Finding counterparts for All-sky X-ray surveys with Nway: a Bayesian algorithm for cross-matching multiple catalogues

TL;DR

This paper introduces NWAY, a Bayesian algorithm that reliably cross-matches all-sky X-ray sources with multi-wavelength catalogs like AllWISE and Gaia, significantly improving counterpart identification accuracy and efficiency.

Contribution

The paper presents NWAY, a novel Bayesian cross-matching algorithm that incorporates priors and observable characteristics, outperforming previous methods in speed and reliability.

Findings

Achieved ~95% reliability in counterpart associations.

Demonstrated 96-99% agreement with likelihood ratio methods.

Enabled faster, automated identification without visual inspection.

Abstract

We release the AllWISE counterparts and Gaia matches to 106,573 and 17,665 X-ray sources detected in the ROSAT 2RXS and XMMSL2 surveys with |b|>15. These are the brightest X-ray sources in the sky, but their position uncertainties and the sparse multi-wavelength coverage until now rendered the identification of their counterparts a demanding task with uncertain results. New all-sky multi-wavelength surveys of sufficient depth, like AllWISE and Gaia, and a new Bayesian statistics based algorithm, NWAY, allow us, for the first time, to provide reliable counterpart associations. NWAY extends previous distance and sky density based association methods and, using one or more priors (e.g., colors, magnitudes), weights the probability that sources from two or more catalogues are simultaneously associated on the basis of their observable characteristics. Here, counterparts have been determined using a WISE color-magnitude prior. A reference sample of 4524 XMM/Chandra and Swift X-ray sources demonstrates a reliability of ~ 94.7% (2RXS) and 97.4% (XMMSL2). Combining our results with Chandra-COSMOS data, we propose a new separation between stars and AGN in the X-ray/WISE flux-magnitude plane, valid over six orders of magnitude. We also release the NWAY code and its user manual. NWAY was extensively tested with XMM-COSMOS data. Using two different sets of priors, we find an agreement of 96% and 99% with published Likelihood Ratio methods. Our results were achieved faster and without any follow-up visual inspection. With the advent of deep and wide area surveys in X-rays (e.g. SRG/eROSITA, Athena/WFI) and radio (ASKAP/EMU, LOFAR, APERTIF, etc.) NWAY will provide a powerful and reliable counterpart identification tool.

Tables

Table 1. Table 1: XMMSL2 vs 2RXS AllWISE association for sources in common

XMMSL2-2RXS		Sources in	Identical Best
Sep.	Mean Sep.	common	AllWISE ctp.
arcsec	arcsec	N	%
$\le$5	3.2	1145	98.5
$\le$10	6.1	3559	98.5
$\le$30	12.4	8202	95.7
$\le$60	15.9	9330	91.6

Equations

$$X/O=log(f_{\mathrm{X}}/f_{\mathrm{opt}})=log(f_{\mathrm{X}})+C+m_{\mathrm{opt}}/2.5=\pm 1$$

$$[W1]=-1.625\ast log{F}_{(0.5-2keV)}-8.8$$

$$P(H)=N_1/\prod _{i=1}^k{N}_i=1/\prod _{i=2}^k{N}_i=1/\prod _{i=2}^k{\nu }_i{\Omega }_i.$$

$$P(H)=c/\prod _{i=2}^k{\nu }_i{\Omega }_1.$$

$$P(H|D)\propto P(H)\times P(D|H).$$

$$P(D|H)=2^{k-1}\frac{\prod {\sigma }_i^{-2}}{\sum {\sigma }_i^{-2}}\exp \left\{-\frac{{\sum }_{i<j}{\varphi }_{ij}{\sigma }_j^{-2}{\sigma }_i^{-2}}{2\sum {\sigma }_i^{-2}}\right\}$$

$$\frac{P(H_1|D)}{P(H_0|D)}$$

$$B$$

$$P(H_1|D)$$

$$\text{p\_any}=1-P(H_0|D)/\sum _{i}P(H_i|D)$$

$$\text{p\_i}=P(H_i|D)/\sum _{i>0}P(H_i|D)$$

$$LR=\frac{q(m)}{n(m)}\times f(r)$$

$$P(D|H)$$

$$=$$

$$P(D_m|H)=\prod \frac{{\int }_m\bar{q}(m)p(m|D_m)dm}{{\int }_m\bar{n}(m)p(m|D_m)dm}$$

Full text

††pubyear: 2017††pagerange: Finding counterparts for all-sky X-ray surveys with Nway: a Bayesian algorithm for cross-matching multiple catalogues–B.6.1

1 Introduction

Active Galactic Nuclei (AGN) play an important role in the evolution of galaxies in the Universe. It is now established that most massive galaxies host a supermassive black hole in their centre, and that the black hole accretion activity and history might have a profound influence on their growth. A comprehensive picture of this link can only be obtained from a complete census of AGN, covering the full luminosity function at any redshift. This is possible solely by merging AGN samples selected at different wavelengths and through complementary criteria (Padovani et al., 2017), and by combining shallow wide-area with deep pencil beam surveys. The broad wavelength coverage is required to identify AGN at all redshifts at the wavelengths where they dominate the Spectral Energy Distribution (SED) of their host galaxy (e.g., Gamma-ray: Armstrong et al. 2015; X-ray: Georgakakis & Nandra 2011; optical: Bovy et al. 2011; Palanque-Delabrouille et al. 2016; Mid-Infrared: Assef et al. 2013; Radio: De Breuck et al. 2002). Pencil beam surveys (e.g., Luo et al., 2017) allow the study of the high-redshift population and the faint end of the luminosity distribution, while shallower wide-area surveys (e.g., Georgakakis et al., 2017; LaMassa et al., 2016) trace the brightest sources and at the same time provide access to rare objects.

The selection of AGN at X-ray energies provides an excellent compromise between completeness and purity of the sample. X-rays are sensitive to all but the most obscured AGN even when hosted in luminous galaxies and have very low contamination from other source populations. Limited by the available datasets, and by the small field of view of the most sensitive imaging telescopes, X-ray selected AGN samples were until now predominantly obtained from deep pencil beam surveys (e.g., COSMOS: Hasinger et al. 2007; Brusa et al. 2010; Civano et al. 2012; Marchesi et al. 2016; CDFS: Luo et al. 2010; Hsu et al. 2014; Luo et al. 2017; AEGIS-X: Nandra et al. 2015; Lockman Hole: Fotopoulou et al. 2012) or limited to the brightest and most extreme sources (e.g., BAT: Baumgartner et al., 2013). Only very recently Stripe82X (LaMassa et al., 2016; Tasnim Ananna et al., 2017) and XMM-XXL (e.g., Pierre et al., 2017; Fotopoulou et al., 2016; Georgakakis et al., 2017) opened access to two shallow, wide areas of $\approx$30 deg2 and $\approx$50 deg2. Still, the total population of X-ray selected and spectroscopically confirmed AGN counts only $\approx 20,000$ objects and continues to be dwarfed by the $\approx 300,000$ optically selected quasars (e.g., DR12Q: Pâris et al., 2017). The new revisions of the ROSAT All-sky Survey (2RXS; Boller et al., 2016) and the second release of the XMM-Newton Slew Survey (XMMSL2111https://www.cosmos.esa.int/web/xmm-newton/xmmsl2-ug) with a total of $\approx 130,000$ X-ray sources may finally provide AGN counts comparable to those found in the SDSS.

So far, the most challenging aspect of the exploitation of these samples was the identification of the multi-wavelength counterparts needed for the source characterization and redshift estimates. This was related to at least two shortcomings. First, the positional uncertainties of all but the brightest sources in these X-ray catalogues are in general too large to assign a single, unambiguous optical counterpart based solely on a simple coordinate match. Second, the multi-wavelength catalogues used for identifying the counterparts lacked depth and homogeneous, contiguous coverage. At least the latter problem can now be addressed with the publicly available AllWISE survey (i.e. the combination of WISE and NEOWISE: Wright et al., 2010; Mainzer et al., 2011, 2014). This survey maps the entire sky at mid-infrared wavelengths from 3.4 to 22 $\mu$m to a depth at which the majority of the point-source populations of 2RXS and XMMSL2 (AGN, stars, star-forming galaxies) is expected to be detected222Note, that the detection of an AGN in the mid-infrared requires the availability of reprocessing dust, i.e. dust free AGN will be missed. Many Compton-Thick AGN will be missed as well. (see § 4.1).

Even with a suitable catalogue at hand, the large X-ray positional uncertainties still require us to recognize the correct counterpart among the many that are possible, avoiding the bias toward the optically brightest sources (e.g., Naylor et al., 2013). The most frequently used technique is based on the Likelihood Ratio (LR) method (Sutherland & Saunders, 1992). Using a primary catalogue (here X-rays) and a secondary catalogue (here mid-infrared) the ratio of the likelihoods of each IR source being the true counterpart to a X-ray or background source is calculated taking into account the coordinates (i.e. their separations), the associated uncertainties, the density of the sources in the two catalogues, and the source magnitudes and distribution. For X-ray sources with large positional uncertainties this limited set of information is often insufficient to reliably identify the counterpart.

For this reason we developed a new code, Nway , that goes beyond the LR approach by simultaneously considering in addition to astrometric information (i.e. position, associated uncertainties and sky density of sources as a function of magnitude), various known source properties (e.g. magnitudes, colours) using Bayesian statistics for each step.

The paper focusses on two main topics: firstly, we increase the sample of bright X-ray selected AGN by identifying and releasing the coordinates of the AllWISE counterparts to the point-like X-ray sources in 2RXS and XMMSL2 all-sky surveys. This will facilitate spectroscopic follow-up and further source characterization (e.g., Dwelly et al., 2017). Secondly, we present the Nway code and release it to the public, together with a detailed user manual. In order to keep the two aspects separated, in the main body of the paper we only provide a short description of Nway (§ 3). Instead we focus on the X-ray catalogues (§ 2), the construction of the prior based on AllWISE photometry (§ 4), the assessment of the reliability of our associations by comparison with the literature (§ 5), and the AllWISE properties of the counterparts (§ 6), in comparison with the results from X-ray pencil-beam surveys. The release of the catalogues is presented in § 7. The detailed description of the Nway algorithm and the verification results are made available in the Appendixes A and B. Test performances of Nway are presented in Appendix C, where we also show the strength of the method and the improvement of simultaneously using two priors instead of one.

Along the paper we assume Vega magnitudes unless differently stated. In order to allow direct comparison with existing works from the literature of X-ray surveys, we adopt a flat $\Lambda$CDM cosmology with $h=H_{0}/[100\mathrm{kms}^{-1}\mathrm{Mpc}^{-1}]=0.7$; $\Omega_{M}$=0.3; $\Omega_{M}$=0.7.

2 The Datasets

In the following we describe the properties of the 2RXS, XMMSL2, and AllWISE catalogues and their preparation for this work.

2.1 ROSAT All-Sky Survey

The first all-sky imaging X-ray survey in the 0.1-2.4 keV band was performed by ROSAT (Truemper, 1982) between 1990 and 1991. Besides a catalogue of extended sources, two catalogues of point-like sources were published: the Bright Source Catalogue (BSC) containing the 18,816 brightest sources (Voges et al., 1999) and the Faint Source Catalogue (FSC) encompassing the 105,924 fainter objects down to a detection likelihood limit of 6.5 (Voges et al., 2000). In view of the launch of SRG/eROSITA (Merloni et al., 2012) and taking advantage of the advancement in technology, data reduction, analysis, and detection algorithms of the last 25 years, the original data have recently been reprocessed by Boller et al. (2016). The newly generated catalogue (2RXS) for point-like X-ray sources has been released to the community333http://www.mpe.mpg.de/ROSATcomparedwith/2RXS and includes $\approx$13,5000 sources.

When comparing with the 1RXS catalog, which combines BSC and FSC, the number of reliable sources in the 2RXS has increased (both bright and faint) while the number of spurious detections has decreased (see Boller et al., 2016, for more details). We select all 2RXS detections which lie within the ‘extragalactic’ part of the sky, i.e. with $|b|>15^{\circ}$ and at least 6 and 3 deg away from the optical centres of the Large and Small Magellanic Clouds, respectively. After this geometric filter, we are left with 106695 2RXS X-ray detections with an estimated coverage of 30,575.9 deg2. Observed in projection outside the crowded Galactic Plane, these sources are predominantly extragalactic. The catalogue is further cleaned by removing 122 sources without estimated positional uncertainty and without listed counts. The well known correlation between X-ray flux444We computed Galactic foreground absorption corrected fluxes following the procedure presented in Appendix A of Dwelly et al. (2017) intensity, positional uncertainty and detection likelihood is shown for the final 106,573 sources in the primary catalogue in the left panel of Fig. 1, with the flux distribution (converted into the 0.5-2 keV band) shown in Fig. 2. 95 % of the sources have a 1$\sigma$ positional error smaller than 29*′′* compared with the 34*′′* found in the extragalactic area of 1RXS.

2.2 XMM-Newton Slew 2 survey

The XMM-Newton European Photon Imaging Camera pn (EPIC-pn) accumulates data during slews between pointed observations. The most recent catalogue derived from this dataset covers 84 % of the sky (release 2.0, 14th March 2017). We extracted all detections from the ‘Clean’ version of the catalogue (which we will henceforth refer to as the XMMSL2 catalogue), which lie in the same area as defined for 2RXS. After this geometric filter, we are left with 22,306 X-ray detections with at least 0.1 s of effective XMMSL2 exposure with an estimated coverage of $\approx$25,500 deg2.

The final catalogue was filtered to remove candidate duplicate detections of identical X-ray sources using the original column UNIQUE_SRCNAME, retaining a total of 17,672 sources with 2,704 sources detected only at 0.2-12 keV, 553 detected only at 0.2-2 keV , and 168 sources detected only at 2-12 keV.

$52.8$ % ( 9,333) of the XMMSL2 sources have at least one 2RXS source within a radius of 60*′′, with 236/21/3/1/1 XMMSL2 sources being associated with 2/3/4/5/6 2RXS sources, respectively. The distribution of the positional uncertainties as a function of the flux in the detection band, colour coded by the likelihood of the detection, is presented in the right panel of Fig. 1. Note, that figure shows the original positional uncertainty augmented by 5′′* in quadrature to account for the systematic uncertainty on attitude reconstruction. The flux distribution (converted into the 0.5-2 keV band) shown in Fig. 2.

2.3 AllWISE catalog

The Wide-field Infrared Survey Explorer (WISE555see also http://irsa.ipac.caltech.edu/Missions/wise.html for a summary and details and on the reactivated mission; Wright et al., 2010), was launched in 2009 and over the course of one year scanned the entire sky at least twice in the 3.4 and 4.6 $\mu$m bands (hereafter W1, W2, respectively) and at least once in the 12 and 22 $\mu$m bands (W3, W4). In the AllWISE data release666Available at http://wise2.ipac.caltech.edu/docs/release/allwise/ (November 13, 2013 Cutri et al., 2013) all the available data are combined, reaching a 5$\sigma$ limiting W1, W2, W3, and W4 magnitudes of better than 17.6, 16.1, 11.5, and 7.9 (all in the Vega system) over 95 % of the extragalactic sky ($|b|>$15∘). The coverage is inhomogeneous, being deepest at the Ecliptic Poles.

We generated two independent catalogues that include all AllWISE sources located within a radius of 120*′′* from an X-ray position listed in the 2RXS and XMMSL2 catalogues, respectively. From each catalogues duplicated sources were removed. No additional filtering was performed. This procedure results in two independent catalogues of 6,252,516 unique entries for 2RXS and 1,288,533 for XMMSL2, covering total unique areas of 368.81 deg2 and 60.79 deg2, respectively.

3 NWAY in a nutshell

Nway has been developed for identifying the multi-wavelength counterparts to X-ray sources to multiple catalogues in a multi-dimensional parameter space (e.g., position and positional uncertainty, density of sources, magnitudes, colours, variability, morphology, etc.) in a Bayesian framework. The code builds on the original work of Budavári & Szalay (2008) who developed the algorithm for simultaneously matching multiple catalogues and enhances it by allowing sources to be present only in a subset of the catalogues (e.g., Pineau et al., 2017). Additionally, Nway can either generate an internal prior for each source parameter following the implementation of the Maximum Likelihood Ratio as presented in e.g., Brusa et al. (2007), or use an external, pre-constructed prior.

Nway has already been successfully applied in a number of studies, e.g., the identification of counterparts to ROSAT (1RXS; Voges et al., 1999, 2000) sources in the pilot SDSS-III/SEQUELS program (Dwelly et al., 2017) using two optical bands, simultaneously; the search for counterparts to Chandra and XMM detections in the Extended Chandra Deep Field South (Hsu et al., 2014) using three independent catalogues (optical, near-infrared and 3.6 $\mu$m) simultaneously and with internally constructed priors (see for all the options the Nway manual at https://github.com/JohannesBuchner/nway/raw/master/doc/nway-manual.pdf). It has also been applied to 1RXS and earlier XMM-Slew Survey (release 1.6, 26th Feb 2014) data on the BOSS imaging footprint (Dwelly et al., 2017), adopting an external, mid-infrared based colour-magnitude prior, similar to the one chosen in this work.

A comprehensive description of Nway is given in Appendix B together with a verification using internally generated priors for XMM-COSMOS (see Appendix C). In the following we focus on the application of the code to the scientific aim of the paper, the AllWISE counterparts to 2RXS and XMMSL2.

The Nway code answers the question: Considering the astrometric information (i.e. distance from the X-ray source, positional uncertainties, and number densities) and priors (e.g. magnitude and colour distribution), what is the posterior probability for each AllWISE source within a given radius from a 2RXS or XMMSL2 detection to be the correct counterpart to the X-ray source?. For the analytical details the reader is referred to Appendix B.5. In short, Nway first computes for each source in the AllWISE catalogue the Bayes factor considering only distance from the X-ray source, positional uncertainties, and number densities. Next, the Bayes factor is weighted by the mid-infrared magnitude-colour information (see § 4). Then, each AllWISE source is associated with the probability p_i of being the right counterpart to a specific X-ray detection. In addition, for each X-ray detection, Nway provides the probability, p_any, that any of the AllWISE sources is the right counterpart. The higher the value of p_any the lower is the probability of a chance association. In the output catalogue of Nway , for a given X-ray source, all the AllWISE within the search area are listed, ranked in decreasing order by their p_i. For comfort Nway flags the first AllWISE source of each group as match_flag=1, this being the best counterpart among those available. A match_flag=2 indicates the AllWISE sources with a ${{\texttt{p}}\_i}/{{\texttt{p}}\_i}_{best}<\alpha$ from the first, $\alpha$ being defined by the user (in this paper it is fixed to 0.5); these are considered secondary possible counterparts. Everything else is flagged as match_flag=0.

4 Application of NWAY to 2RXS and XMMSL2

In this section we motivate the AllWISE colour-magnitude prior, subsequently present the results of the application of Nway to the 2RXS and XMMSL2 catalogue defined in § 2, and finish with the comparison of the associations for sources that are in common to both X-ray catalogues.

4.1 AllWISE colour-magnitude prior

The prior is defined as the probability, given observable information alone, i.e. before considering any positional information, that a counterpart is related to an X-ray source. Given that the X-ray point-source population is an ensemble made of stars, nearby galaxies, and galaxies at unknown redshift hosting an AGN of unknown power, a prior based on a single magnitude distribution is insufficient. This is especially true for X-ray detections with large positional uncertainties. Ideally, the prior would use the entire SED as discriminator (e.g. Roseboom et al., 2009). In practice, the lack of sufficiently deep multi-wavelength coverage of the entire sky requires a compromise.

The AllWISE catalogue provides photometric coverage of the entire sky in the mid-infrared, a regime where the number density of sources is low compared with e.g., the optical bands. At the same time, virtually all point-like X-ray sources found in 2RXS and XMMSL2 are expected to be detected at the depth of the AllWISE survey, as we show later in this section.

To generate the prior we need to start with an X-ray sample that matches the sources expected at the depths of 2RXS and XMMSL2 but with secure counterpart association. Beyond a comparable flux limit this sample also needs to cover a sufficiently large area to include rare and bright objects. Both characteristics are fulfilled by the 3XMM-DR5 (Rosen et al., 2016) with a sky coverage of 877 deg2 and with a flux limit significantly deeper than 2RXS and XMMSL2. Following the same screening procedure outlined in Dwelly et al. (2017), but on the entire footprint of the survey we retained 2,349 sources distributed as in Fig. 2. All the sources selected in this way have a unique AllWISE counterpart within 5*′′, 98% within 3′′*. Given the PSF of WISE, this provides a high confidence that the counterpart association is reliable.

The colour-magnitude distribution of the AllWISE counterparts to the 3XMM-DR5 sources is shown in Fig. 3 together with the properties of the AllWISE field population. The 3XMM-DR5 counterparts are well separated from the bulk of the AllWISE population, suggesting this colour-magnitude distribution to be an efficient prior. As in Dwelly et al. (2017), we generated a grid on the [W2],[W1-W2] plane with steps of 0.25 mag in [W2] and 0.1 mag in [W1-W2] (see Figure 4) and for each bin computed the ratio of the densities of 3XMM-DR5 counterparts and field sources. This two-dimensional distribution of density ratios encodes the prior which we apply to the Bayes factor. The Bayes factor was computed taking into account astrometry (i.e. separation between the sources and respective positional uncertainty) and number density of the sources.

4.2 2RXS and AllWISE association

We applied Nway and the prior discussed in § 4.1 to $\approx 6$  Million AllWISE (see § 2.3) sources within 2′ from the 106,573 2RXS sources (see §  2.1). At least one AllWISE candidate counterpart is found for all but 93 (0.01%) of the 2RXS sources. A histogram of the distribution of p_any is shown as the yellow solid line in the top panel of Fig. 5.

The 93 sources without any AllWISE counterpart (green points in Fig. 6) include a) extended X-ray sources (e.g. 2RXS_J152238.4+083422, a spectroscopically confirmed cluster at z$\approx$0.035); b) X-ray sources with candidate counterparts present in the AllWISE images but not contained in the AllWISE catalog; and c) X-ray sources with very bright optical candidate counterparts not present in the AllWISE catalog.

63,305 2RXS sources ($\approx$59% of the sample) have a p_any$>0.5$ while for 35,571 sources ($\approx$33% of the sample) p_any is lower than 0.3. Interestingly, 60% of the latter are fainter than 14.5 mag in W2. In this region the magnitude distribution of the prior overlaps with the bulk of the field population, indicating that the limit of the disentangling power of the prior has been reached and that the selected AllWISE counterpart could be the result of a chance association. This is partly due to astrometric scatter.

Next we investigate the typical p_any for unreliable associations. We run Nway in the same configuration after a) shifting the coordinates of the 2RXS catalogue by 6′ in Declination, b) extracting the AllWISE sources within 2′ from the new 2RXS positions and c) removing the 2,059 sources (2% of the sample) that after the shift entered in the 2′ radius circles from actual 2RXS sources. As expected, the distribution of p_any (Figure 5 top; yellow long-dashed line) peaks toward low values of p_any, with 78 % of the sample having p_any$<\sim$0.15. This coincides with the idea that in any random sky position very few sources have properties matching the prior. E.g., only 5 % in the randomized 2RXS sample have p_any$>$0.5 and p_i$>$0.8. We can easily imagine that some of these sources are counterparts to actual X-ray sources below the 2RXS flux sensitivity. This hypothesis will be validated with future deeper X-ray data, e.g., from SRG/eROSITA.

A very conservative p_any $>$ 0.5 for a reliable association (thus with $<2$ % probability of chance association; see Figure 5) results in a sample of 62,944 AllWISE counterparts to 2RXS sources. However, we release here the entire catalogue of 2RXS counterparts, leaving to the user to decide the acceptable level of completeness and purity for their application. Figure 5 (bottom) shows the fraction of expected interlopers for any given value of p_any.

If we consider only sources with X-ray detection likelihood (EXI_ML; as defined in Boller et al., 2016) larger than 10, the fraction of sources with p_any$>$0.5 increases to 80 % (40,207/50,544). This means that many of the sources with low p_any are among those with low detection likelihood, indicating that they could be just spurious detections. Alternatively, they could be real sources with poorer positional determination. The distribution of p_any for the sources with EXI_ML$>$10 is shown with the dotted line in Fig. 5.

4.2.1 Multiple associations

There are 17,734 (16.6 %) 2RXS sources with more than one candidate AllWISE counterpart77712321/3681/1177/386/121/34/11/2/1 cases with 2/3/4/5/6/7/8/9/10 AllWISE counterparts within the search area, respectively. Not only do most of these sources have a low p_any, but the candidate counterparts are also faint in W2 and on average at larger distance from the X-ray position. This suggests that the X-ray sources themselves could be spurious. In fact, $>75$ % of this subsample have EXI_ML$<10$. Only 7 % of them have p_any$>0.9.$, with the possible counterparts located in areas well populated by the prior. Given the poor angular resolution of ROSAT, it would not be a surprise if the candidate counterparts belong to distinct X-ray sources, detected as one in 2RXS. This is demonstrated in Figure 7 (top), which shows the colour-magnitude distribution of the AllWISE sources for the 47 % (7 %) of the 12,321 2RXS sources with two candidate counterparts having p_any$<$0.1($>$0.9).

4.3 XMMSL2 and AllWISE association

The analysis done in the previous section was repeated for the XMMSL2-AllWISE association, with the summarizing plot being in the middle and bottom panels of Fig. 5. First of all, the smaller X-ray positional error of XMM translates into a distribution of p_any toward higher values (compare the solid and dashed cumulative curves in the right panel of the figure), with about 76% of the sources having p_any$>$0.5 and p_i$>$0.8. Only 21% of the sources have p_any$<$0.3 with only 8 XMMSL2 sources without any AllWISE candidate counterpart within 2′.

As for 2RXS, we systematically offset the positions of the XMMSL2 catalogue and run Nway with the same setting. Now we find that for only 3% of the cases (571/17665), p_any$>$0.5 and p_i$>$0.8. The smaller positional uncertainty also reduces the fraction of sources with more than one possible counterpart. In total 1210 XMMSL2 sources (6.8%) have more than one possible counterpart8881015/163/25/17/1/1 sources having 2/3/4/5/6/7 possible counterparts. As for 2RXS, we analyze the properties of the XMMSL2 sample with two possible counterparts. Of the 1015 sources belonging to this group, 108 (10%) have p_any$>$0.9 and 739 (73%) have p_any$<$0.1. As for 2RXS, also for XMMSL2 the majority of the sources with multiple associations have low p_any, low magnitude distribution for the possible counterpart and, above all, low X-ray detection likelihood EXI_ML_B8$<10$. Like for 2RXS, we will provide all the associations in the catalogue, leaving the user to decide on the threshold for the reliability.

4.4 2RXS vs. XMMSL2 associations

It is interesting to compare the association found for the sources that are in common to the 2RXS and XMMSL2 catalogues as the smaller positional uncertainties of latter can give insight on the reliability of the associations for the former.

Table 1 summarizes our results for the AllWISE counterparts that are in common for X-ray sources with matching coordinates within 5/10/30/60*′′*. Overall, the agreement between the associations is very good, with the highest fraction of identical counterparts found for the subset of X-ray sources with the smaller separation between 2RXS and XMMSL2.

5 Comparison with literature

Since the release of the first ROSAT all-sky catalogues (Voges et al., 1999, 2000) there have been many attempts to determine the multi-wavelength counterparts. Most of the follow-up of ROSAT point-like sources concentrated on the bright sources (Rutledge et al., 2000; Schwope et al., 2000; Mahony et al., 2010), even if, the adopted methodologies (e.g., association technique, secondary catalogues for the follow-up) changed over time. A direct comparison between those earlier works and the results presented here would only allow comparing the associations without assessing their correctness. In addition, the X-ray positions have changed from 1RXS to 2RXS (see Boller et al. 2016) for more details.

It is for this reason that we decided to test our associations against an astrometric reference sample of 4,524 X-ray sources from XMM, Chandra and Swift in the BOSS footprint, that have reliable counterparts (see Dwelly et al., 2017, for details on the sample). A match on the X-ray positions within 60*′′* provides 1,496 unique identifications in 2RXS, while additional 14 2RXS sources have a second possible match. Of these, 1,418 have identical AllWISE counterparts corresponding to an accuracy of 94.8%, with $\approx$94% of the identical associations having p_any$>$0.5.

The exercise repeated for XMMSL2, results in identical AllWISE counterparts for 533 of the 547 sources that have a match within 30*′′* in the reference catalog, corresponding to 97.4% agreement. 514/533 (96.4 %) have p_any$>$0.5. This attests both the appropriateness of the prior and the reliability of Nway .

6 Source Characterization

For the counterparts of the 2RXS and XMMSL2 all-sky surveys no single survey provides both photometry and spectroscopy over the full sky. However, we can make an educated guess of the type of population by a) matching with Gaia999http://archives.esac.esa.int/gaia (Arenou et al., 2017), b) studying the AllWISE colour distribution of the counterparts and comparing them with literature (e.g., Wright et al., 2010; Nikutta et al., 2014), and, finally 3) comparing Infrared and X-ray properties of the counterparts with those well studied in the COSMOS field (XMM-COSMOS, Chandra-COSMOS, Legacy Chandra-COSMOS; Brusa et al., 2010; Civano et al., 2012; Marchesi et al., 2016, respectively).

6.1 2RXS and XMMSL2 counterparts in Gaia

The release of the first Gaia DR1 catalogue enables us to further characterize the AllWISE counterparts of 2RXS and XMMSL2. In particular, it allows the identification of the sources with proper motion, indicating their Galactic nature. For this purpose we used the Hot Stuff for One Year (HSOY) catalogue (Altmann et al., 2017). HSOY includes 583,001,653 objects with precise astrometry based on the cross-match between the Catalogue of Positions and Proper Motions (PPMXL; Roeser et al., 2010) and Gaia DR1 (Arenou et al., 2017). We find a HSOY match within 3*′′* for 91427/132216 (70%) and 14558/19120 (76%) of all the AllWISE candidate counterparts (i.e. match_flag=1 and match_flag=2) to 2RXS and XMMSL2, respectively. Limiting the search only to the best AllWISE counterparts (i.e. match_flag=1), we obtained a match with Gaia for 80078/106573 (75%) and 14008/17665 (80%). Of these, 10472/80078 (13%) and 2054/14008 (15%) have a measured proper motion (above 5$\sigma$) in the HSOY catalog, identifying them as Galactic objects.

6.2 IR/X-ray properties comparison with COSMOS

Originally, Maccacaro et al. (1988) noted that AGN found in the Einstein Observatory Extended Medium Sensitivity Survey (EMSS; Gioia et al., 1987) were characterized by log(f${}_{\rm x}/{\rm f}_{\rm V})=\pm$1, with M stars and galaxies only marginally overlapping in this region. Since then, this locus was adapted by practically all X-ray surveys, extending the relation to other wavelengths (r, i, K, IRAC/[3.6 $\mu$m]) and X-ray energy bands. The validity of the locus was confirmed with recent works (e.g., Brusa et al., 2007, 2010; Civano et al., 2012) pointing out that the near-infrared (e.g., K) or MIR (e.g., 3.6$\mu$m) bands provide a tighter correlation with X-rays than the optical bands. Here, however, the faintest of the X-ray AGN would fall below the locus (e.g., dashed line in Fig. 8) and thus would overlap more with galaxies and stars.

In this paper we extend the earlier studies by combining the Chandra Legacy-COSMOS survey with 2RXS and XMMSL2. The Chandra Legacy-COSMOS survey (Civano et al., 2016; Marchesi et al., 2016) is preferred as it has a homogenous depth and covers sufficient area to host also some bright and rare sources. In addition, the counterparts are secure and well understood, thanks to the depth and amount of the available ancillary data. Furthermore, the spectroscopic follow-up and the reliable photometric redshifts via SED fitting (Marchesi et al., 2016; Salvato et al., 2011) allow the classification of the sources as Type1 (unobscured) and Type2 (obscured) AGN, Galaxies (sources with ${\rm L_{X}<10^{42}erg/s}$), and stars. Fig. 8 (top) compares the properties of the counterparts in COSMOS, 2RXS, and XMMSL2 in the W1 vs. X-ray flux plane. The AllWISE/W1 total magnitude for the Chandra Legacy-COSMOS sources has been derived from the flux in IRAC/[3.6 $\mu$m] within 1.9*′′* aperture as listed in Laigle et al. (2016) using the conversion factor 0.765 and transforming AB to Vega magnitudes as prescribed by the S-COSMOS documentation available through the Infrared Science Archive (IRAS101010http://irsa.ipac.caltech.edu/data/COSMOS/tables/scosmos/scosmos_irac_200706_colDescriptions.html; see also Sanders et al., 2007). The additional correction of W1 - [3.6] = 0.01 mag, following Stern et al. (2012) was applied.

In the same figure, when plotting the 2RXS and XMMSL2 sources, we considered for clarity only those with a detection likelihood larger than 10, p_any$>$0.5 and with an unique AllWISE counterpart. In the figure, the dashed line correspond to the locus defined by (Maccacaro et al., 1988),

[TABLE]

but using the flux at 0.5-2 keV band and the W1 magnitude, instead of 0.3-3.4 keV and mopt=V-band, respectively. The coefficient C takes into account the different effective central wavelength and width of the filters.

The solid line is empirical and defined as

[TABLE]

This new relation much better separates AGN from galaxies and stars over six orders of magnitude and passes through the bimodal distribution of the counterparts to 2RXS and XMMSL2. As for COSMOS, most of the 2RXS and XMMSL2 sources below this relation are stars with a well measured proper motion as described in § 6.1. A complementary way to visualize this natural separation is to plot the distribution of the sources with with respect to the solid line (Fig. 8 bottom). Here, stars are indicated with a solid line, while non-stars are represented with filled histograms. Interestingly, 98.7% of the spectroscopically confirmed X-ray selected AGN presented in Dwelly et al. (2017) lie above the solid line. Similarly, only 0.02% of all the AllWISE counterparts to 2RXS classified as stars via their proper motion, lie above the relation. We suggest to use this new empirical X-ray/MIR relation as a straightforward mean to separate stars and quasars in samples of point-like X-ray sources. In fact, we show in the next section that most of the sources below the relation, despite not having measured proper motions, are also stars based on their AllWISE colours. Inversely, only 0.03% of the AllWISE counterparts to 2RXS and XMMSL2 that are classified as AGN using the WISE colours as defined by Stern et al. (2012); Assef et al. (2013), lie below the solid line.

6.3 IR properties of 2RXS and XMMSL2 counterparts

The AllWISE colours [W1-W2] and [W2-W3]1111110.02% (0.08%), 0.07% (0.3%) and 20% (20%) are only upper limits in W1,W2,W3 in 2RXS (XMMSL2), respectively. of the candidate counterparts can be used their qualitative characterization, as suggested by Wright et al. (e.g. 2010); Nikutta et al. (e.g. 2014). Fig. 9 shows the AllWISE colours of the 2RXS (top) and XMMSL2 (bottom) counterparts, using in background Fig. 12 of Wright et al. (2010). To the well known loci we added the location of the Fermi/Blazars identified by, e.g. D’Abrusco et al. (2013). That is also the location of most of the X-ray sources that are associated with radio emitters (e.g., NVSS: Condon et al., 1998); 4308 sources in 2RXS and 1307 in XMMSL2, respectively. As suggested by Tsai et al. (2013), the sources in this locus are nearby, jetted objects (z$<$0.5), suggesting indeed the presence of an AGN in their cores (Emonts private communication, Emonts et al, in prep). As expected, the bulk of the X-ray population in 2RXS and XMMSL2 is characterized by QSO, AGN, and stars. In particular, the sources below the relation 2 are concentrated in the stellar locus, while the bulk of the sources with W1 above the relation are in the AGN/QSO loci.

7 catalogues release

We release the AllWISE associations to the sources in the 2RXS and XMMSL2 catalogues outside the Galactic plane. The list and the description of columns are provided for each catalogue in the two following sections. In short, we provide few columns that are keys to the identification of the X-ray sources, simply extracted, without any modification, from their original catalogues. We complement each source with the list of possible AllWISE counterparts and the output columns of Nway that are essential for those users interested in defining more pure and complete subsamples. We provide columns that inform the user on whether or not there is more than one possible counterpart. Finally, the data are complemented with a match to the Gaia DR1 catalog. A simple match with the unique identifier from 2RXS, XMMSL2, AllWISE, 2MASS and Gaia will allow the user to retrieve additional columns from the original catalogues, not listed in our release. The catalogues accompany this paper but will be available also via Vizier and at the dedicated web page http://www.mpe.mpg.de/XraySurveys/2RXS_XMMSL2/.

7.1 2RXS-AllWISE catalogue

Column 1. 2RXS_ID: IAU Identifier from Boller et al. (2016).

Columns 2-3. 2RXS_RA, 2RXS_DEC: 2RXS J2000 Right Ascension and Declination, in degrees.

Column 4. 2RXS_e_RADEC: 2RXS positional error, in arc seconds.

Column 5. 2RXS_ExiML: 2RXS source Detection Likelihood. User should refer to the Boller et al. (2016) for discussion on the fraction of false detections as function of this parameter.

Column 6. 2RXS_Ext: 2RXS source extent in units of image pixels.

Column 7. 2RXS_ExtML: Probability of the 2RXS source extend.

Column 8. 2RXS_SRC_FLUX: 2RXS flux in unit of erg cm*-2* s*-1* (see Dwelly et al., 2017, for details).

Column 9. 2RXS_SRC_FLUX_ERR: 2RXS flux error (see Dwelly et al., 2017, for details).

Column 10. ALLW_ID: WISE All-Sky Release Catalogue name (Cutri et al., 2013).

Columns 11-12. **ALLW_RA, ALLW_DEC **: J2000 AllWISE Right Ascension and Declination, in degrees.

Column 13. ALLW_e_RADEC: AllWISE positional error, in arc seconds.

Columns 14-17. ALLW_w[1234]mpro: AllWISE Vega magnitude in the W1, W2, W3, W4 bands.

Columns 18-21. ALLW_w[1234]sigmpro: AllWISE magnitude error in the W1, W2, W3, W4 bands.

Columns 22-25. ALLW_w[1234]snr: AllWISE signal to noise ratio in the W1, W2, W3, W4 bands.

Column 26. ALLW_cc_flags: AllWISE reliability flag from Cutri et al. (2013).

Column 27. Separation_ALLW_2RXS: Separation between members of this association, in arcsec.

Column 28. dist_bayesfactor: Logarithm of ratio between prior and posterior from distance matching.

Column 29. dist_post: Distance posterior probability comparing this association vs. no association, as in (Budavári & Szalay, 2008).

Column 30. bias_ALLW_COLOURMAG_PIX: Probability weighting introduced by AllWISE prior. 1 indicates no change.

Column 31. p_single: Same as dist_post, but weighted by the AllWISE color-magnitude prior.

Column 32. p_any: For each entry in the X-ray catalogue, the probability that any of the associations is the correct one. The lower p_any, the lower is confidence that a reliable counterpart was found. See § 4.2.

Column 33. p_i: Relative probability of the match, if one exists. The p_i add up to unity for each X-ray source.

Column 34. match_flag: 1 for the most probable match, if existing; 2: almost as good solutions p_i/p_i${}_{\rm best}>$ 0.5).

Column 35-36. GroupID, GroupSize: if the 2RXS source has only one possible AllWISE counterpart, the two columns are blank. Otherwise, the GroupSize value indicate the number of possible counterparts while the GroupID value is the same integer for the group. A sort on the GroupID value, will rank the first non-unique match group together, followed by all the rows in the second non-unique group, etc. All the unique matches are listed last.

Column 37. 1RXS_ID: Source name in the 1RXS catalogues (Voges et al., 1999, 2000).

Column 38. ALLW_2MASS_ID: 2MASS Identifier as listed in the AllWISE catalog.

Columns 39-41. ALLW_[jhk]_m_2mass: 2MASS magnitude in the j,h,k bands, as from AllWISE catalogue.

Columns 42-44. ALLW_[jhk]_msig_2mass: 2MASS magnitude errors in the j,h,k bands, as from AllWISE catalogue.

Columns 45. Gaia_DR1_ID: Solution ID from the original Gaia DR1 catalogue (see Fabricius et al., 2016, for more details).

Columns 46-47. Gaia_DR1_RA, Gaia_DR1_DEC: Gaia J2000 Right Ascension and Declination as computed by Vizier.

Columns 48-49. pmra, pmdec: Proper motion in Right Ascension and Declination as measured by Gaia.

Columns 50-51. pmra_error, pmdec_error: Proper motion errors in Right Ascension and Declination as measured by Gaia.

Columns 52. phot_g_mean_flux: Gaia mean flux in units of e-/s.

Columns 53. phot_g_mean_flux_error: Gaia mean flux error in units of e-/s.

Columns 54. phot_g_mean_mag: Gaia mean magnitude.

7.2 XMMSL2-AllWISE catalogue

Column 1. XMMSL2_ID: Unique identifier from Boller et al. (2016).

Columns 2-3. XMMSL2_RA, XMMSL2_DEC: 2RXS J2000 Right Ascension and Declination, in degrees.

Column 4. XMMSL2_e_RADEC: XMMSL2 original positional uncertainty augmented by 5*′′* in quadrature.

Columns 5-7. XMMSL2_DET_ML_B[876]: XMMSL2 source Detection Likelihood in the respective energy bands.

Column 8-10. XMMSL2_Ext_B[876]: XMMSL2 source extent in units of image pixels, in the respective energy bands.

Column 11-13. XMMSL2_Ext_ML_B[876]: Probability of the XMMSL2 source extend in the respective energy bands.

Column 14-16. XMMSL2_FLUXB[876]: XMMSL2 flux in the respective energy bands, in erg cm*-2* s*-1* units.

Column 17-19. 2RXS_FLUX_B[876]_ERR: XMMSL2 flux errors in the respective energy bands, in erg cm*-2* s*-1* units.

Column 20. ALLW_ID: WISE All-Sky Release catalogue name (Cutri et al., 2013).

Columns 21-22. **ALLW_RA, ALLW_DEC **: J2000 AllWISE Right Ascension and Declination, in degrees.

Column 23. ALLW_e_RADEC: AllWISE positional error, in arc seconds.

Columns 24-27. ALLW_w[1234]mpro: AllWISE Vega magnitude in the W1, W2, W3, W4 bands.

Columns 28-31. ALLW_w[1234]sigmpro: AllWISE magnitude error in the W1, W2, W3, W4 bands.

Columns 32-35. ALLW_w[1234]snr: AllWISE signal to noise ratio in the W1, W2, W3, W4 bands.

Column 36. ALLW_cc_flags: AllWISE reliability flag from Cutri et al. (2013).

Column 37. Separation_ALLW_XMMSL2: Separation between members of this association, in arcsec.

Column 38. dist_bayesfactor: Logarithm of ratio between prior and posterior from distance matching.

Column 39. dist_post: Distance posterior probability comparing this association vs. no association, as in (Budavári & Szalay, 2008).

Column 40. bias_ALLW_COLOURMAG_PIX: Probability weighting introduced by AllWISE prior. 1 indicates no change.

Column 41. p_single: Same as dist_post, but weighted by AllWISE prior.

Column 42. p_any: For each entry in the X-ray catalogue, the probability that any of the associations is the correct one. The lower p_any, lower is confidence that a reliable counterpart was found. See § 4.2.

Column 43. p_i: Relative probability of the match, if one exists. The p_i add up to unity for each X-ray source.

Column 44. match_flag: 1 for the most probable match, if existing; 2: almost as good solutions (p_i/ p_ibest> 0.5).

Column 45-46. GroupID, GroupSize: if the 2RXS source has only one possible AllWISE counterpart, the two columns are blank. Otherwise, the GroupSize value indicate the number of possible counterparts while the GroupID value is the same integer for the group. A sort on the GroupID value, will rank the first non-unique match group together, followed by all the rows in the second non-unique group, etc. All the unique matches are listed last.

Column 47. 1RXS_ID: Source name in the 1RXS catalogues (Voges et al., 1999, 2000).

Column 48. ALLW_2MASS_ID: 2MASS Identifier as listed in the AllWISE catalog.

Columns 49-51. ALLW_[jhk]_m_2mass: 2MASS magnitude in the j,h,k bands, as from AllWISE catalogue.

Columns 52-54. ALLW_[jhk]_msig_2mass: 2MASS magnitude errors in the j,h,k bands, as from AllWISE catalogue.

Columns 55. Gaia_DR1_ID: Solution ID from the original Gaia DR1 catalogue (see Fabricius et al., 2016, for more details).

Columns 56-57. Gaia_DR1_RA, Gaia_DR1_DEC: Gaia J2000 Right Ascension and Declination as computed by Vizier.

Columns 58-59. pmra, pmdec: Proper motion in Right Ascension and Declination as measured by Gaia.

Columns 60-61. pmra_error, pmdec_error: Proper motion errors in Right Ascension and Declination as measured by Gaia.

Columns 62. phot_g_mean_flux: Gaia mean flux in units of e-/s.

Columns 63. phot_g_mean_flux_error: Gaia mean flux erorr in units of e-/s.

Columns 64. phot_g_mean_mag: Gaia mean magnitude.

8 NWAY release

Together with the AllWISE counterparts to the 2RXS and XMMSL2 catalogues, we also also Nway . The Nway software and manual are available at https://github.com/JohannesBuchner/nway. In order to make the user familiar with the code, the release is completed with the catalogues used in the testing phase discussed in Appendix C. We would like to stress that the use of Nway is not limited to finding the counterparts to X-ray sources. With the advent of deep and wide area surveys in X-rays (e.g. eROSITA, Athena) and radio (e.g., ASKAP/ EMU:Norris et al. 2011; LOFAR:van Haarlem et al. 2013; APERTIF:Oosterloo et al. 2010), Nway will provide a powerful and reliable counterpart identification tool.

9 Discussion and Conclusions

We presented the catalogues of secure AllWISE counterparts to the ROSAT/2RXS and XMMSL2 X-ray extragalactic all-sky surveys. Only a small fraction (less than 5%) of the X-ray/AllWISE associations is expected to be due to chance associations. Associations were obtained using a new algorithm, Nway , capable of handling complicated priors. In particular, we have used here a prior based on the WISE colour-magnitude properties of about 2500 X-ray sources from the 3XMM-DR5 catalogue with flux distribution similar to 2RXS and XMMSL2.

Nway can be used for finding the right counterparts to other (not only X-ray) surveys. However, the prior which we apply Nway in this work is tuned to the properties of the input catalogs and thus is not universal. E.g., adopting for 2RXS a similar prior to that used in Dwelly et al. (2017), which was constructed with half of the sources adopted here, the AllWISE counterpart changes for 3% of the sources (3431/106573). The prior is appropriate only as long as it well represents the population. For this reason, the prior adopted for the extragalactic region covered by 2RXS and XMMSL2 cannot be used with the same reliability for finding the correct counterparts of X-ray sources in the Galactic plane, where the X-ray catalogues are dominated by stars. Similarly, it will not be possible to use the same prior with the same reliability for finding the counterparts to X-ray surveys that are significantly shallower or deeper than the two discussed in this work.

9.1 Finding counterparts to eROSITA point-like sources

The design of Nway was dictated by the need of developing a flexible algorithm that could be used with the patchwork of multi-wavelength coverage of the entire sky available for finding the counterparts of eROSITA (Merloni et al., 2012).

eROSITA combines a wide field of view, large collecting area, long survey duration, broad energy bandpass, and good point source location accuracy, making it by far the most powerful X-ray survey instrument ever built. In the soft energy band (0.5–2 keV), the planned four-year eROSITA all-sky survey, will have a median point source flux limit of $10^{-14}$ erg cm*-2* s*-1* (Merloni et al., 2012), approximately 30$\times$ deeper than the ROSAT all-sky survey (for AGN-like X-ray spectra).

In the hard X-ray band (2–10 keV), the predicted flux limit of $2\times 10^{-13}$ erg cm*-2* s*-1* is around 100$\times$ deeper than the only existing all-sky survey conducted at these energies (i.e. the High Energy Astronomy Observatory, HEAO-I: Wood et al., 1984). On completion, the eROSITA survey is expected to detect about 4 millions X-ray sources, with 3/4 of them being AGN. Thankfully, the location accuracy for point-like eROSITA sources is expected to be better than 10*′′* radius (combination of statistical and systematic uncertainties), substantially better than for typical ROSAT sources. This will also be enabled by the availability of Gaia that will allow accurate positional accuracy on eROSITA single frame by tying the two astrometric reference frames. The eROSITA data will also enable better separation between point-like (mostly AGN and stars) and extended (galaxy cluster) sources on the basis of their X-ray properties alone (Merloni et al., 2012). However, due to the fainter X-ray flux limit expected, the optical-IR counterparts to point-like eROSITA sources will typically be several magnitudes fainter than those presented here. Figure 10 illustrates this by showing the colour-magnitude distribution of the counterparts to the X-ray sources in STRIPE82X (LaMassa et al., 2016; Tasnim Ananna et al., 2017) cut at the depth of eROSITA, colour coded as a function of X-ray flux. Given the increasing depth of eROSITA, the counterparts of the sources get progressively fainter, finally overlapping with the bulk of the AllWISE population within 30*′′* of the X-ray position, here in grey (see for comparison the distribution in Fig. 3).

Hence, in order to select the correct counterparts for several million eROSITA sources, we will need to take into account additional information to separate field populations from the true counterparts to X-ray sources. Deeper WISE catalogues, enabled by the co-addition of the ongoing multi-year NEOWISE survey data (Mainzer et al., 2011, 2014; Meisner et al., 2016) with the existing AllWISE data set, should not only probe to fainter $[W1]$ and $[W2]$ limits, but should also have a smaller photometric scatter at the magnitudes currently probed by the AllWISE survey. Such a reduced scatter will allow better separation of the red (in $[W1$-$W2]$) AGN population from the bluer field stars and galaxies. Note however that at the depth of ROSAT, only 0.01% of the AllWISE counterparts had an upper limit in W2, while the number will increase to fainter X-ray flux, even considering the reactivation of NEOWISE (Mainzer et al., 2014; Meisner et al., 2017) post-cryogenic phase expected to reach a depth in W2 of 19.9, when combined with WISE.

In addition, we expect that one of the main drawbacks of relying on any catalogue derived from WISE data will be the relatively broad PSF ($\sim$6 arcsec full width at half maximum in WISE bands 1 and 2), which results in blending problems for close pairs of sources. This problem will inevitably get worse as the co-added WISE data reach to fainter magnitudes, approaching the confusion limit. In addition, once an AllWISE counterpart has been selected for each X-ray source, the final step of optical counterpart selection must still be carried out. This step becomes particularly difficult when WISE detections are blends of multiple astrophysical sources.

The forced photometry techniques and tools described by Lang et al. (2016) avoid many of the problems associated with combining data across multiple wavebands, and have already been exploited successfully, e.g. in the selection of QSO targets for eBOSS (Myers et al., 2015). By cross-matching eROSITA sources with previously compiled forced photometry catalogues (e.g. derived from Gaia in the Galactic plane, SDSS and DECaLS, DES, VHS photometry), we expect to greatly reduce both the impact of source confusion in the IR, and the general problems related to compiling data across multiple optical-IR wavebands.

The high cross-matching success rate for 2RXS and XMMSL2 has demonstrated that our cross-matching routine and priors are relatively robust. However, the dynamic range of the eROSITA catalogue will be much larger than that considered here.

Therefore, it is likely that a single, X-ray flux-independent prior (as adopted in this work) will be a sub-optimal choice for finding counterparts to all eROSITA sources. We also expect a strong dependence in the mixture of object classes which make up the eROSITA sample as a function of Galactic latitude. Thankfully, the XMM, Chandra and Swift/XRT archives already contain large samples of well-measured X-ray reference sources which populate the entire eROSITA flux range, and which can be used to define new X-ray-flux-dependent and/or Galactic-latitude-dependent optical-IR priors.

However, great care will be needed to understand the very complex inhomogeneities/biases/incompletenesses that will be imprinted by such an optimized cross-matching scheme. It is possible that a single cross-matching procedure is not suitable for all eROSITA science projects, and that a number of individually tailored cross-matching schemes will be required, depending on the patch on the sky.

The bulk of the X-ray sources in our study are stars and AGN, which are intrinsically variable objects. However, we have made the simplifying assumption throughout this work that variability (in luminosity and/or in spectral energy distribution) of X-ray sources is not important for the purposes of counterpart selection. This means that we do not take account of extremely interesting, but difficult to handle scenarios such as where an AGN that was bright at the epoch of its X-ray detection (e.g. in ROSAT) has faded substantially (in all wavebands) several years later when the measurement of its longer wavelength counterpart (e.g. WISE or SDSS) was made (e.g. ‘changing-look’ QSOs; LaMassa et al., 2015; Merloni et al., 2015; Runnoe et al., 2016). However, in the future we will use AGN and stellar variability to our advantage when selecting counterparts to eROSITA X-ray sources. With the present (PTF/iPTF/ZTF121212http://www.ptf.caltech.edu/iptfRau et al. 2009; Catalina131313http://crts.caltech.edu/, Pan-STARRS141414https://panstarrs.stsci.edu/: Chambers et al. 2016; etc.) and forthcoming generation of optical time domain surveys, (e.g. as performed by the Large Synoptic Survey Telescope; Gressler et al., 2014), every potential optical counterpart to an X-ray source will also come with robust measurements of optical variability. Such variability metrics, which naturally separate AGN and stars from field galaxies, and can be simply applied as an additional prior in Nway (see, for example, Budavári et al., 2017).

Acknowledgement

The authors are grateful to the referee for the valuable input on the earlier version of this paper. The authors are grateful to M. J. Freyberg, T. Boller and F. Haberl for their help in understanding the old and new ROSAT catalogues. MS and JB are grateful to G.Hasinger and T. Simm for testing and proving feedback on the various versions of Nway . MS thanks F. Guglielmetti, A. Georgakakis and D. Coffey for various discussions over the years that helped in shaping the final work. JB acknowledges support from FONDECYT Postdoctorados grant 3160439 and the Ministry of Economy, Development, and Tourism’s Millennium Science Initiative through grant IC120009, awarded to The Millennium Institute of Astrophysics, MAS. MB acknowledges support from the FP7 Career Integration Grant “eEASy?’ (CIG 321913). This publication makes use of data products from the Wide-field Infrared Survey Explorer, which is a joint project of the University of California, Los Angeles, and the Jet Propulsion Laboratory/California Institute of Technology, funded by the National Aeronautics and Space Administration. This research has made use of data obtained from the 3XMM XMM-Newton serendipitous source catalogue compiled by the 10 institutes of the XMM-Newton Survey Science Centre selected by ESA. This publication makes use of data products from the Two Micron All Sky Survey, which is a joint project of the University of Massachusetts and the Infrared Processing and Analysis Center/California Institute of Technology, funded by the National Aeronautics and Space Administration and the National Science Foundation. This research has made use of data obtained from XMMSL2, the Second XMM-Newton Slew Survey Catalogue, produced by members of the XMM SOC, the EPIC consortium, and using work carried out in the context of the EXTraS project ("Exploring the X-ray Transient and variable Sky", funded from the EU’s Seventh Framework Programme under grant agreement no. 607452). This work has made use of data from the European Space Agency (ESA) mission Gaia (https://www.cosmos.esa.int/gaia), processed by the Gaia Data Processing and Analysis Consortium (DPAC, https://www.cosmos.esa.int/web/gaia/dpac/consortium). Funding for the DPAC has been provided by national institutions, in particular the institutions participating in the Gaia Multilateral Agreement. This research has made use of the Vizier catalogue access tool, CDS, Strasbourg, France. The original description of the VizieR service was published in A&AS 143, 23. This publication makes use of TOPCAT (Taylor, 2005) and STILTS (Taylor, 2006) available at http://www.starlink.ac.uk/topcat/ and http://www.starlink.ac.uk/stilts/, respectively.

Appendix A A brief history of the matching problem

In astrophysics a source can be characterized by its accurate position on the sky, its redshift and its Spectral Energy Distribution (SED). If the positional accuracy is not known at a sub-arc second precision, the source cannot be the target of a spectroscopy study, and/or multi wavelength data cannot be correctly assembled. While sources that are identified in the Optical and Near-Infrared regime usually have the required precision, this is not the case for sources selected at shorter and longer wavelengths. For example in the Far-infrared bands, Herschel reaches 6-7*′′* Point Spread Function (PSF) at 70$\mu$m, increasing up to $\sim$13*′′* at longer wavelength. Similarly, in X-ray the positional measurement error depends on the counts and spatially varying PSF and therefore is not constant between sources. Typical positional uncertainties go from up to \approx 3\text{{}^{\prime\prime}} (Chandra), to 7*′′* (XMM) but reach up to about 29*′′* for 95% of the of ROSAT sources in 2RXS, with the values increasing toward the periphery of the field of view, up to more than 1′ in the extreme cases. This low positional accuracy, together with the fact that sources with different SEDs and different redshift emit the bulk of their energy in different photometric bands, make it difficult to identifying with certainty the same source in different surveys. Additionally, the entire pairing process is done by means of catalogues which can differ in depth, technique for "source detection" (and definition thereof). In the past, the data were so shallow that a simple cross match in coordinates between catalogues was enough for pairing correctly the sources. Now, we reach sources that are so faint that we must adopt a probabilistic approach.

The most used technique is based on the Likelihood Ratio (LR) method (Sutherland & Saunders, 1992). Taking into account source number densities, coordinates (with relative errors) and magnitude distribution of the sources, the method estimates the ratio between the likelihood that a given source from catalogue B is the correct counterpart to a source detected in a catalogue A, and the likelihood of being a source in the background. Different factors are then considered when computing the threshold above which the likelihood ratio assures a reliable association. The procedure is repeated anew for the pairing between the catalogues A-C, A-D, etc. If catalogues are i) from images at similar wavelength and ii) of sufficient depth, for most of the sources in A, the counterpart in catalogues B, C, D etc will be the same, while for a fraction of the sources further considerations based on the shape of the SED need to be taken into account for the counterpart association.

Moving from a generic description to a specific application, let us focus from now on to the case of finding the correct counterpart to X-ray sources. The LR method has been successfully applied on surveys like XMM-COSMOS (Brusa et al., 2007, 2010), CDFS (Luo et al., 2008; Xue et al., 2011; Luo et al., 2017), Chandra-COSMOS (Civano et al., 2012; Marchesi et al., 2016), XXL (Georgakakis et al., 2017), STRIPE-82X (LaMassa et al., 2016; Tasnim Ananna et al., 2017), AEGIS-X (Georgakakis & Nandra, 2011; Nandra et al., 2015) just to mention a few. For each of these surveys, the authors performed the steps described above, pairing X-ray to optical, to near-infrared and to mid-infrared data, independently. Then, they ranked the ancillary data available in order of reliability (i.e. deep and higher resolution data first) for selecting the correct counterpart in those cases where the LR method does not provided a unique solutions.

The Bayesian approach is increasingly favored by the entire community. Contrary to the LR method that is data-driven, the Bayesian approach uses a model for reference (prior) and thus can be applied also to small samples and areas. This is a strength of the method but but a frequent criticism is that the assumption of a model distribution might not represent the reality. These criticisms are legitimate in general but in the specific case of finding the counterpart to X-ray detected sources they are somewhat outdated. In fact, deep Chandra and XMM surveys are so advanced/extended that reliable models of magnitude distribution of the counterparts to sources detected up to a desired depth, can now be constructed empirically. Another virtue of the Bayesian approach is that many priors can be adopted, each independent of the next. So we can adopt a Bayesian form for the probability of a sources to be the right counterpart based on its position, its magnitude, colour etc.

At the basis of many Bayesian cross-matching algorithms is the formalism introduced by Budavári & Szalay (2008)151515However, the work does not correctly account for the sources that for physical reasons (e.g. due to the shape of the SED, redshift value) are missed in some of the catalogues. This has been pointed out by many authors (e.g. Roseboom et al., 2009; Pineau et al., 2011) . This enables simultaneously cross-matching of multiple catalogues and provides the Bayes factor fror the astrometric measures. This Bayes factor from the astrometry is then combined with one (or more) related to physical properties. E.g., Roseboom et al. (2009) search the right counterparts to sub-millimeter sources by computing the photometric redshift and SED fitting of each source within a certain radius circle.

Independently from the adopted method, an additional difficulty arises when the goal is to find the counterparts to X-ray surveys that cover over hundreds of square degrees (e.g., eROSITA: Merloni et al., 2012). In this case, the multi-wavelength catalogues from where to draw the correct identification will not be homogeneously covering the field, but rather a patchwork of different surveys/depths, thus effecting the actual magnitude distribution of the field sources and thus the determination of the real counterpart.

In view of these new challenges, we designed Nway , an algorithm based on two-steps Bayesian approach. In the following we provide the complete description of the code and its application to test cases in COSMOS. The code is released, together with a detailed manual and a set of test data for training purposes. purposes.

Appendix B Matching Methodology

This section lays out in detail the computation Nway performs. Further details and clear explanations on the use of the Nway are presented in the manual and tutorial of the code, distributed via Github at https://github.com/JohannesBuchner/nway.

The features of Nway include:

Matching of N catalogues simultaneously. 2. 2.

Computation of all combinatorially possible matches. 3. 3.

Consideration of partial matches across catalogues, i.e. the absence of counterparts in some catalogues. 4. 4.

Taking into account the positional uncertainties and the source number densities, computation of the probability of each possible match. 5. 5.

Computation of the probability that there is no match. 6. 6.

Incorporating magnitude, colour or other information about the sources of interest, refining the match probabilities.

This is done in several steps:

Finding combinatorially all possible matches. See Section B.1. 2. 2.

Computing each match probability from number densities, separation distances and positional errors alone, taking into account the chance of a random alignment. See Section B.2. 3. 3.

For each source of the primary catalogue (in the application from this paper: for each the X-ray source), compute (a) the probability that this source does not have a counterpart and (b), assuming this source has a counterpart, compute the relative probability for each possible match. See Section B.3. 4. 4.

Refining the probabilities by additional prior information. See Section B.4.

In Nway , only the first catalogue (primary catalogue) has a special role. For every entry in this catalogue, matches are sought in the other catalogues. The entries in the primary catalogue must come with an ID. All catalogues must contain RA, DEC, positional error information, the size of the area of sky covered by the catalogue. The latter information is used to compute the probability of a chance alignment.

B.1 Computing all possible matches