We set these pixels to one in an image otherwise filled with zeros. This is the satur mask, which we need for running PHOTO on the coadded images.

Although closely related, the weight map and the inverse variance image are not the same. The weight map is the inverse variance image multiplied by the geometry mask. We set all masked nonzero pixels to the INTERP flag value (0.0000001). This allows us to keep track of the pixels altered by the masking.

We use the weight map as input for the coaddition process, in which a weighted clipped mean of the data images, all mapped onto the same output image, will be performed. In addition, we coadd the inverse variance images and the satur maps as well, and although for those we use straight sums, the weight map plays a role in the sense that only the pixels that pass the clipping for the data coadd are included (for details, see Section 4.9.2).

The inverse variance map, satur mask, and weight map are all mapped onto the same output image as the data. For the inverse variance and weight images, we apply the same Lanzcos-3 interpolation used for the data in order to replicate the noise correlation. The satur masks were mapped using a nearest neighbor interpolation. This is more suitable than Lanzcos-3 because masked pixels affect only their immediate neighbors.

In the process of performing the wide Lanzcos-3 interpolation of the data, the geometry mask is used to prevent masked pixels from contributing to the interpolation with more than minimal weight. In addition, once the images are mapped onto the output image, any non-overlap is set to zero weight using the geometry mask.

After this mapping procedure, each stack of images is ready for the coaddition process. No scaling has been done at this stage, so the stack for a given output image can be further filtered and/or weighted as needed.

With the four stacks of images (data, inverse variance, weight, and satur) all aligned and cropped to match the output image, we have all the inputs needed to produce the coadded images and run PHOTO on them. We chose to use all the data in our image set. This allows us to maximize the depth of the coadd data.

Alternatively, one could devise a selection criteria to produce a coadd data set tailored for a specific purpose. For example, one could select only images with the best seeing and limit them in number to form a uniform depth, aiming at weak lensing studies.

4.9.1. Weights

The data in our image set are of variable quality and we wish to optimize our coadd, so we designed a weighting scheme. The S/N of the measurement of flux from a star is

$$\begin{equation} {\rm S/N} \propto {{N_{\rm photons}}\over {\sqrt{A_{\rm psf}} \sigma _{\rm sky}}}, \end{equation} \tag{ 8 }$$

where N_photons is the number of photons detected from the source, A_psf is the area in pixels the source subtends, and σ_sky is the sky noise per pixel. As N is proportional to transparency T and A∝FWHM², a reasonable choice for the weights is

$$\begin{equation} w_i = {{T_i^2}\over {{\rm FWHM_i}^2 \sigma _i^2(p_i)}}. \end{equation} \tag{ 9 }$$

We choose these weights because although the usual inverse variance weighting produces the minimum variance image, the S/N for a star depends on the square of seeing. However, due to a typo in the code, we actually used w_i = T_i/(FWHM_iσ_i)². Despite this issue, the weights used give the highest weight to good seeing data taken when the sky is clear and dark. Since this issue was uncovered well after the production run was finished, it was judged as impractical to fix. The weights are calculated individually for the images of each bandpass, so the weights choose the set of best images in seeing, noise, and transparency for each filter. With this weight, the PSF of the coadd is 03 less than the median of the input images in ugri and 02 less in z.

We take the seeing for each field and filter from the tsField file output by PHOTO, which is simply an average over the frame, and use it regardless of how much the frame contributes to a given output coadd frame. The transparency T, in turn, is already available as a product of the photometric calibration discussed in Section 3.2. We have therefore cataloged for each image the transparency, the seeing, and via the inverse variance image, the sky variance. This allows us to proceed to the coaddition.

4.9.2. Weighted Clipped Mean

Figure 3 shows a side-by-side comparison between coadd and single pass data in the r band. The single pass counterpart is one out of 28 images used in the coaddition, which means that it passed all the cuts discussed in the text. This example features run 206, camcol 3, field 505 in the r band. It illustrates the fact that a number of objects bellow the detection threshold of each image can be well detected and measured in the coadd. The seeing, however, is larger for the coadd image in this example.

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Our principal challenge is to measure the PSF of the coadd. The PSF of SDSS images varied in time, corresponding to image rows, due to atmospheric fluctuations. It also varied in space, corresponding to columns, due to camera optics. In the standard single run reduction, PHOTO fits the spatial variation of the PSF by computing a PSF basis set using a Karhunen–Loéve (KL) expansion. The stars in the frame and the two flanking on either side (five in total) are used to determine the KL basis functions B_R(u, v):

$$\begin{equation} P_{i}(u,v) = \sum _{r=1}^{r=n} a^r_{i} B_R(u,v), \end{equation} \tag{ 10 }$$

where P_i is the PSF of the i^th star, u, v are the pixel coordinates relative to the basis function origin, and n sets the number of terms to use in the expansion (we use n = 3). The stars in the image and the flanking half an image on either side (two in total) are used to determine the KL coefficients $a^r_{i}$:

$$\begin{equation} a^r_{i} \approx \sum _{l=m=0}^{l+m\le N} b^r_{lm}x^l_{i}y^m_{i}, \end{equation} \tag{ 11 }$$

where x, y are the coordinates of the center of the i^th star, N is the highest power of x or y included in the expansion (we use N = 2, a quadratic spatial variation of the PSF), and the $b^r_{lm}$ are found by minimizing

$$\begin{equation} \sum _i \left[P_{i}(u,v) - \sum _{r=1}^{r=n} a^r_{i} B_r(u,v)\right]^2. \end{equation} \tag{ 12 }$$

Each of the input runs had already been separately processed through PHOTO, and thus the KL basis set and coefficients had already been determined for them. We use this to calculate the effective PSF at each point in the coadd image as follows. Using the KL basis set for each input run corresponding to the given output frame and the two flanking frames, we compute a weighted sum of the model PSFs (using the weights of Equation (9)) on a two-dimensional grid with a spacing of ∼15. We then fit the KL basis B_R(u, v)'s to these over the coadd frame and the two flanking frames. The computation of the coefficents of the PSF expansion $a^r_{i}$ are done on the coadd frame and the two flanking half-frames as in the single pass runs, with the exception that we set the maximum number of coefficients, n in Equation (10), to n = 4 and the highest power of x, y, N in Equation (11), to N = 3.

PHOTO had to be modified to read in a file containing the weights and effective gains of the images. The effective gain of a coadd image G should be such that it gives bright objects Poisson statistics for the variance associated with a source when scaling the averaged pixel counts back to electrons. Therefore,

$$\begin{equation} \sigma ^2(P) = P/G, \end{equation} \tag{ 13 }$$

where P is the weighted sum, over all exposures i, of the object counts p_i,

$$\begin{equation} P = \sum w_ip_i = \sum w_i T_i o_i, \end{equation} \tag{ 14 }$$

and p_i ≡ T_io_i is the object's flux per exposure o_i times the flux scale factor T_i. The corresponding variance is

$$\begin{eqnarray} \nonumber \sigma ^2(P) & = & \sum w^2_i\sigma ^2(p_i) \\ \nonumber & = & \sum w^2_i T^2_i\sigma ^2(o_i) \\ \nonumber & = & \sum w_i^2T^2_io_i/g_i \\ & = & \sum p_iw_i^2T_i/g_i. \end{eqnarray} \tag{ 15 }$$

Assuming that p_i does not vary much from image to image, we obtain

$$\begin{equation} \sigma ^2(P) \approx \! \langle p_i \rangle \sum w^2_i T_i/g_i = P \sum w^2_i T_i/g_i. \end{equation} \tag{ 16 }$$

From Equation (13) and (16) we conclude that the effective gain is

$$\begin{equation} G = \left[ \sum w_i^2T_i/g_i\right]^{-1}. \end{equation} \tag{ 17 }$$

In the SDSS the gains g_i of all single-run frames going into a single coadd frame are the same (it is always the same CCD), so they may be replaced by a single g outside the sum:

$$\begin{equation} G = \frac{g}{\sum w_i^2T_i}. \end{equation} \tag{ 18 }$$

To process the images with PHOTO we also need to compute the effective sky level S'. Assuming that dark and read noise are accounted for in the Poisson noise of the effective sky, we can easily obtain S' using the effective gain G calculated in Equation (18) and the relation S' = Gσ²(S'), which is analogous to Equation (13). We used the clipped variance of the coadd frame to compute σ²(S'). Alternatively, one could take the sky variances and weights of the input frames and compute it using $\sigma ^2(S^{\prime })=\sum w_i^2T_i^2\sigma ^2(s_i)$.

Using this method we obtained, for each field and for all five filters, the effective sky brightness (S'), sky noise (σ(S')), and gain (G, Equation (18)). With these quantities in addition to the weights (w_i, Equation (9)), we processed all of the coadd fields through PHOTO.

The SDSS code Target is used to apply the calibration to the raw outputs of PHOTO, following Section 3.2. Following Lupton et al. (1999), we convert from fluxes f to mags m using

$$\begin{equation} m(f/f_0) = -\frac{2.5}{\ln 10} \left[{\rm asinh}\left(\frac{f/f_0}{2\,b}\right) +\ln (b)\right], \end{equation} \tag{ 19 }$$

where b is an arbitrary dimensionless softening parameter below which the magnitude scale goes from logarithmic to linear, and f₀ is a reference flux that sets the zero-point, zp ≡ m(0), of the magnitude scale. The values of b that we used for the coaddition are given in Table 3, along with the asinh magnitudes associated with a zero flux object and the magnitudes corresponding to f = 10f₀b. Above this flux, the asinh magnitude and the traditional logarithmic magnitude differ by less than 1% in flux. These values can be compared to their equivalent numbers for the main survey, given in Table 21 of Stoughton et al. (2002). The coadd images were all placed onto a uniform flux scale such that 1 DN corresponds to a flux of 1 picomaggie, corresponding to a logarithmic (and not asinh) AB magnitude of 30.

Table 3. Asinh Magnitude Softening Parameters for the Coadd

Filter	b	zp	m(10b)
u	$1.0\phantom{0}\times 10^{-11}$	27.50	24.99
g	0.43 × 10−11	28.42	25.91
r	0.81 × 10−11	27.72	25.22
i	$1.4\phantom{0}\times 10^{-11}$	27.13	24.62
z	$3.7\phantom{0}\times 10^{-11}$	26.08	23.57

Notes. Values reported by Abazajian et al. (2009). Column zp is the zero-point magnitude, zp ≡ m(0). The final column gives the magnitude (here an AB magnitude) associated with an object for which f/f₀ = 10 b.

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PHOTO uses a variety of measures of the brightness of objects in the image. In particular, every object is fit to the local PSF model, giving a so-called PSF magnitude, and to an exponential or de Vaucouleur's galaxy profile convolved with the PSF (the "model magnitude"; Abazajian et al. 2004). A star will have the same brightness in the two, and in single runs, we have found that stars can effectively be separated from galaxies with the cut |r_psf − r_model| ⩽ 0.145. At r = 21, this gives the correct classification (as calibrated, e.g., from deep Hubble Space Telescope images) 95% of the time.

The coadd data extend to magnitudes at which galaxies significantly outnumber stars (see discussion in Section 6.6) and we found that we need to use a more stringent criterion

$$\begin{equation} |r_{\rm {psf}} - r_{\rm {model}}| \le 0.03 \end{equation} \tag{ 20 }$$

to select stars. The threshold value of 0.03 was chosen empirically by examining the PSF−model magnitude plots. See Section 6.2 for a detailed discussion, including an example of a typical PSF−model magnitude plot. As PHOTO measures every parameter for every object regardless of its determination of its type, this has no effect on the other measurements. We discuss how well this star–galaxy separator works in Section 6.3.

We use the standard star catalog of Ivezić et al. (2007) to verify our photometric calibration. To build that catalog Ivezić et al. (2007) took the median of individual measurements of bright stars from 58 Stripe 82 runs and then applied several corrections to their catalog: (1) a color-term-like correction for the bandpass of each camera column; (2) a purely R.A. flat field correction in r derived by comparing PT data with SDSS data; and (3) a purely decl. flat field correction to the colors relative to the r band from stellar locus colors. We will take the resulting catalog (which is calibrated to 1% accuracy) as the truth and compare with our own measurements. As Ivezić et al. (2007) applied corrections that we did not, the comparison is not completely circular.

We sliced our star catalog into a series of 1 mag bins and matched to the Ivezić et al. (2007) catalog using a 1'' matching radius, discarding objects with more than one match inside that radius. No flags were applied to our star selection; in particular, we did not demand an isolated, well-measured set of stars to start with.

Table 4 summarizes the results of our comparison. We defined Δ_i as the median of the difference between our measurements and the quantities reported in the standard star catalog, for magnitudes (i = m) and colors (i = c). Δ_i is a measure of the zero-point offset. Statistical uncertainty in the zero-points, obtained as the rms of the differences, are of the order a few millimag$/\sqrt{N_{\rm {obj}}}$; as usual in photometry we can expect systematics to dominate this. The offsets from the standard zero-points are less than 5 mmag (1% photometry corresponds to 10 mmag) in all cases.

Table 4. Relative Zero-points for Selected Coadd Catalog Samples

filter	Nobj	mag	Δm	color	Δc	〈Δm〉R.A.	〈Δc〉R.A.	$\langle \Delta _m\rangle _{\rm {{\rm d}ec{\rm l.}}}$	$\langle \Delta _c\rangle _{\rm {d}ec{\rm l.}}$
u	1311	20–21	1.2 ± 1.6	u-g	−4.3 ± 3.6	−21.0 ± 8.2	−12.1 ± 8.1	−21.3 ± 28.8	−12.4 ± 15.3
g	1399	19–20	2.2 ± 1.6	g-r	−1.7 ± 1.4	6.5 ± 4.6	−1.2 ± 3.5	5.8 ± 10.2	−1.1 ± 6.3
r	3703	19–20	3.7 ± 1.5	r-i	−2.9 ± 2.2	3.2 ± 2.5	4.0 ± 2.1	3.2 ± 6.3	4.0 ± 4.1
i	7436	19–20	0.9 ± 2.0	i-z	−1.5 ± 2.3	0.1 ± 2.0	6.3 ± 3.1	1.0 ± 6.2	5.7 ± 6.8
z	3963	18–19	4.3 ± 2.4	...	...	−5.7 ± 3.3	...	−5.3 ± 11.7	...

Notes. Rows correspond to samples created independently for each filter. Magnitude ranges are indicated by the mag column. N_obj is the number of stars in each sample. Δ_i is the median zero-point difference, in millimags, for either magnitudes (i = m) or colors (i = c), and 〈Δ_i〉_j is the median of the mean difference in spatial bins, RA or Dec.

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We also examined the spatial variations of the zero-point offset and its uncertainty. To this end, we first took 50 equally spaced bins of width 22 in R.A., and computed the mean differences in magnitudes and colors for each bin. We repeated this procedure in decl. bins, choosing 30 bins of 5' width. Figure 4 shows these mean zero-point offsets as a function of R.A. and decl., indicating that spatial variations are non-negligible, especially for the u band. As a measure of the overall offsets, we computed the median of those means, 〈Δ_i〉_j, where j refers to either R.A. or decl. bins. These values are also included in Table 4 with the uncertainties estimated as the rms around the means.

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Our comparison to the standard star catalog is quantified in Figure 4 and Table 4. Along the R.A. axis in Stripe 82 the calibration varies by 5 mmag in g, r, i and z. The colors g − r, r − i, and i − z similarly vary by less than 5 mmag. In u and u − g, the variations are larger, but still less than 10 mmag. The variations in Dec are somewhat worse, indicative of optics issues in the camera and in the PT flat-field images. They are <5 mmag in g, r, i, z, g − r, r − i, and i − z, but are 30 millimag for u and 20 mmag for u − g.

While we are able to see variation in the coadd catalog relative to the standard star catalog, we applied no corrections based on these. For one thing, at this level of precision it is not clear that the standard star catalog is right (Schlafly et al. 2012; Tonry et al. 2012; Magnier et al. 2013). Work on finding the best possible calibration for the Stripe 82 data continues.

We verified how well we modeled the PSF in several ways, starting by computing the spatial variations in seeing, which can be written as

$$\begin{equation} {\rm FWHM(arcsec)} = 2 \sqrt{2\ln {2}}\sqrt{{\tt mrrcc}/2}\cdot S, \end{equation} \tag{ 21 }$$

where S = 0396 is the pixel scale and mrrcc is the sum of the second moments of the PSF (see Stoughton et al. (2002) for the formal definition of mrrcc, an adaptive second moment measure of size). We used high S/N stars in each bandpass for this test. Our results, illustrated on the left panel of Figure 5, show that the seeing is best in the redder filters, consistent with a Kolmogorov seeing law with the exception of the r band and i band. We interpret this as an effect of selecting the input frames for the coadd in the r band (see Table 1) associated with an effect of the timescales of the Kolmogorov law. Recall that the data making up a field are taken at different times, ranging over eight minutes from the r band to the g band. Our data support the assumption that the timescales of Kolmogorov seeing are such that the seeing is uncorrelated after ∼one minute, but since i and r lie next to each other in the imager, they can be correlated. The median of the seeing of the single pass images are a few tenths of an arcsecond worse than the points on this plot; this reflects the weighting of the coadd (Equation (9)).

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Figure 5 also shows the seeing as a function of decl., averaging over R.A. The seeing is affected by the camera optics, which causes the upturn at one end of the camera, The N strip (run 206) has about 0075 worse seeing in ugiz, and 015 worse seeing in r, taken on the S strip (run 106), presumably due to the statistics of the seeing in the input images. However, at decl. ≳ +05, as the camera optics begin to dominate, the seeing difference becomes negligible.

Another interesting test of our PSF modeling is to check the reconstructed PSF at the position of stars. In Figure 6 we show the mean ratio of mrrcc for stars and the reconstructed PSF at the locations of stars, as a function of decl. The strong declination dependence seen in Figure 5 is not apparent. This suggests that the modeling is fitting the spatial variation of the PSF well. We also found no correlation between the statistics of star galaxy separation and column number, again suggesting reasonable success in the PSF modeling.

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However, given the importance of the PSF modeling for crucial aspects of the data, such as star/galaxy classification, accurate photometry, and shape measurements, we examine it in more detail. A sensitive test of the PSF modeling is the r-band "PSF minus model" plot, r_psf − r_model versus r_psf. This is shown in Figure 7 for both stars (red) and galaxies (blue) (for details on the star/galaxy separation, see Section 5.4). Stars are expected to be found at a very narrow region around r_psf − r_model = 0, and this is clearly the case for the bright magnitudes in our plot. There is a noticeable trend toward negative values of r_psf − r_model = 0 at the faint end, around magnitude 23, and an upward spread at about magnitude 22. The upward spread is, in fact, stars (see Section 6.3) but is not due, for example, to stars overlapping faint galaxies as one can convince oneself by calculating the surface densities of the object. These two features in the diagram suggest that we have magnitude-dependent PSF fitting problems that may introduce systematics in analyses involving the coadd data.

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While probing this, we found that during the processing described in Section 4.9.2, the coadd incorrectly set the error propagation value to T instead of the correct value of T². This means that the weight actually used was

$$\begin{equation} w = {{1}\over {{\rm FWHM}^2 \sigma ^2}}. \end{equation} \tag{ 22 }$$

This has little effect on the coadded image as we only selected those images with 1/T ⩽ 1.2. The PSF photometry, however, depends so sensitively on the modeled PSF that this may be the root of the problem. The PSFs were constructed using the correct weights (Section 5.1), but these differed from those of the images. In an experiment we ran Photo with input weights that were pure inverse variance instead of SN weight that the images were built with. The resulting PSF model versus PSF plots showed stars with deviations away from zero at 0.05 mags/mag level. The effects we are seeing in Figure 7 are of order 0.01 mags per magnitude, consistent with the small effect the transmission has on the weights. We conclude that the low level PSF problems indicated by the structure in Figure 7 is due to this mismatch of weights between the coadd images and the coadd photometry measurements. This issue was discovered well after the production run was finished, and it was judged as impractical to fix.

These misclassified stars are an issue as they lie in the magnitude–color space of LRGs. The contamination level of the galaxy catalog by misclassified stars was estimated using the morphological-independent redshift survey catalog from the Virmos Very Large Telescope Deep Survey (VVDS; Le Fèvre et al. 2005), in particular the i < 22.5 22 hr field. We selected the objects above 95% confidence level, which resulted in a catalog containing 5158 stars and 4264 galaxies. This catalog served as a truth table, to match galaxies in the coadd at R.A. ∼ −25°; 908 spectroscopic stars and 3438 spectroscopic galaxies matched galaxies in the coadd catalog, indicating a contamination level of 18%. This is ∼three times higher than expected, assuming that the performance of the SDSS Star/Galaxy separator is as good for the coadd as for the single pass data. The field in question is among the lowest galactic latitudes and highest stellar densities in the coadd, so this contamination rate is approximately an upper limit. The actual contamination rate would be roughly proportional to R.A. in the coadd.

We verified that most of the problematic objects are in a localized region of the PSF−model versus magnitude space, specifically inside the triangular region

$$\begin{eqnarray} \begin{array} {@{} lll} r_{\rm {psf}}-r_{\rm {model}} & < & 0.1 \cdot r_{\rm {psf}} - 2.1 \\ r_{\rm {psf}}-r_{\rm {model}} & > & 0.03 \\ 21 < r_{\rm {psf}} & < & 22.5. \end{array} \end{eqnarray} \tag{ 23 }$$

This indicates that an improved star galaxy separator can be designed using morphology and presumably color cuts.

We used the package 2DPHOT (La Barbera et al. 2008) to measure the completeness of the coadd star and galaxy catalogs. This is done by adding simulated objects to the images and computing the recovery rate for both stars and galaxies. The parameters used to generate the objects are taken from the image itself. 2DPHOT detects the objects in the image, performs star–galaxy separation, and measures the photometric and structural (Sérsic) parameters (PHOTO is not used in this process). Then it creates a list of objects that reproduces the magnitude and size distributions of the stars and galaxies found in the image and adds these simulated objects to the image. Finally, it measures the new image and computes the completeness as the fraction of objects recovered in each magnitude bin.

The resulting 2DPHOT completeness versus magnitude curves, C(m), are well fit by a Fermi–Dirac distribution function

$$\begin{equation} C(m) = \frac{f_0}{1+exp((m-\mu)/\sigma)}, \end{equation} \tag{ 24 }$$

where μ is the magnitude limit of the catalog (defined to be the magnitude at which C(m) = 50%), f₀ is a normalization constant, and the parameter σ controls how fast the completeness falls when it reaches the completeness threshold. We use this fitting function to determine the depth of the coadd galaxy and star catalogs, μ_G and μ_S, respectively.

Our results are illustrated in Figure 8. The plots on the first row show the r-band completeness for the same two fields pictured in Figure 3. On the right we have the coadd field (run 206, camcol 3, field 505) and for comparison, on the left, one of the 28 single pass images used as input for that particular field. The coadd reaches r = 24.3 for point sources and r = 23.4 for galaxies, going about 2 mag deeper than a single pass image, as expected. The plots on the second and third rows show the coadd results for the other filters. Table 5 is a compilation of the median and rms values of the coadd magnitude limits, calculated for 50 fields randomly selected.

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Stars populate a well-defined locus in the color–color space almost independent of magnitude. Therefore, color–color diagrams are useful to assess the quality of the photometry. Figure 9 shows the color–color diagrams of an isolated and well-measured sample of coadd stars selected at −5° ⩽ R.A. ⩽0°, in a high galactic latitude in Stripe 82 (see query in the Appendix). The sample was split into 1 magnitude bins in the r band. At brighter magnitudes, the intrinsic thinness of the stellar locus is apparent. At fainter magnitudes statistical noise begins to dominate. Since this is an r-band sample, the u-band stars in the 20 < r < 21 panel are quite faint; one cannot read from these diagrams where the S/N of the data degrades. One can read in these diagrams just how good the photometry can be for clean samples of stars.

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Due to an error in the code, the saturated flag did not properly propagate through PHOTO. With this in mind, the photometry of objects with m < 15.5 should not be trusted.

The number counts of stars and galaxies allow us to assess the depth of the coadd and the success of the star/galaxy separation. Figure 10 shows the i-band star and galaxy counts in patches of area 5 deg² at a variety of galactic longitudes and latitudes along Stripe 82. The galaxy number counts show the Euclidean m^0.6 power law expected at i < 20, the slow change of slope due to cosmological volume and galaxy evolution at fainter magnitudes, and a roll-off at i ≈ 23.5 due to completeness issues. We have seen in panel 5 of Figure 8 that the galaxy catalog is ≈50% complete at i = 23, consistent with the deviation from the slowly rolling power law index seen here. We conclude that this plot shows nothing seriously wrong with the galaxy counts from 16 ⩽ i ⩽ 23.5.

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The star counts are slightly more problematic. They are rough power laws that cross the galaxy counts at i ≈ 20. The expected roll-over from incompleteness is apparent near i ≈ 23.5. The steep rise in counts of stars (seen easiest in the R.A. = +57° patch) is from galaxies being misclassified as stars: the galaxies outnumber the stars by a factor of ≳ 30, so even a small misclassification rate results in a substantial increase in stars. At low S/N, where faint galaxies are compact, the PSF and model magnitudes become similar. Once this happens (Figure 7), stars can no longer be distinguished by galaxies using PSF model alone.

The models are from Trilegal star count modeling (Girardi et al. 2005, online version v1.4). In Figure 10 the galactic model parameters are those found by Jurić et al. (2008) using their SDSS star count tomographic mapping technique. The slope of the rough power law is due to a combination of thin disk at brighter magnitudes and halo stars at fainter magnitudes. This model does not fit the data particularly well in the R.A. =−31° patch, or at lower R.A. Models with a lower exponential scale height, such as those derived from SDSS data using Trilegal model fits by B. Santiago (2011, private communication) or from the SDSS M-dwarf fits of Bochanski et al. (2007) do a good job of describing the Stripe 82 data at R.A. ≲ −31°. Figure 11 shows number counts in all five filters for the R.A. = −31° patch. The bands indicate the range of star counts predicted from the Santiago and Bochanski galactic models. Note the sudden increase in star counts at faint magnitudes in all filters; this is an artifact due to leakage of galaxies into the objects classified as stars. Despite this problem, we see reasonable agreement between the Stripe 82 star counts and the models. For a detailed study of the Milky Way structure in Stripe 82, see, e.g., Sesar et al. (2010).

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