THE CANADA–FRANCE–HAWAII TELESCOPE LEGACY SURVEY: STACKED IMAGES AND CATALOGS

The MegaCam camera is a wide field camera mounted at the prime focus of the CFHT. It is a mosaic of 36 CCDs, each measuring 2048 × 4612 pixels, arranged in a 9 × 4 grid. The field of view is just under 1° on a side. The resolution of MegaCam is 0.187 arcsec pixel⁻¹. It is located behind a wide field corrector and an image stabilizing unit. For further details about the MegaCam/MegaPrime camera, the reader is referred to Boulade et al. (2003).

MegaCam is equipped with five broadband filters: u*g'r'i'z'. The bandpasses are shown in Figure 6. Halfway through the survey, the i' filter was damaged. It was replaced with a new filter with a slightly bluer bandpass than the original. The old and new filters are labeled i'₁ and i'₂, respectively, in the figure.

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All the CFHTLS images were preprocessed using the Elixir pipeline at CFHT before delivery to the CADC. For full details on the Elixir pipeline, the reader is referred to Magnier & Cuillandre (2004) and the CFHT Elixir Web site.² What follows is a summary.

Elixir first applies an overscan correction for each amplifier of each CCD (two amplifiers per CCD). The bias frame is built from 20 to 30 frames, once per observing run. No dark current corrections are applied.

Flat fielding is done using twilight flats. These twilight flats are modified using a photometric superflat to correct for scattered light and changes in the pixel scale and optical transmission across the field of view of the detector. The flats are generated on a run-by-run basis.

For the i' and z' images only, fringes are removed. A master fringe image is constructed from a large number of images, suitably scaled. The master fringe image is scaled and subtracted from each image.

An astrometric calibration is done by using the known plate scale and image orientation to get an approximate plate solution. This is then refined using the USNO catalog (Monet et al. 2003). The final solution is typically good to better than an arcsecond.

Elixir does a photometric calibration based on observations of Smith et al. (2002) standard stars taken at the beginning and end of each night. The accuracy and necessary corrections to this calibration are discussed in Section 5.2.

Elixir provides a pixel mask for MegaCam images indicating the location of bad pixels and columns. It is used in the stacking procedure described in Section 6.

The images were selected from the CFHT archive at the CADC. All images with centers within 005 of the nominal pointing centers were included. For the most part these are CFHTLS images; however, this also includes a number of public images from other programs. The COSMOS field consists of four MegaCam pointings arranged around the D2 Field and centered on it. The u*-band images from the COSMOS program were included. Each of these images covers about a quarter of the D2 Field. Images taken in the new i' filter were not included in the Deep Field stacks. The complete list of CFHT exposures that were included in the stacks is available online.³

The images were checked in a number of ways before being included in the stacks. SExtractor was run on each image. The stellar locus was identified automatically. The properties of objects in the stellar locus were used to set rejection criteria.

The image quality (IQ; i.e., seeing) was measured. If the seeing was worse than 15, the image was rejected. Figure 7 shows the distribution of seeing for the CFHTLS input images split by survey and filter.

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In addition, if the PSF was noticeably trailed (as a result poor tracking), the image was rejected. If a and b are the semimajor and semiminor axes of sources in the stellar locus, the asymmetry index, A, is given by

$$\begin{equation} A=1-b/a. \end{equation} \tag{ 1 }$$

The average of this index over each field, $\overline{A}$, is always slightly greater than 0 owing to the optical aberrations in the MegaPrime optics. This index is crude but is more than sufficient to reject trailed images. Images with an average asymmetry index $\overline{A} >0.2$ are noticeably trailed and were rejected.

Images taken in conditions of poor transparency were also rejected. If the measured zero point was more than 1 mag below the nominal value, the image was rejected.

Finally, a subsection of one chip of each image was examined by eye for other abnormalities. For example, one image flaw that cannot be detected automatically is a discrete jump in telescope position during the exposure. In this case, each source is doubled (and bright stars will leave a trail between the two images) but the PSF will stay the same. This is immediately obvious to the eye, but hard to detect automatically.

Two versions of the Deep Field stacks were produced: the "full" stacks and the "best seeing" stacks. This section describes how the images that went into the "best seeing" stacks were selected.

When making "best seeing" stacks, one must decide how many input images to include and how many to throw away. There is a trade-off between limiting magnitude and IQ. As one includes more images with increasingly bad IQ, the output IQ degrades while the depth increases.

Quantitatively, the IQ of a stack of N images is very well estimated by the median of the IQ of its input images (i.e., the IQ of input image N/2 if the images are sorted by IQ). To estimate the limiting magnitude, two assumptions were made: first the effects of IQ on depth are not important (not quite true) and second all the input images have the same exposure time (for the CFHTLS Deep Fields, quite true). In this simplified case, the limiting magnitude of a stack of N images goes approximately as 1.25log₁₀N.

These two measures were applied to the input images of the CFHTLS Deep Fields. The images were sorted in order of IQ: best seeing first, worst seeing last. As the number of input images increases, the IQ of the stack will get worse, but the limiting magnitude will get deeper. The results are shown in the grid of plots in Figure 8. The expected IQ is plotted against expected limiting magnitude for each of the five filters and four Deep Fields. The black lines show both IQ and limiting magnitude increasing with number of images. The solid points show the locus of the "full" stacks.

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It can be seen that while the median of the IQ of the input images is an excellent predictor of the IQ of the output stacks, the limiting magnitude prediction is less accurate. Deviations of several tenths of a magnitude from the prediction are not uncommon. This is not surprising, given the simplicity of the underlying assumptions.

The question becomes: is there a point on the graphs where an improvement in IQ comes at only a small cost in limiting magnitude? Some of the graphs exhibit kinks where the slope of the line changes. Obviously, it is better to be at or slightly below these kinks. Ideally all the output images would have the same IQ to increase the accuracy of matched aperture photometry.

Instead of picking a single threshold (for example, stacking the best 25% of the images, or stacking only images with IQ = 07 or better), images were included to produce a desired output IQ. This is possible because, as shown by the open dots in Figure 8, the median input IQ is an excellent predictor of the output IQ. Therefore, when choosing input images, one first sorts the images by increasing IQ. Then one goes down the ordered list until an input image with the target IQ is found, and then selects twice that number of images. For example, if one wants a 065 seeing stack, and there are 41 images with seeing better than 065, one should stack the best 82 images.

The chosen target IQ was 065 for the g'r'i'z' filters and 08 for the u* filter. Decreasing the seeing below 0.8 becomes rapidly prohibitive for the u* band. Similarly, the z'- and particularly the g'-band depths decrease rapidly if the target IQ is decreased even slightly. The u*-band target is different than the other bands because targeting IQ = 065 would include no images, and targeting IQ = 08 in the other bands would improve the seeing only slightly relative to the "full" stacks.

These criteria were applied to the input images of the CFHTLS Deep Fields and the chosen images were stacked. The resulting IQ and limiting magnitudes are plotted as solid points in Figure 8. The IQ is consistently on (or very slightly below) the prediction. The limiting magnitudes are usually better than predicted, since the prediction relies on the (not completely correct) assumption that IQ does not affect depth.

SExtractor (Bertin & Arnouts 1996) was used to generate the image source catalog. The source catalog was limited to brighter sources with well-defined centers. The parameters were set such that sources must have five continuous pixels 5σ above the sky level in order to count as a detection (DETECT_MINAREA=5 and DETECT_THRESH=5 in SExtractor parlance). The catalog was cleaned. Cosmic rays were rejected by removing sources that were more compact than stars as measured by their half-light radius. Extended sources were identified and rejected as those sources whose Kron magnitudes were more than 2 mag greater than their 15 radius magnitudes.

A bootstrapping series of astrometric reference catalogs were used. The Naval Observatory Merged Astrometric Dataset (NOMAD; Zacharias et al. 2004) was used as the first external astrometric catalog. NOMAD has a higher source density and slightly better astrometric accuracy than the USNO catalog used by Elixir. Where available, the SDSS DR7 (Abazajian et al. 2009) was used to refine the astrometry. The final reference catalog was generated from a subset of the CFHTLS images themselves. For the Deep Fields a subset of the images whose centers were the farthest away from field center were used. For the D2 Field, some archival COSMOS images were included; for the D3, some W3 images were included. Only i'-band images were used. For the Wide Fields, images from the main pointings were supplemented with images from the "pre-imaging" survey. Here, r'-band images were used. Each of these images was calibrated using the NOMAD/SDSS astrometric solutions. Source catalogs were generated for each image with SExtractor. The catalogs were cleaned as noted in Section 4.1 with a further magnitude cut at mag = 23. The (x, y) positions were converted to (R.A., decl.) using the astrometric solutions. The catalogs for a given survey and field were then merged, forming a final reference catalog for each field. These internal catalogs have smaller astrometric residuals and a higher source density than the external catalogs. Using these internal catalogs left the external astrometric accuracy unchanged (or improved very slightly) but dramatically improved the internal astrometric accuracy.

The first step of the mapping was the initial matching of the observed catalogs (Section 4.1) to the reference catalogs (Section 4.2). For the matching, each CCD of the mosaic was treated independently. The (x, y) coordinates of the observed catalog were converted to (R.A., decl.) using the initial Elixir WCS. The catalogs were shifted in R.A. and decl. with respect to one another until the best match between the two catalogs was found. This simple up–down/left–right shift method was found to be sufficient. More complex methods such as triangle matching or Fourier transform methods were not necessary since the plate scale and image orientation were known beforehand. If there was no good match for a particular CCD (for example, when the initial Elixir WCS was in error), its WCS was replaced with a default WCS and the matching procedure was restarted.

Once the matching was complete, the astrometric fitting could begin. The initial matching was always done with the NOMAD catalog; the higher source density of the SDSS and the internal catalogs would have made matching prohibitively slow. Typically 20–50 sources per CCD were found with this initial matching.

Once the initial match was complete, the first astrometric solution was determined. Using this astrometric solution, the matching was restarted, this time with tighter constraints on what constituted a good match between an observed source and a source in the reference catalogs. This procedure was repeated several times, steadily shrinking the matching radius from 8'' to 05. The order of astrometric solution increased from a linear fit in the first iteration to a third-order solution by the last. This match/map/repeat procedure was done for each CCD individually.

The final astrometric mapping for each image was determined on the scale of the whole mosaic, all 36 CCDs at once. This final mapping consisted of a linear mapping for each CCD: shift and skew for each axis. In terms of FITS WCS keywords, this corresponds to the CRVAL/CRPIX keywords and the CD matrix. The radial distortion correction is modeled as

$$\begin{equation} R=r (1 + a_2 r^2 + a_4 r^4), \end{equation} \tag{ 2 }$$

where R is the true radius and r is the measured radius. This equation is analogous to Equation (3) of Chiu (1976) and the equation in Section 3.1 of Cuillandre et al. (1996). Thus, only two parameters were used to describe the distortion for the whole mosaic. In other systems, a second- or third-order function is needed to describe the distortion, requiring up to 20 parameters per CCD, or a total of 720 parameters per mosaic. The best values of a₂ and a₄ were found through a crude but robust mapping of parameter space. Once determined, the astrometric mapping described by the values a₂ and a₄ was re-expressed in terms of the FITS WCS PVn_n keywords (M. R. Calabretta et al. 2004, in preparation).

The internal and external astrometric uncertainties are estimated to be 002 and 007, respectively, as discussed in Section 8.1.

The Sloan ugriz filters are not identical to the MegaCam filters, as shown in Figure 6. The color terms between the two filter sets can be described by the following equations:

$$\begin{eqnarray} \mbox{$u^*$}&=& u_{{\rm SDSS}} -0.211 \times (u_{{\rm SDSS}} - g_{{\rm SDSS}})\nonumber \\ \mbox{$g^{\prime }$}&=& g_{{\rm SDSS}} -0.155 \times (g_{{\rm SDSS}} - r_{{\rm SDSS}})\nonumber \\ \mbox{$r^{\prime }$}&=& r_{{\rm SDSS}} -0.030 \times (r_{{\rm SDSS}} - i_{{\rm SDSS}})\nonumber \\ \mbox{$i^{\prime }$}&=& i_{{\rm SDSS}} -0.102 \times (r_{{\rm SDSS}} - i_{{\rm SDSS}})\nonumber \\ \mbox{$z^{\prime }$}&=& z_{{\rm SDSS}} +0.036 \times (i_{{\rm SDSS}} - z_{{\rm SDSS}}) \end{eqnarray} \tag{ 3 }$$

after Regnault et al. (2009).

All images lying completely in the SDSS were directly calibrated without referring to other standard stars such as the Smith et al. (2002) standards. The systematics in the SDSS photometry are about 0.02 mag (Ivezić et al. 2004). The presence of at least 1000 usable sources in each square degree reduced the random error to effectively zero. It was possible to calibrate the CCDs of the mosaic individually with about 30 standards in each. For each MegaCam image, the corresponding catalog (in instrumental magnitudes) was matched to the SDSS catalog for that patch of sky. The difference between the instrumental MegaCam magnitudes and the SDSS magnitudes gave the zero point for that exposure or that CCD. The zero point was determined by the median.

There are about 10⁴ SDSS sources per square degree, but when one cuts by stellarity and magnitude this number drops to around 1000. Only stars were used since the above color terms are more appropriate to stars than galaxies. Only objects with 17 < mag < 20 were used; the brighter objects are usually saturated in the MegaCam image and including the fainter objects tends to only increase the noise in the median. This process could be used any night; it was not necessary for the night to be photometric. The D2, D3, and the W3 and part of the other Wide Fields were calibrated in this manner.

For pointings outside the SDSS, the Elixir photometric keywords were used, with modifications. The Elixir zero points were compared to those determined from the SDSS using the procedure above for a large number of archival MegaCam images. There are systematic offsets between the two sets of zero points, particularly for the u* band. These offsets show variations with epoch, which are caused by modifications to the Elixir pipeline (J.-C. Cuillandre 2008, private communication). There are also differential offsets between the CCDs of a single image. For MegaPipe, the offsets are applied to the Elixir zero points to bring them in line with the SDSS zero points.

5.2.1. Differential Offsets

The differential offsets are the offsets between different CCDs of a single MegaCam image, regardless of whether the image was taken under photometric conditions or not. Ideally, the photometric zero point should be the same across the MegaCam field. In practice, differences exist. To investigate the differential zero-point offsets, the MegaCam archive was searched for images matching the following criteria:

• 1.  
  Each image must have an exposure time greater than 60 s.
• 2.  
  Each image must lie within the SDSS.
• 3.  
  No two images from the same camera run should lie within 03 of each other.

The last condition is to ensure that the isolated patches of bad SDSS photometry do not affect the final result.

Each of the 12,000 images matching these criteria was astrometrically calibrated using the standard MegaPipe pipeline. The photometric zero points were computed separately for each CCD. The difference between the individual zero points and the average of all the zero points was computed. Since the images were not necessarily taken under photometric conditions, the Elixir zero point may or may not be valid and therefore was ignored at this stage. The old i' filter (I.MP9701) and the new i' filter (I.MP9702) were treated separately. Note that many of these images (indeed, the majority) are not CFHTLS images, but archival MegaCam images taken for a variety of projects.

The resulting differential zero-point measurements were aggregated by filter and camera run ID (=CRUNID). For each of the CCDs, a median photometric zero-point offset was computed for each filter/CRUNID combination. The individual and median zero-point offsets were plotted. Two examples are shown in Figure 9. In these plots the gray lines show the individual zero-point offsets (SDSS-Elixir) as a function of CCD number. The superimposed black line shows the median zero-point offsets. There is some scatter about the median, indicated by the gray lines and by the error bars (1σ). The pattern of offsets by CCD number reflects the mostly radial pattern noted by Regnault et al. (2009). The amplitude of the pattern varies with filter, but also over time. There is considerable variation in the differential zero-point offsets. For some runs the differential offsets are quite small, as shown by the upper panel of Figure 9. For others it is substantial (0.08 mag or greater), as shown in the lower panel. Going through the plots sequentially, one sees definite changes over time. Further, at least in some cases, the amplitude of the offsets is large enough to warrant correction. Zero-point corrections were determined for every CRUNID and filter combination. These corrections were then applied to every CFHTLS image that was not calibrated using the SDSS.

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5.2.2. Absolute Offsets

The next step was to measure the offsets of the MegaPipe mosaic as a whole with respect to the SDSS. To measure this for a given image, the Elixir zero point determined from the header keywords was compared to the zero point determined by comparing the instrumental magnitudes of stars in the images to the magnitudes in the SDSS. As before, the MegaCam archive was searched for all suitable images. In this case, the criteria were slightly different. In order for the Elixir zero point to be valid, the night must have been photometric.

SkyProbe (Cuillandre et al. 2002) plots were used to determine whether a given night was photometric. SkyProbe is a small camera mounted on CFHT aligned with the telescope. It covers a 5° × 7° field of view down to 12th magnitude. It monitors the flux from several hundred stars over the course of the night. Drops in the fluxes indicate an increase in atmospheric attenuation. Large variations in the atmospheric attenuation indicate that the night was not photometric. Plots for each night are available online.⁴

The patches of bad SDSS photometry are small relative to the size of the MegaPrime field of view so the requirement that all the pointings must be different was relaxed. The new criteria were then as follows:

• 1.  
  Each image must have an exposure time greater than 60 s.
• 2.  
  Each image must lie within the SDSS.
• 3.  
  The night must be photometric.

For each image, the average difference between the SDSS and Elixir zero points over all the CCDs was measured. SExtractor was run on the image with MAG_ZEROPOINT=0. The resulting catalog was astrometrically calibrated using the standard MegaPipe system. The relevant part of the SDSS catalog was retrieved from the web. The SDSS magnitudes were converted to MegaCam magnitudes using the transformations described earlier.

The image catalog was matched to the SDSS catalog. The difference between the instrumental magnitudes and the SDSS magnitudes (transformed into the MegaCam filter system using Equation (3)) yields the SDSS photometric zero point. The Elixir zero point ${\tt ZP_{{\rm Elixir}}}$ was calculated from the MegaCam image header keywords as follows:

$$\begin{eqnarray} {\tt ZP_{{\rm Elixir}}} &=& {\tt PHOT{\underline{\ }}C} + {\tt PHOT{\underline{\ }}K} \times ({\tt AIRMASS}-1) \nonumber\\ &&+ 2.5 \times \log _{10}({\tt EXPTIME}), \end{eqnarray} \tag{ 4 }$$

where PHOT_C is the nightly instrumental zero point determined by Elixir, PHOT_K is the airmass coefficient, AIRMASS denotes the airmass, and EXPTIME is the exposure time.

The differential offset described above was also applied to the Elixir zero points. Note that because differential offsets average to 0 over the whole mosaic (by design), applying these offsets only reduces the scatter of the individual CCD zero points about the average zero point, but does not change the value of the average zero point.

The zero-point differences were then aggregated by filter and CRUNID. The median zero-point difference was computed, as well as the standard deviation about this median. For some filter/CRUNID combinations, no data were taken on photometric nights. For other combinations, data were available for multiple photometric nights during a single run. In a number of cases, there was variation in the zero-point difference between two apparently photometric nights of the same run. The zero-point differences might be stable within a single night, but differences were found from night to night. In addition, there were occasionally variations in the zero-point differences within a single night, indicating that the night was not photometric.

The zero-point differences for the filter/CRUNID combinations were sorted into three categories, labeled "No info," "Excellent," and "Marginal" as follows:

• 1.  
  No info. No data were taken on photometric nights during this camera run with this filter.
• 2.  
  Excellent. Data were taken on at least two photometric nights during this camera run, and at least six images were taken on those nights, and the scatter about the median zero-point difference is less than 0.02 mag.
• 3.  
  Marginal. There is some data, but it does not meet the criteria outlined above. Either data were not taken on more than one night, or less than six images were taken, or the scatter is large.

Figure 10 shows the zero-point history of MegaCam for the u*g'r'i'z' filters. Each point represents a single CRUNID. The "Excellent" cases are shown as filled points. The "Marginal" cases are shown as open points. The "No info" cases are not shown. The Elixir zero point is not always consistent with the SDSS zero point. The amplitude of the variation is typically about 0.05 mag, except for the u* band, for which the amplitude is greater than 0.2 mag.

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Thus, two sets of corrections were applied to the Elixir zero points: a differential correction was applied to each CCD within the mosaic and a global correction was applied to the mosaic as a whole. Both sets of corrections were applied as a function of CRUNID. Applying these corrections greatly increased the consistency of the photometry, as discussed in Section 8.2.

Some of the CFHTLS data neither lie in the SDSS nor were taken on photometric nights. These data cannot be photometrically calibrated by either of the preceding methods. However, if such an image overlaps another that can be calibrated by one of the preceding methods, it in turn can be calibrated.

For the Deep Fields, this was fairly straightforward. Since all the images in a field lie at the same position, in-field photometric standards using images taken on photometric nights were established. Those standards were then used to calibrate all the data for that field. In fact, the self-consistency of the photometry was used to determine which nights were photometric. The two Deep Fields that do not lie in the SDSS, the D1 and the D4, were calibrated in this manner. The initial photometry was computed for each image using the Elixir zero points. Catalogs were generated for each image based on these zero points, and the catalogs were cross-referenced to each other to determine image-to-image variations in photometry. The photometric consistency was checked over each night. Any night showing large variations in photometry from image to image was deemed non-photometric. (Interestingly, some of these nights appear completely photometric based on the SkyProbe measurements.) While the presence of large variations indicates that the night was not photometric, the absence of such variations does not guarantee that the night was photometric. The photometric catalogs on the apparently photometric nights were compared over the run of the survey. Again, if the photometry for a given night was not consistent with that of other nights, it was flagged as non-photometric. In the end, a minimum of five nights per filter and per field were identified as both photometric and consistent. The images from these nights were stacked and a catalog generated from the resulting image. This catalog became the photometric reference for that field and filter.

A similar method was used for the Wide Fields. The pointings within a field overlap at the edges, allowing photometric comparisons. Each filter was processed separately. First, the photometry was homogenized within each pointing. In the simplest case, all the images within a pointing could be directly calibrated using either the (corrected) Elixir zero points or the SDSS. These pointings were flagged as probably photometric (PP). If only some of the images of a pointing could be directly calibrated, these images were used as reference for the others. These pointings were also flagged as PP. If none of the images could be calibrated directly, one of the images was used as a photometric reference for the others. The photometry of the images in this pointing would be self-consistent within itself, but probably not with adjacent pointings. These pointings were flagged as not photometric (NP).

Next, the photometric consistency between pointings was checked. Consistency here means a systematic zero-point difference of less than 0.03 mag. If a pointing previously flagged PP was found to be inconsistent with any other PP or DP pointing, then its status was downgraded to NP. On the other hand, if a pointing was consistent with at least two other adjacent pointings, its status was upgraded to definitely photometric (DP).

Having identified the DP pointings, the next step was to calibrate adjacent pointings. There are typically a few hundred stars in the overlap region between two adjacent pointings. Only pointings that overlap along an edge were calibrated in this manner, not pointings that only overlap at the corners. The random error associated with transferring the zero point in this manner is typically 0.05 mag per star. When 300 stars are used, the random error drops to below 0.002 mag. Once a new pointing was calibrated by overlap, it was checked for consistency with previously calibrated pointings. If it was consistent, it was flagged as DP and used to calibrate other adjacent pointings. Eventually all pointings in a field were calibrated. In principle transferring the zero point in this manner could cause an increase in photometric zero-point error as the number of transfers increases. In practice, all pointings were at most two steps away from a DP pointing. The W1, W2, and W4 Fields were calibrated in this manner.

SExtractor was run on each image to produce a catalog. The images were filtered before detection. The detection criteria were that three adjacent pixels had to have flux levels 1σ above the background. SExtractor's deblend contrast parameter was set to 0.002. Table 1 shows the relevant SExtractor detection parameters. The inverse variance weight maps described in the previous section were used. The resulting catalogs, one per pointing and filter, are available on the web as described in Section 9.

The individual catalogs were masked to identify areas where the detection and photometric measurements of sources may have been compromised. These areas included areas around bright stars, diffraction and bleed spikes from the brightest stars, and satellite/meteor trails. Also, in some cases the dither pattern of the input images was insufficient to provide uniform depth across the pointing. An automatic detection method was used to find the bright stars and the diffraction/bleed spikes. This was supplemented with laborious hand masking.

The areas around bright stars were masked. The Guide Star Catalog 2 (Lasker et al. 2008) was queried for the position and magnitude of stars brighter than 15th magnitude. The stars were masked with a circular mask covering the core of the star and a cross-shaped area covering the diffraction spikes. The size of the mask scaled with the brightness of the star. The top left panel of Figure 11 shows an example. Stars brighter than 13th magnitude often produce bleed trails in the images extending along the y-axis of the CCDs. The masking algorithm followed the bleed trails until they blended into the sky background, extending the cross part of the stellar masks in the vertical direction. The top right panel of Figure 11 shows an example.

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Stars brighter than 9th magnitude also produce a visible pupil image. These halos have a radius of about 1100 MegaCam pixels (approximately 3.5 arcmin). The center of the halos is offset from the stars that created them toward the center of the MegaCam field; the size of the offset is 0.022 times the distance of the star to the center of the field, as noted in Section 3.4 of Erben et al. (2009). The bottom left panel of Figure 11 shows an example of a pupil mask.

In addition, meteor trail residuals were masked. Since meteor trails only show up in one image, and since a median is used to combine the images, these are mostly removed during the stacking process, leaving at most some residual noise. However, in some places (the gaps between CCDs of the MegaCam mosaic and over bad columns) less than the full number of input images were available, and the meteor trails were visible in the output image. The trail masking method described in Section 3.4, item 3 of Erben et al. (2009) was attempted. However, as noted in that section, the method is not 100% reliable. Additional hand masking of meteor trails was required. The bottom right panel of Figure 11 shows an example of meteor trail masking near the edge of a mosaic.

Masks were generated for each image of the survey. The masks were checked by eye and minor modifications made. The masks are available for download as ds9 region files.⁵

To increase the usability of the catalogs, the individual catalogs were merged. Each merged catalog contains measurements in all five of the u*g'r'i'z' MegaCam filters and covers an entire survey (either the Wide or the Deep).

Separate catalogs were generated using each of the five bands as a reference/detection image. Thus, there is a u*-selected, g'-selected, r'-selected, i'-selected, and z'-selected catalog for each survey, each suited to different science goals (e.g., i'-selected for general galaxy population studies, z'-selected for i'-band dropout searches, g'-band for galactic stars, etc.). Although the catalogs are selected in a single filter, measurements are made in all five filters for each catalog, removing the need to consult multiple catalogs. There are thus a total of 5 × 2 = 10 separate catalogs, each complete in itself, but with different detection characteristics.

The procedure to produce the merged catalogs was as follows:

• 1.  
  Individual catalog generation. SExtractor was run in double-image mode on the individual images.
• 2.  
  Catalog masking. Areas where photometric measurements and detections may be compromised were masked as described in Section 7.2.
• 3.  
  Catalog merging. The masked catalogs for each pointing were combined and redundant sources (ones that appear in more than one individual catalog) were removed.

Before settling on this procedure, a number of other methods were considered and ultimately rejected.

Instead of generating catalogs pointing by pointing and then merging the catalogs, one could SWarp together all the images for each Wide Field into a single enormous image and run SExtractor on that image. This has the advantage of eliminating the catalog merging step, which involves a fair amount of bookkeeping. The disadvantage is that these images are unwieldy. They would measure up to 140,000 pixels across and be 100 Gb in size. While it may be possible (although difficult) to run SExtractor on such images, one cannot distribute such images over the web. Users would be presented with catalogs for which they cannot view the source image.

Instead of generating separate catalogs for each filter, one could build some sort of master image containing flux from each band, for example, a χ² image. One could then use SExtractor in double-image mode with the master image as a detection image. This has the advantage of simplicity: it produces a single catalog. However, the five bands are not even in depth. The z' band in particular is much noisier than the other four. While adding the bands together produces a deeper image in theory (you now have photons from all wavelengths), in practice it is also noisier. Leaving out the noisier bands (u*z') means that the resulting catalog is no longer suitable for all purposes.

Instead of publishing a separate catalog for each filter, one could merge the five catalogs into one. One could cross-identify sources common to different catalogs and merge their entries into a single entry. This again has the advantage of producing a single catalog rather than several and (if done correctly) eliminates the disadvantages of the previous master-image method. The trick, of course, is accurately and reliably cross-identifying the sources between filters. For bright, well-separated sources this is easy. The MegaPipe images are registered to very high accuracy, making cross-identification by position fairly accurate. However, the IQ and depth vary between different filters; objects that appear as a single source in one filter may appear as two sources in another filter. Even if the cross-identification can be done correctly, the bookkeeping is non-trivial. That being said, it has worked successfully for the SDSS and this option may be explored in the future.

In short, the easiest and best way to make a clean catalog was

• 1.  
  to merge the different pointings at the catalog level rather than at the image level; and
• 2.  
  not to merge the catalogs for different filters but to generate multiple catalogs, one per filter.

7.3.1. Individual Catalog Generation

7.3.2. Catalog Merging

8.1.1. Internal Astrometric Consistency

The internal accuracy was checked by running SExtractor on each stacked image in every band and obtaining catalogs of object positions. The positional catalogs for each band were matched to each other and common objects identified. If the astrometry was perfect, then the position of an object in each band would be identical. In practice, there are astrometric residuals. Examining these residuals gives an idea of the astrometric uncertainties.

Figure 12 shows checks on the internal astrometry between the g'- and r'-band images for the D2 Field. The top left panel shows the direction and size (greatly enlarged) of the astrometric residuals as line segments. This plot is an important diagnostic of astrometry because, while the residuals are typically quite small, there are outliers in any distribution. As long as these outliers are relatively isolated from each other and pointing in random directions, all is well. Conversely, if there are a large number of residuals in close proximity to each other, all pointing in the same direction, this indicates a systematic misalignment between the two images in question. The figure shows no such misalignments. Similar plots were made for every possible pair of images in the CFHTLS. No misalignments or other problems were found, indicating that there are no systematic astrometric errors for this pair of images.

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The two right panels show the residuals in R.A. and decl. separately, which are about 002 (68%-tile). The bottom left panel shows the astrometric residuals in R.A. and decl. The histograms show the relative distribution of the residuals in both directions. The 68%-tile of the residuals is 003 radially, about $\sqrt{2}$ larger than the residuals in any one direction.

In general, the internal astrometric uncertainties are slightly larger than the example shown in Figure 12. Similar plots were made for every possible pair of images in the CFHTLS. The typical internal astrometric residuals are 002 in any one direction. However, no misalignments or other problems were found, indicating that there are no systematic astrometric errors.

8.1.2. External Astrometric Accuracy

Accurate astrometry relative to external systems is relevant for follow-up observations with other telescopes. External accuracy was checked by matching each catalog for each field and filter back to the astrometric reference catalog. Again, the scatter in the astrometric residuals is a measure of the uncertainty, and the presence of any localized large residuals indicates a systematic shift. Figure 13 shows an example of the astrometric residuals of one of the pointings with respect to the SDSS. The panels have the same meanings as in Figure 12. The external residuals are larger than the internal residuals. Similar plots were generated for all pointings and bands. On average, the astrometric residuals are 007 with respect to the SDSS and 018 with respect to NOMAD. No systematic offsets are visible, indicating that they are smaller than the random residuals, if indeed they exist.

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8.2.1. Internal Consistency

The edges of the Wide pointings overlap. Comparing the photometry of objects in the overlap regions gives a measure of the photometric consistency. Systematic differences in photometry indicate zero-point offsets. Figure 14 shows a histogram of the pointing-to-pointing zero-point differences. In comparing two pointings, the sign of the zero-point difference is irrelevant, so the absolute value of the differences is plotted here. The average offsets are about 0.01 mag, with a tail out to 0.05 mag.

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8.2.2. External Accuracy

At least some of the pointings of each of the Wide Fields as well as the D2 and D3 Fields lie within the SDSS. This makes it possible to directly compare the magnitudes in those fields to an external reference. Figure 15 shows a typical comparison between the SDSS (transformed to the MegaCam system as described by Equation (3)) and the CFHTLS for the five bands. The agreement is very good. At bright magnitudes, for g', r', and i' bands, there are deviations caused by stars saturating in the MegaCam images. There is no evidence for systematic shifts greater than 0.01 mag. There is also relatively little scatter (at least at moderate magnitudes), which argues that the color terms in the SDSS-MegaCam transformation are fairly accurate. The u* band shows greater scatter than the other bands; this is caused by the larger and more uncertain color terms in the transformation between the SDSS and MegaCam photometric systems.

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Similar comparisons have been done for every pointing overlapping the SDSS. Figure 16 summarizes the results. The offsets are typically slightly larger than the internal photometric residuals shown above, ranging from 0.010 mag in i' to 0.018 for the u* band.

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This test is not a completely independent test of photometry because in many cases the photometric zero point was derived from the SDSS. However, for many pointings it is independent: for example, as shown in Figure 5, the "+0" row of the W4 Field lies only partially within the SDSS. These pointings were not directly calibrated using the SDSS but rely on the Elixir calibration and adjacent pointings for their photometric zero points. However, when the CFHTLS magnitudes of objects in the overlap region of these pointings are compared with SDSS magnitudes, they agree to within the errors described above.

As an additional test, the W3 Field (which lies entirely within the SDSS) was reduced a second time, without any reference to the SDSS. The reduction was done using only the modified Elixir photometric calibration as described in Sections 5.2 and 5.3. The zero points thus derived were found to be consistent with the zero points derived from the SDSS to within 0.01 mag rms.

Regnault et al. (2009) did an extremely thorough analysis of the photometric calibration of the SNLS. This analysis is completely independent of the current analysis and ties its photometric zero points to Landolt & Uomoto (2007) standards. They published tertiary standards for the four Deep Fields. The present photometry was compared to these tertiary standards. After correcting for the zero-point difference (the Regnault standards are in a natural magnitude system, which is approximately Vega-based, whereas the MegaPipe photometry is in the AB system), the zero-point offsets were found to be approximately 1%. Again, the offsets in the u* band were found to be slightly larger, approximately 2%.

8.2.3. Magnitude Limits

The limiting magnitudes of the images were determined by adding fake stars and galaxies to the images and then trying to recover them using the same parameters used to generate the real image catalogs.

The fake galaxies used were taken from the images themselves, rather than adding completely artificial galaxies. A set of 40 bright, isolated galaxies were selected out of the field and assembled into a master list. Postage stamps of these galaxies were cut out of the field. The galaxies were faded in both surface brightness and magnitude through a combination of scaling the pixel values and resampling the images.

To test the recovery rate at a given magnitude and surface brightness, galaxy postage stamps were selected from the master list, faded as described above to the magnitude and surface brightness in question and then added to the image at random locations. SExtractor was then run on the new image. The fraction of fake galaxies found gives the recovery rate at that magnitude and surface brightness. An illustration of adding the galaxies is shown in Figure 11 of Gwyn (2008).

To test the false-positive rate, the original image was multiplied by −1; the noise peaks became noise troughs and vice versa. SExtractor was run, using the same detection criteria. Since there are no real negative galaxies, all the objects thus detected are spurious.

The magnitude/surface brightness plot in Figure 17 shows the results of such tests. The black points are real objects. The bottom edge of the black points is the locus of point-like objects. The green points show the false-positive detections, located at faint magnitudes and surface brightnesses. The red numbers show the percentage of artificial galaxies that were recovered at a given magnitude/surface brightness. Superimposed on the numbers are two contour lines in blue, at 50% and 70% completeness.

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Deriving a single limiting magnitude from such a plot is slightly difficult. The cleaner cut in the false positives seems to be in surface brightness. Extended objects become harder to detect at brighter magnitudes, whereas stellar objects are detectable a magnitude or so fainter.

Point source limiting magnitudes were also calculated. Isolated point sources were taken from the images, scaled in magnitude only and added back to the images in a similar manner as the galaxies. For the Deep Fields in particular, the images were effectively crowded. An artificial source added to the image stood a significant chance of ending up close enough to a real source that it would not be detected. To compensate for this, sources were added to two images. The first was just the original image. The second was the original image with all the real sources removed and their pixels replaced with values matching the background noise characteristics, i.e., a blank image. The differences between the two completeness limits are shown in Figure 18. This figure shows the completeness limits for a Wide pointing. The blue line shows the fraction of artificial point sources that can be recovered from the blank image. The red line shows the same for the original image. The blue line is consistently a few tenths of a magnitude deeper than the red line. The black points show the number counts of real sources. The green points show the number counts for false-positive detections. The 50% completeness limit for this image is 25.7 mag. The 50% blank-image completeness limits are the values quoted as limiting magnitudes in Tables 2–4.

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Table 2. Deep Full Stacks: Limiting Magnitudes and Image Qualities

Field	u*		g'		r'		i'		z'
	mlim	IQ ('')	mlim	IQ ('')	mlim	IQ ('')	mlim	IQ ('')	mlim	IQ ('')
D1	27.4	0.94	27.9	0.89	27.6	0.83	27.4	0.78	26.1	0.81
D2	27.5	0.96	27.9	0.92	27.6	0.84	27.3	0.83	26.4	0.83
D3	27.4	0.97	28.0	0.94	27.8	0.83	27.6	0.80	26.4	0.78
D4	27.3	0.96	27.8	0.91	27.7	0.83	27.2	0.80	26.2	0.78

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Table 3. Deep Good Seeing Stacks: Limiting Magnitudes and Image Qualities

Field	u*		g'		r'		i'		z'
	mlim	IQ ('')	mlim	IQ ('')	mlim	IQ ('')	mlim	IQ ('')	mlim	IQ ('')
D1	27.1	0.80	27.2	0.64	27.3	0.64	27.3	0.64	25.9	0.64
D2	27.4	0.80	27.4	0.65	27.4	0.64	27.1	0.64	26.2	0.64
D3	27.1	0.78	27.5	0.65	27.5	0.64	27.3	0.64	26.1	0.64
D4	26.8	0.78	27.0	0.64	27.0	0.64	26.8	0.64	25.9	0.64

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Table 4. Wide Stacks: Limiting Magnitudes and Image Qualities

Pointing	u*		g'		r'		i'		z'
	mlim	IQ ('')	mlim	IQ ('')	mlim	IQ ('')	mlim	IQ ('')	mlim	IQ ('')
W1+0+0	26.1	0.83	26.4	0.86	26.0	0.75	25.8	0.74	24.0	0.96
W1+0+1	25.7	1.01	26.3	0.91	26.1	0.83	26.0	0.64	24.9	0.78
W1+0+2	25.9	0.97	26.5	0.92	25.8	0.86	26.0	0.57	24.6	0.84
W1+0+3	26.0	0.86	26.3	0.90	25.7	0.92	25.7	0.80	24.5	0.90
W1+0-1	26.0	1.06	26.5	0.89	25.9	0.79	25.9	0.55	24.1	1.01
W1+0-2	25.9	1.11	26.8	0.66	25.8	0.84	25.8	0.69	24.7	0.66
W1+0-3	26.2	1.09	26.5	0.91	25.8	0.86	25.8	0.72	24.5	0.78
W1+0-4	26.1	0.84	26.6	0.95	25.9	0.77	25.6	0.71	25.0	0.56
W1+1+0	26.3	0.85	26.6	0.79	25.9	0.73	25.8	0.57	24.5	0.84
W1+1+1	25.8	0.97	26.7	0.92	25.8	0.84	26.0	0.86	25.1	0.79
W1+1+2	25.7	1.03	26.2	0.86	26.1	0.67	25.5	0.81	24.9	0.69
W1+1+3	26.0	0.88	26.5	0.83	25.9	0.95	25.6	0.80	24.9	0.68
W1+1-1	26.0	0.97	26.6	0.98	26.0	0.74	26.3	0.81	24.2	0.95
W1+1-2	26.0	1.10	26.7	0.76	25.9	0.84	26.0	0.81	24.6	0.78
W1+1-3	26.1	1.05	26.5	0.81	25.8	0.82	25.7	0.70	24.7	0.80
W1+1-4	26.1	0.92	26.6	0.93	26.1	0.76	25.9	0.66	24.9	0.58
W1+2+0	25.7	1.01	26.5	0.86	26.0	0.82	25.4	0.74	24.3	0.87
W1+2+1	25.9	1.06	26.3	1.07	25.7	0.88	25.5	0.88	25.0	0.68
W1+2+2	25.9	0.96	26.3	0.91	25.7	0.89	25.9	0.88	24.7	0.82
W1+2+3	26.3	1.00	26.5	0.83	26.2	0.70	25.6	0.78	25.2	0.60
W1+2-1	26.3	0.96	26.6	0.78	26.0	0.72	25.7	0.77	24.5	1.02
W1+2-2	26.1	1.10	26.5	0.77	25.9	0.73	25.8	0.69	24.2	1.00
W1+2-3	26.3	0.81	26.6	0.73	25.8	0.81	25.5	0.73	24.4	0.92
W1+2-4	26.1	0.89	26.7	0.85	25.8	0.77	25.7	0.75	24.9	0.61
W1+3+0	26.3	0.64	26.3	0.92	25.9	0.83	25.7	0.73	24.6	0.86
W1+3+1	26.1	0.90	26.4	0.99	25.8	0.82	25.8	0.87	24.5	0.82
W1+3+2	26.2	1.10	26.3	0.87	25.9	0.83	25.6	0.70	24.6	0.64
W1+3+3	25.7	1.25	26.3	0.90	25.7	0.82	25.3	0.87	24.7	0.62
W1+3-1	25.9	1.05	26.5	0.80	26.1	0.74	25.8	0.77	24.8	0.74
W1+3-2	26.4	0.85	26.6	0.74	25.8	0.74	25.8	0.72	24.6	0.82
W1+3-3	26.2	0.80	26.7	0.67	25.8	0.87	25.7	0.64	24.5	0.90
W1+3-4	26.0	1.04	26.3	0.90	25.8	0.78	26.1	0.66	24.8	0.63
W1+4+0	26.4	0.75	26.5	0.91	25.9	0.74	25.3	0.70	24.7	0.75
W1+4+1	26.3	0.84	26.4	0.87	25.7	0.76	25.3	0.84	25.0	0.49
W1+4+2	26.3	0.79	26.7	0.90	25.9	0.67	25.6	0.94	24.8	0.61
W1+4+3	26.0	1.00	26.4	0.89	25.8	0.78	25.2	0.88	24.2	0.86
W1+4-1	25.9	1.17	26.6	0.89	25.9	0.77	26.0	0.61	24.5	0.85
W1+4-2	26.4	0.74	26.6	0.81	25.9	0.73	25.9	0.73	24.4	0.75
W1+4-3	26.0	0.94	26.5	0.85	25.7	0.92	25.8	0.73	24.5	0.79
W1+4-4	26.2	0.91	26.1	1.00	26.1	0.92	26.0	0.83	24.8	0.62
W1-1+0	26.0	0.97	26.5	0.73	26.0	0.74	25.7	0.64	24.3	0.87
W1-1+1	26.1	1.06	26.7	1.02	25.8	0.83	25.9	0.62	24.5	0.79
W1-1+2	25.9	0.96	26.5	0.92	25.9	0.88	25.7	0.64	25.0	0.72
W1-1+3	26.1	0.80	26.4	1.00	25.8	0.89	25.6	0.81	24.9	0.66
W1-1-1	26.7	0.70	26.9	0.59	25.7	0.90	25.9	0.58	24.1	0.91
W1-1-2	26.3	0.83	26.9	0.62	26.0	0.75	25.6	0.86	24.1	1.03
W1-1-3	26.3	0.76	26.2	1.01	25.8	0.77	25.7	0.71	24.5	1.01
W1-1-4	26.4	0.70	26.7	1.05	25.9	0.76	25.5	0.88	24.8	0.68
W1-2+0	26.5	0.82	26.8	0.64	25.8	0.84	26.1	0.52	24.4	0.96
W1-2+1	26.4	0.81	26.8	0.72	25.9	0.75	25.4	0.89	24.9	0.76
W1-2+2	26.4	0.79	26.2	0.92	25.9	0.73	25.2	0.82	24.4	0.88
W1-2+3	26.2	0.83	26.1	0.90	26.0	0.74	25.3	0.76	24.4	0.67
W1-2-1	26.3	0.79	26.9	0.66	25.9	0.86	26.1	0.50	24.5	1.07
W1-2-2	26.3	0.79	26.6	0.70	26.1	0.76	25.9	0.53	24.8	0.62
W1-2-3	26.4	0.72	26.3	0.81	25.8	0.83	25.8	0.65	24.2	1.05
W1-2-4	26.3	0.92	26.5	0.96	26.0	0.75	26.0	0.58	25.2	0.55
W1-3+0	26.0	0.88	26.3	0.98	25.9	0.85	25.6	0.78	24.7	0.65
W1-3+1	26.3	0.90	26.3	0.86	25.9	0.85	25.6	0.79	24.7	0.59
W1-3+2	26.3	0.86	26.1	0.98	25.7	0.89	25.7	0.77	24.8	0.66
W1-3+3	26.1	0.85	26.7	0.75	25.8	0.68	25.7	0.74	24.6	0.67
W1-3-1	25.9	1.01	26.5	0.80	25.9	0.97	25.6	0.82	24.9	0.52
W1-3-2	25.8	1.10	26.6	0.69	25.8	0.83	25.5	0.75	24.7	0.74
W1-3-3	26.0	0.74	26.6	0.70	25.9	0.75	25.8	0.75	24.9	0.73
W1-3-4	26.3	0.97	26.4	0.89	26.1	0.78	25.8	0.65	25.0	0.55
W1-4+0	25.9	1.05	26.2	0.90	25.8	0.90	26.0	0.61	24.5	0.74
W1-4+1	25.7	1.09	26.2	0.98	25.7	0.88	25.7	0.71	24.7	0.74
W1-4+2	25.8	1.10	26.6	1.05	25.8	0.77	25.6	0.68	24.5	0.88
W1-4+3	26.2	0.85	26.3	0.85	26.0	0.72	25.5	0.74	24.3	0.94
W1-4-1	26.4	0.87	26.4	0.77	25.7	0.94	25.6	0.73	24.0	1.03
W1-4-2	26.5	0.84	26.2	0.93	25.6	1.02	25.7	0.70	24.6	1.06
W1-4-3	26.0	0.88	26.5	0.92	25.5	1.01	25.5	0.69	24.4	0.89
W1-4-4	26.1	0.81	26.5	0.84	25.5	0.90	25.8	0.90	24.6	0.64
W2+0+0	25.9	0.96	26.6	0.90	25.9	0.72	25.7	0.77	24.3	0.91
W2+0+1	26.0	0.94	26.2	0.88	25.7	0.88	25.7	0.61	24.6	0.55
W2+0+2	25.7	1.08	26.4	0.91	26.0	0.63	25.9	0.73	24.4	0.93
W2+0+3	26.1	0.81	26.2	0.74	25.9	0.83	25.6	0.56	24.2	0.84
W2+0-1	25.9	1.20	26.1	0.98	25.7	0.93	25.7	0.66	24.5	1.09
W2+1+0	26.0	0.79	26.6	0.85	25.6	0.92	25.6	0.77	24.6	0.89
W2+1+1	25.9	0.99	26.4	0.81	25.9	0.91	25.7	0.86	24.7	0.85
W2+1+2	25.9	1.25	26.2	0.92	26.0	0.87	25.5	0.83	24.5	0.70
W2+1+3	25.9	1.08	26.2	0.91	26.0	0.81	25.4	0.88	24.5	0.77
W2+1-1	26.3	0.83	26.0	0.92	25.5	1.05	25.5	0.80	24.6	0.74
W2+2+0	25.9	0.97	26.2	0.83	25.7	0.81	25.7	0.78	24.4	1.01
W2+2+1	26.0	1.01	26.3	0.78	25.9	0.79	25.8	0.58	24.7	0.81
W2+2+2	26.1	1.19	26.3	0.80	26.5	0.63	26.0	0.84	24.6	0.79
W2+2+3	26.0	0.88	26.9	0.66	26.0	0.81	25.4	0.86	24.5	0.93
W2+2-1	26.4	0.75	26.3	0.94	25.7	0.78	25.6	0.78	24.6	0.81
W2+3+0	25.9	0.92	26.3	0.85	25.6	0.99	25.9	0.60	24.9	0.82
W2+3+1	26.3	0.88	26.3	0.93	26.0	0.75	25.2	0.89	24.5	0.79
W2+3+2	26.1	1.02	26.3	0.97	25.9	0.80	25.5	0.80	25.0	0.76
W2+3+3	26.4	0.84	26.4	0.99	26.1	0.85	25.5	0.65	24.5	0.90
W2+3-1	26.3	1.01	26.2	0.86	25.7	0.82	25.4	0.83	24.7	0.64
W2-1+0	25.8	1.01	27.0	0.58	25.9	0.73	26.0	0.57	24.6	0.80
W2-1+1	26.0	0.89	26.5	1.02	25.9	0.94	25.9	0.68	24.3	0.89
W2-1+2	25.9	1.16	26.6	0.80	25.7	0.84	25.8	0.55	24.6	0.80
W2-1+3	25.7	1.10	26.7	0.76	26.3	0.67	25.7	0.73	24.5	0.77
W2-1-1	25.6	0.96	26.5	0.98	25.8	0.84	25.8	0.69	24.6	0.78
W3+0+0	25.6	1.03	26.3	0.96	25.5	0.95	25.3	0.83	24.2	0.81
W3+0+1	25.9	0.99	26.4	0.90	25.6	0.91	25.7	0.72	24.7	0.77
W3+0+2	25.6	1.06	26.4	0.85	25.9	0.79	25.6	0.71	24.4	0.85
W3+0+3	26.0	0.96	26.4	0.84	26.1	0.60	25.5	0.64	24.8	0.92
W3+0-1	25.6	0.99	26.0	1.08	25.8	1.13	25.9	0.72	25.2	0.80
W3+0-2	26.3	0.82	26.4	0.84	25.6	1.00	25.5	0.69	24.6	0.75
W3+0-3	26.1	0.75	26.2	0.91	25.9	0.92	25.1	0.80	24.8	0.86
W3+1+0	26.3	0.82	26.4	0.87	25.8	0.90	25.4	0.75	24.6	0.99
W3+1+1	26.0	0.95	26.5	0.83	25.7	0.89	25.7	0.77	24.7	0.77
W3+1+2	26.5	0.88	26.6	0.72	26.0	0.73	25.9	0.69	24.5	0.78
W3+1+3	26.4	0.84	26.3	0.87	26.1	0.77	25.7	0.70	24.7	0.82
W3+1-1	25.6	1.16	26.4	0.90	25.7	0.92	25.6	0.72	24.6	1.00
W3+1-2	25.8	1.17	26.5	0.83	25.5	0.93	25.4	0.79	24.5	0.84
W3+1-3	25.7	1.16	26.7	1.02	26.0	0.87	25.7	0.72	24.6	0.83
W3+2+0	26.0	0.91	26.7	0.89	25.8	0.85	25.6	0.76	24.5	0.64
W3+2+1	26.3	0.76	26.6	0.97	26.0	0.72	25.7	0.85	24.4	0.83
W3+2+2	26.6	0.65	26.4	0.83	25.8	0.81	25.7	0.60	24.5	0.65
W3+2+3	26.0	1.00	26.5	0.74	26.0	0.66	25.6	0.73	25.0	0.72
W3+2-1	26.2	0.90	26.4	0.94	25.9	1.03	25.7	0.74	24.3	0.74
W3+2-2	26.3	0.84	26.6	0.89	25.5	1.02	25.9	0.64	24.7	0.61
W3+2-3	26.3	0.87	26.6	0.97	25.6	0.99	25.5	0.67	24.7	0.61
W3+3+0	26.0	1.02	26.3	1.02	26.0	0.79	25.8	0.69	24.6	0.82
W3+3+1	26.1	0.91	26.4	1.01	25.9	0.80	25.5	0.90	24.3	0.73
W3+3+2	25.9	0.96	26.3	0.94	26.1	0.71	25.6	0.81	24.2	0.74
W3+3+3	25.9	0.95	26.3	0.72	26.1	0.71	25.5	0.81	25.3	0.68
W3+3-1	26.1	0.90	26.7	0.91	25.9	0.83	25.7	0.94	24.7	0.71
W3+3-2	26.3	1.01	26.6	1.02	25.9	0.85	25.6	0.67	24.8	0.66
W3+3-3	26.3	0.89	26.6	0.95	25.9	0.85	25.6	0.82	24.7	0.72
W3-1+0	26.1	0.71	26.3	1.07	26.0	0.75	25.4	0.60	24.1	0.89
W3-1+1	26.1	0.89	26.5	0.78	25.7	0.89	25.5	0.85	24.4	0.88
W3-1+2	25.7	1.03	26.7	0.77	25.8	0.87	25.6	0.72	24.5	0.78
W3-1+3	26.1	0.98	26.3	0.89	25.7	0.82	26.1	0.89	24.6	0.68
W3-1-1	25.9	1.03	26.7	0.80	25.6	0.83	25.6	0.60	24.3	0.88
W3-1-2	25.3	0.90	26.3	0.91	26.0	0.75	25.4	0.72	24.6	0.73
W3-1-3	25.7	0.77	26.5	0.88	25.5	0.90	25.7	0.72	24.7	0.66
W3-2+0	26.2	0.72	26.5	0.75	26.2	0.73	25.1	0.82	24.8	0.62
W3-2+1	26.1	0.88	26.6	0.89	25.8	0.91	25.7	0.87	24.7	0.74
W3-2+2	25.9	0.94	26.8	0.68	25.8	0.86	25.6	0.63	24.5	0.88
W3-2+3	26.2	0.89	26.5	0.86	26.1	0.78	26.0	0.90	25.1	0.77
W3-2-1	25.8	0.95	26.1	1.02	25.9	0.70	26.0	0.72	24.8	0.60
W3-2-2	26.1	0.79	26.3	0.94	25.9	0.80	25.6	0.71	24.6	0.71
W3-2-3	25.9	1.22	26.4	0.99	25.8	0.84	25.5	0.76	24.2	0.90
W3-3+0	26.1	0.75	26.5	0.77	26.1	0.67	25.8	0.58	24.7	0.68
W3-3+1	26.3	0.70	26.8	0.78	25.7	0.92	26.1	0.92	24.7	0.69
W3-3+2	25.8	1.08	26.7	0.70	26.1	0.88	25.4	0.72	24.9	0.64
W3-3+3	26.2	1.05	26.5	0.81	26.0	0.75	25.6	0.71	24.9	0.75
W3-3-1	25.6	1.00	26.0	0.88	25.8	0.74	25.6	0.79	24.8	0.63
W3-3-2	26.1	0.67	26.5	0.69	25.6	0.78	25.7	0.58	24.8	0.59
W3-3-3	26.1	0.74	26.3	0.84	25.5	0.96	25.8	0.54	24.8	0.64
W4+0+0	25.7	1.14	26.3	0.85	26.1	0.63	25.6	0.74	25.0	0.74
W4+0+1	25.9	0.99	26.4	0.97	26.1	0.62	25.5	0.70	24.6	0.60
W4+0-1	26.2	0.80	26.7	0.86	26.0	0.75	25.9	0.59	24.7	0.56
W4+0-2	26.3	0.75	26.8	0.79	25.9	0.75	25.9	0.66	24.8	0.70
W4+1+0	25.9	0.91	26.4	0.70	25.8	0.75	25.7	0.56	24.7	0.62
W4+1+1	26.2	0.80	26.8	0.60	25.8	0.77	25.9	0.59	24.7	0.60
W4+1-1	25.9	0.92	26.1	0.90	25.9	0.71	26.0	0.67	24.5	0.71
W4+1-2	25.9	0.91	26.2	0.89	26.0	0.67	25.5	0.80	24.8	0.56
W4+2+0	25.9	0.85	26.3	0.73	26.0	0.69	25.9	0.63	24.1	0.86
W4+2-1	25.5	1.07	26.3	0.83	25.8	0.72	25.6	0.73	24.9	0.79
W4+2-2	25.7	1.15	26.3	0.79	25.8	0.68	26.0	0.76	24.8	0.82
W4-1+0	25.7	0.72	26.5	0.90	26.0	0.70	25.7	0.71	24.4	0.81
W4-1+1	26.1	0.80	26.8	0.86	25.9	0.69	25.9	0.87	24.6	0.74
W4-1+2	26.2	0.83	26.7	0.71	25.8	0.66	25.9	0.58	24.3	1.14
W4-1+3	26.0	0.97	26.5	0.85	25.7	0.83	26.0	0.55	24.1	0.94
W4-1-1	26.1	0.75	26.9	0.90	25.8	0.78	26.1	0.77	24.8	0.55
W4-1-2	26.5	0.64	26.6	0.87	25.9	0.84	25.6	0.80	24.9	0.55
W4-2+0	25.9	1.01	26.5	0.78	25.8	0.82	25.9	0.80	24.6	0.68
W4-2+1	25.4	1.09	26.0	0.91	26.1	0.66	25.2	0.62	24.4	0.87
W4-2+2	25.9	1.01	26.5	0.80	25.7	0.73	25.6	0.74	24.2	0.83
W4-2+3	26.2	0.85	26.5	0.88	25.6	0.91	25.7	0.78	24.3	0.69
W4-3+0	26.0	0.99	26.7	0.74	25.8	0.76	25.4	0.56	24.5	0.87
W4-3+1	26.1	1.06	26.6	0.78	25.8	0.73	25.6	0.48	24.3	0.87
W4-3+2	26.0	0.89	26.4	0.82	25.7	0.66	25.1	0.74	24.5	0.77
W4-3+3	26.0	0.93	26.6	1.06	25.6	0.83	25.9	0.63	24.5	0.71

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