11institutetext: 1 Department of Astronomy, University of Geneva, ch. d’Écogia
16, 1290 Versoix, Switzerland
2 University Observatory, Faculty of Physics, Ludwig-Maximilians-Universität,
Scheinerstr. 1, 81679 Munich, Germany
3 Max Planck Institute for Extraterrestrial Physics, Giessenbachstr. 1, 85748
Garching, Germany
4 Max-Planck-Institut für Astronomie, Königstuhl 17, 69117 Heidelberg, Germany
5 Université Paris-Saclay, Université Paris Cité, CEA, CNRS, Astrophysique,
Instrumentation et Modélisation Paris-Saclay, 91191 Gif-sur-Yvette, France
6 Department of Astronomy & Physics and Institute for Computational
Astrophysics, Saint Mary’s University, 923 Robie Street, Halifax, Nova Scotia,
B3H 3C3, Canada
7 Aix-Marseille Université, CNRS, CNES, LAM, Marseille, France
8 Leiden Observatory, Leiden University, Niels Bohrweg 2, 2333 CA Leiden, The
Netherlands
9 Université Paris-Saclay, CNRS, Institut d’astrophysique spatiale, 91405,
Orsay, France
10 ESAC/ESA, Camino Bajo del Castillo, s/n., Urb. Villafranca del Castillo,
28692 Villanueva de la Cañada, Madrid, Spain
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Portsmouth PO1 3FX, UK
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Piero Gobetti 93/3, 40129 Bologna, Italy
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Universitá di Bologna, via Piero Gobetti 93/2, 40129 Bologna, Italy
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Arnesano, 73100, Lecce, Italy
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USA

#  Euclid preparation. XXXI. The effect of the variations in photometric
passbands on photometric-redshift accuracy

Euclid Collaboration: S. Paltani1<EMAIL_ADDRESS>J. Coupon1 W. G.
Hartley1 A. Alvarez-Ayllon1 F. Dubath1 J.J. Mohr2,3 M. Schirmer4 J.-C.
Cuillandre5 G. Desprez1,6 O. Ilbert7 K. Kuijken8 N. Aghanim9 B. Altieri10 A.
Amara11 N. Auricchio12 M. Baldi13,12,14 R. Bender3,15 C. Bodendorf3 D.
Bonino16 E. Branchini17,18 M. Brescia19,20 J. Brinchmann21 S. Camera22,23,16
V. Capobianco16 C. Carbone24 V. F. Cardone25,26 J. Carretero27,28 F. J.
Castander29,30 M. Castellano25 S. Cavuoti20,31 R. Cledassou32,33 G. Congedo34
C.J. Conselice35 L. Conversi10,36 Y. Copin37 L. Corcione16 F. Courbin38 M.
Cropper39 A. Da Silva40,41 H. Degaudenzi1 J. Dinis41,40 M. Douspis9 X. Dupac10
S. Dusini42 S. Farrens5 S. Ferriol37 P. Fosalba30,29 M. Frailis43 E.
Franceschi12 P. Franzetti24 S. Galeotta43 B. Garilli24 W. Gillard44 B.
Gillis34 C. Giocoli12,45 A. Grazian46 S. V. Haugan47 H. Hoekstra8 A.
Hornstrup48,49 P. Hudelot50 K. Jahnke4 M. Kümmel15 S. Kermiche44 A.
Kiessling51 M. Kilbinger5 T. Kitching39 R. Kohley10 B. Kubik37 M. Kunz52 H.
Kurki-Suonio53,54 S. Ligori16 P. B. Lilje47 I. Lloro55 E. Maiorano12 O.
Mansutti43 O. Marggraf56 K. Markovic51 F. Marulli13,12,14 R. Massey57 D.C.
Masters58 S. Maurogordato59 H. J. McCracken60 E. Medinaceli12 S. Mei61 M.
Melchior62 M. Meneghetti12,14 E. Merlin25 G. Meylan38 M. Moresco13,12 L.
Moscardini13,12,14 E. Munari43 S.-M. Niemi63 J. Nightingale57 C. Padilla27 F.
Pasian43 K. Pedersen64 W.J. Percival65,66,67 V. Pettorino5 G. Polenta68 M.
Poncet32 L. A. Popa69 F. Raison3 R. Rebolo70,71 A. Renzi72,42 J. Rhodes51 G.
Riccio20 E. Romelli43 M. Roncarelli12 E. Rossetti73 R. Saglia15,3 D. Sapone74
B. Sartoris15,43 P. Schneider56 A. Secroun44 C. Sirignano72,42 G. Sirri14 J.
Skottfelt75 L. Stanco42 J.-L. Starck76 C. Surace7 P. Tallada-Crespí77,28 I.
Tereno40,78 R. Toledo-Moreo79 F. Torradeflot77,28 I. Tutusaus80,52 E. A.
Valentijn81 L. Valenziano12,14 T. Vassallo43 Y. Wang58 G. Zamorani12 J.
Zoubian44 S. Andreon82 H. Aussel76 S. Bardelli12 M. Bolzonella12 A. Boucaud61
D. Di Ferdinando14 M. Farina83 J. Graciá-Carpio3 V. Lindholm53,54 D.
Maino84,24,85 N. Mauri86,14 C. Neissner27 V. Scottez50,87 E. Zucca12 C.
Baccigalupi88,89,43,90 M. Ballardini91,92,12 A. Biviano43,89 A. Blanchard80 S.
Borgani93,89,43,90 A. S. Borlaff94 C. Burigana91,95,96 R. Cabanac80 A.
Cappi12,59 C. S. Carvalho78 S. Casas97 G. Castignani13,12 K. Chambers98 A. R.
Cooray99 H.M. Courtois100 O. Cucciati12 S. Davini101 G. De Lucia43 H. Dole9 J.
A. Escartin3 S. Escoffier44 F. Finelli12,96 S. Fotopoulou102 K. Ganga61 K.
George2 G. Gozaliasl53 H. Hildebrandt103 I. Hook104 A. Jimenez Muñoz105 B.
Joachimi106 V. Kansal76 E. Keihanen107 C. C. Kirkpatrick107 A.
Loureiro34,106,108 J. Macias-Perez105 G. Maggio43 M. Magliocchetti83 R.
Maoli109,25 S. Marcin62 M. Martinelli25 N. Martinet7 S. Matthew34 L. Maurin9
R. B. Metcalf13,12 P. Monaco93,43,90,89 G. Morgante12 S. Nadathur11 A.A.
Nucita110,111,112 L. Patrizii14 J. E. Pollack61 V. Popa69 C. Porciani56 D.
Potter113 A. Pourtsidou34,114 L. Pozzetti12 M. Pöntinen53 P. Reimberg50 A.G.
Sánchez3 Z. Sakr80,115,116 E. Sefusatti43,90,89 M. Sereno12,14 A. Spurio
Mancini39 J. Stadel113 J. Steinwagner3 R. Teyssier117 C. Valieri14 J.
Valiviita53,54 S. E. van Mierlo81 A. Veropalumbo84 M. Viel88,89,43,90 J. R.
Weaver118

(Received 25 May 2023 / Accepted 26 September 2023)

The technique of photometric redshifts has become essential for the
exploitation of multi-band extragalactic surveys. While the requirements on
photometric redshifts for the study of galaxy evolution mostly pertain to the
precision and to the fraction of outliers, the most stringent requirement in
their use in cosmology is on the accuracy, with a level of bias at the sub-
percent level for the Euclid cosmology mission. A separate, and challenging,
calibration process is needed to control the bias at this level of accuracy.
The bias in photometric redshifts has several distinct origins that may not
always be easily overcome. We identify here one source of bias linked to the
spatial or time variability of the passbands used to determine the photometric
colours of galaxies. We first quantified the effect as observed on several
well-known photometric cameras, and found in particular that, due to the
properties of optical filters, the redshifts of off-axis sources are usually
overestimated. We show using simple simulations that the detailed and complex
changes in the shape can be mostly ignored and that it is sufficient to know
the mean wavelength of the passbands of each photometric observation to
correct almost exactly for this bias; the key point is that this mean
wavelength is independent of the spectral energy distribution of the source.
We use this property to propose a correction that can be computationally
efficiently implemented in some photometric-redshift algorithms, in particular
template-fitting. We verified that our algorithm, implemented in the new
photometric-redshift code Phosphoros, can effectively reduce the bias in
photometric redshifts on real data using the CFHTLS T007 survey, with an
average measured bias $\Delta z$ over the redshift range $0.4\leq z\leq 0.7$
decreasing by about 0.02, specifically from $\Delta z\simeq 0.04$ to $\Delta
z\simeq 0.02$ around $z=0.5$. Our algorithm is also able to produce corrected
photometry for other applications.

###### Key Words.:

Galaxies: distances and redshifts – Cosmology: observations – Surveys –
Techniques: photometric – Techniques: miscellaneous

##  1 Introduction

Multi-megapixel cameras with large fields of view have revolutionised
extragalactic astrophysics and observational cosmology by enabling photometric
surveys of large sky areas in several optical and near-infrared bands. The
Dark Energy Survey (DES; The Dark Energy Survey Collaboration 2005), the Kilo-
Degree Survey (KiDS; de Jong et al. 2013), and the Hyper Suprime-Cam Strategic
Survey Program (HSC-SSP; Aihara et al. 2018) are recent examples of
photometric surveys with areas exceeding 1000 deg2. These surveys enable the
measurement of the cosmic shear, which is the distortion of the images of
distant objects caused by the propagation of light rays through inhomogeneous
matter (Blandford et al. 1991). The cosmic shear allows the reconstruction of
dark-matter maps at different redshifts, from which the distribution of matter
and its evolution can be inferred. Modern cosmological surveys have
established cosmic shear as one of the main modern cosmological probes and
have already provided important cosmological constraints (e.g. Abbott et al.
2018 for DES; Asgari et al. 2021 for KiDS; and Hamana et al. 2020 for HSC-
SSP).

Euclid (Laureijs et al. 2011) is a mission of the European Space Agency that
will perform a survey over 15 000 deg2 of extragalactic sky (Euclid
Collaboration: Scaramella et al. 2022) with optical and near-infrared imaging,
as well as with slitless multi-object spectroscopy in the near-infrared. The
main scientific probes of Euclid are the cosmic shear and galaxy clustering.
They are supported by two instruments: The VIS optical camera (Cropper et al.
2014) will provide us with high-resolution images of galaxies for the
determination of the cosmic shear; and the Near Infrared Spectrometer and
Photometer near-infrared instrument (NISP; Maciaszek et al. 2016) will perform
near-infrared photometry in three bands to support cosmic shear determination,
as well as near-infrared spectroscopy for the study of three-dimensional
galaxy clustering. Compared to current ground-based surveys, the determination
of the cosmic shear with Euclid will greatly benefit from high-resolution
imaging and near-infrared photometry from space, but also from the
significantly larger sky area (three times the area of DES) and depth.

The measurement of redshifts for a large fraction of the surveyed galaxies is
an indispensable step in the cosmological study of the cosmic shear. While
redshift measurements can be performed through spectroscopy, spectroscopy is
more challenging than photometry at very faint fluxes, which limits the number
of measurable redshifts, even when efficient multi-object spectrographs are
available. In the case of Euclid, spectroscopic redshifts will be determined
for some 30 million galaxies, while the total number of galaxies for which
sufficiently precise shapes can be measured will exceed one billion (Laureijs
et al. 2011). The technique of photometric redshifts, that is, using
photometric observations only, is currently the only way to determine
redshifts on such a huge scale. The limited precision of photometric redshifts
is not an issue for cosmic shear because the efficiency of lensing is a slowly
varying function of the redshift. Photometric redshifts have thus become an
essential tool of modern observational cosmology.

Photometric-redshift determination requires photometric observations of
galaxies in several wavelength bands, which defines a multi-dimensional flux
or colour space111Here we consider colour spaces only.. The redshift is then
obtained from the construction of a mapping between the position of an object
in this colour space and the redshift. We can in principle determine this
mapping through different approaches, either based on real objects with known
redshifts or on simulated objects. The determination of redshifts of galaxies
using a small number of broad-band photometric measurements was pioneered six
decades ago by Baum (1962). The template-fitting (TF) technique, which
involves the comparison of the magnitudes or fluxes of a galaxy with those of
simulated objects, was first developed by Loh & Spillar (1986) and
subsequently exploited in several codes that are still in use today (e.g.
Arnouts et al. 1999; Bolzonella et al. 2000). A more recent approach using
machine-learning (ML), whose first applications to photometric-redshift
calculations were made by Firth et al. (2003) and Tagliaferri et al. (2003)
using artificial neural networks, is now becoming more and more popular, and
virtually every ML approach has been investigated, for instance support vector
machine (Wadadekar 2005), decision trees (Carrasco Kind & Brunner 2013), and
Gaussian processes (Almosallam et al. 2016). A review of the challenges of
photometric-redshift determination in the context of cosmological surveys has
been presented in Newman & Gruen (2022).

The usefulness of any quantity for a scientific application is bound to meet
the requirements on the quality of its determination. In the case of
photometric redshifts, this quality is usually expressed with three
parameters. When a photometric-redshift determination is successful, the
measurement is distributed around the true value; the dispersion $\sigma_{z}$
of this distribution is a measure of the precision of the determination.
However, it sometimes happens that the photometric-redshift determination
fails completely and the predicted redshift is found to lie very far from the
true value; the probability of such a failure is called the ‘outlier
fraction’. Finally, the bias $\Delta z$ is the location of the peak of the
distribution of the differences between predictions and true redshift values
and determines the accuracy of the predictions.

The Euclid requirements on the precision of photometric redshifts (Amara &
Réfrégier 2007) are not extremely demanding [$\sigma_{z}=0.05(1+z)$ and an
outlier fraction of $10$% at a magnitude 24.5] compared with what can be
achieved on small fields; for instance, in the COSMOS field, Weaver et al.
(2022) obtained, at similar depths, $\sigma_{z}$ better than $0.02(1+z)$ and
an outlier fraction better than $5$%. Meeting the Euclid requirements
nevertheless remains challenging over the full Euclid wide survey because of
the difficulty to obtain deep-enough photometry over a field almost 10 000
times larger (Euclid Collaboration: Desprez et al. 2020). However, the biggest
difficulty in using photometric redshifts for cosmology is the accuracy in the
photometric-redshift determination that is necessary for the cosmic-shear
probe; in Euclid, the requirement is expressed as the bias on the mean
redshift in each of the approximately ten tomographic redshift bins at
redshift $z$, which needs to be less than $\Delta z/(1+z)=0.002$. Such a
stringent requirement demands that some bias correction takes place after the
photometric-redshift determination, for instance by calibrating the bias
directly in the colour space occupied by galaxies (Masters et al. 2015). This
study was the main motivation behind the C3R2 project to gather a gold-
standard set of spectroscopic redshifts over the full colour space of galaxies
(Masters et al. 2017, 2019; Euclid Collaboration: Guglielmo et al. 2020;
Stanford et al. 2021; Euclid Collaboration: Saglia et al. 2022). It is however
essential to remove, as much as possible, any bias before the calibration step
if we want to meet this requirement. We focus here on one specific source of
bias that is inherent to the photometric measurement and that is linked to the
time and spatial variations of the photometric passbands. Spatial variations
are dependent on the positions of the sources in the field of view and thus
induce spatial variations unrelated to the source in the photometric redshifts
that, in turn, introduce a spurious signal into the correlation function of
the cosmic shear. Such variations may have an effect on the dispersion,
outlier fraction, and bias; we focus here on the bias since this is the most
stringent requirement on photometric redshifts for Euclid.

In this paper, we first review the current knowledge about passband variations
for surveys relevant to Euclid. We then discuss the problems caused by these
variations and demonstrate the effect quantitatively using idealised
simulations. We finally propose an efficient implementation that can remove
most of the bias resulting from this issue, and validate it using a real
photometric survey. While we focus on the spatial dependence in this paper,
the correction also applies to time variations of the passbands.

##  2 Photometry

Photometric measurements are performed over ranges of wavelengths called
‘passbands’ that are defined by a transmission curve. This curve, denoted
$T(\lambda)$, is a function of the wavelength $\lambda$ that describes the
fraction of photons (or of the energy) entering the telescope that is
ultimately recorded on the detector. The main component affecting the
transmission curve is an optical filter that restricts the range of accessible
wavelengths; other components include the detector quantum efficiency and the
transmission through the other optical elements. Usually $T(\lambda)$
transmissions are designed to approximate more or less a top-hat filter. One
defines the AB flux $F_{T}$ of a source with rest-frame spectral energy
distribution (SED) $L(\lambda_{0})$, with $\lambda_{0}=\lambda/(1+z)$ where
$z$ is the redshift, through the photometric transmission curve $T(\lambda)$
as (Oke & Gunn 1983)

$F_{T}=\frac{\int_{0}^{\infty}s(\lambda)\,T(\lambda)\,\lambda\,\mathrm{d}\lambda}{\mathrm{c}\int_{0}^{\infty}T(\lambda)\,\frac{\mathrm{d}\lambda}{\lambda}}\;,$
(1)

where $c$ is the speed of light and

$s(\lambda)=\frac{L(\lambda_{0})\,10^{-0.4E_{B-V}k_{\lambda}}}{4\pi\,(1+z)\,D_{\mathrm{L}}^{2}}$
(2)

is the observed flux as a function of wavelength, with $D_{\mathrm{L}}$ the
luminosity distance, $E_{B-V}$ the Galactic reddening and $k(\lambda)$ the
extinction law. The above expression assumes that the detector is counting
photons, which is the case for the vast majority of modern detectors in the
optical and near-infrared range. The AB convention provides the flux of a
source with flat SED (when expressed per unit of frequency) that would leave
the same number of counts on the detector as the source under study when
passing through the transmission curve $T(\lambda)$. We point out that, with
the definition of Eq. (1), the normalisation of $T(\lambda)$ is arbitrary. We
assume here that $T(\lambda)$ is normalised so that
$\int_{0}^{\infty}T(\lambda)\,\mathrm{d}\lambda=1$.

Photometric extragalactic surveys use a (small) set of $N$ passbands with
transmission curves $T_{i}(\lambda)$, $i=1,\ldots,N$, covering distinct, and
usually very slightly overlapping, wavelength ranges. The $ugriz$ set of
passbands of the Sloan Digital Sky Survey Photometric Camera (Gunn et al.
1998) is now the most common system used for extragalactic surveys, with
slight differences in the filters depending on the cameras and telescopes. The
Euclid photometric survey consists of the very broad $I_{\scriptscriptstyle\rm
E}$ passband of the VIS optical instrument, which covers roughly the $riz$
bands and which we will not consider further here, and the three passbands in
the near-infrared of the NISP instrument, the $Y_{\scriptscriptstyle\rm E}$,
$J_{\scriptscriptstyle\rm E}$, and $H_{\scriptscriptstyle\rm E}$ passbands
(see Sect. 3.4). The full set of passbands used in this paper, which we denote
$\cal T$, are shown in Fig. 1.

Figure 1: Set of transmission curves ${\cal
T}=\\{ugriz$$Y_{\scriptscriptstyle\rm E}$$J_{\scriptscriptstyle\rm
E}$$H_{\scriptscriptstyle\rm E}$} used for the Euclid mission (from left to
right). The $ugriz$ passbands are only fiducial, since different sets will be
used by Euclid; those represented here are from SDSS. The
$Y_{\scriptscriptstyle\rm E}$$J_{\scriptscriptstyle\rm
E}$$H_{\scriptscriptstyle\rm E}$ passbands are from NISP on board Euclid. Only
the filter transmissions are shown, without atmospheric, telescope, and
detector quantum efficiency effects.

##  3 Spatial variation of photometric passbands

A consistent photometric system implicitly assumes that the transmissions
$T_{i}(\lambda)$, $i=1,\ldots,N$ are the same for all objects. However, small
variations from object to object are possible. In the presence of passband
variations, the colours of the different sources occupy different colour
spaces, and cannot be compared any more. Passband variations can occur due to
several effects as a function of time or of position on the detector. Time
dependence can be introduced by atmospheric effects and by the evolution or
degradation of the properties of the optical elements, filters, or detectors
in a way that depends on the wavelength. Spatial dependence of the passbands
is another issue that affects differently, and systematically, the sources in
the field of view. Such effects can be introduced by non-uniformities of the
filters or detectors, or by the fact that the optical beams of off-axis
sources hit the filter at angles that are different from those in the case of
on-axis sources. The latter effect can in principle be predicted. In order to
obtain photometric measurements with the highest possible accuracy, several
teams have measured passband variations across the field of view of their
camera. We briefly review studies of the spatial variations of the passbands
below, in order to demonstrate that the issue is general, and not limited to a
particular survey.

Passband variations have been measured for several photometric systems.
Denoting $E(x)$ the average of any function $x(\lambda)$ over the transmission
$T(\lambda)$, that is,

$E(x)=\frac{\int_{0}^{\infty}x(\lambda)\,T(\lambda)\,\mathrm{d}\lambda}{\int_{0}^{\infty}T(\lambda)\,\mathrm{d}\lambda},$
(3)

we characterise the measured variations using the changes in the first four
moments of the normalised transmission curve $T(\lambda)$ in different
locations in the field of view: the mean $\mu=E(\lambda)$; the variance
$\sigma^{2}=E\left((\lambda-\mu)^{2}\right)$; the skewness
$\Sigma=E\left(\left(\frac{\lambda-\mu}{\sigma}\right)^{3}\right)$; and the
kurtosis $\kappa=E\left(\left(\frac{\lambda-\mu}{\sigma}\right)^{4}\right)$.
We point out that the skewness and the kurtosis are the standardised third and
fourth moments, respectively. We will also use the dispersion $\sigma$,
instead of the second moment $\sigma^{2}$, as it is more intuitively
understandable. We stress that these moments do not depend on the SED of the
source.

###  3.1 Sloan Digital Sky Survey Photometric Camera

The response of the Sloan Digital Sky Survey Photometric Camera (Gunn et al.
1998), which is mounted on the Sloan Foundation Telescope at Apache Point
Observatory, has been characterised in extensive detail in Doi et al. (2010).
The camera is composed of 30 CCDs in a matrix of 5 rows and 6 columns. The
passband variations are determined empirically using a series of monochromatic
1 nm-wide dome flats (Rheault et al. 2012). They measured in particular the
response for the six columns of the imager for the five $ugriz$ bands, in
order to estimate the column-to-column passband variation. Figure 2 (top)
shows the six $r$ passbands obtained in one campaign222The data for all
filters can be found at
http://www.ioa.s.u-tokyo.ac.jp/~doi/sdss/SDSSresponse.html. Stronger
variations due to temperature and ageing are however reported by Doi et al.
(2010); they quote variations in the passband effective wavelengths from about
3 nm in the best case ($g$ and $i$ passbands) to 12 nm in the worst case ($z$
passband, which is strongly affected by the CCD quantum efficiency), and 3.5
nm in the $r$ passband. Unfortunately, the measurements obtained at other
periods do not seem to be available.

Figure 2: Passband variations of the SDSS $r$ filter. _Top_ : Measured
variations of the SDSS $r$ filters from Doi et al. (2010). Measurements have
been performed for each of the six different CCD columns. The transmissions
have renormalised so that the maximum of the transmission at Position index 1
is 1. The inset shows a zoom on the cut-off of the transmissions. The legend
gives the position indices associated to each colour. _Bottom_ : The first
four moments of the six SDSS $r$ passbands. Each transmission is identified
with a specific colour, identical in the top and bottom panels.

Figure 2 (bottom) shows the first four moments of the $r$ passbands. One
detector column (3) shows particularly large variations of mean wavelength and
dispersion. The overall shape remains however very similar, with very minor
change in skewness ($<0.01$) and kurtosis (0.007).

###  3.2 Dark Energy Camera

The Dark Energy Survey (The Dark Energy Survey Collaboration 2005) is a $griz$
photometric survey of about 5000 deg2 with the 2.2 deg-diameter Dark Energy
Camera (Flaugher et al. 2015) located on the Blanco telescope at Cerro Tololo
Observatory. Passband variations have been studied in detail by Li et al.
(2016). They measured the change of the passband as a function of the distance
to the centre of the detector up to the edge of the field-of-view around 1.1
deg. The $i$ filter was found to show the largest variation, with a wavelength
shift of the cut-on wavelength of about 6 nm. Figure 3 (top) shows the
variation of the $r$ passband. As in the case of SDSS, there is little
variation in the shape of the passband.

Figure 3: Passband variations of the DES $r$ filter. _Top_ : Measured
variations of the DES $r$ filters from Li et al. (2016). Six measurements have
been performed at different off-axis positions; larger position indices
indicate larger off-axis distances. The transmissions have renormalised so
that the maximum of the transmission at Position index 1 is 1. The inset shows
a zoom on the cut-off of the transmissions. The legend gives the position
indices associated to each colour. _Bottom_ : The first four moments of the
six DES $r$ passbands. Each transmission is identified with a specific colour,
identical in the top and bottom panels.

Figure 3 (bottom) shows the first four moments of the $r$ passbands; larger
position indices indicate larger off-axis distances. The mean wavelength is
very stable, with an amplitude of variation of about 1 nm compared to the
passband in the centre of the field of view, which we refer to in the
following as the ‘central passband’, without clear dependence on the off-axis
angle; the dispersion variation is very comparable to that of SDSS. Again very
small changes in skewness and kurtosis are observed.

###  3.3 MegaCam

The MegaCam instrument (Boulade et al. 2003) is a 1-deg2 imaging camera
located on the prime focus of the Canada-France-Hawaii Telescope in Hawaii.
The detailed calibration of the camera has been performed in the framework of
the Supernova Legacy Survey project (Guy et al. 2010) by Betoule et al.
(2013). They found the location of the source as a function of the distance to
the centre of the detector plane to be a major driver of passband variations,
with transmissions moving towards shorter wavelengths. This is expected for
optical interference filters (see discussion in, e.g., section 3.2 of Euclid
Collaboration: Schirmer et al. 2022, and references therein). Figure 4 (top)
shows the variation of the $r$ passband, where the position index is
correlated with the off-axis distance, with every shift in the position index
corresponding to about an additional 5 arcmin distance from the centre of the
field (See also Sect. 6).

Figure 4: Passband variations of the MegaCam $r$ filter. _Top_ : Measured
variations of the MegaCam $r$ filters from Betoule et al. (2013). Ten
measurements have been performed at different off-axis positions; larger
position indices indicate larger off-axis distances. The transmissions have
renormalised so that the maximum of the transmission at Position index 1 is 1.
The inset shows a zoom on the cut-on of the transmissions. The legend gives
the position indices associated to each colour. _Bottom_ : The first four
moments of the ten MegaCam $r$ passbands. Each transmission is identified with
a specific colour, identical in the top and bottom panels.

Figure 4 (bottom) shows the first four moments of the $r$ passbands. The
changes in the mean wavelength are much more significant, with a peak-to-peak
amplitude of about 8 nm, although the change in the dispersion is not
significantly larger than in the case of SDSS or DES. Similar changes in mean
wavelengths do occur with other SDSS or DES passbands. The amplitude of
skewness and (especially) kurtosis variations are several times larger than
for SDSS, indicating the presence of more significant changes in the shape of
the passband.

###  3.4 Euclid NISP

Euclid NISP (Maciaszek et al. 2016) is equipped with 3 photometric filters
covering the Euclid $Y_{\scriptscriptstyle\rm E}$$J_{\scriptscriptstyle\rm
E}$$H_{\scriptscriptstyle\rm E}$ bands, shown in Fig. 1. The NISP photometric
system is described in detail in Euclid Collaboration: Schirmer et al. (2022).
The passbands were computed as a function of position in the NISP field of
view, based on local filter passband measurements and full ray tracing of the
Euclid NISP optical system to account for angle-of-incidence variations on the
filter surface, reaching an accuracy of 0.8 nm. The study focuses on the
determination of cut-on and cut-off wavelengths, but does not addresses other
variations of the transmission. Polynomial expressions have been provided to
compute the cut-on and cut-off wavelengths at any position (see figure 8 in
Euclid Collaboration: Schirmer et al. 2022). These measurements showed the
existence of a blue shift that depends on the off-axis distance, which can be
explained by the different incident angles of the incoming light. Blue shifts
between 2.5 nm and 6.1 nm have been observed for the cut-on or cut-off
wavelengths of the three filters, with the $Y_{\scriptscriptstyle\rm E}$ and
$H_{\scriptscriptstyle\rm E}$ filters being the least and the most affected,
respectively (see figure 9 in Euclid Collaboration: Schirmer et al. 2022).

##  4 Consequences of passband variations

###  4.1 Effect on the photometry

Photometric observations require a calibration in order to derive, for each
observation frame, the so-called zero-point, i.e. the relation between the
physical flux (or magnitude) and the count rate on the detector. This is
generally achieved by using reference stars located in the field of view. The
AB definition from Eq. (1) provides an exact relation only for sources that
have the same SED as the calibration stars. In order to cope with galaxies
with different SEDs, the calibration is sometimes refined with the addition of
a colour term (e.g., Padmanabhan et al. 2008; de Jong et al. 2015), which is a
correction based on the ratio of the counts in adjacent photometric bands,
providing a coarse approximation for the true shape of $s(\lambda)$. This
method calibrates all bands simultaneously, which can be done only for the
bands that are observed with the same telescope. It is therefore not well
adapted to the Euclid survey, which will use a combination of telescopes, and
thus colour-term correction is not used.

With regard to passband variations, the fact that a unique zero-point
correction is computed for each frame implies that any spatial variations of
the passband across the field-of-view are ignored. The calibration is
therefore formally valid only for some average passband. In the case of
Euclid, a second step is performed which corrects for any spatially dependent
systematic deviations of the reconstructed fluxes. This step makes the fluxes
independent of possible changes in the normalisation of the passband, provided
they only depend on the location in the field of view. In the case of passband
variations, this correction would remove any effect linked to a change in the
effective area of the transmission. Atmospheric absorption also results in
passband variations; however, they can be considered as spatially uniform over
a single observation. Hence the calibration process will absorb this effect in
the zero point, although in reality the correction depends on the colour of
the object. For extended objects, the passband could in principle change
across the object, which would affect photometric extraction in a complicated
way; however, in cosmological applications useful galaxies are very small
compared to the scale on which passband variations are measured, so we can
safely ignore this effect.

As a result of the Euclid photometric calibration, variations that are not
limited to a change in normalisation do impact the photometric measurements.
In their very detailed analysis of the photometric stability of the SDSS
camera, Doi et al. (2010) found that column-to-column variations of the
passbands induce errors on the $g$, $r$, $i$ and $z$ fluxes of up to 1% (0.01
mag). Based on the ‘Scientific Challenge 8’ simulations of the Euclid
performance, Euclid Collaboration: Schirmer et al. (2022) found that the
effect is of the order of a few millimags in the Euclid near-infrared bands.
Such bias would impact the performance of photometric-redshift determination.
In order to remove this bias, it would be necessary to include the correct
transmission curve $T(\lambda)$ in Eq. (1); however, the bias would also
depend on the a priori unknown SED of the object, so that the correction
cannot be performed on isolated frames.

###  4.2 Effects on photometric-redshift determination

####  4.2.1 First-order effect

We build here a toy model to allow us to estimate the amplitude of the effect
of passband variations on the photometric redshift using a simplistic SED
consisting of a step function (when expressed per wavelength) at the
wavelength of the Balmer break, which we set here to be exactly 400 nm. We
consider a system of three top-hat transmission curves $UGR$, with
$U(\lambda)=1$ if 300 nm $<\\!\lambda\\!<$ 400 nm, $G(\lambda)=1$ if 400 nm
$<\\!\\!\lambda\\!\\!<$ 500 nm, and $R(\lambda)=1$ if 500 nm
$<\\!\\!\lambda\\!\\!<$ 600 nm, all transmission curves being 0 outside of
these ranges. From Eq. (1), we find that the fluxes $f_{U}$ and $f_{R}$ are
constant if we consider only redshifts $z<0.25$, since the Balmer break
remains within the $G$ band. As the redshifted Balmer breaks moves across the
$G$ passband, $f_{G}$ changes as a function of $z$ according to Eq. (1) (see
Fig. 5), so that there is a direct relationship between $f_{G}$ and the
redshift. Because of the $\lambda$ term in the numerator of Eq. (1), the
relationship is not exactly linear.

Figure 5: Flux in the $G$ passband as a function of redshift (blue solid
line), assuming $f_{U}=1$ and $f_{R}=2$ in arbitrary units per Hz. The red
dashed line shows a linear fit to the relation.

We consider a varying $G$ passband where the only possible variation is a
shift in the mean wavelength of the passband by an amount of $\delta$, i.e.
$G(\lambda)=1$ if 400 nm $<\\!\lambda-\delta\\!<$ 500 nm, and
$G(\lambda)\\!=\\!0$ otherwise. Obviously, the mean $\mu$ of $G$ is shifted by
the same amount $\Delta\mu=\delta$. As discussed in Sect. 3, typical values of
$\delta$ can be of the order of a few nanometres. Figure 6 shows the bias that
results from a shift of $\Delta\mu$ of the passband for a true redshift
$z=0.15$.

Figure 6: Bias $\Delta z$ resulting from a shift of $\Delta\mu$ in the $G$
passband for $z=0.15$ (solid blue line). The dashed red line is the
‘theoretical’ bias $\Delta\mu/400$ nm, where $\Delta\mu$ is the error on the
location of the Balmer break. The grey area shows the domain of variations of
the mean wavelengths of the measured MegaCam $r$ passbands (from 0 to 8 nm).

From Fig. 6 we found that the bias $\Delta z$ is almost a linear function of
$\mu$, with $\Delta z=-0.0033\Delta\mu/1$ nm. This is close, but not
identical, to the expected bias if one is able to locate the Balmer break with
an error of $\Delta\mu$. In such a case, we would have: $\Delta
z=\Delta\mu/400$ nm${=-0.0025\Delta\mu/1}$ nm. The larger slope is due to the
stronger weight of long wavelengths in Eq. (1). This very simplistic analysis
shows nevertheless that a shift of 1 nm leads to a bias that is of similar
amplitude to Euclid’s requirement on the bias, $\Delta z/(1+z)=0.002$, which
shows that changes in passband variations need to be taken into account in the
computation of photometric redshifts. In Fig. 6, we also showed the range of
$\Delta\mu$ observed in Sect.3. We found that, because of the tendency of off-
axis transmissions to move towards the blue, photometric redshifts are
expected to be biased positively if the central passbands are used.

####  4.2.2 Consequences for photometric-redshift algorithms

Passband variations add a major complexity in the process, because potentially
each source is observed with a (slightly) different set of passbands, which
means that each source lives in a different colour space (which we assume is
known through the measurements of its actual passband). Therefore each source
requires a different mapping from colour space to redshift. The two main
approaches to photometric-redshift determination face significant, but
distinct, issues when dealing with multiple colour spaces.

##### Template-fitting

The TF approach to determine photometric redshifts (e.g., Arnouts et al. 1999;
Bolzonella et al. 2000, ; Paltani et al. in preparation) involves the
knowledge of galaxy SEDs, the so-called templates, which are assumed to be
known at all relevant rest-frame wavelengths. The source fluxes are compared
with reference fluxes that are computed by integrating the templates through
the passbands. Hence, it is straightforward to compute the reference fluxes in
the colour space of the source of interest. However, the calculation of the
reference fluxes implies integrations through passbands and becomes
computationally expensive for any reasonably large survey, and impossible for
catalogues of billions of sources such as the future Euclid Wide survey. In
the case where all sources occupy the same colour space, this is solved by
first calculating a single grid of fluxes for all models at all redshifts. If
each passband has $n$ variations, the model grid becomes $n$ times larger,
which could become quite cumbersome.

##### Machine-learning approaches

ML is a vast class of algorithms that share the basic principle to infer the
desired relation (in this case between the source’s fluxes and the redshift)
in a purely data-driven manner. ML approaches almost always involve a training
phase, where the algorithms build up internally (learn) the colour-redshift
relation. After the training phase, photometric redshifts can be very
efficiently computed through this newly determined relation. However, the
training phase of ML algorithms can be a quite computationally intensive
process, depending on the algorithm. This is not an issue in the case where
all the sources and reference objects occupy the same colour space, since it
needs to be performed only once. However, in the presence of passband
variations, all sources can in principle occupy distinct colour spaces, so
that the training phase would need to be performed many times, which may
become computationally difficult. Another difficulty lies in finding enough
training objects in each colour space, so that the colour-redshift relation
can be accurately learned. As a matter of fact, finding a reference sample
that covers entirely a single colour space of galaxies is already extremely
difficult (Masters et al. 2015).

##  5 Implementation in template-fitting algorithms

The TF algorithm is based on the assumption that the SEDs of the real objects
are drawn from a known set of SEDs, so that we are able to compute the
predicted model colours in any colour space. However, associating to every
source the full passband information, including the variations specific to
this source, is quite demanding in terms of data management. Furthermore, this
results in a much larger model grid size. We thus propose here a simplified
correction that only takes into account the shifts in the mean wavelengths of
the passbands, and we then validate our approach with simulations.

###  5.1 Template-fitting likelihood

TF algorithms compare the source fluxes with those obtained from simulated
objects with known parameters $\alpha$. These parameters are typically the set
of reference SEDs used to match the observed SED of the source, the redshift
of the source, the internal reddening law (e.g., Prevot et al. 1984, Calzetti
et al. 2000, etc.), and the value of internal reddening
$E_{B-V}^{\mathrm{int}}$. The likelihood of the match as a function of
$\alpha$ is given by $\exp{\left(-\chi^{2}_{\alpha}/2\right)}$, with

$\chi^{2}_{\alpha}=\sum_{T\in{\cal
T}}\frac{\left(f^{\mathrm{s}}_{T}-ar^{\alpha}_{T}\right)^{2}}{{\sigma^{\mathrm{s}}_{T}}^{2}},$
(4)

where the sum runs over all passbands ${T\in{\cal T}}$, $f^{\mathrm{s}}_{T}$
is the source flux through passband $T$, $r^{\alpha}_{T}$ is the reference
flux of the simulated object with parameters $\alpha$ obtained from Eq. (1),
and $\sigma^{\mathrm{s}}_{T}$ are the uncertainties of the source fluxes in
passband $T$. Finally, $a$ is a scale factor that is left free to minimise
$\chi^{2}_{\alpha}$ in Eq. (4); alternatively, $a$ can be included in
$\alpha$, so that it can be marginalised upon or its posterior can be obtained
(this is especially useful for the determination of physical parameters; see,
e.g., Phosphoros333Available using Anaconda; see
https://anaconda.org/astrorama/phosphoros; Paltani et al., in preparation).

###  5.2 Correction factor for passband variations

The likelihood in Eq. (4) does not take into account the possibility of filter
variations. If the source flux is measured through a variation $T^{\prime}$ of
passband $T$, and the simulated flux with parameters $\alpha$ is computed
using passband $T$, we introduce a correction factor $C^{\alpha}_{T\rightarrow
T^{\prime}}$ to be applied to $r^{\alpha}_{T}$ in order to obtain
$r^{\alpha}_{T^{\prime}}$ to be used in Eq. (4) as

$r^{\alpha}_{T^{\prime}}=C^{\alpha}_{T\rightarrow
T^{\prime}}\,r^{\alpha}_{T}\;,$ (5)

so that we get a new equation for $\chi^{2}_{\alpha}$:

$\chi^{2}_{\alpha}=\sum_{T\in{\cal
T}}\frac{\left(f^{\mathrm{s}}_{T^{\prime}}-a\,C^{\alpha}_{T\rightarrow
T^{\prime}}\,r^{\alpha}_{T}\right)^{2}}{{\sigma^{\mathrm{s}}_{T^{\prime}}}^{2}}\;.$
(6)

Since we know the SED $s^{\alpha}(\lambda)$ and the passbands $T$ and
$T^{\prime}$, we can use Eq. (1) to determine the correction factor
$C^{\alpha}_{T\rightarrow T^{\prime}}$ :

$C^{\alpha}_{T\rightarrow
T^{\prime}}=\frac{r^{\alpha}_{T^{\prime}}}{r^{\alpha}_{T}}=\frac{\int_{0}^{\infty}s^{\alpha}(\lambda)\,T^{\prime}(\lambda)\,\lambda\,\mathrm{d}\lambda}{\int_{0}^{\infty}s^{\alpha}(\lambda)\,T(\lambda)\,\lambda\,\mathrm{d}\lambda}\leavevmode\nobreak\
\frac{\int_{0}^{\infty}T(\lambda)\,\frac{\mathrm{d}\lambda}{\lambda}}{\int_{0}^{\infty}T^{\prime}(\lambda)\,\frac{\mathrm{d}\lambda}{\lambda}}\;.$
(7)

The assumption that the main parameter affecting the bias is the mean
wavelength of the passband allows us to propose a very important
simplification. With $\Delta\lambda$ being the difference of mean wavelength
between $T$ and $T^{\prime}$,

$\Delta\lambda=\int_{\lambda}\left[T^{\prime}(\lambda)-T(\lambda)\right]\,\lambda\,\mathrm{d}\lambda\;,$
(8)

the correction factor can be expressed as a function of $\Delta\lambda$, that
is, $C^{\alpha}_{T\rightarrow T^{\prime}}=C^{\alpha}_{T}(\Delta\lambda)$. We
can thus compute $C^{\alpha}_{T}(\Delta\lambda)$ for different wavelength
shifts of the passband $T$ using Eq. (7). We note that $\Delta\lambda$ is
simply the difference between the mean wavelengths of the two passbands, and,
crucially, does not depend on the SED. In the case of multiple exposures with
different variations of the passband, the resulting $\Delta\lambda$ is defined
as the exposure time-weighted average of the individual $\Delta\lambda_{i}$ of
the different exposures. We point out that this approach is very similar to
the correction of the Galactic extinction using the full knowledge of the SED
as developed by Galametz et al. (2017) (see Appendix B.1 for more details),
which is implemented in Phosphoros (Paltani. et al., in preparation).

Appendix A describes in detail how $C^{\alpha}_{T}(\Delta\lambda)$ can be
approximated with an analytical function of $\Delta\lambda$. We found that we
get excellent approximations of $C^{\alpha}_{T}(\Delta\lambda)$ for all sets
of parameters $\alpha$, all redshifts, and all four CWW templates using a
second-order polynomial with the constant term fixed to 1:

$C^{\alpha}_{T}(\Delta\lambda)\approx
a^{\alpha}_{T}\Delta\lambda^{2}+b^{\alpha}_{T}\Delta\lambda+1\;.$ (9)

Since $a^{\alpha}_{T}$ and $b^{\alpha}_{T}$ depend only on the passband $T$
and on the set of parameters $\alpha$, we can precompute grids of these
parameters.

###  5.3 Simulations

In order to validate our approach, we performed idealistic, noiseless
simulations to estimate the bias resulting from passband variations. We used
the four CWW templates (Elliptical, Sab, Sbc, Irregular; Coleman et al. 1980)
in order to estimate the bias over a range of galaxy types. We simulated
MegaCam $ugriz$ and Euclid $Y_{\scriptscriptstyle\rm
E}$$J_{\scriptscriptstyle\rm E}$$H_{\scriptscriptstyle\rm E}$ photometry of
objects modelled with the CWW templates ignoring internal reddening and
photometric uncertainty over the redshift range 0–3. We chose MegaCam because
of its rather large passband variations; in addition, MegaCam $u$ and $r$
passbands will be used for the northern part of the Euclid wide survey (Euclid
Collaboration: Scaramella et al. 2022). Objects were simulated in each band
with randomly chosen instances of its ten possible MegaCam variants (see Sect.
3.3). For the Euclid NISP passbands, we created ten arbitrary passbands by
shifting the nominal passband by 0 to 8 nm. We then determined the redshift
using TF using only the central passbands, ignoring passband variations. In
absence of passband variations, this setup would produce perfect photometric
redshifts, without any uncertainty, nor bias, so that any uncertainty or bias
is entirely due to the mismatch between the passbands used to determine the
source and reference fluxes, respectively. We performed three different tests
of increasing complexity: firstly, only the $r$ passband can vary; secondly
all passbands can vary independently; and finally fluxes are measured using a
stack of four exposures, each of them having random sets of passband
variations.

Figure 7 shows the resulting bias in the three configurations. When the
central passbands are used to determine the photometric redshifts, some bias
is clearly present at a level that is of the same order as the accuracy
requirement for Euclid in all three configurations. When only the $r$ passband
is randomised, the bias is completely concentrated in a specific redshift
range, which matches quite well that where the Balmer break, which is present
to different extents in all four CWW templates, falls into the $r$ passband.
When all passbands vary, the bias is present at all redshifts and reaches
about $0.007(1+z)$ in the worst case. Considering an average passband shift of
5 nm, in our toy model the bias reaches about $0.011(1+z)$ at $z=0.15$, which
is about 50% larger than the bias we found in the simulations. This difference
was expected because the features in the CWW templates are not as sharp as the
step function we used in our toy model; the fact that we are at redshift
$\sim\\!0.5$ instead of 0.15 further smooths the transition. When multiple
exposures were stacked, the bias was practically identical to that in the
single-exposure case because using four exposures makes the effective
transmission less variable, without changing its average.

Figure 8 shows plots similar to Fig. 7, but for the dispersion. We see again
that using wrong passbands has an effect on the quality of the photometric-
redshift predictions. This effect, which reaches at most $0.005(1+z)$ is
however quite small compared to the Euclid requirement on the dispersion
[$\sigma_{z}=0.05(1+z)$] in all configurations. We note that, in the case of
four exposures, the dispersion is a factor 2 lower, as expected from the
averaging of four exposures.

Figure 7: Bias in the photometric redshifts due to passband variations as a
function of redshift in the MegaCam $ugriz$ \+ Euclid
$Y_{\scriptscriptstyle\rm E}$$J_{\scriptscriptstyle\rm
E}$$H_{\scriptscriptstyle\rm E}$ configuration. Each plot shows the bias for
the four CWW templates, as indicated. _Left_ : only the $r$ passband is
variable; _centre_ : all passbands are variable; _right_ : all passbands are
variable, and four exposures have been stacked. In the left plot, the vertical
dashed lines indicate the redshifts where the Balmer break enters and exits
the $r$ passband. The different curves show the bias when the passband
variations are ignored (black line), or when the full central passbands
corrected for the shifts in mean wavelengths (blue line) are used. The dashed
red lines show the bias when the second-order polynomial correction on the
flux presented in Sect. 5.2 is used. We point out that the blue and red lines
are very close, so that the blue line is often barely visible.

Figure 8: Dispersion in the photometric redshifts due to passband variations
as a function of redshift in the MegaCam $ugriz$ \+ Euclid
$Y_{\scriptscriptstyle\rm E}$$J_{\scriptscriptstyle\rm
E}$$H_{\scriptscriptstyle\rm E}$ configuration. Each plot shows the dispersion
for the four CWW templates, as indicated. _Left_ : only the $r$ passband is
variable; _centre_ : all passbands are variable; _right_ : all passbands are
variable, and four exposures have been stacked. The different curves show the
bias when the passband variations are ignored (black line), or when the full
central passbands corrected for the shifts in mean wavelengths (blue line) are
used. The dashed red lines show the bias when the second-order polynomial
correction on the flux presented in Sect. 5.2 is used. We point out that the
blue and red lines are very close, so that the blue line is barely visible.

When we applied the $C^{\alpha}_{T}(\Delta\lambda)$ correction factors, we
found that the bias was significantly reduced, such that it always remained
within the requirements (see Fig. 7). It is even in general below
$0.0005(1+z)$, except in the case of the ‘Irregular’ SED, where a peak at
about $0.0015(1+z)$ remains. This is probably due to the presence of sharp
features, such as strong emission lines, in this SED. In Fig. 8 we see that
using only the shifts in mean wavelengths was also able to reduce the
dispersion to a large extent, except in the case of the ‘Irregular’ SED, where
some residual dispersion remains. This is again probably an effect of the
presence of sharp features, such as strong emission lines, in this SED. As a
conclusion, for realistically varying passbands, the knowledge of the full
passbands for each objects is not necessary; it is sufficient to know the mean
wavelengths of the passbands, which is a quantity that is much easier to
handle by photometric-redshift algorithms, and in particular TF. The bias and
dispersion found when using the second-order polynomial approximation of
$C^{\alpha}_{T}(\Delta\lambda)$ were extremely close to those involving the
full $C^{\alpha}_{T}(\Delta\lambda)$, demonstrating that the second-order
polynomial approximation provides a very good representation of the correction
factors.

##  6 Application to real data

We verified the capability of the method developed here to reduce the bias in
the photometric-redshift determination by applying it to real data. We used
the seventh (final) data release of the Legacy Survey performed at the Canada-
France-Hawaii Telescope
(CFHTLS444http://terapix.calet.org/terapix.iap.fr/rubrique5c64.html?id_rubrique=268).
We used only the W1 wide-field, which we matched with the VIPERS Public Data
Release 2 (VIPERS-PDR2; Scodeggio et al. 2018) spectroscopic-redshift
catalogue obtained with the VIMOS multi-object spectrograph on the ESO-VLT;
VIPERS is colour-selected to include mostly sources at redshifts $0.5<z<1.2$
(Garilli et al. 2014). The CFHTLS catalogue contains the MegaCam versions of
the usual $ugriz$ passbands, with some objects being observed with a different
$i$ passband denoted $y$555This $y$ band mostly overlaps the $i$ band, and
should not be confused with the usual $y$ or Euclid $Y_{\scriptscriptstyle\rm
E}$ bands, which lie at the limit between optical and near-infrared.. Since we
were mostly interested in the bias, we selected sources brighter than
$r=21.5$, in order to remove as much as possible the statistical uncertainties
from the photometric-redshift determinations. From the VIPERS-PDR2 catalogue,
we only kept very secure objects with flags either 3.5 or 4.5, excluding stars
(identified with a spectroscopic redshift of 0). The match between the two
catalogues resulted in 4915 objects. With the cut at bright magnitudes, we
found that most of the sources are found in the redshift range $0.4<z<0.7$.

The calibrated data we were using do not contain any information regarding the
atmospheric effects on the passband; consequently the only effect that we
could take into account are the spatial variations of the passbands. As
discussed in Sect. 3.3, the transmission curves of the MegaCam instrument have
been measured at ten different off-axis distances (Betoule et al. 2013). Using
the pixel scale and pixel size of MegaCam, we could convert these physical
distances into off-axis angles. The pointing strategy of CFHTLS is such that a
given object is observed with the same off-axis angles in all passbands, which
maximises the effect of passband variations. The shifts in the mean
wavelengths of the passbands as a function of off-axis angle are shown in Fig.
9. The off-axis dependence is present in all passbands, but with quite
different amplitudes. We note however that the relations are more complex than
expected from purely geometric considerations, which means that the shift is
spatially dependent in a more complex way that what we can model here.

Figure 9: Mean wavelength shifts of the six MegaCam transmissions as a
function of the off-axis angle compared to the on-axis transmissions. The $y$
transmission has been measured at slightly different off-axis distances from
the other transmissions.

We first computed the photometric redshifts of the 4915 sources with a fully
standard TF approach using Phosphoros. We used the 31 COSMOS templates used in
Ilbert et al. (2009) and applied internal reddening up to
$E_{B-V}^{\mathrm{int}}=0.5$ on the templates of spiral and starburst galaxies
only using either the SMC extinction curve from Prevot et al. (1984), or the
Calzetti et al. (2000) extinction law for star-forming galaxies, with the
addition of a bump at 2175 Å introduced by Massarotti et al. (2001). Standard
emission lines in Phosphoros have been added to the COSMOS templates based on
the Kennicutt relation and the line flux ratios observed in sources in the
SDSS-III/Baryonic Oscillation Spectroscopic Survey (Kennicutt 1998; Thomas et
al. 2013, ; see Paltani et al., in preparation, for details). We do not
apply any refinement in the algorithm, such as a luminosity prior, brightness
prior, or zero-point correction, because we focussed on the determination of
the bias, and not on the production of the best possible catalogue. Figure 10
shows the overall quality of the photometric redshifts. While the details of
the performance are not important, we obtained very good predictions even with
this limited analysis, with a dispersion (measured with the normalised median
absolute deviation, NMAD) of 0.038 and an outlier fraction of 2.9%. The bias
appears significant, with the region with the highest density of sources lying
$\Delta z\sim 0.04$ above the 1:1 relation.

Figure 10: Photometric-redshift predictions for the W1 CFHTLS catalogue of
bright galaxies with secure redshifts presented as a density plot in the
photometric-redshift–spectroscopic-redshift plane. The grey solid line is the
1:1 line. The red lines show the limits $0.85(1+z)$ and $1.15(1+z)$,
respectively, traditionally separating good predictions from outliers.

Using the off-axis angles of each source, we obtained wavelength shifts for
each passband by interpolating the relations shown in Fig. 9. We used then
Phosphoros in the exact same configuration as above, but this time taking into
account the correction for wavelength shifts presented in Sect. 5.2. We
obtained a normalised median absolute deviation NMAD of 0.039 and an outlier
fraction of 3.2%. Both values are very close to, but slightly worse than,
those obtained without the application of the correction for the mean
wavelength shifts. The additional noise could result from the too simplistic
assumption we made here that the wavelength shifts only depend on the off-axis
angle. Figure 11 compares the densest part of the photometric-redshift
prediction plots with and without mean wavelength shifts. The bulk of the
sources very clearly moved towards the 1:1 relationship when the shifts are
applied, indicating a reduction in the bias, although some significant bias
remains.

Figure 11: Comparison of the photometric-redshift predictions presented as
density plots in the photometric-redshift–spectroscopic-redshift plane without
(left) and with (right) mean wavelength corrections. The grey solid line is
the 1:1 line. The red lines show the limits $0.85(1+z)$ and $1.15(1+z)$,
respectively.

We quantified more precisely the reduction in the bias by computing the median
difference in redshift between the photometric-redshift predictions and the
true redshifts, in bins of redshift. Figure 12 compares the bias in the
predictions from the original catalogue and from our computations involving
the mean wavelength shifts over the redshift range $0.45<z<0.65$, where the
bias can be reliably estimated. From Fig. 12, we found that the bias, which is
positive everywhere, was reduced over this redshift interval by 0.018 in
average. As expected from the field-of-view dependence of the transmissions,
passband variations induce a positive bias on the photometric redshifts. This
reduction matches very well the expected bias due to passband variations for
an average shift of the order of 5 nm, which is the value expected in the case
of the MegaCam passbands. However, at redshift $z=0.48$, where the reduction
reaches a maximum, the residual bias is about 0.016 and still exceeds the
Euclid requirement of $\Delta z/(1+z)<0.002$ by a factor about 8 (Laureijs et
al. 2011).

Figure 12: Median bias in photometric-redshift predictions as a function of
redshift for the original catalogue (black line) and when taking into account
the correction in the mean passband wavelengths (red line). The dashed blue
line shows the reduction in the bias between the two solid curves. The grey
area indicates the Euclid requirement. While it is clear that the passband
correction reduces the bias significantly, there is still a need for a post-
processing calibration step in order to meet the Euclid requirements.

##  7 Discussion

###  7.1 Origin of the residual bias in photometric-redshift algorithms

The method we proposed here is able to remove a significant fraction of the
bias due to the variations of the passbands. We designed the method to cope
with spatial variations of the passbands, although in principle the same
approach could be used for, for instance, passband variations due to
atmospheric effects. In practice, this might be made difficult by the
calibration process, which performs a colour-independent correction. In
Appendix B we describe other processes that can lead to (at least apparent)
passband variations. However, the residual bias is still larger than the
requirements by a large factor. This is due to the fact that there are other
sources of bias that are not due to the changes in the passbands. This could
be due to other issues in the photometry. Any defect in the calibration
process (for instance a wrong zero-point) will lead to biased determinations
of the photometric redshift. In the latter case, a zero-point correction
(Ilbert et al. 2006) can be introduced in the TF algorithms, and has been
proven to be quite effective to reduce the bias. We did not use this
correction here on purpose, since we wanted to focus on the effect of passband
shifts. Other, in particular non-linear, issues in the photometry may not be
easily removed. One example is the imperfect PSF homogenisation that needs to
be applied to the different photometric bands.

Another source of bias is the mismatch between the SED models and reality. For
instance, SEDs are too few, or they do not match accurately enough the SEDs of
true galaxies. To alleviate these issues, some photometric-redshift codes
include linear combination of templates (EAZY; Brammer et al. 2008), or SED
templates can be adapted to match better with the observed colours (Coupon et
al. 2009). We point out that the latter method may also be able to remove some
bias inherent to the photometry. Other model errors that cannot be corrected
by these methods may contribute to the bias, such as, for instance, imprecise
estimate of the Galactic reddening, incorrect Galactic-reddening attenuation
curve, inaccurate intrinsic reddening laws, or wrong assumptions on the IGM
absorption.

Even if the models were correct, the priors could be incorrect, leading to
biased estimates. In algorithms implementing Bayesian statistics the choice
(or not) to apply a specific prior (e.g., luminosity function or similar, or
the colour-space coverage of the SED models) leads to biases. Algorithms using
maximum-likelihood are even more subject to bias, since the maximum-likelihood
solution of an optimisation problem is biased when the error distribution is
not symmetric. Even if one does not consciously use Bayesian statistics, some
choices act as priors and affect the determination of the photometric
redshifts, such as, for example, the choice of a maximum value for the
internal reddening or for a galaxy luminosity.

ML algorithms have in principle significant advantages over TF algorithms with
respect to the bias, as they can learn the defects in the photometry, although
we argued in Sect. 4.2.2 that passband shifts are difficult to take into
account. They also do not rely on our imperfect knowledge of the Universe.
However, the quality of the ML model depends a lot, and in a very complicated
way, on how well the training sample matches the target data set. In
particular the requirement to get a spectroscopic redshift biases the training
samples towards bright emission-line galaxies (see Hartley et al. 2020, for an
in-depth discussion). Thus ML algorithms are also affected by model
imperfections and incorrect priors, which are hard to determine objectively.

Ultimately, a bias correction remains necessary, since not all sources of bias
can be identified precisely enough to be corrected. Different approaches have
been proposed to this end. In the case of Euclid, a direct calibration in
colour-space based on the approach developed in Masters et al. (2015) will be
implemented. Such calibration at the level of the Euclid requirements remains
nevertheless extremely challenging (Wright et al. 2020), so that any well-
understood source of bias, such as the passband variation, should be removed
beforehand as far as possible.

###  7.2 Comparison with the colour-term approach

Using an approach similar to the colour-term calibration (see Sect. 4.1),
Betoule et al. (2013) implemented a correction for the effect of passband
variations based on a colour term. Using their notation (see their equation
6), the corrected magnitude $m_{|x_{0}}$ of a source with magnitude $m_{|x}$
is given by

$m_{|x_{0}}\,\approx\,m_{|x}\,-\,\delta k(x)\,(c_{|x_{0}}-c^{0}(x)),$ (10)

where $x_{0}$ refers to the centre of the field of view, $x$ is the location
of the source in the field of view, $\delta k(x)$ is the position-dependent
colour term, $c_{|x0}$ is the colour of the object, and ${c^{0}(x)}$ is the
reference colour for the affine transformation; we ignore here the flat-field
and zero-point corrections, as they are not relevant for our discussion.
Betoule et al. (2013) chose $c_{|x0}=(g-i)_{|x_{0}}$, which is a good
temperature indicator for stars. In the limit of small passband variations,
$\delta k(x)$ is small, which alleviates the need to obtain accurate colours;
consequently, Betoule et al. (2013) ignore the effect of passband variations
in the determination of $(g-i)_{|x_{0}}$.

We can compare the above equation with the correction in Eq. (5). Both
approaches use an estimate of the SED of the source and some position-
dependent factor to determine corrected flux or magnitude. A noticeable
difference is that our correction is expressed as a function of a property,
the mean wavelength shift, that is independent of the intrinsic properties of
the object, contrarily to the observed colour. The colour-term correction is
also based on a very crude approximation of the SED of the source based on a
single colour, which may be unsuitable for the estimation of the SED for
passbands that are far from the $g$ and $i$ passbands, such as the Euclid
$Y_{\scriptscriptstyle\rm E}$$J_{\scriptscriptstyle\rm
E}$$H_{\scriptscriptstyle\rm E}$ near-infrared passbands. By contrast, in our
method the SED is determined over all passbands based on all the available
photometry simultaneously using empirically or physically motivated templates.
Eq. (10) also introduces an unwanted correlation with the $g$ and $i$ bands.

One additional advantage of our approach is that the SED is not needed during
the photometric extraction and calibration stages, which makes it easier to
combine photometric catalogues from different surveys. In the case of the
Euclid survey of the southern hemisphere, the Euclid near-infrared photometry
will be complemented with the DES survey (and with the Rubin LSST later on;
see Guy et al. 2022). The northern sky will be even more complicated, with the
optical survey consisting of observations from several telescopes (Euclid
Collaboration: Scaramella et al. 2022): the CFHT (Canada-France Imaging
Survey; Ibata et al. 2017) for the $u$ and $r$ bands; Subaru with the Hyper
Suprime-Cam (Miyazaki et al. 2018) for the $g$ and $z$ band; and Pan-STARRS
(Chambers et al. 2016) for the $i$ band. In order to constrain colour terms,
the photometric-calibration approach would require the simultaneous analysis
and calibration of all these photometric data, which can become impractical.
In addition, it is now commonly accepted that the team that conducted a survey
is able to provide the best calibration, so that a new scientific analysis
rarely starts from the raw data, but rather from calibrated stacks, if not
directly from the published photometric catalogues. Our approach can be
applied in a straightforward way to any catalogue assembled from distinct
catalogues while benefiting from the best possible calibration and passband
variation correction.

###  7.3 Photometry corrected for passband variations

The $C^{\alpha}_{T}(\Delta\lambda)$ correction factors can be computed only
with TF algorithms, because these are the only photometric-redshift algorithms
for which the SEDs of the reference objects are known in general. ML
algorithms require only a spectroscopic redshift. However, TF algorithms can
be used to create corrected photometry that can later be used for any purpose,
including the computation of photometric redshifts using any other algorithms,
simply by applying the $C^{\alpha}_{T}(\Delta\lambda)$ correction factor to
the photometric flux and uncertainty through passband $T$ for the best-fit
$\alpha$ parameters and the measured $\Delta\lambda$ wavelength shift. The
possibility to provide corrected photometric measurements has been implemented
in Phosphoros.

A more advanced implementation would involve the creation of posteriors for
the $C^{\alpha}_{T}(\Delta\lambda)$ correction factors. This would have the
advantage of relying less on a specific best-fit solution, and is probably
more robust. However, this makes the output more cumbersome to use, since the
flux and uncertainty are replaced by a full posterior distribution. A
convenient way to deal with these distributions, including the correlations
between the correction factors for different passbands, is to provide a
sampling of the $C^{\alpha}_{T}(\Delta\lambda)$ posteriors.

We note finally that relying on TF to determine the
$C^{\alpha}_{T}(\Delta\lambda)$ does not lead to degeneracies. Indeed, even if
two comparable solutions at very different redshifts exist, they would have by
definition similar spectral shapes over the range of wavelengths covered by
the passbands. The fact that the templates do not match exactly the SEDs of
real objects is not a serious problem either, especially if the full
posteriors of the correction factors are used, because what is most relevant
is the range of colours provided by the templates.

###  7.4 Passband variations in ML algorithms

It is not straightforward to apply passband variation corrections to ML
algorithms. The main advantage of these approaches is indeed that they do not
rely on the knowledge of any SED, which is necessary to compute the
photometric corrections. It is possible to determine the SEDs of the objects
in the training set with TF, making use of the fact that their true redshifts
are known. Passband variations can be taken into account using a procedure
where the SED and the photometric corrections are determined iteratively until
the procedure converges on consistent SED and corrections.

Once the SEDs of the training set objects are obtained, it is possible to
correct their photometric measurements to match the colour space of any source
in the target sample. However, as anticipated in Sect. 4.2.2, most ML
algorithms would require a potentially computationally expensive training
phase to match every single colour space of the target sample. One notable
exception is the $k$-nearest neighbours algorithm ($k$-NN; Cover & Hart 1967),
which does not require any training. The algorithm finds the $k$ closest
reference objects to a given source, and compute the value of interest using
some (weighted or not) average over theirs. NNPZ (Tanaka et al. 2018) is an
example of an implementation of the $k$-NN algorithm for the determination of
photometric redshifts. In such a case, it is straightforward apply the $k$-NN
by correcting first, for each target, the fluxes of the training set objects
based on the correction factors of the target.

The $k$-NN algorithm relies on a definition of the distance between two
objects. One possibility is to adopt a $\chi^{2}$ distance, with the
likelihood given by $\exp{\left(-\chi^{2}_{\alpha}/2\right)}$, where
$\chi^{2}_{\alpha}$ is given by Eq. (4), with the exception that $\alpha$ runs
over the training set, instead of a grid of models. In such a case, the
correction procedure and all equations from Sect. 5.2 can be used with minimal
and straightforward modifications.

##  8 Conclusion

We studied the effects of variations of photometric passbands in the process
of photometric-redshift determination. Biases due to passband variations are
expected to be positive due to the usual shift of transmissions towards
shorter wavelengths for off-axis sources. We found that, when taking into
account the SED of the objects, it is sufficient to consider only the change
in the mean wavelengths of the passbands in order to get a very accurate
determination of the correction to be applied to the observed flux. Crucially,
the mean wavelength can be determined irrespectively of the SED of the source.
This simplification allowed us to propose an accurate and efficient correction
that can be readily implemented in any TF algorithm, and has already been done
in Phosphoros.

While the method we proposed here is able to remove effectively some bias in
the photometric-redshift predictions, a significant bias remains. In the
application to the CFHTLS photometric catalogue, we found a reduction by a
factor 2 at $z=0.5$, although it remained a factor 6 higher than the Euclid
requirements. Remaining sources of bias were discussed. Bias may originate
from the photometry itself. It could also be due to the inadequacy of the
model; for instance, the templates might not represent accurately the SEDs of
galaxies, or they may lack diversity. But the bias could also be due to the
application of wrong priors, in particular hidden priors (Schmidt et al.
2020). These sources of bias could in principle be alleviated with deeper
understanding of the properties of galaxies, but this remains a very
challenging task. ML algorithms are immune to some photometric issues and to
model-dependent effects, but are particularly sensitive to the hidden priors
resulting from the construction of the training sample.

TF has different strategies to cope with biases inherent to the photometry
itself, such as corrections for offset in the zero-points (Ilbert et al.
2006), or template adaptations (Coupon et al. 2009). Each of these methods
should be used to remove the bias as much as possible. In many situations, ML
algorithms should be largely insensitive to such biases. Passband variations,
however, induce biases in the photometry that depend on the object, and taking
them into account in ML algorithms would require an extremely complex training
sample. Passband variations, if not corrected, can also imprint spurious
spatial patterns that can bias cosmological parameter estimation; this is
especially a concern for photometric galaxy clustering.

The method proposed here is able to remove most of the bias due to time and
spatial variations of the passbands. While this is only one of the many biases
affecting photometric-redshift determination, it is important to perform these
corrections, because the cosmological requirements on the accuracy of
photometric redshifts are so stringent for a mission such as Euclid that a
post-processing calibration step is required; starting this step with the
smallest possible bias is necessary if we want this calibration step to
succeed. As a bonus, our method is able to provide corrected photometry, so
that it can be used for other applications, for instance, the computation of
new photometric-redshift predictions based on any other algorithm, in
particular using ML.

###### Acknowledgements.

The Euclid Consortium acknowledges the European Space Agency and a number of
agencies and institutes that have supported the development of Euclid, in
particular the Academy of Finland, the Agenzia Spaziale Italiana, the Belgian
Science Policy, the Canadian Euclid Consortium, the French Centre National
d’Etudes Spatiales, the Deutsches Zentrum für Luft- und Raumfahrt, the Danish
Space Research Institute, the Fundação para a Ciência e a Tecnologia, the
Ministerio de Ciencia e Innovación, the National Aeronautics and Space
Administration, the National Astronomical Observatory of Japan, the
Netherlandse Onderzoekschool Voor Astronomie, the Norwegian Space Agency, the
Romanian Space Agency, the State Secretariat for Education, Research and
Innovation (SERI) at the Swiss Space Office (SSO), and the United Kingdom
Space Agency. A complete and detailed list is available on the Euclid web site
(http://www.euclid-ec.org).
Based on observations obtained with MegaPrime/MegaCam, a joint project of CFHT
and CEA/IRFU, at the Canada-France-Hawaii Telescope (CFHT) which is operated
by the National Research Council (NRC) of Canada, the Institut National des
Science de l’Univers of the Centre National de la Recherche Scientifique
(CNRS) of France, and the University of Hawaii. This work is based in part on
data products produced at Terapix available at the Canadian Astronomy Data
Centre as part of the Canada-France-Hawaii Telescope Legacy Survey, a
collaborative project of NRC and CNRS.

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##  Appendix A Computation of the correction functions

We describe here how the correction factors $C^{\alpha}_{T}(\Delta\lambda)$
can be computed for the transmission curve $T(\lambda)$, in order to correct
the fluxes in Eq. (6), with $\alpha$ being an index that runs through all the
model parameters. In the case of real reference objects with known SEDs,
$\alpha$ is just a integer sequence. In the case of TF, $\alpha$ runs through
all points in the computed grid, that is,
$\alpha\in\\{z_{i},\mathit{SED}_{j},{E_{B-V}^{\mathrm{int}}}_{k}\\}$, where
$i$, $j$ an $k$ are integer sequences that enumerate all redshifts, all SEDs,
and all internal reddening values, respectively, in the grid.

For each passband and each $\alpha$, we compute the fluxes
$F^{\alpha}_{T}(\Delta\lambda)$ over a sequence of $\Delta\lambda$ that
encompasses (but should not largely exceed) the range of shifts in the mean
wavelengths for the passband of interest. We use here 20 values between $-10$
and $+10$ nm. The correction is therefore:

$C^{\alpha}_{T}(\Delta\lambda)=F^{\alpha}_{T}(\Delta\lambda)/F^{\alpha}_{T}(0)\;.$
(11)

We then approximate $C^{\alpha}_{T}(\Delta\lambda)$ with a polynomial
expression. In general, we would like to minimise the number of parameters
used to represent $C^{\alpha}_{T}$, because the number of models $\alpha$
might be very large, and because high-degree polynomials might cause unwanted
fluctuations. Figure 13 shows a few examples of correction factors as a
function of $\Delta\lambda$ for different values of $T$ and $\alpha$.
Investigating the $C^{\alpha}_{T}(\Delta\lambda)$ functions, we find that a
second-degree polynomial always provides a very good approximation, requiring
three parameters for each $(T,\alpha)$. However, we have the constraint
$C^{\alpha}_{T}(0)=1$, so that there are effectively only two free parameters.
Thus we define

$\hat{C}^{\alpha}_{T}(\Delta\lambda)=\frac{C^{\alpha}_{T}(\Delta\lambda)-1}{\Delta\lambda}\;.$
(12)

We point out that $\hat{C}^{\alpha}_{T}(\Delta\lambda)$ is well defined for
$\Delta\lambda\neq 0$. We can then approximate
$\hat{C}^{\alpha}_{T}(\Delta\lambda)$ with a linear function:

$\hat{C}^{\alpha}_{T}(\Delta\lambda)\approx
a^{\alpha}_{T}\Delta\lambda+b^{\alpha}_{T}$ (13)

The coefficients $a^{\alpha}_{T}$ and $b^{\alpha}_{T}$ can be obtained using a
simple least-square minimisation. Finally, we have, using Eq. (12):

$C^{\alpha}_{T}(\Delta\lambda)\approx
a^{\alpha}_{T}\Delta\lambda^{2}+b^{\alpha}_{T}\Delta\lambda+1\;,$ (14)

The second-order approximation from Eq. (14) is shown on Fig. 13 for four
arbitrary values of the passband $T$ and of the model parameter $\alpha$.
Figure 7 shows that using the approximated correction Eq. (14) is practically
indistinguishable from computing the flux using the full shifted passband.

Figure 13: Correction factors $C^{\alpha}_{T}(\Delta\lambda)$ (blue points)
and the corresponding approximation with a second-degree polynomial (red line)
for four arbitrary values of the passband $T(\lambda)$ and of the coordinates
$\alpha$ in the model grid.

##  Appendix B Other effects leading to passband variations

Several other effects may affect the passband, and induce passband variations.
However, depending on the details of the effects, they may be treated
differently. We discuss three such effects below.

###  B.1 Galactic reddening

Dust in our Galaxy scatters the light emitted by extragalactic sources. The
induced, extrinsic, so-called Galactic reddening depends on the direction in
the sky, and its amplitude, parameterised by the value of the reddening
$E_{B-V}$, can be determined either from the observation of stars or
extragalactic sources with well known intrinsic SEDs, such as quasars or
passive galaxies (e.g., Schlegel et al. 1998; Schlafly et al. 2010; Schlafly &
Finkbeiner 2011; Mörtsell 2013), or from direct measurement of the quantity of
dust, for instance by combining Planck data with IRAS $100\,\mu$m data (Planck
Collaboration et al. 2014). Galactic reddening is also strongly wavelength-
dependent (hence the name), and affects the colours of the observed objects,
so that its wavelength dependence can be determined from observations of well-
characterised objects (Fitzpatrick 1999).

Photometry is often corrected for Galactic reddening by computing a colour
term that depends only on the amount of dust in the line of sight. However,
the (unknown) SED of the source needs to be taken into account in order to
compute an accurate correction. Galactic reddening could be applied to all
models in the grid, in order to derive, for all passbands, the shifts in the
mean wavelength that are induced considering the model and the reddening
$E_{B-V}$; however, when computing these shifts, the flux corrections are
obtained at the same time. Therefore, in Galametz et al. (2017) a more direct
approach has been adopted, where the flux corrections are derived directly,
without computing the wavelength shifts. Thus, while Galactic reddening could
be corrected by computing the wavelength shifts following the approaches we
used here, in Phosphoros the two corrections are computed separately. We point
out again that the work of Galametz et al. (2017) is very similar to the
approach described here, and has been the inspiration for the wavelength-shift
correction presented in this paper.

###  B.2 Absorption in the intergalactic medium

The intergalactic medium (IGM) is filled with hydrogen that very efficiently
absorbs the ultraviolet emission of objects at cosmological distances either
through Lyman bound-bound transitions, or through the Lyman bound-free
transition below 912 Å. Because of the clumpy distribution of matter in the
Universe, each line of sight is subject to a different amount of absorption.
Contrarily to the case of Galactic reddening, the amount of gas in the line of
sight cannot be measured, and the SED cannot be recovered exactly. However, an
average IGM transmission curve $G_{z}(\lambda_{0})$, which depends strongly on
the redshift $z$ and on the rest-frame wavelength $\lambda_{0}=\lambda/(1+z)$,
can be determined based on an analytic modelling of the distribution of gas
clumps (Madau 1995; Inoue et al. 2014), or on cosmological numerical
simulations (Meiksin 2006).

The IGM transmission curve $G_{z}(\lambda_{0})$ can be seen as modifying the
transmission curve $T(\lambda)$, such that the object is observed through an
effective curve $T^{\prime}(\lambda)=G_{z}(\lambda/(1+z))\,T(\lambda)$.
However, for TF algorithms, the IGM transmission depends only on the
parameters $\alpha$ of the model, so that it can be applied directly to the
model SED when computing the predicted fluxes. Therefore, there is no need
express the IGM absorption as an effect of passband variation.

###  B.3 Effect of the motion of the Earth

Light from extragalactic sources is affected by Doppler effects due to the
velocity $v$ of the observer. Several components contribute to $v$: the motion
of the observatory due to the Earth’s rotation, or, in the case of Euclid, due
to the motion of the spacecraft around the second Lagrangian point $L_{2}$;
the revolution of the Earth, or of $L_{2}$, around the Solar system
barycentre; and finally the peculiar motion of the Solar system barycentre
with respect to the Universe’s comoving frame. Velocities due to Earth’s
rotation and revolution can be very easily calculated (e.g., using the
barycorrpy Python package666https://pypi.org/project/barycorrpy/; Kanodia &
Wright 2018). Earth’s rotation induces maximum velocities in the direction of
the Earth’s equatorial plane below 1 $\mathrm{km}\text{\,}{\mathrm{s}}^{-1}$,
that is, $v/c\sim 10^{-6}$, which have negligible effect. Likewise, the motion
of Euclid around $L_{2}$ has velocities below 1
$\mathrm{km}\text{\,}{\mathrm{s}}^{-1}$. Earth’s revolution velocity is
however of the order of $30$ $\mathrm{km}\text{\,}{\mathrm{s}}^{-1}$, that is,
$v/c\sim 10^{-4}$, reaching a maximum along the ecliptic plane. Finally, the
peculiar motion of the Solar system barycentre can be obtained from the dipole
of the cosmic microwave background (CMB), and implies a peculiar velocity of
$371\pm 1$ $\mathrm{km}\text{\,}{\mathrm{s}}^{-1}$ towards Galactic
coordinates $(\ell,b)=($$\pm 0.15,$$\pm 0.15)$, or $v/c\sim 10^{-3}$ (Fixsen
et al. 1996); this direction is also in the ecliptic plane. Velocities must be
added as vectors, and only the resulting radial velocity matters (at these
velocities, transverse Doppler effect can be ignored). Even though the Euclid
survey will avoid the ecliptic plane because of the zodiacal light (Euclid
Collaboration: Scaramella et al. 2022), the avoidance angle of $None$ means
that velocities up to 98% of the maximal velocities of the Earth’s revolution
and Solar-system peculiar motion will occur.

The resulting radial velocities affect the photometric fluxes through Doppler
boosting, which is the combination of three effects: relativistic aberration,
time dilation, and blue- or redshifting of the spectrum. To first order, the
first two effects lead to an achromatic amplification $(1-3v/c)$, where a
positive velocity indicates a motion away from the source. The amplification
or attenuation can reach a level of a few $10^{-3}$, or a few millimagnitudes,
but it is removed by the calibration process, as it affects all sources in an
exposure in the same way.

The third effect results in a shift of the observed wavelengths by a term
$(1+v/c)$ (with the same definition of the sign of $v$ as above). This induces
approximately a translation of the passband of $(1+v/c)$ times the mean
wavelength of the passband. Considering the mean wavelength of the MegaCam $r$
filter is 630 nm, the maximum shift due to these velocities is about 0.8 nm,
which represents about 10% of the wavelength shift due to the off-axis
dependence of the filter, and is therefore non-negligible. Since the radial
velocity can be computed easily, the resulting wavelength shift can be applied
to the photometric-redshift determinations using the algorithm presented here.