WISE2MBH: A scaling-based algorithm for probing supermassive black hole masses through WISE catalogs

Supermassive Black Holes (SMBHs) are commonly found at the centers of massive galaxies. Estimating their masses ( 𝑀 BH ) is crucial for understanding galaxy-SMBH co-evolution. We present WISE2MBH, an efficient algorithm that uses cataloged Wide-field Infrared Survey Explorer (WISE) magnitudes to estimate total stellar mass ( 𝑀 ∗ ) and scale this to bulge mass ( 𝑀 Bulge ), and 𝑀 BH , estimating the morphological type ( 𝑇 Type ) and bulge fraction ( 𝐵 / 𝑇 ) in the process. WISE2MBH uses scaling relations from the literature or developed in this work, providing a streamlined approach to derive these parameters. It also distinguishes QSOs from galaxies and estimates the galaxy 𝑇 Type using WISE colors with a relation trained with galaxies from the 2MASS Redshift Survey. WISE2MBH performs well up to 𝑧 ∼ 0 . 5 thanks to K-corrections in magnitudes and colors. WISE2MBH 𝑀 BH estimates agree very well with those of a selected sample of local galaxies with 𝑀 BH measurements or reliable estimates: a Spearman score of ∼ 0.8 and a RMSE of ∼ 0.63 were obtained. When applied to the ETHER sample at 𝑧 ≤ 0 . 5, WISE2MBH provides ∼ 1.9 million 𝑀 BH estimates (78.5% new) and ∼ 100 thousand upper limits. The derived local black hole mass function (BHMF) is in good agreement with existing literature BHMFs. Galaxy demographic projects, including target selection for the Event Horizon Telescope, can benefit from WISE2MBH for up-to-date galaxy parameters and 𝑀 BH estimates. The WISE2MBH algorithm is publicly available on GitHub.


INTRODUCTION
Supermassive black holes (SMBH) are characterized by having masses ( BH ) ranging from ∼ 10 5 to 10 10  ⊙ and are believed to be located at the centers of all galaxies with a bulge, including the Milky Way (e.g., Ferrarese & Ford 2005;Graham 2016).The presence of SMBH is inferred from observations of stellar and gas motions in galactic nuclei (e.g., Genzel et al. 2010;Saglia et al. 2016), as well as powerful nuclear X-ray to radio emission (e.g., Broderick et al. 2015;Liu et al. 2022).The Event Horizon Telescope (EHT) has provided the most direct evidence of the existence of SMBH (Event Horizon Telescope Collaboration et al. 2019, 2022).SMBH play a crucial role in shaping the evolution and structure of galaxies, as they can affect the surrounding stars and gas through 'feedback', and they are expected to co-evolve with their host galaxies (Kormendy & Ho 2013).
Active galactic nuclei (AGN) are a manifestation of SMBH that are powered by a luminous accretion process at the centers of numerous galaxies.AGN emission can cover the entire electromagnetic spectrum (Padovani et al. 2017).At infrared (IR) wavelengths, the emission is primarily attributed to a toroidal arrangement of dust, ★ Contact e-mail: jheryev@gmail.com which absorbs the radiation emitted by the central accretion disk and re-radiates it at IR (Netzer 2015;Hickox & Alexander 2018).
Quasars (QSO) are a type of extremely luminous AGN (with bolometric luminosity, log  bol , in the range of 44 to 48 erg s −1 ) that are powered by high accretion rates onto SMBH ( bol / Edd ∼ 10 −2.9 to 10 1.8 , Kong & Ho 2018).Given these high accretion rates, QSOs are among the most luminous objects in the universe, making it difficult (though not completely impossible) to discriminate the morphology of their host galaxies at any redshift (Dunlop et al. 2003).QSOs are among the earliest and most distant observable objects (e.g.,  ∼7.64, Wang et al. 2021), and are important probes of the early universe and the formation and growth of galaxies and their SMBHs (Inayoshi et al. 2020).
The Wide-field Infrared Survey Explorer (WISE, Wright et al. 2010) was a NASA IR−Wavelength astronomical space telescope that surveyed the entire sky in four IR bands, with central wavelengths at 3.4 m, 4.6 m, 12 m, and 22 m (W1, W2, W3 and W4, respectively).These bands can be used to study several features of a galaxy.The shorter wavelength bands (W1 and W2) predominantly capture emission from stars (e.g., Jarrett et al. 2011;Jarrett et al. 2012;Norris et al. 2014) and warm dust (e.g., Lyu et al. 2019;Noda et al. 2020;Li & Shen 2023), providing insights into stellar populations in galaxies (e.g., Kettlety et al. 2018).The longer wavelength bands (W3 and W4) are sensitive to emission from cooler dust, revealing A simple algorithm that makes use of WISE cataloged data and a spectroscopic redshift to estimate the stellar mass, morphological type, bulge fraction, and  BH of an extragalactic source.Solid (dotted) lines represent the main path to estimate a value (or upper limit), or to reject an object from the algorithm.Orange (blue) boxes show the input (derived) quantities; boxes with both colors can either be provided to or are estimated by the algorithm.Inputs in dashed boxes are optional.WISE magnitudes with their respective mean photometric errors (σ   ) are used to generate random normal samples of size 10 4 for a Monte Carlo approach to error propagation.
regions of colder dust associated with older stars (e.g., Singh et al. 2021;Li et al. 2023), allow estimation of star formation rates (SFRs, e.g., Lee et al. 2013;Cluver et al. 2017) and potentially highlighting the presence of AGN (e.g., Lyu & Rieke 2022b;Hviding et al. 2022).By analyzing the relative intensities and the spatial distribution of emission across these bands, it is possible to obtain the temperature, composition, and distribution of a galaxy's dust and stars, thereby unraveling its evolutionary history and physical characteristics.
One of the most relevant physical characteristics that can be es-timated with WISE photometry is the total stellar mass of a galaxy ( * ).As stated by Cluver et al. (2014, hereafter C14) the W1 band is the most sensitive to the light emitted by the bulk of stellar population in galaxies (e.g., Meidt et al. 2012;Norris et al. 2014), thus allowing a determination of the mass attributed to the mass-dominant stellar population by using a mass-to-light (M/L) ratio (e.g., Kettlety et al. 2018).The M/L ratio can be constrained using the W1−W2 color.More recently, Jarrett et al. (2023, hereafter J23) have refined this process, developing a more stringent method to obtain stellar mass estimates from only the W1 band and assuming a global M/L ratio of ∼0.35 for all galaxies.Since early-type galaxies have higher M/L (∼0.8) ratios, WISE-based estimates obtained using individually tailored M/L ratios give results that are significantly different from the W1-only estimate.In instances of low uncertainty in WISE colors, the M/L ratio can be obtained through the use of the W1−W2 or W1−W3 color, and hence better correct for the range of M/L that is observed from early to late-types.
Morphology and its evolution is another relevant property of galaxies and (e.g., Abraham et al. 1994;Abraham & van den Bergh 2001;Willett et al. 2013).Many studies (e.g., Abraham et al. 1996;Whyte et al. 2002;Pahre et al. 2004) suggest that IR morphological classifications of galaxies can be superior to optical classifications, due to the physical properties that can be studied in the mid-IR, e.g., SFR and stellar populations, which evolve with the morphological type of the galaxy.Using WISE, distinct populations of early-type and late-type galaxies shown clearly different IR colors (e.g., Wright et al. 2010;Lee et al. 2017;Yao et al. 2020).Recently, Jarrett et al. (2019) showed a Hubble sequence-like morphological evolution with the W2−W3 color for a sample of well-known nearby galaxies, where early-type galaxies showed redder (stellar-dominated) colors compared to latetype (ISM + stellar colors) galaxies, with a clear sequence from the early-type to late-type.When combined with  * estimates (e.g., C14, J23), the most massive galaxies were shown to be dominated by high fractions (≥ 0.8) of spheroid-like galaxies.These authors found the same behavior when exploring the bulge-to-total ratios (/): galaxies with / ≥ 0.9 are dominantly at W2−W3 ≤ 1.5, which was set as the cutoff between spheroid galaxies and intermediate disks.Other studies have also shown that early-type and late-type galaxies showed distinct distributions of W2−W3 color (e.g., Sadler et al. 2014;Cluver et al. 2020), with the former being once again the most massive galaxies.In the sample of Cluver et al. (2020) at log  * ≥ 11.2, the percentage of early-type galaxies reaches ∼ 80%.
Powerful AGN are characterized by their high IR luminosity, resulting in strong WISE detections.WISE can differentiate between these AGN and galaxies using WISE color-color criteria (e.g., Jarrett et al. 2011;Hviding et al. 2022).This differentiation is important as AGN play a crucial role in the evolution of galaxies: they have the capability to heat the surrounding gas and dust, suppress the formation of new stars, and ultimately quench star formation.Stern et al. (2012) proposed a simple W1−W2 cutoff to successfully identify AGN to a depth of W2 ∼ 15.This cutoff was exploited, improved, and used for the creation of the WISE AGN catalog (Assef et al. 2018) and to derive new criteria to identify low luminosity AGN (log  bol in the range of 40 to 44 erg s −1 ; Hviding et al. 2022).
WISE colors are useful in identifying AGN, being notably good for type 2 obscured AGN (e.g., Hviding et al. 2022), but are poor in identifying relatively weak AGN and highly obscured AGN at modest to large redshifts (z> 2 e.g., Hickox & Alexander 2018;Lyu & Rieke 2022a;Lyu et al. 2023).The latter may constitute a significant fraction (≥40%) of the QSO population at high redshifts (Lyu et al. 2023).For complete identification of AGN, observations at multiple wavelengths are required (e.g., Yao et al. 2022).In this work AGN identification is not a goal in itself; it is used to evaluate and correct AGN contamination to WISE W1 and W2 based host galaxy stellar mass estimates.Since highly obscured (and relatively weak) AGN are expected to contribute only a small fraction of the total (AGN plus galaxy) flux at wavelengths shorter than 5m (e.g., Lyu & Rieke 2022a, Fig. 6) they are not expected to significantly contaminate the host galaxy stellar mass estimates calculated here.
Empirically, a strong correlation is observed between the  BH and the bulge mass ( Bulge ) in elliptical galaxies, as well as in spi-ral galaxies with pseudo-bulges and classical bulges (e.g., Häring & Rix 2004;Kormendy & Ho 2013;Schutte et al. 2019) with previous theoretical approaches supporting this idea (e.g., Croton 2006).Combining this with the observed evolution of / with the  Type of a galaxy (e.g., Wang et al. 2019;Dimauro et al. 2022;Quilley & de Lapparent 2022), it is possible to exploit the WISE magnitudes and colors to estimate  Bulge and then  BH , from  * .
Combining all of the above, WISE photometry can be used to distinguish between different types of extragalactic objects, e.g.QSOs, ULIRGS, AGN, and galaxies, and then for galaxies, to estimate the total stellar mass and morphology (using colors between the W1, W2, and W3 bands, e.g., Wright et al. 2010, C14, J23) and thus estimate  Bulge and  BH .
In this work, we introduce a new algorithm, WISE2MBH (see Fig. 1), which takes advantage of existing relationships derived from WISE data, the proportionality between the masses of the galaxy bulge and its SMBH, and new scaling relationships derived here.The WISE2MBH algorithm is capable of classifying regular galaxies, estimating their morphological type, and thus their bulge to total mass ratio, and estimating the mass of the SMBH.Additionally, it can identify QSOs from WISE colors; due to the AGN contamination, the algorithm provides upper limit values for these.This algorithm and the resulting sample of SMBH masses are relevant to the study of individual sources using powerful instruments such as the EHT and its next-generation upgrade (ngEHT, Johnson et al. 2023;Doeleman et al. 2023), as well as to studies of SMBH populations and evolution.In Sect. 2 we introduce the data used as input to the algorithm, in Sect. 3 to 9 we explain, in detail, the main steps of the algorithm.In Sect. 10 we present the main results and statistics of the WISE2MBH final sample generated by the algorithm, in Sect.11 we briefly discuss the results, their relevance, main assumptions and limitations, and lastly in Sect.12 we present our conclusions.

ETHER sample
The Event Horizon and Environs sample (ETHER) aims to be the definitive sample and database from which to choose targets for the EHT and ngEHT.The database, its algorithms, and references for data sources, was first presented in Ramakrishnan et al. (2023, hereafter R23).ETHER has since been expanded by including the following literature and database samples: (a) all galaxies in the HyperLeda database (Makarov et al. 2014) Massaro et al. 2015); (f) the 2MRS sample (Huchra et al. 2012); and (g) the ∼ million galaxies with SDSS and WISE photometry in the catalog of Chang et al. (2015).Several other individual black hole masses and radio fluxes from the literature have also been incorporated.Full details on the updated ETHER sample will be published in Nagar et al. (in prep).
Given the above updates and after consolidating multiple entries from the same source, the ETHER sample currently contains 3.8 million extragalactic sources, of which 233 have  BH measurements,  including methods such as stellar dynamics (e.g., Thater et al. 2019), gas kinematics (e.g., Boizelle et al. 2021), and reverberation mapping (Bentz & Katz 2015), and ∼860,000 have  BH estimates.Of the estimates, ∼331,000 are from the M-sigma relationship, ∼525,000 are from 'single-epoch' spectroscopy and using scaling relationships from reverberation mapping (e.g., Dalla Bontà et al. 2020;Rakshit et al. 2021), ∼3,000 are from M-L bulge estimations, and ∼600 are from other 'fundamental-plane' type relationships (see R23 for more details).
Astroqueries to the NASA Extragalactic Database (NED) and SIMBAD are used to incorporate and update positions, spectroscopic redshifts (thus luminosity and angular distances for objects at D ≥ 50 Mpc), object types, morphological types ( Type ), AGN classifications, and radio to X-ray fluxes.The object type comes directly from NED source classifications, which have shown great precision (≥ 80%) for nearby sources ( ≤ 11 Mpc, Kuhn et al. 2022).The object types in ETHER, incorporated from NED, are as follows: 'Galaxies' cover regular galaxies over the range from elliptical to spiral galaxies, 'QSOs' denote galaxies with significant nuclear activity thus luminosity, 'Radio Sources' (RS) refer to sources detected in the radio regime, without any distinction between galaxies or QSOs.If a source lacks any NED classification, it is designated as an 'Unknown' object type.Through visual inspection of sub-samples using the Sloan Digital Sky Survey (SDSS), and the color-color criteria using WISE colors (see Section 4), we established that this NED source classification was sufficiently accurate for our needs.
Morphological types ( Type ) are available for a significant fraction of the sample (22.4% via NED and SIMBAD queries and from individual samples e.g., Huchra et al. 2012;Makarov et al. 2014).These are predominantly E (−6 to −4) and Sc (4-5), representing 37.3% and 23.8% of all sources with available morphology, respectively.The large fractions in these two  Type bins is primarily due to the binary classification of  Type in some ingested samples (e.g., Dobrycheva 2013).When available, this  Type is used as an input to the algorithm; when not available, the  Type is estimated from the W2−W3 color (see Section 5).All ETHER sources at  ≤ 0.5 (∼2.3 M) form the parent sample for this work.

WISE catalogues
The WISE mission, funded by NASA as a Medium-Class Explorer mission, features a space-based infrared telescope with megapixel cameras cooled by a two-stage solid hydrogen cryostat.This telescope conducted an all-sky survey which simultaneously captured images in four broad spectral bands: W1, W2, W3, and W4, centered on 3.4, 4.6, 12, and 22 m with an angular resolution of 6.1 ′′ , 6.4 ′′ , 6.5 ′′ and 12.0 ′′ , and achieving sensitivities of 0.08, 0.11, 1 and 6 mJy, respectively.In this work, we use both the AllWISE catalog (Cutri et al. 2021) and the WISE Extended Source Catalog (WXSC, Jarrett et al. 2019).The latter includes mid-infrared photometry and measured global properties of the 100 largest (in angular size) galaxies in WISE.

AllWISE catalog
The AllWISE catalog, which combines data from the WISE cryogenic and NEOWISE (Mainzer et al. 2011) catalogues, contains almost 750 million sources, including galaxies, stars, brown dwarfs, and asteroids, making it one of the most comprehensive IR catalogues ever created.This catalog provides photometric quality flags (qph) for a given detection for all sources, making it the best option to extract data for our purposes; we accepted only sources with quality A, B, C or U in the first three bands, which translates to detections with signal-to-noise ratios (S/N) in the ranges of S/N ≥ 10, 10 > S/N ≥ 3, 3 > S/N ≥ 2 and S/N < 2, respectively, with the last flag considered an upper limit with 95% confidence.Extension flags (ex) are also provided in this catalog, allowing easy differentiation between extended and point sources.This flag is directly related to the 2MASS Extended Source Catalog (XSC, Jarrett et al. 2000), ranging from 5 for completely extended sources to 0 for point sources.This flag contributes to the overall quality flag of our estimates and thus allows the selection of high-quality sub-samples (e.g., using only extended sources) from our overall sample.In this work, sources with ex equal to 4 or 5 are considered extended, while those with values 0 to 3 are considered point sources.Thus, in our definition, a WISE extended source is one whose position falls within 5 ′′ of the central position of a 2MASS XSC source, and a WISE point source is one that is not associated with a 2MASS XSC source, or is offset by ≥ 5 ′′ from the central position of a 2MASS XSC source even if it falls within its isophotes.

WISE Extended Source Catalog (WXSC)
The AllWISE photometric catalogues are optimized for the characterization of point-sources 1 .For highly extended sources, source detection and extraction may not include all the extended components of the galaxy in a single source, thus leading to an underestimation of true brightness.
The WXSC (Jarrett et al. 2019) provides full source characterization in all four WISE bands for the 100 largest galaxies (in angular extent) in the sky.WXSC uses new mosaics with native resolution allowing for precise measurements of both the target galaxy and its local environment.These mosaics are further resampled with 1" pixels, greatly enhancing analysis accuracy.When a galaxy is present in WXSC, we use these magnitudes instead of those from AllWISE.

WISE2MBH parent sample
All sources in the ETHER sample at  ≤ 0.5 were cross-matched with the AllWISE and WXSC catalogues with a cone search radius of 3 ′′ , preferring WXSC matches over AllWISE to consider better photometric values.After deleting duplicates, 95.2% of the ETHER targets matched the two WISE catalogues.
Given that the AllWISE catalog has a high density of sources (on average ∼ 5 sources per arcmin 2 ) and the ETHER sample at z < 0.5 is also relatively large (∼2.3 million galaxies, with a bias towards northern galaxies), it is important to address the probability of false matches in a 3 ′′ radius.We generate random samples of 10 5 right ascension and declination coordinate pairs, divide them into subsamples in ( between ±15) and outside the Galactic plane, and test for random matches and possible contamination of W1 magnitudes due to confusion.When matching these random samples to the AllWISE 1 AllWISE Explanatory Supplement: Cautionary Notes catalog, we find a 4% (∼ 5%) chance of matching a random position source out of the plane (in the plane) of the galaxy with an AllWISE source.The distributions of W1 magnitudes for real ETHER galaxies matched to WISE are consistently brighter than the magnitudes obtained in the random matches.The W1 magnitude distributions of both AllWISE sources matched to the random position catalogues ( = 17 mag) and to the ETHER galaxies ( = 14 mag), are significantly fainter than those of Galactic AGB stars in AllWISE ( = 8 mag; Suh 2021).
The differentiation between early-type galaxies and stars in All-WISE catalogues, in the absence of redshifts, remains a topic of ongoing discussion.Machine learning classification techniques have exhibited significant achievements in this area (e.g., Kurcz et al. 2016).In our case we do not expect to be affected by this problem for two reasons (a) ETHER has both precise (typically sub-arcsec accuracy) coordinates and includes only known extragalactic objects (when redshifts exist these are spectroscopic); (b) the low probability of random matches as explored in the previous paragraph.
A summary of the resulting WISE2MBH parent sample is given in Table 1 where we list the total number of sources of each object type (), and the total number of sources of the WISE2MBH parent sample and ETHER.The other columns represent a percentage of completeness.The distribution of sources in redshift by object type and ex can be seen in Fig. 2.

K-CORRECTIONS FOR WISE MAGNITUDES AND COLORS
Jarrett et al. ( 2023) used spectral energy distribution (SED) templates of galaxies with different morphological types (e.g., Brown et al. 2014) to calculate K-corrections for the W1 magnitude, and the W1−W2 and W2−W3 colors, for each available  Type over the range  = 0 − 3.These K-correction lookup tables are shown in Figs. 9 and 10 of J23, but only for  ≤ 0.5.Due to the uncertain morphological type and AGN contamination of objects classified as 'RS', and given the tendency of QSOs to continue to reside in the AGN/QSO area of a WISE color-color diagram independent of redshift (e.g., Mateos et al. 2012;Stern et al. 2012), we do not apply K-corrections to the RS and QSO object types.
Of the remaining objects (i.e., galaxies or unknown) only those which reside in the 'Estimate zone' of Fig. 3 (see Sect. 4) are Kcorrected.This selection is made to avoid a false K-correction for AGN-contaminated sources.
Our algorithm applies the J23 K-corrections for three  Type ranges; ellipticals (−5 to −3), lenticulars (from −3 to 0) and spirals (from 0 to 8).The existing  Type of an individual object, if available, is used to select between these three lookup tables.If no  Type is available, the lookup table to be used is decided as follows.Mateos et al. (2012, see their Fig. 5) use galaxy templates of various morphological types, and with varying amounts of AGN contamination, to derive the redshift evolution of a galaxy's WISE colors over the redshift range 0 and 2.Although the WISE colors vary with redshift (especially for late-type galaxies), they find that, in the absence of a large AGN contamination, WISE colors can distinguish between early-type and late-type galaxies over this full redshift range.We thus define the following cutoff in the observed W2−W3 color to distinguish between elliptical and spiral galaxies (without AGN contamination) up to  ∼ 1.5, (1) where  0 and  0 are the zero magnitude flux density and observed flux density in the respective band, respectively.We set the logarithmic ratio of observed flux densities in Eq. 1 to −0.1, since Fig. 5 of Mateos et al. (2012) shows that this value clearly separates elliptical and spiral galaxies over their full redshift range.This translates to a cutoff value of W2−W3 Limit ∼ 1.58 to distinguish elliptical and spiral galaxies (values larger than this imply a spiral galaxy).Note that this cutoff is similar to the values previously derived by Jarrett et al. (2019).The Lenticular lookup table is not used in this case, since lenticulars and spirals are not easily distinguishable in WISE color-color diagrams (see e.g., Sect.5).These K-correction look up tables have been shown to be reliable for galaxies at  ≤ 0.5 (e.g., Jarrett et al. 2023;Karademir et al. 2023).

DISTINGUISHING GALAXIES AND POWERFUL AGN/QSOS USING WISE
If an object type is not already available (which is the case for 76% of the parent sample; Sect.2), the algorithm uses WISE color-color criteria to distinguish galaxies from powerful AGN/QSOs, and to determine whether the derived values are estimates or upper limits (due to contamination from an AGN).
The algorithm is first tested on targets with an available object type.As mentioned in Sect.3, we use cutoffs in the observed W1−W2 and W2−W3 colors to identify objects that will be K-corrected.The same cutoffs are now applied to the K-corrected colors to determine if this target falls within the estimate, upper limit or rejection zones in the WISE color-color plot shown in Fig. 3.These objects then, respectively, follow the estimate, upper limit or reject paths of the algorithm shown in Fig. 1 and described in Sect.7.
Objects with object type QSO and RS are likely to have significant AGN contamination in their W1 mag, which will likely increase their  * estimates, independent of their extension in WISE.For these objects, we therefore immediately consider the WISE2MBH  * values as upperlimits.The horizontal dashed line in Fig. 3, which separates the upper limit zone from the estimate zone, is a combination of the widely used W1−W2 = 0.8 limit to separate AGN/QSOs from galaxies (e.g., Stern et al. 2012;Michalik & Lindegren 2016;Guo et al. 2018), together with a wedge region between W2−W3 ∼2.2 -4.4 motivated by previously defined AGN/QSOs regions (Jarrett et al. 2011;Hviding et al. 2022).We slightly modified the wedge region defined by Jarrett et al. (2011) by optimizing the WISE color classification of known QSOs in our sample.The wedge region we chose is defined as follows ( 2) which now fully extends to the blue-side of the diagram thanks to the W1−W2 = 0.8 cutoff.With this overall cut, almost 95% of the known QSOs in our sample are classified as QSOs by this color criterion.
The vertical limit at W2−W3 = 4.4 in Fig. 3, which separates the Reject zone from the other two zones, is also set by us.At W2−W3 ≥ 4.4, the WISE2MBH  Type (see Sect. 5) is 8 or larger, and the estimated / would be very low (Sect.6).While this would be correct for, e.g., irregular galaxies, it is incorrect for, e.g., extremely dusty, hybrid-starburst-AGN galaxies (Tsai et al. 2015) or newborn AGN (Arevalo et al. in prep) 2 .
The WISE color-color distributions of different object types (including unknown types) in our sample are shown in the top (WISE point sources) and bottom (WISE extended sources) panels of Fig.

3.
The top panel of Fig. 3 shows that most QSOs and RS reside at high W1−W2 colors, in an area populated by a variety of AGN and also ULIRGs, making it a challenge to correctly identify them using only WISE data.On the other hand, known galaxies, even if pointlike for WISE, show clear overdensities in the region where elliptical and spiral galaxies are expected to reside.
For the bottom panel of Fig. 3 (WISE extended sources), all (∼ 99%) known galaxies lie in the Estimate zone, at WISE colors expected of elliptical and spiral galaxies, and away from the regions of starbursts, LINERs and (U)LIRGs.Only a few QSOs and RS are present in this figure: the QSOs (orange dots) do not clump in the expected QSO area, but are instead distributed over a large range of colors, overlapping with regions of galaxies, ULIRGs, and Seyferts; RS (green dots) are predominantly situated in the galaxies area, with The red line in the right panel marks the threshold number of galaxies in a bin in order for that bin's median to be used for the fit (filled black circles in the main panel).The black dot-dashed line shows the best fit logit function to the filled black circles: this is used for the W2−W3 to  Type conversion when  Type is previously unknown.The estimated  Type is limited to the range −5 and 8; when a galaxy's W2-W3 color lies beyond the range of the logit function shown, the  Type is clipped at these values.The colored areas distinguishing morphologies listed in the inset are from Jarrett et al. (2019).Given the similarity of the color dispersions in the three bins at each extreme end of the -axis, we define two vertical dashed lines which delineate galaxies we refer to as high bulge fractions (HBF; bulge fractions between 0.4 and 1) and low bulge fractions (LBF; bulge fractions between 0.1 and 0.3).
a few in the LINER area.This tendency for both QSOs and RS to not only reside in the expected wedge region described above is not unexpected; since these extended sources are most likely weaker AGNs whose emission is not sufficient to change the galaxy color so as to be classified as a QSO.This was shown in Mateos et al. (2012) who tested different percentages of AGN activity, and showed that only powerful AGNs (80% fraction) up to  = 2 reside in an area similar to the wedge shaped region defined here or in previous studies.
The bulk of sources with unknown type (76% of the parent sample; red contours in both panels) sit in the Estimate zone, mostly following the expected distribution of galaxies.The clear separations seen for known object types give us confidence that the classification of these unknown types as galaxies is reliable.

USING W2-W3 COLOR TO ESTIMATE T-TYPE
While  Type is available for some (∼ 22%) sources in the WISE2MBH parent sample, a reliable estimation of this is required for most of the sample.To obtain these  Type , we exploit the fact that the W2−W3 color shows clearly separated regions where elliptical and spiral galaxies reside (e.g., Wright et al. 2010;Jarrett et al. 2019;Cluver et al. 2020).Although these regions partially overlap with other classifications based on star formation activity, the trend is sufficient to estimate the morphology of the galaxy, i.e.,  Type .
Our conversion of W2−W3 color to  Type is trained using ∼18,000 galaxies from the 2MRS catalog (Huchra et al. 2012) for which manually classified  Type are available.The median W2−W3 colors of 2MRS galaxies in each  Type bin between −5 and 8 (open and filled black circles in the main panel of Fig. 4) show a clear S-shape curve.The number of galaxies in each  Type bin is shown in the right histogram of Fig. 4. Given the S-shape, i.e., the lack of differentiation in the -axis for the three to four bins at each extreme end of W2-W3 colors, a sufficiently large number of objects per bin is required for a clear result.We therefore use statistical power analysis to define the required sample size threshold; details of this analysis can be found in Appendix B. For this power analysis, a power  = 0.8 and a significance threshold  = 0.05 were assigned.The effect size (  ) was calculated for each set of two consecutive bins in W2−W3 (in order of increasing  Type ), and the resulting median   (0.15) implies that the sample size per bin must be  ≳ 700.Therefore, all bins with a sample size greater than this  were accepted (filled black circles in Fig. 4).
A logit function was fit to these accepted median values, providing us with our W2−W3 to  Type conversion.Since the logit function's domain goes from 0 to 1, the W2−W3 color is shifted and normalized (W2−W3 SN ) before fitting: 71 this logit function is shown as a dashed-dotted line in Fig. 4. Despite leaving out three late-type  Type bins from the fit ( Type = 6, 7 and 8), the logit function fits almost perfectly to all medians.The morphological limits of Fig. 10 of Jarrett et al. (2019) are presented for comparison as colored regions in the figure.While the overall fit is S-shaped, we find two clearly separated regions in the graph: the high bulge fraction (HBF) region, which is delimited by the 84 th percentile values of the bins centered on  Type = −5 to −3, and the low bulge fraction (LBF) region, which is delimited by the 16 th percentile values of the bins centered on  Type = 7 and 8. Our HBF region limit (W2−W3 = 1.58; vertical dashed line in the figure) is similar to the cutoff used by Jarrett et al. (2019) to distinguish between spheroids and intermediate disks (the division between pink and green regions in the figure), and also similar to the value of W2−W3 Limit which we use to classify galaxies with unknown morphological type into elliptical and spiral galaxies in order to select the K-correction lookup table to be used (see Eq. 1 and Sect.3).
The logit function in Eq. 3 is used whenever a source does not have an available  Type or if the available  Type comes from a binary classification (e.g., Dobrycheva 2013).The WISE2MBH algorithm uses  Type in the range −5 to 8. Available  Type values outside this range are clipped to the closest limit value, in case these really define a morphology, e.g., −9 is often used to define a QSO (ZCAT convention), so those values are not clipped.If a measured W2−W3 color is outside the range of the logit function presented here, the estimated  Type is also clipped to the closest limit value  Type .This is most relevant for elliptical galaxies, whose W2−W3 colors are often less than 0.7, which the algorithm converts to  Type = −5.
This  Type estimator in WISE2MBH is an auxiliary function: it prefers an input value but will provide a WISE-based value when necessary.The S-shape curve makes the classifications at extreme  Types very uncertain at the moment of distinguishing between consecutive  Types , but the distinction between bulge-dominated and disk-dominated galaxies is clear.

BULGE-TO-TOTAL RATIO FROM T-TYPE
The morphological type of a galaxy within the Hubble sequence has been shown to be a good proxy of /.This inverse behavior (recently discussed in Quilley & de Lapparent 2022, 2023) shows that early-type galaxies tend to be almost pure bulges (/ ∼ 1), while very late-type galaxies and irregulars tend to have small to null bulge fractions (/ ∼ 0.01).This inverse behavior also supports the posited scenarios of galaxy bulge growth via mergers: late-type galaxies merge consecutively until lenticular, elliptical, and finally, the brightest cluster galaxies (BCGs) are formed (Edwards et al. 2020), leading not only to the formation of the most massive galaxies (Bluck et al. 2014), but also to the most massive SMBHs (Mezcua et al. 2018).
By nature, elliptical galaxies are expected to have quenched their star formation, leading to a decrease in their SFR and specific SFR (sSFR), despite environmental effects (e.g., Casado et al. 2015).Recently, Ge et al. (2018) showed that galaxies with lower sSFR tend to be more massive and have higher / (≥ 0.7) compared to galaxies with higher sSFR, and also found a trend with galaxy age where the oldest galaxies have higher /.Morell et al. (2020) showed similar results, showing that their passive galaxy sample (made up of 70% ellipticals and 15% lenticulars) is the one with higher / (∼ 0.7).
Massive elliptical galaxies are the most relevant sources for future EHT observations, and while varying their estimated / between 0.8-1 does not considerably affect the final  Bulge estimate, a misclassification of  Type could result in the incorrect use of a spiral-like / (≤ 0.2), leading to an incorrectly low  Bulge , thus  BH (Bluck et al. 2014).Despite many references pointing to a / ≤ 0.8 for early-type galaxies (e.g., Ge et al. 2018;Morell et al. 2020), we will impose a limit of / = 1 for  Type = −5 (e.g., Caramete & Biermann 2010;Quilley & de Lapparent 2023), with an exponential decrease with  Type as shown in Fig. 5.We note that Caramete & Biermann (2010) derived correction factors to convert near-IR luminosities to  BH for galaxies of a variety of  Type in the nearby universe ( ≤ 0.025) with similar impositions for elliptical galaxies; our / ratios are analogous to their conversion factors but calibrated with a larger sample.
We calibrate the / ratio as a function of  Type using several samples from the literature.Mendel et al. (2014) provide total, bulge, and disk masses, for a large sample of SDSS galaxies.Their values, combined with  Type from ETHER, give us distributions of / over a wide range of  Type , although only −5 and 5 have enough statistics to be considered robust.The ETHER  Type unfortunately comes primarily from the binary classification of Dobrycheva ( 2013), so we expect that the relationship between  Type and / is biased, i.e., underestimated for ellipticals and overestimated for spirals.Sofue (2016) provide both bulge and disk masses for a small sample of nearby galaxies ( ≤ 0.03) for which we obtained  Type from NED. Morell et al. (2020) provide an average value of / for a sample of passive galaxies, which are composed of specific fractions of ellipticals, lenticulars and spirals, mostly dominated by the first two.
The  Type value we used in this case is weighted by these fractions.
From Gao et al. (2020) we take values of / only for  Type equal to −1, 1 and 2 (S0, Sa and Sab) and consider  Type equal to 8 (Sdm) as a limit to secure reasonable bulge masses even for very late-type galaxies.
A plot of  Type as a function of / for all these samples is shown in Fig. 5.A direct fit to all these points is shown with the dashed gray line in the figure.When we force a value of / = 1 for  Type = −5, our final fit is functionally similar to that of Caramete & Biermann (2010, Fig. 1), but now for significantly larger and more recent datasets.The final fit is, which results in a 13% increase in / at  Type = −5 compared to the direct fit to all datapoints, while for  Type ≥ −1 the difference is negligible.

STELLAR AND BULGE MASS FROM WISE PHOTOMETRY
The process of converting WISE photometry to stellar mass is described in C14, who take advantage of the fact that W1 is an exceptional tracer of the bulk of stellar population in galaxies and that the W1−W2 color can constrain the M/L ratio.For their (and our) calculation, the W1 absolute magnitude of the Sun is taken from Willmer (2018).J23 have presented an updated  * estimator valid across a larger redshift range, making use of multi-color criteria and K-corrections.We use the K-corrections of the latter (see Section 3) for W1 and W1−W2 colors together with the stellar mass estimator of C14 to derive the total stellar mass ( * ).
To avoid excessive (and likely erroneous) M/L values estimated from W1−W2 colors, our algorithm limits the input W1−W2 values to the range −0.2 to 0.6 (corresponding to high and low M/L).Any source with a W1−W2 color outside this range is clipped to the nearest limit M/L, i.e., the distribution of W1−W2 (generated by the random normal samples of W1 and W2, see Fig. 1 and Sect.9) is shifted until the median reaches the closest limit value.
Once  * and its errors are calculated following the process outlined above, the value is stored unless the estimated mass is log  * ≤ 6.5 or log  * ≥ 13.This range is more strictly constrained at the low mass end than other catalogues, and more lax at the high mass end (e.g., Dimauro et al. 2018;Durbala et al. 2020).The flexibility at the high mass end is in order to not lose extremely rare extreme SMBHs, often called ultra massive black holes (UMBHs, e.g, Runge & Walker 2021) and SMBH upper-limits, e.g., QSO with high M/L, which produce very high  BH estimates, but which are flagged as upper limits.The lower limit value is extracted directly from J23.  correlations and scatters between these samples did not demonstrate a preference.Appendix C contains a more detailed discussion of this WISE to SDSS comparison.
The  Bulge estimations are obtained by combining / with  * .Effectively, the estimated / allows the estimation of both  Bulge and  Disk .In the WISE2MBH final catalog, we provide only  * and / for simplicity.Figure 7 presents a comparison of WISE2MBH bulge masses with those derived by Mendel et al. (2014) in a low redshift sample: once again, the agreement is relatively good, but now shows an increased scatter.This can be explained by our assumption of a simple evolution of / with  Type , which does not consider other important factors in galaxy evolution such as gas availability, molecular gas content, size distribution, stellar age, and the impact of bars and bulges (e.g., Laurikainen et al. 2007;Fisher & Drory 2011;Koyama et al. 2019).

BLACK HOLE MASS FROM WISE PHOTOMETRY
The value of  Bulge calculated in the previous section is used to derive a first estimate of  BH using the  BH −  Bulge relationship of Schutte et al. (2019).These first  BH estimates were compared with a control sample presented in Table 2.The control sample consists of 152 galaxies with directly measured  BH from different methods (e.g., Saglia et al. 2016; van den Bosch 2016) and 647 galaxies for which high-quality stellar velocity dispersions () were available, obtained via observations with the Hobby-Eberly Telescope (HET; van den Bosch et al. 2015).Using these values of  allowed us to accurately estimate  BH using the M- relationship of Saglia et al. (2016).
The control sample was selected according to the following criteria: (a) the value of  BH is flagged as a measurement or high-quality estimate from , (b) the  BH estimate from WISE2MBH is not an upper limit, (c) the source has an ex flag equal to 5 and (d) the control sample source must have log  BH ≤ 10.32.While points (a) and (b) are self-explanatory, (c) is required to consider only completely   Linear regression done to the control sample  BH (dependent variable) and WISE2MBH  BH estimates (independent variable) revealed a slope of 0.9 and an intercept of 0.98.A t-test was then used to check the statistical significance of these results compared to a linear regression close to the equality line (expected slope of 1 and intercept of 0) with a p-value of  = 0.05.The results showed that the slope did not differ significantly from the equality line, while the intercept was significantly different from zero, suggesting the need for a compensation factor.These findings demonstrate the presence of subtle, yet systematic, discrepancies between the WISE2MBH  BH estimates (when the Schutte et al. (2019) scaling is used) and the control sample values.To address these systematic offsets empirically, a compensation factor (  ) is defined as follows,  2020), and these populations do not necessarily follow the several scaling relations used in WISE2MBH.This also means that estimates slightly higher than  BH = 5 should be treated with caution.

ALGORITHM
The WISE2MBH algorithm was conceived with the purpose of addressing the lack of  BH estimates for more than 80% of the ETHER sample (see Sect. 11.4).Nevertheless, it provides a simple and uniform tool with wide-ranging applications in studies of morphology and galaxy and black hole evolution.It is useful for both generation of large samples from existing data, and of sub-samples for future observations and monitoring with observational facilities.This main steps of the algorithm are summarized in Fig. 1.In summary, the process is as follows: • W1, W2 and W3 magnitudes and a spectroscopic redshift are used to calculate the K-corrected W1 absolute magnitude, W1−W2 and W2−W3 colors with the use of lookup tables from J23 (see Sect. 3).
• The K-corrected W1 absolute magnitude and the W1−W2 color are used to estimate  * of the source with the process described in C14.For that calculation, the W1 absolute magnitude of the Sun is taken from Willmer (2018) (see Sect. 7).
• The source is placed into the estimate or upper limit zone, making use of several inputs (see Sect. 4).
-If a source is placed in the estimate zone and its  Type is available, it continues in the algorithm.
-If a source is placed in the estimate zone and no  Type is available, the latter is estimated from the W2−W3 color (see Sect. 5).
-If a source is placed in the upper limit zone, its  Type is not considered if available, nor calculated if not available.In this case, values calculated later in the algorithm are considered upper limits.
• The  Type of the source is used to estimate /; for sources in the upper limit zone / is set to 1 (see Sect. 6).
•  Bulge is estimated using the previously derived  * and / (see Sect. 7).
• A first  BH estimate is obtained using the  BH − Bulge relation of Schutte et al. (2019).
• The final estimate of  BH is obtained after adding   to the first estimate.Sources with  BH in the IMBH range are removed.
Not every source enters the algorithm.There are two main reasons for a source to be rejected during the algorithms pre-processing: (a) the source does not have a spectroscopic  available, or (b) the quality of the WISE magnitudes are not considered 'usable', i.e., qph from the AllWISE catalog is not equal to A, B, C or U, for the W1, W2 and W3 bands.A source can be dropped during an intermediate step of the algorithm if  * or  BH are considered outliers (described in Sects.7 and 8).
The algorithm uses a Monte Carlo approach to estimate errors.It generates random normal samples (using arrays of size 10 4 ) for the W1, W2, and W3 magnitudes and respective mean photometric errors for each source.These distributions are propagated through the algorithm, where errors in the scaling relations used, if available, are considered, thus delivering a final  BH (or other estimated quantity) with asymmetric error bands for each source that was not rejected.The   is applied to the final distribution of values and not only to the nominal value.The nominal, low, and high values reported are the median and 1 percentile values of the final distribution, respectively.Two examples of the final error distributions can be seen in Fig. A1.

WISE2MBH final sample
When used with the WISE2MBH parent sample, the algorithm generates ∼2 million  BH estimates and upper limits, rejecting only 3.9% of the parent sample.A summary of the statistics in the final sample is shown in Table 3.
Percentages listed represent the percentage with respect to the total value of a given object type.New WISE2MBH estimations are those for which no previous measurement / estimate of  BH existed in the ETHER sample.Almost 80% of the final sample are first-time  BH estimates or upper limits, most of which come from galaxies and unknown object-type sources that reside in the estimate zone of Fig.

3.
We provide a table, in FITS format, of our WISE2MBH final sample and its derived quantities.The table provides the source name, RA and DEC in degrees from the AllWISE catalog, spectroscopic redshift, object type,  Type used (either from ETHER or calculated by the algorithm), plus the estimates of  * , /, median plus 1 values of  BH as estimated by the algorithm and a quality flag for each source.An excerpt of the online table is shown in Table 4.
The 7-digit quality flag provided in the table stores information useful for the selection of subsamples.The first four digits of this flag sequentially report the photometric quality of the measurements in the W1, W2, and W3 bands, as well as the extension flag of the source of AllWISE.The fifth digit serves as a binary indicator for the upper limit condition, where 0 denotes an estimate and 1 an upper limit.The sixth digit characterizes the K-correction quality; a value of 0 denotes an optimal correction, 1 is a suboptimal correction, and 2 indicates that K-correction was not applied.Lastly, the seventh digit denotes the origin of the  Type estimate.A value of 0 implies a source with a known  Type , 1 indicates a  Type estimated by the algorithm, and 2 indicates that  Type was not estimated by the algorithm due to an upper limit condition.In particular, it is important to clarify that the seventh digit does not inherently rank the quality of the  Type estimate as superior or inferior, as the available  Type values can come from binary classifications, as described in Dobrycheva (2013), or consider all morphological categories, as in the case of 2MRS.This is discussed in Sect.11.3.
The highest quality sources (HQS) are classified as AAA500 in the first six digits, while the lowest quality sources (LQS) have the fourth and fifth digits equal to 01.Examples of HQS and LQS sources  can be seen in Fig. A2.These sources illustrate the wide range of sources which the algorithm deals with, with HQS being mostly nearby galaxies large enough to have top quality in W1, W2, W3 and also being considered completely extended, while LQS are mostly QSO and compact very late-type galaxies.While HQS is not a strict proxy of the best  BH estimates, we recommend using this subsample (118367 HQS, more than half of them (82812) also classified as galaxies) when a high reliability sample is required.Most QSO and RS are classified as LQS (all of their  BH are upper limits), due to the resolution limits of WISE and 2MASS, and/or AGN contamination (see Sect. 11.1).
The errors in the estimates tend to be smaller in HBF sources than in LBF sources, since the extension of the sources allows for better quality in the WISE magnitudes.In Fig. 9 it is possible to see that behavior for a few tens to thousands of sources at LBF. Final distributions of  * and  BH for each object type can be seen in Fig. 10.It is clear that QSO and RS appear to have more massive  * and  BH estimates: this is due to these being upper limits (i.e.assuming / = 1).Galaxies are primarily estimates: the / ratio is used here to estimate  Bulge and  BH .More than 80% of sources with unknown type are estimates: their distribution is almost the same as galaxies but very different from QSO and RS.

Local black hole mass function
The local black hole mass functions (BHMFs) produced using the  BH estimated from the WISE2MBH final sample can be seen in Fig. 11  The symbols in the figure show two cases of the BHMF for BHs at  ≤ 30 Mpc: (a) using only estimates present in ETHER before the WISE2MBH algorithm was used (black symbols), and (b) using ETHER estimates, and in the absence of these, using estimates from WISE2MBH (red symbols).The difference between these two cases, and the WISE2MBH BHMF at  ≤ 30 Mpc (darkest blue line) is clear.At low masses (log  BH ≤ 7) the ETHER-only BHMF is significantly lower than the other two.The ETHER+WISE (red symbols) and WISE2MBH BHMF agree well between each other and  primarily sit on or between previously derived literature expectations.At the highest masses, the ETHER and ETHER+WISE BHMFs are above previous literature predictions: of the 17 SMBH with log  BH ≥ 9, the majority are M- estimates with  from LAMOST and SDSS, so their true  BH are sensitive to both errors in  and in the high-mass M- relationship.These BHMFs in the highest mass bins should thus be treated with caution, and will be explored in more detail in Hernández-Yévenes et al. (in prep).
At the high mass end, the WISE2MBH BHMFs in all distance shells except for the closest one, drop significantly below the ETHER and ETHER+WISE BHMFs and also the literature expectations.That is, when considering median values of  BH , pure WISE-based estimates do not find the few massive known black holes in the local universe.The increasing incompleteness of the WISE2MBH BHMFs with distance, especially at lower masses, is due to the redshift distribution in the sample.
Why does WISE2MBH not recover the shape of the BHMF at the massive end?We explain this as a result of not considering the intrinsic scatter of the relationship between the bulge mass and the SMBH mass.Indeed, when we consider the predicted probability density function (PDF) of each  BH estimate from WISE2MBH (the dispersion of each PDF is dominated by the dispersion in the bulge-to-SMBH mass conversion) and randomly select a value from this PDF (instead of its median), the high-mass black hole populations are recovered and the WISE2MBH BHMFs are even higher than previous literature BHMFs (right panel of Fig. 11).We do the above exercise 10 times in each distance shell to create different PDF-sampled BHMFs.Even through the closest distance shell shows considerable fluctuations in its ten PDF-sampled BHMFs, all three shells consistently show BHMFs close to, but higher than, previous literature BHMFs.
The combination of less massive  * estimates and low / for spiral galaxies in the sample can lead to differences at the low mass end when comparing WISE2MBH BHMFs with other BHMFs in the literature.In any case the ETHER-only BHMF is lower than literature expectations due to the construction of the sample: galaxies with an estimated black hole mass of log  BH ≤ 5.5 were eliminated from the sample, but note that galaxies without black hole masses remained in the sample and for these WISE2MBH could later add a mass estimate.The ETHER+WISE BHMF is in good agreement with other BHMF that obtain similar densities.
The BHMFs from Gallo & Sesana (2019) and Yao et al. (2023) are mostly focused in lower mass ranges.The former used the stellar mass function derived from Galaxy And Mass Assembly (GAMA) together X-ray imaging from the Chandra Observatory, to constrain the BHMF down to low-mass regimes (log  BH ≥ 4).The latter was derived by studying tidal disruption events (TDE), which are less frequent in SMBH with higher masses (log  BH ≥ 7); they argue that their BHMF over log  BH 5 to 7 could be considered as an upper limit.With that information, a limit was established and our BHMF is in good agreement with that.
The BHMF of Shankar et al. (2016) is the most widely used for comparisons, due to the compensated phenomenology used to derive it, as compared to the BHMF previously derived in Shankar et al. (2009).The Shankar et al. (2016) BHMF corrects the low-mass range, now showing a clear downward trend, but it is still limited in  BH range.Pesce et al. (2021) used the Shankar et al.BHMF to build an updated BHMF which extrapolates to higher mass ranges, but does not update the low-mass end BHMF.The WISE2MBH BHMFs are lower at both the high and low mass end as compared to Shankar et al. (2016) and Pesce et al. (2021).
Given recent interest in black hole binaries (BHB) over a wide range of masses (for a review, see De Rosa et al. 2019), our predicted BHMF favors recent studies.Sato-Polito et al. ( 2023) discussed the need for a larger population of inspiraling supermassive BHB (SMBHB) compared to current predictions to explain the stochastic gravitational wave background found by pulsar timing array (PTA) collaborations.Their prediction agrees with our closest WISE2MBH PDF-sampled BHMF; favoring the hypothesis.Izquierdo-Villalba et al. ( 2024) predict a similar local BHMF when trying to connect LISA-detectable binaries (log  BH < 7) to SMBHB populations: our ETHER+WISE and most-local PDF-sampled BHMF presents almost perfect agreement with the predicted low-mass population in their work, while our WISE2MBH PDF-sampled BHMF is almost one dex lower at the high mass end (see their Fig.B1).

Assumptions and limitations
The algorithm makes several assumptions and approximations, potentially introducing biases, particularly for certain types of galaxies.As highlighted by Jarrett et al. (2019) and Cluver et al. (2020), despite the reduced sensitivity of the W3 band compared to W1 and W2, leading to fewer detections (as noted by Jarrett et al. 2013), the color W2−W3 has been shown to be an effective indicator of morphology.Therefore, we choose to use this color to estimate the morphology.The galaxy-averaged W1−W2 color (rather than the colors of the individual components, e.g., bulge and disk) is assumed to correctly trace the averaged M/L ratio as C14 and J23 suggest, and is effectively used for this purpose.
Our estimations of  * come from the method of C14, which presents a different dependence of / on W1−W2 compared to J23, which was the method for which the K-corrections were developed.Despite this, our most important goal with respect to the ETHER sample is not  * estimates, but rather  BH estimates.The latter showed better agreement with measured  BH values in the literature when using the method of C14.
We initially used the empirical relationship of Schutte et al. (2019) to convert the WISE-based bulge masses to a SMBH mass, given that this is the most recent study of the relationship, and extends to relatively low black hole masses.For this conversion, a comparison of WISE2MBH  BH to the control sample of SMBH measurements and high quality estimates shows a slight offset from equality.The best fit compensation factor (  ) is positive for an initial  BH estimate with log  BH < 9.42.When the first estimate of log  BH is close to the realm of IMBH, this can lead to a large compensation factor.For example, when the first  BH estimate is log  BH = 4.6, thus a   = 0.5, then a final  BH estimate of log  BH = 5.1 is stored.In this example, if   was not applied, the source would have been dropped from the final sample.
When combining the Schutte et al. scaling relationship with the compensation factor (Eq. 6; see also Fig. 12), it is clear that the final scaling relationship we derive and use lies in-between (Fig. 12) the  Bulge −  BH scaling relations of Kormendy & Ho (2013) and Schutte et al. (2019), being more in agreement with the former (latter) at lower (higher) masses.This shows that our final estimates effectively show a population of SMBH with masses larger than that predicted by Schutte et al.. Given that we use only galaxies with detections in all of W1, W2, and W3, and that SMBH with final mass estimates at log  BH < 5 are deleted, the WISE2MBH algorithm is highly incomplete in estimates for dwarf galaxies, so the compensation factor and thus the final  Bulge −  BH scaling relationship are calibrated using only more massive SMBH.
The compensation factor obviously biases the final WISE2MBH  BH estimates to predict a population of SMBH mostly similar to the few local sources that have measurements or reliable estimates of  BH .Given that we use this same   for all redshifts in WISE2MBH, this can lead to systematic errors at higher redshifts.This underlines the need for more and better direct measurements of  BH over a range of redshifts, plus a review of systematics in previous measurements (e.g., Liepold et al. 2023;Osorno et al. 2023).
The algorithm is limited by the relatively low angular resolution of WISE (≥ 6.1 ′′ ), i.e., its limited capacity to constrain the extension or morphology of compact sources.This factor, combined with a certain degree of AGN contamination in some cases, exacerbates the classification of some targets as upper limits (rather than estimates) using the color-color criteria described in Sect.3, causing us to use unnecessarily large values of  Bulge (due to fixing / = 1) and thus providing overestimated  BH values as an upperlimit.

Using WISE2MBH at higher redshifts
In this work we have presented and discussed the use of the WISE2MBH algorithm up to redshift 0.5.This since K-corrections to the WISE photometry are relatively reliable up to this redshift, and the scaling relationships used are derived primarily in local galaxy samples.
In principle, the WISE2MBH algorithm can be extended to the redshift range 0.5-3 by using the K-correction lookup tables from J23.However, the results over this redshift range have significantly larger uncertainties both due to the larger uncertainties in the Kcorrections and due to the several other scaling relationships used being derived and calibrated for lower redshift samples ( ≤ 0.5).
Specifically, the K-corrections, as presented in J23, have been made available for redshifts up to  = 3, but have been shown to be reliable only at  ≤ 0.5.The zone selection and object criteria explained in Sect.4, have been applied systematically to every source within the WISE2MBH parent sample.Object selection and upper limit or rejection criteria are based on K-corrected fluxes and are thus more reliable at  ≤ 0.5.The  Type estimator, an essential component of the algorithm, was calibrated using the 2MRS sample, whose sources are  ≤ 0.15.Lastly, the method used to estimate  * , as explained in C14, has been calibrated for  ≤ 0.12.
To test the algorithm at higher redshifts, we use WISE2MBH to derive estimates for ETHER galaxies at redshifts 0.5-3.Figure 13 compares WISE2MBH  * estimates with literature values of  * for galaxies at  ≥ 0.5 from a control sample from multiple catalogues (Tacconi et al. 2013;Liu et al. 2019;Bacon et al. 2023;López et al. 2023;Mei et al. 2023;Poitevineau et al. 2023).In this figure, the filled markers show sources with WISE  * estimates and the unfilled markers show sources with WISE  * upper limits.All literature values are  * estimates.
In the context of these comparisons between estimates and upper limits against the control sample, the following observations hold: • WISE-based estimates, having undergone prior K-correction, predominantly show a slight underestimation in their  * values.
• WISE-based upper limits, which did not undergo K-correction, typically show an overestimation in their WISE2MBH  * values.This tendency is consistent with the possible AGN contamination of these sources and the absence of K-correction.Since these are explicitly marked as upper limits in our algorithm, this does not lead to errors.
The WISE2MBH algorithm has been shown to be reliable in recognizing sources that are contaminated by powerful AGN emission.To err on the safe side, and given that these sources have insignificant color-color evolution across redshift (e.g., Mateos et al. 2012), we do not use K-corrections for these sources, and mark their mass estimates as upper limits.Nevertheless, it should be noted that the algorithm has a significantly decreased accuracy in estimating  * at redshifts higher than  = 0.5.

Building on the algorithm and heterogeneity
The WISE2MBH algorithm can be considered as an auxiliary tool for obtaining  BH estimates from a homogeneous dataset.This dataset provides consistent estimates of  BH using WISE data for most sources and a consistent set of relations, following accepted ideas and scaling relations applicable to the majority of extragalactic sources.Although the WISE2MBH final sample is heterogeneous in its composition (i.e.different extragalactic objects), users can define subsamples to recover homogeneity.
The use of a single, well-calibrated dataset from observations with the same instrument and method can ensure consistency and comparability of results, but may limit their generalizability and bias the final conclusions.The process described in this work does not necessarily rely solely on WISE data.WISE is used to obtain physical properties of extragalactic sources in the ETHER sample, but external data, such as source classification and morphological types, are also employed in some cases.
Independently derived physical properties, such as  * or  Bulge can be input to an intermediate stage of WISE2MBH in order to obtain  BH estimates.This approach injects heterogeneity into the algorithm and its results.
Despite the use of a homogeneous parent sample, both  BH (lowz) and  * (low-and high-z) were compared with heterogeneous control samples, the former being more important to discuss.The  BH control sample is described in Table 2.The primary methods used to measure  BH in the control sample are stellar dynamics, gas dynamics, and reverberation mapping (RM); all known for their reliability and precision in measuring  BH .Poor quality , single-epoch RM, and other methods used to obtain  BH in R23 were excluded.Despite the heterogeneity of the control sample, the ratio between the literature measured or estimated  BH and the WISE2MBH  BH estimate was calculated in every case, and no prominent differences or scatter was found.Furthermore, the WISE-based bulge luminosity to black hole mass scaling closely follows the relationship of Kormendy & Ho (2013), which is widely used in the literature.
Authors who wish to work with a completely homogeneous subsample of the WISE2MBH parent sample which is based only on WISE data and spectroscopic redshift, could define the subsample as follows: (a) only sources with object type galaxy or unknown, and (b) only sources with a quality flag ending in 2. These constraints ensure that authors work with estimates that only used WISE data for the classification of upper limits and omit estimates that made use of previously known  Type .

Relevance for the EHT and ngEHT
The EHT (and future ngEHT) is the best facility for the imaging of the innermost environments of black holes in terms of sensitivity and resolution (∼10 mJy and ∼15 as, Doeleman et al. 2019;Pesce et al. 2022;Johnson et al. 2023;Doeleman et al. 2023) for the next decade, opening the possibility of imaging (and making movies of) tens to hundreds of SMBH in the nearby Universe.Relevant science goals include testing general relativity, the role of magnetic fields in black hole accretion and jet formation.Recently, Pesce et al. (2021) have demonstrated that with current EHT facilities (at 230GHz), we can expect to resolve ∼5 new SMBH shadows, while with ngEHT observing at 345GHz, this number can be increased by factor ∼3.The challenge is to identify these very rare sources.
In the context of scientific exploitation of the EHT, R23 have developed the ETHER sample and database.Combining ≳3M sources in a parent sample of galaxies and AGN, comprehensive radio to X-ray observed spectral energy densities (SEDs), and jet and accretion flow SED modelling to predict the expected EHT flux, ETHER can provide target samples for the EHT for any given science goal.WISE2MBH was originally developed to fill large gaps in the ETHER sample: delivering both black hole mass estimates and upper limits, and galaxy morphologies.Its accuracy and ease of use, combined with its relevance to galaxy evolution studies, especially at high redshift, motivate its publication as a separate entity from ETHER.
This algorithm does not necessarily intend to replace previous estimates of  BH in the literature (except for values based on poor quality  or relatively unreliable 'fundamental planes'), but rather to increase the completeness of the ETHER sample.As more precise estimations or measurements become available (e.g., from SDSS Black Hole Mapper, Kollmeier et al. 2017, Sect. 2.2), WISE  BH estimates can be replaced in ETHER.
In R23, the authors detail a methodology that requires  BH estimates (or upper limits) and X-ray flux data, primarily from the Chandra Source Catalog (CSC) to predict the EHT-observable flux of an SMBH.With the release of the eROSITA all-sky survey (eRASS, Merloni et al. 2024) DR1 an estimated ∼ 1 million new hard X-ray flux measurements have become available.Integrating eROSITA, WISE2MBH and ETHER, will allow SED fitting, thus radio flux estimates, for almost all sources in eROSITA.This is a critical step for target selection for the ngEHT.

SUMMARY
This work presents a simple and new algorithm to obtain stellar masses ( * ), morphological types ( Type ), bulge-to-total ratios (/), and black hole masses ( BH ) estimates of a galaxy.The WISE2MBH algorithm is publicly available at GitHub3 .This algorithm, which only requires WISE catalog data, classifies sources as galaxies or QSOs, and estimates multiple physical quantities such as  * and  BH .The algorithm uses previously derived scaling relations and our own derived relations to obtain a final  BH estimate or upper limit.Using a parent sample of ∼2.1 million sources from the ETHER sample post-cross-match to the AllWISE and WISE extended source (WXSC) catalogues, a final sample of ∼1.9 million  BH estimates and ∼100 thousand upper limits were calculated.Among the estimates (i.e.not considering upper limits), ∼78.5% are first-time estimates of known galaxies or unclassified sources.QSOs and radio sources (RS), as classified by NED, are also part of the sample, but due to the nature of their emission and the quality or extension of these sources in WISE, all of their final values of  BH are marked as upper limits.The final sample table is available online via CDS.
The detailed manual morphological classifications ( Type ) of galaxies in the 2MASS redshift survey (2MRS) were used to derive a relation between  Type and WISE W2−W3 color, with the objective of estimating  Type for sources that do not have one previously assigned in the literature.All available and estimated  Type are used to obtain / using an exponential relation described in Sect.6 that is consistent with previous studies.The obtained / are used to calculate  Bulge from a WISE derived  * .Finally, we use our  BH −  Bulge scaling relation -which lies between accepted literature scaling relationships -to estimate  BH .All uncertainties are propagated through the algorithm using a Monte Carlo approach, delivering the 1 (upper and lower) errors of the final distributions as low and high values, respectively.
The  BH estimates were compared to a control sample of  BH measurements and reliable estimates, showing a significant difference in the linear regression analysis with respect to the equality line, i.e., an offset that causes some values to be overestimated and others underestimated.To compensate for this offset, we implement a compensation factor (  ) derived with the use of the control sample.After compensation, the comparison achieves a Spearman score of ∼ 0.78 and a RMSE of ∼ 0.63.The final scaling relationship used (eqn.6) lies between the relationships of Kormendy & Ho (2013) and Schutte et al. (2019), being more consistent with the former at low SMBH masses and the latter at high SMBH masses.The mean uncertainty was calculated for the  BH estimates, considering a simple mean between low and high errors and then taking the mean value of the distribution of means, obtaining a value of ∼ 0.5 dex, showing more scatter in low bulge fraction (LBF) sources, compared to high bulge fraction (HBF) sources (see Fig. 9).
The black hole mass function (BHMF) of the WISE2MBH final sample is in good agreement with other previously and independently derived BHMFs.The ETHER-only sample has few low mass estimates (log  BH ≤ 6), while the WISE2MBH final sample can provide this population of sources, and the overall combination of both samples generates the most complete BHMF.We find evidence that high mass SMBH are more common than predicted by literature BHMFs.
When using the WISE2MBH algorithm or the final sample described in this work, it is important to take into account the assumptions and associated limitations.The algorithm provided on GitHub can be easily modified to change the scaling relations used or incorporate new ones, tailored to the user's requirements.
Regarding the final sample, we recommend not considering all  * or  BH estimates if the main goal is to study only a few sources and/or restrict the sample to only high quality sources (HQS).In case of population studies, almost the complete final sample can be used, depending on the redshift limit, quality flag, and the requirements of the user.
The final sample was generated in a homogeneous manner, i.e. all estimates come from relations that make use of WISE cataloged data to derive physical quantities, except for the use of  Type from the literature in some cases.This gives confidence that the derived values are consistent from one to another and no externally derived physical parameters were used to obtain the final  BH .
The WISE2MBH final sample is already incorporated into the ETHER sample, providing almost 3 million new  BH estimates and upper limits that, and it will be used iteratively to provide upto-date values in case new sources are ingested into the ETHER sample.These estimates are crucial for the selection of samples of interest for the Event Horizon Telescope (EHT) and the nextgeneration EHT (ngEHT), and are used on each update of the sample.Its high percentage of success in estimating a new  BH , combined with spectral fitting of accretion and jet models to hard X-ray data from Chandra, XMM, and eROSITA, allows one to predict radio fluxes from the accretion inflow and jets, and thus obtain a first selection of sources detectable with the EHT or ngEHT.

APPENDIX B: COMPARISON FOR VARIOUS SAMPLES IN W2-W3 COLOR TO T-TYPE CONVERSION
Our conversion of W2−W3 color to  Type was calibrated with galaxies in the 2MRS sample (Sect.5).To explore the reliability and errors of this conversion, we used the same method of Sect.5, but for other samples present in Hyperleda (Makarov et al. 2014).
The GZ2 sample exhibits a clear bimodality in T (bottom row of Fig. B1).The authors provide detailed classifications for spiral galaxies, allowing clear differentiation from Sa to almost Sd classifications (1 to 6 in  Type ).However, in the morphological range from lenticular to elliptical, the level of detail is completely lost: all except 530 (i.e.98%) of these sources are classified as  Type = −5.
Dobrycheva classified their sample galaxies into two bins:  Type equal −5 and 5 (i.e.ellipticals and spirals; bottom row of Fig. B1).Although this binary classification has demonstrated efficacy for machine learning training and subsequent classification of different samples (e.g., Vavilova et al. 2021Vavilova et al. , 2022)), it does not provide the level of detail required for our W2−W3 color to  Type calibration.Nevertheless, we use this sample here only for comparison purposes.
In contrast, the 2MRS sample shows a relatively smooth and well populated distribution of  Type values (bottom row of Fig. B1).
The distributions (median values and 1 intervals) of W2−W3 for each  Type in 2MRS and GZ2 are presented in the top panel of Fig. B1.For Dobrycheva, only the medians are shown for comparison.The GZ2 fit shows a similar trend to 2MRS in the range of  Type 0 to 5, covering the same range but with smaller error bars.At the upper end, the fits differ by approximately 0.2 in W2−W3, which has a negligible impact on the final  Bulge estimates when comparing the GZ2 fit to 2MRS fit.At the lower end, there is a difference of almost 0.4 in W2−W3, leading to significant variations in the estimated /.For a source with a W2−W3 color of 1, the 2MRS fit gives a / ∼ 0.5 estimate, while the GZ2 fit results in a / ∼ 1, corresponding to a difference of 0.3 dex in the  Bulge estimates, which are the first estimates affected by the / value.These systematic changes in fit have significant implications for the final estimates and the overall statistics of the sample.Due to the lack of detail and biased representation, the GZ2 sample and its fit were discarded from our analysis.For the Dobrycheva sample, elliptical galaxies exhibit distinct shifts compared to both 2MRS and GZ2, primarily due to the binarity of the classification used.
Therefore, it is crucial to use a sample that is large enough to accurately discriminate between  Type and then to rely on the trend shown by that specific sample.We defined this 'large enough' sample size to be , and determined it to be approximately  ∼ 700 per bin by statistical power analysis, which is a widely used statistical tool for sample size determination in meta-analyses (e.g., Borenstein et al. 2009;Grundler et al. 2022).In statistical power analysis, the three parameters to be set are statistical power, significance threshold, and effect size.The statistical power () is often defined as the probability that a study can correctly detect a real effect (i.e., probability of avoiding a Type II error).The significance threshold () is defined as the highest level of acceptable risk in terms of incorrectly rejecting a null hypothesis that is actually true (i.e., Type I error probability).Effect size (  ) is a standardized technique available to measure the expected strength of the results in a study, regardless of the sample size.This   can only be calculated for two samples, so in multisample scenarios this has to be calculated for each pair of samples in a predefined order, e.g., if the samples represent an evolution from 0 to 10 in a quantity, the pairs to calculate   must be 0−1, 1−2, and so on.
For this analysis, we used the median values of the W2−W3 distribution for each  Type value in 2MRS, with parameters;  = 0.8,  = 0.05 and   = 0.15.However, the value of   varies for each pair or consecutive distribution of W2−W3 following the order of  Type , as shown in the middle plot of Fig. B1.The median value of   was approximately 0.17, which corresponds to  ∼ 550.To ensure stricter statistical power, we reduced   to 0.15.For the lower and upper ends of  Type ,   can go as low as 0.001, requiring a sample size of  ≥ 10 7 to confidently claim that the consecutive distributions are, in fact, two distinct populations and not two samples from the same parent sample.However, the chosen value of   is sufficiently strict to clearly discriminate between each distribution of W2−W3 for  Type ranging from -3 to 6.
In the bottom plots of Fig. B1 one can see that the required  (black dots) is surpassed in most cases by the three samples (colored histograms), but notoriously larger samples are needed for the most extreme values of  Type .It is clear that the results obtained from the 2MRS statistical power analysis are not directly applicable to limit the use of other samples, but similar median values are expected for the whole population of galaxies and the different  Type between samples, thus needing similar , independent of the sample used.
We decided to use 2MRS over the other samples tested due to the completeness in the lenticular-spiral regime and since both 2MRS and GZ2 samples showed similar behavior in the late-type regime, even though 2MRS is almost 10 times smaller in overall sample size.

APPENDIX C: CORRELATIONS AND SCATTER BETWEEN WISE2MBH AND SDSS ESTIMATES OF STELLAR MASS
Since deriving  * is the first step towards estimating  BH , the WISE-estimated  * must be accurate as compared to estimates obtained with different homogeneous methods applied to very large samples.We found that using only the C14 method, or only the J23 method, did not lead to optimal results when comparing to literature values.Instead combining the  * estimation from C14 with prior K-corrections derived by J23 gave the most accurate results.
To test for bias and consistency, we compare the WISE2MBH derived  * estimates to those of three different value added catalogues (VACs) of  * available via SDSS: Portsmouth group (Maraston et al. 2013) who used SED fitting with passive and star-forming templates with the Kroupa initial mass function (IMF, Kroupa 2001), Wisconsin group (Chen et al. 2012) who used the principal component analysis (PCA) method in the optical rest-frame spectral region (3700−5500 Å) with two different single stellar population models from Bruzual & Charlot (2003, BC03) and Maraston & Ström-bäck (2011, M11), and finally the Granada group (Montero-Dorta et al. 2016) who used the flexible stellar population synthesis (FSPS) method for early and wide formation times.
The correlation and scatter matrices considering all catalogues mentioned above and WISE2MBH for three different redshift bins are presented in Fig. C1.
For correlations (upper row of Fig. C1), Spearman scores range from ∼ 0.6 − 0.9 for almost every method considered in the lowest redshift bin ( < 0.1).When considering a larger redshift bin ( < 0.3), all correlation scores decreased, particularly for methods that take into account completely different methodologies, such as Port_SF and Gra_Early.When considering the entire redshift range ( < 0.5), the scores decreased once again, now down to 0.34 for the case of WISE2MBH and Gra_Wide.
The worst scores are found between Gra_Wide and Port_SF in the  < 0.1 (score = 0.62) and  < 0.3 (score = 0.47) bins.Port_SF performs the worst in intercomparisons in the  < 0.3 bin.WISE2MBH performs well compared to SDSS VACs except in the highest ( < 0.5) redshift bin.
When considering the scatter between multiple methods (lower row of Fig. C1) WISE2MBH shows the lowest RMSE across all the methods and redshift bins, i.e., it is the method which overall most closely agrees with all VACs on average.For WISE2MBH and Port_Passive, the RMSE is 0.24 for  < 0.1 and even lower (0.22) when considering the entire redshift range.However, for WISE2MBH and Gra_Early, the RMSE increases with redshift.
For the specific case of WISE2MBH and Gra_Early, both correlation and scatter got worse with increasing redshift.The correlation starts reasonably well (score = 0.76) in the lowest redshift bin, then becomes one of the worst scores in the entire  < 0.5 bin (score = 0.44).The same pattern is seen in the scatter, which increases from RMSE of 0.32 to 0.48, the latter being one of the highest RMSE in the entire matrix.In Fig. 6 it is clear that there is a systematic shift with redshift that causes the low correlation score.It is the only pair that shows this strong tendency to get worse in both metrics with increasing redshift.

Figure 1 .
Figure 1.WISE2MBH:A simple algorithm that makes use of WISE cataloged data and a spectroscopic redshift to estimate the stellar mass, morphological type, bulge fraction, and  BH of an extragalactic source.Solid (dotted) lines represent the main path to estimate a value (or upper limit), or to reject an object from the algorithm.Orange (blue) boxes show the input (derived) quantities; boxes with both colors can either be provided to or are estimated by the algorithm.Inputs in dashed boxes are optional.WISE magnitudes with their respective mean photometric errors (σ   ) are used to generate random normal samples of size 10 4 for a Monte Carlo approach to error propagation.

Figure 2 .
Figure 2. Redshift distribution of the WISE2MBH parent sample.Left: Separated by object type.Right: Separated by source extension (as listed in the AllWISE catalog).Galaxies and unknown type sources dominate by a few orders of magnitude over other object types, while point-like (3-0) sources dominate at z ≳ 0.1.

Figure 3 .
Figure 3. WISE color-color plot showing the location of our sample objects and defining the areas over which the stellar mass estimation is an upperlimit (Upper limit zone), is not estimated by the algorithm (Reject zone), or estimated by the algorithm (Estimate zone); these zones are separated by black dashed lines.Note that QSO and RS object types are considered as stellar mass upper limits independent of whether they fall in the "Upper limit" or "Estimate" zone.For clarity, we plot separately the point sources (top panel) and extended sources (bottom panel; see Sect. 2).The background-filled colors and labels are from Fig. 12 of Wright et al. (2010), and are shown for reference.Contours, in colors following the color legend on top of the figure, show the number density of object types in our sample.In the bottom panel, the object types QSO and RS are shown as colored points (instead of contours) of the corresponding color.

Figure 4 .
Figure 4.  Type as a function of W2−W3 color for galaxies in the 2MRS sample.For each  Type bin we plot the median value (black circle) and one sigma dispersion (horizontal bar) of the W2−W3 colors of galaxies in the bin.Distributions of W2−W3 and  Type are shown in the panels at the top and right of the figure, respectively.The red line in the right panel marks the threshold number of galaxies in a bin in order for that bin's median to be used for the fit (filled black circles in the main panel).The black dot-dashed line shows the best fit logit function to the filled black circles: this is used for the W2−W3 to  Type conversion when  Type is previously unknown.The estimated  Type is limited to the range −5 and 8; when a galaxy's W2-W3 color lies beyond the range of the logit function shown, the  Type is clipped at these values.The colored areas distinguishing morphologies listed in the inset are fromJarrett et al. (2019).Given the similarity of the color dispersions in the three bins at each extreme end of the -axis, we define two vertical dashed lines which delineate galaxies we refer to as high bulge fractions (HBF; bulge fractions between 0.4 and 1) and low bulge fractions (LBF; bulge fractions between 0.1 and 0.3).

Figure 5 .
Figure 5. Distributions of the bulge-to-total mass ratio (/) as a function of  Type for different literature samples.A decreasing exponential is fitted to the data points: the gray dashed curve is the original fit and the black dashed curve is the fit when one fixes / = 1 for  Type = −5.For clarity, small shifts on the x-axis are used to avoid overlapping symbols and error bars.Data points and the horizontal pink dashed line are from Caramete & Biermann (2010); Mendel et al. (2014); Sofue (2016); Morell et al. (2020); Gao et al. (2020) following the colors listed in the inset.

Figure 6 .
Figure 6.WISE2MBH  * compared to the low redshift ( ≤ 0.5) galaxy samples of Chang et al. (2015, Top left) and Mendel et al. (2014, Top right) and the SDSS Value Added catalogues of the Portsmouth (Maraston et al. 2013, Middle), Granada (Chen et al. 2012, Bottom right) and Wisconsin (Montero-Dorta et al. 2016, Bottom left) groups.The black dashed line shows the line of equality and red dots with error bars represent the median and 1 dispersion of the difference between the respective masses for bins of 0.05 in redshift.Colors represent the counts following the color bar to the right of each panel.

Figure 7 .
Figure 7. Same as Fig. 6, but now comparing  Bulge from WISE with its equivalent value in the low redshift ( ≤ 0.4) sample of Mendel et al. (2014).

Figure 8 .
Figure 8.A comparison of measured black hole masses (Top, crosses) and highly reliable black hole mass estimates (Middle, boxes) vs. WISE2MBH  BH values in the WISE2MBH final sample.Each data point is marked with its 1 error bars.Subsamples of HBF and LBF galaxies are distinguished by color following the inset.Gray dotted lines are the RMSE scatter bands.The bottom panel shows the distribution of the mass ratios for subsamples of measured and estimates and a KDE for the complete distribution; the mean ratio and 1  dispersion (1.00 ± 0.08) of the latter are shown with red and gray dashed lines.

Figure 9 .
Figure9.Mean  BH errors as a function of /.In LBF, there are few cases with larger error compared to the mean errors for both HBF and LBF.This comes directly from the detection of small sources that have worst quality than the bulk of sources detected by WISE in the AllWISE catalog, producing greater errors due to error propagation.
From the ETHER sample Nagar et al. (in prep); see R23 for a description of the compilations.extended sources, and (d) is necessary since van den Bosch et al. (2015) contains a few very large  values.To avoid those, we consider the maximum value from van den Bosch (2016), who used the same observations from HET as part of the study to measure  BH , as a limit for our control sample.The control sample covers a mass range from log  BH of ∼6 to ∼10, this being almost the complete mass range for SMBH.The heterogeneity of this control sample is discussed in Sect.11.3.

Figure 10 .
Figure 10.Distribution of  * (left panel) and  BH (middle panel) for the WISE2MBH final sample.For Galaxies and Unknown, two distributions are shown: the dashed line for estimates and the solid line for both estimates and upper limits.The distributions change slightly from one panel to the other, due to the use of / to obtain  Bulge .The right panel shows a stacked histogram illustrating the number of  BH estimates and upper limits for each object type.
for shells of increasing distance out to 300 Mpc.These WISE2MBH final sample BHMFs are compared to the local BHMF in R23, and other BHMFs from independent methods: Shankar et al. (2016, compensated  BH -), Gallo & Sesana (2019, X-ray), Pesce et al. (2021,  BH - * ) and Yao et al. (2023, TDE).The WISE2MBH final sample BHMF was derived using only estimates from WISE.At log  BH > 10, the WISE2MBH final sample has only a few (elliptical) galaxies, so that the derived BHMFs shows large fluctuations given the small number statistics.For this reason, the BHMFs at log  BH ≳ 10 are shown as dashed lines.

Figure 11 .
Figure 11.Left: Solid lines show the black hole mass function (BHMF) in the WISE2MBH final sample for shells of width 30 Mpc ending at distances of 30 to 300 Mpc, following the color bar at right; at high black hole masses, the BHMFs are uncertain and are shown with dashed lines.ETHER+WISE and ETHER points (see inset) represent the closest BHMF (0 − 30 Mpc) present in ETHER with and without considering WISE  BH estimates.Right: Same as left, but now for three shells ending at 30, 180 and 300 Mpc, showing the BHMF using median values (solid) and ten curves showing the BHMF created using random sampling from the PDF of each  BH estimate (dashed; see text).For reference both panels show four BHMFs independently derived by Shankar et al. (2016); Gallo & Sesana (2019); Pesce et al. (2021); Yao et al. (2023).

Figure 12 .
Figure 12.Comparison of different  BH −  Bulge scaling relations from the literature, including Kormendy & Ho (2013); Saglia et al. (2016); Schutte et al. (2019) and the modified scaling presented in this work.Grey area represents the limit of the WISE2MBH algorithm for log  BH < 5, where it drops all estimates.

Figure 13 .
Figure 13.WISE2MBH-derived  * compared to the high redshift ( ≥ 0.5) samples of Tacconi et al. (2013); Liu et al. (2019); Bacon et al. (2023); López et al. (2023); Mei et al. (2023); Poitevineau et al. (2023), with colors specified in the inset.The black dashed line is the line of equality.Datapoints are colored by redshift following the color bar on the right (blue is  = 0.5 and red is  ∼ 3).Filled and unfilled markers are used for estimates and upper limits of WISE2MBH  * , respectively; all literature values are estimates.

Figure A1 .
Figure A1.Top: WISE two-color images with FOV of 10' of NGC 7626 (left; a HBF galaxy) and NGC 7773 (right; an LBF galaxy): blue represents the W2 band and red the W3 band.The W2−W3 color clearly distinguishes between the HBF and LBF galaxies.In each panel, the SDSS DR16 image of the galaxy is shown as an insert in the upper right corner for reference.Bottom: The corresponding WISE2MBH  BH probability density function (PDF) provided by our algorithm for each source in the top row.The red vertical line denotes our final (median)  BH value and the dashed vertical lines represent the 1 of the distribution; the reported values for the lower and upper values of  BH .The HBF galaxy has smaller  BH uncertainties as compared to the LBF galaxy; a trend seen in general for LBF galaxies, e.g., Fig. 9.

Figure B1 .
Figure B1.Top: As in Fig. 4 but for all samples described in Appendix B. Blue dots are slightly shifted in the Y-axis to distinguish between error bars.Colors follow the legend in the top left of the panel.Middle: Effect size (  ) for every consecutive distribution of W2−W3 color, following the order of  Type .The position on the X-axis is the middle value between consecutive  Type .Bottom:  Type distributions of the three samples shown in the top panel are shown in colored histograms of the corresponding color.Black connected dots denote the sample sizes ( ) required to establish a distinction between consecutive  Type bins, as derived from the medians of the 2MRS W2−W3 distributions in each  Type bin.

Table 1 .
Statistics of the WISE2MBH parent sample.

Table 2 .
Control sample of  BH measurements and reliable M−  estimations.
Schutte et al. (2019)ted as the need for a specific   for LBF sources or a misbehavior of previous steps for these types of galaxies, e.g.underestimation of / or  * .The  BH −  Bulge scaling relation of Schutte et al. can be combined with our   , to obtain the following relationship, which is effectively that used in WISE2MBH: Finally, in case the algorithm estimates a log  BH ≤ 5, the source is dropped.While the  BH −  Bulge relation ofSchutte et al. (2019)can reach log  BH ≤ 5, such black holes go down to the limits of intermediate mass black holes (IMBH).IMBHs and their host populations are an active topic of research (for a review, seeGreene et al. 104 log  BH + 0.98(5)and added to the estimate.After this empirical correction, a Spearman score of 0.78 and a root mean squared error (RMSE) of 0.63 dex (see Fig.8) were calculated for the set of compensated estimates and control sample.Since the majority of the control sample are HBF, the small offset of LBF shown at lower  BH ranges does not affect the overall comparison

Table 3 .
Statistics of the WISE2MBH final sample.

Table 4 .
Excerpt of the WISE2MBH final sample.Some values are rounded for table presentation.The full version of this table is available online via CDS.