AGN Populations in Large-volume X-Ray Surveys: Photometric Redshifts and Population Types Found in the Stripe 82X Survey

Figure 1 shows the regions covered by the Stripe 82X survey. There are two regions with contiguous XMM-Newton coverage awarded to our team in XMM-Newton cycles 10 and 13 (AO10 and AO13), and the rest are from archival XMM-Newton and Chandra pointings, as described in LaMassa et al. (2013a, 2013b, 2016b). The AO10 and AO13 surveys have slightly different exposure times: 6–8 ks for AO13 and 4–6 ks for AO10. The nonuniform exposure times occur due to mosaicking. Key ancillary data sets are indicated in the figure (see caption for details).

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The brighter flux limit of Stripe 82X presents several challenges. First, the flux distribution of the AGN population sampled by Stripe 82X is not well represented in pencil-beam surveys for which photometric redshifts have been computed in previous works. Figure 2 shows histograms of soft X-ray flux (0.5–2 keV) for deeper, smaller-volume surveys compared to Stripe 82X. The bright AGNs typically found in Stripe 82X are rare in COSMOS and practically absent in the CDFS.

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Conversely, the bottom panel of Figure 2 shows that eROSITA is expected to have a flux limit similar to that of our XMM-Newton AO10 and AO13 surveys. Photometric redshifts for other large-volume X-ray surveys, such as the 3XMM Serendipitous Catalog (880 deg²; Rosen et al. 2016) and the Chandra Source Catalog, are not available yet, but the object types will likely be similar to those of Stripe 82X.

We resolve the first challenge by selecting a reduced set of templates that represents the population of Stripe 82X, as described in Section 3.2 and Appendix B. These templates should also be applicable to the other wide shallow fields, such as 3XMM and eROSITA. In fact, the template library we compiled for this work has already been used successfully in Georgakakis et al. (2017) to calculate photometric redshifts for XMM-XXL sources (50 deg²; Menzel et al. 2016) with X-ray fluxes (soft band) ${F}_{{\rm{X}}}\lt {10}^{-13}$ erg s⁻¹ cm⁻².

The second challenge is that Stripe 82X contains both archival observations of varying depths—6 deg² from Chandra and 7.4 deg² from XMM-Newton (LaMassa et al. 2013a, 2013b, 2016b)—and relatively uniform XMM-Newton imaging awarded to our team in AO10 and AO13 (PI: Urry; LaMassa et al. 2013a; LM16). After removing periods of high background, the latter allocations resulted in 4.6 and 15.6 deg² of independent area, respectively. We dealt with the varying depths by calibrating each subsample separately for each field, i.e., optimizing the spectral energy distribution (SED) libraries independently, but in any case we converged on the same set of templates for all four fields (Section 3.2).

The third challenge is that for areas covering more than a few square degrees with a variety of observations taken by independent teams with different instruments, multiwavelength catalogs are assembled by simple matches in position. Instead, for smaller fields, such as COSMOS (S09; S11; Hsu et al. 2014), CANDELS GOODS-S (Guo et al. 2013), and ECDFS MUSYC (Cardamone et al. 2010), the multiwavelength catalogs are obtained in a homogeneous manner, including the registration of the images at the same reference and taking into account the individual point-spread function (PSF). For Stripe 82X, we had to reach a compromise while dealing with catalogs that have different types of fluxes or magnitudes (e.g., aperture-corrected, Petrosian) and very different spatial resolution, which is discussed further in Section 3.4.

A common challenge for all the X-ray surveys is finding the correct and consistent multiwavelength counterpart of each X-ray source. LM16 presented a combined multiwavelength catalog for all Stripe 82 X-ray fields, with identifications done pairwise using a statistical maximum likelihood estimator (MLE; described in detail in Section 2.3) algorithm (Sutherland & Saunders 1992). That is, each ancillary data set (GALEX, SDSS, VHS, UKIDSS, AllWISE) was matched separately to the master X-ray catalog. Once the counterpart was found, a match in coordinates was performed for retrieving radio and infrared information.

In LM16, running MLE on each band returned the same counterpart across multiwavelength catalogs for approximately 4943 (80%) cases. If we only consider the catalogs/wavebands that this work and LM16 have in common—GALEX, SDSS, VHS, UKIDSS, and AllWISE—487 (8%) X-ray sources in LM16 have conflicting associations (i.e., distinctly different sources are chosen as counterparts in different bands; see Figure 3 for an example). In such cases, LM16 left it to the reader to decide which ancillary band provides the correct X-ray counterpart. We note that LM16 found no associations for 12% of the sample because the potential counterparts were below an empirically determined reliability threshold (10%) or because no candidate counterparts were found within the search radius around the X-ray source (2%).

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In this paper, we update the multiwavelength catalog, based in part on deeper and more homogeneous ancillary data than were previously available. Specifically, using new SDSS coadded catalogs from Fliri & Trujillo (2016; hereafter FT16) and MIR data from Spitzer (Papovich et al. 2016; Timlin et al. 2016), we repeated the MLE matching (for a detailed description of the process, see Section 2.3). Then, in cases of conflicting counterparts at different wavelengths, we established a procedure for identifying the same, correct counterpart at each wavelength (see Section 2.5), which is necessary if we are to use the SED to calculate photometric redshifts.

The optical data are from SDSS, which repeatedly imaged a 300 deg² area in the $u,g,r,i$, and z broadband filters, with 70–90 exposures at each location. The SDSS single-epoch images were combined to create deeper coadded catalogs (FT16; Jiang et al. 2014, hereafter J14) that reach ∼2.5 mag deeper than the single-epoch data (or the depth of the larger SDSS survey) and increase the likelihood of finding an optical association for an X-ray source. The FT16 catalog is 0.2 mag deeper in the SDSS i band than the earlier J14 coadded catalog because it eliminates data of poor quality. We use FT16 when available and J14 when the X-ray source is outside the footprint of the FT16 catalog.

We use near-infrared (NIR) data from the VHS (McMahon et al. 2013), which primarily covers the southern hemisphere but includes Stripe 82 (the equatorial coverage goes up to a declination of +15). VHS has a $5\sigma \,{K}_{\mathrm{AB}}$ depth of 20.3 mag and provides broadband data in the $J,H$, and K filters. We also use UKIDSS (Lawrence et al. 2007) NIR data when constructing SEDs, but not in the phase of counterpart identification (explained in Section 2.8).

MIR data comes from the WISE AllWISE catalog and the Spitzer Space Telescope Infrared Array Camera (IRAC). These two instruments have slightly different broadband filters that cover approximately the same wavelengths in the first two channels. AllWISE has a resolution of 1875 pixel^–1 and 95% completeness at a depth of 19.8 mag (AB) at 3.4 μm.¹⁷ This means that nearby objects are often blended together, and the positional accuracy is lower than that for VHS and SDSS. IRAC has much better spatial resolution than AllWISE (08 pixel^–1 in CH1 at 3.6 μm).

The Spitzer IRAC Equatorial Survey (SpIES) catalog (Timlin et al. 2016) has a 95% completeness at ∼20.34 mag (AB) in CH1. In the same band, the Spitzer-HETDEX Exploratory Large Area Survey (SHELA) catalog (Papovich et al. 2016) has a 95% completeness at 20.0 mag (AB). Because of their depth and resolution, for determining the counterparts, we use IRAC data rather than AllWISE when possible.

The MLE has been used to identify the correct association between sources from different catalogs, in particular for counterparts to point-like X-ray sources (e.g., Brusa et al. 2007; Georgakakis & Nandra 2011; Rovilos et al. 2011; Xue et al. 2011; Civano et al. 2012; LaMassa et al. 2013a, 2016b; Nandra et al. 2015; Marchesi et al. 2016). In this work, we highlight the major steps of the matching procedure, while the reader can refer to the more detailed description in Naylor et al. (2013).

We performed MLE matching for six different optical, NIR, and MIR catalogs; the total number of matches for each catalog is given in Table 1. The MLE method considers three factors of the ancillary waveband to determine correct counterparts to the X-ray sources: (i) the area-normalized density of background sources in the field as a function of magnitude (or flux), (ii) the area-normalized density of sources in the vicinity of the X-ray source per unit magnitude after subtracting the background distribution, and (iii) the positional errors associated with each catalog (astrometric offsets between catalogs must be corrected in advance).

The most critical point is the estimate of the magnitude distribution of the background sources, so that we can distinguish between background sources and X-ray counterparts. For example, an oversubtraction of sources in point (ii) will reduce the likelihood that a faint source is identified as the right counterpart to an X-ray source, with the effect being stronger for shallow X-ray surveys like Stripe 82X. Also, many of the Stripe 82 counterparts will be stars, bright nearby objects, or quasars, which will outshine some fainter sources, reducing the effective depth (i.e., this part of the sky will appear shallower than other, random locations in the sky lacking bright objects). To put it another way, the background near our X-ray objects has fewer visible faint sources than the field as a whole, so the background estimated from elsewhere has more faint objects than the region of interest. This was already noticed by Rutledge et al. (2000) and Brusa et al. (2007) and is further explained in Naylor et al. (2013).

To mitigate the inaccuracy at faint magnitudes in Stripe 82, we use the following process. First, we find all the objects close to the X-ray positions, i.e., within circles of radius 5'' for Chandra fields or 7'' for XMM-Newton fields; this includes the true counterparts plus the background. The area-normalized histogram of the magnitudes of these objects is what we call the total magnitude distribution. Then, we estimate a preliminary area-normalized background tallying all the objects in a 2 deg² X-ray surveyed area. The total magnitude distribution includes fewer sources, is noisier, and has fewer objects than the estimated background at faint magnitudes. Therefore, we replace the total magnitude distribution at $r\gt 23.5$ mag with the estimated background at those faint magnitudes. Using these two distributions, we tentatively identify the counterparts of all the X-ray sources using the MLE method. We then refine the background estimate by removing a circle of 2'' radius around each counterpart in the 2 deg² area (a few hundred sources) and repeat the counterpart identification. The magnitude distribution of these counterparts will be correct at the bright end but does not include objects fainter than r = 23.5 mag because they are considered background objects.

We still need to determine the background at faint magnitudes. We define a control sample of 18,000–20,000 objects with the same magnitude distribution as the bright counterparts (this is the distribution of counterpart magnitudes with $r\lt 23.5$ mag identified in the previous step) but located far enough away from the X-ray source positions to not be counterparts (though still in the X-ray surveyed region). We then measure the background near these objects within an annulus of inner radius 1'' and outer radius 5'' (for Chandra fields) or 7'' (for XMM-Newton fields) around these objects. This empirically measures the background in regions where faint objects are masked by bright objects, similar to what occurs around X-ray sources. This corrected background agrees with the previously estimated background at bright magnitudes but no longer exceeds the total magnitude distribution at faint magnitudes (at $r\gtrsim 23$ mag). We refer to Section 2.2 in Brusa et al. (2007) for a demonstration.

We then repeat the identification process using the corrected background and the original total magnitude distribution. In addition, we make an astrometric correction by finding the median offset between the X-ray and ancillary catalogs and correcting the X-ray position after each iteration of the MLE identification (following Hsu et al. 2014). Typical offsets between X-ray and SDSS coordinates are $\sim 1^{\prime\prime}$ for XMM-Newton and $\sim 0\buildrel{\prime\prime}\over{.} 02$ for Chandra. In the final output catalog, we provide the original R.A. and decl. of the X-ray sources without the offsets applied.

As discussed above, we use the FT16 coadded SDSS catalog ($r\lt 25$ mag) where possible; otherwise, we use the J14 coadded catalog (u, r, and z bands). For the NIR, we use the VHS K band (McMahon et al. 2013), as it is deeper than UKIDSS and has better photometric precision. For the MIR, we predominantly use the SpIES and SHELA Spitzer IRAC surveys (3.6 μm), since they have better spatial resolution than AllWISE and deeper coverage. However, for archival pointings outside the SpIES or SHELA footprints, we use AllWISE to find infrared counterparts (W1).

To determine whether we have identified the same source in two different catalogs, we consider the separation between the two positions. For separations $\lt 1^{\prime\prime}$ (for SDSS and VHS) and $\lt 2^{\prime\prime}$ (for IRAC), discrete sources cannot be resolved and are identified as single objects in their respective catalogs. There could be accidental alignments with a random faint source, because they have the largest space density.

For SDSS and VHS, at the faintest magnitudes, the closest pairs lie at $\sim 1\buildrel{\prime\prime}\over{.} 2$, so we are safe in assuming that counterparts within 1'' are the same object and those with larger separations are distinct objects. For IRAC, the smallest separation between faint sources is $\sim 2\buildrel{\prime\prime}\over{.} 3$, so an error radius of 2'' is appropriate; that is, IRAC sources should be within 2'' of the cataloged SDSS or VHS position. Source blending within the error radius can make coordinates inaccurate, as illustrated by Figure 4. Even though SDSS, VHS, and IRAC all identify the same bright counterpart, the IRAC position is less accurate because it is affected by blending with the object above. Therefore, we allow a bigger matching radius for IRAC (2'') to account for blending.

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As an extra precaution to avoid contamination, we visually inspected all 157 cases (2%) where the separation between the SDSS/VHS and IRAC counterparts is between 1'' and 2'' and added comments in our final catalog (Appendix A). Not surprisingly, we found that these cases incorrectly appear extended either due to very crowded fields that cause blending, or due to saturation, or are real extended sources where each band points to a different part of the same object, leading to errors in pinpointing the correct position of the source. For crowded fields, where several objects are too close to each other, we cannot obtain clean photometry, and this will affect the accuracy of the photometric redshifts. Thus, the "nearby_neighbor_sextractor" and the "manual_check" columns in our final catalog are provided to assist the user. The descriptions of all columns in the final catalog are given in Appendix A.

Comparing the positions of all SDSS and VHS counterparts for the XMM AO10 and AO13 X-ray sources, we found that 86% are within 1'' of each other and the median distance between them is 019 (Figure 5). Because the offsets between the SDSS and VHS coordinates are so small, and as both the SDSS and VHS catalogs have sufficiently high resolution to spatially distinguish between sources that are $\sim 1\buildrel{\prime\prime}\over{.} 2$ apart, we conclude that if the counterparts chosen by SDSS and VHS are more than 1'' apart, they must be two different objects.

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Due to the vastly different resolution of AllWISE, those counterparts are not considered unless we do not have a match for a source in any other catalog (66 cases). We do not discard an association completely if it is only detected in one band, because very obscured objects may only be identifiable in the MIR and can potentially be more interesting, e.g., obscured and/or high-redshift AGNs, than bright objects identified in all bands. We provide quality flags and reliability classes to indicate the level of uncertainty in counterpart identification (Section 2.5).

We find at least one possible counterpart within the search radius of an X-ray source, either in the optical or in the NIR and MIR wavelength, for 97.8% of the Stripe 82X sample. However, not all of the associations are correct, because a fraction of them will be chance associations. We use the likelihood ratio (LR) results from MLE to guide us as to whether a counterpart is reliable. The LR is the probability that the correct counterpart is found within the search radius divided by the probability that the association is a chance coincidence with a background source. We determined a critical threshold (${\mathrm{LR}}_{\mathrm{th}}$) for each band, below which we assumed that a counterpart is not reliable (these are identified in the master catalog as having a "sub-threshold" reliability class) following the method elaborated in Civano et al. (2012) and Marchesi et al. (2016), which we summarize below.

The ${\mathrm{LR}}_{\mathrm{th}}$ is the LR value at which (reliability + completeness)/2 is maximum, where reliability is defined by $R={N}_{\mathrm{ID}}/{N}_{\mathrm{LR}\gt {L}_{\mathrm{th}}}$ (ratio of the sum of all the reliabilities of the candidate counterparts and total number of sources above threshold) and completeness $C={N}_{\mathrm{ID}}/{N}_{X}$ (ratio of the sum of reliabilities of all the sources identified as possible counterparts and the total number of X-ray sources; Civano et al. 2012; Marchesi et al. 2016). After determining ${\mathrm{LR}}_{\mathrm{th}}$, we look closely at cases with more than one reliable counterpart identified in any given band (about 10% of the X-ray sources have multiple counterparts above this threshold in at least one band). For each band, we look at the ratio of LR values for the most and the next most reliable counterparts, i.e., ${\mathrm{LR}}_{12}={\mathrm{LR}}_{1}/{\mathrm{LR}}_{2}$. If that ratio is above the median value for all ${\mathrm{LR}}_{12}$ in that band, we assign it a "secure" reliability class. If the ratio falls below the median, we put that source in the "ambiguous" reliability class. Note that this classification is done among counterparts identified within a single band of multiwavelength data for each band on which we ran MLE.

Next, we compare the results of the analysis described above among different bands and find that for 12% of X-ray sources, different bands choose different objects as the most reliable counterpart (Section 2.4). If the two counterparts have different reliability classes (secure, ambiguous, or sub-threshold), we choose the more secure case. If both counterparts have the same reliability class, we choose the one with a higher likelihood ratio. Table 2 lists the reliability class for all counterparts.

Table 2.  Summary of Matches within Each Band

Band	Total	LR > LRth	Secure	Ambiguous
SDSSa	5825 (94%)	5518 (89%)	5358 (87%)	160 (2.6%)
VHS Kb	4253 (69%)	4123 (67%)	3853 (62%)	270 (4.4%)
IRAC CH1c	4728 (76%)	4354 (70%)	4102 (66%)	252 (4.1%)

Notes.

^a FT16 r band has ${\mathrm{LR}}_{\mathrm{th}}$ of ≈0.6–0.7 and median ${\mathrm{LR}}_{12}\approx 3\mbox{--}4$. ^bVHS K band has ${\mathrm{LR}}_{\mathrm{th}}$ of ≈1–1.5 for XMM AO10 and AO13 and ≈0.1–0.2 for archival XMM and Chandra. Median ${\mathrm{LR}}_{12}\approx 1.5\mbox{--}4.5$. ^cIRAC Ch1 band has ${\mathrm{LR}}_{\mathrm{th}}$ of ≈0.8–2.0 and median ${\mathrm{LR}}_{12}\,\approx 1\mbox{--}2.5$ (${\mathrm{LR}}_{12}\,\gt \,{\mathrm{LR}}_{\mathrm{th}}$ in each field).

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As we resolve conflicting associations, we assign quality flags to reflect the confidence with which we identify the correct counterpart.

• Quality Flag (QF) 1: For 4524 X-ray sources (73.2%), there is a unique counterpart in at least two bands, and their positions agree, so we consider the identifications unambiguous. For 28 of these sources, the LR is sub-threshold in all bands, but the same counterpart is identified in more than one band; therefore, we consider these unambiguous as well.
• Quality Flag 1.5: For 174 sources (2.8%), there are ambiguous counterparts in at least one band, but the other bands help resolve the ambiguity; therefore, these associations are also considered relatively secure.
• Quality Flag 2: For 190 sources (3.0%), there are counterparts that disagree, of which one is either sub-threshold or very close to the threshold ($\mathrm{LR}\lt {\mathrm{LR}}_{\mathrm{th}}+2$), while the other is well above the threshold ($\mathrm{LR}\gt {\mathrm{LR}}_{\mathrm{th}}+5$). We choose the latter case regardless of band but flag the association as not quite as secure.
• Quality Flag 3: For 416 sources (6.7%), there are different counterparts in multiple bands with comparable LR but different reliability classes (this occurs because ${\mathrm{LR}}_{\mathrm{th}}$ and ${\mathrm{LR}}_{12}$ vary between bands). Here we choose the association with the more secure reliability class, assigning it QF 3 and the secondary counterpart QF −1.
• Quality Flag 4: For 740 sources (12%), there is a counterpart in only one waveband. Of these, 631 sources have $\mathrm{LR}\gt {\mathrm{LR}}_{\mathrm{th}}$, which is considered reliable; 109 sources are below the LR threshold and therefore possibly spurious matches.
• Quality Flag 0, −1: A QF 0 indicates that we found no association with that X-ray source in any multiwavelength catalog or that these are sources taken from LM16. There are 137 (2.2%) such sources. QF −1 is an alternate to the association given for QF 3. This means that in the final catalog, one X-ray source can be listed twice, QF 3 being the more secure counterpart and QF −1 being the less secure alternative, according to our MLE analysis.

For 173 X-ray sources, there are only matches below threshold (QF 1:28, QF 2:1, QF 3:35, QF 4:109). This totals 6181 X-ray sources to which we assigned a quality flag.

To summarize, of the 6181 unique X-ray sources, 4524 (73%) have unambiguous counterparts in one or more bands (QF 1), 780 (12%) have different counterparts in different bands that we resolve with three levels of confidence (QF 1.5, 2, 3), and 740 (12%) have counterparts in only one band (QF 4). These are considered reliable when they fall above LR_th (631 cases), and 137 (2.2%) have no counterparts from our association analysis (QF 0). For 173 (∼3%) sources, there are sub-threshold counterparts (at various QF values). Other than the 29 sub-threshold sources, we take QF 1, 1.5, and 2 sources to be secure.

The flow chart in Figure 6 summarizes the steps in the identification process. We note that some identifications may be improved when deep Subaru HyperSuprimeCam optical imaging becomes available. In our final output catalog (Appendix A), we report the multiwavelength associations and photometry for all objects, along with their reliability classes and quality flags.

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There is some overlap between the X-ray observations (between all fields except AO10 and AO13 and AO10 and archival Chandra, as shown in Figure 1), resulting in 6181 unique sources and 185 duplicates. The identification of duplicates was done in LM16. We have independently determined the counterparts for all 185 duplicates and found that, even though in most cases duplicate X-ray observations were assigned the same multiwavelength counterpart, in 11 cases (23 observations—10 of these sources have two duplicate observations each, but one source has three different observations), objects identified as the same X-ray source were assigned two different counterparts. This occurs due to slight differences in X-ray coordinates, which in turn changes the surrounding area in which MLE looks for ancillary counterparts.

In order to determine which X-ray source to rely on in case of disagreement of counterparts, we choose the Chandra X-ray source over the XMM X-ray source due to the high positional accuracy of the former (20 observations of 10 sources). When deciding between duplicate Chandra observations (two or three observations of the same source), we choose the source with the lowest positional uncertainty. In the final output catalog, we use a "duplication_flag" to indicate which source we preferred and sources for which counterparts disagreed. The flag is described in Appendix A.

There are some differences between this catalog and LM16 because we use a deeper, updated optical catalog (FT16) and new IRAC catalogs, and because we use a different calculation of the background magnitude distribution used for MLE matching. For 5329 (86%) cases, there is at least one band in which our chosen association agrees with LM16 within 1''. For 40 X-ray sources (0.6%), our chosen counterpart is different from the previous catalog in all bands. There are also 658 new associations (11%) in our catalog due to deeper data. Of these, 507 are above ${\mathrm{LR}}_{\mathrm{th}}$ in at least one band. The LM16 catalog purposely did not include any sub-threshold sources, whereas our catalog does (with reliability class marked as sub-threshold) because they are used to resolve conflicting counterparts.

In addition, the LM16 catalog has 20 SDSS counterparts that were not detected in the coadded catalogs that we used. Seven of these sources were not reported in SDSS DR13 (LM16 used DR12), suggesting that they are likely spurious, and we found no counterparts in other bands. We did not include them in our final catalog. Thirteen of the 20 are in the SDSS DR13 catalog because they were detected in one or more bands at one or more epochs. We include these 13 in our final catalog ("Association" field says "LM16"). However, two of these are not detected in any other multiwavelength catalogs (VHS, IRAC, AllWISE, coadded SDSS) and so should be considered with caution.

Another five have detections in only one of those wavebands, which is better but also requires caution. Finally, five have detections in five or more wavebands and have coadded SDSS data as well, but after the offset correction that we applied, they fall outside the X-ray error radius (5''–7''), so our MLE analysis ignored these possible associations. There is one source from LM16 for which we get two bands of data, in the VHS J and H bands, but as we do not run MLE in these bands, we miss this association. We annotate these 13 objects with the appropriate "manual_check" in our final catalog.

Before constructing broadband SEDs, we identify NIR counterparts in the UKIDSS catalog (Lawrence et al. 2007) and ultraviolet (UV) counterparts in the GALEX catalog (Martin et al. 2005) using nearest-neighbor matching. These catalogs are not used for association analysis, but once we have identified a counterpart with an appropriate QF, nearest-neighbor matching with an error radius is sufficient to identify that counterpart in ancillary catalogs. Table 3 shows the error radius allowed between different ancillary catalogs to construct the final multiwavelength catalog (Appendix A). For the nearest-neighbor match, the counterpart coordinate comes from the catalog with the most accurate astrometry (as in Section 2.5); for example, if the SDSS and IRAC counterparts for an X-ray source agree, we choose the SDSS coordinates because of their better positional accuracy.

Table 3.  Search Radius Allowed for Nearest-neighbor Matches between Ancillary Catalogs

Catalog	SDSS and VHS	IRAC
SDSS DR12 redshifts	10	20
GALEX	20	20
SDSS (FT16)	10	20
SDSS (J14)	10	20
VHS	10	20
UKIDSS	10	20
SpIES	20	10
SHELA	20	10
AllWISE	10	10

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Like VHS, UKIDSS provides broadband data in the $J,H$, and K filters but is shallower (and thus with larger photometric errors; see Figure 7), with a $5\sigma \,{K}_{\mathrm{AB}}$ depth of 20.1 mag. We still include UKIDSS when constructing SEDs (but do not use it in the analysis in Sections 2.3–2.5) to check if having additional NIR data leads to more robust photometric redshifts; we discuss the results in Section 3.5. Roughly 58.7% of the X-ray sources have UKIDSS counterparts. Because the GALEX survey of Stripe 82 is relatively shallow, we find UV counterparts for only 20% of the X-ray sources.

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We correct the UV, optical, and NIR data for extinction using the Galactic extinction values from SDSS DR13 (SDSS Collaboration et al. 2016) and color excess from VHS (McMahon et al. 2013). The corrected magnitudes can then be compared to template SEDs. (The transformation of template SEDs to the observed photometric system is discussed in Section 4.1 of S09.)

To summarize, the SEDs consist of far- and near-ultraviolet (FUV and NUV, respectively) from ${GALEX};u,g,r,i$, and z from SDSS; $J,H$, and K from VHS/UKIDSS; CH1 and CH2 from IRAC; and W1 and W2 from AllWISE. We also provide AllWISE W3 and W4 magnitudes in our final catalog, but it is not used for SED construction. The bands used for SED fitting are discussed in Section 3.5.

We computed photometric redshift via the SED fitting technique using the code LePhare (Arnouts et al. 1999; Ilbert et al. 2006), which has been extensively used and tested both for normal galaxies and for AGNs. LePhare can accommodate user-specified SED templates, extinction law(s), and extinction value(s). In addition, it allows the use of luminosity priors to reject unlikely redshifts. Finally, LePhare corrects for intergalactic absorption, an important effect for high-redshift sources.

Since the mix of active nucleus and host galaxy can be different in every object, the fitting process is more complicated than for normal, inactive galaxies. This added complexity manifests as degeneracy in the results, as small parameter shifts cause more than one template to be a good fit to the photometric SED.

In order to break this degeneracy, we use secondary information to limit the parameter space of the redshift solution. The full-width at half maximum (FWHM) of sources in ground-based optical images, for example, can tell us whether the source is at low redshift: sources classified as "extended" in ground-based images cannot be at redshift $z\gt 1$, and their SEDs will have a large host galaxy contribution.

In contrast, a point-like source will either have a high redshift or be a star or a quasar, and these considerations translate into templates and luminosity priors that increase the accuracy of the derived photometric redshifts. Accordingly, we treat point-like and extended sources differently, as shown in S09.

Photometric catalogs usually provide morphology information, or typical FWHMs of stars/point-like objects in different bands. We rely on ground-based morphology provided by FT16, J14, and McMahon et al. (2013). The exact procedure for the classification into these two subsamples is discussed in Section 3.3. Note that the use of morphological classification done on lower-resolution, ground-based images rather than on Hubble Space Telescope (HST) images, as in S09, will affect our results, as we explain in Section 3.3.

Photometric redshifts are calibrated using the objects that have spectra (see Section 3.6 for more details). We start by selecting a limited set of SED templates, large enough to represent the full sample yet small enough to avoid degeneracy in ${\chi }^{2}$ fitting. We do this separately for the two subsamples (extended and point-like; described in Section 3.3). Using trial and error, we try to achieve photometric redshifts that have small normalized median absolute deviations,

$$\begin{eqnarray}&&{\sigma }_{\mathrm{nmad}}=1.48\times \mathrm{median}(| ({z}_{\mathrm{spec}}-{z}_{\mathrm{phot}})/(1+{z}_{\mathrm{spec}})| ),\end{eqnarray} \tag{ 1 }$$

and a low outlier fraction (η), where outliers are sources that have

$$\begin{eqnarray}&&| ({z}_{\mathrm{spec}}-{z}_{\mathrm{phot}})/(1+{z}_{\mathrm{spec}})| \gt 0.15\end{eqnarray} \tag{ 2 }$$

with respect to the spectroscopic redshifts.

The process of choosing correct luminosity priors, extinction laws, and color excesses was very similar to that of S09. We used the same extinction law (Prevot et al. 1984) and values for the extinction $E(B-V)=(0,0.1,0.2\mbox{--}0.5)$ and luminosity priors as in S09, as these settings produced the most reliable results by trial and error. The luminosity priors used for each subset are the same as those used in S09: an absolute magnitude range of −24 mag $\lt \,{M}_{g}\,\lt \,-8$ mag for extended and −30 mag $\lt \,{M}_{g}\,\lt \,-20$ mag for (optically) point-like objects. We fit the SEDs across a redshift range of 0.03–7.0 in steps of 0.01.

The best-fit photometric redshift is based on the minimum ${\chi }^{2}$ of the fit. Two example SED fits are shown in Figure 8. In the first case, the photometric redshift solution is unique, while in the second case, there are two possible solutions at different redshifts, obtained with different templates. In addition to the photometric redshift (${z}_{\mathrm{phot}}$) for each source, LePhare provides the redshift probability distribution function, P(z). We define a quality metric, PDZ, as the probability that the true redshift is within $\pm 0.1(1+{z}_{\mathrm{phot}})$ of ${z}_{\mathrm{phot}}$:

$$\begin{eqnarray}&&\mathrm{PDZ}=100\times \displaystyle \frac{{\int }_{{z}_{\mathrm{phot}}-0.1(1+{z}_{\mathrm{phot}})}^{{z}_{\mathrm{phot}}+0.1(1+{z}_{\mathrm{phot}})}P(z){dz}}{{\int }_{0}^{\infty }P(z){dz}},\end{eqnarray} \tag{ 3 }$$

where

$$\begin{eqnarray}&&P(z)=\displaystyle \frac{\exp [-\tfrac{{\chi }^{2}\min (z)}{2}]}{\exp [-\tfrac{{\chi }^{2}\min (z\mathrm{best})}{2}]}.\end{eqnarray} \tag{ 4 }$$

In LePhare, we set very low values of MIN_THRES (0.005) and DZ_WIN (0.025; see LePhare manual) so that secondary peaks in the redshift probability distributions could be identified. Our output catalogs and SED fit plots report these secondary redshifts, as well as the ${\chi }_{\sec }^{2}$ and ${\mathrm{PDZ}}_{\sec }$, as shown in Figure 8 (top right and bottom right panels).

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The selection of template library and extinction laws was done using all the spectroscopy from SDSS DR13 and DR12. Afterward, we added in more redshifts from DR12, DR10, and additional surveys listed in Section 1 (∼20% of the whole spectroscopic sample). Therefore, we were treating the initial 80% as our training set and the 20% we added in later as a test set. The two subsets (initial data and added data) have similar outlier fractions and overall accuracy. After this paper was submitted, SDSS DR14 was released (Abolfathi et al. 2017). We present the results for this new spectroscopic sample in Section 4.

As discussed in Section 2.1, our survey has a larger volume and is shallower than fields for which photometric redshifts have previously been calculated. The difference between populations in this work and previous deeper photometric redshift works is summarized in Table 4.

Table 4.  Population Type Comparison between Different Fields^a

Field	Source Countb	Extended Source Countc	Point-like Source Countd	${F}_{0.5\mbox{--}2\mathrm{keV}}\,\gt$ 1e−14 erg cm−2 s−1
S82X	6044e	2486 (41%)	3289 (54%)	2304 (38%)
Lockman Hole (Fotopoulou et al. 2012)	388	134 (34.5%)f	140 (36%)	19 (5%)
CDFS (Hsu et al. 2014)	744	591 (79%)	153 (21%)	7 (1%)
ECDFS (Hsu et al. 2014)	1207	974 (81%)	233 (19%)	19 (2%)
Chandra-COSMOS (Marchesi et al. 2016)	4016g	2023 (50%)	1726 (43%)	221 (5.5%)
XMM-COSMOS (S09)	1542	464 (30%)	1032 (67%)	179 (11.6%)

Notes.

^aMore details about each of these fields is given in Table 8. This table does not include fields for which z_phot were only calculated for point-like sources. ^bSources that had associations or were categorized as extended and point-like sources. ^cExtended sources tend to be nearby and galaxy-dominated. ^dPoint-like sources tend to be AGN-dominated and variable. ^e6044 out of 6181 have associations, and 269 could not be categorized as point-like or extended. ^f103 are too faint to resolve, and 12 are blended with nearby sources. ^gNot all objects could be categorized as point-like or extended due to lack of optical counterpart.

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In particular, Stripe 82X tends to have more high-luminosity quasars and AGN-dominated emission, as evidenced by the higher fraction of high soft flux (flux in 0.5–2 keV band) sources. As a result, we could not use the template set optimized for the deeper fields; instead, we began with a large selection of templates from which we selected a smaller set best suited to Stripe 82X. In Appendix B, we describe the process for selecting suitable templates for SED fitting, and we give an example of how we rejected a particular template.

We started with templates used by Ilbert et al. (2009), S09, S11, and Hsu et al. (2014) for galaxies and AGNs in deep, pencil-beam surveys (CDFS, ECDFS, Chandra-COSMOS, and XMM-COSMOS). Hsu et al. (2014) constructed hybrids combining AGN templates with semi-empirical star-forming galaxy templates from Noll et al. (2009). We tested each of these libraries on our spectroscopic sample, noted which templates were frequently providing us with the best fits and fewest outliers, and, by trial and error, selected a small set of templates that minimized the outlier fraction and ${\sigma }_{\mathrm{nmad}}$.

In general, spectroscopy is only possible for brighter objects (see Figure 9), and photometric redshifts are needed for the faintest sources in a sample. So there is always a possibility that the relatively bright population on which the photometric redshifts are calibrated is systematically different from the faint sources. We took this into account by including templates that would fit the AGN-dominated sources (mostly Type 1 or starburst spectra) and templates that would fit fainter objects not represented in the spectroscopic sample (Type 2 templates). Newly released SDSS DR14 (Abolfathi et al. 2017) spectra made it possible to test our accuracy using a blind sample of objects fainter than our training set. The results of this test are discussed in Section 4.

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The final list of templates is given in Table 5. As we mentioned in Section 2.1, our template library has also been used to calculate photometric redshifts for XMM-XXL (Georgakakis et al. 2017), providing a better accuracy than the original library defined in S09. Thus, we believe this library would be more appropriate for the eROSITA all-sky survey, which has a depth similar to that of Stripe 82X and XMM-XXL (see bottom panel of Figure 2). This library should also be considered when computing the ${z}_{\mathrm{phot}}$ of AGNs in wide surveys, such as the Dark Energy Survey (DES), Euclid, and, more importantly, LSST.

S09, S11 demonstrated how morphological information, combined with a prior in absolute magnitude, improves the reliability of photometric redshifts for AGNs. Ideally, as in COSMOS and other deep pencil-beam surveys, the morphological information comes from HST without seeing effects deforming the images. For Stripe 82X, we get morphological information from ground-based SDSS images. Besides the lower spatial resolution of the images, we have to deal with the fact that when the seeing is poor, the stars used for defining the PSF are also smeared. This means sources that ought to be classified as extended are wrongly classified as point-like. A quantification of this effect was given in Hsu et al. (2014).

As mentioned in Section 2, FT16 created deep stacked images from which low-quality data were removed, improving on the overall photometry and morphological classification. We determined that the FT16 classification as point-like or extended is the most reliable compared to other available morphological information because it gives the smallest number of outliers in the comparison of photometric and spectroscopic redshifts. Using the J14 morphological information, our results had around 25% outliers, whereas using FT16 helped us reduce that to around 14%.

We summarize the classification process in Figure 10. When FT16 classification was not available (18% of objects), we used the information from J14, ideally in the r band (for 6% of the objects); failing that, we used the information from the $i,z,g$, and u bands, in that order (4%). If no optical information was available, we used the NIR morphology from VHS data (2%); specifically, if PGAL $\gt 0.5$ (McMahon et al. 2013), we consider the object extended; otherwise, we consider it point-like.

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Objects that are not extended in the SDSS or VHS data are grouped with the point-like objects but flagged as uncertain. In the end, 40% (2486) of the Stripe 82 X-ray sources are clearly extended, 53% (3289) are clearly point-like, and 5% (269) are tentatively point-like but flagged uncertain. The remaining 2% (137) do not have a multiwavelength counterpart.

We indicate the final morphology assigned to a source using the "classification" column in the final output table, as described in Appendix A. Once objects have been classified as point-like or extended, we apply a different set of templates (summarized in Table 5) and priors to these two morphological classes to calculate photometric redshifts.

For 1156 objects with saturated photometry or bright nearby neighbors (∼16%), we calculate photometric redshifts but flag them as uncertain. We discuss the accuracy and outlier fractions of objects with and without bright nearby neighbors separately, as the photometry of the former is not reliable.

The comparison of photometric and spectroscopic redshifts for point-like and extended AGNs in the XMM AO10 and AO13 samples (to differentiate from the XMM archival sample) is shown in Figure 11 . A census of all of the sources is given in Table 6.

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Table 6.  Summary of Available Photometric Redshifts by Subsample for the 6181 Stripe 82X Sources^a

Type	XMM AO10 + AO13	Archival XMM	Chandra	Total
Total	3613	1607	1146	6366
After removing duplicates	3541	1496	1144	6181
Objects with associations	3516 (99%)	1450 (97%)	1061 (93%)	6027 (98%)
All point-like	1976 (56%)	757 (51%)	546 (48%)	3279 (53%)
With zspec	793 (23%)	292 (20%)	246 (22%)	1331 (22%)
All extended	1404 (40%)	641 (43%)	434 (38%)	2479 (40%)
With zspec	288 (8%)	138 (9%)	108 (9%)	534 (9%)
Cannot classifyb	136 (4%)	52 (3%)	81 (7%)	269 (5%)
No associations	25 (1%)	46 (3%)	83 (7%)	154 (2%)
Reporting ${z}_{\mathrm{phot}}$ forc	3501 (100%)	1418 (95%)	1025 (91%)	5944 (96%)

Notes.

^aAll percentages relative to the total number of nonduplicate sources and rounded to the nearest integer. ^bThere are multiwavelength associations in at least one band for these sources, but we do not have enough SDSS or VHS information for proper classification. We calculate photometric redshifts for these sources assuming they are point-like. ^cNumber of X-ray sources for which we are reporting photometric redshifts. Photometric redshifts calculated with less than two bands of data should not be considered reliable. To assist the user, we provide the "num_filt" column in our final catalog with the number of bands of data available for a source.

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The success of photometric redshifts, especially for AGNs, is determined to a large extent by the homogeneity of the optical and infrared photometry from different surveys. While photometry in pencil-beam areas is relatively homogeneous—e.g., CANDELS fields (Hsu et al. 2014), the entire COSMOS (S09, S11), and AEGIS surveys (Nandra et al. 2015)—the situation is worse for wide-field areas.

In particular, Stripe 82X has NIR photometry reported for different apertures in images of different resolution, i.e., sampling different amounts of the host galaxies. SpIES and SHELA provide aperture and SEXTRACTOR (Bertin & Arnouts 1996) AUTO fluxes (a Kron-like flux) for Spitzer IRAC sources, while the VHS catalog reports Kron, PSF, and Petrosian magnitudes. The photometry reported in GALEX also provides AUTO magnitude, whereas the AllWISE catalog provides a total flux (though this is affected by source blending). Optical photometry is available for different aperture sizes, as well as Petrosian and AUTO magnitude.

Under these circumstances, computing and correcting for systematic zero-point effects (e.g., Ilbert et al. 2009; Hsu et al. 2014; S09) is at best difficult and at minimum unreliable. For this reason, we came up with a compromise approach: instead of applying systematic shifts, we tested different types of photometry in order to find the combination that provides the highest accuracy and the smallest fraction of outliers.

Accordingly, after some experimentation, we decided to use AUTO magnitudes (where available) for both extended and point-like sources. Only for the NIR, we decided on a 28 radius aperture magnitude for point-like sources and 57 radius aperture magnitude for extended sources after testing several aperture sizes for each sample. For AllWISE, we provide mpro magnitudes in all four bands, although we only use $w1{mpro}$ and $w2{mpro}$ to construct SEDs.

In Section 2.8, we noted that we tried including UKIDSS and AllWISE photometry in the fitted SEDs, even though VHS/IRAC bands cover the same wavebands with slightly better accuracy, because we wanted to determine whether having additional bands of data over the same range of wavelengths led to better photometric redshifts. We found that excluding UKIDSS and AllWISE in cases where we have VHS/IRAC marginally improves the results by lowering outliers by ∼1% in the overall sample and leads to fewer cases where we are completely unable to compute photometric redshifts.

Therefore, we do not use UKIDSS photometry if we have VHS data available. Similarly, we ignore AllWISE data if we have IRAC data available. Other than these exceptions, we use all available bands from FUV to W2/CH2 for SED fitting. We report AllWISE W3 and W4 magnitudes in the final catalog but do not use them for ${z}_{\mathrm{phot}}$ calculation. Note that we do use UKIDSS and/or AllWISE bands to compute photometric redshifts for objects that do not have VHS/IRAC data.

The accuracy of photometric redshifts is based on a sample of secure spectroscopic redshifts. In practice, the fitting procedure, classifications, templates—all the aspects of determining photometric redshifts—are adjusted to produce the most accurate results with respect to the spectroscopic sample. Therefore, to ensure the most accurate results, we excluded any spectroscopic redshifts flagged as uncertain.

SDSS DR12Q (Pâris et al. 2017) presents quasar redshifts that have been visually inspected and are therefore reliable. For cases where we do not find any redshifts from DR12Q, we look for redshifts in SDSS DR13 (SDSS Collaboration et al. 2016). For DR13, no public visual inspection information is yet available, and the pipeline sometimes returns multiple redshifts for the same object. This occurs because the reliability of a redshift is dependent upon the signal-to-noise ratio of the spectrum (discussed in detail in Menzel et al. 2016) and the nature of the source.

We found that for 300 cases, the same object was associated with more than one redshift in DR13, with no warning flag (zwarning = 0). Among these, 289 sources have multiple redshifts that agree down to two decimal places. A difference on the third decimal place corresponds to a velocity of about 1000 km s⁻¹, which is below our photometric redshift sensitivity of 0.01 (due to a limited number of bands). For these cases, we select the redshift with the lowest error. For the 11 cases where the difference is significant or two different object types (AGN/star) are identified by two different spectra, we proceeded to a visual inspection and determined the correct redshift. We made use of Dwelly et al. (2017) to visually inspect all DR13 spectra reported in this work.

SDSS DR10, DR12Q, and DR13 provide redshifts for 1577 cases, and additional spectroscopy (sources listed in output table and LM16) provides redshifts for another 288 sources; the latter have also all been visually inspected.

AGNs vary on timescales of hours to years. In Stripe 82, the SDSS survey includes 80–100 repeated images over a period of 3 yr, so it is sensitive to variability on timescales of months. Palanque-Delabrouille et al. (2011) used color cuts and morphology (point-like) from SDSS DR12 to select quasar-like objects from Stripe 82 and calculate their strength of variability using a neural network code. As all of our objects are not necessarily quasars, only 3052 of our X-ray objects (49%) overlap with their sample.

The neural network output parameter from Palanque-Delabrouille et al. (2011), nnv, which reflects the strength of quasar-like variability, varies between 0 and 1. (A value of −1 indicates that not enough epochs of data were available to draw reliable conclusions.) Most of these sources—2432 objects, which is 39% of our sample—have ${nnv}\gt 0.5$, meaning highly variable. Of these, 204 have extended morphology and are probably nearby Type 1 AGNs, while 2228 have point-like morphologies and thus could be variable stars or quasars. Unlike S09, we divided the variable sources according to their morphology. For the extended variable sources, we have spectra for 45 sources, and we achieved an outlier fraction of 29% and ${\sigma }_{\mathrm{nmad}}=0.0669$. For the point-like variable objects, we have spectra for 1013 sources, and we get 14.9% outliers and ${\sigma }_{\mathrm{nmad}}=0.0561$.

AGN variability can have an effect on the photometric redshifts, particularly when data at different wavelengths were obtained at different times. However, for the sources dominated by stellar light (3387 sources; 54.8% overall, of which 1044 objects, or 16.9%, are fainter than $r\gtrsim 22.5$ mag), AGN variability is unlikely to affect the photometric redshifts. For the most luminous AGNs (2329 objects; 37.7%), variability can affect the shape of the SED, but most of these (1216 objects; 19.7%) have spectroscopic redshifts. So at most, AGN variability could affect up to 1113 objects (18% of the sample); of these, 471 (7.6%) have ${nnv}\gt 0.5$ and 188 (3%) do not have sufficient information to calculate nnv. These sources can be easily identified in the final catalog.

Table 8 compares the quality of our photometric redshifts with those in other X-ray surveys. Not surprisingly, the surveys with intermediate-band or narrowband photometry generally have more accurate photometric redshifts and fewer outliers, because narrowband photometry can easily detect the emission lines of the galaxy component. In addition, narrowband and intermediate-band photometry fills the gaps between the broadband filters. This allows the detection of characteristic features like the $4000\,\mathring{\rm A}$ and Lyα breaks that can be missed in broadband photometry.

Table 8.  Comparison of Photometric Redshifts for X-Ray Surveys with Different Depths (of X-Ray and Multiwavelength Data) and Number of Photometric Bands

Field	Ref.	Area	Number of Sources	Mean Exp Time	Soft Flux Limit	Bandwidth	Accuracy	Outliers
		(deg2)		(ks)	(${F}_{0.5\mbox{--}2\mathrm{keV}}$)	N/I/Ba	(${\sigma }_{\mathrm{nmad}}$)	η (%)
S82 Archival Chandra		6	1146	5.7–71.2	$8\times {10}^{-16}$	B	0.0627	17.1
S82 Archival XMM		7.4	1607	17–65	$7\times {10}^{-17}$	B	0.0609	13.2
S82 XMM AO10+AO13		20.2	3613	4–8	$2\times {10}^{-15}$	B	0.0595	12.8
Lockman Hole	Fotopoulou et al. (2012)	0.20	388b	185	$1.9\times {10}^{-16}$	B	0.069	18.3
CDFS	Hsu et al. (2014)	0.13	740	2000	$9.1\times {10}^{-18}$	I	0.014	6.73
ECDFS	Hsu et al. (2014)	0.3	762	250	$1.1\times {10}^{-16}$	I	0.016	10.14
AEGIS-XD	Nandra et al. (2015)c	0.29	1325	200	$5.3\times {10}^{-17}$	N/B	0.022	2.8
Chandra-COSMOS	Marchesi et al. (2016)	0.90	4016	200	$1.9\times {10}^{-16}$	N/I/B	0.015	6
XMM-COSMOS	S09	2.13	1887	60	$1.7\times {10}^{-15}$	N/I/B	0.015	6.3d

Notes.

^aN, I, and B stand for narrowband, intermediate-band, and broadband, respectively. ^bFor X-ray-detected sources (3.9σ significance), which are more similar to our sample. ^cThis paper only includes photometric redshifts for point sources, but we report the total number of sources in this table. ^dFor the QSOV sample.

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Our results are most similar to the Fotopoulou et al. (2012) results for the Lockman Hole, where only a limited number of broadband filters was used. Our results are slightly better due to a template library that is better optimized for bright AGNs. It is worth noting that, contrary to other surveys, the published photometry for Stripe 82X was not analyzed homogeneously, as explained in Section 3.4. Reprocessing the data uniformly across multiwavelength catalogs—in particular, defining consistent total magnitudes—would likely reduce the photometric uncertainties and thus improve the accuracy of the photometric redshifts.

This is the first photometric redshift study of a large, bright, X-ray-selected AGN sample. In particular, this is the largest investigation of AGN-appropriate templates for sources with ${F}_{0.5\mbox{--}2\mathrm{keV}}\gt {10}^{-14}$ erg cm⁻² (38%; 2426 objects). The well-studied pencil-beam surveys have far fewer objects at these bright fluxes; for example, CDFS has seven sources (1%) and AEGIS has 22 sources (2%) brighter than this limit. Even XMM-COSMOS, with 2 deg² of coverage, has only 221 (6%) sources of this type. We need to quantify the fitting quality at these bright fluxes, because a point-like object with high soft flux is most likely a quasar with a nearly perfect power-law spectrum. Such SEDs lack features such as the 4000 Å break, which are important for template fitting. As such, we expect that our results should deteriorate with increasing soft X-ray flux.

We tested the accuracy of photometric redshifts for three X-ray bins brighter than this limit, as shown in Figure 13 and summarized in Table 9. For point-like objects, the fraction of outliers and ${\sigma }_{\mathrm{nmad}}$ increases for higher soft flux samples, while the opposite happens for extended samples. This is likely because the X-ray-brightest extended objects are nearby and their SEDs are dominated by host galaxy light, whereas the brightest point-like objects are quasars dominated by pure power laws. Note that it is the outlier fraction that sees the biggest change with flux limit; the accuracy of the photometric redshifts changes much less because it is the median value.

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Table 9.  Outlier Fractions for Different Soft Flux Cuts^a

Object Type	${F}_{0.5\mbox{--}2\mathrm{keV}}\gt \,{10}^{-14}$ erg cm−2 s−1	${F}_{0.5\mbox{--}2\mathrm{keV}}\,\gt$ $3.5\times {10}^{-14}$ erg cm−2 s−1	${F}_{0.5\mbox{--}2\mathrm{keV}}\,\gt$ 10−13 erg cm−2 s−1
Point-like	16.4% outlier	25.5% outlier	34.3% outlier
	${\sigma }_{\mathrm{nmad}}=0.0605$	${\sigma }_{\mathrm{nmad}}=0.0665$	${\sigma }_{\mathrm{nmad}}=0.0822$
	695 objects	149 objects	29 objects
Extended	9.8% outlier	4.3% outlier	0% outlier
	${\sigma }_{\mathrm{nmad}}=0.0619$	${\sigma }_{\mathrm{nmad}}=0.0555$	${\sigma }_{\mathrm{nmad}}=0.0633$
	193 objects	69 objects	20 objects

Note.

^aOnly for objects without bright nearby neighbors.

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Since Stripe 82 is fully covered by SDSS, others have calculated photometric redshifts for subsets of optically or spectroscopically selected quasars using a variety of techniques. Here we compare results for the Stripe 82X sources that are in common with those studies, as summarized in Table 10. The samples that exclusively select quasars from optical data are expected to (and in some cases do) perform better than our approach. This is because of the difference in population type: our sample includes Type 1 and Type 2 AGNs, starburst galaxies, and some spiral and elliptical galaxies. The diversity in object type requires a large number of models in our template library, which leads to degeneracy and increases our outlier numbers and inaccuracy.

Table 10.  Comparison to Photometric Redshifts in the Literature for Sources Overlapping with Our Work^a

Photoz Catalog	Source Counts	Outliers	${\sigma }_{\mathrm{nmad}}$
Peters et al. (2015; Quasars)	565	16.11%	0.0475
This work		10.8%	0.0529
Richards et al. (2015; Quasars)	920	6.8%	0.0236
This work		16%	0.0601
Bovy et al. (2012; Quasars)	945	8.15%	0.0312
This work		13.76%	0.0559

Note.

^aThe values of ${\sigma }_{\mathrm{nmad}}$ and fraction of outliers are from a comparison of photometric and spectroscopic redshifts for the sources common to both works.

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Bovy et al. (2012; hereafter the XDQSO model) uses spectroscopically confirmed quasars from SDSS DR7 with $z\gt 0.3$ to model the color–redshift relationship. The XDQSO model uses this color–redshift relationship, along with an apparent magnitude–dependent redshift prior, to calculate a probability distribution of photometric redshift for each object. They found accurate photometric redshifts ($| \bigtriangleup z| \lt 0.3$) for 97% of the quasars (we do not have their spectroscopic sample available to calculate η and ${\sigma }_{\mathrm{nmad}}$). There are 945 objects in the Stripe 82X sample that overlap with the XDQSO work and also have spectra, so we compared our results.

Peters et al. (2015) used a Bayesian technique to select quasar candidates from SDSS and Stripe 82 and used the method described in Weinstein et al. (2004) to find photometric redshifts for these objects. This work also optically selects quasars only. This approach uses astrometric parameters and optical photometry and reaches an accuracy of $| \bigtriangleup z| \lt 0.3$ for 76.9% of the quasars. Using the spectroscopic sample of that work, this translates to η = 23.3% and ${\sigma }_{\mathrm{nmad}}=0.035$.

Richards et al. (2015) used the same approach as Peters et al. (2015) but with more bands (adding GALEX and IRAC to the SDSS, VHS, and AllWISE data). More bands improves results: this work has an accuracy of $| \bigtriangleup z| \lt 0.3$ 93% of the time (when NIR is present), which translates to η = 7.03% and ${\sigma }_{\mathrm{nmad}}=0.018$ for the entire spectroscopic sample of that work.

A machine-learning approach to calculate SDSS quasar luminosity was taken by Brescia et al. (2013). However, the spectroscopic redshifts used to train the data were from SDSS DR7, and, while trying to compare our results, we found that with the updated pipeline in SDSS DR12/DR13, many of these objects have changed spectroscopic redshift. As a consequence, our results were not comparable.

Following the method explained in S09, we identified stars by SED fitting using the following criteria on point-like sources: ${\chi }_{\mathrm{best}}^{2}\geqslant 1.5\times {\chi }_{\mathrm{star}}^{2}$. Here ${\chi }_{\mathrm{best}}^{2}$ corresponds to the ${\chi }^{2}$ value returned by the best-fit AGN/galaxy template, and ${\chi }_{\mathrm{star}}^{2}$ corresponds to the best-fit star template. The appropriate multiplicative factor depends on photometry and usually falls between 1 and 2. Figure 14 compares the quality of fit for galaxy/quasar and stellar templates. The spectroscopically confirmed AGNs are shown as red points, all of which fall below a ratio of 1.5.

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Generally, if a star has a close neighbor, it could mistakenly be classified as extended, in which case a galaxy template will fit about as well as the stellar template. This is the biggest cause of misclassification of stars. Truly point-like objects can easily be distinguished as stars or AGNs/quasars, as shown in Figure 8 (and confirmed by their spectra).

To investigate if we can improve the accuracy of classification, we looked into a color-based method of stellar identification presented in LaMassa et al. (2016a): $R-W1\,=(0.998\pm 0.02)(R-K)+0.18$, where all magnitudes are in the Vega magnitude system. We found that the SED fitting method is more accurate in correctly identifying stars based on objects for which we have spectroscopic information. We summarize our findings in Table 11.

One caveat of finding stars using template fitting is that if the object is misclassified as extended due to close nearby neighbors, we use incorrect priors and cannot correctly classify such objects. Scranton et al. (2002) discussed some challenges that arise in morphological classification due to close nearby neighbors. We found five cases where a close nearby object causes a star to appear extended. These stars were misclassified into the extended sample. Without better-resolution data, we cannot correct for nearby neighbors distorting FWHM information. Nearby elliptical galaxies are very well fitted by stellar templates, so we cannot apply the ${\chi }_{\mathrm{best}}^{2}/{\chi }_{\mathrm{star}}^{2}\gt 1.5$ threshold on the extended sample without misclassifying a lot of galaxies as stars. This is why we search for stars only among the point-like sources. One of these five stars is recognized as a star by the color–color test, one is miscategorized as "extragalactic," and, for the rest, we do not have color diagnostic information.

As we discussed in Section 1, Stripe 82X provides information on a population that was barely represented in COSMOS due to its smaller volume. Figure 15 shows the luminosity–redshift distributions of these two surveys, demonstrating that the larger Stripe 82X volume samples ∼15 times higher luminosities.

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Template fitting gives us a rough idea of the demographics of the Stripe 82 X-ray sources. Table 12 presents numbers of each type of object based on the best-fit SED template. Of course, since most of the "galaxies" have X-ray luminosities in excess of ${10}^{42.5}$ erg s⁻¹, these are really obscured AGNs. Together with the explicitly identified Type 2s, obscured AGNs make up roughly one-third of the extragalactic objects. The observed ratio of Type 2 to Type 1 AGNs is roughly 4:5; the intrinsic ratio can be much higher, because obscured AGNs are fainter than unobscured AGNs (for the same underlying luminosity). In Figure 16, we present the overall distribution in luminosity and redshift for each type of object.

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Table 12.  Comparison of the Accuracy of the SED Fitting Method and Color–Color Method for Identifying Stars for Objects with Spectra

Best-fit SED Type	Template Number	Description	Number of Objects
Stars	${\chi }_{\mathrm{best}}^{2}\geqslant 1.5\times {\chi }_{\mathrm{star}}^{2}$	As explained in Section 4.5	230 (3.7%)
Ellipticals	33–35 (From Table 5)	Elliptical galaxies	797 (12.9%; 4 at z > 3)
Spirals	14, 36–40	Spiral galaxies	1000 (16.2%; 6 at z > 3)
Starbursts	1, 2, 13, 25, 26, 29–32	M82, Mrk231, I22491-Type 1 hybrids	1474 (23.8%; 24 at z > 3)
		with $\gt \,=\,50 \%$ I22491 contributions
Type 2	15–24	Any Type 2 hybrid	123 (2%; 9 at z > 3)
Type 1 hybrid	5–12, 27, 28	Type 1 hybrids with >50% Type 1 contribution	1655 (26.8%; 56 at z > 3)
High-luminosity quasar	2, 3	100% quasars	682 (11%; 17 at z > 3)
Total			5961 (96.4%; 114 at z > 3)

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The most interesting question is whether there are many obscured sources at high luminosity. Following Brusa et al. (2010), we look for an obscured population using R–K Vega magnitudes and the 2–10 keV X-ray–to–r-band flux ratio (X/O), as shown in Figure 17. Objects with $\mathrm{log}(X/O)\gt 1$ and R–K > 5 are obscured AGN candidates. In the Stripe 82X sample, 368 objects have $\mathrm{log}(X/O)\gt 1$. Of the 368 objects, 78 also have ${(R\mbox{--}K)}_{\mathrm{Vega}}\gt 5$ (∼2.5 deg⁻²), and their SEDs are well fitted by starburst, spiral, elliptical, and Type 2 templates, consistent with what we expect for heavily obscured AGNs/quasars.

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For comparison, Brusa et al. (2010) found 105 objects in this region of color–color space in the XMM-COSMOS survey (∼52.5 deg⁻²). Nine of our 78 candidate obscured objects have redshift $z\gt 2$, and one has redshift $z\gt 3;$ independent of color, 815 (129) objects have $z\gt 2$ ($z\gt 3$). Of the 129 sources (2% of the entire sample) that have redshift $z\gt 3$, only 32 (<1%) are spectroscopically confirmed. Among the 97 without spectroscopic confirmation, only 25 have $\mathrm{PDZ}\gt 90 \%$ and can be considered reliable. The low PDZ of the remaining sources is due to the limited number of photometric points and/or their large photometric errors.

Using slightly different criteria, Perola et al. (2004) and Civano et al. (2005) found that objects with $\mathrm{log}(X/O)\gt 1$ and ${F}_{(2\mbox{--}10)\mathrm{keV}}\geqslant {10}^{-14}$ erg (cm² s)^–1 are high-luminosity obscured quasar candidates. Of the Stripe 82X sources, 375 meet these criteria (11.981 deg⁻²). We have a dedicated spectroscopic follow-up program targeting candidate obscured quasars (LaMassa et al., submitted), and we are focusing particularly on obscured and/or high-redshift candidates.

Finally, we report the total distribution of photometric redshifts and X-ray full-band luminosities (0.5–10 keV) by taking into account the probability distribution of redshifts for each object, and not just the best-fit value, in Figures 18 and 19, respectively. Typical probability distributions of the redshift of individual objects look similar to the bottom panels of Figure 8. The total redshift distribution as shown in Figure 18 was calculated by summing over the probability distributions of all objects.

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As expected, the Stripe 82X luminosity distribution is skewed to higher luminosities, as shown in Figure 19 (calculated using the full photometric redshift probability distributions). These probability distributions of individual objects can also be used to derive luminosity functions, clustering, and other redshift-dependent results, as shown by Allevato et al. (2016), which will be the subject of future work.

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