# Color Transformations of Photometric Measurements of Galaxies in Optical and
Near-Infrared Wide-Field Imaging Surveys111An online color transformation
calculator is available at: https://colors.voxastro.org/

Victoria A. Toptun European Southern Observatory, Karl Schwarzschildstrasse 2,
85748, Garching bei München, Germany  Sternberg Astronomical Institute, M.V.
Lomonosov Moscow State University, Universitetsky pr., 13, 119234, Moscow,
Russia  Department of Physics, M.V. Lomonosov Moscow State University, Moscow,
Russia  Igor V. Chilingarian Center for Astrophysics – Harvard and
Smithsonian, 60 Garden St. MS09, Cambridge, MA 02138, USA  Sternberg
Astronomical Institute, M.V. Lomonosov Moscow State University, Universitetsky
pr., 13, 119234, Moscow, Russia  Kirill A. Grishin Université Paris Cité,
CNRS, Astroparticule et Cosmologie, F-75013 Paris, France  Sternberg
Astronomical Institute, M.V. Lomonosov Moscow State University, Universitetsky
pr., 13, 119234, Moscow, Russia  Ivan Yu. Katkov New York University Abu
Dhabi, Abu Dhabi, UAE  Center for Astro, Particle, and Planetary Physics, NYU
Abu Dhabi, Abu Dhabi, UAE  Sternberg Astronomical Institute, M.V. Lomonosov
Moscow State University, Universitetsky pr., 13, 119234, Moscow, Russia

###### Abstract

Over the past 2 decades, wide-field photometric surveys in optical and
infrared domains reached a nearly all-sky coverage thanks to numerous
observational facilities operating in both hemispheres. However, subtle
differences among exact realizations of Johnson and SDSS photometric systems
require one to convert photometric measurements into the same system prior to
analysis of composite datasets originating from multiple surveys. It turns out
that the published photometric transformations lead to substantial biases when
applied to integrated photometry of galaxies from the corresponding catalogs.
Here we present photometric transformations based on piece-wise linear
approximations of integrated photometry of galaxies in the optical surveys
SDSS, DECaLS, BASS, MzLS, DES, DELVE, KiDS, VST ATLAS, and the near-infrared
surveys UKIDSS, UHS, VHS, and VIKING. We validate our transformations by
constructing $k$-corrected color-magnitude diagrams of non-active galaxies and
measuring the position and tightness of the ‘red sequence.’ We also provide
transformations for aperture magnitudes and show how they are affected by the
image quality difference among the surveys. We present the implementation of
the derived transformations in python and idl and also a web-based color
transformation calculator for galaxies. By comparing DECaLS and DES, we
identified systematic issues in DECaLS photometry for extended galaxies, which
we attribute to the photometric software package used by DECaLS. As an
application of our method, we compiled two multi-wavelength photometric
catalogs for over 200,000 low- and intermediate-redshift galaxies originating
from CfA FAST and Hectospec spectral archives.

Color equation (269); Photometric systems (1233); Galaxy photometry (611);
Galaxy colors (586)

##  1 Introduction

Over the past decades, significant progress has been achieved in the quality
and depth of wide-field imaging sky surveys in virtually all wavelength
domains. However, different instruments implement even ‘standard’ photometric
systems slightly differently, and methods of data processing and analysis also
vary. Therefore, for many scientific applications which require multiple
sources of photometric data (e.g. to search astrophysical sources based on
specific features of their spectral energy distributions or compare physical
properties of objects of the same type originating from several surveys), one
needs to homogenize and post-process these datasets. Homogeneous photometric
and spectroscopic datasets possess a great discovery potential for finding new
statistical properties of populations of astronomical objects of a given type
(e.g. a universal color–color–magnitude relation of non-active galaxies found
by Chilingarian & Zolotukhin, 2012) or using sophisticated data mining
techniques to identify rare sources with a frequency among the total
population of 1:10,000 (e.g. compact elliptical galaxies Chilingarian &
Zolotukhin, 2015 or intermediate-mass black holes Chilingarian et al., 2018)
or even 1:100,000 (e.g. diffuse extended post-starburst galaxies Grishin et
al., 2021).

A few years ago our team released the Reference Catalog of Spectral Energy
Distributions of galaxies (RCSED; Chilingarian et al., 2017), which contains
the results of analysis of 800 k spectra of non-active galaxies accompanied by
homogenized photometric data from GALEX (Martin et al., 2005), SDSS (Abazajian
et al., 2009), and UKIDSS (Lawrence et al., 2007). The photometric
measurements in RCSED originated from multiple sources, however, measurements
in every photometric band came only from one particular survey. This situation
changed drastically in RCSEDv2, which contains a substantially larger sample
of galaxy spectra, about 4.7 million, over the entire sky and photometric data
from over a dozen of different wide-field sky surveys originating from
different telescopes and instruments. However, all optical and near-infrared
measurements were made in the photometric systems implementing either SDSS
(Fukugita et al., 1996) or Johnson (Johnson et al., 1966).

Creating a photometric catalog of galaxies, which covers almost the entire sky
requires a combination of several photometric surveys, hence, leading to the
problem of converting all the measurements to a ‘standard’ photometric system.
Color transformations for all the surveys used in RCSEDv2 were described in
the literature, however, they were calculated using stars (Shanks et al.,
2015; Abbott et al., 2021; Dey et al., 2019), which cover a limited range of
colors. Hence, even though they work reasonably well for aperture and PSF
magnitudes of point sources, it is not surprising that if one applies them to
galaxies having different colors and extended shapes, some serious systematic
errors will emerge. We should consider all the emerging systematics for
extended targets and, therefore, the calculations from ideally corrected point
sources will not suit our purposes. For example, aperture corrections become
very important if one starts from point sources. They are dependent on the
image quality, which differs quite substantially from one area of the sky to
another even within each particular ground-based survey. Moreover, aperture
corrections for galaxies require the knowledge of their seeing-deconvolved
surface brightness profiles. This is the main motivation for us to use total
magnitudes of galaxies to re-compute the transformations. This should also
significantly mitigate the problem of varying depths delivered by different
surveys we included in this study.

Figure 1: A color–magnitude diagram for low-redshift ($0.02<z<0.2$) galaxies
constructed from SDSS (gray shaded image), DES (blue contours), and BASS
(green contours). DES and BASS data were converted into the SDSS photometric
system using the published transformations. The red sequence as defined by the
RCSED catalog is shown by the red solid line and its r.m.s. by the red dotted
lines.

As an ultimate quality test of the color conversions in the optical bands one
can use a unique property of non-active galaxies, their color bi-modality, in
particular the very tight ‘red sequence’ formed mostly by early-type galaxies
in the color–magnitude diagrams constructed from $u$ or $g$ (blue) and $r$,
$i$, or $z$ (red) data (Visvanathan & Sandage, 1977; Sandage & Visvanathan,
1978). In order for this test to work properly, we need to know the redshifts
and compute accurate $k$-corrections, that is a correction of a photometric
measurement of an extragalactic source, which brings it to either rest-frame
or to a specifically chosen redshift value. If we use galaxy flux measurements
(Petrosian magnitudes, (Petrosian, 1976)) from Dark Energy Survey and
published color conversions between DES and SDSS from Abbott et al. (2021) for
a sample of low-redshift ($z<0.2$) galaxies from RCSEDv2, there will be a
systematic offset of the red sequence in the $g-r$ color by about $-0.05$ mag,
and the slope will be shallower. On the other hand, if we use BASS, the
Northern part of DESI imaging surveys, the red sequence turns up redder by
$+0.06$ mag, also with a somewhat different slope from that derived using SDSS
data (see Figure 1). Therefore, the intrinsically thin (0.04 mag) red sequence
in the combined dataset will puff up to $>0.1$ mag hampering many scientific
applications, which require SED fitting, e.g. photometric redshift estimates,
search for members of specific galaxy types, inferring star formation
histories.

This brings our main motivation to re-compute color transformations between
several variants of the optical SDSS ($ugriz$) and near-infrared Johnson
($YJHK$) photometric systems using a carefully selected sample of low-to-
intermediate redshift galaxies with spectroscopic redshifts, all measured in a
similar fashion in terms of fluxes (Petrosian magnitudes). We can then test
the derived transformations by reconstructing the color–magnitude diagrams of
galaxies in different bands and measuring the position and thickness of the
red sequence.

##  2 Data

Figure 2: Distributions of redshift (left) and major axis (right) for RCSEDv2
(blue) and RCSED (red). A large amount of sources on z=0.6 originates from
eBOSS. Figure 3: Coverage footprints of optical (right) and infrared (left)
photometry for the parent sample of 4.7M galaxies.

The input photometric catalog is based on 4.7M galaxies with redshifts in
publicly available wide-field surveys covering almost completely the northern
and southern hemispheres of the sky. Redshifts are needed for further
validation of photometry through building a red sequence (see 5.2), as well as
for quickly separating galaxies from stars. To exclude the stars, we selected
only objects with redshifts greater than 0.0016 (corresponding to radial
velocity $\sim$ 480 km/s) or those classified as galaxies in the Hyperleda
database (Makarov et al., 2014). The sample comprises galaxies originating
from the following spectroscopic surveys and catalogs: SDSS DR16 (Ross et al.,
2020), 2dFGRS (Colless et al., 2001), 6dFGS (Jones et al., 2004), FAST (Mink
et al., 2021), Hectospec (Fabricant et al., 2005), LAMOST (Yao et al., 2019),
LEGA-C (van der Wel et al., 2021), DEEP2/DEEP3 (Newman et al., 2013), WiggleZ
(Parkinson et al., 2012), GAMA DR4 (Bellstedt et al., 2020), 2dFLenS (Blake et
al., 2016).

To investigate the differences in photometric systems and calculate the
conversions, we used publicly available wide-field surveys in the optical and
infrared spectral domains with photometric systems similar to or implementing
those of SDSS and UKIDSS. The final photometric catalog was assembled from the
following sky surveys: SDSS, DESI Legacy Surveys (Dey et al., 2019), Dark
Energy Survey (Abbott et al., 2021), DELVE (Drlica-Wagner et al., 2022),
PanSTARRS PS1 (Flewelling et al., 2020), VST ATLAS (Shanks et al., 2015), KiDS
(de Jong et al., 2017) (optical), UKIDSS, UHS (Dye et al., 2018), VHS (Cross
et al., 2012), VIKING (Edge et al., 2013) (NIR) and unWISE (Schlafly et al.,
2019) (NIR/mid-IR). The datasets were retrieved using Virtual Observatory
access services. We list all the data sources in Table 1.

The unWISE data are not used to calculate conversions, but to remove quasars
and strong type-I AGN from the sample. We exclude quasars, which often show
significant broadband variability in optical and NIR domains exceeding 0.2 mag
and therefore are not suitable for calibration purposes by using an empirical
filter based on unWISE magnitudes: $w1-w2<0.2$ (Assef et al., 2013). In
addition, to exclude sources with poor quality of photometry, we did not use
measurements with statistical uncertainties exceeding 0.3 mag.

Table 1: List of surveys used in this study Name of survey | Instrument | Coverage | Bands | Type | Apertures | Source
---|---|---|---|---|---|---
|  | (deg2) |  |  | Diameter (′′) |
SDSS DR17 | SDSS | 14,555 | ugriz | Petrosian | 2, 3 | SDSS CasJobs222https://skyserver.sdss.org/CasJobs/
DES DR2 | DECam | 5000 |  griz (DECam) | Petrosian |  2.5cm0.49, 0.97, 1.46, |
1.95, 2.92, 3.90, |  |  |  |  |  |
4.87, 5.84, 6.82, |  |  |  |  |  |
7.79, 11.69, 17.53 | Astro Data Lab333https://datalab.noirlab.edu/ |  |  |  |  |
DELVE DR2 | DECam | 17,000 |  griz (DECam) | Petrosian | none | Astro Data Lab
2.5cmLegacy Survey DR9 |  |  |  |  |  |
(DeCALS, MzLS+BASS) |  2.5cmDECam |  |  |  |  |
90Prime |  |  |  |  |  |
Mosaic-3 | 19,000 | grz | Model |  2.5cm1, 1.5, 2, 3, 4, |  |
7, 10, 14 | Astro Data Lab |  |  |  |  |
PanSTARRS DR1 | PanSTARRS | 30,000 |  griz (PS) | Kron, PSF | none | Vizier TAP444https://tapvizier.u-strasbg.fr/adql/
VST ATLAS DR3 | VST OmegaCAM | 4500 |  ugriz (VST) | Petrosian | 2, 2.8, 5.7 | Vizier TAP
KiDS DR3 | VST OmegaCAM | 1350 |  ugriz (VST) | Kron | none | Vizier TAP
UKIDSS DR11 | WFCAM | 3700 |  YJHK (UKIRT) | Petrosian | 2, 2.8, 5.7 | Astro Data Lab
UHS DR1 | WFCAM | 12,700 |  YJHK (UKIRT) | Petrosian | 2, 2.8, 5.7 | WFCAM Science Archive555http://wsa.roe.ac.uk/
VHS DR6 | VIRCAM | 16,730 |  YJHK (VISTA) | Petrosian | 2, 2.8, 5.7 | Astro Data Lab
VIKING DR5 | VIRCAM | 1500 |  YJHK (VISTA) | Petrosian | 2, 2.8, 5.7 | VISTA Science Archive666http://horus.roe.ac.uk/vsa/

##  3 Method description

###  3.1 Data preprocessing

Before starting data homogenization, it is necessary to reduce all the
magnitudes to common zero-points and aperture radii, as well as clear the
sample from obviously incorrect values that will clog up the statistics. In
the beginning, magnitudes with NaN values such as – 9999, 999, etc. (the exact
value depends on the survey) were removed from the sample. Surveys that are
not originally in the AB system are converted to AB using published zero-point
differences: Hewett et al. (2006) for the UKIRT surveys UKIDSS and UHS
(+0.634, +0.938, +1.379, +1.9 for YJHK respectively) and González-Fernández et
al. (2018) for the VISTA surveys VHS and VIKING (+0.6, +0.916, +1.366, +1.827
for YJHK respectively). Also, to fit the data we use measurements only for
galaxies with small angular sizes having radii containing 50% of Petrosian
flux smaller than 2.′′5 to minimize the effects of potentially poor sky
subtraction for very extended targets.

Figure 4: Example of a piecewise linear regression in the Legacy-MzLS $g$ band
(left) and KiDS $i$ band (right). Black dashed lines show the breakdown into
linear regression segments. The color indicates the weights. The red line is
the fitting result. The Y-axis shows the difference between the correcting
band and the corresponding SDSS band, the X-axis shows the color in the
correcting survey.

###  3.2 Calculation of color transformations

After preprocessing we convert all magnitudes to some “standard” photometric
system: this is a crucial step when one needs to use photometric measurements
originating from different sources. Despite the same name, e.g. “SDSS g,” the
actual implementations of a given photometric systems by different survey have
significantly different filter throughput curves. Therefore, one needs to
adopt one reference photometric system and convert all the available
measurements into it. We adopted the filter sets from SDSS in the optical and
UKIDSS in the near-infrared as our internal reference. One has to pay special
attention to the fact that “integrated” photometry from different surveys may
have different magnitude types, e.g. Petrosian in SDSS vs Kron (Kron, 1980) in
PanSTARRS. Also, Petrosian radii depend on the survey depth, therefore they
are not the same for, e.g. SDSS and DES, leading to inconsistent integrated
flux measurements. Nevertheless, for the computation of color transformations
we used total magnitudes because the main goal of our work was to be able to
build integrated spectral energy distributions of galaxies using data from
different surveys, which would be consistent among them. We also discuss the
applicability of our transformations to aperture magnitudes, which do not
depend on the survey depth, but do depend on the image quality.

We approximate a relation between the magnitude difference of the survey being
converted and the reference survey and an observed color of a galaxy. A simple
$\chi^{2}$ fitting of a 2D scattered dataset with the weights of each point
corresponding to the uncertainties of original measurements will not produce
satisfactory results because densely populated regions of the color–magnitude
space will dominate the statistics and the solution will ‘ ‘ignore” sparsely
populated “edges.” Chilingarian & Zolotukhin (2012) proposed a weighted
approach to fit 2D and 3D scattered datasets from SDSS and GALEX to infer the
parameterisation of the color–color–magnitude relation for non-active
galaxies: a weight assigned to each measurement depends on the local density
of data points in the parameter space inferred from 2D or 3D histograms. Here
we use a variant of this approach, which includes 2 steps.

The first step is to compute a 2D color–magnitude histogram in the survey we
are converting, e.g. $g$ vs $(g-r)$ if we correct DES $g$ into SDSS $g$ using
DES $g-r$ color. We keep 98% of data along each dimension by first computing
1D distributions of all data points along color and magnitude and then
rejecting 1% of the strongest outliers on each side of the distribution. Then
the remaining color–magnitude space is split into 50$\times$50 bins to compute
a 2D histogram. Then the 1st-step weight is computed as the inverse of a count
in the 2D histogram to account for the non-uniform distribution of galaxies in
the observed color–magnitude space. For the subsequent analysis we reject the
measurements, which fall into the bins of a 2D histogram containing fewer than
15 data points.

On the second step we compute a set of 1D distributions of magnitude
difference between the survey being converted (e.g. DES $g$) and the reference
(e.g. SDSS $g$) in 17 bins on the observed color (e.g. DES $g-r$) assigned as
follows: (i) the 80%-ile of the data is split into 11 equally sized bins; (ii)
the 80%-98%-ile is split into 3 bins on each side of the 90%-ile distribution;
(iii) the remaining 1% of outliers by color on either side are rejected. Then
in each of the 17 bins we compute a 1D histogram on the magnitude difference
(e.g. $g_{\mathrm{DES}}-g_{\mathrm{SDSS}}$) in 150 bins rejecting 10% of the
strongest outliers on each side, i.e. keeping 80%-ile of the data. This
apparently high number is needed to properly handle the central very dense
part of the distribution. Then, each of these 1D histograms is normalized by
its maximum and the 2nd-step weights for each photometric measurement is
computed as the inverse of the counts in the 1D distributions.

At the end, we multiply 1st- and 2nd-step weights to assign a final weight to
each measurement and use a linear regression to compute the color
transformation in the form:

$mag_{\mathrm{target}}=mag_{\mathrm{ref}}+a+b\cdot color_{\mathrm{ref}}$

In most cases, a linear regression cannot properly describe the photometric
transformation in the entire range of observed colors. In such cases, we use a
piece-wise linear regression with up-to three color sub-ranges (see Figure 4).
We preferred a piece-wise linear fit to a nonlinear (e.g. 2-nd or 3-rd degree
polynomial) solution because the latter case could lead to very high residuals
in the tails of the color distribution. We do not impose any constraints on
magnitude range and compute the regression for the color dependence only,
because the result does not show any magnitude dependent effects (see Figure
5).

To calculate the quality of the derived transformation, we could not use a
standard r.m.s. because some of the photometric measurements used in the
fitting have rather large uncertainties of up-to 0.3 mag (see the previous
section). Therefore, when we compute a histogram of the fitting residuals, we
see a ‘background’ level looking like a large Gaussian with a sharp peak on
top of it: if we compute an r.m.s. value without taking this into account, we
will severely overestimate the r.m.s. of the residuals. Therefore, the values
of residuals presented in the next section of the paper correspond to the
moments of the peak on top of the background distribution shown in Figure 6.

Figure 5: Magnitude residuals for corrected VST Atlas and SDSS as a function
of the magnitude used in the fitting. The result does not show any magnitude-
dependent effects.
Figure 6: A histogram of the fitting residuals for the VST Atlas $i$-band
approximated by two Gaussians, one for a ‘background’ level and another one
for a sharp peak.

##  4 Results

In this section we present color transformations of optical integrated
photometric measurements of galaxies from SDSS into DES, DELVE, DESI Legacy
Surveys, VST KiDS, and VST ATLAS surveys and vice versa. We discuss the
applicability of our method to the PanSTARRS data where the direct computation
of transformations did not yield accurate results. We also provide
transformations of near-infrared measurements from UKIDSS/UHS to VIKING/VHS
and vice versa. Each set of transformations is provided in a form of piece-
wise linear relations. To illustrate the quality of photometric
transformations, we present the plots of both
$\mathrm{mag_{converted}}-\mathrm{mag_{reference}}$ (Figure 7) and color
difference (Figure 8) between the converted photometry and the measurements
for the same galaxies in the reference survey (e.g.
$(g-r)_{\mathrm{DES}}-(g-r)_{\mathrm{SDSS}}$ for the DES$\rightarrow$SDSS
conversion).

###  4.1 DES and DELVE

We used a piece-wise linear regression of 2 or 3 segments to compute the
transformations between DES and SDSS measurements. Overall, the fitting
quality is very high with the residuals at a few-% level for Petrosian
magnitudes. The DELVE survey despite being shallower than DES was conducted
with the same instrument using the same (physical) filters, and all the data
reduction, processing, and analysis algorithms were identical (DES Data
Management system pipeline; Morganson et al., 2018). We applied the color
transformations obtained for DES to DELVE data and they demonstrated full
consistency (see the discussion about the reconstruction of the red sequence
below).

$\displaystyle\mathrm{DES\rightarrow SDSS}:$ $\displaystyle\scriptstyle
g-r<0.656:\qquad
g_{\mathrm{sdss}}=g_{\mathrm{des}}+0.072(g_{\mathrm{des}}-r_{\mathrm{des}})+0.033$
$\displaystyle\scriptstyle 0.656<g-r<1.187:\qquad
g_{\mathrm{sdss}}=\scriptstyle
g_{\mathrm{des}}+0.198(g_{\mathrm{des}}-r_{\mathrm{des}})-0.049$
$\displaystyle\scriptstyle g-r>1.187:\qquad
g_{\mathrm{sdss}}=g_{\mathrm{des}}-0.005(g_{\mathrm{des}}-r_{\mathrm{des}})+0.193$
$\displaystyle\scriptstyle\sigma_{\mathrm{g_{\mathrm{sdss}}}}=0.048\;mag$
$\displaystyle\scriptstyle g-r<0.792:\qquad
r_{\mathrm{sdss}}=r_{\mathrm{des}}+\scriptstyle
0.052(g_{\mathrm{des}}-r_{\mathrm{des}})+0.074$ $\displaystyle\scriptstyle
0.792<g-r<1.148:\qquad
r_{\mathrm{sdss}}=r_{\mathrm{des}}+0.098(g_{\mathrm{des}}-r_{\mathrm{des}})+0.037$
$\displaystyle\scriptstyle g-r>1.148:\qquad
r_{\mathrm{sdss}}=r_{\mathrm{des}}+0.208(g_{\mathrm{des}}-r_{\mathrm{des}})-0.088$
$\displaystyle\scriptstyle\sigma_{\mathrm{r_{\mathrm{sdss}}}}=0.041\;mag$
$\displaystyle\scriptstyle r-z<0.621:\qquad
r_{\mathrm{sdss}}=r_{\mathrm{des}}+0.034(r_{\mathrm{des}}-z_{\mathrm{des}})+0.089$
$\displaystyle\scriptstyle 0.621<r-z<1.284:\qquad
r_{\mathrm{sdss}}=r_{\mathrm{des}}+0.381(r_{\mathrm{des}}-z_{\mathrm{des}})-0.126$
$\displaystyle\scriptstyle r-z>1.284:\qquad
r_{\mathrm{sdss}}=r_{\mathrm{des}}-0.058(r_{\mathrm{des}}-z_{\mathrm{des}})+0.438$
$\displaystyle\scriptstyle\sigma_{\mathrm{r_{\mathrm{sdss}}}}=0.042\;mag$
$\displaystyle\scriptstyle i-z<0.237:\qquad
i_{\mathrm{sdss}}=i_{\mathrm{des}}+0.122(i_{\mathrm{des}}-z_{\mathrm{des}})+0.048$
$\displaystyle\scriptstyle 0.237<i-z<0.400:\qquad
i_{\mathrm{sdss}}=i_{\mathrm{des}}+0.407(i_{\mathrm{des}}-z_{\mathrm{des}})-0.019$
$\displaystyle\scriptstyle i-z>0.400:\qquad
i_{\mathrm{sdss}}=i_{\mathrm{des}}+0.174(i_{\mathrm{des}}-z_{\mathrm{des}})+0.074$
$\displaystyle\scriptstyle\sigma_{\mathrm{i_{\mathrm{sdss}}}}=0.046\;mag$
$\displaystyle\scriptstyle r-z<0.803:\qquad
z_{\mathrm{sdss}}=z_{\mathrm{des}}+0.060(r_{\mathrm{des}}-z_{\mathrm{des}})+0.031$
$\displaystyle\scriptstyle r-z>0.803:\qquad
z_{\mathrm{sdss}}=z_{\mathrm{des}}-0.099(r_{\mathrm{des}}-z_{\mathrm{des}})+0.159$
$\displaystyle\scriptstyle\sigma_{\mathrm{z_{\mathrm{sdss}}}}=0.061\;mag$
$\displaystyle\mathrm{SDSS\rightarrow DES}:$ $\displaystyle\scriptstyle
g-r<0.797:\qquad
g_{\mathrm{des}}=g_{\mathrm{sdss}}-0.132(g_{\mathrm{sdss}}-r_{\mathrm{sdss}})-0.001$
$\displaystyle\scriptstyle 0.797<g-r<1.043:\qquad
g_{\mathrm{des}}=\scriptstyle
g_{\mathrm{sdss}}-0.238(g_{\mathrm{sdss}}-r_{\mathrm{sdss}})+0.083$
$\displaystyle\scriptstyle g-r>1.181:\qquad
g_{\mathrm{des}}=g_{\mathrm{sdss}}-0.165(g_{\mathrm{sdss}}-r_{\mathrm{sdss}})+0.007$
$\displaystyle\scriptstyle\sigma_{\mathrm{g_{\mathrm{des}}}}=0.045\;mag$
$\displaystyle\scriptstyle g-r<0.600:\qquad
r_{\mathrm{des}}=r_{\mathrm{sdss}}+0.027(g_{\mathrm{sdss}}-r_{\mathrm{sdss}})-0.117$
$\displaystyle\scriptstyle g-r>0.600:\qquad
r_{\mathrm{des}}=r_{\mathrm{sdss}}-0.093(g_{\mathrm{sdss}}-r_{\mathrm{sdss}})-0.044$
$\displaystyle\scriptstyle\sigma_{\mathrm{r_{\mathrm{des}}}}=0.040\;mag$
$\displaystyle\scriptstyle r-z<0.659:\qquad
r_{\mathrm{des}}=r_{\mathrm{sdss}}-0.040(r_{\mathrm{sdss}}-z_{\mathrm{sdss}})-0.083$
$\displaystyle\scriptstyle 0.659<r-z<1.783:\qquad
r_{\mathrm{des}}=r_{\mathrm{sdss}}-0.276(r_{\mathrm{sdss}}-z_{\mathrm{sdss}})+0.072$
$\displaystyle\scriptstyle r-z>1.783:\qquad
r_{\mathrm{des}}=r_{\mathrm{sdss}}-0.235(r_{\mathrm{sdss}}-z_{\mathrm{sdss}})-0.0003$
$\displaystyle\scriptstyle\sigma_{\mathrm{r_{\mathrm{des}}}}=0.036\;mag$
$\displaystyle\scriptstyle i-z<0.216:\qquad
i_{\mathrm{des}}=i_{\mathrm{sdss}}-0.162(i_{\mathrm{sdss}}-z_{\mathrm{sdss}})-0.040$
$\displaystyle\scriptstyle 0.216<i-z<0.583:\qquad
i_{\mathrm{des}}=i_{\mathrm{sdss}}-0.237(i_{\mathrm{sdss}}-z_{\mathrm{sdss}})-0.024$
$\displaystyle\scriptstyle i-z>0.583:\qquad
i_{\mathrm{des}}=i_{\mathrm{sdss}}-0.102(i_{\mathrm{sdss}}-z_{\mathrm{sdss}})-0.102$
$\displaystyle\scriptstyle\sigma_{\mathrm{i_{\mathrm{des}}}}=0.038\;mag$
$\displaystyle\scriptstyle r-z<0.336:\qquad
z_{\mathrm{des}}=z_{\mathrm{sdss}}+0.421(r_{\mathrm{sdss}}-z_{\mathrm{sdss}})-0.231$
$\displaystyle\scriptstyle 0.336<r-z<0.868:\qquad
z_{\mathrm{des}}=z_{\mathrm{sdss}}+0.035(r_{\mathrm{sdss}}-z_{\mathrm{sdss}})-0.101$
$\displaystyle\scriptstyle r-z>0.909:\qquad
z_{\mathrm{des}}=z_{\mathrm{sdss}}+0.123(r_{\mathrm{sdss}}-z_{\mathrm{sdss}})-0.177$
$\displaystyle\scriptstyle\sigma_{\mathrm{z_{\mathrm{des}}}}=0.063\;mag$

Figure 7: Fitting residuals between several surveys (converted magnitudes) and
the SDSS as a function of the color used in the approximation. SDSS and DES
magnitudes are Petrosian, DESI Legacy Surveys magnitudes are model-based. The
red line shows a median position in each bin, the blue line shows the median
absolute deviation in each bin; their boundaries are marked with gray vertical
lines. The surveys are labeled in the bottom right corner.

Figure 8: Same as Figure 7, but the residuals are shown for the converted
colors.

###  4.2 DESI Legacy Surveys

The DESI Legacy Surveys (Dey et al., 2019) include the two parts, Northern and
Southern, carried out using three different telescopes and instruments: the
Northern part consists of the Beijing–Arizona Sky Survey (BASS; $g$ and
$r$-band observations collected at the 2.3 m telescope using the 90Prime
camera) and the Mayall $z$-band Sky Survey (MzLS; $z$-band observations
collected at the 4 m Mayall telescope using the Mosaic-3 camera); the Southern
part includes the data collected with the DECam instrument at the 4 m Blanco
telescope (same as DES/DELVE). Therefore, one needs to derive separate sets of
photometric transformations for the Northern and Southern parts of DESI Legacy
Surveys data. It turns out that the DES/DELVE color transformations cannot be
applied to the LS-South measurements despite being collected with the same
telescope and instrument and even sometimes being derived from the same raw
data, because of the differences the original observations were reduced and
analyzed (see details in the next section). Therefore, we derived specific
transformations for LS-South to/from SDSS photometry. The LS-South
transformations yield higher residuals than those for DES/DELVE data because
of higher systematic errors in the DECaLS photometry: (e.g.,
$\sigma_{r_{\mathrm{decals}}}=0.087\;mag$ versus
$\sigma_{r_{\mathrm{des}}}=0.044\;mag$ and
$\sigma_{r_{\mathrm{vst}}}=0.041\;mag$). Transformations for LS-South (DECaLS)
and LS-North (MzLS+BASS) yield similar r.m.s. of the fitting residuals.

$\displaystyle\mathrm{DECaLS\rightarrow SDSS}:$ $\displaystyle\scriptstyle
g-r<0.712:\qquad
g_{\mathrm{sdss}}=g_{\mathrm{decals}}+0.014(g_{\mathrm{decals}}-r_{\mathrm{decals}})+0.055$
$\displaystyle\scriptstyle 0.712<g-r<0.984:\qquad
g_{\mathrm{sdss}}=g_{\mathrm{decals}}+0.404(g_{\mathrm{decals}}-r_{\mathrm{decals}})-0.223$
$\displaystyle\scriptstyle g-r>0.984:\qquad
g_{\mathrm{sdss}}=g_{\mathrm{decals}}+0.049(g_{\mathrm{decals}}-r_{\mathrm{decals}})+0.126$
$\displaystyle\scriptstyle\sigma_{\mathrm{g_{\mathrm{sdss}}}}=0.072\;mag$
$\displaystyle\scriptstyle g-r<0.732:\qquad
r_{\mathrm{sdss}}=r_{\mathrm{decals}}+0.038(g_{\mathrm{decals}}-r_{\mathrm{decals}})+0.084$
$\displaystyle\scriptstyle 0.732<g-r<0.951:\qquad
r_{\mathrm{sdss}}=r_{\mathrm{decals}}+0.252(g_{\mathrm{decals}}-r_{\mathrm{decals}})-0.073$
$\displaystyle\scriptstyle g-r>0.951:\qquad
r_{\mathrm{sdss}}=r_{\mathrm{decals}}+0.082(g_{\mathrm{decals}}-r_{\mathrm{decals}})+0.089$
$\displaystyle\scriptstyle\sigma_{\mathrm{r_{\mathrm{sdss}}}}=0.087\;mag$
$\displaystyle\scriptstyle r-z<0.629:\qquad
r_{\mathrm{sdss}}=r_{\mathrm{decals}}+0.061(r_{\mathrm{decals}}-z_{\mathrm{decals}})+0.099$
$\displaystyle\scriptstyle 0.629<r-z<1.280:\qquad
r_{\mathrm{sdss}}=r_{\mathrm{decals}}+0.359(r_{\mathrm{decals}}-z_{\mathrm{decals}})-0.088$
$\displaystyle\scriptstyle r-z>1.280:\qquad
r_{\mathrm{sdss}}=r_{\mathrm{decals}}+0.054(r_{\mathrm{decals}}-z_{\mathrm{decals}})+0.302$
$\displaystyle\scriptstyle\sigma_{\mathrm{r_{\mathrm{sdss}}}}=0.073\;mag$
$\displaystyle\scriptstyle r-z>0.779:\qquad
z_{\mathrm{sdss}}=z_{\mathrm{decals}}+0.191(r_{\mathrm{decals}}-z_{\mathrm{decals}})-0.023$
$\displaystyle\scriptstyle r-z<0.779:\qquad
z_{\mathrm{sdss}}=z_{\mathrm{decals}}-0.066(r_{\mathrm{decals}}-z_{\mathrm{decals}})+0.177$
$\displaystyle\scriptstyle\sigma_{\mathrm{z_{\mathrm{sdss}}}}=0.079\;mag$
$\displaystyle\mathrm{SDSS\rightarrow DECaLS}:$ $\displaystyle\scriptstyle
g-r<0.657:\qquad
g_{\mathrm{decals}}=g_{\mathrm{sdss}}-0.034(g_{\mathrm{sdss}}-r_{\mathrm{sdss}})-0.035$
$\displaystyle\scriptstyle 0.657<g-r<0.973:\qquad
g_{\mathrm{decals}}=g_{\mathrm{sdss}}-0.356(g_{\mathrm{sdss}}-r_{\mathrm{sdss}})+0.176$
$\displaystyle\scriptstyle g-r>0.973:\qquad
g_{\mathrm{decals}}=g_{\mathrm{sdss}}-0.113(g_{\mathrm{sdss}}-r_{\mathrm{sdss}})-0.060$
$\displaystyle\scriptstyle\sigma_{\mathrm{g_{\mathrm{decals}}}}=0.071\;mag$
$\displaystyle\scriptstyle g-r<0.662:\qquad
r_{\mathrm{decals}}=r_{\mathrm{sdss}}+0.007(g_{\mathrm{sdss}}-r_{\mathrm{sdss}})-0.112$
$\displaystyle\scriptstyle 0.662<g-r<0.968:\qquad
r_{\mathrm{decals}}=r_{\mathrm{sdss}}-0.214(g_{\mathrm{sdss}}-r_{\mathrm{sdss}})+0.034$
$\displaystyle\scriptstyle g-r>0.968:\qquad
r_{\mathrm{decals}}=r_{\mathrm{sdss}}-0.057(g_{\mathrm{sdss}}-r_{\mathrm{sdss}})-0.117$
$\displaystyle\scriptstyle\sigma_{\mathrm{r_{\mathrm{decals}}}}=0.065\;mag$
$\displaystyle\scriptstyle r-z<0.619:\qquad
r_{\mathrm{decals}}=r_{\mathrm{sdss}}-0.036(r_{\mathrm{sdss}}-z_{\mathrm{sdss}})-0.107$
$\displaystyle\scriptstyle 0.619<r-z<1.787:\qquad
r_{\mathrm{decals}}=r_{\mathrm{sdss}}-0.247(r_{\mathrm{sdss}}-z_{\mathrm{sdss}})+0.022$
$\displaystyle\scriptstyle r-z>1.787:\qquad
r_{\mathrm{decals}}=r_{\mathrm{sdss}}-0.202(r_{\mathrm{sdss}}-z_{\mathrm{sdss}})-0.056$
$\displaystyle\scriptstyle\sigma_{\mathrm{r_{\mathrm{decals}}}}=0.068\;mag$
$\displaystyle\scriptstyle r-z>0.787:\qquad
z_{\mathrm{decals}}=z_{\mathrm{sdss}}-0.093(r_{\mathrm{sdss}}-z_{\mathrm{sdss}})-0.041$
$\displaystyle\scriptstyle 0.787>r-z>0.893:\qquad
z_{\mathrm{decals}}=z_{\mathrm{sdss}}+0.050(r_{\mathrm{sdss}}-z_{\mathrm{sdss}})-0.154$
$\displaystyle\scriptstyle r-z<0.893:\qquad
z_{\mathrm{decals}}=z_{\mathrm{sdss}}+0.2356(r_{\mathrm{sdss}}-z_{\mathrm{sdss}})-0.320$
$\displaystyle\scriptstyle\sigma_{\mathrm{z_{\mathrm{decals}}}}=0.144\;mag$
$\displaystyle\mathrm{MzLS+BASS\rightarrow SDSS}:$ $\displaystyle\scriptstyle
g-r<0.664:\qquad
g_{\mathrm{sdss}}=g_{\mathrm{bass}}-0.072(g_{\mathrm{bass}}-r_{\mathrm{bass}})+0.071$
$\displaystyle\scriptstyle 0.664<g-r<1.102:\qquad
g_{\mathrm{sdss}}=g_{\mathrm{bass}}+0.241(g_{\mathrm{bass}}-r_{\mathrm{bass}})-0.137$
$\displaystyle\scriptstyle g-r>1.102:\qquad
g_{\mathrm{sdss}}=g_{\mathrm{bass}}+0.017(g_{\mathrm{bass}}-r_{\mathrm{bass}})+0.110$
$\displaystyle\scriptstyle\sigma_{\mathrm{g_{\mathrm{sdss}}}}=0.066\;mag$
$\displaystyle\scriptstyle g-r<0.732:\qquad
r_{\mathrm{sdss}}=r_{\mathrm{bass}}-0.053(g_{\mathrm{bass}}-r_{\mathrm{bass}})+0.120$
$\displaystyle\scriptstyle 0.732<g-r<0.951:\qquad
r_{\mathrm{sdss}}=r_{\mathrm{bass}}+0.265(g_{\mathrm{bass}}-r_{\mathrm{bass}})-0.099$
$\displaystyle\scriptstyle g-r>0.951:\qquad
r_{\mathrm{sdss}}=r_{\mathrm{bass}}+0.048(g_{\mathrm{bass}}-r_{\mathrm{bass}})+0.108$
$\displaystyle\scriptstyle\sigma_{\mathrm{r_{\mathrm{sdss}}}}=0.061\;mag$
$\displaystyle\scriptstyle r-z<0.619:\qquad
r_{\mathrm{sdss}}=r_{\mathrm{bass}}+0.040(r_{\mathrm{bass}}-z_{\mathrm{mzls}})+0.102$
$\displaystyle\scriptstyle 0.619<r-z<1.302:\qquad
r_{\mathrm{sdss}}=r_{\mathrm{bass}}+0.262(r_{\mathrm{bass}}-z_{\mathrm{mzls}})-0.035$
$\displaystyle\scriptstyle r-z>1.302:\qquad
r_{\mathrm{sdss}}=r_{\mathrm{bass}}+0.081(r_{\mathrm{bass}}-z_{\mathrm{mzls}})+0.201$
$\displaystyle\scriptstyle\sigma_{\mathrm{r_{\mathrm{sdss}}}}=0.072\;mag$
$\displaystyle\scriptstyle r-z<0.692:\qquad
z_{\mathrm{sdss}}=z_{\mathrm{mzls}}+0.235(r_{\mathrm{bass}}-z_{\mathrm{mzls}})-0.053$
$\displaystyle\scriptstyle 0.692<r-z<0.797:\qquad
z_{\mathrm{sdss}}=z_{\mathrm{mzls}}+0.047(r_{\mathrm{bass}}-z_{\mathrm{mzls}})+0.078$
$\displaystyle\scriptstyle r-z<0.797:\qquad
z_{\mathrm{sdss}}=z_{\mathrm{mzls}}-0.076(r_{\mathrm{bass}}-z_{\mathrm{mzls}})+0.175$
$\displaystyle\scriptstyle\sigma_{\mathrm{z_{\mathrm{sdss}}}}=0.081\;mag$
$\displaystyle\mathrm{SDSS\rightarrow MzLS+BASS:}$ $\displaystyle\scriptstyle
g-r<0.689:\qquad
g_{\mathrm{bass}}=g_{\mathrm{sdss}}+0.007(g_{\mathrm{sdss}}-r_{\mathrm{sdss}})-0.025$
$\displaystyle\scriptstyle 0.689<g-r<0.845:\qquad
g_{\mathrm{bass}}=g_{\mathrm{sdss}}-0.455(g_{\mathrm{sdss}}-r_{\mathrm{sdss}})+0.293$
$\displaystyle\scriptstyle g-r>0.845:\qquad
g_{\mathrm{bass}}=g_{\mathrm{sdss}}-0.107(g_{\mathrm{sdss}}-r_{\mathrm{sdss}})-0.001$
$\displaystyle\scriptstyle\sigma_{\mathrm{g_{\mathrm{bass}}}}=0.067\;mag$
$\displaystyle\scriptstyle g-r<0.603:\qquad
r_{\mathrm{bass}}=r_{\mathrm{sdss}}+0.007(g_{\mathrm{sdss}}-r_{\mathrm{sdss}})-0.106$
$\displaystyle\scriptstyle 0.603<g-r<0.930:\qquad
r_{\mathrm{bass}}=r_{\mathrm{sdss}}-0.188(g_{\mathrm{sdss}}-r_{\mathrm{sdss}})+0.011$
$\displaystyle\scriptstyle g-r>0.930:\qquad
r_{\mathrm{bass}}=r_{\mathrm{sdss}}-0.030(g_{\mathrm{sdss}}-r_{\mathrm{sdss}})-0.135$
$\displaystyle\scriptstyle\sigma_{\mathrm{r_{\mathrm{bass}}}}=0.060\;mag$
$\displaystyle\scriptstyle r-z<0.592:\qquad
r_{\mathrm{bass}}=r_{\mathrm{sdss}}-0.045(r_{\mathrm{sdss}}-z_{\mathrm{sdss}})-0.091$
$\displaystyle\scriptstyle 0.592<r-z<1.240:\qquad
r_{\mathrm{bass}}=r_{\mathrm{sdss}}-0.198(r_{\mathrm{sdss}}-z_{\mathrm{sdss}})-0.001$
$\displaystyle\scriptstyle r-z>1.240:\qquad
r_{\mathrm{bass}}=r_{\mathrm{sdss}}-0.218(r_{\mathrm{sdss}}-z_{\mathrm{sdss}})+0.023$
$\displaystyle\scriptstyle\sigma_{\mathrm{r_{\mathrm{bass}}}}=0.068\;mag$
$\displaystyle\scriptstyle r-z<0.309:\qquad
z_{\mathrm{mzls}}=z_{\mathrm{mzls}}-0.349(r_{\mathrm{sdss}}-z_{\mathrm{mzls}})+0.036$
$\displaystyle\scriptstyle 0.309<r-z<1.015:\qquad
z_{\mathrm{mzls}}=z_{\mathrm{mzls}}-0.077(r_{\mathrm{sdss}}-z_{\mathrm{mzls}})-0.047$
$\displaystyle\scriptstyle r-z<1.015:\qquad
z_{\mathrm{mzls}}=z_{\mathrm{mzls}}+0.247(r_{\mathrm{sdss}}-z_{\mathrm{mzls}})-0.377$
$\displaystyle\scriptstyle\sigma_{\mathrm{z_{\mathrm{mzls}}}}=0.078\;mag$

###  4.3 VST ATLAS and KiDS

VST ATLAS is a wide-field survey of the Southern hemisphere of the sky; KiDS
covers a smaller area, but at a higher depth. Both surveys were conducted with
the 2.6 m ESO VST telescope and include the $u$ band in addition to $griz$
even though the $u$ band images are very shallow, especially in VST ATLAS.
Both surveys have a small overlap with SDSS, what makes the analysis more
difficult because of a much smaller number of measurements, which we can use
to fit color transformations. Nevertheless, it was still possible to calculate
the transformations to/from SDSS. We use VST ATLAS DR3 data because it is
accessible via Virtual Observatory protocols through CDS Vizier. However, we
compared DR3 versus DR4 photometric measurements and found no statistically
significant differences between them, therefore the color transformations
presented here can be used for VST ATLAS DR4. The VST ATLAS catalog contains
Petrosian magnitudes, while only Kron magnitudes are provided by KiDS, which
introduces additional systematic uncertainties to the transformations of KiDS
to/from SDSS. The fitting residuals are rather small for the $griz$ bands
(0.039 to 0.065 mag; similar to DES) and much larger for the $u$ band in the
KiDS survey (0.13 mag). We do not provide $u$-band transformations between VST
ATLAS and SDSS because of the limited depth of both surveys in the u band for
galaxies as compared to stars. The VST ATLAS color transformations published
in Shanks et al. (2015) provide a very good quality of the converted color. We
find a very small difference about 0.02 mag in the mean $r$-band magnitudes.
Nevertheless, here we provide color transformations obtained using our
approach for consistency with other surveys.

$\displaystyle\mathrm{VST\;ATLAS\rightarrow SDSS:}$ $\displaystyle\scriptstyle
g-r<0.309:\qquad
g_{\mathrm{sdss}}=g_{\mathrm{vst}}-0.051(g_{\mathrm{vst}}-r_{\mathrm{vst}})+0.040$
$\displaystyle\scriptstyle 0.309<g-r<1.059:\qquad
g_{\mathrm{sdss}}=g_{\mathrm{vst}}+0.010(g_{\mathrm{vst}}-r_{\mathrm{vst}})+0.021$
$\displaystyle\scriptstyle g-r>1.059:\qquad
g_{\mathrm{sdss}}=g_{\mathrm{vst}}-0.128(g_{\mathrm{vst}}-r_{\mathrm{vst}})+0.169$
$\displaystyle\scriptstyle\sigma_{\mathrm{g_{\mathrm{sdss}}}}=0.052\;mag$
$\displaystyle\scriptstyle g-r<0.248:\qquad
r_{\mathrm{sdss}}=r_{\mathrm{vst}}+1.206(g_{\mathrm{vst}}-r_{\mathrm{vst}})-0.284$
$\displaystyle\scriptstyle[0.248<g-r<0.821:\qquad
r_{\mathrm{sdss}}=r_{\mathrm{vst}}+0.080(g_{\mathrm{vst}}-r_{\mathrm{vst}})-0.005$
$\displaystyle\scriptstyle g-r>0.821:\qquad
r_{\mathrm{sdss}}=r_{\mathrm{vst}}+0.021(g_{\mathrm{vst}}-r_{\mathrm{vst}})+0.042$
$\displaystyle\scriptstyle\sigma_{\mathrm{r_{\mathrm{sdss}}}}=0.041\;mag$
$\displaystyle\scriptstyle r-i<0.024:\qquad
i_{\mathrm{sdss}}=i_{\mathrm{vst}}+0.636(r_{\mathrm{vst}}-i_{\mathrm{vst}})-0.138$
$\displaystyle\scriptstyle 0.024<r-i<0.310:\qquad
i_{\mathrm{sdss}}=i_{\mathrm{vst}}+0.547(r_{\mathrm{vst}}-i_{\mathrm{vst}})-0.136$
$\displaystyle\scriptstyle r-i>0.310:\qquad
i_{\mathrm{sdss}}=i_{\mathrm{vst}}-0.004(r_{\mathrm{vst}}-i_{\mathrm{vst}})+0.035$
$\displaystyle\scriptstyle\sigma_{\mathrm{i_{\mathrm{sdss}}}}=0.047\;mag$
$\displaystyle\scriptstyle i-z<-0.297:\qquad
z_{\mathrm{sdss}}=z_{\mathrm{vst}}+1.011(i_{\mathrm{vst}}-z_{\mathrm{vst}})+0.071$
$\displaystyle\scriptstyle-0.297<i-z<0.356:\qquad
z_{\mathrm{sdss}}=z_{\mathrm{vst}}+0.504(i_{\mathrm{vst}}-z_{\mathrm{vst}})-0.079$
$\displaystyle\scriptstyle i-z>0.356:\qquad
z_{\mathrm{sdss}}=z_{\mathrm{vst}}+0.079(i_{\mathrm{vst}}-z_{\mathrm{vst}})+0.072$
$\displaystyle\scriptstyle\sigma_{\mathrm{z_{\mathrm{sdss}}}}=0.067\;mag$
$\displaystyle\mathrm{SDSS\rightarrow VST\;ATLAS:}$ $\displaystyle\scriptstyle
g-r<1.056:\qquad
g_{\mathrm{vst}}=g_{\mathrm{sdss}}-0.079(g_{\mathrm{sdss}}-r_{\mathrm{sdss}})+0.038$
$\displaystyle\scriptstyle g-r>1.056:\qquad
g_{\mathrm{vst}}=g_{\mathrm{sdss}}+0.051(g_{\mathrm{sdss}}-r_{\mathrm{sdss}})-0.099$
$\displaystyle\scriptstyle\sigma_{\mathrm{g_{\mathrm{vst}}}}=0.041\;mag$
$\displaystyle\scriptstyle g-r<0.544:\qquad
r_{\mathrm{vst}}=r_{\mathrm{sdss}}+0.044(g_{\mathrm{sdss}}-r_{\mathrm{sdss}})-0.058$
$\displaystyle\scriptstyle g-r>0.544:\qquad
r_{\mathrm{vst}}=r_{\mathrm{sdss}}-0.043(g_{\mathrm{sdss}}-r_{\mathrm{sdss}})-0.010$
$\displaystyle\scriptstyle\sigma_{\mathrm{r_{\mathrm{vst}}}}=0.036\;mag$
$\displaystyle\scriptstyle
i_{\mathrm{vst}}=i_{\mathrm{sdss}}+0.128(r_{\mathrm{sdss}}-i_{\mathrm{sdss}})-0.071$
$\displaystyle\scriptstyle\sigma_{\mathrm{i_{\mathrm{vst}}}}=0.041\;mag$
$\displaystyle\scriptstyle i-z<0.089:\qquad
z_{\mathrm{vst}}=z_{\mathrm{sdss}}+0.492(i_{\mathrm{sdss}}-z_{\mathrm{sdss}})-0.194$
$\displaystyle\scriptstyle 0.089<i-z<0.399:\qquad
z_{\mathrm{vst}}=z_{\mathrm{sdss}}+0.413(i_{\mathrm{sdss}}-z_{\mathrm{sdss}})-0.187$
$\displaystyle\scriptstyle i-z>0.399:\qquad
z_{\mathrm{vst}}=z_{\mathrm{sdss}}+0.561(i_{\mathrm{sdss}}-z_{\mathrm{sdss}})-0.246$
$\displaystyle\scriptstyle\sigma_{\mathrm{z_{\mathrm{vst}}}}=0.061\;mag$
$\displaystyle\mathrm{KiDS\rightarrow SDSS:}$ $\displaystyle\scriptstyle
u-g<1.809:\qquad
u_{\mathrm{sdss}}=u_{\mathrm{KiDS}}+0.037(u_{\mathrm{KiDS}}-g_{\mathrm{KiDS}})-0.105$
$\displaystyle\scriptstyle u-g>1.809:\qquad
u_{\mathrm{sdss}}=u_{\mathrm{KiDS}}-0.460(u_{\mathrm{KiDS}}-g_{\mathrm{KiDS}})+0.795$
$\displaystyle\scriptstyle\sigma_{\mathrm{u_{\mathrm{sdss}}}}=0.130\;mag$
$\displaystyle\scriptstyle
g_{\mathrm{sdss}}=g_{\mathrm{KiDS}}-0.005(g_{\mathrm{KiDS}}-r_{\mathrm{KiDS}})+0.030$
$\displaystyle\scriptstyle\sigma_{\mathrm{g_{\mathrm{sdss}}}}=0.052\;mag$
$\displaystyle\scriptstyle
r_{\mathrm{sdss}}=r_{\mathrm{KiDS}}-0.002(g_{\mathrm{KiDS}}-r_{\mathrm{KiDS}})+0.035$
$\displaystyle\scriptstyle\sigma_{\mathrm{r_{\mathrm{sdss}}}}=0.039\;mag$
$\displaystyle\scriptstyle r-i<0.477:\qquad
i_{\mathrm{sdss}}=i_{\mathrm{KiDS}}+0.129(r_{\mathrm{KiDS}}-i_{\mathrm{KiDS}})-0.020$
$\displaystyle\scriptstyle r-i>0.477:\qquad
i_{\mathrm{sdss}}=i_{\mathrm{KiDS}}-0.231(r_{\mathrm{KiDS}}-i_{\mathrm{KiDS}})+0.152$
$\displaystyle\scriptstyle\sigma_{\mathrm{i_{\mathrm{sdss}}}}=0.046\;mag$
$\displaystyle\mathrm{SDSS\rightarrow KiDS:}$ $\displaystyle\scriptstyle
u-g<1.253:\qquad
u_{\mathrm{KiDS}}=u_{\mathrm{sdss}}-0.549(u_{\mathrm{sdss}}-g_{\mathrm{sdss}})+0.679$
$\displaystyle\scriptstyle 1.253<u-g<1.579:\qquad
u_{\mathrm{KiDS}}=u_{\mathrm{sdss}}+0.444(u_{\mathrm{sdss}}-g_{\mathrm{sdss}})-0.565$
$\displaystyle\scriptstyle u-g>1.579:\qquad
u_{\mathrm{KiDS}}=u_{\mathrm{sdss}}-0.622(u_{\mathrm{sdss}}-g_{\mathrm{sdss}})+1.118)$
$\displaystyle\scriptstyle\sigma_{\mathrm{u_{\mathrm{KiDS}}}}=0.131\;mag$
$\displaystyle\scriptstyle g-r<0.872:\qquad
g_{\mathrm{KiDS}}=g_{\mathrm{sdss}}-0.142(g_{\mathrm{sdss}}-r_{\mathrm{sdss}})+0.083$
$\displaystyle\scriptstyle 0.872<g-r<1.452:\qquad
g_{\mathrm{KiDS}}=g_{\mathrm{sdss}}-0.019(g_{\mathrm{sdss}}-r_{\mathrm{sdss}})-0.023$
$\displaystyle\scriptstyle g-r>1.452:\qquad
g_{\mathrm{KiDS}}=g_{\mathrm{sdss}}-0.175(g_{\mathrm{sdss}}-r_{\mathrm{sdss}})+0.201)$
$\displaystyle\scriptstyle\sigma_{\mathrm{g_{\mathrm{KiDS}}}}=0.049\;mag$
$\displaystyle\scriptstyle g-r<1.153:\qquad
r_{\mathrm{KiDS}}=r_{\mathrm{sdss}}-0.005(g_{\mathrm{sdss}}-r_{\mathrm{sdss}})-0.026$
$\displaystyle\scriptstyle g-r>1.153:\qquad
r_{\mathrm{KiDS}}=r_{\mathrm{sdss}}+0.022(g_{\mathrm{sdss}}-r_{\mathrm{sdss}})-0.058$
$\displaystyle\scriptstyle\sigma_{\mathrm{r_{\mathrm{KiDS}}}}=0.037\;mag$
$\displaystyle\scriptstyle g-r<0.335:\qquad
i_{\mathrm{KiDS}}=i_{\mathrm{sdss}}+0.147(r_{\mathrm{sdss}}-i_{\mathrm{sdss}})-0.096$
$\displaystyle\scriptstyle g-r>0.335:\qquad
i_{\mathrm{KiDS}}=i_{\mathrm{sdss}}+0.193(r_{\mathrm{sdss}}-i_{\mathrm{sdss}})-0.111$
$\displaystyle\scriptstyle\sigma_{\mathrm{i_{\mathrm{KiDS}}}}=0.045\;mag$

###  4.4 PanSTARRS

PanSTARRS data demonstrate a highly nonlinear structure of magnitude
differences from SDSS magnitudes as a function of color. Therefore, the method
of piecewise linear regression described above could not be applied to the
PanSTARRS data with a reasonable number of segments, and we used color
transformations based on PSF measurements from the literature Finkbeiner et
al. (2016). These transformations yield high systematic differences for the
corrected values of Kron magnitudes compared to SDSS Petrosian magnitudes (see
Figure 9), much higher than those we observed in the case of VST KiDS. We
notice that the differences with SDSS increase with the size of galaxy.
PanSTARRS also contains Petrosian magnitudes, which can be better for
comparison with SDSS data, but we do not consider them because they are not
available through Virtual Observatory services which we used to access the
data.

Figure 9: Color residuals for PanSTARRS and the SDSS as a function of the
color and Petrosian radius after the conversions from Finkbeiner et al.
(2016). The symbols are the same as in Figure 7. SDSS magnitudes are
Petrosian, PanSTARRS magnitudes are Kron.

###  4.5 VHS and VIKING

The wide-field infrared surveys VHS and VIKING conducted at the 4 m ESO VISTA
telescope were translated to the UKIRT system using the same method as for the
optical surveys. Both surveys use identical prescriptions for data reduction,
processing and analysis and, hence, were treated as a single dataset in our
study. For the approximation, we used a piecewise linear regression with 2
segments over the entire color range. The fitting residuals are somewhat
higher than for optical surveys (0.07–0.1 mag) but the residuals do not show
any trends with the color.

$\displaystyle\mathrm{VISTA\rightarrow UKIRT:}$ $\displaystyle\scriptstyle
Y_{\mathrm{ukirt}}=Y_{\mathrm{vista}}-0.342(Y_{\mathrm{vista}}-J_{\mathrm{vista}})-0.226$
$\displaystyle\scriptstyle\sigma_{\mathrm{Y_{\mathrm{ukirt}}}}=0.077\;mag$
$\displaystyle\scriptstyle
J_{\mathrm{ukirt}}=J_{\mathrm{vista}}-0.288(J_{\mathrm{vista}}-Y_{\mathrm{vista}})-0.372$
$\displaystyle\scriptstyle\sigma_{\mathrm{J_{\mathrm{ukirt}}}}=0.102\;mag$
$\displaystyle\scriptstyle
H_{\mathrm{ukirt}}=H_{\mathrm{vista}}-0.239(H_{\mathrm{vista}}-Ks_{\mathrm{vista}})-0.358$
$\displaystyle\scriptstyle\sigma_{\mathrm{H_{\mathrm{ukirt}}}}=0.099\;mag$
$\displaystyle\scriptstyle
K_{\mathrm{ukirt}}=Ks_{\mathrm{vista}}-0.091(Ks_{\mathrm{vista}}-H_{\mathrm{vista}})-0.320$
$\displaystyle\scriptstyle\sigma_{\mathrm{Ks_{\mathrm{ukirt}}}}=0.103\;mag$
$\displaystyle\mathrm{UKIRT\rightarrow VISTA:}$ $\displaystyle\scriptstyle
Y-J<0.216:\qquad
Y_{\mathrm{vista}}=Y_{\mathrm{ukirt}}+0.078(Y_{\mathrm{ukirt}}-J_{\mathrm{ukirt}})+0.260$
$\displaystyle\scriptstyle Y-J>0.216:\qquad
Y_{\mathrm{vista}}=Y_{\mathrm{ukirt}}-0.109(Y_{\mathrm{ukirt}}-J_{\mathrm{ukirt}})+0.300$
$\displaystyle\scriptstyle\sigma_{\mathrm{Y_{\mathrm{vista}}}}=0.074\;mag$
$\displaystyle\scriptstyle
J_{\mathrm{vista}}=J_{\mathrm{ukirt}}+0.713(J_{\mathrm{ukirt}}-Y_{\mathrm{ukirt}})+0.122$
$\displaystyle\scriptstyle\sigma_{\mathrm{J_{\mathrm{vista}}}}=0.086\;mag$
$\displaystyle\scriptstyle H-K<0.200:\qquad
H_{\mathrm{vista}}=H_{\mathrm{ukirt}}-0.069(H_{\mathrm{ukirt}}-Ks_{\mathrm{ukirt}})+0.354$
$\displaystyle\scriptstyle H-K>0.200:\qquad
H_{\mathrm{vista}}=H_{\mathrm{ukirt}}-0.332(H_{\mathrm{ukirt}}-Ks_{\mathrm{ukirt}})+0.407$
$\displaystyle\scriptstyle\sigma_{\mathrm{H_{\mathrm{vista}}}}=0.098\;mag$
$\displaystyle\scriptstyle H-K<-0.218:\qquad
K_{\mathrm{vista}}=Ks_{\mathrm{ukirt}}+0.490(Ks_{\mathrm{ukirt}}-H_{\mathrm{ukirt}})+0.375$
$\displaystyle\scriptstyle H-K>-0.218:\qquad
K_{\mathrm{vista}}=Ks_{\mathrm{ukirt}}+0.210(Ks_{\mathrm{ukirt}}-H_{\mathrm{ukirt}})+0.314$
$\displaystyle\scriptstyle\sigma_{\mathrm{Ks_{\mathrm{vista}}}}=0.101\;mag$

Figure 10: Same as Figure 8 for the difference between VIKING and UKIDSS. All
the magnitudes are Petrosian.

###  4.6 Web-based Service and Code Availability

We implemented the derived transformations on the project web-site
https://colors.voxastro.org/ in the form of a simple calculator that can
convert individual measurements. We also provide the conversion formulae in
the form of code snippets in idl and python available from the same website.
Both implementations yield identical results provided the same input data. The
input parameters for the function calc_color_transf are: (i) a name of a
filter added to the name of a survey for the input data, e.g. for magnitudes
in DES g band this argument is ‘g_des’; (ii) the same as (i) for an output
values; (iii) a list of input total magnitude measurements; (iv) a list of
input color values. The size of an output array equals to the size of input
lists in (i) – (ii), which should also be equal to each other.

##  5 Discussion

In the previous section we derived photometric transformations for galaxies in
several wide-field surveys in the optical and infrared domains to the SDSS and
UKIRT photometric systems. Here we discuss the application of the derived
transformations to aperture magnitudes, test them on colors of early-type
galaxies, explain the shortcomings of the DESI Legacy Surveys photometric
measurements and propose a solution on how to improve them, and apply the
derived photometric transformations to spectroscopic samples of nearby and
intermediate-redshift galaxies.

###  5.1 Applying the transformations to aperture magnitudes

While our main goal was to correct integrated photometric measurements to
build galaxy SEDs consistent among different surveys, it is also important to
be able to correct aperture photometric measurements, e.g. to compute the
photometry in the apertures matching fiber sizes in spectroscopic surveys.

We applied the transformations derived for total magnitudes in Section 4 to
magnitudes extracted from circular apertures with the diameters
$1.^{\prime\prime}5$ (Hectospec fibers), $2^{\prime\prime}$,
$3^{\prime\prime}$ (SDSS fibers) and $5.^{\prime\prime}7$, the largest
aperture in VST ATLAS and WFCAM/VIRCAM infrared surveys. For photometric
surveys that do not include measurements in these apertures, the magnitudes
were calculated using quadratic interpolation from the available aperture
magnitudes. In Table 1 we list the available aperture diameters for each of
the surveys. Because the extrapolation beyond the maximum available aperture
is subject to biases, we use the converted optical photometry in the
$1.^{\prime\prime}5$, $2^{\prime\prime}$ and $3^{\prime\prime}$ apertures
(estimating $1.^{\prime\prime}5$ aperture magnitudes also requires an
extrapolation “inwards,” however it is done into a smaller aperture than the
original measurements and therefore is only affected the shape of the inner
part of a galaxy light profile convolved with the atmospheric seeing). Then,
we compare the converted aperture measurements from each survey to those in
either SDSS or UKIDSS/UHS. Because we do not have SDSS magnitudes in the
$5.^{\prime\prime}7$ aperture, we compare the VST and DES surveys where such
measurements are available with each other (see Figure 12).

The aperture values calculated using transformations for total magnitudes
demonstrate different offsets from the reference survey depending on the size
of the aperture without noticeable color trends in most color/survey/aperture
combinations. At the same time, the offset values vary a lot for different
surveys even for the same aperture size and reference photometric band, e.g.,
3′′ DES photometry in the $z$-band is offset with respect to the SDSS by 0.187
mag, and the 3′′ VST magnitudes are offset by 0.312 mag. The offset values
were taken into account to correctly translate the aperture photometry and are
presented in Table 2. These offsets are caused primarily by the difference in
the mean seeing quality of the imaging surveys we consider, and also by the
difference in depth between the surveys, which are absorbed by our color
transformations of integrated magnitudes and have to be taken back from the
transformations when dealing with aperture fluxes, which should not be
seriously affected by the depth.

The VST Atlas survey includes two types of an aperture magnitude, “Default
point source aperture corrected magnitude” and “Default extended source
aperture magnitude.” We provide offsets only for “Default extended source
aperture magnitude” since this is more correct for extended objects.

As one can see in Figure 11, 13, 19, 21, the aperture magnitude differences
for small-sized apertures ($1.^{\prime\prime}5$ and $2^{\prime\prime}$) have
substantial dispersion about a mean value, it is higher for
$1.^{\prime\prime}5$ than for $2^{\prime\prime}$, and the dispersion drops
significantly for $3^{\prime\prime}$ values getting close to that of total
magnitudes. This effect is trivially explained by the mean image quality
difference among the considered surveys when measuring aperture fluxes of
extended sources having a great variety of intrinsic light profiles. For SDSS
the average seeing FWHM is $1.^{\prime\prime}7$ (2009ApJS..182543A), while it
is close to $1^{\prime\prime}$ for DECaLS (Dey et al., 2019) and is
subarcsecond for VST ATLAS (Shanks et al., 2015). Therefore, the smaller the
aperture size, the larger fraction of light will be lost for a compact object
in a survey with a worse delivered image quality. On the other hand, for
extended galaxies without much central light concentration, the seeing quality
will have a much lesser effect, dropping to zero in the limiting case of a
constant surface brightness object. If we look at non-variable stars, which
are constant point sources, they will form very tight sequences without color
dependence having the mean value, which can be calculated by integrating two-
dimensional Gaussians corresponding to the typical values of seeing quality in
the two surveys inside the apertures and then converting the flux ratio into a
magnitude difference. Larger aperture sizes, which significantly exceed the
seeing FWHM (e.g. $3^{\prime\prime}$), enclose a higher fraction of the
central light concentration of a galaxy, therefore the mean offset becomes
smaller and the dispersion of points around the mean values also decreases.
This explanation is consistent with the results shown in Figure 12, which
demonstrates a zero difference between the VST and DES in $5.^{\prime\prime}7$
apertures computed using the transformations for total magnitudes. This is
because (i) $5.^{\prime\prime}7$ is a relatively large aperture to being
affected, and (ii) the VST and DES surveys have similar seeing FWHM (sub-
arcsecond and arcsecond), while the SDSS has a FWHM of $1.^{\prime\prime}7$.

Table 2: Mean offset values in different apertures used for correction of
aperture magnitudes into SDSS bands (Survey value - SDSS value).

Aperture | g | r | i | z
---|---|---|---|---
| mag | mag | mag | mag
DES $1.^{\prime\prime}5\>^{*}$ | -0.051 | -0.239 | -0.317 | -0.308
DES $2^{\prime\prime}\>{}^{*}$ | -0.193 | -0.334 | -0.393 | -0.379
DES $3^{\prime\prime}\>{}^{*}$ | -0.100 | -0.168 | -0.200 | -0.187
VST $1.^{\prime\prime}5\>^{*}$ | -0.187 | -0.233 | -0.330 | -0.359
VST $2^{\prime\prime}$ | -0.209 | -0.239 | -0.312 | -0.330
VST $3^{\prime\prime}\>{}^{*}$ | -0.115 | -0.123 | -0.158 | -0.159
DECaLS $1.^{\prime\prime}5$ | -0.055 | -0.124 |  | -0.197
DECaLS $2^{\prime\prime}$ | -0.190 | -0.241 |  | -0.284
DECaLS $3^{\prime\prime}$ | -0.078 | -0.110 |  | -0.112
MzLS+BASS $1.^{\prime\prime}5$ | 0.133 | -0.038 |  | -0.261
MzLS+BASS $2^{\prime\prime}$ | -0.040 | -0.168 |  | -0.325
MzLS+BASS $3^{\prime\prime}$ | 0.023 | -0.052 |  | -0.127
Aperture | Y | J | H | K
| mag | mag | mag | mag
VISTA $1.^{\prime\prime}5\>^{*}$ | 0.015 | 0.048 | 0.033 | 0.059
VISTA $2^{\prime\prime}$ | 0.012 | 0.044 | 0.023 | 0.050
VISTA $3^{\prime\prime}\>{}^{*}$ | 0.011 | 0.040 | 0.009 | 0.035

${}^{*}\>$Magnitudes were calculated using quadratic interpolation from the
available aperture magnitudes (see Table 1)

In Figure 11 we see a clump that consists of a large number of data points in
the $g$ and $r$ bands residuals between DES and SDSS in the 3′′ aperture at
the mean color $g-r=1.8$ mag, which lies substantially off the principal
sequence formed by the rest of the galaxies in the sample. This clump includes
exclusively intermediate-redshift ($z>0.4$) luminous red galaxies (LRGs) and
it disappears in the residuals in the $i$ and $z$ bands. The most prominent
difference is seen in the $r$ band and we attribute it to difference among the
filter throughput curves: the DES $r$ extends sufficiently further into the
red ($\lambda_{r,\mathrm{cutoff,DES}}=720$ nm versus
$\lambda_{r,\mathrm{cutoff,SDSS}}=670$ nm). The difference in the $g$ band has
an opposite sign and a similar mean absolute offset but larger spread because
the SDSS $g$-band photometry has very poor quality for LRGs because they are
red and therefore faint in $g$ band. As a result, the color difference
$\Delta(g-r)$ in the converted colors remains very close to zero. The clump
originates from the sharp increase of the number of objects in the sample at
$z>0.4$ because of the LRG selection in eBOSS. They present at colors
$(g-r)<1.8$ mag but in relatively small quantities, however the LRG selection
kicks in at around $z=0.4$. LRGs have a very strong redshift dependence of
$g-r$ and it is very sensitive to the difference between the exact filter
throughput curves in SDSS and DES, and one needs a redder color, e.g. $r-z$ to
properly describe the trend. For luminous red galaxies at redshifts $z>0.4$,
the $g-r$ color saturates at $\sim$1.7 mag and therefore it cannot be used for
parameterization of the color transformation. Therefore, for very red galaxies
we recommend using the $r-z$-based transformations (see color differences for
the $r-z$ parameterization in Figure 13.)

Figure 11: Magnitude difference between DES corrected using our
transformations for total magnitudes and SDSS system as a function of color
for 1.5′′, 2′′ and 3′′ apertures. All colors are computed in the same
apertures. The zero level corresponding to the offsets from Table 2 is shown
by a magenta line. Figure 12: Residuals in the $r$-band between the magnitudes
from VST ATLAS and DES in 5.7′′ apertures after color transformations into the
SDSS system.

Figure 13:  The magnitude difference between DES and DESI Legacy Surveys
corrected using our transformations for total magnitudes and SDSS system as a
function of color for total, 1.5′′, 2′′ and 3′′ apertures for $r$-band
corrected with $r-z$ color. All colors are computed in the same apertures. DES
and SDSS total magnitudes are Petrosian, DESI Legacy Surveys total magnitudes
are model based. Figure 14: Optical color–magnitude diagram for extinction and
k- corrected magnitudes from different surveys. The red line shows the red
sequence parametrization from Chilingarian et al. (2017). A number of objects
used for each histogram is shown in the left corner of each plot.

###  5.2 Red sequence in different catalogs

As we mentioned in the Introduction, the optical color–magnitude diagram
constructed from integrated extinction and $k$-corrected photometric
measurements represents the ultimate test of the quality of the obtained
photometric relations: early-type galaxies form a very narrow ‘red sequence’ –
its position and tightness reflects the quality of the input photometric data.
It is also important to know the red sequence shape in different colors for
practical astrophysical purposes, such as selecting early-type candidate
members in galaxy clusters using photometric data. Chilingarian et al. (2017)
published the red sequence shape derived for different filter combinations in
the SDSS DR7.

In Figure 14 we present the $(g-r)$ versus $M_{r}$ color–magnitude plots the 5
surveys we used in our combined catalog transformed into the SDSS photometric
system using our results and the PanSTARRS survey converted using the
literature transformations. We restricted the redshift range to $0.02<z<0.2$.
We applied $k$-corrections to the converted measurements according to
Chilingarian et al. (2010); Chilingarian & Zolotukhin (2012) and corrections
for the Galactic extinction (Schlafly & Finkbeiner, 2011). We overplot the
polynomial approximation for the red sequence presented in Chilingarian et al.
(2017). It turns out that the red sequence has the same slope and location in
all surveys except PanSTARRS, which validates the photometric transformations
we derived in this work. The red sequence in DES, DELVE, and KiDS has a
similar or smaller scatter compared to SDSS, and also a very small number of
very red outliers ($>+2\sigma$ from the mean red sequence position). The
distributions of galaxies in the $(g-z)$ versus $M_{z}$ and $(g-i)$ versus
$M_{i}$ color spaces behave similarly: DES, DELVE, and KiDS show fully
consistent results with the SDSS as presented in the RCSED catalog paper
(Chilingarian et al., 2017).

Figure 15: Positions of an edge-on galaxy with a dust lane
(J010252.85+050631.7, red star) and a BCG (J232653.85-524149.5, yellow star)
in the center of a massive galaxy cluster. The stars symbols show their
positions on the diagrams for DES (left) and DESI Legacy Surveys (right). The
cutouts were made using data from DESI Legacy Surveys Data Release 9.

The red sequence based on the DESI Legacy Surveys data, both from DECaLS and
MzLS demonstrates a larger scatter than those from DES and DELVE despite being
based on the data from the same facility, however, for the most luminous late-
type galaxies, the scatter value is similar or even smaller compared to DES
likely because of the better number statistics. There is also a substantial
number of very red objects above $+2\sigma$ off the red sequence in the
intermediate-luminosity range – we see a similar behavior in the VST ATLAS
data. We have inspected some of the objects off from the red sequence by
$>4\sigma$ and they all settle within 1$\sigma$ of the DES red sequence (see
Figure 15). The first object (marked by red star) probably has a segmentation
problem related to the presence of the dust lane. The second galaxy (marked by
yellow star) is the central galaxy in a cluster and it has issues with
deblending caused by the presence of numerous satellites in its vicinity.
These examples illustrate that compared to other surveys, total magnitudes in
the DESI Legacy Surveys have a higher intrinsic scatter, which we investigate
in the next subsection.

The red sequence based on PanSTARRS total magnitudes converted into SDSS using
the transformations from the literature is wider than that in SDSS and has the
same tilt as SDSS only for small and spatially unresolved objects. It ‘bends
up’ for spatially extended galaxies, which is probably caused by imperfect sky
subtraction for extended targets. The red sequence is also substantially
thicker and there is a large number of red outliers.

###  5.3 DESI Legacy Surveys integrated photometry

Figure 16: Comparison of the total magnitude measurements from DES (red) and
DECaLS (blue) catalogs with those from Source Extractor output for the same
region of the sky. We plot the difference between total magnitudes in the
catalog and in the Source Extractor output as a function of Petrosian radius.

While computing photometric transformation for different imaging surveys
carried out with the DECam instrument we noted that while DES-based
transformations yielded excellent results for DELVE, they do not work properly
for DESI Legacy Surveys: the residuals displayed strong systematics with a
nonlinear color dependence. This was the reason why we used other
transformations for DESI Legacy Surveys, which work well for it as shown at
Figures 7 and 8. DELVE and DES used the same data processing and photometric
calibration tools while DESI Legacy Surveys used totally different solutions.

There are two possible reasons for this behavior: (i) difference in the
pipeline processing of raw images, e.g. a sky subtraction algorithm clipping
outer parts of extended targets in one of the pipelines; (ii) difference in
the source extraction and deblending algorithms. To identify the primary
source of inconsistencies, we decided first to test the source extraction. DES
and DELVE used Source Extractor (Bertin & Arnouts, 1996) while DECaLS used
tractor (Lang et al., 2016) implemented in python. To perform the test, we
downloaded a $16^{\prime}\times 16^{\prime}$ region of the sky in the $r$-band
centered on the galaxy cluster A 168 from DECaLS using Astro Data Lab. Galaxy
clusters are convenient for such a comparison because they include a large
number of galaxies at the same redshift, many of which reside on the red
sequence and have regular elliptical morphology, which simplifies source
extraction and measurement. Then we ran Source Extractor with a default
configuration on this image and compared the results with those presented in
DES for about 100 sources. In Figure 16 we present the comparsion of Source
Extractor-based DECaLS photometry and the published values derived by tractor
against DES. It turns out that using Source Extractor on DECaLS images yields
fully consistent results with DES with a zero mean deviation and an r.m.s of
0.044 mag compared to 0.075 mag for DECaLS photometry. We also see fewer
‘catastrophic outliers’ especially among extended galaxies. In the insets of
Figure 16 we present two examples of such outliers. Both seem to have problems
with deblending: one galaxy has a red star projected on its outer part, which
was not separated from it by tractor and caused the unphysically red color of
this object in DECaLS; the second galaxy was not separated from a satellite in
DECaLS. The increased r.m.s. is probably due to a less stable algorithm used
in tractor to define a surface brightness threshold for extended sources.

This test provides an explanation why the color–magnitude plot from DESI
Legacy Surveys shows a puffed-up red sequence and a high number of red
outliers (see the discussion in 5.2). We also see a way to improve extended
object photometry in DECaLS: the images from DECaLS should be analyzed using
Source Extractor to bring the photometry to the consistency with DES and
DELVE.

###  5.4 Photometry for CfA public spectroscopic datasets

As an example use case for our color transformation, we compiled integrated
photometric measurements in the optical and NIR bands for the galaxy samples
from the two CfA public archives for FAST (73,849 spectra for 42,694 objects)
and Hectospec (211,075 spectra for 164,439 objects). Both datasets originate
from data archives and cover a very wide area that extends beyond the SDSS
footprint, so that a combination of SDSS, DELVE, DES, and BASS/MzLS was needed
to compile an optical photometric catalog.

The CfA FAST archive (Mink et al., 2021) contains 141,531 spectra for 72,247
objects and include observations of nearby galaxies, supernovae, various types
of stars, and planetary nebulae collected between 1994 and 2021 using the FAST
spectrograph operated at the 1.5 m Tillinghast telescope at the Fred Lawrence
Whipple Observatory. The spectra were automatically processed with the FAST
data reduction pipeline. We kept 42,694 objects which we identified as
extragalactic targets using either their redshifts or a cross-identification
in Hyperleda and for which we could find photometric measurements in wide-
field surveys.

The CfA Hectospec archive includes the results of spectroscopic observations
from the multi-fiber spectrograph Hectospec operated at the 6.5 m MMT
(Fabricant et al., 2005). These are the results of publicly released programs
typically older than 5 yr processed with the Hectospec pipeline. The sample
includes observations of low-to-intermediate redshift galaxies, extragalactic
star clusters, bright stars, H ii regions, and planetary nebulae in nearby
galaxies as well as some distant quasars. We kept 164,439 distinct objects,
which we identified as galaxies with available photometric measurements and
excluded extragalactic star clusters and planetary nebulae as well as Galactic
and Local group stars.

To compile a photometric catalog, we converted the Petrosian magnitudes from
DES, DELVE, and BASS/MzLS, VHS, and VIKING into the SDSS and UKIDSS
photometric systems, and corrected them for the foreground extinction. We
complemented these measurements with extinction-corrected GALEX and unWISE
photometry. Some examples of multi-wavelength SEDs for Hectospec galaxies are
shown in Figure 17: we demonstrate the differences of the magnitudes from the
original SDSS measurements before (red) and after (black) applying our color
transformations. The converted DES measurements show excellent agreement with
SDSS. Our photometric catalog will be included into the CfA data archive in
the form of additional properties for each extragalactic object.

As an additional quality check, we computed $k$-corrections for the converted
optical magnitudes using prescriptions from Chilingarian & Zolotukhin (2012)
and built fully corrected color-magnitude diagrams for FAST and Hectospec
galaxies, which we show in Figure 18. The red sequence position and tightness
based on DES and DELVE data perfectly match those from SDSS published by the
RCSED project (Chilingarian et al., 2017).

Figure 17: Spectral energy distributions for three galaxies from the CfA
Hectospec archive using SDSS, uncorrected and converted DES optical
photometry. The data in the FUV and NUV bands are from GALEX DR7, YJHK are
from UKIDSS, w1 and w2 are from unWISE. Lower panels: difference between SDSS
and original (red) or converted (red) DES. Figure 18: Extinction and
$k$-corrected color–-magnitude diagrams for FAST (left) and Hectospec (right)
galaxies. Photometric measurements taken from SDSS are shown by a grey
2D-histogram, DELVE and DES photometry are shown by blue and red dots
respectively.

###  5.5 On the quality of integrated photometry of galaxies in optical
surveys

By applying photometric transformations derived in this work to the data
originating from various surveys and comparing them against SDSS and each
other, and also quantifying the properties of the color-magnitude diagrams
(e.g. red sequence location and tightness), we can make the following
assessment of the quality of integrated photometric measurements of galaxies:

  * •

DES provides the highest quality of optical photometry of galaxies compared to
all other wide-field surveys and can be used as a ‘gold standard’ for many
scientific applications and future projects, e.g. to check the quality of
photometric calibration from the forthcoming Rubin Legacy Survey of Space and
Time. It also allows for refined photometric tasks, e.g. color-based search
for rare galaxies (Chilingarian & Zolotukhin, 2015; Grishin et al., 2021) or
simultaneous fitting of spectral and photometric information (Chilingarian &
Katkov, 2012) to infer detailed star formation histories of galaxies using
complex parametric stellar population models (see e.g. Grishin et al., 2019).

  * •

DELVE is a shallower equivalent of DES with similar photometric quality and
low systematic errors even though higher statistical uncertainties because of
the lower depth. Its main advantage is a very wide sky coverage of 17,000 deg2
vs. 5000 deg2 in DES.

  * •

VST ATLAS and KiDS conducted at the 2.6 m ESO VST also have minimal systematic
photometric errors and are complementary to DES and DELVE. In addition, they
supply $u$-band photometry not available in the surveys done with DECam
providing a Southern hemisphere to SDSS.

  * •

Both parts of the DESI Legacy Surveys, DECaLS and BASS/MzLS exhibit
significant systematics in the extended source photometry, which exceeds
statistical uncertainties by a factor of a few. Nevertheless, the photometric
measurements can be adequately transformed into the SDSS photometric system
and can be used for certain applications of the photometric/SED-based
selection even though the photometric uncertainties in any type of an SED
fitting using DESI Legacy Surveys data should be artificially increased to
account for the systematics. We attribute it mostly to the source extraction
algorithm, so that a complete re-processing of DECaLS data using a different
software package (e.g. Source Extractor) should drastically improve the
quality of photometry of galaxies.

  * •

PanSTARRS photometry deviates from SDSS nonlinearly with color, and the
deviations in all bands show significant trends with the galaxy size. This is
noticeable not only by the dependence between residuals and sizes, but also by
the red sequence – it has a correct slope and is tight only for spatially
unresolved objects. Further investigations are needed to fully understand the
sources of this systematics, but it will likely be improved when using a
different source extraction software (see e.g. Makarov et al., 2022, for an
application of Source Extractor to PanSTARRS data).

## Acknowledgements

V.T., I.C. and K.G. acknowledge the RScF grant No. 19-12-00281 for supporting
the analysis of the photometric data and the Interdisciplinary Scientific and
Educational School of Moscow University “Fundamental and Applied Space
Research.” I.K. acknowledges the RScF grant No. 21-72-00036 for supporting the
development of the web-site for online transformations of photometric
measurements. IC’s research is supported by the Telescope Data Center at
Smithsonian Astrophysical Observatory. This research has made use of TOPCAT,
developed by Mark Taylor at the University of Bristol; Aladin, developed by
the Centre de Données Astronomiques de Strasbourg (CDS).

Funding for the SDSS and SDSS-II has been provided by the Alfred P. Sloan
Foundation, the Participating Institutions, the National Science Foundation,
the U.S. Department of Energy, the National Aeronautics and Space
Administration, the Japanese Monbukagakusho, the Max Planck Society, and the
Higher Education Funding Council for England. The SDSS Web Site is .

The DESI Legacy Surveys consist of three individual and complementary
projects: the Dark Energy Camera Legacy Survey (DECaLS; Proposal ID
#2014B-0404; PIs: David Schlegel and Arjun Dey), the Beijing-Arizona Sky
Survey (BASS; NOAO Prop. ID #2015A-0801; PIs: Zhou Xu and Xiaohui Fan), and
the Mayall z-band Legacy Survey (MzLS; Prop. ID #2016A-0453; PI: Arjun Dey).
DECaLS, BASS and MzLS together include data obtained, respectively, at the
Blanco telescope, Cerro Tololo Inter-American Observatory, NSF’s NOIRLab; the
Bok telescope, Steward Observatory, University of Arizona; and the Mayall
telescope, Kitt Peak National Observatory, NOIRLab. Pipeline processing and
analyses of the data were supported by NOIRLab and the Lawrence Berkeley
National Laboratory (LBNL). The DESI Legacy Surveys project is honored to be
permitted to conduct astronomical research on Iolkam Du’ag (Kitt Peak), a
mountain with particular significance to the Tohono O’odham Nation.

NOIRLab is operated by the Association of Universities for Research in
Astronomy (AURA) under a cooperative agreement with the National Science
Foundation. LBNL is managed by the Regents of the University of California
under contract to the U.S. Department of Energy.

This project used data obtained with the Dark Energy Camera (DECam), which was
constructed by the Dark Energy Survey (DES) collaboration. Funding for the DES
Projects has been provided by the U.S. Department of Energy, the U.S. National
Science Foundation, the Ministry of Science and Education of Spain, the
Science and Technology Facilities Council of the United Kingdom, the Higher
Education Funding Council for England, the National Center for Supercomputing
Applications at the University of Illinois at Urbana-Champaign, the Kavli
Institute of Cosmological Physics at the University of Chicago, Center for
Cosmology and Astro-Particle Physics at the Ohio State University, the
Mitchell Institute for Fundamental Physics and Astronomy at Texas A&M
University, Financiadora de Estudos e Projetos, Fundacao Carlos Chagas Filho
de Amparo, Financiadora de Estudos e Projetos, Fundacao Carlos Chagas Filho de
Amparo a Pesquisa do Estado do Rio de Janeiro, Conselho Nacional de
Desenvolvimento Cientifico e Tecnologico and the Ministerio da Ciencia,
Tecnologia e Inovacao, the Deutsche Forschungsgemeinschaft and the
Collaborating Institutions in the Dark Energy Survey. The Collaborating
Institutions are Argonne National Laboratory, the University of California at
Santa Cruz, the University of Cambridge, Centro de Investigaciones
Energeticas, Medioambientales y Tecnologicas-Madrid, the University of
Chicago, University College London, the DES-Brazil Consortium, the University
of Edinburgh, the Eidgenossische Technische Hochschule (ETH) Zurich, Fermi
National Accelerator Laboratory, the University of Illinois at Urbana-
Champaign, the Institut de Ciencies de lÉspai (IEEC/CSIC), the Institut de
Fisica dÁltes Energies, Lawrence Berkeley National Laboratory, the Ludwig
Maximilians Universitat Munchen and the associated Excellence Cluster
Universe, the University of Michigan, NSFś NOIRLab, the University of
Nottingham, the Ohio State University, the University of Pennsylvania, the
University of Portsmouth, SLAC National Accelerator Laboratory, Stanford
University, the University of Sussex, and Texas A&M University.

BASS is a key project of the Telescope Access Program (TAP), which has been
funded by the National Astronomical Observatories of China, the Chinese
Academy of Sciences (the Strategic Priority Research Program “The Emergence of
Cosmological Structures” Grant # XDB09000000), and the Special Fund for
Astronomy from the Ministry of Finance. The BASS is also supported by the
External Cooperation Program of Chinese Academy of Sciences (Grant #
114A11KYSB20160057), and Chinese National Natural Science Foundation (Grant #
12120101003, # 11433005).

The DESI Legacy Surveys team makes use of data products from the Near-Earth
Object Wide-field Infrared Survey Explorer (NEOWISE), which is a project of
the Jet Propulsion Laboratory/California Institute of Technology. NEOWISE is
funded by the National Aeronautics and Space Administration.

The Legacy Surveys imaging of the DESI footprint is supported by the Director,
Office of Science, Office of High Energy Physics of the U.S. Department of
Energy under Contract No. DE-AC02-05CH1123, by the National Energy Research
Scientific Computing Center, a DOE Office of Science User Facility under the
same contract; and by the U.S. National Science Foundation, Division of
Astronomical Sciences under Contract No. AST-0950945 to NOAO.

Based on data products from observations made with ESO Telescopes at the La
Silla Paranal Observatory under program IDs 177.A-3016, 177.A-3017 and
177.A-3018, and on data products produced by Target/OmegaCEN, INAF-OACN, INAF-
OAPD and the KiDS production team, on behalf of the KiDS consortium. OmegaCEN
and the KiDS production team acknowledge support by NOVA and NWO-M grants.
Members of INAF-OAPD and INAF-OACN also acknowledge the support from the
Department of Physics & Astronomy of the University of Padova, and of the
Department of Physics of Univ. Federico II (Naples).

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Figure 19: Magnitude difference between VST Atlas corrected by transformations
for total magnitudes and SDSS for 1.5′′, 2′′ and 3′′ apertures. All colors are
computed in the same apertures. The zero corresponding to the offsets from
Table 2 is shown by a magenta line.

Figure 20:  Same as Figure 19 for magnitude difference in the VIKING system
and UKIDSS.

Figure 21: Magnitude difference between DESI Legacy Surveys corrected by
transformations for total magnitudes and SDSS system for 1.5′′, 2′′ and 3′′
apertures. All colors are computed in the same apertures. The zero
corresponding to the offsets from Table 2 is shown by a magenta line.