Dust-to-neutral gas ratio of the intermediate and high velocity HI clouds derived based on the sub-mm dust emission for the whole sky

We derived the dust-to-HI ratio of the intermediate-velocity clouds (IVCs), the high-velocity clouds (HVCs), and the local HI gas, by carrying out a multiple-regression analysis of the 21cm HI emission combined with the sub-mm dust optical depth. The method covers over 80 per cent of the sky contiguously at a resolution of 47arcmin and is distinguished from the absorption line measurements toward bright galaxies and stars covering a tiny fraction of the sky. Major results include that the ratio of the IVCs is in a range of 0.1--1.5 with a mode at 0.6 (relative to the solar-neighbourhood value, likewise below) and that a significant fraction, ~20 per cent, of the IVCs include dust-poor gas with a ratio of<0.5. It is confirmed that 50 per cent of the HVC Complex C has a ratio of<0.3, and that the Magellanic Stream has the lowest ratio with a mode at ~0.1. The results prove that some IVCs have low metallicity gas, contrary to the previous absorption line measurements. Considering that the recent works show that the IVCs are interacting and exchanging momentum with the high-metallicity Galactic halo gas, we argue that the high-metallicity gas contaminates a significant fraction of the IVCs. Accordingly, we argue that the IVCs include a significant fraction of the low metallicity gas supplied from outside the Galaxy as an alternative to the Galactic-fountain model.


INTRODUCTION
The high-velocity clouds (HVCs) and their slower-moving counterparts, the intermediate-velocity clouds (IVCs), are H i clouds with large velocities that a simple model of the Galactic rotation can not explain.Typically observed at positions distant from the Galactic plane and predominantly in a negative radial velocity range.The early discoveries were made over a half-century ago (e.g., Munch 1952;Münch & Zirin 1961;Muller et al. 1963;Smith 1963;Blaauw & Tolbert 1966; see a review by Wakker et al. 2004 for research history up to 1999).There have been several efforts to constrain the distance to the IVCs/HVCs using the absorption-line bracketing technique.The IVCs are located relatively close, and have typical  heights of ∼ 1-2 kpc (e.g., Wakker et al. 2008;Lehner et al. 2022, see also compilations by Wakker 2001;van Woerden & Wakker 2004 and references therein), and are likely to be the disc-halo interface objects, while the major HVCs are further away, several to ∼ 10 kpc above the disc (e.g., van Woerden et al. 1999;Thom et al. 2006Thom et al. , 2008;;Wakker et al. 2007Wakker et al. , 2008;;Smoker et al. 2011;Richter et al. 2015;Lehner et al. 2022) and belong to the inner halo.One of the important questions is whether the IVCs and HVCs are different halocloud populations (i.e., gas comes from the disc, and the one fuels the disc) or just the same population of objects with different heights and radial velocities.
The metallicity is a crucial parameter for pursuing the origin and has traditionally been measured by observing absorption lines.The previous works presented that the prominent HVCs in the Complexes ★ E-mail: t.hayakawa@a.phys.nagoya-u.ac.jp (TH) A and C have sub-solar metallicities / ⊙ ∼ 0.1-0.3(Richter et al. 2001b;Tripp et al. 2003;Collins et al. 2003Collins et al. , 2007;;Sembach et al. 2004;Fox et al. 2023, see also compilations by Wakker 2001) and are of extra-galactic origin.The IVCs have been reported to have nearsolar metallicity of ∼ 0.5-1.0solar (Wakker 2001 and references therein;Richter et al. 2001a,b;Sembach et al. 2004) and are usually explained by the so-called Galactic-fountain model in which hot gas ejected from the disc by the stellar feedback falls back in the form of neutral clouds (Shapiro & Field 1976;Bregman 1980).However, the measured metallicities often have rather large uncertainties due to such factors as ionization states and interstellar depletion (van Woerden & Wakker 2004 summarised the possible problems in the metallicity determination in their section 5).In addition, these measurements are made at a limited number of locations because of a tiny number of bright background objects and do not lead to firmly establish the metallicity of the IVC.
An alternative method for measuring metallicity is to use dust emissions.It is a reasonable proxy for absorption measurements because the dust consists of a significant fraction of heavy elements, perhaps comparable to that in the gas phase (see also discussion in Section 5.1).For this purpose, the 100 m emission obtained with IRAS is often used.However, it is not a valid measure of the interstellar dust mass.In the Rayleigh-Jeans regime, all components having various temperatures in a line-of-sight contribute to the emission proportionally to their masses, whereas lower-temperature components contribute little in the Wien regime.Generally, only in the Rayleigh-Jeans regime can we measure all dust mass.In interstellar dust, the peak wavelength of the modified Planck function is around 100 m, and the sub-mm wavelength satisfies the Rayleigh-Jeans regime.The Planck Collaboration obtained such sensitive sub-mm dust emission over the whole sky at a 5 arcmin resolution and derived dust optical depth at 353 GHz (850 m),  353 , by fitting the modified Planck function at four wavelengths from 100 to 850 m obtained by Planck and IRAS (Planck Collaboration XI 2014;Planck Collaboration Int. XLVIII 2016).
Previous works by Fukui et al. (2014, 2015, F14 andF15 hereafter), Okamoto et al. (2017, O17), Hayashi et al. (2019a, H19), and Hayashi et al. (2019b) showed that  353 characterises well the linearity of the dust emission in a number of regions of the Milky Way.The scheme was also applied to the Large Magellanic Cloud (LMC).As such, Fukui et al. (2017, F17) used the linear relationship between the velocity-integrated intensity of H i line,  H i , and  353 , and estimated that the H i ridge including R136 has a factor of two lower dust-to-H i ratio than the optical stellar Bar region in LMC.In addition, Tsuge et al. (2019, T19) presented that the N44 region near the H i Ridge in the LMC has a ∼ 30 per cent lower dust-to-H i ratio than that of the Bar region.Most recently, Fukui et al. (2021, F21 hereafter) applied the method to IVC 86−36 in the Pegasus-Pisces (PP) Arch and derived a dust-to-H i ratio relative to the low-velocity (LV) ISM (considered to be in the solar neighbourhood) of ∼ 0.2, strongly suggesting that PP Arch originated in a dust-poor environment but not in the disc.Their measurements of the dust-to-gas ratio significantly improved the preceding metallicity measurements by optical/ultraviolet atomic absorption lines of the background stellar spectrum in the IVC (Fitzpatrick & Spitzer 1997).Although the absorption-line results suggest the subsolar metal abundance, the abundance values vary by ∼ 0.5 dex depending on the atomic species, leaving the metallicity unquantified.Considering the whole above, we judge the dust-to-H i ratio measurements (F17; T19; F21) to be most appropriate and adopt it in the present work with an aim to extend the metallicity measurement of the H i gas over the whole sky outside the Galactic plane.
The aim of the present paper is to use  353 and the 21 cm H i emission and to derive the dust-to-H i ratio in the IVCs and possibly the other H i components.It is not yet examined well if the low column density H i gas in the IVCs and HVCs have similar dust properties to the Galactic disc, and the extension of the dust emission to the halo H i gas is a challenge.Nonetheless, we believe that the paper benefits the community by reporting possible new aspects of the dust emission in these H i gases as well as the evolutionary trend of the Galactic ISM.The H i is an important mass reservoir, while the ionized gas may carry a large mass similar to the neutral H i gas.We focus on the neutral H i gas in the present paper because the dust-togas ratio in H i is one of the essential factors giving information on the chemical evolution.The paper is organized as follows: Section 2 describes the datasets used, and Section 3 gives the dust and gas properties analysed by the present work.Section 4 shows the dust-to-H i ratio distribution in middle/high latitudes including the individual outstanding components.Section 5 gives a discussion focusing on the implications of the IVC dust-to-H i ratio.Section 6 concludes the present work.

DATASETS
We used archival H i data from the HI4PI full-sky survey (HI4PI Collaboration 2016) and Planck PR2 dust data (Planck Collaboration Int.XLVIII 2016).Both are on the HEALPix 1 (Górski et al. 1 http://healpix.sourceforge.net/2005) projection, and we performed the following analyses without reprojection.The HEALPix is a framework for the pixelization of the data on the sphere.Unlike general regular grid maps, the resolution of a HEALPix map is defined by the resolution parameter  side (generally a power of 2).The map has 12 side2 pixels of the same area over the whole sky, i.e., each pixel has a solid angle of 4/(12 side 2 ) sr. See section 4 of Górski et al. (2005) for further details, and refer to table 1 in the same publication for the correspondence between  side and mean pixel spacing.

H i data
The HI4PI data combines those from the first release of the Effelsberg-Bonn H i Survey (EBHIS, Winkel et al. 2016) and the third revision of the Galactic All-Sky Survey (GASS, Kalberla & Haud 2015).The brightness temperature noise level is ∼ 43 mK (rms) at a velocity resolution of 1.49 km s −1 .The velocity coverage (with respect to the Local Standard of Rest, LSR) is The FWHM angular resolution of the combined map is 16.2 arcmin, twice better than the Leiden/Argentine/Bonn (LAB) Survey (Kalberla et al. 2005).The data are divided into 19 parts, and each of them is presented as a FITS binary table containing spectra on the HEALPix grid with  side = 1024 (the mean pixel spacing is 3.4 arcmin).

Dust optical depth data
Planck Collaboration Int.XLVIII (2016) used Planck 2015 data release (PR2) maps and separated Galactic thermal dust emission from cosmic infrared background anisotropies by implementing the generalized needlet internal linear combination (GNILC) method.The GNILC dust maps have a variable angular resolution with an effective beam FWHM varying from 5 to 21.8 arcmin (see fig. 2 of Planck Collaboration Int.XLVIII 2016).The authors then produced the dust optical depth, temperature, and spectral index maps by fitting a modified blackbody model to the GNILC dust maps at 353, 545, 857 GHz, and IRAS 100 m map.We used the  353 data released version R2.01 in the HEALPix format with  side = 2048 (the mean pixel spacing is 1.7 arcmin).The median relative uncertainty in  353 is ( 353 )/ 353 = 0.037 in || > 15 • .Note that O17, H19 2 , F17, T19 and F21 used Planck 2013 data release (PR1) dust data released version R1.20 (Planck Collaboration XI 2014), whereas Hayashi et al. (2019b) and the present study used the Planck Collaboration Int.XLVIII (2016) data.We compared the two datasets and found a good consistency between them (see Appendix A).

Pre-processing of the data
Figure 1 shows the Galactic longitude-LSR velocity diagrams for H i in both low latitudes (|| < 15 • ) and middle/high latitudes (|| > 15 • ).The diagrams reveal the Galactic-rotational motion in the former while indicating negligibly observable rotation in the latter.Therefore, we used the LSR velocity to define the IVCs in the The H i data were integrated into the five velocity ranges (summarized in Table 1) and combined into a single HEALPix format data for each velocity range.The HI4PI H i and the Planck dust data have different resolutions, and we matched them as follows: (1) if the resolution of the Planck data is 21.8 arcmin at a data point, the H i data were smoothed, (2) otherwise (5.0, 8.7, 11.7, or 15.0 arcmin), the dust data were smoothed to 16.2 arcmin resolution.Then both were degraded to  side = 256 (the mean pixel spacing is 13.7 arcmin) to reduce the computational cost.In the following sections, we used these processed data unless otherwise noted.

NEUTRAL GAS AND DUST OPTICAL DEPTH PROPERTIES
Fig. 2 shows the spatial distributions of  H i in the negative (i.e., blueshifted) intermediate velocity (NIV,  LSR = −100-−30 km s −1 ), positive (red-shifted) intermediate velocity (PIV,  LSR = +30-+100 km s −1 ), and LV (| LSR | < 30 km s −1 ) components, respectively.Those of the negative-and positive high-velocity (NHV and PHV, | LSR | > 100 km s −1 ) components are also shown as silhouettes.Fig. 3 shows the distribution function of the apparent amount (the product of the column density and the solid angle) of H i gas, as a function of the column density, where  = NHV, NIV, ).The only small apparent amount of H i gas is in the positive velocity ranges, except for the Magellanic System.The LV components covering the whole sky are considered to be the local volume gas within 300 pc of the sun.They have an H i column density range of 10 20 -10 21 cm −2 with a peak at 3-4 × 10 20 cm −2 , which is one order of magnitude higher than IVCs (1 × 10 19 to 2 × 10 20 cm −2 ) and HVCs (≲ 1 × 10 20 cm −2 ).Fig. 4 shows the spatial distribution of  353 , and Fig. 5 shows the correlation plots between  353 and  H i in the || > 15 • skies with masking described in Section 4.1.Here, the  H i values are velocity-decomposed, but the  353 values are not.The good correlation between the two quantities is found only in the LV component, while the other components show a rather poor correlation.The plots demonstrate that the LV component dominates  353 in almost every single direction, consistent with the remarkably similar spatial distribution of the LV H i gas (Fig. 2(c)) and  353 (Fig. 4).d The total apparent amount of H i gas of the pixels with  (  )/ soln better than 0.1 and the percentage of above.

THE ESTIMATE OF THE DUST-TO-H i RATIO
The present work uses  H i and  353 to derive the dust-to-H i ratio in the LV, IVC and HVC components separately.The basic concept introduced in the previous studies (F17; T19; F21) is that the regression coefficient of a single regression between  H i and  353 (or the gradient of the best-fit line in the  H i - 353 plane) gives the dustto-H i ratio.However, in the framework of the present study,  353 is the sum of the contribution from the five velocity components, and it is impossible to estimate each ratio by a single regression between, e.g.,  H i,NIV and a velocity-decomposed  353,H i,NIV .Assuming that  353 is a linear combination of multiple  H i, -terms (see Section 4.2), we can estimate the dust-to-H i ratios of all components simultaneously as the partial coefficients of a multiple regression, or the gradient of the best-fit "plane" in an ( + 1)-dimensional space,  • 1, +67.• 6) (the black sphere and the red ones, respectively).The radius of each sphere is proportional to the 1/3 power of the weight given by equation ( 9) (decays with the angular distance in the sky from the regression point, maximum is 1.0 and truncated below 10 −3 ).The  H i,NHV ,  H i,PIV , and  H i,PHV are small enough for these data points, and the contribution from these components to  353 can be regarded as zero.The best-fit surface given by equation ( 8) with  NIV (150.• 1, +67.where  is the number of the  H i, terms and  = 5 in the present study. The dust-to-H i ratio of each velocity component is not uniform across the sky, and we, therefore, used the geographically weighted regression (GWR) technique (Brunsdon et al. 1996;Fotheringham et al. 2002) which allows us to derive the spatial distribution of the dust-to-H i ratio, whereas the general regression estimates a set of global and spatially-invariant coefficients.The GWR technique is an outgrowth of the ordinary least-squares (OLS) regression.It is an advanced extension of the moving-window technique and estimates local coefficients at each pixel using a distance-decay weighting function.
Figure 6 is a visualization of the regression-analysis process at a regression point (a pixel where we want to estimate the local coefficients) (, )=(150.• 1, +67.• 6), in a three-dimensional ( H i,LV - H i,NIV - 353 ) space4 .We find that the "plane" 5 immediately gives the two gradients  353 - H i,LV and  353 - H i,NIV , and these gradients give the dust-to-H i ratios of the LV and NIV component, respectively.
As presented in Section 3,  353 is dominated by the LV component, with ∼ 1-2 orders smaller contributions from the other components.One might wonder whether extracting such faint and reduced contributions is possible.The issue could be severe if we subtract the the following subsections, but  H i,NHV ,  H i,PIV and  H i,PHV are negligible in many cases. 5The best-fit "plane" is slightly curved due to the small non-linearity of the sub-mm emission by the dust growth as expressed by an exponent  = 1.2 (see Section 4.2).The black-filled ellipses cover nearby galaxies taken from the catalogue by Karachentsev et al. (2013) and samples studied by Wang et al. (2016).The light-grey shadow in || > 15 • presents the areas where LV H i emission is saturated.
contribution from the LV component and analyze the remaining, as done in F21.In the present method, if we explain using the above model, the gradients in the  H i,NIV and  H i,LV directions are mutually independent, and it does not matter that the LV is dominant in  353 .The uncertainty of  353 (( 353 ), typically on the order of 10 −8 in the directions analyzed) probably provides the extraction limit in the present method.The HVCs, Magellanic Stream and some IVCs (such as PP-Arch) are close to the limit (typically estimated to be several × 10 −8 ), whereas the IVC complexes IV Arch/Spur have large enough  353 (≳ 10 −7 ) (see also Section 4.3).

Masking
The pixels which meet any of the criteria (a)-(c) were masked before the regression analysis; • 0) where the LV H i emission is judged to be saturated.In both samples, the H i integrated intensity of the NHV, NIV, PIV, and PHV components are small enough ( H i, < 5.5 K km s −1 or  * H i, < 1 × 10 19 cm −2 ), and their contribution to  353 is approximately zero.
(a) Galactic latitude is || < 15 • in order to eliminate contamination by the Galactic-disc components far from us.
(b) CO emission is detected at the 3 level (1.4 K km s −1 ), where the molecular gas is not negligible compared to the atomic gas.We used the Planck PR2 type 2 CO(1-0) map (Planck Collaboration X 2016).The median uncertainty of the map is ( CO ) = 0.44 K km s −1 in || > 15 • .
(c) The areas covering nearby galaxies listed in either or both the catalogue by Karachentsev et al. (2013) and the samples studied by Wang et al. (2016).Fig. 7 summarises the masked areas.

Formulation
The observed  353 is the sum of the contribution from each component, As it is not feasible to simultaneously estimate both   (  ,   ) and  H i, (  ,   ) as free parameters in a regression model due to the compounded relationship arising from their multiplication, we assume that under the optically thin approximation of the H i emission ( H i, ≪ 1) (see Section 4.3 for exception handling for optically thick cases).Then, we set up the regression equation using equations (4), ( 5) and (7) as The constant term  353,0 (  ,   ) is the amount of  353 unaffected by variation in  H i, , e.g., the zero-level offsets (see also discussion in Section 5.2), and has been found empirically to take far off from zero if H i emission is saturated due to the optical depth effect (Fig. 8).

Regression analysis and resulsts
We estimated  (  ,   ) at each pixel using the GWR technique.We used a truncated-Gaussian weighting function6 where    is the great-circle angular distance between the regression point and the -th (  = 1, • • • , ) pixel (  ,   ),  bw is the bandwidth of the weighting function, and  tr is the truncation radius.We used  bw = 20 arcmin (FWHM = 47 arcmin), and  tr = 1.2 degree (truncates below   (  ,   ) = 1 × 10 −3 ).The  cannot take negative values; negative  does not make physical sense and will overestimate the other coefficients.We thus introduced a non-negative leastsquares (NNLS) regression technique.In the case either or both of the dust-to-H i ratio and the column density of the component  is  2).The grey shadow presents the areas with no valid  NHV or  (  NHV ) values.
Fig. 7 shows the saturated H i area.Note that saturation should not be an issue for the IVCs and HVCs due to their low column densities.This procedure was applied only to the LV component.Fig. 9 shows the spatial distribution of estimated / soln , where the solar neighbourhood dust-to-H i ratio  soln = 6.29 × 10 −26 cm 2 is an empirical constant determined so that the LV component has a mode value of  LV / soln = 1.0 and / soln present the ratios relative to the solar-neighbourhood value.Figs.10-13 show closeup views of / soln and their standard error ( )/ soln focusing on four regions-of-interest (ROIs), NIV-Galactic North (GN), NIV-Galactic South (GS), NHV-GN, and the Magellanic Stream (MS).Fig. 14 shows the distribution functions of / soln and ( )/ soln in the LV and the four ROIs for all the pixels with valid values and also the fraction determined with ( )/ soln better than 0.1.We find that in the IVCs and HVCs in the ROIs, the fraction with significant determination is ∼ 50 per cent.This fraction is significantly larger than the previous results when the coverage in the sky is considered, particularly in terms of the solid angle.We give a detailed description of the individuals in the following subsection.

The LV component
Fig. 14(a) shows the distribution function of  LV / soln .The ratio is determined in most pixels with a sufficiently high confidence level better than (  LV )/ soln = 0.1.The peak of the distribution is normalised to 1.0 as intended, and a Gaussian function well approx-imates the distribution.The measurements of the metallicity of G dwarfs were made for the local volume within 25 pc by many authors in the last five decades (e.g., Rocha-Pinto & Maciel 1996).These works indicate that the dispersion of metallicity is 0.2-0.3dex with a peak at −0.2 dex.The dispersion is somewhat larger than the present one.Considering the much longer age of G dwarfs, several Gyr, than the dynamical timescale of H i gas, ∼ Myr, the difference seems reasonable, and the H i gas in the local volume seems to be well mixed probably by the turbulent motion.

The NIV-GN region
The NIV-GN region ( = 90 • -270 • ,  > 15 • ) contains a large apparent amount of IVC in the complexes IV Arc/Spur, LL IV Arch and Complex M (Complex M straddles the −100 km s −1 boundary, part of which is in the NHV range in the present study).
The  NIV / soln in this region is peaked at 0.6 with a broad continuous shape ranging from less than 0.1 to high values beyond 1.5 (Fig. 14(b)).The absorption measurements toward bright background objects indicated the metallicity is near solar (see Table 2), i.e., not much different from the LV components.The present dust-to-H i ratio distribution gives a significantly different trend; there exists a significant amount of gas having  NIV / soln < 0.5 (of the fractions with significance better than the (  NIV )/ soln = 0.1 threshold, 10 per cent display  NIV / soln < 0.3 and 20 per cent  NIV / soln < 0.5).This difference is likely caused by the tiny fraction of the IVCs observed by the absorption measurements.2).The grey shadow presents the areas with no valid  or  (  ) values.

The NIV-GS region
The most prominent object in the NIV-GS region ( = 45 • -180 IVCs are in low latitudes, some of which appear to be connected to the Galactic disc.The  NIV / soln values in the NIV-GS region exhibit a gently flat distribution ranging from 0.1 or below to 1.5 and above (Fig. 14(c)).

The NHV-GN region
The NHV-GN region ( = 60 • -180 There have been several attempts to detect thermal dust emission associated with HVCs (e.g., Wakker & Boulanger 1986;Saul et al. 2014), some possible detections (e.g., Peek et al. 2009;Lenz et al. 2016), but no confident one have been reported8 .The difficulty is due to the low column density and the low metallicity.In the present work, barely sufficient column density in the scattered small areas allows the dust-to-H i estimation (Fig. 12).Fig. 14(d) indicates that  NHV / soln in this region peaked at 0.1-0.2 with a relative uncertainty of (  NHV )/  NHV ∼ 1 and is even lower than  NIV / soln in the NIV-GN region (Fig. 14

The Magellanic Stream
The Magellanic Stream is a well-known giant filamentous structure, starting at the Magellanic Clouds and trailing over 100 degrees, having a large velocity gradient from +180 km s −1 near the Clouds to −450 km s −1 at the tip.Fig. 13 is a patchwork of the NHV, NIV, PIV, and PHV maps showing the spatial distribution of / soln and Fig. 14(e) shows the distribution function.The MS has the lowest dust-to-H i ratio among the four ROIs, with a mode at ∼ 0.1 and a tail extending to ∼ 0.5.

Comparison of the present dust-to-H i ratio with the absorption-line measured metallicity
The present work revealed the dust-to-H i ratio distribution which covers a large fraction of the ISM, the IVCs, and the HVCs.Dust only traces solid-phase heavy elements but is often, and also in the present study, used as a proxy for the gas-phase heavy elements assuming a nearly constant dust-to-metal (DTM) ratio.Recent studies of nearby galaxies show that well-evolved galaxies with metallicities 12 + log(O/H) ≳ 8.2 (cf. the solar value is 8.7, e.g., Asplund et al. 2009) have a more or less constant DTM ratio (e.g., De Vis et al. 2019).In the context of the low-velocity gas of the solar neighbourhood, these results support the assumption that the DTM ratio is constant and that the dust-to-H i ratio is a good proxy for metallicity.We demonstrate in Fig. 15 that the present estimates are not far from the previous absorption-line measurements in Table 2, whereas the discrepancies toward PG 0804+761 (LL IV Arch), RX J2043.1+0324(the Smith Cloud) and Fairall 9 (the MS) suggest potential variations in the DTM ratio.Based on the current limited observational data, it is hard to evaluate how generally the abovementioned assumption holds for IVCs and HVCs.

Impact of the dust associated with the warm ionized medium on the estimates of the dust-to-H i ratio
The diffuse warm ionized medium (WIM) outside of localized H ii regions is another significant component of the ISM.The absorptionline studies revealed that HVCs and IVCs are associated with the ionized components (e.g., Sembach et al. 2003;Shull et al. 2009;Lehner & Howk 2011).The velocity-resolved high-sensitivity surveys of diffuse H  emission using Wisconsin H  Mapper (WHAM) showed that the neutral and ionized components trace each other well, though the detailed structure is not identical (e.g., Tufte et al. 1998;Haffner et al. 2001;Barger et al. 2012Barger et al. , 2017)).The estimated mass of the associated ionized component is roughly comparable to that of the neutral counterpart.
The observational evidence for the WIM-related dust is reported (Howk & Savage 1999;Dobler et al. 2009;Werk et al. 2019), but the extent to which it contributes to the FIR/submillimeter optical depth is poorly understood.Previous works attempted to decompose the dust emission intensity into the WIM-related and H i-related components.Lagache et al. (1999Lagache et al. ( , 2000) ) claimed that they made a decomposition for the first time and that the FIR dust emissivity associated with WIM is close to the one with H i. On the contrary, Odegard et al. (2007) reported that the WIM-related component is consistent with zero within the uncertainties.Casandjian et al. (2022) reported that the FIR emission of the dust associated with a Reynolds layer of ionized hydrogen is below their detection limit.These authors used regression models expressing dust emission intensity as a linear combination of  H i and H  emission intensity ( H ).We performed an alternative analysis by taking another approach.
In Section 4, we estimated the dust-to-H i ratio  by making a least-squares fit to the regression model of equation ( 8).We refer to this as the "without-WIM-terms model" in this section and set up another "with-WIM-terms model" under the assumption that there is some contribution to  353 from the dust associated with the WIM ( 353,H + , ).The dust optical depth is a function of column density, whereas  H is the line-of-sight integral of the electron temperature  e , density  e and ionized hydrogen density where  is the line-of-sight path length over which the electrons are recombining.The distribution of  e and  e ≃  H + in a line of sight is   Kunth et al. (1994), 2 Wakker (2001), 3 Collins et al. (2003), 4 Collins et al. (2007), 5 Tripp et al. (2003), 6 Fox et al. (2023), 7 Richter et al. (2001b), 8 Sembach et al. (2004), 9 Fox et al. (2016), 10 Cashman et al. (2023), 11 Danly et al. (1993), 12 Spitzer & Fitzpatrick (1993), 13 Fitzpatrick & Spitzer (1997), 14 Fox et al. (2010), 15 Fox et al. (2013), 16 Gibson et al. (2000), 17 Richter et al. (2013) f Estimated dust-to-H i ratio relative to the solar-neighbourhood value, toward the nearest neighbour of the background object (this work) g With ionization correction h The average of the −66.usually unknown.Assuming that they are constant over the emitting region, we can approximate equation ( 11) as and  H + is given as where  = ∫ .Then, we assume that the WIM term is proportional to  H 1/2 .The WHAM Sky Survey (Haffner et al. 2003(Haffner et al. , 2010) ) is the sole velocity-resolved all-sky H  survey data presently available but provides us with no information on the high-velocity (| LSR | ≳ 100 km s −1 ) WIM.If  353,H + ,NHV and  353,H + ,PHV are tiny fraction, close to  353,H i,NHV and  353,H i,PHV (on the order of 10 −8 or less), then approximating them to 0 would not significantly affect the results.In summary, the WIM term is given as We emphasize that the coefficients   in the WIM terms are not dustto-WIM ratios, unlike the coefficients   in the H i terms.Using  H instead of ( H ) 1/2 almost reproduces the same results described below, indicating that the reality of the WIM terms in the model is not a very important issue in this discussion.Fig. 16, the H  intensity ( H ) maps in the NIV, LV, and PIV velocity ranges, shows that the data suffer from zero-level offsets (typically ∼ 0.1-0.2R, where 1 R = 10 6 /4 photons cm −2 s −1 sr −1 ) and block discontinuity, probably due to data-reduction issues.We, therefore, used the data in  = 90 • -150 • and  > 45 • in the following analysis, where the WHAM data are of acceptable quality (judged by the eye), and the optically-thick LV H i occupies a small fraction of the area.The WHAM-SS is undersampled with a 1 deg beam at a 1 deg spacing (see Haffner et al. 2003, section 2.2), having a ∼ 3-12 times lower resolution than the HI4PI and Planck data, and we convolved the  H i and  353 data with the WHAM beam centred on each WHAM pointing.Following Finkbeiner (2003), we approximated the WHAM beam to be a smoothed top-hat function where  is the angular distance from the beam center,  0 and  s are set to 0.5 and 0.025 deg, respectively.Fig. 17(a) compares the estimates of  NIV obtained using the "without-WIM-terms model" and those obtained using the "with-WIM-terms model", and Fig. 17(b) compares  LV .The two models produce statistically similar results, although minor discrepancies exist in individual data points.Fig. 18 shows a clear tendency that the constant term  353,0 in the "with-WIM-terms model" is somewhat smaller (note that this is only a comparison of the results from the two models and does not prove or measure  353,H + , , as it is uncertain how realistic the WIM terms in the model are).
In multiple regression analysis, generally, the constant term represents the portion of the dependent variable ( 353 in the present study) which is not significantly correlated with the independent variable ( H i, in the "without-WIM-terms model").The contribution to  353 from dust associated with WIM, if any, is included in the constant term in the "without-WIM-terms model" and has negligible impact on the estimates of  in the present study unless it is highly correlated with the H i spatial distribution on a scale of ∼ 1 degree.

The dust-to-gas ratio of the H i components
The derived dust-to-H i ratio in the LV component and the four ROIs (Section 4) exhibits the following characteristics.
(i) The ratio in the LV component has a distribution peaked at  LV / soln = 1.0 with a Gaussian shape having dispersion somewhat smaller but similar to the metallicities of G dwarfs in the local volume.
(ii) The NIV-GN region has a distribution with a significant fraction (25 per cent) of gas with a low ratio of  NIV / soln < 0.5.In the NIV-GS region, PP Arch and IVC 105−24 exhibit a low  NIV / soln ≲ 0.3.
(iv) The Magellanic Stream has the lowest distribution peaked at / soln < 0.1 and more than 80 per cent of the fraction have ratios / soln < 0.2.We find that the LV, NHV-GN and the Magellanic Stream results are reasonably consistent with the previous absorption measurements, whereas the IVC results differ significantly from the previous results.
The conventional interpretation of the IVC is the Galactic fountain model, i.e., the stellar explosion in the plane feeds the high-metallicity gas to the Galactic halo to form the IVCs.We argue that the present results require a model that the extragalactic low metallicity gas also supplies a significant fraction of the IVCs.Otherwise, the significant fraction (20 per cent) of the gas with  NIV / soln < 0.3-0.5 cannot be explained.Further, the interaction of the low metallicity IVCs with the high metallicity gas in the Galactic halo is an important mechanism to increase the metallicity in the IVCs.Gritton et al. (2017) and Henley et al. (2017) have recently discussed this possibility.It is very likely that the IVCs are dynamically interacting with the Galactic halo gas, as shown in IVC 86−36 by F21.The interaction causes deceleration of the IVCs along with the accretion and mixing with the halo gas.The interaction likely produces a trend that the metallicity, as well as the cloud mass, is increased along with the deceleration in the sense that the higher metallicity gas has a lower velocity.The  (Haffner et al. 2003(Haffner et al. , 2010) ) in the NIV velocity range shown in the same projection as Fig. 2. The southern limit of the WHAM northern sky survey (Haffner et al. 2003),  J2000 = −30 • , is presented by thick broken lines.The contours show  H i,NIV = 30 K km s −1 in the same velocity range, and the cyan silhouette shows the NHV H i components with  H i,PHV > 10 K km s −1 .The grey solid lines indicate || = 15 • .The circular sectors bounded by thick solid lines present the two regions analyzed in Section 5.2.(b) Same as (a) but for the PIV velocity range.The contours show  H i,PIV = 30 K km s −1 and the magenta silhouette shows the PHV H i components with  H i,PHV = 10 K km s −1 .
effect is approximately quantified as the mass increasing by a factor of two, accompanying a velocity decrease by a factor of two due to momentum conservation.Concerning the metallicity, the interaction between an infalling cloud with a metallicity of 0.2 and the halo gas with a metallicity of 1.0 having the same mass will result in a metallicity of 0.6, which is closer to the halo metallicity than the infalling cloud, when the two gases are fully mixed.The interaction has a strong impact on the observed metallicity of the IVCs.

CONCLUDING REMARKS
We conducted a multiple-regression analysis combining 21 cm H i emission data with sub-mm dust optical depth  353 to investigate the dust-to-H i ratio in the H i gas outside the Galactic plane.We have resulted in a comprehensive dust-to-H i ratio distribution for the IVCs, HVCs, Magellanic Stream, and the low-velocity component in the solar neighbourhood at an effective resolution of 47 arcmin, which differs from the conventional optical absorption line measurements toward bright galaxies and stars that cover a tiny fraction of the gas.The main conclusions are summarised below.
(i) The present study allowed us to derive the dust-to-H i ratio over a far greater portion of the H i gas than the previous studies.The major results include that the dust-to-H i ratio of the IVCs (relative values to the solar-neighbourhood value) varies from < 0.2 to > 1.5 and that a significant fraction, 20 per cent, of the IVCs includes the dust-poor gas of < 0.5 with a mode at 0.6.In addition, it is shown that more than 50 per cent of the HVCs in Complex C have ratios of < 0.3 and that the Magellanic Stream has the lowest ratio with a mode at ∼ 0.1.
(ii) We argue that a large fraction of the dust-poor IVC indicates that the infalling external H i gas of low metallicity is significant in the IVCs.A possible picture is that the low metallicity IVCs are falling onto the Galactic plane and are interacting with the high metallicity gas in the Galactic halo.Such interaction is evidenced by the kinematic bridge features connecting the IVCs with the disc as observed, for instance, in IVC 86−36 in PP Arch, and will cause accretion of the halo gas onto the IVCs.The accretion increases the metallicity and mass of the IVCs and decelerates their infall velocity, as shown by the theoretical work.Therefore, the observed fraction of the low metallicity IVCs gives a secure lower limit for the fraction.By the interaction, the IVC mass will be doubled accompanying a decrease of the infall velocity by a factor of 2 by the momentum conservation as well as the increase of the metallicity.This suggests that the fraction of the observed low metallicity IVCs should be doubled to 40 per cent prior to the onset of the collisional interaction in the halo.Consequently, we suggest an overall picture that both the external infall and the Galactic fountain work to produce the IVCs, where the mass of the IVCs involved is comparable between the two mechanisms.
(iii) To pursue further the implications of the IVCs, we need more systematic studies of the H i gas over the whole sky by focusing on the individual regions and the overall properties of the IVCs and HVCs.The forthcoming new instruments, including SKA and ngVLA, are expected to provide important opportunities in these studies by covering the Local Group galaxies and nearby and more distant galaxies.
Fig. 5 also presents nearly vertical features displaced or broadened along the  353 axis in the high-and intermediate-velocity components.It indicates that dust-poor HV/IV H i overlaps with the dust-rich LV H i.

Figure 3 .
Figure 3. Distribution of the apparent amount of H i gas (the product of the column density and the solid angle,  * H i, ) as a function of the column density under the optically thin approximation  * H i (equation (2)) in the NHV, NIV, LV, PIV and PHV velocity ranges (from top to bottom), excluding toward molecular clouds or nearby galaxies (see Section 4.1).The panels in the left column are for  > +15 • , and those in the right column are for  < −15 • .The black lines in panels (a)-3 and (b)-3 further exclude the saturated LV H i emission (see Section 4.3).The shades in magenta show the contribution from the Magellanic Stream.

Figure 4 .
Figure 4. Spatial distribution of 353 GHz dust optical depth ( 353 ) (Planck Collaboration Int.XLVIII 2016) shown in the same projection as Fig. 2. The overlaid silhouette outlines the molecular clouds with  CO > 1.4 K km s −1 (Planck Collaboration X 2016).

Figure 5 .Figure 6 .
Figure 5. Density plots showing the correlation between  353 and  H i in the NHV, NIV, LV, PIV and PHV velocity ranges (from top to bottom), excluding toward molecular clouds or nearby galaxies (see Section 4.1).The plots in panels (a)-3 and (b)-3 further exclude the saturated LV H i (see Section 4.3).The  H i values are obtained by integrating within each velocity range, but  353 values are the total amount on the line-of-sights (not velocity-decomposed).The panels in the left column are for  > +15 • and those in the right column are for  < −15 • .The contours panels (a)-3 and (b)-3 contain 50, 75 and 90 per cent of data points.The solid lines indicate the solar-neighbourhood dust-to-H i ratio ( soln ) line in this work  H i (K km s −1 ) = (1.59× 10 25 )/(1.82× 10 18 ) ×  353 Figure6.A toy model of the regression analysis.The 3D scatter plot in the  H i,NIV - H i,LV - 353 space shows the data points at and in the neighbour of a regression point (, ) = (150.• 1, +67.• 6) (the black sphere and the red ones, respectively).The radius of each sphere is proportional to the 1/3 power of the weight given by equation (9) (decays with the angular distance in the sky from the regression point, maximum is 1.0 and truncated below 10 −3 ).The  H i,NHV ,  H i,PIV , and  H i,PHV are small enough for these data points, and the contribution from these components to  353 can be regarded as zero.The best-fit surface given by equation (8) with  NIV (150.• 1, +67.• 6)/ soln = 0.70 ± 0.09 and  LV (150.• 1, +67.• 6)/ soln = 1.40 ± 0.08 is also shown.
or H I emission is saturated W CO > 1.4 K km s −1 Nearby galaxies b > 0°b < 0°F igure 7. The masked and saturated H i areas in the present study (Sections 4.1 and 4.3) shown in the same projection as Fig. 2. The low latitude zone || < 15 • is delimited by thick solid lines.The dark-grey shadow outlines the area with  CO > 1.4 K km s −1 (Planck Collaboration X 2016).
Figure 8.(a) Scatter plot showing the  353 - H i,LV correlation for data points at and in the neighbour (within 1.2 deg) of a regression point (, ) = (345.• 4, +35.• 7) where the LV H i emission is judged to be optically thin (not saturated).The filled circle is the data point at the regression point, and the open circles are those in the neighbour.The radius of each circle is proportional to the 1/2 power of the weight given by equation (9), (decays with the angular distance in the sky from the regression point, maximum is 1.0 and truncated below 10 −3 ).The horizontal bars attached to the data points show  (  353 ) (Planck Collaboration Int.XLVIII 2016).The solid lines show local regression lines described by equation (8) at the position.The dashed line indicates the solar-neighbourhood dust-to-H i ratio ( soln ) line in this work (see Section 4).(b) Same as (a) but for data points within 1.2 deg of (, ) = (185.• 0, −39.• 0) where the LV H i emission is judged to be saturated.In both samples, the H i integrated intensity of the NHV, NIV, PIV, and PHV components are small enough ( H i, < 5.5 K km s −1 or  * H i, < 1 × 10 19 cm −2 ), and their contribution to  353 is approximately zero.

Figure 9 .
Figure 9. (a) Spatial distribution of the relative dust-to-H i ratio  NIV / soln encoded as colour hue and  H i,NIV presented by brightness, shown in the same projection as Fig. 2. The grey shadow presents the areas with no valid  NIV values.The white contours in the lower panel outline the Magellanic Stream.

Figure 9 -
Figure 9 -continued (d) Same as (a) but for the PIV.

)Figure 10 .)Figure 11 .
Figure 10.(a) A close up view of Fig. 9(a) focusing on the NIV-GN region containing IV Arch/Spur, LL IV Arch, and Complex M. (b) Same as (a) but the colour hue shows the standard error  (  NIV )/ soln .The filled circles indicate the positions of the background objects where absorption measurements were made (see Table2).The grey shadow presents the areas with no valid  NIV or  (  NIV ) values.

Figure 12 .
Figure 12.(a) A close up view of Fig. 9(b) focusing on the NHV-GN region including Complexes A/C/M.(b) Same as (a) but the colour hue shows the standard error  (  NHV )/ soln .The filled circles indicate the positions of the background objects where absorption measurements were made (see Table2).The grey shadow presents the areas with no valid  NHV or  (  NHV ) values.

Figure 13 .
Figure 13.(a) A patchwork of the NHV, NIV, PIV, and PHV maps showing the spatial distribution of the dust-to-H i ratio  / soln (colour) and  H i, (brightness) in the Magellanic Stream.(b) Same as (a) but the colour hue shows the standard error  (  )/ soln .The filled circles indicate the positions of the background objects where absorption measurements were made (see Table2).The grey shadow presents the areas with no valid  or  (  ) values.

Figure 14 .
Figure 14.(a)-1 Density plot showing the two-dimensional probability density with respect to  LV / soln (horizontal axis) and its standard error  (  LV )/ soln (vertical axis) for the LV component, weighted by the apparent amount of H i gas  * H i,LV  of each pixel.The contours contain 50, 75, and 90 per cent of the total amount.The horizontal line indicates  (  LV )/ soln = 0.1.(a)-2 The filled histogram shows (i) a projection of the above onto the horizontal axis, and the solid line histogram shows (ii) the subset better than  (  LV )/ soln = 0.1.The overlaid curves depict the cumulative percentage of (ii) (the scale is displayed on the right-hand vertical axis).(a)-3 The ratio of (ii) to (i).

Figure 15 .
Figure 15.The correlation between the ion abundances / ⊙ obtained by absorption measurements (see Table 2) and  / soln (this work).The dashed line shows the line where both have the same value.The filled and open symbols represent that the / ⊙ correspond to measurements from O i and S ii observations, respectively.The arrows attached to the filled circles indicate that the / ⊙ values are the lower limit.
Figure 16.(a) The greyscale image shows the spatial distribution of H  intensity  H(Haffner et al. 2003(Haffner et al. , 2010) ) in the NIV velocity range shown in the same projection as Fig.2.The southern limit of the WHAM northern sky survey(Haffner et al. 2003),  J2000 = −30 • , is presented by thick broken lines.The contours show  H i,NIV = 30 K km s −1 in the same velocity range, and the cyan silhouette shows the NHV H i components with  H i,PHV > 10 K km s −1 .The grey solid lines indicate || = 15 • .The circular sectors bounded by thick solid lines present the two regions analyzed in Section 5.2.(b) Same as (a) but for the PIV velocity range.The contours show  H i,PIV = 30 K km s −1 and the magenta silhouette shows the PHV H i components with  H i,PHV = 10 K km s −1 .
continued (c) Spatial distribution of  H in the LV.The contours show  H i,LV = 30, 100, and 300 K km s −1 .

Figure 17 .Figure 18 .Figure A1 .Figure B1 .Figure B2 .Figure B3 .
Figure 17.(a)-1 The black and red lines show the  * H i,NIV -weighted probability density with respect to  NIV / soln obtained using the "without-WIM-terms" and "with-WIM-terms" models, respectively (see Section 5.2).(a)-2 Scatter plot showing a correlation between  NIV / soln (without-WIM-terms model) and  NIV / soln (with-WIM-terms model).The solid line shows the line where both have the same value.(b) Same as (a) but for  LV / soln .
• • • , PHV represents the velocity ranges listed in Table 1, (  ,   ) are the galactic coordinates of the -th

Table 1 .
Summary of the velocity ranges in the present study.
* H i  (|| > 15 • ) c  * H i  (with confident  ) d Name a (km s −1 ) (cm −2 sr) (cm −2 sr)a Identification name and its abbreviation form in parentheses.bMinimum and maximum velocities with respect to the LSR.c The total apparent amount of H i gas obtained by equation (1) in || > 15 • skys excluding the Magellanic Stream.
353 (  ,   ) = ∑︁   353,H + , (  ,   ) + ∑︁   353,H i, (  ,   ) + ∑︁   353,H 2 , (  ,   ), + , ,  353,H i, , and  353,H 2, are the contribution from the dust associated with ionized, atomic, and molecular gas in the velocity range , respectively.The molecular fraction can be approximated to be zero in the unmasked regions.If we assume that the contribution from the ionized component is negligibly small (see also discussion in Section 5.2), then equation (3) can be rewritten as 353 (  ,   ) = ∑︁   353,H i, (  ,   ).(4)The contribution from a component  is expressed as a function of its H i column-density  H i, , introducing a dust-to-H i ratio parameter ,  353,H i, (  ,   ) =   (  ,   ) H i, (  ,   ) Hayashi et al. (2019b)el having a nonlinear relationship with  353 found by O17 and H19.These authors used the 21 cm H i data with  353 following F14 and F15 by taking into account the optical depth effect of the 21 cm H i emission.O17 derived a  353 - H i relationship with  = 1.3 for the H i gas in the Perseus region, and H19 obtained  = 1.2 in the Chamaeleon molecular cloud complex.The value of  greater than 1.0 was suggested to be due to the dust evolution effect byRoy et al. (2013)who derived the non-linearity with  = 1.3 from (the far infrared optical depth)-(near infrared colour excess) relationship in Orion.Hayashi et al. (2019b)made a Fermi-LAT ray analysis and confirmed the non-linearity with  ∼ 1.4.In the following analyses, we adopted  = 1.2.The H i column density is a function of  H i, and H i optical depth  H i, , > 15 • ) is PP Arch.The present result reveals that the head of PP Arch (IVC 86−36) has a low  NIV / soln distribution peaked at ∼ 0.2-0.3,consistent with the (F21) results, and the tail exhibits even lower values.Another head-tail structure IVC elongated from (, ) = (105 • , −24 • ) parallel to PP Arch (referred to as IVC 105−24 for convenience) shows similar  NIV / soln values, suggesting that they are the same type of object.Many other unidentified minor • ,

Table 2 .
List of the absorption measurements.
Table 2) and  / soln (this work).The dashed line shows the line where both have the same value.The filled and open symbols represent that the / ⊙ correspond to measurements from O i and S ii observations, respectively.The arrows attached to the filled circles indicate that the / ⊙ values are the lower limit.