# Measurement of AGN dust extinction based on the near-infrared flux
variability of WISE data

Shoichiro Mizukoshi,1 Takeo Minezaki,1 Shoichi Tsunetsugu,1 Atsuhiro Yoshida,1
Hiroaki Sameshima,1 Mitsuru Kokubo,2 and Hirofumi Noda3
1Institute of Astronomy, Graduate School of Science, The University of Tokyo,
2-21-1 Osawa, Mitaka, Tokyo 181-0015, Japan
2Department of Astrophysical Sciences, Princeton University, Princeton, New
Jersey 08544,USA
3Department of Earth and Space Science, Graduate School of Science, Osaka
University, 1-1 Machikaneyama, Toyonaka, Osaka 560-0043, Japan E-mail:
<EMAIL_ADDRESS>
(Accepted XXX. Received YYY; in original form ZZZ)

###### Abstract

We present the measurement of the line-of-sight extinction of the dusty torus
for a large number of obscured active galactic nuclei (AGNs) based on the
reddening of the colour of the variable flux component in near-infrared (NIR)
wavelengths. We collected long-term monitoring data by Wide-field Infrared
Survey Explorer (WISE) for 513 local AGNs catalogued by the Swift/BAT AGN
Spectroscopic Survey (BASS) and found that the multi-epoch NIR flux data in
two different bands (WISE $W1$ and $W2$) are tightly correlated for more than
90% of the targets. The flux variation gradient (FVG) in the $W1$ and $W2$
bands was derived by applying linear regression analysis, and we reported that
those for unobscured AGNs fall in a relatively narrow range, whereas those for
obscured AGNs are distributed in a redder and broader range. The AGN’s line-
of-sight dust extinction ($A_{V}$) is calculated using the amount of the
reddening in the FVG and is compared with the neutral hydrogen column density
($N_{\rm{}H}$) of the BASS catalogue. We found that the $N_{\rm{}H}/A_{V}$
ratios of obscured AGNs are greater than those of the Galactic diffuse
interstellar medium (ISM) and are distributed with a large scatter by at most
two orders of magnitude. Furthermore, we found that the lower envelope of the
$N_{\rm{}H}/A_{V}$ of obscured AGNs is comparable to the Galactic diffuse ISM.
These properties of the $N_{\rm{}H}/A_{V}$ can be explained by increase in the
$N_{\rm{}H}$ attributed to the dust-free gas clouds covering the line of sight
in the broad-line region.

###### keywords:

infrared: galaxies – galaxies: nuclei – galaxies: Seyfert – quasars: general –
galaxies: evolution

††pubyear: 2021††pagerange: Measurement of AGN dust extinction based on the
near-infrared flux variability of WISE data–Measurement of AGN dust extinction
based on the near-infrared flux variability of WISE data

##  1 Introduction

The unified model (e.g. Antonucci, 1993) has been interpreted as the structure
of an active galactic nucleus (AGN) in which the dusty torus surrounds the
supermassive black hole and accretion disc. The dusty torus is important for
explaining the differences in optical spectra of AGNs, i.e., the Seyfert type,
caused by a difference at the viewing angle to the AGN with the common
structure.

The dusty torus can function as a mass reservoir for the central engine,
supplying dust and gas. However, the radiation from the central engine will
blow out most of the accreting materials into the circumnuclear region or
intergalactic space as the AGN feedback (Fabian, 2012, for review). This
radiation is considered to blow out not only the accreting material but also
the interstellar medium in its host galaxy and finally suppress its star
formation activity (Combes, 2017; Harrison, 2017), in addition to the growth
of the central black hole. The feeding and feedback processes between the AGN
and its host galaxy may account for their co-evolution over cosmic time, thus
resulting in correlations between AGN and host–galaxy properties (e.g.,
Kormendy & Richstone, 1995; Magorrian et al., 1998; Kormendy & Ho, 2013).

Fabian et al. (2009) suggested that radiation pressure on dusty gas in an AGN
depends on the dust abundance, hence it is one of the important parameters to
yield some hints about the co-evolution. The optical extinction $A_{V}$
characterises the amount of dust in the line of sight, which is commonly
attributed to the dusty torus. Optical spectroscopic analyses based on the
flux ratios of two or more optical emission lines (e.g., Balmer decrement,
Baker & Menzel, 1938; Ward et al., 1987; Gaskell, 2017) or the luminosity
ratio of the broad H$\alpha$ line and the hard X-ray (Shimizu et al., 2018)
have been performed to measure the $A_{V}$ of the dusty torus for more than
ten thousands of less-obscured AGNs (e.g., Jun et al., 2021, and citation
therein). However, because of strong obscuration in the optical band, these
methods based on optical observations are difficult to apply to obscured AGNs.

Emission lines in the near-infrared (NIR) band (e.g., the Paschen series) have
been used to estimate the $A_{V}$ of several dozens of obscured AGN targets
(e.g. Ward et al., 1987; Maiolino et al., 2001a; Schnorr-Müller et al., 2016).
Alternatively, Burtscher et al. (2016) measured the colour temperature of NIR
continuum emission for 29 AGNs—about a half of which are obscured AGNs—by the
spectral fitting for the $K$-band spectrum. They considered the average
temperature measured for type-1–1.9 AGNs as the "intrinsic" value, without any
intervening dust extinction, and estimated the $A_{V}$ of AGN samples by
measuring the temperature decrease from the intrinsic value. They examined the
ratio of the neutral hydrogen column density $N_{\rm{H}}$ that was derived
from X-ray spectral analysis (see Section 2.1) and the $A_{V}$, and reported
that $N_{\rm{H}}/A_{V}$ is usually larger than the typical value of the
Galactic diffuse interstellar medium (ISM) with large scatters, by at most two
orders of magnitude, as has been presented in other studies (e.g. Maiolino et
al., 2001a, b; Imanishi, 2001). Xu et al. (2020) derived the $A_{V}$ of 175
AGNs using the strength of the 9.7 $\mu$m silicate feature and presented that
the silicate strength is weakly correlated with $N_{\rm{}H}$ with large
scatters.

Winkler et al. (1992) presented the optical flux time variation of an AGN as
another clue for measuring the $A_{V}$ without spectroscopic analyses. They
calculated the flux variation gradient (FVG) using the monitoring data from
$UBV$ bands, which is the ratio of the amplitude of flux variations in two
different bands. They used regression analysis on the flux–flux plot to
measure the FVG, which takes the flux data from two bands at the same epoch on
the vertical and horizontal axes. They found that the FVGs for unobscured AGNs
are nearly blackthe same value in every target, which can be considered as the
intrinsic FVG. The $A_{V}$ for each target was then calculated by measuring
the reddening of the FVG relative to the intrinsic FVG. This method has the
advantage of ignoring host-galaxy emission because it only uses the variable
components of the optical flux that are attributed to the AGN.

The flux of the NIR continuum emission is also time variable, responding to
that of the UV-optical continuum emission from the accretion disc (e.g.,
Suganuma et al., 2006; Koshida et al., 2014; Minezaki et al., 2019; Lyu et
al., 2019; Yang et al., 2020). Such NIR emission is considered to be emitted
particularly from the innermost region of the dusty torus, in which hot dust
is heated to 1000–2000 K by the strong emission from the central engine (e.g.,
Barvainis, 1987; Lyu et al., 2017; Baskin & Laor, 2018). Glass (2004)
investigated the colours of the variable NIR flux components of 41 AGNs—six
are obscured AGNs—using monitoring data in JHKL bands. He demonstrated that
the intrinsic NIR FVGs fall within moderately narrow ranges as for the optical
FVG reported by Winkler et al. (1992). Accordingly, Glass (2004) mentioned
that the FVG in NIR bands may be useful to estimate the dust extinction due to
the outer torus for obscured AGNs, and applied his method to NGC1068.

In this study, we used NIR monitoring data from two bands to obtain the NIR
FVGs of nearby obscured and unobscured AGNs. We estimated their line-of-sight
optical extinction $A_{V}$ from the reddening of the NIR FVGs. Furthermore, by
comparing these $A_{V}$ to $N_{\rm{H}}$ determined using X-ray spectral
analyses (Koss et al., 2017), we examined the correlation between them and
compared it with the typical $N_{\rm{}H}/A_{V}$ of the Galactic diffuse ISM.
This paper is organised as follows: We describe the data, the AGN catalogue,
and our sample selection in Section 2. Section 3 presents the flux–flux plot
and the derivation of the NIR FVG in detail. We also present the results of
our samples’ NIR FVG properties here. Section 4 explains how to convert the
NIR FVG to the $A_{V}$ and demonstrates the results for our samples. We
compare our results with the typical $N_{\rm{}H}/A_{V}$ of the Galactic
diffuse ISM in Sections 3 and 4. In Section 5, after comparing our $A_{V}$
estimate with other studies, the possible scenarios for explaining the
distribution of the $N_{\rm{}H}/A_{V}$ are discussed. We then provided our
summary in Section 6. Throughout this study, we adopted the cosmology
$H_{0}=70\ \mathrm{km\ s^{-1}\ Mpc^{-1}}$, $\Omega_{0}=0.30$, and
$\Omega_{\Lambda}=0.70$.

##  2 targets and data

###  2.1 BASS AGN catalogue

Our targets are obtained from the catalogue of the BAT AGN Spectroscopic
Survey (BASS, Koss et al., 2017; Ricci et al., 2017), which was originally
provided in the Swift/BAT 70-month catalogue (Baumgartner et al., 2013). This
catalogue contains 836 nearby AGNs detected using the Swift/BAT 14–195 keV
band with a redshift peak of $z\sim 0.05$.

Koss et al. (2017) analysed the optical spectra of about 77% of the BASS AGNs
and classified almost all of them as Seyfert galaxies. They classified 539 of
them into Seyfert types, whereas the BASS website111https://www.bass-
survey.com affords the Seyfert types of 594 AGNs. The latter’s Seyfert types
are used in this study.

In the BASS AGN catalogue, the values of $N_{\rm{}H}$ were primarily obtained
by X-ray spectral analysis using not only 14–195 keV data but also soft X-ray
data at 0.3–10 keV, which was obtained from XMM-Newton (Jansen et al., 2001),
Chandra (Weisskopf et al., 2000), Suzaku (Mitsuda et al., 2007), or Swift/XRT
(Burrows et al., 2005). In the $N_{\rm{}H}$ measurement, the Galactic
absorption was considered in advance using the value from the Hi maps of
Kalberla et al. (2005). The $N_{\rm{}H}$ of AGNs were then measured in which
photoelectric absorption and Compton scattering were considered using the
ZPHABS and CABS models, respectively (Ricci et al., 2017). The lower limit
measured by the analysis was a $\log N_{\rm{H}}\ [\rm{cm}^{-2}]=20$ (Koss et
al., 2017; Ricci et al., 2017), which is lower than the typical value of the
Galactic diffuse ISM. Ricci et al. (2017) estimated the $N_{\rm{}H}$ of 75
Compton-thick AGNs using torus modelling to precisely determine them. We note
that Burtscher et al. (2016) used the $N_{\rm{}H}$ in the BASS AGN catalogue
for their investigation as in this study.

###  2.2 Details about WISE data

We used the NIR photometric data obtained by the Wide-field Infrared Survey
Explorer (WISE, Wright et al., 2010). WISE was launched in 2009 and performed
its cryogenic all-sky survey for about a year in four bands: $W1$ ($3.4\
\mu$m), $W2$ ($4.6\ \mu$m), $W3$ ($12\ \mu$m), and $W4$ ($22\ \mu$m). This
all-sky survey is called the ALLWISE programme. WISE was reactivated without
cryogen after a two-year hibernation period to conduct an all-sky monitoring
survey in the $W1$ and $W2$ bands to examine near-Earth objects (NEOWISE,
Mainzer et al., 2011, 2014). Currently, WISE provides the all-sky monitoring
data in $W1$ and $W2$ bands spanning for more than ten years, which comprises
at most three epoch data from ALLWISE and typically 15 epoch data from
NEOWISE. Each epoch data comprises typically a few dozen of single-exposure
data.

Typically, WISE observes each target once every six months. We averaged the
photometric data in each epoch after excluding outliers that deviated from the
average by more than $3\sigma$ of the distribution of the other flux data in
the same epoch. We excluded monitoring data points from the epoch that did not
have $W1$\- or $W2$-band photometric data. The analysis of the NIR FVG
described in the following sections demonstrates that it falls in the range
corresponding to the power-law spectrum of $F_{\nu}\propto\nu^{\alpha}$ with
$-4<\alpha<0$. We, therefore, adopt the zero magnitude flux densities of
306.682 and 170.663 Jy for the $W1$ and $W2$ bands, respectively, which are
those for $F_{\nu}\propto\nu^{-2}$ (Wright et al., 2010). In this study, the
uncertainty of the derived flux density because of the difference of zero
magnitude flux density is at most about 3% (Wright et al., 2010; Jarrett et
al., 2011), which is smaller than the typical uncertainty of the NIR FVG in
this study. We corrected for the Galactic extinction in the flux calculation.
We adopted the Galactic extinction for each target from NASA/IPAC
Extragalactic Database (NED), which is based on Schlafly & Finkbeiner (2011).
We demonstrate examples of the light curve using the WISE data in Fig. 1 (1a)
and (2a) for the typical type-1 and type-2 AGNs, respectively.

Figure 1: 1a: The NIR light curve of LEDA 126226, as an example of the typical
type-1 AGN. The blue and the red circles indicate the $W1$\- and $W2$-band
fluxes, respectively. The error bar of each data point represents $\pm
1\sigma$ error of the flux data. 1b: The $W1$-band flux to $W2$-band flux plot
of LEDA 126226 as an example. The blue circles indicate NEOWISE data and the
orange triangles indicate ALLWISE data. The black solid line indicates the
best-fit regression line for the data points, and the red thin lines show the
400 samples from the posterior distribution of the regression. 2a: The same
figure as 1a for LEDA 2730634 as an example of the typical type-2 AGN. 2b: The
same figure as 1b for LEDA 2730634 as an example of the typical type-2 AGN.

###  2.3 Target selection

First, we selected 594 AGNs in the BASS catalogue with the Seyfert type. We
then excluded 25 AGNs that are classified as blazars in the Roma Blazar
Catalogue (BZCAT, Massaro et al., 2009). Targets with the Galactic extinction
$A_{V}(\mathrm{Gal.})>2$ mag were excluded to decrease the uncertainty of the
estimated $A_{V}$ caused by that of the Galactic extinction. Typically,
obscured AGNs have $\log N_{\rm{}H}\ [\mathrm{cm^{-2}}]\gtrsim 22$, which
corresponds to $A_{V}\gtrsim 5$ mag assuming the typical $N_{\rm{}H}/A_{V}$ of
the Galactic diffuse ISM (Predehl & Schmitt, 1995; Nowak et al., 2012). We
excluded 26 targets here.

Next, we set the target redshift to $z<0.5$ so that the thermal emission from
the hot dust in the innermost region of the dusty torus dominated the NIR
observed fluxes. Because most of the BASS AGNs are nearby sources, only two
AGNs were excluded.

We then selected targets with good-quality WISE data, which are defined by the
data quality flags as good frame quality (qual_frame$\,\geq 5$), good signal-
exposure image (qi_fact$\,=0.5,\,1$), no charged particle hits (saa_sep$\,\geq
5$), no moon effect (moon_masked$\,=0$), and no effect from artifacts
(cc_flags$\,=\,$’0000’). Here we excluded the targets with no good-quality
data from our samples. Moreover, we excluded targets that had at least one
saturated pixel in more than 10% of the observational data across all epochs,
whereas we used WISE profile-fitting photometry, which used only unsaturated
pixels. We excluded 28 targets here in total. Therefore, we selected 513 AGNs
for subsequent analysis. Although some targets lacked usable data from the
ALLWISE programme, we did not exclude them because our NIR FVG analysis could
be adequately conducted only with the data from the NEOWISE programme.

Figs. 1 (1b) and (2b) show the $W1$-band flux to $W2$-band flux plot for the
targets, LEDA 126226 and LEDA 2730634, as examples of typical type-1 and
type-2 AGNs, respectively. In the horizontal axis of the flux–flux plot, we
consider the difference from the mean $W2$ flux value to approximately
minimise the uncertainty of the intercept of the vertical axis. The data
points spread in the flux–flux plot were then fitted using the linear
relation,

$f_{W1}=\alpha+\beta\,(f_{W2}-\langle f_{W2}\rangle),$ (1)

where $f_{W1}$ and $f_{W2}$ are $W1$\- and $W2$-band fluxes, respectively, and
$\langle f_{W2}\rangle$ is the average of the $W2$-band fluxes. We set the
intercept $\alpha$ and slope $\beta$ as free parameters, and the NIR FVG for
the target is estimated as $\beta$ of the best-fit linear regression.

Before performing linear regression analysis, we calculated the correlation
coefficient $r$ between the $W1$ and $W2$ flux data to evaluate the strength
of the correlation between flux variations in $W1$ and $W2$ bands. We reported
that the flux variations in the two NIR bands were highly correlated for the
majority of the targets ($r\geq 0.9$ for 83% of the 513 targets), which is
consistent with the results of Glass (2004). Moreover, our result is based on
an order of magnitude larger sample, including multiple obscured AGNs.

The targets having a high correlation coefficient were subjected to linear
regression analysis. However, there are some targets with high correlation
coefficients for which the NEOWISE programme data points are clustered in a
small area on the flux–flux plot, thus suggesting small flux variation and
weak correlation for the NEOWISE data. A few data points from the ALLWISE
programme located separately may produce a strong correlation in such cases.
In this study, we focus on the flux variation of the thermal radiation on a
timescale of a few to ten years; hence, our suitable targets should
demonstrate the clear flux variation even in the NEOWISE data alone.
Consequently, we excluded targets with weak correlation for the NEOWISE data
based on the correlation coefficient, calculated using only data from the
NEOWISE programme ($\equiv r_{\rm{}NEO}$). 466 AGNs were selected for which
both $r$ and $r_{\rm{}NEO}$ were larger than 0.7 because 0.64 is the value of
the correlation coefficient indicating a 99% confidence level for a sample
size of 15. Many targets eliminated here show not only the small flux
variation but also the large $N_{\rm{}H}$ with the peak of $\log
N_{\mathrm{H}}\ [\mathrm{cm^{-2}}]\sim 23.5$. This indicates that these
targets may be heavily obscured sources.

##  3 near-infrared flux variation gradient

###  3.1 Linear regression analysis of the flux–flux plot

We performed linear regression analysis on the flux–flux plots of the 466
targets. Because data points in the flux–flux plot have errors on both axes,
we used a Python port of a Bayesian linear regression routine called
Linmix_err in IDL developed by Kelly (2007), called Linmix222Linmix:
http://linmix.readthedocs.org/, which normally incorporates distributed errors
in dependent and independent variables. Linmix assumes the distribution of
independent variables with multiple Gaussians, and we herein set the number of
Gaussian $K$ as $K=2$, which is the smallest number we can set to make Linmix
work properly. In this study, we included the intrinsic scatter in our
calculation.

Figs. 1 (1b) and (2b) show the best-fit regression line and samples from the
posterior distribution of the regression for each target. We estimated the NIR
FVGs with the good precision of the FVG error ($\equiv\sigma_{\beta}$) being
smaller than 0.2 for the 463 targets. This represents more than 90% (463/513)
of the local non-blazar BASS AGNs with known Seyfert type, small Galactic
extinction, and good WISE data. We selected these 463 targets to estimate the
line-of-sight dust extinction using the NIR FVG described in the following
sections. Table 1 lists the estimated NIR FVGs and errors for the target AGNs.

Figure 2: Left: The histograms of the NIR FVG $\beta$ for the 463 AGN samples.
From top to bottom panels, they represent the histograms for the Seyfert
type-1 (blue), type-1.2 (cyan), type-1.5 (green), type-1.9 (orange), and
type-2 (red) targets, respectively. Right: The histograms of the error of the
NIR FVG $\sigma_{\beta}$ for the same samples. The top to bottom panel order
and colour code is the same as the left panels.

###  3.2 Results on the near-infrared FVG

Fig. 2 shows the distribution of the NIR FVG $\beta$ (the left panel) for the
463 targets in a sequence by Seyfert type. These targets comprise 40, 93, 83,
70, and 177 targets of Seyfert type 1, 1.2, 1.5, 1.9, and 2, respectively. We
found that the NIR FVGs for less obscured (type-1–1.5) AGNs were distributed
around $\beta\sim 0.9$ in a relatively narrow range as reported by Glass
(2004). However, we found a clear reddening trend of the NIR FVGs for type-2
AGNs. They were broadly distributed in a range of $\beta\sim 0.4$–0.9 with
many targets at $\beta<0.4$. Observed flux time variation in the $W1$ band is
more likely to be suppressed than that in the $W2$ band because of dust
extinction, which makes the regression line flatter or $\beta$ smaller in
obscured AGNs. The NIR FVG of type-1.9 AGNs is distributed around $\beta\sim
0.8$ with a small number of the targets at $\beta<0.5$.

Fig. 2 shows the distribution of the FVG error $\sigma_{\beta}$ (the right
panel). The distribution of the FVG error differs little between the targets
of different Seyfert types. The average FVG error for all targets is
$\langle\sigma_{\beta}\rangle=0.07$ with only 18% (84/463) of targets having
relatively large FVG errors of $\sigma_{\beta}>0.1$.

In Fig. 3, the NIR FVG is compared with $N_{\rm{}H}$ measured by X-ray
absorption. The colours of the data points are coded in a sequence by Seyfert
types. We successfully obtained the NIR FVGs for both unobscured and obscured
AGNs whose hydrogen column densities ranged from $\log N_{\rm{}H}\
[\rm{cm}^{-2}]=20$ (lower limit of the measurement) to $\log N_{\rm{}H}\
[\rm{cm}^{-2}]\sim 25$. We found that the NIR FVGs for less obscured AGNs of
$\log N_{\rm{}H}\ [\rm{cm}^{-2}]\lesssim 22$ fall in a relatively narrow
range. These AGNs are dominated by type-1–1.5 AGNs. However, we found that the
NIR FVGs show clear reddening for obscured AGNs of $\log N_{\rm{}H}\
[\rm{cm}^{-2}]\gtrsim 22$, which are dominated by Seyfert type-1.9–2 AGNs.

We note that there is a large difference in the NIR FVG at fixed $N_{\rm{}H}$
for obscured AGNs of $\log N_{\rm{}H}\ [\rm{cm}^{-2}]\gtrsim 22$; accordingly,
there is a large difference in $N_{\rm{}H}$ by at most two orders of magnitude
at fixed NIR FVG for obscured AGNs of Seyfert type 1.9–2. This indicates that
there is a significant scatter in the ratio of dust reddening to the hydrogen
column density for obscured AGNs as has been reported (e.g., Maiolino et al.,
2001a; Burtscher et al., 2016). This will be discussed in the following
sections.

Figure 3: The NIR FVG $\beta$ plotted against the line-of-sight hydrogen
column density $N_{\rm{}H}$. The filled circles represent the data points with
$\sigma_{\beta}<0.1$, and the open circles represent those with
$0.1\leq\sigma_{\beta}<0.2$. The colours of circles indicate the different
Seyfert types of the targets in the same way as in Fig. 2. The black segment
in the lower left represents the mean $\pm 1\sigma$ error of the near-IR FVGs,
$\langle\sigma_{\beta}\rangle=0.07$. The grey band represents the typical
relationship between the NIR FVG and the hydrogen column density of the
Galactic diffuse ISM (see Section 4.3).

##  4 LINE-OF-SIGHT DUST EXTINCTION

###  4.1 Calculation of dust extinction from near-infrared FVG

The clear trends of reddening of the NIR FVG for obscured AGNs in Figs. 2 and
3 show that the line-of-sight dust extinction of AGNs can be estimated from
the amount of reddening of the NIR FVG. Here we explain the formula for
estimating the dust extinction from the NIR FVG.

The time-variable components of $W1$\- and $W2$-band fluxes, $\tilde{f}_{W1}$
and $\tilde{f}_{W2}$, respectively, are expressed as follows:

$\tilde{f}_{W1}=\tilde{f}_{W1,0}\times 10^{-2/5\times A_{W1}},$ (2)

$\tilde{f}_{W2}=\tilde{f}_{W2,0}\times 10^{-2/5\times A_{W2}},$ (3)

where $\tilde{f}_{W1,0}$ and $\tilde{f}_{W2,0}$ are the $W1$\- and $W2$-band
time-variable fluxes without extinction, respectively, and $A_{W1}$ and
$A_{W2}$ are the magnitudes of extinction in the $W1$ and $W2$ bands,
respectively. Taking the logarithm of the ratio of Equations (2) and (3),
black the relationship between the $W1$-band to $W2$-band FVG
$\beta=\tilde{f}_{W1}/\tilde{f}_{W2}$, $A_{W1}$, and $A_{W2}$ can be expressed
as follows:

$\log\beta=\log\beta_{0}-\frac{2}{5}\,(A_{W1}-A_{W2}),$ (4)

where $\beta_{0}=\tilde{f}_{W1,0}/\tilde{f}_{W2,0}$ is the $W1$-band to
$W2$-band FVG for the AGNs without extinction. Equation (4) can be expressed
using the magnitude of the $V$-band dust extinction $A_{V}$ and the dust
extinction curve as follows:

$\log\left(\frac{\beta}{\beta_{0}}\right)=-\frac{2}{5}\,(k_{W1}-k_{W2})A_{V},$
(5)

where $k_{W1}=A_{W1}/A_{V}$ and $k_{W2}=A_{W2}/A_{V}$. We assumed the $A_{V}$
as the rest-frame $V$-band extinction in the AGN, while $A_{W1}$ and $A_{W2}$
are the extinction at the rest-frame wavelengths of $\lambda_{W1}/(1+z)$ and
$\lambda_{W2}/(1+z)$ at the target redshift of $z$, respectively, where
$\lambda_{W1}=3.4\,\mu$m and $\lambda_{W2}=4.6\,\mu$m. Finally, we can
calculate the amount of dust extinction in the AGN based on the reddening of
the NIR FVG by using the following equation,

$A_{V}=-\frac{5}{2(k_{W1}-k_{W2})}\log\left(\frac{\beta}{\beta_{0}}\right)$
(6)

We assumed the standard extinction curve of the Galactic diffuse ISM
(Fitzpatrick, 1999) for $k_{W1}$ and $k_{W2}$, and $R_{V}=3.1$ to obtain
$k_{W1}=0.064$ and $k_{W2}=0.045$, respectively, when $z=0$. We assumed a
foreground screen geometry for the obscurer because the NIR emitting region is
much smaller than the parsec-scale outer torus (Burtscher et al., 2015, 2016;
Lyu et al., 2019; Minezaki et al., 2019; Noda et al., 2020; Gravity
Collaboration et al., 2020; Gámez Rosas et al., 2022).

The uncertainty of the NIR FVG for each target $\sigma_{\beta}$ and that for
the AGN without extinction $\sigma_{\beta_{0}}$ are transferred to that of
$A_{V}$ for the target as follows:

$\sigma_{A_{V}}=\frac{5}{2\ln(10)\times(k_{W1}-k_{W2})}\sqrt{\left(\frac{\sigma_{\beta}}{\beta}\right)^{2}+\left(\frac{\sigma_{\beta_{0}}}{\beta_{0}}\right)^{2}}.$
(7)

###  4.2 Near-infrared FVG for unobscured AGNs

As discussed in Section 3.2, the NIR FVGs for less obscured AGNs fall in a
relatively narrow range. Then, we estimated the NIR FVG for unobscured AGNs
$\beta_{0}$ to investigate the dust extinction of obscured AGNs. We selected
the NIR FVGs of 95 optically- and X-ray-unobscured AGN targets. They have
Seyfert types of type 1–1.5 and their hydrogen column densities are $\log
N_{\rm{}H}\ [\rm{}cm^{-2}]\leq 20$. They have relatively small uncertainties
of the NIR FVG ($\sigma_{\beta}<0.1$).

Fig. 4 shows the NIR FVGs against the redshifts for these unobscured targets.
Their NIR FVGs fall in a relatively narrow range regardless of Seyfert type
but appear to be spread more than expected from the measurement errors,
indicating some amount of the target-to-target variation in the NIR FVGs for
unobscured AGNs, or the intrinsic scatter $\sigma_{\beta_{0}}$. To incorporate
the possible redshift dependence, we assumed the following relation:

$\beta_{0}(z)=\beta_{0}(z=0)+b\,\log(1+z),$ (8)

and we fitted Equation (8) to the data by the weighted least squares method,
adding the intrinsic scatter of the NIR FVG $\sigma_{\beta_{0}}$ to the error
of the NIR FVG by root-sum-square,
$\sqrt{\sigma_{\beta}^{2}+\sigma_{\beta_{0}}^{2}}$, so that the reduced
$\chi^{2}$ reaches unity. The best fit parameters are obtained as
$\beta_{0}(z=0)=0.86\pm 0.02$ and $b=-0.12\pm 0.57$ with
$\sigma_{\beta_{0}}=0.10$. Accordingly, we used

$\beta_{0}=0.86-0.12\log(1+z),$ (9)

$\sigma_{\beta_{0}}=0.10,$ (10)

for calculating the dust extinction and its error from the NIR FVG in the
following sections.

Assuming a blackbody spectrum, $\beta_{0}(z=0)=0.86$ and
$\sigma_{\beta_{0}}=0.10$ corresponds to the colour temperature of $T=1077\pm
140\ \rm{}K$, which is consistent with that of hot dust considered to be
located in the innermost dusty torus (e.g., Mor et al., 2009; Netzer, 2015;
Hönig, 2019; Lyu & Rieke, 2021). $T=1077\pm 140\ \rm{}K$ is slightly lower
than those estimated for the NIR $H-K$ and $K-L$ FVG colours for the local
type-1 AGNs by Glass (2004), which is possibly attributed to the difference in
observed wavelengths. The $W1-W2$ FVG colour tends to be more sensitive to
lower-temperature dust thermal emission than the $H-K$ and $K-L$ FVG colours.

As shown in Fig. 4, the NIR FVGs of unobscured AGNs are almost unchanged
systematically black with redshift. The FVG is constant regardless of redshift
when the spectral energy distribution (SED) of the flux variation is a power-
law spectrum, whereas it becomes redder in higher redshift when the SED is a
blackbody spectrum of single-temperature dust. Therefore, such a weak redshift
dependence of the NIR FVG suggests that the SED of the flux variation in 3–5
$\mu$m shows a bump much broader than a blackbody spectrum, as is found in the
typical SED of quasars (e.g., Elvis et al., 1994; Hernán-Caballero et al.,
2016; Lyu et al., 2017; Hickox & Alexander, 2018). This weak redshift
dependence of NIR FVG and possible broad bump in the NIR SED of AGNs black can
be explained by a composite of thermal radiation from multi-temperature dust.
By assuming a distribution of the amount of multi-temperature dust as a power-
law of temperature $N(T)\propto T^{k}$ with a maximum temperature of 2000 K,
the $W1$\- to $W2$-band flux ratio of the composite SED of the blackbody
radiation from the multi-temperature dust matches the best-fit
$\beta_{0}(z=0)$ of 0.86 when $k\sim-3.1$. We show the redshift dependence of
the flux ratio in Fig. 4 with the blue line. The reddening of the flux ratio
blackof multi-temperature blackbody radiation is weaker than that of the
single-temperature blackbody radiation in higher redshift, and this result
suggests that the SED of the blackbody radiation from multi-temperature dust
shows a broader bump in 3–5 $\mu$m than that from single-temperature dust.

black Then, we compared the $\beta_{0}$ with $W1$\- to $W2$-band flux ratios
derived from composite SEDs of quasars. The flux ratio from the SED of Hernán-
Caballero et al. (2016) is 0.73, which is redder than $\beta_{0}$ even if the
intrinsic scatter $\sigma_{\beta_{0}}$ is considered. We also compared the
$\beta_{0}$ with the flux ratio derived from the composite SED of warm-dust
deficient (WDD) quasars that are defined by Lyu et al. (2017). WDD quasars
have a broad NIR bump in their SEDs as normal quasars, but they show weaker
mid-infrared emission, which is attributed to the reduction of the warm dust
component in the dusty torus. We derived the flux ratio from the composite SED
of the WDD quasars as 0.81. We found that it falls within the intrinsic
scatter of $\sigma_{\beta_{0}}=0.10$ from the best-fit NIR FVG of
$\beta_{0}(z=0)=0.86\pm 0.02$, while the $\beta_{0}(z=0)$ is still bluer than
the flux ratio for the WDD quasars. This result indicates that the variable
component of flux in 3–5 $\mu$m are dominated by the thermal emission of hot
dust in the innermost region of the dusty torus, which is so compact that the
thermal reradiation response to the flux variation of the central engine is
larger than that in the outer torus. This strengthens the view of a foreground
screen geometry for the obscurer to measure the line-of-sight extinction based
on the reddening of the NIR FVG, as described in Section 4.1.

Figure 4:  The NIR FVG $\beta$ of the unobscured targets and their redshift.
The colours of the data points indicate the different Seyfert types of the
targets, in the same way as in Fig. 2, and the error bars represent $\pm
1\sigma$ errors of the $\beta$. The black solid line represents the best-fit
regression line to the data. The black dashed line represents the power-law
model (power-law index $\alpha=-0.5$), which has the same $\beta$ at $z=0$.
The black dotted line represents the single-temperature blackbody model
($T=1077$ K), which has the same $\beta$ at $z=0$. The blue line represents
the multi-temperature blackbody model with the maximum temperature of 2000 K
and the distribution of the multi-temperature dust of $N(T)\propto T^{-3.1}$.

###  4.3 Results of dust extinction

Table 1 lists the dust extinction estimated from the NIR FVG and the error
based on Equations (6), (7), and (9). We estimated the dust extinction of
obscured AGNs up to $A_{V}\sim 65$ mag. The mean uncertainty of $A_{V}$ of our
targets is $\sigma_{A_{V}}=8.3$ mag.

Figure 5: The comparison between the $A_{V}$ and $N_{\rm{H}}$ of our samples.
We adopt the same colours as in Fig. 2 for the same Seyfert types. The filled
circles and open circles also represent the same samples as in Fig. 3. We show
the typical $N_{\rm{H}}/A_{V}$ of the Galactic diffuse ISM with the grey band.
The black segment in the upper right indicates the average of the $1\sigma$
error of the $A_{V}$ estimate, $\langle\sigma_{A_{V}}\rangle=8.3$ mag. Dashed
grey segments indicate the variable $N_{\rm{}H}$ for the common samples to
Burtscher et al. (2016). The data for the variable $N_{\rm{}H}$ is obtained
from Burtscher et al. (2016) and the citation therein.

Fig. 5 shows the dust extinction compared with the hydrogen column density
from the X-ray absorption. As reported from Fig. 3, almost all samples with
$\log N_{\rm{H}}\ [\rm{cm^{-2}}]\lesssim 22$ distribute around $A_{V}\sim 0$
mag, and these targets are dominated by Seyfert type-1–1.5 AGNs. Fig. 6 shows
the distribution of $A_{V}$ of samples with $\log N_{\mathrm{H}}\
[\mathrm{cm^{-2}}]\leq 20$. The average and the standard deviation of them are
calculated as $-0.74\pm 0.65$ mag and 8.0 mag, which would be consistent with
a distribution of the average $A_{V}=0$ with the typical measurement error for
them of $\langle\sigma_{A_{V}}\rangle(\log N_{\mathrm{H}}\leq 20)=8.0$ mag.
Therefore, negative $A_{V}$ values for the same targets in Figs. 5 and 6 are
considered to be mostly caused by the measurement error.

Figure 6: The histogram of the dust extinction $A_{V}$ for the samples with
$\log N_{\mathrm{H}}\ [\mathrm{cm^{-2}}]\leq 20$. The colours indicate the
Seyfert types of the samples in the same way as in Fig. 2.

black In contrast, the typical $A_{V}$ of the samples shows a broad
correlation with $N_{\rm{H}}$ within $A_{V}\lesssim 65$ mag in the range of
$\log N_{\rm{H}}\ [\rm{cm^{-2}}]\gtrsim 22$. These samples with high
$N_{\rm{}H}$ are dominated by type-1.9–2 AGNs. They show a target-to-target
difference in $N_{\rm{H}}/A_{V}$ that spreads over about two orders of
magnitude. In Fig. 5, we draw the relation between $N_{\rm{}H}$ and $A_{V}$
for the standard $N_{\rm{}H}/A_{V}$ of the Galactic diffuse ISM
($N_{\rm{H}}/A_{V}=(1.79$–$2.69)\times 10^{21}\ \rm{cm}^{-2}\ mag^{-1}$,
Predehl & Schmitt, 1995; Nowak et al., 2012) for comparison. The data points
for obscured AGNs are located above the Galactic curve, which indicates that
these obscured AGNs have typically larger $N_{\rm{H}}/A_{V}$ than the Galactic
diffuse ISM. These trends for obscured AGNs agree with the results reported in
the literature (e.g., Maiolino et al., 2001a; Burtscher et al., 2016).
Although there are some data points below the Galactic curve at low
$N_{\mathrm{H}}$ ($\log N_{\mathrm{H}}\lesssim 22$), they are considered to be
mostly consistent with small $A_{V}$ values on the Galactic curve with their
measurement errors (see Fig. 6). black Furthermore, the lower envelope of the
data distribution of obscured AGNs is almost consistent with the Galactic
curve, which is reported by Burtscher et al. (2016) for a smaller number of
data.

black In Fig. 3, we draw the $N_{\rm{}H}-\beta$ relation for the standard
Galactic diffuse ISM. We found that the obscured AGNs with $\beta\lesssim
0.65$ and $\log N_{\rm{}H}\ [\rm{}cm^{-2}]\gtrsim 22$ have larger $N_{\rm{}H}$
than the expected from the NIR FVG assuming the Galactic diffuse ISM. This
trend is similar to that in Fig. 5.

Table 1: Main properties of our final samples and the results of our analyses. (1) | (2) | (3)a | (4)a | (5)a | (6) | (7) | (8) | (9)b | (10) | (11)b
---|---|---|---|---|---|---|---|---|---|---
Source | Counterpart | Redshift | Sy type | $\log N_{\rm{H}}$ | $r$ | $r_{\rm{NEO}}$ | $\beta$ | $\sigma_{\beta}$ | $A_{V}$ | $\sigma_{A_{V}}$
|  |  |  | (cm-2) |  |  |  |  | (mag) | (mag)
SWIFTJ0001.0-0708 | 2MASX J00004876-0709117 | 0.0375 | 1.9 | 22.19 | 0.96 | 0.879 | 0.71 | 0.07 | 9.6 | 8.2
SWIFTJ0001.6-7701 | 2MASX J00014596-7657144 | 0.0584 | 1.9 | 20.00 | — | 0.866 | 1.04 | 0.09 | $-10.5$ | 8.0
SWIFTJ0003.3+2737 | 2MASX J00032742+2739173 | 0.0397 | 2 | 22.86 | — | 0.996 | 0.87 | 0.03 | $-1.1$ | 6.7
SWIFTJ0006.2+2012 | Mrk 335 | 0.0258 | 1.2 | 20.48 | 0.981 | 0.943 | 1.03 | 0.06 | $-10.0$ | 7.5
SWIFTJ0009.4-0037 | SDSS J000911.57-003654.7 | 0.0733 | 2 | 23.56 | — | 0.922 | 0.52 | 0.08 | 25.0 | 9.8
SWIFTJ0021.2-1909 | LEDA 1348 | 0.0956 | 1.9 | 21.98 | 0.98 | 0.977 | 0.89 | 0.05 | $-2.3$ | 6.8
SWIFTJ0026.5-5308 | LEDA 433346 | 0.0629 | 1.9 | 20.00 | 0.983 | 0.982 | 0.93 | 0.06 | $-4.6$ | 7.1
SWIFTJ0029.2+1319 | PG 0026+129 | 0.142 | 1.2 | 20.00 | 0.962 | 0.905 | 0.94 | 0.11 | $-4.6$ | 8.0
SWIFTJ0034.6-0422 | 2MASX J00343284-0424117 | 0.213 | 2 | 23.45 | 0.992 | 0.892 | 0.41 | 0.06 | 30.7 | 8.2
SWIFTJ0042.9-2332 | NGC 235A | 0.0222 | 1.9 | 23.5 | 0.939 | 0.942 | 0.36 | 0.04 | 47.0 | 8.6
a Data are taken from BASS AGN catalogue (Koss et al., 2017; Ricci et al., 2017). |
b The $1\sigma$ error. |
(This table is available in its entirety in machine-readable form.) |

##  5 discussion

###  5.1 Comparison of extinction estimates in different methods

Burtscher et al. (2016) and Shimizu et al. (2018) measured the dust extinction
in different methods for some of our target AGNs. In this section, we compare
our estimates of dust extinction with those of these two studies.

Burtscher et al. (2016) estimated the dust extinction of 29 AGNs based on the
colour temperature of the $K$-band continuum emission from the unresolved AGN
core. They obtained the amount of reddening of the NIR thermal emission from
hot dust in the innermost dusty torus, which is attributed to the dust
extinction of the outer dusty torus. In Fig. 7, we compared our estimates of
the dust extinction with those of Burtscher et al. (2016), $A_{V,\rm{}B16}$
for the common 15 AGNs. In Fig. 7, $A_{V,\mathrm{B16}}$ gets larger as $A_{V}$
increases, and $A_{V}$ is consistent with $A_{V,\mathrm{B16}}$ for the 11
targets in $A_{V}<20$ mag. However, for the four targets in $A_{V}>20$ mag,
$A_{V}$ is systematically larger than $A_{V,\mathrm{B16}}$ by about 15 mag.

One possible reason for smaller $A_{V,\mathrm{B16}}$ than $A_{V}$ is the
underestimation of the intrinsic colour temperature in the $K$ band assumed in
Burtscher et al. (2016). Burtscher et al. (2016) adopted the intrinsic
$K$-band colour temperature of $T=1311\pm 129$ K, which is the mean colour
temperature of their 13 type-1–1.9 AGNs. This is somewhat lower than the
intrinsic colour temperature suggested by $K-L$ or $H-K$ colour in other works
(e.g., Glass, 2004; Kishimoto et al., 2007), although the $H-K$ colour may
suggest a hotter colour temperature than that in the $K$ band.

black In the case of heavily obscured AGNs, black another possible reason is
the effect of blackthe emission from cool extended dusty structure. black
Burtscher et al. (2016) assumed that the observed $K$-band emission came from
the hot dust region in the innermost region of the dusty torus through the
outer torus. blackHowever, recent IR interferometric observations of NGC 1068
(Gámez Rosas et al., 2022) indicates that the $LM$-band emission come from
cooler and extended structure shifted above the hot dust in the innermost
dusty torus, and the $K$-band emission is also the same. We consider that this
interpretation would explain the difference between $A_{V}$ of Burtscher et
al. (2016) and those of this study: single-epoch NIR emission for heavily
obscured AGNs may be contaminated by the cooler extended component, which
causes relatively low dust extinction, while most of the time-variable flux
component would come from the compact hot dust region in the innermost dusty
torus, which causes relatively high dust extinction. Furthermore,
blackalthough we use the NIR emission in the same way as Burtscher et al.
(2016), the observed wavelengths of $W1$ and $W2$ bands are longer than that
of the $K$ band, and so the former is more sensitive to the emission from the
heavily obscured regions than the latter. Consequently, the extinction
estimates in this study are expected to be less affected by the black emission
from such an extended structure than those in Burtscher et al. (2016), and the
difference of $A_{V}$ occurs for heavily obscured AGNs.

black Shimizu et al. (2018) estimated the dust extinction of the BASS AGNs by
measuring the attenuation of the broad H$\alpha$ emission compared to the
14–150 keV hard X-ray emission, assuming that the luminosity ratio between
them for unobscured AGNs is constant. Because optical H$\alpha$ emission is
more sensitive to dust extinction, their method is expected to be advantageous
for measuring a small amount of dust extinction for less obscured AGNs, as
they reported that the typical uncertainty was estimated as
$\sigma_{A_{V}}=1.2$ mag. In Fig. 7, we compared our estimates of dust
extinction $A_{V}$ with those of Shimizu et al. (2018), $A_{V,\rm{}S18}$. For
comparison, we calculated the $A_{V,\rm{}S18}$ for all of our targets whose
broad H$\alpha$ luminosity is available. We note that the value of
$A_{V,\rm{}S18}$ presented in Shimizu et al. (2018) is reproduced for common
AGNs. We found that both estimates were roughly consistent with one another
for $A_{V}\lesssim 20\ \rm{}mag$ within the error. However, $A_{V,\rm{}S18}$
was smaller than our estimates for $A_{V}\gtrsim 20\ \rm{}mag$. A similar
trend was observed by Xu et al. (2020) compared with their dust extinction
estimates based on the mid-IR silicate absorption feature. We suspect that
$A_{V,\rm{}S18}$ for obscured AGNs is underestimated because of the
contribution of the broad H$\alpha$ flux scattered in the polar region, as
suggested by Xu et al. (2020). blackAlthough typical degrees of H$\alpha$
polarization for AGNs are very small ($\sim 1\%$, see e.g. Ramos Almeida et
al., 2016), the scattered H$\alpha$ emission could be a significant
contributor when the nuclear emission is attenuated by a factor of more than
100 ($\gtrsim 5$ mag). black This polarized H$\alpha$ emission is often
observed for Seyfert type-2 AGNs by spectropolarimetry (e.g., Antonucci &
Miller, 1985).

blackAlthough the uncertainty of our $A_{V}$ estimate is larger than that of
Burtscher et al. (2016) and Shimizu et al. (2018), our method can estimate
$A_{V}$ for more-heavily obscured AGNs by using longer NIR wavelength, and can
be applied to large number of AGNs easily by using the all-sky NIR monitoring
data by WISE. These features of our method enables us to proceed statistical
study of dust extinction for obscured AGNs.black

Figure 7: The comparison of our dust extinction estimates $A_{V}$ with those
by Burtscher et al. (2016) ($A_{V,\rm{}B16}$) and Shimizu et al. (2018)
($A_{V,\rm{}S18}$). The black diamonds with error bars represent the data for
$A_{V,\rm{}B16}$ and $A_{V}$, the coloured dots represent the data for
$A_{V,\rm{}S18}$ and $A_{V}$, and the black dashed line represents $A_{V}$
equals $A_{V,\rm{}B16}$ or $A_{V,\rm{}S18}$. The colours of dots indicate the
different Seyfert types of the targets in the same way as in Fig. 2, and the
error bars in the middle left represent the $\pm 1\sigma$ error for the
coloured dots.

###  5.2 Distribution of dust extinction to neutral hydrogen absorption

We here discuss possible origins for the excess of $N_{\mathrm{H}}/A_{V}$
ratios for obscured AGNs over the standard value for the Galactic diffuse ISM.
There are two possible explanations for the $N_{\mathrm{H}}/A_{V}$ excess over
the Galactic value: (1) the decrease of the $A_{V}$ caused by the depletion of
the small dust grains (e.g. Maiolino et al., 2001a, b), or (2) the increase of
the $N_{\mathrm{H}}$ caused by dust-free gas clouds covering the line of sight
in the broad-line region (BLR) of the AGN (e.g. Granato et al., 1997;
Burtscher et al., 2016).

In Scenario (1), the size distribution of dust grains in the dusty torus is
larger than that of Galactic diffuse ISM, which decreases the observed dust
reddening and extinction for fixed $N_{\rm{}H}$. The lack of the 2175Å carbon
bump in the extinction curve of a typical type-1 AGN suggests the absence of
small dust grains in AGN circumstances (e.g., Maiolino et al., 2001a, b;
Gaskell et al., 2004; Czerny et al., 2004). Two mechanisms are proposed for
depleting small dust grains in the dusty torus: (i) the coagulation or
aggregation of small dust grains and their growth into larger dust grains
(e.g., Maiolino et al., 2001a, b), and (ii) the selective destruction of small
dust grains because of dust sublimation (e.g., Baskin & Laor, 2018) or the
Coulomb explosion (Tazaki et al., 2020).

In Scenario (2), dust-free gas clouds are assumed to be located in the dust-
free zone within the dust sublimation radius, such as in BLR, and their line-
of-sight crossing causes a temporary increase in $N_{\rm{H}}$ without an
increase in dust extinction. This is supported by the time variation of the
$N_{\rm{H}}$ in time scales of months to years, which is observed in some AGNs
(e.g., Risaliti et al., 2002; Burtscher et al., 2016). Burtscher et al. (2016)
demonstrated the $N_{\rm{}H}$ time variation of 15 AGNs, which vary between
$22\lesssim\log N_{\rm{}H}\ [\rm{cm^{-2}}]\lesssim 25$ in several years. This
fluctuating range roughly corresponds to the scattering range of obscured AGNs
in Fig. 5, which strengthens this scenario. Furthermore, the target-to-target
difference of $N_{\rm{}H}/A_{V}$ for AGNs with $\log N_{\rm{}H}\
[\rm{}cm^{-2}]\gtrsim 22$ can be explained by the difference in the extinction
level because of the different gas density or number of dust-free gas clouds
in the line of sight.

We here support the $N_{\rm{}H}$ excess of the dust-free gas clouds as a major
origin to explain the behaviour of $N_{\rm{}H}/A_{V}$ of obscured AGNs,
because (a) the dust-free gas scenario is easier to explain the large target-
to-target difference in $N_{\rm{}H}/A_{V}$, (b) the extinction by the dusty
torus with the Galactic dust properties can explain the lower envelope of the
distribution of $N_{\mathrm{H}}/A_{V}$ data that is consistent with that of
the Galactic diffuse ISM, and (c) the dust-free gas scenario easily explains
the $N_{\rm{}H}$ time variation. For the common ten target AGNs with Burtscher
et al. (2016), we overwrite the range of time variation of $N_{\rm{}H}$ in
Fig. 5 (Burtscher et al., 2016, and citation therein). The amplitude of
$N_{\rm{}H}$ variation is comparable to the $N_{\rm{}H}$ range of the
distribution of the obscured AGNs.

In addition, large dust grains may influence the $N_{\mathrm{H}}/A_{V}$
distribution in a minor manner. We note that the lower envelope of the
distribution of obscured AGNs in the $A_{V}\gtrsim 40$ mag region may appear
to move slightly upward from that in the $A_{V}\lesssim 40$ mag region in Fig.
5. This might indicate the effect of large dust grains in the mid-plane region
of the dusty torus that might be observed from an edge-on view. Wada et al.
(2019, 2021) suggest that the emission from the central engine does not
effectively reach the mid-plane of the dusty torus and that the snow line is
located around several parsecs from the centre. They concluded that, in such a
case, small ice grains might grow into larger dust grains through aggregation.

##  6 Conclusion

In this study, we estimated the dust extinction of X-ray-selected AGNs using
long-term NIR monitoring data obtained by WISE, and examined the relationship
between the dust extinction and hydrogen column density. Our results and
conclusions are as follows:

1\. We measured the flux variation gradient (FVG) using the flux data in $W1$
and $W2$ bands ($\equiv\beta$) of 513 samples and obtained it with an
uncertainty of $\sigma_{\beta}<0.2$ for more than 90% (463/513) of these
samples.

2\. We compared the NIR FVGs with Seyfert types and the line-of-sight neutral
hydrogen column density $N_{\rm{}H}$. For AGNs with $\log N_{\rm{H}}\
[\rm{cm^{-2}}]\lesssim 22$, the majority of which are Seyfert type 1–1.5 AGNs,
the NIR FVGs are distributed in a relatively narrow range regardless of the
Seyfert types and the $N_{\rm{}H}$. However, for AGNs with $\log N_{\rm{H}}\
[\rm{cm^{-2}}]\gtrsim 22$, the majority of which are Seyfert type 1.9–2 AGNs,
the NIR FVGs show clear reddening when $N_{\rm{}H}$ increases.

3\. We determined the intrinsic NIR FVG by fitting the NIR FVGs for unobscured
AGNs, and then we calculated the dust extinction for the 463 AGNs based on the
reddening of their NIR FVGs. This novel method is advantageous for measuring
the dust extinction for a large sample of obscured AGNs because it uses the
variable flux component attributed to the AGN, allows us to ignore the
correction for host-galaxy emission, and because the thermal emission of hot
dust in the NIR is observable even for obscured AGNs.

4\. The $N_{\rm{H}}/A_{V}$ of our obscured AGN samples is typically greater
than that of the Galactic diffuse ISM, and the lower envelope of the
distribution of these $N_{\rm{H}}/A_{V}$ is consistent with that of the
Galactic diffuse ISM. They have target-to-target scatter that spans
approximately two orders of magnitude as has been reported in the literature.
These behaviours of obscured AGNs can be explained by the scenario in which
dust-free gas clouds in the BLR crossing the line of sight cause both the
increase of $N_{\rm{}N}$ and its time variation.

## Acknowledgements

SM is supported by JST SPRING, Grant Number JPMJSP2108. HN is supported by
Japan Society for the Promotion of Science (JSPS) KAKENHI with the Grant
number of 19K21884, 20H01941, and 20H01947. HS is supported by Japan Society
for the Promotion of Science (JSPS) KAKENHI with the Grant number of 19K03917.

This publication has made 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 publication
also makes use of data products from NEOWISE, which is a project of the Jet
Propulsion Laboratory/California Institute of Technology, funded by the
Planetary Science Division of the National Aeronautics and Space
Administration.

We also acknowledge the use of public data from the BAT AGN Spectroscopic
Survey.

This research has made use of the NASA/IPAC Extragalactic Database (NED),
which is operated by the Jet Propulsion Laboratory, California Institute of
Technology, under contract with the National Aeronautics and Space
Administration.

## data availability

The WISE data used in this study are publicly available in the NASA/IPAC
Infrared Science Archive (https://irsa.ipac.caltech.edu/Missions/wise.html).
The data of BASS AGN catalogue used in this study are also publicly available
from the BASS website (https://www.bass-survey.com). The all data derived by
the analyses of flux-flux plots in this study are available in the online
version of this paper.

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