Piercing through Highly Obscured and Compton-thick AGNs in the Chandra Deep Fields. I. X-Ray Spectral and Long-term Variability Analyses

We use MYTorus-based models (Murphy & Yaqoob 2009) to fit the observed-frame 0.5–7 keV spectra of the full sample in order to identify heavily obscured sources. Due to limited counts, we do not bin the spectra because they may lose some key information of the sources. We use the Cash statistic (Cstat in XSPEC; Cash 1979) as our spectral fitting statistic. Cstat has a similar probability distribution to χ² statistics and has been proved to be more appropriate in the low-count regime. Since Cstat is not appropriate for the background-subtracted spectra, we simultaneously fit the source and background spectra, with the latter (with 1642 median counts) being fit with the cplinear model (Broos et al. 2010) that properly describes the observed Chandra background by a continuous piecewise-linear function in 10 energy segments (an example is shown in Figure 2). In this way, we are able to maximize the usage of information relevant to the sources.

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We adopt two models to fit the source spectra with different components and degrees of freedom (dof) in order to find a statistically robust best-fit one:

• 1.  
  The MYTorus baseline model: phabs × (MYTZ × zpow + f_ref × MYTS + f_ref × gsmooth (MYTL)).
• 2.  
  The soft-excess model: phabs × (MYTZ × zpow + f_ref ×MYTS + f_ref × gsmooth (MYTL) + ${\text{}}{f}_{\mathrm{exs}}$ × zpow).

These models include all the typical spectral features found in highly obscured AGNs (see Figure 3). The phabs component models the Galactic photoelectric absorption. The MYTZ ×zpow term represents the zeroth-order transmitted power-law continuum across the torus that takes into account both photoelectric absorption and Compton scattering processes. MYTS and gsmooth (MYTL) stand for the reflection component and the broadened Fe Kα, Kβ fluorescent emission lines, respectively. A second power law is added to represent the soft-excess component often found in AGN spectra, possibly originating from the zeroth-order continuum being scattered by the extended CN materials in obscured AGNs (Guainazzi et al. 2005; Bianchi et al. 2006; Corral et al. 2011). During spectral fitting, the inclination angle θ is fixed at 75°, which is the average value within a range of 60°–90° where the torus intercepts our line of sight (LOS). The nominal normalizations of MYTS, MYTL, and the second power law are set to be the same as that of the first power law, and we use the constants ${\text{}}{f}_{\mathrm{ref}}$ and ${\text{}}{f}_{\mathrm{exs}}$ to represent the real normalizations. ${\text{}}{f}_{\mathrm{exs}}$ is allowed to vary between 0 and 0.1, and we assume a 200 keV high-energy cutoff throughout. One thing to mention is that the best-fit ${N}_{{\rm{H}}}$ values in MYTorus represent the equatorial values, and we always use the LOS values ${N}_{{\rm{H}},\mathrm{LOS}}={(1-4{\cos }^{2}\theta )}^{\tfrac{1}{2}}{N}_{{\rm{H}}}$ in this work.

Several previous works suggested that the inclusion of the soft-excess and reflection components in the spectral models is crucial for us to correctly estimate ${N}_{{\rm{H}}}$ of highly obscured sources (e.g., Brightman et al. 2014; Lanzuisi et al. 2015). However, the low counts of many sources do not allow us to apply complex models with free parameters since it could lead to large degeneracies. Therefore, we choose the model components and the parameter spaces according to the following criteria:

• 1.  
  We fix Γ at 1.8 (e.g., Tozzi et al. 2006; Marchesi et al. 2016) for the 851 sources with counts less than 300. For the 301 sources with counts larger than 300, we set Γ free to find a best-fit value.
• 2.  
  For all sources, we first fit the spectra with a free ${f}_{\mathrm{ref}}$. If ${\text{}}{f}_{\mathrm{ref}}$ is less than 10⁻⁵, we then fix it to 10⁻⁸, which is an arbitrary value set as the lower limit for ${\text{}}{f}_{\mathrm{ref}}$ (i.e., indicating a negligible reflection component); otherwise, we fix ${\text{}}{f}_{\mathrm{ref}}$ at 1, which is the default value adopted in MYTorus.
• 3.  
  To determine whether we should add a soft-excess component, we compare the Cstat between models with and without considering soft-excess emission. The best-fit model is chosen to be the one that has a statistically robust low Cstat value. More specifically, adding a new soft-excess component should improve the Cstat at least for ΔC = 3.84 with 1 more dof. This criterion is based on the fact that ΔC approximately follows the χ² distribution; thus, ΔC = 3.84 is roughly consistent with a >95% confidence level (Tozzi et al. 2006; Liu et al. 2017).
• 4.  
  In order to avoid extremely untypical Γ caused by the degeneracy between Γ and ${N}_{{\rm{H}}}$, we re-fit the spectra of those sources with Γ pegged at 1.4 (i.e., the lowest value permitted by MYTorus) or Γ > 2.4 (i.e., the typical maximum photon index for high-${\lambda }_{\mathrm{Edd}}$ AGNs; e.g., Wang et al. 2004; Fanali et al. 2013) by fixing their Γ at 1.8. If ΔC = C_fix − C_free > 3.84, we adopt the free Γ value; otherwise, we adopt Γ = 1.8.

From now on, for convenience, we refer model 1 (2) with negligible ${\text{}}{f}_{\mathrm{ref}}$ as model A (B) and model 1 (2) with ${\text{}}{f}_{\mathrm{ref}}$ fixed at 1.0 as model C (D), as summarized in Table 1. In order to ensure the consistency of the models used for subsequent bias corrections (see Section 6) as much as possible, we have to assume fixed parameters in case of low counts. We justify our models in the Appendix by validating that the usage of fixed parameters does not significantly affect our results. We also compare our fitting results with several previous works (see Section 4.1) and those obtained from the Borus model (Baloković et al. 2018; see Section 5.3) and find good consistency.

Table 1.  Model Definitions Used in This Work

Model	${\text{}}{f}_{\mathrm{ref}}$	${\text{}}{f}_{\mathrm{exs}}$
A	10−8	0
B	10−8	$0\lt {f}_{\mathrm{exs}}\lt 0.1$
C	1.0	0
D	1.0	$0\lt {f}_{\mathrm{exs}}\lt 0.1$

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We note that the simple absorbed power-law model (e.g., ${phabs}\times {zwabs}\times {zpow};$ hereafter model z) has been widely used in the literature to obtain rough estimates of AGN parameters. However, such a model is not appropriate for highly obscured AGNs since zwabs only models photoelectric absorption and does not take into account the Compton scattering process, which is particularly important in the high-${N}_{{\rm{H}}}$ regime. Therefore, we also fit the spectra using model z, aiming at directly testing how accurately such a simple model reproduces the main spectral parameters.

Thanks to the increased exposure time, the improved data reduction procedure (see Table 1 in Xue et al. 2016 for a summary), the updated redshift measurements, and the usage of physically appropriate spectral fitting models for highly obscured AGNs, we are able to extract higher-quality spectra and obtain more robust parameter constraints than previous works in the same field (e.g., Tozzi et al. 2006), especially for faint sources. Before performing further analyses, we first make direct comparisons with several works that presented spectral fitting parameters in the CDF-S to understand how different methods influence the spectral fitting results. The works used for comparisons are as follows:

• 1.  
  Liu et al. (2017) (L17): L17 performed an extensive X-ray spectral analysis for the bright sources (hard-band counts >80) in the 7 Ms CDF-S using the ${wabs}\times ({plcabs}\,\times {power}\,-\,{law}\,+\,{zgauss}\,+\,{power}\,-\,{law}\,+\,{plcabs}\,\times {pexrav}\times {constant})$ model. The background spectrum was also fitted with the cplinear model. The redshift information used in the two works is the same.
• 2.  
  Brightman et al. (2014) (B14): B14 presented X-ray spectral fitting results in the 4 Ms CDF-S using the BNtorus model (Brightman & Nandra 2011). A total of 112 sources are found in common between the two works; 24 of them have redshifts different by more than 0.2 (Δz > 0.2) compared with the values adopted in the updated 7 Ms catalog and are neglected in the following comparison.
• 3.  
  Buchner et al. (2015) (B15): The 4 Ms CDF-S spectra were analyzed using a physically motivated torus model and a Bayesian methodology to estimate spectral parameters. A total of 114 sources are found in common between the two works, and 30 of them with Δz > 0.2 are excluded.

As shown in Figure 4, our measured ${N}_{{\rm{H}}}$ and ${L}_{{\rm{X}}}$ values are in general agreement with previous works, despite that for individual sources the usage of different model configurations and data may result in large discrepancies. The largest distinction happens between B14 and our work, in that 10 highly obscured AGNs in our sample are reported to be unobscured in B14. To understand the discrepancy, we refit their spectra using the same model and method as described in B14, and the results again confirm their heavily obscured nature. Therefore, the large discrepancy could be due to the adopted data of different depths, as well as the different data reduction and spectral extraction methods used in the two works.

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The comparisons between the results obtained through MYTorus and model z are shown in Figure 5. Note that since MYTorus does not allow ${N}_{{\rm{H}}}$ to vary below ${10}^{22}\ {\mathrm{cm}}^{-2}$, a number of unobscured sources thus have best-fit ${N}_{{\rm{H}}}={10}^{22}\ {\mathrm{cm}}^{-2}$. Therefore, we only consider sources with MYTorus $\mathrm{log}\,{N}_{{\rm{H}}}\gt 22.2\ {\mathrm{cm}}^{-2}$ in the comparison.

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The ${N}_{{\rm{H}}}$ and the observed 0.5–7 keV flux derived by the two models are generally consistent. But at a given ${N}_{{\rm{H}}}$, the absorption-corrected intrinsic 0.5–7 keV flux ratio and the intrinsic ${L}_{{\rm{X}}}$ ratio between the two models dramatically increase with ${N}_{{\rm{H}}}$. It is obvious that model z underestimates the intrinsic luminosity owing to neglect of the Compton scattering process, and the discrepancy is already evident at ${N}_{{\rm{H}}}\sim {10}^{23}\ {\mathrm{cm}}^{-2}$ (see also Burlon et al. 2011). Therefore, such models with only photoelectric absorption taken into consideration (e.g., zwabs, zphabs, ztbabs) must be used with caution in the highly obscured regime.

The photon index distribution for the 62 highly obscured sources with free Γ during the fitting process is shown in Figure 6. The mean value of the distribution is 1.82 ± 0.04, in agreement with previous X-ray spectral analyses in the CDF-S (e.g., Tozzi et al. 2006; Liu et al. 2017) and COSMOS (e.g., Lanzuisi et al. 2013). Note that our Γ distribution peaks at Γ = 1.4 instead of the mean value 1.8. This is because the photon index is restricted within 1.4–2.6 in MYTorus. If we use MYTZ alone, which does not limit the Γ range to fit the spectra, most sources with Γ pegged at 1.4 will have smaller Γ, and thus the distribution will appear more symmetric with a larger dispersion.

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We also try to search the potential correlations among Γ, ${N}_{{\rm{H}}}$, and ${L}_{{\rm{X}}}$ using the Spearman rank correlation test. No correlation between Γ and redshift (Spearman's ρ = 0.00, p-value = 0.99) is detected, indicating that the inner disk and corona structures have little evolution across cosmic time. The Spearman tests also suggest no significant correlation between Γ and ${N}_{{\rm{H}}}$ (${L}_{{\rm{X}}}$).

It is noteworthy that there are eight sources with Γ = 1.4 or Γ > 2.5 (see Table 4). Unlike some of the high-count sources for which we artificially fix Γ at 1.8 (see Section 3.2), these sources have significantly worse fits if we do not allow Γ to vary freely (the average improvement of ΔC is 9.0 if we set Γ free). Sources with very flat photon indices are often considered to be reflection-dominated CT candidates, and their extremely obscured nature may not be revealed by our best-fit ${N}_{{\rm{H}}}$ (Georgantopoulos et al. 2009). Moreover, since we are dealing with the stacked spectra here, sources that possessed large spectral variability may also exhibit untypical spectral shape, as might be the case for XID 249 (see Section 7.4). Additionally, we cannot rule out the possibility that the extreme photon indices of some sources are wrongly measured owing to insecure photometric redshifts.

Table 4.  Information of Sources with Extreme Photon Indices

Field	ID	Γ	NH	Counts	ztype	zqual
CDF-S	98	1.40	${0.24}_{-0.01}^{+0.02}$	3689	zphot	insecure
	135	2.53	${1.59}_{-0.24}^{+0.21}$	617	zspec	insecure
	172	1.40	${0.33}_{-0.08}^{+0.07}$	534	zphot	...
	243	1.40	${0.12}_{-0.01}^{+0.01}$	372	zspec	secure
	249	1.40	${0.18}_{-0.02}^{+0.02}$	830	zphot	...
	597	1.40	${0.24}_{-0.04}^{+0.03}$	351	zspec	secure
	760	1.40	${0.72}_{-0.07}^{+0.08}$	351	zspec	insecure
CDF-N	546	1.40	${0.11}_{-0.01}^{+0.01}$	456	zspec	secure

Note. Columns are the same as in Table 2. The zqual column represents the quality of the spectroscopic redshift (secure or insecure). For sources with insecure spectroscopic redshifts, the X-ray source catalogs may choose to adopt the photometric redshifts as ZFINAL (see Section 4.4 in Xue et al. 2011 for details).

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For highly obscured AGNs, the significant absorption and scattering of soft X-ray photons may lead to large hardness ratio (HR); thus, the HR may be used as an indicator of obscuration level (e.g., Wang et al. 2004). In Figure 7 we show the observed HR (here we adopt the definition of HR as (H−S)/(H+S), where H stands for the observed-frame 2–7 keV count rate and S stands for the 0.5–2 keV count rate) as a function of redshift for highly obscured sources and CT candidates confirmed by the spectral fitting, as well as less obscured AGNs (best-fit ${N}_{{\rm{H}}}$ < ${10}^{23}\ {\mathrm{cm}}^{-2}$) that contaminate our sample. Heavily obscured sources have significantly larger HR than less obscured sources as expected. We also calculate the effective photon index Γ_eff (obtained by fitting the spectra using XSPEC model phabs × zpow with ${N}_{{\rm{H}}}$ fixed at the Galactic value) for the HOS. Ninety percent of them have Γ_eff < 1.0, and the median Γ_eff is only −0.38 for CT AGNs, which again verify their heavily obscured nature.

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The anticorrelation between HR and z is due to the fact that for high-redshift sources the observed soft band actually corresponds to a much harder band in the rest frame and the photons may not be absorbed significantly. Note that the HRs of the most heavily obscured CT AGNs are not always the largest at a given redshift. This can be ascribed to the additional soft-excess component softening the CT AGN spectra, and CN AGNs with intrinsically flat photon indices are also plausible to have larger HRs than CT AGNs.

To test whether a simple HR value can be used to select heavily obscured AGNs, we calculate the evolution of HR as a function of redshift for typical X-ray spectral parameters. Since our purpose is to find the critical HR for highly obscured AGNs as a function of redshift, we simply consider the threshold condition: a source with ${N}_{{\rm{H}}}={10}^{23}\ {\mathrm{cm}}^{-2}$. Additionally, we set ${\text{}}{f}_{\mathrm{ref}}$ at 1.0, set the high-energy cutoff at 200 keV, and fix the photon index Γ of the two power laws at 2.0. The constant ${\text{}}{f}_{\mathrm{exs}}$ is set to 5%, which is typical for AGNs with ${N}_{{\rm{H}}}\,\sim {10}^{23}\ {\mathrm{cm}}^{-2}$ (e.g., Brightman et al. 2014). The reason why we adopt ${\rm{\Gamma }}=2.0$ instead of the mean value 1.8 is that ∼96% of the sources with a well-constrained photon index in L17 have Γ ≲ 2.0. Therefore, with this value we expect that our derived HR curve will be able to successfully select out most of the highly obscured AGNs. The model is shown in Figure 2(a), and the derived model HRs at different reshifts are shown in green in Figure 7 (curve A).

It is worth noting that not all CT AGNs have larger HRs than CN AGNs. We also show the HR curve derived for a CT AGN with ${N}_{{\rm{H}}}={10}^{24}\ {\mathrm{cm}}^{-2}$. This time ${\text{}}{f}_{\mathrm{exs}}$ is chosen to be 1% owing to the fact that ${\text{}}{f}_{\mathrm{exs}}$ decreases with increasing ${N}_{{\rm{H}}}$ (see Section 5.2). The model is shown in Figure 2(b), and the simulated HR curve is also shown in Figure 7 (curve B). It is clear that at low redshifts, even a very small soft-excess fraction can easily dominate the soft X-ray spectra that could make CT AGNs even softer than CN AGNs. Therefore, to avoid missing a large number of high-${N}_{{\rm{H}}}$ sources at low redshifts, we use curve B at $z\leqslant 0.8$ and curve A at z > 0.8 as our selection curve in the following analysis (i.e., the combined curve).

For a source with given HR and redshift values, we consider it as a heavily obscured candidate if the observed HR lies above the combined HR curve. By applying this curve to our full sample, 480 sources will be selected as highly obscured candidates (hereafter the candidate sample). Eighty percent (382/480) of sources in the candidate sample are indeed highly obscured as confirmed from spectral fitting, i.e., the accuracy is 80%. The parameter distributions for the selected candidates and confirmed sources are shown in Figure 8 (top panel). Only 20% of the candidate sources are contaminated by less obscured sources, with most of them having ${N}_{{\rm{H}}}$ only slightly smaller than ${10}^{23}\ {\mathrm{cm}}^{-2}$ and lying close to the boundary curve, as expected. The redshift, luminosity, and soft-excess fraction distributions for the selected candidates are very similar to sources that are confirmed to have ${N}_{{\rm{H}}}\gt {10}^{23}\ {\mathrm{cm}}^{-2}$.

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In addition, we define the completeness fraction as the ratio of the number of highly obscured sources (confirmed) that lie above the selection curve to the total number of highly obscured sources (i.e., all red and blue points in Figure 7). The completeness fractions as a function of several parameters are shown in the second row in Figure 8. The average completeness fraction for the selection curve is 88% (382/436) and remains >90% for most high-redshift and high-luminosity bins, while at lower redshifts and larger ${\text{}}{f}_{\mathrm{exs}}$, the completeness fraction significantly drops to <80%, since the soft excess dominates the observed spectrum, which makes highly obscured sources lie below the selection curve.

We also show the result after changing the soft-excess fraction for the model CT AGN to 5% (see curve C in Figure 7). This time the completeness fraction at low redshift is improved to 93%, but the accuracy dramatically drops to 68% since a large amount of less obscured AGNs are mistakenly included. As a trade-off, we choose to adopt the combined curve A and B as our selection curve. The functional form of this curve can be written as

$$\begin{eqnarray}&&\mathrm{HR}=\left\{\begin{array}{l}-0.240{z}^{3}+0.230{z}^{2}+0.680z-0.200,\\ \ (z\leqslant 0.8);\\ -0.004{z}^{3}+0.077{z}^{2}-0.530z+0.740,\\ \ (z\gt 0.8).\end{array}\right.\end{eqnarray} \tag{ 1 }$$

We note that the conclusions in the following sections remain largely unchanged if we use the HR-selected sample instead, suggesting that a simple HR value can be used to identify highly obscured AGNs and select a representative highly obscured sample as long as the redshift and soft-excess effects are properly taken into account while deriving the boundary curve.

In Figure 9 we show the redshift distribution of the HOS in the bottom right panel. A total of 191 sources have spectroscopic redshifts, and 245 sources have photometric redshifts. The peak appears at z ∼ 1–2.5, which is known as the peak epoch of both star formation and black hole accretion activities (e.g., Aird et al. 2015). The source amount slightly decreases down to z ∼ 0.5 and significantly drops at z < 0.5. Since Chandra only pointed at small patches of sky in these two surveys, the small number of sources detected at z < 0.5 is mainly due to the small volume probed and the insufficient penetrability of the corresponding rest-frame energies at low redshifts; thus, the detection probability is severely limited in finding nearby highly obscured AGNs. The downward trend toward high redshifts is primarily due to the flux limits and may be partly due to the fact that photoelectric absorption in the soft band of mildly obscured (${N}_{{\rm{H}}}\sim {10}^{23}\ {\mathrm{cm}}^{-2}$) sources shifts out of the Chandra soft bandpass at z ∼ 2.5, making it more difficult to constrain ${N}_{{\rm{H}}}$ (Vito et al. 2018).

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The ${L}_{{\rm{X}}}$ distribution for the HOS is displayed in the top left panel of Figure 9. The $\mathrm{log}\,{L}_{{\rm{X}}}$ (${L}_{{\rm{X}}}$ in units of $\mathrm{erg}\ {{\rm{s}}}^{-1}$) peaks in the range of 43.5–44.0, with a mean value of 43.65 ± 0.03. The luminous sources with $\mathrm{log}\,{L}_{{\rm{X}}}$ > 44.0 (44.5) account for 32% (11%) of the whole sample. CT AGNs, in particular, have even higher luminosities, with a mean value of 44.10 ± 0.06, because the minimum detectable ${L}_{{\rm{X}}}$ for a flux-limited survey significantly increases with increasing ${N}_{{\rm{H}}}$ (see Figure 16). The correlations among ${L}_{{\rm{X}}}$, ${N}_{{\rm{H}}}$, and redshift are also plotted in Figure 9. The well-known Malmquist bias can be clearly seen in the lower left ${L}_{{\rm{X}}}$–z corner, and we will discuss the influence of this bias on our results in Section 6.4.

The observed distributions of the best-fit ${N}_{{\rm{H}}}$ in six luminosity bins of the HOS are shown in Figure 10. It appears that the amount of high-NH sources in lower-Lx bins is significantly lower than that in the higher-Lx bins, due to the fact that the minimum detectable luminosity significantly increases with NH (see Section 6.4). We show the normalized cumulative ${N}_{{\rm{H}}}$ distributions in Figure 11(a). Sources with best-fit ${N}_{{\rm{H}}}$ > $1.0\times {10}^{24}\ {\mathrm{cm}}^{-2}$ account for ∼25% (108/436) of the HOS. The distribution obtained in Brightman et al. (2014), which utilized the 4 Ms CDF-S data, is shown in the same plot for comparison. The K-S tests imply that the ${N}_{{\rm{H}}}$ distributions from the CDF-S and the CDF-N are consistent with p-value = 0.96 but are different from that in Brightman et al. (2014) with p-value $\ll 0.001$, due to the fact that we identify more sources between $\mathrm{log}\,{N}_{{\rm{H}}}$ = 23.5 and 24.0 cm⁻², while the distribution in Brightman et al. (2014) peaks at $\mathrm{log}\,{N}_{{\rm{H}}}$ = 23.0–23.5 cm⁻².

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Although we have significantly improved the photon statistics for each source using the deepest data, we should caution that most of the CT AGNs discovered still have counts <200 and their ${N}_{{\rm{H}}}$ might be poorly constrained. Therefore, we adopt a more conservative criterion that a source will be regarded as a CT candidate only if it has the best-fit ${N}_{{\rm{H}}}\gt {10}^{24}\ {\mathrm{cm}}^{-2}$ and the 1σ lower limit on ${N}_{{\rm{H}}}$ is constrained to be greater than $5\times {10}^{23}\ {\mathrm{cm}}^{-2}$. This criterion selects 102 sources, with 66 from CDF-S and 36 from CDF-N, accounting for ∼23% of the HOS, to be CT candidates. Note that there are some well-studied CT candidates in the literature that are not classified as CT AGNs in our sample owing to their best-fit ${N}_{{\rm{H}}}$ < ${10}^{24}\ {\mathrm{cm}}^{-2}$, such as XID 328, XID 551 (XID 153 and XID 202 in Comastri et al. 2011, respectively), XID 539 (XID 403 in Gilli et al. 2011), and XID 375 (BzK8608 in Feruglio et al. 2011) in the CDF-S. However, all these sources have best-fit ${N}_{{\rm{H}}}$ > 8 × ${10}^{23}\ {\mathrm{cm}}^{-2}$, which shows good consistency with previous studies; thus, we confirm their heavily obscured nature using deeper Chandra data.

We note that at ${f}_{2-10\mathrm{keV}}\gt {10}^{-15}\ \mathrm{erg}\ {\mathrm{cm}}^{-2}\ {{\rm{s}}}^{-1}$, the number of our CT AGNs in the CDF-N (22 sources) is higher than that found by Georgantopoulos et al. (2009), which presented 10 CT candidates in the same field. This may be because Georgantopoulos et al. (2009) mainly focused on searching reflection-dominated CT AGNs and possibly missed some transmission-dominated sources. We also fit the 22 CT candidates using the BNtorus model (Brightman & Nandra 2011) and find that 20 sources have best-fit ${N}_{{\rm{H}}}\gt {10}^{24}\ {\mathrm{cm}}^{-2}$ and ${f}_{2-10\mathrm{keV}}$ > 10⁻¹⁵ $\mathrm{erg}\ {\mathrm{cm}}^{-2}\ {{\rm{s}}}^{-1}$, which confirms the MYTorus result.

We calculate the observed number counts (log N−log S) for CT candidates selected in Section 4.5. The observed log N−log S is defined as the observed source amount divided by survey area. However, as we will discuss in Section 6.2, the sky coverage is not a constant value across the whole flux and ${N}_{{\rm{H}}}$ ranges; hence, we assign a weighting factor 1/ω_area (see Section 6.2 for details) for each source while calculating their cumulative number counts. The corrected results are shown in Figure 11(b), and several number count measurements presented in previous works in the 4 Ms CDF-S (Brightman & Ueda 2012), X-UDS (Kocevski et al. 2018), and COSMOS (Lanzuisi et al. 2018) fields are also displayed for comparison. Note that Brightman & Ueda (2012) reported the result in the 0.5–8 keV band and we convert the 0.5–8 keV flux into 2–10 keV by assuming a ${\rm{\Gamma }}=-0.4$ power law, which is the median effective photon index for our CT AGNs.

The corrected number counts for the CDF-S and CDF-N are generally consistent within the error bars at higher fluxes, despite that we obtain six CT AGNs in the CDF-N but only three in the CDF-S at ${f}_{2-10\mathrm{keV}}$ > $4\times {10}^{-15}\ \mathrm{erg}\ {\mathrm{cm}}^{-2}\ {{\rm{s}}}^{-1}$, possibly caused by the low number statistics and the cosmic variance due to the small sky coverage of CDFs (e.g., Harrison et al. 2012). At the faint end, both CDF-S and CDF-N number counts flatten owing to the limited detection ability, and the CDF-N number counts are significantly smaller than the CDF-S, because the three times shallower exposure limits its detectability of faint CT sources (see Figure 11(c)). Our results are also in agreement with the 4 Ms CDF-S result at the faint end, the COSMOS result at the bright end, and the X-UDS result at moderate fluxes.

We also compare the observed log N−log S with several CXB model predictions in z = 0–5 (Gilli et al. 2007; Ballantyne et al. 2011; Akylas et al. 2012; Ueda et al. 2014). For the Gilli et al. (2007),²⁰ Akylas et al. (2012),²¹ and Ueda et al. (2014)²² CXB models, the luminosity range used to derive the predicted number counts is $\mathrm{log}\,{L}_{{\rm{X}}}=42\mbox{--}45.5$ $\mathrm{erg}\ {{\rm{s}}}^{-1}$, similar to our sample; for the Ballantyne et al. (2011) model, the predicted result is presented in $\mathrm{log}\,{L}_{{\rm{X}}}=41.5\mbox{--}48\ \mathrm{erg}\ {{\rm{s}}}^{-1}$. As can be seen from Figure 11(b), our observed log N−log S prefers the moderate CT number counts as predicted by Akylas et al. (2012) and Ueda et al. (2014), while other models more or less overestimate or underestimate the number counts.

In summary, the observed parameter distributions and relationships presented in this section provide a basic description of the highly obscured AGN population and are crucial for distinguishing various CXB models. However, these observed distributions, in particular, the observed ${N}_{{\rm{H}}}$ distribution, are influenced by several biases. We will discuss more details about these biases and reconstruct the intrinsic ${N}_{{\rm{H}}}$ distribution representative for the highly obscured AGN population in Section 6.

We collect optical classification results for our CDF-S sample by cross-matching their optical/near-IR/IR/radio counterpart positions presented in the X-ray source catalogs (CP_RA and CP_DEC) with the Silverman et al. (2010) E-CDF-S optical spectroscopic catalog using a 05 matching radius. Among the 55 matched sources, 31 sources have been classified. To our surprise, 19% (6/31) of them are labeled as broad-line AGN (BLAGN). Detailed information on these sources is given in Table 5.

Table 5.  Information of X-Ray Highly Obscured BLAGNs in the CDF-S

XID	Model	${\text{}}{f}_{\mathrm{ref}}$	Oclass	z	ztype	zqual	counts	$\mathrm{log}\,{N}_{{\rm{H}}}$ (${\mathrm{cm}}^{-2}$)	z-range	zx	log NH,x (${\mathrm{cm}}^{-2}$)
(1)	(2)	(3)	(4)	(5)	(6)	(7)	(8)	(9)	(10)	(11)	(12)
882	A	$\ll 1$	BLAGN	3.19	zspec	Secure	80	23.1	⋯	⋯	⋯
944	C	1	BLAGN	0.96	zphot	Insecure	69	23.9	all	2.00	24.1
16	A	$\ll 1$	BLAGN	3.10	zphot	Insecure	409	23.7	z > 1.4	1.66	23.2
399	C	1	BLAGN	1.73	zspec	Secure	1077	23.2	⋯	⋯	⋯
968	C	1	BLAGN	2.03	zspec	Secure	500	23.8	⋯	⋯	⋯
977	C	1	BLAGN	4.64	zspec	Insecure	225	23.9	z > 1.2	3.25	23.9

Note. Column (4): optical classification result from Silverman et al. (2010). Thirty-one X-ray highly obscured AGNs in our CDF-S sample have optical classification results, and six of them are identified as BLAGNs. Column (10): redshift range for sources with insecure redshifts to be determined as being X-ray highly obscured. Column (11): best-fit X-ray redshift by setting redshift as a free parameter during spectral fitting. Column (12): best-fit $\mathrm{log}\,{N}_{{\rm{H}}}$ when adopting the X-ray best-fit redshift. Other columns have the same meaning as in Table 2.

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Most of the six sources have sufficient counts and reliable redshift measurements to constrain ${N}_{{\rm{H}}}$. Three sources have insecure spectroscopic redshifts, although they are labeled as BLAGN, and the 7 Ms main catalog chooses to adopt the photometric redshift as ZFINAL for two of them. This could be due to the fact that they have low-S/N optical spectra or only one emission line is available for sources within particular redshift ranges, which makes it difficult to conclusively determine the line nature, thus giving an insecure redshift. For three sources with insecure redshifts, we set redshift as a free parameter in the spectral fitting and obtain the best-fit X-ray redshift and corresponding ${N}_{{\rm{H}}}$. All of them are still best fitted by ${N}_{{\rm{H}}}\gt {10}^{23}\ {\mathrm{cm}}^{-2}$. The redshift range in which the source will remain X-ray highly obscured is also listed. In general, the ${N}_{{\rm{H}}}$ estimates for the six sources are robust. This result suggests that the heavy X-ray absorption in a fraction of sources is largely an LOS effect caused by some compact clumpy clouds obscuring the central X-ray-emitting region, but the global CF for the high-${N}_{{\rm{H}}}$ materials is limited, and thus the BLR is not blocked; alternativley, the heavy X-ray obscuration may be produced by the BLR itself.

There are 80 sources in our sample that require a second power law to fit the soft-excess component. The origin of this component is still a puzzle. Many different models, e.g., warm Comptonization or blurred ionized reflection from the disk, partially ionized absorption in a wind, or power-law continuum in other directions scattered into the LOS from large-scale CN matter, are proposed to explain the diverse situations found in different sources (e.g., Boissay et al. 2016). However, for highly obscured AGNs, the scattered mechanism is preferred since either blurred reflection or warm Comptonization from the accretion disk will be significantly attenuated by the obscuring materials.

We show the ${\text{}}{f}_{\mathrm{exs}}$, which represents the relative normalization of the soft component with respect to the intrinsic power law, as a function of ${N}_{{\rm{H}}}$ in Figure 12. We find a significant anticorrelation between the two parameters: Spearman's ρ = −0.66, p-value =$\ll 0.001$. The mean ${\text{}}{f}_{\mathrm{exs}}$ is 3.7% for the total sample. The scattered fraction decreases from 6.1% to 4.3% to 1.3% for ${N}_{{\rm{H}}}$ <5 × ${10}^{23}\ {\mathrm{cm}}^{-2}$, 5 × ${10}^{23}\ {\mathrm{cm}}^{-2}$ < ${N}_{{\rm{H}}}$ < 1.5 × ${10}^{24}\ {\mathrm{cm}}^{-2}$, and ${N}_{{\rm{H}}}$ > 1.5 × ${10}^{24}\ {\mathrm{cm}}^{-2}$, respectively.

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To verify that this anticorrelation is not caused by the parameter degeneracy that sources with a large soft-excess fraction and a high ${N}_{{\rm{H}}}$ might be misclassified as low ${N}_{{\rm{H}}}$, low ${\text{}}{f}_{\mathrm{exs}}$, and low intrinsic luminosity sources owing to the low S/N, we assume that CT AGNs have exactly the same ${\text{}}{f}_{\mathrm{exs}}$ as CN AGNs (here we simply choose ${\text{}}{f}_{\mathrm{exs}}$ = 5%). Then, we generate fake spectra using model D for a sample of simulated sources, which have redshifts and ${N}_{{\rm{H}}}$ uniformly distributed between z = 0 and 4 with an interval of Δz = 0.5 and $23\ {\mathrm{cm}}^{-2}\,\lt \mathrm{log}\,{N}_{{\rm{H}}}\lt 25\ {\mathrm{cm}}^{-2}$ with an interval of ${\rm{\Delta }}\mathrm{log}\,{N}_{{\rm{H}}}=0.25$. For each source we simulate 100 spectra with 0.5–7 keV counts of ∼50, 100, and 150, respectively. We then fit the fake spectra and perform a Spearman rank correlation test for the output ${N}_{{\rm{H}}}$ and ${\text{}}{f}_{\mathrm{exs}}$. The Spearman's ρ ≈ 0.1 (insensitive to the simulated spectral counts), indicating no apparent correlation; thus, the observed anticorrelation is intrinsic.

By assuming that the second power law originates from the backscattered continuum, Brightman & Ueda (2012) found that the scattered fraction depends on the opening angle, i.e., sources with small opening angles have lower ${\text{}}{f}_{\mathrm{exs}}$. Moreover, Ueda et al. (2015) and Kawamuro et al. (2016) found that the ratios of [O iii] and [O iv] to hard X-ray luminosity are lower in low scattering fraction sources, inferring that the low-${\text{}}{f}_{\mathrm{exs}}$ AGNs are buried in small opening angle torus. Therefore, the anticorrelation between ${\text{}}{f}_{\mathrm{exs}}$ and ${N}_{{\rm{H}}}$ indicates that high-${N}_{{\rm{H}}}$ sources might preferentially reside in high-CF tori. This is also consistent with the results from Brightman & Ueda (2012) and Lanzuisi et al. (2015), who found a similar anticorrelation between ${\text{}}{f}_{\mathrm{exs}}$ and ${N}_{{\rm{H}}}$ and explained it as highly obscured AGNs being also heavily geometrically obscured (but see the discussion in the next section).

Unlike local CT AGNs in which the prominent neutral Fe lines and reflection hump (hereafter the reprocessed components) are prevalently detected in the high-quality hard X-ray spectra and can serve as unambiguous signatures of being CT (e.g., Tanimoto et al. 2018; Marchesi et al. 2019; Zhao et al. 2019), 41% of our highly obscured sources and 38% of the CT candidates have negligible ${\text{}}{f}_{\mathrm{ref}}$, even for some high-count sources or CT AGNs. (see Table 3). It should be noted that the low detection rate of reprocessed components is not in conflict with their highly obscured nature. The poor S/N for low-count sources and the narrow Chandra spectral coverage make it challenging to detect narrow iron lines and to distinguish the transmission and reflection components, which may cause a significant overestimation of the fractions.

But we note that it is possible that highly obscured AGNs may have weak reprocessed components if the global CFs of high-${N}_{{\rm{H}}}$ materials are low or the central AGN is geometrically fully buried by CT materials such that even the reflected photons cannot escape (e.g., Brightman et al. 2014). To distinguish different scenarios, we use the Borus model (Baloković et al. 2018),²³ which allows CF to vary freely, to fit the spectra for all sources with counts >200. This time we treat photon index as a free parameter in order to make a direct comparison between the two models. The distribution of the derived ${\mathrm{CF}}_{\mathrm{tor}}$, which is defined as the cosine value of the opening angle θ_tor measured from the symmetry axis toward the equatorial plane, is shown in Figure 13(a). We also show the comparisons of the main spectral fitting parameters with those obtained from the MYTorus model and find good consistency.

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For sources with negligible reprocessed components in the MYTorus model (${f}_{\mathrm{ref}}\ll 1$), the Borus results are consistent with the weak torus scenario (${\mathrm{CF}}_{\mathrm{tor}}\approx 0.1$), which might suggest that their heavy obscuration is simply an LOS effect (i.e., some high column density clouds on various scales along our sight line may obscure the compact X-ray emitter), without the necessity to invoke a strongly buried nuclear environment. In contrast, sources with ${\text{}}{f}_{\mathrm{ref}}$ = 1.0 are mostly best fitted by a highly covered model ($0.8\lt {\mathrm{CF}}_{\mathrm{tor}}\lt 1.0$) that has torus covering angles (90° − θ_tor) between 65° and 90°. We emphasize that although the LOS and the global torus ${N}_{{\rm{H}}}$ are linked in the spectral fitting, those fully buried sources do not need to be covered by very high ${N}_{{\rm{H}}}$ materials in all directions, since the best-fit ${N}_{{\rm{H}}}$ will more likely converge to the values determined by the LOS component because the photoelectric absorption is a much stronger spectral feature. Moreover, we find that the average ${\mathrm{CF}}_{\mathrm{tor}}$ for ${\text{}}{f}_{\mathrm{exs}}\lt 0.05$ sources is 0.60, while for ${\text{}}{f}_{\mathrm{exs}}\gt 0.05$ sources the $\overline{{\mathrm{CF}}_{\mathrm{tor}}}$ is 0.32. This might provide evidence that the lower soft-excess fraction in some sources is caused by a geometrically more buried structure (see Section 5.2).

However, we caution here that the current S/N and spectral coverage do not allow us to constrain the CF from X-ray spectral analysis. If we fix ${\mathrm{CF}}_{\mathrm{tor}}$ at 0.1 for sources that are best fitted by a fully buried model, we may still obtain acceptable fits with Cstat values only slightly increasing (an example is shown in Figure 13(b)). Conclusively disentangling several scenarios is beyond the scope of this work, and multiwavelength data are needed to further shed light on the nature of the obscuring materials (J. Y. Li et al. 2019, in preparation).

To consider the uncertainty of the spectral fitting, we perform a resampling procedure to the best-fit ${N}_{{\rm{H}}}$. Given the asymmetric errors, we assume that the errors on ${N}_{{\rm{H}}}$ obey the "half-Gaussian" distribution. We first generate 1000 ${N}_{{\rm{H}}}$ for each source with the σ of the half-Gaussian distribution equal to the lower 1σ error, and then we generate another 1000 ${N}_{{\rm{H}}}$ with σ equal to the upper 1σ error. The mean value μ is set to the best-fit ${N}_{{\rm{H}}}$. The resampled ${N}_{{\rm{H}}}$ distribution (calculated by averaging the 2000 resampled distributions and also corrected for the sky coverage effect; see Section 6.2) is shown in the shaded region of Figure 18. It has an extended tail down to ${N}_{{\rm{H}}}\lt {10}^{23}\ {\mathrm{cm}}^{-2}$, which contains only 4.0% of the resampled data, suggesting that even if considering the spectral fitting errors, most of our sources are still consistent with being highly obscured.

Due to Chandra's instrument features, the point-source PSF size increases and the effective exposure time dramatically decreases toward large off-axis angles. The sensitivities of detecting faint sources reduce prominently at the outskirt of the field, leading to small sky coverage while the observed flux is low. Therefore, a correction must be made.

We calculate the energy-flux-to-count-rate conversion factor (ECF) by assuming the soft-excess model (model D) with Γ, ${\text{}}{f}_{\mathrm{ref}}$, and ${\text{}}{f}_{\mathrm{exs}}$ fixed at 1.8%, 1.0%, and 1%, respectively. Combining ECF and the exposure map, we build a sensitivity map that represents the flux limits corresponding to the 20-count cut. Using this map, we calculate the sky coverage as a function of observed 0.5–7 keV flux, ${N}_{{\rm{H}}}$, and redshift as shown in Figure 14. Then, we measure the sky coverage for each source based on their observed flux, ${N}_{{\rm{H}}}$, and redshift obtained from spectral fitting. We define ω_area as the ratio between the source sky coverage and the maximum sky coverage of the two Chandra surveys (484.2 and 447.5 arcmin² in the CDF-S and the CDF-N, respectively). To correct for the sky coverage effect, we simply weigh each source by 1/ω_area while resampling the observed ${N}_{{\rm{H}}}$ distribution. To avoid extremely large weights, we apply another cut for the 22 sources with sky coverage less than 100 arcmin², setting their weighting factors to the median factor 1.3, since these sources lie close to the count cut and we cannot rule out the possibility that their extremely small sky coverage may be a result of inappropriate spectral modeling.

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Considering the measurement error of net counts, sources with intrinsically low counts may exceed our count cut (20 photons), and sources with high counts may be missed in our sample. Since the numbers of faint and bright sources are not equal, this error leads to the Eddington bias. To correct for this bias, we convolve the cumulative count distribution for AGNs in the two Chandra survey catalogs with the count errors using Poisson sampling in order to obtain the intrinsic count distribution. We follow the "pseudo-deconvolved" method proposed in L17 (see Section 5.1.3 of L17 for details), by shifting the observed curve leftward with the value equal to the displacement between the observed and convolved curves. We then obtain the deconvolved curve that represents the intrinsic cumulative count distribution.

The difference between the deconvolved and the observed curves can be treated as the number of sources missed or misincluded (depending on the adopted count cut) in our sample. As shown in Figure 15, at a 20-photon count cut, we miss 11.6 and 6.5 sources in the CDF-S and CDF-N, respectively. This is due to the fact that although sources with counts <20 may outnumber the bright sources, such faint sources are not detected in the source catalogs. By controlling the sample size (i.e., the cumulative source number in the y-axes) to be the same, we obtain the effective net counts of 22.4 and 21.1 for the CDF-S and CDF-N, respectively. Thus, the 20-photon count cut will be replaced by the new effective count cut while performing other corrections.

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So far we have only obtained the corrected observed ${N}_{{\rm{H}}}$ distribution for our sample. In order to derive the intrinsic ${N}_{{\rm{H}}}$ distribution for the highly obscured AGN population, we should also consider the undetectable part of the Chandra survey, which is known as the Malmquist bias that sources with lower luminosities will be missed by a flux-limited survey. More specifically, given the count rate limit of the survey (the effective count cut in our situation), whether a source is detectable or not depends on its redshift, spectral shape, column density, intrinsic luminosity, and position in the field, since large off-axis angles lead to significantly lower sensitivities.

To quantify this bias, we generate fake X-ray spectra to find out the minimum ${L}_{{\rm{X}}}$ (L_cut, corresponding to our effective count cut) in different redshift and ${N}_{{\rm{H}}}$ bins, which can be detected at a specific position in the field. We divide the two CDFs into 10 annuli, with each corresponding to an interval of 1'. While running XSPEC simulations, we use the averaged exposure time, rmf file, and arf file calculated from all sources in each annulus. Model D is used in the simulation with the parameters set the same as in Section 6.2.

We show some of the detectable boundary curves in Figure 16. We define the detectable region as the area between the boundary curve and the maximum log ${L}_{{\rm{X}}}$ = 46 $\mathrm{erg}\ {{\rm{s}}}^{-1}$. As we can clearly see, sources with high ${N}_{{\rm{H}}}$ and large off-axis angles have significantly small detectable regions. We follow Equation (3) in T06:

$$\begin{eqnarray}\begin{array}{rcl}F({N}_{{\rm{H}}}){{dN}}_{{\rm{H}}} & = & f({N}_{{\rm{H}}}){{dN}}_{{\rm{H}}}{\int }_{0}^{{z}_{\max }}\displaystyle \frac{{dV}}{{dz}}{dz}\\ & & \times {\int }_{{L}_{\mathrm{cut}({N}_{{\rm{H}}},z)}}^{{L}_{\max }}N({L}_{X},z)\,{{dL}}_{X},\end{array}\end{eqnarray} \tag{ 2 }$$

where f(${N}_{{\rm{H}}}$) represents the intrinsic ${N}_{{\rm{H}}}$ distribution, F(${N}_{{\rm{H}}}$) represents the observed ${N}_{{\rm{H}}}$ distribution, and N(${L}_{{\rm{X}}}$, z) is the X-ray LF. The detectable source fraction for each ${N}_{{\rm{H}}}$ bin relative to the $\mathrm{log}\,{N}_{{\rm{H}}}$ = 23 cm⁻² bin in a given redshift interval can be calculated as

$$\begin{eqnarray}&&f=\displaystyle \frac{{\int }_{{L}_{\mathrm{cut}({N}_{{\rm{H}}},z)}}^{{L}_{\max }}N({L}_{{\rm{X}}},z)\,{{dL}}_{{\rm{X}}}}{{\int }_{{L}_{\mathrm{cut}(23,{\rm{z}})}}^{{L}_{\max }}N({L}_{{\rm{X}}},z)\,{{dL}}_{{\rm{X}}}}.\end{eqnarray} \tag{ 3 }$$

Thus, for the bin we choose

$$\begin{eqnarray}f=\left\{\begin{array}{lr}1, & \mathrm{if}\ 23.0\ {\mathrm{cm}}^{-2}\leqslant \mathrm{log}\,{N}_{{\rm{H}}}\lt 23.25\ {\mathrm{cm}}^{-2},\\ \lt 1, & \mathrm{if}\ \mathrm{log}\,{N}_{{\rm{H}}}\geqslant 23.25\ {\mathrm{cm}}^{-2}.\end{array}\right.\end{eqnarray} \tag{ 4 }$$

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Note that T06 derived the above Equation (2) by assuming that f(N_H, L_X, z), which represents the possibility of detecting a source with column density in the range of ${N}_{{\rm{H}}}$ to ${N}_{{\rm{H}}}$ + d ${N}_{{\rm{H}}}$ for a given L_X and z, varies slowly as ${L}_{{\rm{X}}}$ and z change. There is evidence that the obscured fraction of AGNs evolves with both luminosity and redshift. However, we do not consider such a complicated evolution and simply extract f(N_H, L_X, z) from the integral. Therefore, by multiplying 1/f by the observed ${N}_{{\rm{H}}}$ distribution F(N_H) dN_H in each bin, sources with different ${N}_{{\rm{H}}}$ are corrected to cover the same observable space with respect to the $\mathrm{log}\,{N}_{{\rm{H}}}$ = 23–23.25 cm⁻² bin, and the Malmquist bias is corrected. To avoid the case of the corrected distribution being dominated by sources with extremely small detectable fractions, we apply a cut that while f < 0.25 the weighting factor 1/f will be simply set to 1/0.25. Adopting a lower (0.2) or higher (0.3) detectable fraction cut will lead to ∼2% systematic difference of the final intrinsic CT/highly obscured fraction.

Apparently, the correction depends on both spectral models and the detailed shape of X-ray LFs. Therefore, we compare the results using different X-ray LFs. We combine the z < 3 LF from Miyaji et al. (2015) and z > 3 LF from Georgakakis et al. (2015) as our first LF model, which is the same as in L17. As for the second model, we adopt Aird et al. (2015) LFs for obscured AGNs (22 cm⁻² < ${N}_{{\rm{H}}}$ < 24 cm⁻²). We do not use their LFs for CT AGNs, since they obtained a systematically lower CT fraction than other works and our result in Section 4.6 prefers higher CT fractions (but see the discussion in Section 6.4 of Aird et al. 2015). The third LF we used is from Ueda et al. (2014). The three LF models are shown in Figure 17. We note that the adopted obscured AGN LFs may not be suitable for CT AGNs, but an extensive exploration of this issue is beyond the scope of this work.

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Figure 18 shows the intrinsic ${N}_{{\rm{H}}}$ distributions in five redshift bins after correcting for all the aforementioned biases. We define the intrinsic CT/highly obscured source fraction ${f}_{\mathrm{CH}}$ as

$$\begin{eqnarray}&&{f}_{\mathrm{CH}}=\displaystyle \frac{N(\mathrm{log}\,{N}_{{\rm{H}}}\geqslant 24)}{N(\mathrm{log}\,{N}_{{\rm{H}}}\geqslant 23)}\end{eqnarray} \tag{ 5 }$$

and list its evolution with redshift in Table 6.

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Table 6.  Evolution of the Observed (f_obs) and Corrected Intrinsic (f_int) CT/Highly Obscured Fraction with Redshift

z	${f}_{\mathrm{obs},\mathrm{CH}}$	${f}_{\mathrm{int},\mathrm{CH}}$	$\overline{\mathrm{log}\,{L}_{{\rm{X}}}}$ ($\mathrm{erg}\ {{\rm{s}}}^{-1}$)
(1)	(2)	(3)	(4)
Total Sample
0.0–0.8	0.32	0.57 ± 0.07	42.9
0.8–1.6	0.24	0.50 ± 0.05	43.5
1.6–2.4	0.24	0.50 ± 0.05	43.7
2.4–3.0	0.29	0.54 ± 0.06	43.9
3.0–5.0	0.18	0.50 ± 0.09	44.1
0.0–5.0	0.25	0.52 ± 0.06	43.6
Subsample
1.0–2.0	0.25	0.50 ± 0.05	43.7
2.0–3.0	0.26	0.51 ± 0.05	43.8

Note. The values listed are calculated by averaging the results from the three LF models. Top: results for the total sample. Bottom: results for two subsamples with $43.0\ \mathrm{erg}\ {{\rm{s}}}^{-1}\lt \mathrm{log}\,{L}_{{\rm{X}}}\lt 44.5\ \mathrm{erg}\ {{\rm{s}}}^{-1}$ at z = 1.0–2.0 and at z = 2.0–3.0, which have similar ${L}_{{\rm{X}}}$ distributions (see Figure 9).

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As we can see, different LFs give consistent results. It is obvious that the corrected ${N}_{{\rm{H}}}$ distributions are quite different from the observed ones, especially in the CT regime. This highlights the importance of carefully correcting for the observational biases in order to uncover the underlying distributions. Though the intrinsic ${N}_{{\rm{H}}}$ distributions are different between high-redshift and low-redshift bins, the CT/highly obscured fraction is in accordance with no evident redshift evolution given the uncertainties (i.e., f_CH ≈ 0.52 for all redshift bins; see Table 6). Note that Buchner et al. (2015) also claimed that the CT fraction is constant across cosmic time, but the CT fraction in their work is defined as the number of CT AGNs to the total AGN population. To avoid possible biases induced by the fact that different redshift bins are sensitive to different luminosity ranges, we also select two subsamples with $43.0\ \mathrm{erg}\ {{\rm{s}}}^{-1}\lt \mathrm{log}\,{L}_{{\rm{X}}}\lt 44.5\ \mathrm{erg}\ {{\rm{s}}}^{-1}$ at z = 1–2 and z = 2–3, respectively, which have similar ${L}_{{\rm{X}}}$ distributions (see Figure 9), and perform the same corrections. The results are listed in Table 6 and are still consistent with no redshift evolution.

Vito et al. (2018) presented the ${N}_{{\rm{H}}}$ distribution for the high-redshift (z > 3) AGNs also in the CDFs. Their analyses were based on modeling the X-ray spectra using the wabs ×zwabs × zpow model for the z > 3 sources by carefully taking into account the photometric redshift errors. The comparison of the two works is shown in the z = 3.0–5.0 bin in Figure 18. Our result shows a higher CT fraction, which might be ascribed to the different redshift values adopted in the two works, as well as the large weighting factors while correcting for the undetectable part (see Section 6.4).

However, we note that all these aforementioned corrections are based on sources we do observe, including the correction of the undetectable part in a given ${N}_{{\rm{H}}}$ bin for which we have to make assumptions regarding the unknown AGN population. The series of corrections can be simply understood as multiplying an evolved weighting factor by the observed distribution, which has several limitations. First of all, as mentioned before, different redshift bins sample different luminosity ranges. Several authors have found that the obscured fraction changes with luminosity (Lusso et al. 2013; Buchner et al. 2015); thus, the ${N}_{{\rm{H}}}$ distribution might be luminosity dependent. However, in Equation (2) we ignore this effect and only report the result for the full luminosity range, owing to the limited CT AGN amount detected in faint-L_X bins. Second, for the extremely buried sources (${N}_{{\rm{H}}}$ > ${10}^{25}\ {\mathrm{cm}}^{-2}$), or intrinsically very faint sources, the corrections cannot be made since there are hardly any such sources observed owing to the limit of the survey and the restrictions imposed by our sample selection count cut. Therefore, the part of the ${N}_{{\rm{H}}}$ distribution of sources that are under the detection limit is actually still missed in our final result. It is plausible that ${N}_{{\rm{H}}}$ > ${10}^{25}\ {\mathrm{cm}}^{-2}$ sources may contribute significantly to the heavily obscured AGN population (Risaliti et al. 1999), but such sources are missed in our sample, and the corrections in this part are beyond our attainment. Therefore, the ${N}_{{\rm{H}}}$ distribution at the highest-${N}_{{\rm{H}}}$ bin displayed in Figure 18 is highly uncertain and incomplete, so that the CT/highly obscured fractions in Table 6 should be better treated as lower limits.

We use the same spectral fitting models as described in Section 3.1 to perform spectral variability analyses, except for removing the constants ${\text{}}{f}_{\mathrm{ref}}$ and ${\text{}}{f}_{\mathrm{exs}}$ in the model and untying the normalizations of the intrinsic power law and other spectral components. We simultaneously fit the spectra in each epoch using the best-fit model for each source obtained in Section 4, but this time we allow norm_ref to vary freely to search for possible variation in the scattered continuum. Considering the relatively low counts in each epoch, the fitting strategies are adopted as follows:

• 1.  
  The uncertainties on Γ can cause large degeneracies if we set it free in all epochs. Given that some sources may have large variability in Γ, we first let Γ in each epoch vary freely and obtain C_free. Then, for the first two adjacent epochs (i.e., epoch1 and epoch2), we link Γ and measure the ΔC with respect to C_free. If ΔC > 3.84 (for ${\rm{\Delta }}\mathrm{dof}=1$), Γ is considered to be different between the two epochs at >95% confidence level, and we will set Γ free; otherwise, we will link Γ. Then for the second epoch pair (i.e., epoch2 and epoch3), we link their Γ and compare the Cstat value with the last step. After traversing all the epoch pairs, if no variability is detected, Γ is linked together.
• 2.  
  The reflection and soft-excess components are often considered to be produced in large-scale clouds, e.g., torus, for highly obscured AGNs. Since the time span in the rest-frame is (1+z) times less than in the observed frame and the typical redshift of our sample is relatively high, it is reasonable to assume that in such a short timescale the large-scale components may not vary dramatically. To better constrain the normalization of the intrinsic power law, we tie the normalizations of the reflection component and the soft-excess component in each epoch, respectively, unless setting them free, leading to ΔC > 3.84. Finally, only one source shows a significant change in the reflection component, and the soft-excess fluxes remain constant for all sources.
• 3.  
  For those sources with a weak and constant reflection component confirmed in the last step, we delete the MYTS and MYTL components from the model and only use MYTZ to fit the spectra, in order to reduce the number of free parameters. This procedure may break the self-consistency of the MYTorus model to some extent but will not influence the variability analysis, since we only remove a small constant component in each epoch, which has very little influence on the calculated χ².
• 4.  
  The ${N}_{{\rm{H}}}$ and normalization of the intrinsic power law always vary freely.

The simultaneous fitting yields the best-fit model parameters Γ, ${{norm}}_{\mathrm{ref}}$, and norm_scat. Then, we fit the spectra in each epoch with Γ_i, norm_ref,i, and norm_scat,i fixed at the best-fit value obtained in the simultaneous fitting to obtain the ${N}_{{\rm{H}}}$, observed flux, intrinsic flux, and corresponding errors in each epoch. The spectral fitting parameters of each epoch are listed in Table 8. No significant photon index variability is detected for all sources.

Table 8.  Multiepoch Spectral Fitting Results for the Variability Sample

XID	Γ	${N}_{{\rm{H}}1}$ (a)	NH2	NH3	${L}_{{\rm{X}}1}$ (b)	${L}_{{\rm{X}}2}$	${L}_{{\rm{X}}3}$	flux1 (c)	flux2	flux3
CDF-S
63	1.54	${0.06}_{-0.01}^{+0.01}$	${0.14}_{-0.02}^{+0.02}$	${0.19}_{-0.03}^{+0.04}$	0.47	0.66	0.41	${16.0}_{-1.8}^{+1.9}$	${16.6}_{-1.5}^{+1.5}$	${8.8}_{-0.9}^{+0.9}$
73	1.81	${0.64}_{-0.07}^{+0.08}$	${0.56}_{-0.11}^{+0.11}$	${0.67}_{-0.07}^{+0.07}$	3.37	1.87	3.78	${2.7}_{-0.3}^{+0.3}$	${1.7}_{-0.3}^{+0.3}$	${2.9}_{-0.3}^{+0.3}$
91	2.08	${0.16}_{-0.03}^{+0.03}$	${0.12}_{-0.03}^{+0.03}$	${0.14}_{-0.03}^{+0.03}$	1.29	1.03	1.25	${2.3}_{-0.3}^{+0.3}$	${2.1}_{-0.3}^{+0.3}$	${2.4}_{-0.3}^{+0.3}$
240	1.40	${1.80}_{-0.15}^{+0.17}$	${1.41}_{-0.15}^{+0.17}$	${1.34}_{-0.11}^{+0.12}$	1.76	1.21	1.04	${3.7}_{-0.4}^{+0.5}$	${3.4}_{-0.5}^{+0.5}$	${3.3}_{-0.3}^{+0.3}$
249	1.51	${0.50}_{-0.12}^{+0.14}$	${0.28}_{-0.06}^{+0.08}$	${0.14}_{-0.01}^{+0.02}$	0.25	0.16	0.18	${2.1}_{-0.3}^{+0.3}$	${2.4}_{-0.3}^{+0.3}$	${4.2}_{-0.3}^{+0.3}$
328	2.12	${1.91}_{-0.12}^{-1.02}$	${1.31}_{-0.10}^{+0.11}$	${1.57}_{-0.06}^{+0.06}$	4.37	1.69	4.94	${4.2}_{-0.4}^{+0.4}$	${3.4}_{-0.4}^{+0.5}$	${6.0}_{-0.4}^{+0.4}$
367	2.07	${0.42}_{-0.02}^{+0.03}$	${0.38}_{-0.02}^{+0.02}$	${0.47}_{-0.04}^{+0.06}$	0.14	0.13	0.07	${3.7}_{-0.3}^{+0.2}$	${3.8}_{-0.3}^{+0.3}$	${2.2}_{-0.2}^{+0.2}$
399	1.40	${0.24}_{-0.02}^{+0.02}$	${0.23}_{-0.03}^{+0.03}$	${0.32}_{-0.02}^{+0.02}$	0.38	0.13	0.32	${2.9}_{-0.3}^{+0.3}$	${1.5}_{-0.3}^{+0.3}$	${2.6}_{-0.2}^{+0.2}$
551	1.84	${1.56}_{-0.10}^{+0.11}$	${1.62}_{-0.14}^{+0.15}$	${1.54}_{-0.09}^{+0.09}$	4.02	2.31	3.89	${2.0}_{-0.2}^{+0.2}$	${1.5}_{-0.2}^{+0.2}$	${1.9}_{-0.2}^{+0.2}$
621	2.11	${0.18}_{-0.02}^{+0.02}$	${0.15}_{-0.02}^{+0.02}$	${0.17}_{-0.02}^{+0.02}$	0.40	0.32	0.30	${2.3}_{-0.2}^{+0.2}$	${2.0}_{-0.3}^{+0.3}$	${1.8}_{-0.2}^{+0.2}$
658	1.67	${0.15}_{-0.02}^{+0.02}$	${0.18}_{-0.03}^{+0.03}$	${0.15}_{-0.02}^{+0.02}$	0.54	0.57	0.52	${2.0}_{-0.2}^{+0.2}$	${2.0}_{-0.2}^{+0.2}$	${2.0}_{-0.2}^{+0.2}$
733	1.84	${0.21}_{-0.03}^{+0.04}$	${0.17}_{-0.03}^{+0.03}$	${0.12}_{-0.02}^{+0.02}$	0.94	0.96	0.92	${1.8}_{-0.2}^{+0.2}$	${2.0}_{-0.2}^{+0.2}$	${2.2}_{-0.2}^{+0.2}$
752	2.24	${0.15}_{-0.02}^{+0.02}$	${0.15}_{-0.02}^{+0.02}$	${0.11}_{-0.01}^{+0.01}$	0.28	0.33	0.25	${4.6}_{-0.7}^{+0.7}$	${5.4}_{-0.7}^{+0.7}$	${5.0}_{-0.5}^{+0.5}$
785	1.97	${0.54}_{-0.05}^{+0.05}$	${0.56}_{-0.04}^{+0.05}$	${0.53}_{-0.02}^{+0.03}$	4.73	5.97	9.26	${16.2}_{-2.7}^{+2.8}$	${18.8}_{-2.1}^{+2.1}$	${29.7}_{-1.8}^{+1.8}$
818	1.97	${0.14}_{-0.02}^{+0.02}$	${0.11}_{-0.03}^{+0.03}$	${0.19}_{-0.02}^{+0.03}$	0.61	0.46	0.93	${1.1}_{-0.1}^{+0.1}$	${0.9}_{-0.1}^{+0.1}$	${1.5}_{-0.1}^{+0.1}$
840	1.60	${0.58}_{-0.04}^{+0.05}$	${0.59}_{-0.04}^{+0.04}$	${0.40}_{-0.02}^{+0.02}$	0.53	0.49	0.59	${3.9}_{-0.3}^{+0.3}$	${3.8}_{-0.3}^{+0.3}$	${5.1}_{-0.3}^{+0.3}$
846	1.56	${0.14}_{-0.02}^{+0.03}$	${0.11}_{-0.02}^{+0.03}$	${0.15}_{-0.03}^{+0.03}$	0.84	0.86	0.90	${2.1}_{-0.2}^{+0.2}$	${2.2}_{-0.2}^{+0.2}$	${2.2}_{-0.2}^{+0.2}$
CDF-N
66	1.92	${0.34}_{-0.06}^{+0.09}$	${0.24}_{-0.02}^{+0.02}$	${0.29}_{-0.03}^{+0.03}$	1.19	1.49	1.10	${7.6}_{-1.0}^{+1.0}$	${12.7}_{-0.7}^{+0.7}$	${8.1}_{-0.7}^{+0.7}$
143	1.46	${0.15}_{-0.02}^{+0.02}$	${0.15}_{-0.01}^{+0.01}$	${0.11}_{-0.01}^{+0.01}$	2.07	2.79	2.42	${10.2}_{-0.9}^{+0.9}$	${13.6}_{-0.7}^{+0.6}$	${13.1}_{-0.7}^{+0.7}$

XID	Γ	${N}_{{\rm{H}}1}$	${N}_{{\rm{H}}2}$	NH3	NH4	${L}_{{\rm{X}}1}$	${L}_{{\rm{X}}2}$	${L}_{{\rm{X}}3}$	${L}_{{\rm{X}}4}$	flux1	flux2	flux3	flux4
CDF-S
49	1.66	${0.19}_{-0.02}^{+0.02}$	${0.15}_{-0.03}^{+0.04}$	${0.30}_{-0.03}^{+0.04}$	${0.16}_{-0.02}^{+0.02}$	3.90	1.85	2.69	3.44	${8.3}_{-0.6}^{+0.6}$	${4.4}_{-0.6}^{+0.6}$	${4.7}_{-0.4}^{+0.4}$	${7.8}_{-0.4}^{+0.4}$
81	1.83	${0.17}_{-0.02}^{+0.02}$	${0.14}_{-0.02}^{+0.02}$	${0.17}_{-0.02}^{+0.02}$	${0.19}_{-0.02}^{+0.02}$	7.32	9.21	5.15	6.37	${7.8}_{-0.4}^{+0.5}$	${10.4}_{-0.6}^{+0.6}$	${5.4}_{-0.3}^{+0.4}$	${6.5}_{-0.3}^{+0.3}$
98	0.98	${0.18}_{-0.02}^{+0.03}$	${0.17}_{-0.02}^{+0.02}$	${0.18}_{-0.02}^{+0.02}$	${0.17}_{-0.01}^{+0.01}$	1.06	1.18	1.29	1.11	${10.3}_{-0.8}^{+0.8}$	${11.4}_{-1.0}^{+1.0}$	${12.4}_{-0.8}^{+0.8}$	${11.1}_{-0.6}^{+0.6}$
186	1.79	${0.22}_{-0.02}^{+0.02}$	${0.27}_{-0.03}^{+0.03}$	${0.22}_{-0.02}^{+0.02}$	${0.23}_{-0.02}^{+0.02}$	3.92	4.68	3.20	4.23	${5.2}_{-0.3}^{+0.4}$	${5.7}_{-0.5}^{+0.4}$	${4.3}_{-0.3}^{+0.3}$	${5.6}_{-0.3}^{+0.2}$
214	1.87	${0.26}_{-0.03}^{+0.03}$	${0.35}_{-0.04}^{+0.05}$	${0.30}_{-0.03}^{+0.03}$	${0.30}_{-0.03}^{+0.03}$	9.25	9.36	8.61	8.81	${6.3}_{-0.4}^{+0.4}$	${5.6}_{-0.4}^{+0.5}$	${5.5}_{-0.4}^{+0.4}$	${5.7}_{-0.3}^{+0.2}$
236	1.62	${0.15}_{-0.02}^{+0.03}$	${0.20}_{-0.03}^{+0.03}$	${0.22}_{-0.03}^{+0.04}$	${0.21}_{-0.03}^{+0.03}$	1.68	2.16	1.61	1.65	${3.6}_{-0.3}^{+0.3}$	${4.2}_{-0.5}^{+0.5}$	${3.0}_{-0.3}^{+0.3}$	${3.1}_{-0.2}^{+0.2}$
308	1.69	${0.21}_{-0.01}^{+0.02}$	${0.24}_{-0.02}^{+0.03}$	${0.21}_{-0.01}^{+0.02}$	${0.22}_{-0.02}^{+0.02}$	1.45	0.90	1.11	0.76	${11.9}_{-0.7}^{+0.5}$	${6.8}_{-0.6}^{+0.6}$	${9.2}_{-0.5}^{+0.5}$	${6.2}_{-0.3}^{+0.3}$
458	1.83	${0.19}_{-0.02}^{+0.02}$	${0.25}_{-0.02}^{+0.02}$	${0.31}_{-0.02}^{+0.02}$	${0.27}_{-0.02}^{+0.02}$	0.96	1.75	2.11	1.57	${3.8}_{-0.4}^{+0.4}$	${5.7}_{-0.5}^{+0.5}$	${4.7}_{-0.3}^{+0.3}$	${3.7}_{-0.2}^{+0.2}$
746	1.68	${0.41}_{-0.04}^{+0.04}$	${0.46}_{-0.06}^{+0.06}$	${0.45}_{-0.04}^{+0.04}$	${0.55}_{-0.03}^{+0.04}$	5.17	4.75	5.81	6.93	${4.9}_{-0.3}^{+0.2}$	${4.2}_{-0.4}^{+0.4}$	${5.2}_{-0.4}^{+0.3}$	${5.4}_{-0.3}^{+0.3}$
876	1.87	${0.15}_{-0.02}^{+0.02}$	${0.16}_{-0.02}^{+0.02}$	${0.15}_{-0.02}^{+0.02}$	${0.16}_{-0.02}^{+0.02}$	7.18	7.98	8.99	8.84	${6.9}_{-0.4}^{+0.4}$	${7.6}_{-0.5}^{+0.5}$	${8.7}_{-0.4}^{+0.4}$	${8.3}_{-0.3}^{+0.3}$
933	1.71	${0.31}_{-0.03}^{+0.04}$	${0.23}_{-0.03}^{+0.03}$	${0.25}_{-0.02}^{+0.02}$	${0.23}_{-0.02}^{+0.02}$	1.81	1.46	1.77	1.29	${5.6}_{-0.6}^{+0.6}$	${5.3}_{-0.5}^{+0.5}$	${6.2}_{-0.4}^{+0.4}$	${4.7}_{-0.3}^{+0.3}$
760	1.49	${0.81}_{-0.10}^{+0.12}$	${0.67}_{-0.10}^{+0.10}$	${0.68}_{-0.06}^{+0.06}$	${0.59}_{-0.05}^{+0.05}$	9.18	6.51	8.99	7.33	${6.5}_{-0.8}^{+0.9}$	${5.4}_{-0.8}^{+0.9}$	${7.0}_{-0.6}^{+0.6}$	${6.5}_{-0.4}^{+0.4}$

Note. (a) in units of ${10}^{24}\ {\mathrm{cm}}^{-2}$. (b) $2\mbox{--}10\,\mathrm{keV}$ intrinsic luminosity, in units of ${10}^{44}\ \mathrm{erg}\ {{\rm{s}}}^{-1}$. (c) observed 0.5–7 keV flux, in units of ${10}^{-15}\ \mathrm{erg}\ {\mathrm{cm}}^{-2}\ {{\rm{s}}}^{-1}$.

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We identify ${L}_{{\rm{X}}}$, ${N}_{{\rm{H}}}$, and observed flux-variable sources based on the classical χ² test. For illustration, we explain how we determine ${L}_{{\rm{X}}}$-variable sources. First, we use the cflux model in XSPEC to calculate the observed and absorption-corrected 0.5–7 keV fluxes and corresponding errors. Then, we derive the intrinsic rest-frame 0.5–7 keV luminosity using equation ${L}_{0.5-7\mathrm{keV}}=4\pi {d}_{L}^{2}{f}_{0.5-7\mathrm{keV},\mathrm{int}}\,{(1+z)}^{{{\rm{\Gamma }}}_{\mathrm{int}}-2}$, where the "int" subscript represents the intrinsic value. Since the photon index and redshift are all fixed during spectral fitting of each single epoch, the error on ${L}_{0.5-7\mathrm{keV}}$ is totally attributed to the error of the intrinsic flux ${f}_{0.5-7\mathrm{keV},\mathrm{int}}$. Thus, the ${\chi }^{2}$ of the rest-frame ${L}_{0.5-7\mathrm{keV}}$ actually equals the χ² of the observed-frame ${f}_{0.5-7\mathrm{keV},\mathrm{int}}$. If a source does not vary, i.e., the intrinsic luminosity (or ${N}_{{\rm{H}}}$, flux) is constant, assuming that the errors obey the Gaussian distribution, the χ² value calculated from

$$\begin{eqnarray}&&{\chi }^{2}=\displaystyle \sum _{i=1}^{N}\displaystyle \frac{{({x}_{i}-\overline{x})}^{2}}{{\sigma }_{x,i}^{2}}\end{eqnarray} \tag{ 6 }$$

should obey the χ² distribution with (N − 1) dof, where x represents the tested parameter (${L}_{{\rm{X}}}$, ${N}_{{\rm{H}}}$, or flux), σ_x,i means the 1σ error in the ith epoch, and N is the number of epochs. The χ² test results are listed in Table 9. When the source with three (four) binning epochs satisfies χ² > 7.8 (9.8), we regard it as being variable at >98% confidence level.²⁴ We also check the ${L}_{{\rm{X}}}$- and ${N}_{{\rm{H}}}$-variable identification results using the Akaike information criterion (AIC) method as described in Y16, which does not need to assume the Gaussian errors (see Section 3.2.3 of Y16 for details). The ${\rm{\Delta }}\mathrm{AIC}$ values for each source are also listed in Table 9. Sources with ${\rm{\Delta }}\mathrm{AIC}\gt 4.0$ are assigned as being variable. It can be seen that 30/31 and 30/31 of sources have consistent results in identifying ${L}_{{\rm{X}}}$ and ${N}_{{\rm{H}}}$ variability while using the χ² test and the AIC method, respectively.²⁵

Table 9.  The Chi-Squared Value ${\chi }^{2}$, the ${\rm{\Delta }}\mathrm{AIC}$ Value Δ, and the Normalized Excess Variance ${\sigma }^{2}$ of the Observed 0.5–7 keV Flux, Intrinsic Rest-frame ${L}_{0.5-7\mathrm{keV}}$, and ${N}_{{\rm{H}}}$

XID	${\chi }_{\mathrm{flux}}^{2}$	${\chi }_{{L}_{{\rm{X}}}}^{2}$	${{\rm{\Delta }}}_{{L}_{{\rm{X}}}}$	${\chi }_{{N}_{{\rm{H}}}}^{2}$	${{\rm{\Delta }}}_{{N}_{{\rm{H}}}}$	${\sigma }_{\mathrm{flux}}^{2}$	${\sigma }_{{L}_{{\rm{X}}}}^{2}$	${\sigma }_{{N}_{{\rm{H}}}}^{2}$
CDF-S Four Epochs
49	53.3	21.9	11.9	17.3	11.3	0.074	0.056	0.073
81	69.1	23.7	13.9	3.1	−3.2	0.058	0.037	0
98	3.6	2.9	−3.1	0.4	−5.6	0	0	0
186	13.4	9.6	2.6	2.5	−3.9	0.007	0.009	0
214	2.9	0.4	−6.0	3.0	−3.3	0	0	0
236	7.0	2.7	−3.6	4.4	−2.1	0.008	0	0
308	86.7	44.9	32.4	1.4	−4.7	0.066	0.054	0
458	21.7	28.4	13.5	7.5	2.8	0.024	0.056	0.013
746	6.8	7.7	2.0	7.8	1.9	0.004	0.010	0.003
876	12.0	5.5	−0.9	0.4	−5.6	0.005	0.002	0
933	10.3	10.3	2.2	4.9	−1.4	0.002	0.004	0.005
760	2.6	3.0	−3.1	5.6	−1.7	0	0	0
CDF-S Three Epochs
63	34.2	7.2	2.8	65.2	29.45	0.054	0.026	0.147
73	7.7	5.7	0.7	0.9	−3.27	0.029	0.032	0
240	0.6	1.6	−0.9	1.6	−0.52	0	0	0
249	31.0	2.2	−2.9	130.8	14.38	0.086	0	0.150
328	20.8	18.3	8.5	2.4	1.14	0.050	0.095	0
367	33.8	9.6	1.4	1.4	−0.38	0.044	0.046	0
399	13.5	14.6	8.3	2.6	−1.31	0.051	0.118	0
551	3.6	2.4	−2.2	0.1	−3.95	0.004	0	0
658	0.1	0.2	−3.8	0.8	−3.14	0	0	0
752	0.7	1.9	−2.3	6.8	1.16	0	0	0.002
785	26.0	10.6	4.3	0.2	−3.87	0.062	0.057	0
840	12.0	1.7	1.1	16.1	9.59	0.013	0	0.012
846	0.3	0.1	−3.9	1.1	−2.96	0	0	0
621	2.4	2.0	−2.0	1.3	−2.83	0	0	0
91	0.7	1.0	−3.1	1.1	−2.92	0	0	0
818	13.9	12.8	8.1	3.2	−0.7	0.035	0.067	0.004
733	2.0	0.04	−3.9	4.9	0.2	0	0	0.012
CDF-N Three Epochs
66	25.8	4.2	−0.4	8.6	0.29	0.050	0	0
143	11.8	6.8	2.4	8.8	4.28	0.012	0.008	0.011

Note. Negative values of ${\sigma }^{2}$ are set to 0. Sources with χ² > 7.8 for three epochs, χ² > 9.8 for four epochs, or ${\rm{\Delta }}\mathrm{AIC}\gt 4.0$ will be regarded as being variable at >98% confidence level (Y16).

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Based on the χ² test results, sources that show observed flux variability account for 55% ± 13% (17/31) of the entire sample. The resulting ${L}_{{\rm{X}}}$ and ${N}_{{\rm{H}}}$ variable source fractions are 29% ± 10% (9/31) and 19% ± 8% (6/31), respectively. For the majority of sources, we do not find any correlation between L_X and ${N}_{{\rm{H}}}$ variability patterns, suggesting that the main reason that causes the variation of ${N}_{{\rm{H}}}$ is likely not the change of ionization parameter induced by the variable ${L}_{{\rm{X}}}$, but is likely the occultation of the clumpy clouds moving in/out of our LOS.

We use the normalized excess variance ${\sigma }_{\mathrm{nxv}}^{2}$ (nxv) to estimate the variability amplitude. ${\sigma }_{\mathrm{nxv}}^{2}$ is defined as

$$\begin{eqnarray}&&{\sigma }_{\mathrm{nxv}}^{2}\,=\displaystyle \frac{1}{N\langle x{\rangle }^{2}}\displaystyle \sum _{i=1}^{N}[{({x}_{i}-\langle x\rangle )}^{2}-{({\rm{\Delta }}{x}_{i})}^{2}],\end{eqnarray} \tag{ 7 }$$

where N is the number of binning epochs (three or four), x_i and Δx_i are the best-fit parameters (observed flux, ${L}_{{\rm{X}}}$, ${N}_{{\rm{H}}}$) and their 1σ errors, and $\langle x\rangle$ is the unweighted mean of the calculated parameter. The error on ${\sigma }_{\mathrm{nxv}}^{2}$ is estimated as ${s}_{{\rm{D}}}/(\langle x{\rangle }^{2}\sqrt{N})$, where

$$\begin{eqnarray}&&{s}_{{\rm{D}}}^{2}=\displaystyle \frac{1}{N-1}\displaystyle \sum _{i=1}^{N}{[{({x}_{i}-\langle x\rangle )}^{2}-{({\rm{\Delta }}{x}_{i})}^{2}-{\sigma }_{\mathrm{nxv}}^{2}\langle x{\rangle }^{2}]}^{2}.\end{eqnarray} \tag{ 8 }$$

We calculate ${\sigma }_{\mathrm{nxv}}^{2}$ and corresponding errors on the intrinsic rest-frame 0.5–7 keV luminosity (${\sigma }_{\mathrm{nxv}}^{2}$_,L), observed 0.5–7 keV flux (${\sigma }_{\mathrm{nxv}}^{2}$_,F), and ${N}_{{\rm{H}}}$ (${\sigma }_{\mathrm{nxv}}^{2}$_,N) for each source. The results are listed in Table 9. To check whether our calculation of ${\sigma }_{\mathrm{nxv}}^{2}$ is affected by the limited counts (e.g., Zheng et al. 2017, hereafter Z17), we perform Spearman rank correlation tests and find no significant correlation between ${\sigma }_{\mathrm{nxv}}^{2}$_,F, ${\sigma }_{\mathrm{nxv}}^{2}$_,N, ${\sigma }_{\mathrm{nxv}}^{2}$_,L, and counts with Spearman's ρ = 0.07, p-value = 0.70; ρ = −0.11, p-value = 0.56; and ρ = 0.32, p-value = 0.08, respectively.

We also calculate the fractional root mean square (frms) variability amplitude, which is defined as $\sqrt{{\sigma }_{\mathrm{nxv}}^{2}}$ for sources having ${\sigma }_{\mathrm{nxv}}^{2}$ > 0 and can be treated as the percentage of the variability amplitude. It is obvious that our sample lacks sources that display large variability amplitudes. The mean frms values of the total sample for the observed flux, intrinsic luminosity, and ${N}_{{\rm{H}}}$ are 12%, 10%, and 6%, respectively. For sources having positive nxv, the mean (maximum) frms values are 17% (29%), 19% (34%), and 16% (39%), respectively.

In addition, we do not find any correlation between ${\sigma }_{\mathrm{nxv}}^{2}$_,F, ${\sigma }_{\mathrm{nxv}}^{2}$_,L, and ${L}_{{\rm{X}}}$ (Figure 19) whereas other studies suggested a negative trend, i.e., luminous AGNs show relatively small variations (Y16; Z17; Paolillo et al. 2017). We perform a Spearman correlation test for the 25 common sources using the ${\sigma }_{\mathrm{nxv}}^{2}$_,F derived in Z17, which is obtained from the light-curve analysis; the result is still consistent with no correlation between ${L}_{{\rm{X}}}$ and ${\sigma }_{\mathrm{nxv}}^{2}$_,F (Spearman's ρ = −0.31, p-value = 0.13). As pointed out by Allevato et al. (2013), the calculation of ${\sigma }_{\mathrm{nxv}}^{2}$ is biased at low counts and uneven cadence and should be restricted to large samples. Therefore, the limited source number and the broad redshift span may prevent us from detecting such a correlation. But beyond that, since Y16 and Z17 mainly focused on the AGNs with largest counts available in the CDF-S rather than highly obscured AGNs, the variability behavior and its underlying physical drivers of our sources and those in Y16 and Z17 may be intrinsically different.

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Recently, González-Martín (2018) found a new X-ray variability plane for AGNs that links the characteristic break frequency of the PSD (f_break) with bolometric luminosity and ${N}_{{\rm{H}}}$. This makes ${N}_{{\rm{H}}}$ play a non-negligible role in understanding AGN variability. As we will show in the next section, although Z17 argued that the ${N}_{{\rm{H}}}$ variation does not contribute significantly to the total variability in their sample, the ${N}_{{\rm{H}}}$ variability does play a substantial role for highly obscured AGNs. Nevertheless, the small ${\sigma }_{\mathrm{nxv}}^{2}$ values indicate that the state of highly obscured AGNs does not vary dramatically while concerning the average source properties on a 17 yr timescale in the observed frame.

There are some sources in our sample that show significant variability of the observed flux, intrinsic luminosity, or ${N}_{{\rm{H}}}$. In this section, we perform detailed analyses to study their variability behavior aiming at shedding light on the leading mechanism that drives the large variabilities, and try to investigate the typical location and size of the obscuring clouds.

7.4.1. XID 63

This source has 848 counts available, and its data are binned to three epochs. It is a moderately luminous source (${\bar{L}}_{{\rm{X}}}=5.1\times {10}^{43}\ \mathrm{erg}\ {{\rm{s}}}^{-1}$) at z = 0.737. The best-fit model of this source is the absorbed power law (${\rm{\Gamma }}={1.54}_{-0.22}^{+0.21}$) plus an additional soft-excess component. Γ and norm_soft do not show variations and are linked during all the epochs. The unfolded spectra, light curves, and 1σ and 2σ confidence contours of ${N}_{{\rm{H}}}$ and normalization are displayed in the left column of Figure 20. It has ${\chi }_{{N}_{{\rm{H}}}}^{2}=65.2$, ${\chi }_{f,\mathrm{obs}}^{2}=34.2$, and ${\chi }_{L}^{2}=7.2$, indicating significant variabilities. Although the photon index does not vary, the observed spectral shape changes prominently owing to the large variation in column density. The absorption is weak in epoch1 with ${N}_{{\rm{H}}}$ $=\,5.8\times {10}^{22}\ {\mathrm{cm}}^{-2}$. Then, it transforms into a highly obscured state with ${N}_{{\rm{H}}}$ $=\,1.4\times {10}^{23}\ {\mathrm{cm}}^{-2}$ in epoch2, and its ${N}_{{\rm{H}}}$ continues to rise to $1.9\times {10}^{23}\ {\mathrm{cm}}^{-2}$ in epoch3. The intrinsic flux increases by about a factor of 1.4 from epoch1 to epoch2 and decreases by about a factor of 1.6 from epoch2 to epoch3. Due to the large variability amplitude in ${N}_{{\rm{H}}}$, the observed flux remains constant in the first two epochs but significantly declines in epoch3 by a factor of 1.9. The ${N}_{{\rm{H}}}$ variability may be caused by the clumpy cloud moving into the LOS, since the variations of N_H and L_X are not correlated. By assuming this transition as an eclipse event, we can roughly estimate the distance and angular size of the cloud using the same method in Y16, thus to constrain the cloud size. We use the empirical relation between the inner torus radius r and 14-195 keV luminosity from Koshida et al. (2014) to estimate the distance. The relation can be described as $\mathrm{log}\,r=-1.04\,+0.5\,\mathrm{log}\,({L}_{14-195\mathrm{keV}}/{10}^{44})$, where r and ${L}_{14-195\mathrm{keV}}$ are in units of pc and $\mathrm{erg}\ {{\rm{s}}}^{-1}$, respectively. The ${L}_{14-195\mathrm{keV}}=2.5\,\times {10}^{44}\,\mathrm{erg}\ {{\rm{s}}}^{-1}$ at epoch1 is obtained by extrapolating the power law to the higher-energy band. Thus, the estimated r from the inner torus to the central black hole is ≈0.14 pc. By assuming Keplerian motion, the orbital period of the cloud ${t}_{\mathrm{orbit}}\,=2\pi \tfrac{{r}^{\tfrac{3}{2}}}{{({M}_{\mathrm{BH}}G)}^{\tfrac{1}{2}}}\approx 519\,{\left(\tfrac{{M}_{\mathrm{BH}}}{{10}^{8}{M}_{\odot }}\right)}^{-\tfrac{1}{2}}$ yr. The angular size (viewed from the BH) of the cloud is estimated as $\tfrac{{t}_{\mathrm{tran}}}{{t}_{\mathrm{orbit}}}\times 360^\circ \,\lt 4\buildrel{\circ}\over{.} 1{\left(\tfrac{{M}_{\mathrm{BH}}}{{10}^{8}{M}_{\odot }}\right)}^{\tfrac{1}{2}}$, where the upper limit value of t_tran is calculated as the time span (∼6.0 yr in the rest-frame) between epoch1 and epoch2. By multiplying by r, we estimate the cloud size to be $\lt 0.01\,{\left(\tfrac{{M}_{\mathrm{BH}}}{{10}^{8}{M}_{\odot }}\right)}^{\tfrac{1}{2}}\,\mathrm{pc}$.

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7.4.2. XID 249

7.4.3. XID 328

7.4.4. XID 49

7.4.5. XID 458

7.4.6. Remaining Sources