PHOTOMETRIC REDSHIFTS IN THE HAWAII-HUBBLE DEEP FIELD-NORTH (H-HDF-N)

We collect the U-, B-, V-, R-, I-, z'-, and HK'-band images from C04, the J- and H-band images from Keenan et al. (2010), and the K_s-band image from Wang et al. (2010, W10 hereafter), respectively. We also make use of an independently observed z'-band image from Ouchi et al. (2009). The IRAC 3.6, 4.5, 5.8, and 8.0 μm images were obtained from the Spitzer Heritage Archive processed via Super-Mosaic pipeline version 2.0, calibration pipeline version S18.25.0, and MOPEX (for mosaic processing) versions 18.5.4 (for 3.6, 4.5, and 5.8 μm) and 18.5.6a (for 8.0 μm), while another set of IRAC 3.6 and 4.5 μm images were taken from the Spitzer Extended Deep Survey (SEDS) presented in Ashby et al. (2013). The image information is listed in Table 1 and the filter transmission curves are plotted in Figure 2. We show the R-band image overlaid with various coverages in the H-HDF-N in Figure 3. The GOODS-N, CDF-N, and K_s-band coverages are surrounded by the blue, cyan, and red rectangles, respectively. The yellow rectangle indicates the region (i.e., the central H-HDF-N field) where C04 derived their catalogs. We find that, different from other images, all IRAC images have some residual background, which could bias the PSF model building (see Section 4); therefore, we run SExtractor to subtract the background before further analyses.

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Table 1. Imaging Data

Band	Depth	PSF Size	Zero Point	Solid Angle	Group	Galactic Extinction	Epoch	Reference
(1)	(2)	(3)	(4)	(5)	(6)	(7)	(8)	(9)
U	26.3	163	31.369	0.42	II	−0.048	2002	Capak et al. (2004)
B	26.3	113	31.136	0.31	I	−0.042	2001	Capak et al. (2004)
V	25.8	156	34.707	0.39	II	−0.031	2001	Capak et al. (2004)
R	26.0	160	34.676	0.39	II	−0.025	2001	Capak et al. (2004)
I	25.1	108	33.481	0.39	I	−0.018	2001–2002	Capak et al. (2004)
z' (zc)	24.9	108	33.946	0.39	I	−0.014	2000–2001	Capak et al. (2004)
z' (zo)	25.7	120	33.020	0.33	I	−0.014	2001–2007	Ouchi et al. (2009)
J	24.5	111	23.900	0.22	I	−0.008	2006	Keenan et al. (2010)
H	22.9	132	23.900	0.36	I	−0.005	2008	Keenan et al. (2010)
Ks	23.7	108	23.900	0.36	I	−0.004	2006–2008	Wang et al. (2010)
HK'	22.3	120	30.132	0.11	I	−0.005	1999–2002	Capak et al. (2004)
3.6 μm (A1)	25.1	253	21.581	0.33	III	−0.002	2004–2011	Ashby et al. (2013)
3.6 μm (S1)	24.5	240	21.581	0.33	III	−0.002	2004–2006	Spitzer Archive
4.5 μm (A2)	24.6	253	21.581	0.33	III	−0.002	2004–2011	Ashby et al. (2013)
4.5 μm (S2)	24.2	243	21.581	0.31	III	−0.002	2004–2006	Spitzer Archive
5.8 μm (S3)	22.6	296	21.581	0.33	III	−0.002	2004–2006	Spitzer Archive
8.0 μm (S4)	22.7	3.24	21.581	0.28	III	−0.002	2004–2006	Spitzer Archive

Notes. Column 1: band name. Column 2: 5σ limiting AB magnitude estimated with a 21 diameter (7 pixels) aperture, based on background noise estimation detailed in Section 5.3. Column 3: PSF size that is calculated based on the PSF models built in Section 4.1, which is defined as the aperture diameter that encircles half of the total flux. Column 4: zero point in units of AB magnitude. Column 5: solid-angle coverage in units of deg². We only consider the areas located in the H-HDF-N field. Column 6: group name that is classified based on the PSF size of a band (see Section 5.1). Column 7: galactic extinction in units of AB magnitude (see Section 5.5). Column 8: observational epoch. Column 9: reference of imaging data. The Spitzer Archive data can be retrieved at the Spitzer Heritage Archive.

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For convenience in describing the PSF-matching technique (see Section 4), we denote PSF size as the aperture diameter that encircles half of the total flux. Hereafter, we refer to Ouchi's z' band as zo band, and Capak's z' band as zc band. The filters of zo and zc are identical, but the images were taken during different observational epochs (C04; Ouchi et al. 2009). The 3.6 μm and 4.5 μm bands in Ashby et al. (2013) are referred to as A1 and A2 bands, respectively, while the 3.6, 4.5, 5.8, and 8.0 μm bands from the Spitzer Heritage Archive are referred to as S1, S2, S3, and S4 bands, respectively. Filters of A1 and S1, A2 and S2 are identical, and the observational epochs of S1 and S2 were also used to derive the A1 and A2 images (Ashby et al. 2013; also see Table 1). Although the S1and S2 images are shallower, we still incorporate them in our z_phot estimation, because the deeper A1 and A2 images were obtained by stacking more observations and thus were potentially more blurred (see Table 1 for a comparison between PSF sizes). Indeed, we find that the inclusion of the S1 and S2 bands improves slightly the z_phot quality (see Section 2.2 for the indicators of z_phot quality). The application of the rest-frame template error function by EAzY gives much lower weights to mid-IR than optical bands, therefore the use of duplicate 3.6 μm and 4.5 μm bands should not affect the z_phot estimation significantly (see Section 6), as found above. We do not make use of the HST data in our z_phot estimation for the following reasons. First, the HST images are restricted in the GOODS-Nregion, the area of which is only ≲20% of that of the H-HDF-N (see Figure 3). Second, the HST images have much smaller PSF sizes than our images, but this great advantage would be compromised if we apply the same PSF-matched photometry-extraction procedures (see Section 5) to them. Finally, the 3D-HST team has derived photometry and photometric redshifts (Skelton et al. 2014) for the GOODS-N region to greater depths utilizing the HST data, and their z_phot catalog is available now and complements our catalog effectively.

The z_spec data are collected from a number of references, with relevant information listed in Table 2. We only adopt secure z_spec data that include at least two spectral features. Note that the z_spec data mainly come from Barger et al. (2008) because a significant fraction of their data were compiled from previous works, e.g., Wirth et al. (2004). To remove duplicate z_spec entries, z_spec sources from different references are matched with each other using a 05 matching radius, except for those from Barger et al. (2003) that have already been matched by the authors to the 2 Ms CDF-N main X-ray catalog (A03; see Section 7.4). If one source has multiple z_spec values and any two of those values are inconsistent (i.e., |z_spec1 − z_spec2|/(1 + z_spec1) > 0.01), we then discard all z_spec values of that source (less than 2% of the z_spec values are discarded this way). If a source has only one z_spec value, we simply keep that z_spec for the source. Most of the z_spec sources (∼80%) are in the GOODS-N region. In Figure 4, we plot in the top panel the R-band magnitude (derived from this work; see Section 5; upper limits not included) distribution of all non-stellar z_spec sources, which peaks around R =23.5 mag and declines rapidly beyond that; and we plot in the bottom panel the z_spec distribution of all non-stellar z_spec sources, the vast majority of which have z_spec ≲ 1.6.

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We adopt the widely used normalized median absolute deviation σ_NMAD (e.g., B08) to evaluate our z_phot quality (see Section 6.4), defined as

$$\begin{equation} \rm \sigma _{NMAD}\,=\,1.48\times median \left(\left|\frac{\Delta {z}- \rm median(\Delta {z})}{1+{z}_{\rm spec}}\right| \right), \end{equation} \tag{ 1 }$$

where Δz = z_phot − z_spec. Additionally, we also examine outlier fractions of z_phot results, with outliers being defined as sources having |Δz|/(1 + z_spec) > 0.15.

For each image, we match the detected sources with the stars in the GSC 2.3 catalog (Lasker et al. 2008), and use SExtractor's output flags to discard saturated, blended, and/or near-border ones. We then check source profiles, morphology, and contamination from nearby sources to further filter out surviving galaxies and saturated stars, and find that the profiles of the remaining stars resemble each other. About 20 stars in each image are selected this way to build the PSF models. We create a fixed-size thumbnail image centered on each star and normalize its flux to unity. We then construct the PSF model in the form of an image of the same size, whose pixel values are assigned as the median values of the corresponding pixels of those standard stars. Thus the PSF models represent the typical PSFs of corresponding images, with one single PSF model for each image.

We make use of Lucy's procedure (IRAF task; Lucy 1974) to build the kernel to smooth one PSF to another PSF. Assuming there are two PSF thumbnail images, i.e., PSF A and PSF B as inputs (the corresponding images are denoted as image A and image B), Lucy's procedure will approximate the solution iteratively, which can then be used as the kernel to smooth image A to the PSF level of image B. The merit of Lucy's procedure is that it does not require that the PSFs are of specific modeled shapes such as Gaussian. It performs well when the target PSF size is much larger. In Figure 6, we demonstrate the effectiveness of Lucy's procedure when smoothing the J-band PSF (having the size of 111) to the U-band PSF (having a larger size of 163).

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The PSF sizes can be divided into three groups (see Table 1):

• 1.  
  The B-, I-, zc-, zo-, J-, H-, K_s-, and HK'-band images have the smallest PSF sizes, i.e., ≲13;
• 2.  
  The U-, V-, and R-band images have moderate PSF sizes, i.e., ≈16;
• 3.  
  The A1-, A2-, S1-, S2-, S3-, and S4-band images have the largest PSF sizes, i.e., ≳24.

If we smooth all images to the largest PSF, then the quality of the images with small PSF sizes, such as group I bands, will drop significantly, leading to much larger photometry errors. Therefore, we adopt the following approach to obtain accurate colors, and preserve the quality of the images maximally at the same time, which yields three catalogs (i.e., R-, zc-, and K_s-selected) that are finally merged into one based on the Q_z criterion (see Section 6.3).

5.1.1. zc and K_s Catalogs

Here we describe how we perform PSF matching in order to obtain the zc- and K_s-selected catalogs. For group I images, we smooth the B-, I-, zc-, zo-, J-, K_s-, and HK'-band images to the PSF level of the H-band image (the largest PSF size in group I); therefore, the smoothed images are consistent with the H-band image, and we extract the photometry of the detected sources. For the images in groups II and III, we adopt the strategy of aperture correction (e.g., Cardamone et al. 2010), which we describe below. First, we smooth image M (M stands for the detection band, i.e., zc or K_s; see Section 6.3) to image N (N stands for any band in groups II and III), and perform our photometry procedures on image N and the smoothed image M. Then we define an aperture correction factor, cor_aper, for each detected source, as

$$\begin{equation} \rm cor_{\rm aper}=\frac{flux(M\ at\ PSF\ level\ of\ {H})}{flux(M\ at\ PSF\ level\ of\ N)}, \end{equation} \tag{ 2 }$$

where flux is the total ADUs (Analog-to-Digital Units, i.e., the values of pixels in the image matrix) encircled by the aperture whose diameter equals 1.5 times the corresponding PSF sizes. Finally we obtain the flux of N at the PSF level of the H-band image for a source as

$$\begin{equation} \rm flux(N)= flux(N\ not\ smoothed) \times cor_{\rm aper}. \end{equation} \tag{ 3 }$$

5.1.2. R Catalog

We use SExtractor to detect sources and extract photometry. At first, we detect many more sources in the K_s-band image (>200,000 sources in total) than those in the zc- and R-band images (each having <100,000 sources) when using the same SExtractor parameters, but find that many of the K_s-band sources are false detections through visual inspection. The reason is that during the SExtractor runs, the zc- and R-band images are supplied with corresponding rms maps, while the K_s-band image with a weight map. Internally, SExtractor treats rms maps as absolute noise levels, while weight maps are treated as relative noise levels; it then scales weight maps to absolute rms maps via an internal algorithm. However, this algorithm tends to underestimate the noise level. To avoid the vast majority of false detections, W10 adopted the most stringent cleaning procedure, i.e., setting CLEAN_PARAM = 0.1.¹⁷ This configuration reduces effectively the number of K_s-band detected sources to ≈90,000. Visual inspection shows that false detections are rare in this case. The other parameters in W10 were also chosen carefully and are therefore reliable, and, consequently, we adopt most of their parameters and run SExtractor on the dual-image mode. Table 3 lists the main SExtractor parameters adopted for the K_s-band image. There are only minor changes in SExtractor parameters for other images. Specifically, we lower the detection thresholds by setting DETECT_MINAREA = 2 and CLEAN_PARAM = 1 for the zc and R bands due to the difference between using their rms maps and using the K_s-band weight map as stated above. Generally these detection thresholds are sufficiently low to detect very faint sources. However, this might potentially lead to some false detections. Users of our catalog should be aware of this issue (see Section 7.5).

SExtractor assumes pixel-uncorrelated errors, and neglects background noise contributed by faint sources below the detection threshold. Therefore its derived photometric errors are underestimated. As pointed out by, e.g., Dahlen et al. (2013), accurate photometric errors are crucial in deriving accurate z_phot. To obtain a more accurate background noise estimate for each source, we place around the source 100 apertures of the same size used for photometry extraction. These apertures are placed avoiding sources shown in the SExtractor-provided segmentation checkimage, and do not overlap with each other. Then we calculate the standard deviation of the flux in those apertures, and use it to replace the error given by SExtractor. The new errors, typically being several times larger than those provided by SExtractor, result in much better z_phot quality. Furthermore we also consider additional errors introduced by our PSF-matching procedure. We thus introduce err_psf: if we smooth band P to the PSF level of band Q, then err_psf is defined as

$$\begin{equation} {\rm err}_{{\rm psf}}=\frac{ |{\rm FL}({\rm smoothed}\ P) - {\rm FL}(Q)| }{{\rm FL}(Q)}, \end{equation} \tag{ 4 }$$

where FL means the fraction of light encircled by the photometry aperture in the PSF image. err_psf is a relative quantity, and is multiplied by the flux before being added quadratically to the aforementioned new error. For most bands, err_psf is ≲ 4%. Typically, the background error dominates the err_psf. For those objects whose flux in one band is less than the corresponding error, an upper limit is assigned by setting the flux to the value of the error and is included in the z_phot derivation (see Section 6).

We note that one photometric band might have two images (e.g., zc and zo bands). If large photometric differences exist between the two images, it will be impossible to fit both sets of photometry well with templates in the z_phot estimation. We find that even a single inconsistent band could often ruin z_phot quality, regardless of how perfect the other bands are. To reduce inconsistency, we first eliminate the systematic offset (≲ 0.03 mag) between the two fluxes by adjusting the zero point of either band in order to meet the condition of median(mag1−mag2) = 0. This adjustment is adopted only to facilitate discarding inconsistent photometry and is not applied to the photometry for z_phot estimation. Then, for each source, if

$$\begin{equation} \rm |mag1-mag2| > 3\times max(err\_mag1, err\_mag2), \end{equation} \tag{ 5 }$$

both sets of photometry will be discarded (≲ 5% of sources have inconsistent fluxes in at least one band). The main reasons for inconsistent photometry are blending effects in crowded fields and contamination from nearby bright sources.

The H-HDF-N field, located at high Galactic latitude, was initially chosen to be subject to minimal Galactic extinction. We apply Galactic extinction corrections obtained from an online utility provided through the NASA/IPAC Infrared Science Archive.¹⁸ As shown in Table 1, the corrections, as expected, are small, ranging from 0.002 mag for the A1, A2, S1, S2, S3, and S4 bands to 0.048 mag for the U band.

The above procedures such as PSF matching and aperture correction are all likely to introduce systematic errors into the photometry. We derive zero-point corrections to account for this factor by fitting the photometry for spectroscopic stars (see Section 2.2) with a set of 235 stellar templates at z = 0. The stellar templates are taken from the LePhare photometric redshift and simulation package (Arnouts et al. 1999; Ilbert et al. 2006), which consist of 231 stellar spectra that include all normal spectral types and luminosity classes (Pickles 1998; Chabrier et al. 2000) and 4 white dwarf spectra (Bohlin et al. 1995). Here we only consider the bands bluer than the IRAC ones because the stellar templates of Pickles (1998) do not cover wavelengths beyond 2.5 μm.

We first tentatively match our sources with the z_spec catalogs (see Section 2.2 and Figure 4) using a 05 matching radius and then remove the systematic astrometry difference (≈008) before matching them again. The false-matching rate is estimated to be ≲ 3% by systematically shifting source coordinates and then recorrelating them. We then obtain the new zero points as follows, using a total of ≈130 spectroscopic stars that are not saturated or blended based on visual inspection,¹⁹

$$\begin{equation} \rm new\_zp\,=\,old\_zp - 2.5\times log \left(median \left(\frac{fitting\_flux}{observed\_flux} \right) \right). \end{equation} \tag{ 6 }$$

The results are listed in Table 4. Note that for the three detection bands (i.e., K_s, zc and R), the zero-point corrections are not exactly the same. This should be due to the fact that the photometry from different catalogs is derived from different sets of images (see Section 5.1) and there are some uncertainties associated with the kernels used in smoothing (see Section 5.3) when obtaining those images.

We plot VRzc color–color plots in Figure 7 to show the effect of zero-point corrections. For the case without correcting for zero points (i.e., the left panel of Figure 7), overall, our star colors do not agree well with template star colors. However, for the case with zero-point corrections applied (i.e., the right panel of Figure 7), our star colors are in good agreement with template star colors. Therefore, we conclude that zero-point correction is effective in eliminating systematic errors and adopt the new zero points (i.e., the corrected photometry) for subsequent analyses.

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Our source-detection approach recovers 46,914 sources out of a total of 48,858 sources with ⩾5σ significance in the C04 catalog using a 05 matching radius after removing any systematic astrometry offsets (46914/48858 = 96%). For the K_s-band selected catalog of W10, the recovered number of sources is 53,544 out of a total of 56,967 sources with ⩾5σ significance (53544/56967 = 94%). Through visual inspection, we find that nearly all the unmatched sources are very faint or blended with other sources, and thus source detection and position determination are more uncertain.

To estimate the completeness level of our catalog, we compare our magnitude-dependent source density with that of the GOODS-N catalog (Giavalisco et al. 2004), and plot the source density histograms for our zc band and the HST z(F850lp) band of the GOODS-N catalog in Figure 8. For both catalogs, we only count sources with ⩾3σ significance. Assuming that the GOODS-N z(F850lp) catalog is complete at least down to z ≈25.5 mag, our catalog is then ≈100% and ≈80% complete down to z ≈24.5 mag and 25.5 mag, respectively. Note that in some magnitude (≲ 24 mag) bins, our number counts appears slightly higher than that of HST z(F850lp) band. This might be due to differences in filters and instruments.

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When detecting sources, SExtractor might fail to separate two very close sources. To estimate the importance of this effect in our catalog, we make use of the GOODS-N catalog (Giavalisco et al. 2004) that is based on HST observations with superb angular resolutions of ≈005. We only estimate the effect for our R band, the one with the largest PSF size among the three detection bands, to assess an approximate upper limit of this effect. We then only consider sources with R ≲ 26 mag (i.e., 5σ detection limit of our R band; see Table 1). Through visual inspection of both our R-band and the HST F606W-band images (the HST F606W band is similar to our R band and both bands are centered at ≈6000 Å), we find that the detection generally fails if the angular separation of two sources is less than 12. We also find that ≈10.8% of the HST F606W ≲ 26 mag sources in the GOODS-N catalog have neighboring sources within a radius of 12. We therefore conclude that about 10.8%/2 = 5.4% of our R-band sources might be in fact two sources that can only be separated distinctly in images with higher angular resolution.

Another issue caused by blending is photometry contamination by nearby sources. We assume that photometry of one source is subject to contamination if another source is present within a radius of 1.5 × PSF size. For non-IRAC bands, we estimate the effect for our U band, the one with the largest PSF size of 163, to assess an approximate upper limit of this effect. We find that ≈28% of the U-band sources might suffer from such photometry contamination. However, the actual contamination effect should be much less severe and could be totally negligible in many cases. For example, bright sources are minimally affected (if at all) by their faint neighbors. Furthermore, SExtractor can reduce this contamination effect when computing photometry (see page 40 of the SExtractor manual for version 2.13). For IRAC bands, the photometry contamination is expected to be worse considering their large PSF sizes. However, this situation is largely alleviated given that IRAC data have much lower weights than optical and near-infrared data in z_phot estimation (see Section 2.1).

Sources that suffer from the above blending issues would generally have inaccurate photometry and thus z_phot of poor quality. Such sources typically have large Q_z (the redshift quality parameter defined in Section 6.3) values, e.g., for sources with ⩾5σ significance, ≈56% of the (likely) blended sources have Q_z > 1, in contrast to ≈23% of the non-blended sources having Q_z > 1. In such cases, their z_phot are generally not recommended for use (see Sections 6.3 and 7.5).

We use the EAzY code (B08) to estimate photometric redshifts. EAzY (version 1.00) can fit with linear combinations of template SEDs. There are two novel features of EAzY. First, the default template set (see Figure 9) is based on semi-analytical models rather than spectroscopic samples (usually these are highly biased). B08 used the "nonnegative matrix factorization" algorithm to extract a template set from the library of ∼3000 PÉGASE models (Fioc & Rocca-Volmerange 1997) with ages ranging from 1 Myr to 20 Gyr and having a variety of star formation histories. Second, B08 designed a rest-frame template error function to estimate template uncertainty. This function assigns different wavelength regimes different weights (being largest, moderate, smallest for the rest-frame optical, ultraviolet, and near-infrared bands, respectively), and ensures that the formal redshift uncertainties are realistic (B08).

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6.1.1. Templates

6.1.2. Other Parameters

In Section 5.6, we correct our photometry based on SED fitting of spectroscopic stars. However, the corrected photometry might still have systematic differences compared to the expected photometry based on galaxy templates that are used in z_phot calculation. In order to achieve better z_phot quality, we derive zero-point offsets to eliminate such photometry differences by fitting the photometry at z_spec (see Section 5.6 for details of matching our sources with the z_spec catalogs) with the eight adopted templates (see Figure 9). We compute the zero-point offsets by

$$\begin{equation} \rm zp\_offset \,=\, - 2.5\times log \left(median\left(\frac{fitting\_flux}{observed\_flux}\right)\right). \end{equation} \tag{ 7 }$$

After three iterations, the zero-point offsets converge (vary by ≲0.01 mag). The zero-point offsets are listed in Table 5.

The zero-point offsets for the S3 and S4 bands indicate that our photometry has fluxes higher than template fluxes at >5μm wavelengths. However, the S3 − S4 color of stars without zero-point offsets applied appears more consistent with that of Stern et al. (2005). Furthermore, the inclusion of these two offset bands in our z_phot calculation yields a worse z_phot quality (see Section 6.4). Therefore we conclude that the zero-point offsets for the S3 and S4 bands are unreliable, likely due to the absence of polycyclic aromatic hydrocarbon emission features in our templates. We thus discard zero-point offsets for these two bands and do not use their photometry for z_phot estimation; however, for completeness, we still provide their photometry in the final catalog (see Section 7).

We find that our star colors generally become inconsistent with template star colors after applying the above zero-point offsets that are derived with galaxy templates. Such zero-point offsets are apparently template-dependent. Therefore, we adopt these zero-point offsets only in z_phot estimation, but do not apply them to the photometry presented in the final catalog (see Section 7.6).

As described above, we have three catalogs derived from detections in the K_s-, zc-, and R-band images, respectively. When merging the K_s-, zc-, and R-band catalogs, we use a matching radius of 10 because the positions of a same source in different detection images could sometimes have shifts exceeding 05 even after astrometry correction, due to the fact that such sources (a total of ≈5000, i.e., ≲ 4% of the final catalog; see the latter part of this section) are typically very faint and/or blended to some degree in certain bands and thus have inaccurate positions therein. About 60% of the sources are detected in more than one band; their z_phot values derived from different catalogs are generally very similar even though we use different approaches to produce the three catalogs. However, in some cases these sources do have apparently different z_phot values from different catalogs because the z_phot qualities for given same source present in all three catalogs might be different: (1) if a source appears faint in one detection image, then its position and aperture correction factor (see Section 5.1) are likely to be determined relatively poorly in that image, which might further lead to large uncertainties in photometry; and (2) different catalogs are in fact obtained from differently processed image sets (see Section 5.1), and technically speaking, noise levels and blending conditions differ among those image sets. Therefore, proper choices among the three catalogs should enhance the overall z_phot quality.

Here we adopt the following criterion for source selection. First we calculate the redshift quality parameter Q_z (see Equation (8) of B08) for each source in multiple catalogs as

$$\begin{equation} Q_z=\frac{\chi ^2}{N_{\rm filt}-3} \frac{{z}_{\rm up}^{99}-{z}_{\rm lo}^{99}}{P_{\delta z =0.2}}, \end{equation} \tag{ 8 }$$

where χ² is obtained from template fitting; N_filt − 3 is the degrees of freedom; ${z}_{\rm up}^{99}-{z}_{\rm lo}^{99}$ is the 99% confidence level interval that represents the z_phot scatter (Mobasher et al. 2007); and P_δz is the fraction of the total integrated probability that lies within ±(1 + zp)δz of the z_phot estimate, designed to identify sources that have broad and/or multi-modal probability distributions (Benítez 2000). Then if a source appears in more than one catalog (i.e., multiple detections), we regard the detection with the lowest Q_z value as the most reliable one and adopt its corresponding photometry and z_phot. The relation between Q_z and |Δz|/(1 + z_spec) (where Δz = z_phot − z_spec) is plotted in the top left panel of Figure 10. In Figure 10, we also plot Q_z histograms for spectroscopic and all sources, respectively, in the top right panel (as expected, the spectroscopic sources typically have smaller Q_z values indicating better z_phot quality than average), and present two typical SED fitting results that correspond to two Q_z values in the two bottom panels. The first SED is fitted well by the template with Q_z = 0.09, and its z_phot probability distribution shows a single peak. In contrast, the second SED is fitted relatively poorly by the template with Q_z = 3.31, and its z_phot probability distribution shows double peaks.

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Following the above source-selection criterion, we obtain a total of 131,678 distinct sources, among which 46,447, 25,478, and 59,753 sources are selected from the K_s, zc, and R catalogs, respectively. Typically, a value of Q_z ≲ 1 indicates reliable z_phot quality (i.e., |Δz|/(1 + z_spec) ≲ 0.05). There are 67,415 sources with Q_z < 1 in our merged catalog, among which 15,322, 16,224, and 35,869 sources are detected in the K_s, zc, and R bands, respectively. Figure 11 shows histograms of magnitudes in the detection bands for all sources and those with Q_z < 1, respectively. Typically, sources with lower Q_z values tend to be brighter, e.g., the median magnitudes are K_s = 22.6, zc = 23.8, and R = 24.4 mag for sources with Q_z < 1, and K_s = 23.5, zc = 24.1, and R  =  24.8 mag for all sources, respectively.

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We find 2845 matches between our sources and the z_spec catalogs for non-X-ray objects (see Table 2; also see Section 5.6 for our matching approach). Then we use the z_spec data to evaluate our z_phot quality (for evaluation of X-ray objects, see Section 7.4). We obtain σ_NMAD = 0.029 for those 2845 sources. The median value of Δz/(1 + z_spec) is −0.013 and there are 156 (156/2845 = 5.5%) outliers. More specifically, we find σ_NMAD = 0.024 with 2.7% outliers for sources brighter than R = 23 mag, σ_NMAD = 0.035 with 7.4% outliers for sources fainter than R = 23 mag, σ_NMAD = 0.026 with 3.9% outliers for sources having z < 1, and σ_NMAD = 0.034 with 9.0% outliers for sources having z > 1. Figure 12 demonstrates our z_phot quality. The top three panels are, from left to right, z_phot versus z_spec, histogram of Δz/(1 + z_spec), and Δz/(1 + z_spec) versus R-band magnitude for the 2845 sources, respectively. The bottom three panels are the same as the top panels but are limited to the sources with Q_z < 1. From comparison between the top and bottom panels, we find the criterion of Q_z < 1 filters out many outliers. The z_phot quality is also related to the number of available bands, e.g., |Δz|/(1 + z_spec) ∼ 0.041 when 12 bands (upper limits not counted) are available while |Δz|/(1 + z_spec) ∼ 0.027 when all 15 bands are available.

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We compare our work with Rafferty et al. (2011), which is the most relevant work where they estimated z_phot for sources in the central H-HDF-N area based on broadband photometry from C04 and some other broadband photometry, using the ZEBRA template-fitting code (Feldmann et al. 2006). The major differences of our work from Rafferty et al. (2011) are (1) we include additional high-quality data that have become available only recently (e.g., the K_s band from W10 as well as the IRAC data from the Spitzer Heritage Archive and Ashby et al. 2013); (2) we derive uniform PSF-matched photometry rather than compiling a number of magnitudes of various origins and different derivation methodologies; (3) our catalog consists of 131,678 sources, many more than the 48,858 sources in Rafferty et al. (2011), due to our larger solid-angle coverage (i.e., the entire H-HDF-N field), deeper data, different catalog-construction approach, and inclusion of lower-significance sources; and (4) our z_phot estimation does not involve apparent training procedures. Rafferty et al. (2011) reached σ_NMAD = 0.025 and an outlier fraction of 5.0% (outliers defined as having |Δz|/(1 + z_spec) > 0.20 therein) with training procedures applied. However, their blind-test results showed that their real z_phot quality should be worse by a factor of a few, indicating that our z_phot quality has an overall improvement over their work.

Our approach of PSF-matched photometry extraction is designed to obtain accurate colors rather than absolute fluxes, and it underestimates the fluxes because the aperture diameter is fixed at 1.5 times the PSF size (∼70% of light encircled for point-like sources) for all sources in each band. To convert the aperture photometry to the absolute one, we make use of FLUX_AUTO in SExtractor's output of the detection band, i.e., the K_s, zc, or R band. The algorithm of FLUX_AUTO adopts a flexible aperture size for each source (Kron 1980), and FLUX_AUTO has been widely used to obtain absolute photometry (e.g., Cardamone et al. 2010; W10). If a source has detection in the X (representing K_s, zc, or R) band, then we obtain absolute flux of Y (representing any of the 17 bands), f_{Y, final}, as

$$\begin{equation} f_{\rm {Y}, final}=c\times f_{\rm {Y,aper}}\times \frac{f_{\rm {X,AUTO}}}{f_{\rm {X,aper}}}, \end{equation} \tag{ 9 }$$

where c is the correction factor to convert FLUX_AUTO to absolute flux (c = 1.06 in our case, according to Page 39 of the SExtractor manual for version 2.13). The above procedure conserves colors, which indicates that the same z_phot results would be obtained with either the relative or absolute photometry.

We compare our K_s-band absolute photometry (without applying zero-point offsets) with that of W10 for the common sources that have ⩾5σ significance, and find a median offset of ∼0.10 mag and a scatter of 0.15 mag. No straightforward comparison can be made between our absolute photometry and that of C04, because C04 adopted a different approach to obtain absolute photometry, i.e., using isophotal fluxes of SExtractor rather than auto fluxes that we adopted.

In the astrometry correction procedure (see Section 3), we align all images to the astrometric frame of the C04 images that were well aligned with each other thus being optimal for obtaining accurate colors. However, this astrometry might have subtle systematic errors. To account for this, we match our sources with those detected by the VLA 1.4 GHz observations (Morrison et al. 2010) using a 10 matching radius. Then we use geomap and geoxytran (IRAF tasks) to correct our astrometry so that it is more consistent with that of the VLA data. As in Section 3, we also adopt a 4th-order polynomial correction that is applied to the entire catalog. The median values of the coordinate offsets applied are 028 (R.A.) and −017 (decl.).

We classify a source by fitting its photometry with the set of 235 stellar templates introduced in Section 5.6 at z = 0 and another set of 259 galaxy templates at z = z_phot. The galaxy templates are the PÉGASE2.0 templates (259 in total; Grazian et al. 2006) taken from the EAzY package. For the purpose of star/galaxy separation, the fitting is done via the single template mode (STM) rather than the linear combination mode (LCM) of EAzY, because the former yields slightly more consistent results with the BzK method (Daddi et al. 2004) that separates effectively stars from galaxies. In this mode (STM), we use a 5% error in place of the template error function, since the template error function is designed for the default template set in the LCM. We discard all IRAC data for the purpose of star/galaxy separation because the stellar templates of Pickles (1998) do not cover wavelengths longer than 2.5 μm. We classify a non-spectroscopic source as a star only if it satisfies the following two criteria: (1) $\chi _{\rm star}^{2}<\chi _{\rm gal}^{2}$; and (2) if the source has S/N >5, we then require additionally that its major axis to minor axis ratio (from SExtractor) be in the range of 1.0–1.5; where $\chi _{\rm star}^{2}$ is the fitted χ² using the 235 stellar templates at z = 0 and $\chi _{\rm gal}^{2}$ is the fitted χ² using the 259 galaxy templates at z = z_phot.

Finally, our method identifies a total of 4959 star candidates (with the 229 spectroscopic stars being counted). Among them, 25 are best fitted by a white dwarf template. To verify the accuracy of our method, we also make a BzK diagram. In Figure 13, we only plot sources that have reasonably accurate photometry in the relevant three bands (i.e., $\rm err\_mag_{B}+err\_mag_{zc}< 0.5$ and $\rm err\_mag_{zc}+err\_mag_{K_s}< 0.15$). We find that our template fitting method is consistent with the BzK star/galaxy classification scheme. As expected, if the sources with larger photometric errors are also plotted in Figure 13, then the star/galaxy separation is not as good, but it is still reasonable. For spectroscopic sources, we misclassified 13 galaxies as stars (out of a total of 3126; 13/3126 = 0.4%) and 24 stars as galaxies (out of a total of 229; 24/229 = 10.5%) according to the above criteria. Of those 24 misclassified stars, about one half are very bright thus suffering from issues of bad pixels and/or saturation, and the other half or so are blended with nearby sources to some degree.

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We match our sources (using the optical and near-infrared positions as well as the K_s-band magnitude distribution) with the 2 Ms CDF-N (A03; see Figure 3 for coverage) main X-ray source catalog (using X-ray positions) utilizing the likelihood-ratio matching technique presented in Luo et al. (2010). There are 462 non-stellar X-ray sources matched (13 additional X-ray stars matched), with 281 having z_spec (see Table 2). The false matching rate is estimated to be ≈2%. The z_phot quality of those X-ray sources is as follows: σ_NMAD = 0.037, an outlier fraction of 16.7% (i.e., 47 outliers), and median(Δz/(1 + z_spec)) = −0.014; such z_phot quality is worse than that of non-X-ray sources (see Section 6.4).

There are two main reasons for the worse z_phot quality of X-ray sources. First, the vast majority (≳75%; based on studies of deep X-ray surveys; e.g., A03; Luo et al. 2008; Xue et al. 2011) of these X-ray sources are active galactic nuclei (AGNs). Typically, many AGNs have different SEDs to normal galaxies and therefore it may not be correct to estimate z_phot for AGNs with normal galaxy templates, especially in the case of QSOs whose SEDs are dominated by the central engine. Second, fluxes of AGNs might vary non-periodically on timescales from minutes to decades (e.g., Salvato et al. 2009). Salvato et al. (2009) took AGN variability into account when deriving z_phot based on multi-epoch observations (i.e., one filter has observations in different epochs). However, all our bands were not observed in the same epoch (see Table 1) and we do not have multi-epoch data for a specific band. Therefore our photometry might differ from a snapshot SED, thus being likely to be subject to uncertainties due to AGN variability.

To improve the z_phot quality of X-ray sources, we add three additional QSO templates to the previous template set when estimating z_phot for X-ray sources. The first two are the BQSO and TQSO templates from the SWIRE library (Polletta et al. 2007), both of which are type 1 QSO but with different IR/optical flux ratios; the third one is the optical-to-near-infrared composite QSO template from Glikman et al. (2006), and is built from spectra of QSOs at different redshifts. As in Section 7.3, we use a 5% error instead of the template error function, because the template error function is designed for normal galaxies rather than AGNs (B08). Other EAzY parameters are the same as those in Section 6.1.2. Notably, the LCM algorithm naturally mixes the QSO and stellar light. The resulting z_phot quality is improved appreciably: σ_NMAD = 0.035, an outlier fraction of 12.5% (i.e., 35 outliers), and median(Δz/(1 + z_spec)) = −0.014. Figure 14 demonstrates this improvement in z_phot quality by showing plots of z_phot quality before and after the introduction of the above three QSO templates. Overall, our z_phot quality of X-ray sources appears comparable to those presented in other similar works, e.g., Rafferty et al. (2011).

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It can also be seen in Figure 14 that a small fraction of X-ray sources that were fit well (and thus have good z_phot) without adding the QSO templates are no longer fit well when adding the QSO templates, and vice versa. This is due to degeneracy. To examine the effect of degeneracy introduced by adding these three QSO templates, we compare z_phot results obtained for the 462 X-ray sources, using only the eight galaxy templates shown in Figure 9 and using both the eight galaxy templates and the three QSO templates, respectively. We find a nominal σ_NMAD = 0.015 and a nominal outlier fraction of 6.5%, indicating that the effect of this degeneracy is insignificant.

The users of our catalog should be careful given that we do not apply a significance cut to the sources that are included in our catalog, which might lead to some false detections. Instead, we choose to provide both the significance level (i.e., S/N) and the redshift quality parameter Q_z in our final catalog (see Section 7.6), and recommend the users to apply an appropriate S/N and/or Q_z cut according to their specific scientific interests. We note that, as expected, there is a general anti-correlated trend between S/N and Q_z in spite of significant scatter.

Additionally, the users of our catalog should exercise extra caution in making use of the sources lying in the extended H-HDF-N field (i.e., the regions outside the yellow rectangle in Figure 3), given that, for the sake of completeness, we provide all the sources in the entire H-HDF-N field in our catalog, rather than only those lying within the C04 central H-HDF-N field. The sources in the extended field might potentially suffer from issues such as additional astrometry distortions and large background noise (see Figure 3). The latter could cause large photometric errors and additional false detections (C04). Unfortunately, it is rather difficult to assess z_phot quality in the extended field where there are not any z_spec data. We thus provide a flag indicating whether a source is located in the central or extended H-HDF-N field in our final catalog (see Section 7.6) so that the users could proceed with known caveats.

The H-HDF-N photometric catalog is presented in Table 6. The details of the 53 columns are given below.

• 1.  
  Column 1 gives the source sequence number (i.e., ID, ranging from 1 to 131,678). We list sources in order of increasing right ascension.
• 2.  
  Columns 2 and 3 give the J2000.0 right ascension and declination, respectively. They are consistent with the VLA radio astrometry (see Section 7.2).
• 3.  
  Column 4 gives the z_phot value, which corresponds to the highest peak in the corresponding z_phot probability distribution output by EAzY.
• 4.  
  Column 5 gives the alternative z_phot value denoted as z_alt if available. In the z_phot probability distribution, if a source has other peaks in addition to the highest one (corresponding to z_phot) and satisfies the following two conditions, i.e.,

  $$\begin{eqnarray} &&\frac{|{z_{\rm peak}}- z_{\rm phot}|}{1 + z_{\rm phot}} > 0.15, {\rm and} \end{eqnarray} \tag{ 10 }$$

  $$\begin{eqnarray} &&p({z_{\rm peak}}) > 0.5 \times p(z_{\rm phot}), \end{eqnarray} \tag{ 11 }$$

  where p(z_peak) and p(z_phot) are the values in the probability distribution quoted at redshifts of z_peak and z_phot, respectively, we then define z_alt as the redshift that corresponds to the highest peak among those peaks. Of all sources in our catalog, 34% have z_alt values, while for sources with Q_z < 1 the fraction is 13%. We set z_alt = 0 for stars and z_alt = −1 for sources whose z_alt values are not available.
• 5.  
  Columns 6–11 give the 68% (1σ), 95% (2σ), and 99.7% (3σ) lower and upper limits on z_phot, respectively, which are calculated as

  $$\begin{eqnarray} \alpha /2 &\,=\,& \int _0^{z_\mathrm{low}} \! p(z) \, \mathrm{d}z,\nonumber \\ \alpha /2 &\,=\,& \int _\mathrm{{z}_{up}}^8 \! p(z) \, \mathrm{d}z, \end{eqnarray} \tag{ 12 }$$

  where α = 0.317, 0.046, and 0.003, respectively, and p(z) is the probability distribution of redshift. These limit values are output by EAzY. Note that our z_phot corresponds to the peak value of p(z), thus in some cases z_low might be greater than z_phot, or z_up might be lower than z_phot.²⁰ A total of ∼7% of sources in our catalog have z_low > z_phot or z_up < z_phot. For those sources, the abnormal z_low or z_up value should not be used.
• 6.  
  Column 12 gives the source type: values of −2, −1, 0, and 1 indicate white dwarfs (25 sources), other stars (4934 sources; see Section 7.3), non-X-ray sources (126,257 sources), and X-ray sources (462 sources; see Section 7.4), respectively. If a source is classified as a star, then we do not count it as an X-ray nor non-X-ray source, set its z_phot and corresponding confidence ranges as 0, and set Q_z as −1. We refer whoever interested in X-ray stars (a total of 14 in our catalog) to both this column and Column 18.
• 7.  
  Column 13 gives the redshift quality parameter Q_z (see Section 6.3). Generally, lower Q_z values indicate better z_phot quality. We suggest a criterion of Q_z < 1 for reliable z_phot. There are 67,415 sources with Q_z < 1 in our catalog.
• 8.  
  Column 14 gives the detection significance or S/N, defined as

  $$\begin{equation} \rm S/N = \frac{flux}{background\ noise}, \end{equation} \tag{ 13 }$$

  where flux and background noise (see Section 5.3) are for the detection band (see Section 6.3). The users of our catalog are recommended to apply an appropriate cut based on Q_z and/or S/N to filter out sources of poor photometry and thus z_phot quality, in order to fulfill effectively their specific science goals.
• 9.  
  Column 15 gives the detection band: letters of K, Z, and R indicate K_s-, zc-, and R-band detections, respectively. If a source is detected in multiple bands, the first letter indicates the adopted detection catalog according to the lowest Q_z criterion (see Section 6.3). After applying this criterion, the numbers of sources selected from the K_s, zc, and R catalog are 46447, 25478, and 59753, respectively.
• 10.  
  Columns 16 and 17 give the z_spec value and its reference index, respectively, if available. We only include secure z_spec in this work. The reference indexes are listed in Table 2. Situations where a source has more than one reference are dealt with in Section 2.2. If z_spec for a source is not available, then both values are set to −1. There are a total of 3355 sources having z_spec, including 2845 non-X-ray sources, 281 X-ray sources, and 229 stars.
• 11.  
  Column 18 gives the source index in the A03 2 Ms CDF-N main catalog if it has a match therein (see Section 7.4). For the sources not matched to A03, the value of this column is set to −1. The number of sources matched to A03 is 475, including 462 non-stellar X-ray sources and 13 X-ray stars.
• 12.  
  Column 19 gives a flag indicating whether the source is in the C04 central H-HDF-N region: 0 stands for being outside of the central region (i.e., in the extended region; 53,636 sources) and 1 stands for being in the central region (78,042 sources). The properties of the sources in the central region are of better overall quality than those outside (see Section 5.7).
• 13.  
  Columns 20–53 give the photometry and corresponding errors in magnitudes. This is the corrected photometry derived in Section 5.6 that has further been corrected to the absolute photometry based on FLUX_AUTO algorithm in SExtractor (see Section 7.1). We do not apply the zero-point offsets derived in Section 6.2 in the final catalog. The order is U band, error of U band, B band, error of B band, and so forth (the band order is the same as that in Table 1). If the photometry of a source in one band is not available (due to, e.g., being outside of the field of view or saturated), then we set corresponding columns to values of −99. If the flux is less than its error (both in units of μJy), we apply an upper limit, i.e., set the flux to the value of the error (see Section 5.3). For completeness, the S3- and S4-band photometry, which is not used in z_phot calculation (see Section 6.2), is also presented.