MNRAS
496,
1941–1958 (2020)
doi:10.1093/mnras/staa1572
Advance Access publication 2020 June 5
The C-Band All-Sky Survey: total intensity point-source detection over
the northern sky
R. D. P. Grumitt
,
1
‹
Angela C. Taylor,
1
Luke Jew
,
1
Michael E. Jones
,
1
C. Dickinson
,
2
,
3
A. Barr,
2
R. Cepeda-Arroita
,
2
H. C. Chiang,
4
S. E. Harper,
2
H. M. Heilgendorff,
5
J. L. Jonas,
6
,
7
J. P. Leahy,
2
J. Leech,
1
T. J. Pearson
,
3
M. W. Peel
,
8
,
9
A. C. S. Readhead
3
and J. Sievers
4
1
Sub-department of Astrophysics, University of Oxford, Denys Wilkinson Building, Keble Road, Oxford OX1 3RH, UK
2
Jodrell Bank Centre for Astrophysics, Alan Turing Building, Department of Physics and Astronomy, School of Natural Sciences, The University of
Manchester, Oxford Road, Manchester M13 9PL, UK
3
Cahill Centre for Astronomy and Astrophysics, California Institute of Technology, Pasadena, CA 91125, USA
4
Department of Physics, McGill University, 3600 Rue University, Montr
́
eal QC H3A 2T8, Canada
5
Astrophysics & Cosmology Research Unit, School of Mathematics, Statistics & Computer Science, University of KwaZulu-Natal, Westville Campus, Pri
vate
Bag X54001, Durban 4000, South Africa
6
Department of Physics and Electronics, Rhodes University, Grahamstown 6139, South Africa
7
South African Radio Astronomy Observatory, 2 Fir Road, Observatory, Cape Town 7925, South Africa
8
Instituto de Astrof
́
ısica de Canarias, E-38205 La Laguna, Tenerife, Spain
9
Departamento de Astrof
́
ısica, Universidad de La Laguna (ULL), E-38206 La Laguna, Tenerife, Spain
Accepted 2020 May 29. Received 2020 May 28; in original form 2019 October 18
ABSTRACT
We present a point-source detection algorithm that employs the second-order Spherical
Mexican Hat wavelet filter (SMHW2), and use it on C-Band All-Sky Survey (C-BASS)
northern intensity data to produce a catalogue of point sources. This catalogue allows us
to cross-check the C-BASS flux-density scale against existing source surveys, and provides
the basis for a source mask that will be used in subsequent C-BASS and cosmic microwave
background (CMB) analyses. The SMHW2 allows us to filter the entire sky at once, avoiding
complications from edge effects arising when filtering small sky patches. The algorithm is
validated against a set of Monte Carlo simulations, consisting of diffuse emission, instrumental
noise, and various point-source populations. The simulated source populations are successfully
recovered. The SMHW2 detection algorithm is used to produce a 4
.
76 GHz northern sky
source catalogue in total intensity, containing 1784 sources and covering declinations
δ
≥
−
10
◦
. The C-BASS catalogue is matched with the Green Bank 6 cm (GB6) and Parkes-MIT-
NRAO (PMN) catalogues over their areas of common sky coverage. From this we estimate the
90 per cent completeness level to be approximately 610 mJy, with a corresponding reliability
of 98 per cent, when masking the brightest 30 per cent of the diffuse emission in the C-BASS
northern sky map. We find the C-BASS and GB6 flux-density scales to be consistent with one
another to within approximately 4 per cent.
Key words:
catalogues – surveys – methods: data analysis – cosmic background radiation –
radio continuum: general – cosmology: observations.
1 INTRODUCTION
C-Band All-Sky Survey (C-BASS) is an experiment to observe the
whole sky in total intensity and polarization at 4
.
76 GHz and at
45 arcmin angular resolution (Jones et al.
2018
). The primary pur-
pose of the experiment is to provide data on synchrotron emission
E-mail:
richard.grumitt@physics.ox.ac.uk
for cosmic microwave background (CMB) polarization experiments
at low frequencies compared to the peak of the CMB, whilst avoid-
ing significant de-polarization due to Faraday rotation. C-BASS will
also provide improved measurements of low-frequency emission
components, enabling a detailed study of the Galactic Magnetic
Field. The northern sky survey has now been completed, with
detailed analysis of these data currently underway. In this paper, we
present a 4
.
76 GHz northern sky point-source catalogue, produced
using a point-source detection algorithm employing the SMHW2.
C
2020 The Author(s)
Published by Oxford University Press on behalf of the Royal Astronomical Society
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1942
R. D. P. Grumitt et al.
Extragalactic radio sources are a key contaminant in CMB
studies, with their detection and removal being an important step in
CMB component separation (Brandt et al.
1994
;Tayloretal.
2001
;
Waldram et al.
2003
,
2007
; de Zotti et al.
2010
). For future CMB
experiments, it has been shown that point-source contamination has
the potential to significantly complicate measurements of the
B
-
mode polarization power spectrum on scales
50 (Mesa et al.
2002
; Curto et al.
2013
;Puglisietal.
2018
; Remazeilles et al.
2018
). Given the large beam of C-BASS, the instrument is not
well suited to point-source detection, with detections of all but
the brightest sources (
1 Jy) being limited by source confusion
and diffuse emission. For the C-BASS analysis, performing our
own dedicated source detection allows us to carry out important
data quality checks on the C-BASS flux-density scale and pointing
accuracy, by comparing the C-BASS catalogue with pre-existing
catalogues around 4
.
76 GHz. The resulting catalogue will also allow
us to construct an accurate mask of bright source emission in the C-
BASS maps, which will be key to avoid biasing CMB component
separation analyses with C-BASS. Fainter sources can either be
subtracted using deeper, pre-existing source catalogues, or can be
treated statistically at the power spectrum level. Assuming sources
down to some flux density
S
max
have been removed, and that sources
are Poisson-distributed on the sky, the point-source contribution to
the power spectrum is given by
C
PS
=
S
max
0
d
N
d
S
S
2
d
S,
(1)
where
S
is the flux density and d
N
/d
S
is the differential source count,
i.e. the number of sources per unit flux density, per unit steradian
(Tegmark & Efstathiou
1996
).
Numerous source-detection algorithms have been developed for
CMB experiments (see e.g. Tegmark & de Oliveira-Costa
1998
;
Vielva et al.
2003
; Gonz
́
alez-Nuevo et al.
2006
;L
́
opez-Caniego et al.
2006
;Arg
̈
ueso et al.
2009
; Herranz et al.
2009
;Carvalhoetal.
2012
;
Bennett et al.
2013
; Planck Collaboration XXVIII
2014
; Planck Col-
laboration XXVI
2016c
). The Planck Catalogue of compact sources
(PCCS) used an algorithm based on filtering small patches of sky
using the flat-sky second-order Mexican Hat wavelet as an approxi-
mation to a matched filter, followed by threshold detection based on
the signal-to-noise ratio (SNR). Here, we employ a similar scheme
but using the equivalent spherical function on the whole sky at
once, implemented in the Healpix pixelization scheme (G
́
orski et al.
2005
). Previous applications of the first-order SMHW (SMHW1) to
searches for non-Gaussianity can be found in Cay
́
on et al. (
2001
),
Mart
́
ınez-Gonz
́
alez et al. (
2002
), Curto, Mart
́
ınez-Gonz
́
alez &
Barreiro (
2011
), and to point-source detection in Vielva et al. (
2003
).
The details of this implementation are discussed in Section 3.
The outline of this paper is as follows. In Section 2, we summarize
pre-existing source catalogues directly relevant to our C-BASS anal-
ysis. In Section 3, we discuss the C-BASS point-source detection
algorithm, giving an overview of the method and the Monte Carlo
simulations used to validate the algorithm. In Section 4, we discuss
the C-BASS total intensity, northern sky point-source catalogue
obtained using our detection algorithm. We compare the C-BASS
catalogue with the GB6 and PMN catalogues and calculate the
differential source counts for the C-BASS sources as a cross-check
on the statistical properties of the bright source population. We sum-
marize our results in Section 5. Whilst analysis of polarized sources
is important for future CMB polarization studies, this falls beyond
the scope of this paper where we focus on the C-BASS total intensity
results. In addition, due to the low level of source polarization, only
a small number of polarized sources (
O
(10)) will be detected.
2 PRE-EXISTING SOURCE CATALOGUES
In order to construct an accurate template for masking out the point
sources in the C-BASS maps, it is necessary to construct a point-
source catalogue based on the C-BASS observations themselves and
any useful information that can be gleaned from other more sensi-
tive, higher resolution surveys. Relevant to our work in this paper
are the GB6 (Gregory et al.
1996
), PMN (Griffith & Wright
1993
;
Griffith et al.
1994
,
1995
; Wright et al.
1994
,
1996a
), Effelsberg
S5 (Kuehr et al.
1981
), RATAN-600 (Mingaliev et al.
2007
), and
Combined Radio All-Sky Targeted Eight-GHz Survey (CRATES;
Healey et al.
2007
) catalogues. These catalogues are primarily used
in this paper to make comparisons with the C-BASS catalogue,
for the purposes of data validation and estimation of the statistical
properties of the C-BASS catalogue. However, the catalogues are
also useful for characterizing the faint point-source population in
any C-BASS analysis (i.e. sources with flux-densities below
∼
1Jy),
where reliable extraction from the C-BASS map becomes more
challenging. These radio surveys are summarized in Table
1
.
The GB6 and PMN 4
.
85 GHz source catalogues cover declina-
tions
−
87.5
◦
≤
δ
≤
75
◦
. The GB6 catalogue was produced using the
NRAO seven-beam receiver on the 91 m telescope, and the PMN
catalogue was produced using the Parkes 64 m radio telescope. The
GB6 catalogue has a flux-density limit of approximately 18 mJy,
whilst the PMN catalogue has an average flux-density limit of
approximately 35 mJy over the sky. The GB6 and PMN catalogues
provide far deeper flux-density coverage than can be achieved with
C-BASS, given the differing resolutions and hence confusion levels.
However, it is still necessary for us to obtain our own source
catalogue, such that we can construct an accurate mask for the
brightest sources in the C-BASS maps, and account for source
variability between the C-BASS and GB6/PMN surveys. The GB6
and PMN catalogues remain useful in accounting for fainter sources
(below
∼
1 Jy) in any C-BASS analysis.
Source catalogues covering the North Celestial Pole (NCP) region
are more limited, with GB6 only covering declinations up to
δ
=
75
◦
. The Effelsberg S5 catalogue covers declinations
δ
≥
70
◦
and
is complete down to 250 mJy (Kuehr et al.
1981
). In comparing to
the S5 catalogue, there are significant issues from the variability of
flat-spectrum sources, given the large separation in time between
the S5 survey and the C-BASS survey. The RATAN-600 catalogue
includes measurements at 4
.
8 GHz, and observed 504 sources in the
NCP region with NRAO VLA Sky Survey (NVSS) flux-densities,
S
1
.
4 GHz
≥
200mJy (Mingaliev et al.
2001
,
2007
). The RATAN-600
catalogue was used in a previous analysis of diffuse emission in the
NCP region with C-BASS in Dickinson et al. (
2019
). Whilst this
catalogue provides deeper flux-density coverage than C-BASS, it
was produced by pre-selecting sources for study from the NVSS
catalogue at 1
.
4 GHz (Condon et al.
1998
). This can potentially
miss rising-spectrum sources that would otherwise be observable
in the C-BASS catalogue. Given the C-BASS flux-density limit of
approximately 500 mJy, we can use the NVSS source counts and
the 1
.
4–4
.
85 GHz spectral index distributions in Tucci et al. (
2011
)
to estimate that there may be
O
(1) sources that could be observed
by C-BASS, whilst being missed by RATAN-600.
It is also worth noting the CRATES catalogue (Healey et al.
2007
). CRATES is an 8
.
4 GHz catalogue of flat-spectrum sources
with flux-densities
S
ν
=
4
.
85 GHz
>
65 mJy, covering Galactic latitudes
|
b
|
>
10
◦
. The catalogue therefore serves as a useful proxy for
flat-spectrum sources in the C-BASS catalogue, which are the
primary contributor to source variability. Healey et al. (
2009
)made
additional observations of the NCP region at declinations,
δ>
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C-BASS northern sources
1943
Table 1.
Summary of radio surveys relevant to the work in this paper. In stating the sky coverage,
δ
denotes declination, and
b
denotes Galactic latitude.
Survey name
Reference
Frequency
Sky
FWHM
Flux Limit
Number of
(GHz)
Coverage
(arcmin)
(mJy)
sources
C-BASS
Jones et al. (
2018
)4.76
−
90
◦
≤
δ
≤
90
◦
a
45
∼
500
b
1784
GB6
Gregory et al. (
1996
)4.850
◦
≤
δ
≤
75
◦
3
∼
18
75,162
PMN
Wright et al. (
1996b
)4.85
−
87.5
◦
≤
δ
≤
10
◦
5
∼
35
50,814
Effelsberg S5
Kuehr et al. (
1981
)4.970
◦
≤
δ
≤
90
◦
2.7
∼
250
476
RATAN-600
Mingaliev et al. (
2007
)
4.8 (1.1
−
21.7)
c
75
◦
≤
δ
≤
88
◦
0.67
×
6.6
d
(200)
e
504
CRATES
Healey et al. (
2007
)
8.4 (4.85)
f
|
b
|
>
10
◦
2.4
∼
65
11,131
a
Here, we state the sky coverage of the whole C-BASS experiment. The results concerning point sources in this paper were obtained for the C-BASS norther
n
intensity data, covering declinations
−
10
◦
≤
δ
≤
90
◦
.
b
For C-BASS, we state the value 3.5
σ
,where
σ
is the mean background fluctuation level found across the map, applying the CG30 mask described in
Section 3.5. Details on the estimation of background fluctuation levels are given in Section 3.1.
c
The RATAN-600 catalogue covers six frequencies from 1
.
1to21
.
7 GHz, including 4
.
8 GHz.
d
We state FHWM
RA
×
FWHM
δ
for RATAN-600, determined from the values given in Kovalev et al. (
1999
).
e
The RATAN-600 catalogue was produced by pre-selecting NVSS sources with
S
1
.
4 GHz
>
200 mJy.
f
The CRATES catalogue is primarily at 8
.
4 GHz, with 4
.
85 GHz sources used as the basis for observations of 8
.
4 GHz counterparts. The properties of these
4
.
85 GHz sources are provided with the CRATES catalogue. Further observations were also made at 4
.
85 GHz to fill in gaps at
δ>
88
◦
.
88
◦
to supplement the original CRATES catalogue. The purpose
of this was to bring the flux-density limit in this region down to
the CRATES flux-density limit of
∼
65 mJy. Three sources were
observed in this region at 4
.
85 GHz, with flux densities of 67, 58,
and 142 mJy.
3 C-BASS SOURCE DETECTION ALGORITHM
To detect sources in a sky map we need to remove obscuring diffuse
emission and noise. For a source with a known point spread function
(PSF) embedded in additive noise, the matched filter (MF) is the
optimal filter that can be applied to maximize the source SNR. The
matched filter is given by
MF
(
k
)
=
2
π
d
kk
τ
2
(
k
)
P
(
k
)
−
1
τ
(
k
)
P
(
k
)
,
(2)
where
k
is the Fourier wavenumber,
τ
(
k
) is the source profile, and
P
(
k
) is the power spectrum of the unfiltered map (Tegmark & de
Oliveira-Costa
1998
;L
́
opez-Caniego et al.
2006
). However, the
calculation of the MF involves a number of complications. Chiefly,
we are required to make a noisy estimate of the power spectrum from
our unfiltered map, and integrate it. In constructing the PCCS, it was
found that the Mexican Hat wavelet of the second kind (MHW2)
achieved similar performance to the MF (L
́
opez-Caniego et al.
2006
;
Planck Collaboration XXVIII
2014
; Planck Collaboration XXVI
2016c
). For the present analysis, we adapt the
Planck
algorithm,
using instead the SMHW2 in place of the flat-space MHW2. The
SMHW2 is straightforward to calculate, enables us to filter the
entire sky at once, and allows us to optimize a few free parameters
as opposed to a noisy estimate of the full noise power spectrum.
3.1 Source detection algorithm
Given a sky map consisting of point sources, diffuse emission and
instrumental noise, we can enhance the SNR of point sources in
the map by filtering with the SMHW2. We perform this filtering
at a range of filter scales
R
, to maximize the number of sources we
extract from the sky maps. In changing
R
, we effectively change
the extent to which we down-weight large-scale
-modes that are
dominated by diffuse emission, and also small scales dominated by
instrumental noise. This is particularly important in regions close to
the Galactic plane where diffuse emission is very strong, meaning
we must down-weight large-scale modes harshly in order to extract
point sources. Conversely, in regions with little diffuse emission
we may wish to be less extreme in our down-weighting, so that
we do not excessively reduce point-source power and subsequently
miss detection of fainter sources in these regions. We discuss the
form of the SMHW2 filter and the effect of the filter scale,
R
in
detail in Section 3.2.
The source-detection algorithm is described below, with the
algorithm flowchart displayed in Fig.
1
.
(i) We begin by taking the spherical harmonic transform of our
sky map, obtaining the
a
m
coefficients. We filter the map by
weighting the
a
m
coefficients with the SMHW2 window function,
w
S
2
(
R
)
at some user-defined set of filter scales, i.e. we calculate
f
m
(
R
)
=
a
m
w
S
2
(
R
)
.
(3)
The effect of this filtering is demonstrated in Fig.
2
. Here, we display
a simulated sky consisting of diffuse emission, instrumental noise
and point-sources, along with the corresponding filtered sky with
scale
R
=
1. The details of the production of this simulated sky are
discussed in Section 3.3. The filter has removed diffuse emission
and enhanced the SNR of point-sources across the sky.
(ii) We break the maps, filtered at a range of
R
scales, into
overlapping 10
◦
×
10
◦
patches on the sky and calculate the
corresponding SNR patches. We used
R
scales in the range 0.5
≤
R
≤
2 in steps of 0.1. We choose 10
◦
×
10
◦
patches such that the
flat-sky approximation holds, whilst capturing a sufficient sample
of the sky background in the given sky region. We set the overlap
to 5
◦
in each direction to ensure that we obtain accurate positions
for sources detected close to patch edges. The typical background
fluctuation level in each patch is estimated by calculating the median
absolute deviation (MAD) of the pixels, defined as
MAD(
X
)
=
median
(
|
X
i
−
median(
X
)
|
)
,
(4)
where
X
is the set of pixel values. The MAD estimator avoids sen-
sitivity to outlier pixel values, which can be caused by the presence
of point sources in the sky patch. For each sky patch, we then select
the filter scale that gives the maximal SNR for the patch i.e. the
maximum of the peak pixel value divided by the MAD estimate.
(iii) Having selected the value of
R
that maximizes the patch
SNR, an initial SNR threshold is applied to identify potential point
sources. This produces a patch of tagged and un-tagged pixels. The
tagging is performed on these small, overlapping regions so as to
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496,
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R. D. P. Grumitt et al.
Figure 1.
Flowchart for the C-BASS point-source detection algorithm. The SMHW2 filter is applied over the entire sky, whilst identification of point sources
is performed on 10
◦
×
10
◦
sky regions.
account for the varying background properties across the sky i.e.
variations in residual diffuse emission and noise.
(iv) Given our patch of tagged pixels, we then use the O
PEN
CV
Simple Blob Detector
1
to obtain candidate source locations. The
Simple Blob Detector algorithm works by applying thresholds to
an input image and determining the centres of connected pixels, or
sources. In addition to providing candidate source locations, this
software allows for the filtering of detected sources according to
their geometric properties i.e. their size, circularity, convexity, and
inertia ratio. For our purposes, we set a maximum size limit to help
prevent spurious tagging of bright diffuse emission and disallow
detections within one beam FWHM of one another. For a detailed
discussion of O
PEN
CV and the Simple Blob Detector algorithm, we
refer the reader to Kaehler & Bradski (
2016
). The processing steps
involved in studying these small sky regions are illustrated in Fig.
3
,
where we display the typical output from the steps outlined above.
(v) Given a set of tentative source locations obtained over the
whole sky, we now repeat steps (ii) and (iii), this time centring
on the tentative source locations. In order to refine the source
position estimates, we fit an elliptical Gaussian to a 20 arcmin
square region of the wavelet-filtered map, centred on the candidate
location obtained using O
PEN
CV. Whilst the C-BASS beam and
SMHW2 filter are not Gaussian, by fitting to this small region
around the candidate location the Gaussian fit is sufficient to make
the necessary refinements (
∼
1 arcmin) to our estimates of source
peak locations.
(vi) A final SNR thresholding is applied by dividing the peak
value of the source in the wavelet filtered map by the MAD value
of pixels in an annulus centred on the source, with an inner radius
of 3
◦
and outer radius of 5
◦
. The annulus was chosen to avoid
the first sidelobe in the C-BASS beam, whilst capturing the local
background fluctuations around the source. Sources are retained if
this new SNR estimate exceeds some final threshold. The SMHW2
is constructed so as to preserve point-source amplitudes after being
1
https://opencv.org/
convolved with the C-BASS beam. As with the PCCS, we convert
these wavelet amplitudes to Jy and report them as auxiliary flux-
density (DETFLUX) estimates alongside the primary estimates
from aperture photometry described in step (vii). The DETFLUX
estimates are discussed in more detail in the Appendix.
(vii) The detected source locations from all of the filtered maps
are combined and duplicates are removed from the catalogue,
defined here to be any reported source positions that are within the
beam FWHM of one another. For matched sources in our internal
catalogue, we retain the source with the greatest detected SNR.
(viii) Given the final set of source positions, source flux densities
are obtained. As its primary method, the detection pipeline obtains
flux densities using aperture photometry, adapting the method from
Planck Collaboration XXVIII (
2014
)andG
́
enova-Santos et al.
(
2015
). We begin by converting the map to Jy pix
−
1
, and define
an aperture of radius 45 arcmin around the source position, and an
annulus of inner radius 3
◦
and outer radius 5
◦
. The observed flux
density is then given by
S
obs
=
κ
S
ap
−
̄
S
ann
N
ap
,
(5)
where
κ
≈
1.34 is a correction applied to account for flux density
missing from the aperture (calculated using equation (A.2) of Planck
Collaboration XXVIII
2014
),
S
ap
is the total flux density in the
aperture,
̄
S
ann
is the median flux density in the annulus and
N
ap
is the
number of pixels in the aperture. The uncertainty was estimated as
σ
(
S
obs
)
≈
MAD(
X
ann
)
N
ap
N
ap
,
(6)
where MAD(
X
ann
) is the MAD value of pixels in the annulus and
N
ap
is the number of beams inside the aperture. The scaling applied
to MAD(
X
ann
) is to account for correlations in the background
emission, approximately on the scale of the beam (G
́
enova-Santos
et al.
2015
).
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C-BASS northern sources
1945
Figure 2.
Panel (a): An input 4
.
76 GHz simulated sky consisting of diffuse emission generated using P
Y
SM, instrumental noise generated from a white-noise
realization of the C-BASS sensitivity map, along with a point-source population generated by scattering the GB6, PMN, and RATAN-600 sources at rando
m
positions over the sky. Alongside this we display a zoom-in centred on the location of the black cross on the map. Panel (b): The same map after being conv
olved
with the SMHW2 filter, removing the large scale diffuse emission and leaving behind a sky filled with point sources. As before, we display a zoom-in of the
filtered map alongside it, centred on the location of the black cross. It is important to note here that not all the diffuse emission has been removed at lo
w
Galactic latitudes. In regions of strong diffuse emission the source detection algorithm is significantly less reliable. In panel (b), it is apparent
that the SMHW2
filtering can still leave a significant sidelobe around bright sources. This can lead to issues with the spurious tagging of sidelobe peaks as sources, p
articularly
for the brightest sources on the sky (i.e. those with flux-densities
20 Jy). This problem can largely be side-stepped by allowing for larger exclusion zones
around the brightest sources, as was the case with catalogues such as GB6, removing apparent detections in the first sidelobe of such sources.
3.2 The spherical Mexican Hat Wavelet
The Mexican Hat Wavelet (MHW) of the
n
th kind is defined in
R
2
(i.e. two-dimensional Euclidean space) as
n
(
x
,R
)
∝
n
exp
−
|
x
|
2
2
(
Rσ
)
2
,
(7)
where
x
is the position in the image plane,
is the usual Euclidean
Laplacian operator,
σ
is the Gaussian standard deviation, and
R
is
the filter scale factor. In changing
R
, we change the characteristic
scale of the filter and hence the extent to which the filter down-
weights large- and small-scale modes. Applying this expression we
can arrive at the real-space expression for the MHW2, given by
2
(
x
,R
)
∝
8
(
Rσ
)
4
−
8
(
Rσ
)
2
|
x
|
2
+
|
x
|
4
exp
−
|
x
|
2
2
(
Rσ
)
2
.
(8)
In applying the MHW2 to producing the PCCS, the sky was
divided up into small patches, taking flat projections and applying
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R. D. P. Grumitt et al.
Figure 3.
Images illustrating the pixel tagging process. Panel (a): An example 10
◦
×
10
◦
SNR patch, produced by dividing the filtered sky patch by the MAD
value of the pixels in the patch. Panel (b): The corresponding patch of tagged and un-tagged pixels. The tagged pixels are shown in
black
and un-tagged pixels
in
white
, obtained by applying a threshold of SNR
≥
3.5 to the SNR patch. Panel (c): An image showing the detected source locations for this patch. We use
the O
PEN
CV blob detection algorithm to obtain preliminary source locations from this patch. These blob detections are shown as black circles centred on the
preliminary source locations. These position estimates are then refined by fitting an elliptical Gaussian at the preliminary source locations, with t
he source
positions obtained from the Gaussian fit shown as blue crosses.
the MHW2 filter to them. SNR thresholding was applied to the
filtered patches to extract the source positions and flux densities
were then obtained for the detected sources. Here, we employ the
SMHW2, the equivalent of the MHW2 on
S
2
(i.e. the two-sphere).
For a given filter scale, this allows us to filter the entire sky at
once, avoiding complications from edge effects when filtering small
patches. In dealing with the C-BASS northern sky map, we apply
a5
◦
cos
2
apodization at the edge of our mask to mitigate edge
effects.
In Antoine & Vandergheynst (
1999
), it was shown the continuous
wavelet transform on
S
2
can be constructed by taking the inverse
stereographic projection of the
R
2
wavelet. This preserves the
properties of the
R
2
wavelet and tends to the MHW in the small-
angle limit. The spherical MHW of the
n
th kind is given by
(
n
)
S
2
∝
1
cos
4
θ
2
n
|
x
|
≡
2tan
θ
2
,R
,
(9)
where
θ
is the colatitude on the sphere (Cay
́
on et al.
2001
;Mart
́
ınez-
Gonz
́
alez et al.
2002
; Vielva et al.
2003
; Curto et al.
2011
).
Substituting the MHW2 into equation (9), we obtain the SMHW2
as follows:
(
2
)
S
2
(
θ,R
)
∝
1
cos
4
θ
2
(
Rσ
)
4
−
4
(
Rσ
)
2
tan
2
θ
2
+
2tan
4
θ
2
exp
−
2tan
2
θ
2
(
Rσ
)
2
.
(10)
We normalize the SMHW2 such that we preserve point-source
amplitude after convolving with the filter, i.e.
S
2
B
(
θ
)
(
2
)
S
2
(
θ,R
)
sin
θ
d
θ
d
φ
=
1
,
(11)
where
B
(
θ
) is the C-BASS beam profile.
Given the analytic expression for the SMHW2 in real space on
S
2
, we can obtain the SMHW2 window function by calculating the
Legendre transform of equation (10), i.e. we calculate
w
S
2
(
R
)
=
2
π
π
0
(
2
)
S
2
(
θ,R
)
P
(
cos
θ
)
sin
θ
d
θ,
(12)
where
P
(cos
θ
) is the Legendre polynomial of order
.
In Fig.
4
, we show the real-space SMHW2 and the corresponding
window function, calculated for a
σ
value corresponding to a beam
FWHM of 45 arcmin, with
R
=
{
0.85, 0.95, 1.05, 1.15
}
.The
primary effect of the SMHW2 is to down-weight large scales, which
are dominated by diffuse emission. In addition to this, the SMHW2
also acts to down-weight small-scale modes where instrumental
noise becomes more significant. In the context of C-BASS, large-
scale modes correspond to modes significantly larger than the C-
BASS beam scale,
100, whilst small scales correspond to scales
significantly smaller than the C-BASS beam scale,
1000.
3.3 Algorithm validation
We validate the algorithm by running it over 100 simulations
of the sky at 4.76 GHz. The simulated skies were generated at
NSIDE
=
1024, where the number of equal-area pixels on the
Healpix sphere is given by 12
×
NSIDE
2
. This high resolution
was chosen to allow for the precise determination of source
positions, with the C-BASS PSF oversampled by a factor of
∼
6.5.
The simulated sky includes diffuse emission generated by P
Y
SM
(Thorne et al.
2017
) and an instrumental noise realization. The noise
maps were created by generating white noise realizations of the C-
BASS sensitivity map (including a simulated sensitivity map for
the southern sky survey). This allows us to include the declination
dependent effects from the C-BASS scan strategy.
To the diffuse emission simulation we add a source population to
the sky. These are created by taking source flux-densities from the
PMN, GB6 and RATAN-600 surveys, summarized in Section 2, and
scattering the sources at random positions over the sky. Carrying this
out 100 times we obtain 100 Monte Carlo simulations of the sky,
consisting of the same diffuse background but with their source
populations randomly distributed on the sky, and differing noise
realizations. The simulated skies are all convolved with the C-BASS
beam.
The diffuse components included in the simulation are as
follows:
(i) Synchrotron radiation: This was generated using the de-
sourced, re-processed 408 MHz Haslam map (Haslam et al.
1981
,
1982
; Remazeilles et al.
2015
) as a template and scaled using a
spectral index and curvature model. This corresponds to the P
Y
SM
s3 model. Pixel values at 4
.
76 GHz range from
∼
3000
μ
Kto
∼
8
×
10
5
μ
K, with a mean value of
∼
2
×
10
4
μ
K.
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C-BASS northern sources
1947
Figure 4.
Panel (a): The real space SMHW2, shown for
σ
corresponding to a Gaussian beam FWHM of 45 arcmin and
R
∈
{
0.85, 0.95, 1.05, 1.15
}
. Panel
(b): The corresponding window functions. The SMHW2 down-weights large and small scales, with changes in
R
determining the severity with which we
down-weight large- and small-scale emission. Whilst the C-BASS beam is not exactly Gaussian, by constructing the SMHW2 around these
∼
45 arcmin scales
we are able to filter the large- and small-scale modes around the characteristic beam scale, enhancing point-source SNR in the C-BASS sky map.
(ii) Free-free emission: This was generated with the analytic
model used in the
Planck
C
OMMANDER
analysis (Draine
2011
;
Planck Collaboration IX
2016a
; Planck Collaboration X
2016b
).
The free–free template is scaled using a single power law. This
corresponds to the P
Y
SM f1 model. Pixel values at 4
.
76 GHz
range from
∼
90
μ
Kto
∼
5
×
10
6
μ
K, with a mean value of
∼
1
×
10
4
μ
K.
(iii) Anomalous microwave emission (AME): This was gener-
ated by modelling emission caused by the sum of two spinning-
dust populations. Scaling is calculated using the S
P
D
UST
2 code
(Ali-Ha
̈
ımoud, Hirata & Dickinson
2009
). This corresponds to the
P
Y
SM a2 model. Pixel values at 4
.
76 GHz range from
∼
5
μ
Kto
∼
2
×
10
5
μ
K, with a mean value of
∼
300
μ
K.
(iv) Thermal dust: This was generated by modelling the emission
caused by a single modified blackbody. This corresponds to the
P
Y
SM d1 model (Planck Collaboration X
2016b
). Pixel values at
4
.
76 GHz range from
∼
0
.
01 to
∼
20
μ
K, with a mean value of
∼
0
.
4
μ
K.
(v) CMB: A lensed CMB realization was generated using the
Taylens code (Næss & Louis
2013
), as part of P
Y
SM. This
corresponds to the P
Y
SM c1 model. Pixel values at 4
.
76 GHz range
from
∼−
300 to
∼
300
μ
K, with a mean value of
∼
0
.
1
μ
K.
One of the simulated skies is shown in Fig.
2
, along with the same
skyfilteredatthescale
R
=
1. After running our detection algorithm
over the simulated skies, we compare the detected sources to our
input catalogues. From this, we calculate the completeness level of
our detected sources, averaged over all the simulations, and record
the absolute deviations of detected sources from their true positions.
The validation results are presented in Section 3.5, produced using
an initial SNR threshold of 2.5 to tag candidate sources on the
first loop over the sky, and retaining sources with SNR
≥
3.5 after
looping over the candidate sources. These thresholds were selected
to give a catalogue reliability
90 per cent away from the Galactic
plane. Here, we define catalogue completeness as the fraction of
sources above some given flux-density threshold that are recovered
by the detection algorithm. The catalogue reliability is defined as
the fraction of detected sources that are matched with at least one
source in the input catalogue.
3.4 Catalogue matching: likelihood ratios
The simulated validation catalogue is produced for a map with a
resolution of 45 arcmin. Given the large number of sources used for
our input catalogue, source confusion presents a major complication
when trying to match the output catalogues with the corresponding
input catalogues from our Monte Carlo simulations. One source in
the output catalogue will likely result from the blending of multiple
sources in the input catalogue, and a single source in the input
catalogue can contribute to more than one source in the output
catalogue. We face identical issues when matching the real C-BASS
catalogue to higher resolution catalogues such as GB6 and PMN.
To find matches between our input and output catalogues we use
a likelihood ratio test. This has previously been used in matching
point-source catalogues, particularly when when dealing with prob-
lems from source confusion (see e.g. Richter
1975
; Sutherland &
Saunders
1992
;Mannetal.
1997
; Rutledge et al.
2001
; Chapin et al.
2011
;Wangetal.
2014
). The core of the likelihood ratio test consists
in calculating the ratio of the probability of a match between an input
and output source being a true match, to the probability of the match
being a random association. The likelihood ratio is given by
LR
=
q
(
S
f
)
f
(
r
)
2
π
rρ
(
S
f
)
,
(13)
where
q
(
S
f
) is the probability density function (PDF) for a given
match being a true match as a function of the fractional flux-density
difference (
S
f
) between the two sources,
f
(
r
) is the PDF for true
matches as a function of the absolute positional offset between the
two sources,
r
,2
π
r
is the PDF for random associations assuming
a uniform spatial distribution of background sources, and
ρ
(
S
f
)
is the PDF for random associations as a function of
S
f
.
For the positional PDF of true input-output matches, we assume
a Gaussian distribution over the orthogonal coordinate axes. This
gives a Rayleigh distribution for the PDF of true input-output
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1948
R. D. P. Grumitt et al.
matches as a function of
r
,i.e.
f
(
r
)
=
r
σ
r
exp
−
r
2
2
σ
2
r
,
(14)
where
σ
r
is a scaling parameter. The position and flux-density
PDF parameters are estimated by comparing our output source
catalogues with the corresponding input catalogues. We determine
f
(
r
) by searching for all candidate matches between the input and
output catalogues, using a search radius of 45 arcmin around each
output source. Candidate matches will consist of true input–output
matches, along with random background associations. Assuming
a uniform distribution of background sources the random matches
will follow a linear trend, scaling with the sky area enclosed by a
given search radius. We therefore fit
f
(
r
) plus a linear trend to the
histogram of absolute positional offsets for candidate input–output
matches to obtain an estimate for
f
(
r
).
To determine
ρ
(
S
f
), we generate randomized output catalogues
by randomly distributing our output sources over the sky and
randomizing source flux-densities from the parent distribution. We
then find candidate matches between input and randomized output
sources, again using a 45 arcmin search radius. We bin the candidate
matches as a function of the fractional flux-density difference
between the input and output source, given by
S
f
=
S
out
−
S
in
S
in
,
(15)
where
S
out
is the output source flux density and
S
in
is the input
source flux density. This gives us the number of random matches
as a function of
S
f
and hence our estimate for
ρ
(
S
f
). To
obtain
q
(
S
f
) we carry out the same procedure as above, but
with the real output catalogues. We then estimate
q
(
S
f
) from the
difference between the number of matches as a function of
S
f
for the real output catalogues, and the number of matches for the
randomized catalogues, i.e. from the excess matches in the real
output catalogues.
To determine the LR threshold for declaring a true match, we
calculate the likelihood ratios for all candidate matches between
the input catalogues and randomized output catalogues. From this,
we determine the value of LR at which we would declare 10 per cent
of our randomized matches to be true matches. This value is used
as the LR threshold at which we declare a true input–output match.
This is sufficient for limiting the fraction of false identifications with
the input catalogue, whilst ensuring a low probability of spuriously
identifying a source in the output catalogue as a true positive, given
that matched output sources are typically associated with multiple
input sources.
3.5 Validation: completeness, position, and flux-density
accuracy
We obtain the catalogue completeness and reliability, averaged
over our Monte Carlo simulations, to confirm the ability of the
source-detection algorithm to accurately recover an input source
population. Throughout our analysis, we use the mask shown
in Fig.
5
to characterize the catalogue properties away from the
brightest diffuse emission along the Galactic plane. The mask was
produced by smoothing the C-BASS northern sky map with a 10
◦
FWHM Gaussian and masking the brightest 30 per cent of pixels.
We also mask all pixels corresponding to declinations below
δ<
−
10
◦
in order to match the sky area observed by the C-BASS
northern-sky survey. We refer to this mask as the CG30 mask.
Figure 5.
The combined Galactic plane and C-BASS North mask used in
our analysis of the C-BASS catalogue, shown here in Galactic coordinates.
The mask was constructed to cover the brightest 30 per cent of the northern
sky observed by C-BASS, along with declinations
δ<
−
10
◦
. We refer to
this mask as the CG30 mask.
In determining the positional offset of the detected source from
the input sources, we compare its position to the photocentre of
matched input sources. That is, after matching to input sources
using the LR test, we determine the weighted average position of
the matched input sources, using the input source flux-densities
as weights. In comparing flux densities, we sum flux densities for
matched sources, weighted according to the C-BASS beam profile,
i.e.
S
match
ν
=
i
S
(
i
)
ν
B
(
|
x
i
−
x
det
|
)
,
(16)
where the sum runs over matched sources in the input catalogues,
S
(
i
)
ν
is the flux density of the
i
th matched source,
x
i
is the position
of the
i
th matched source,
x
det
is the position of the detected source,
and
B
is the C-BASS beam profile.
In Fig.
6
, we show maps of the completeness and reliability in
NSIDE
=
4 pixels obtained for the validation simulations. We can
see that close to the Galactic plane the 90 per cent completeness
level is higher and the catalogue reliability is lower. Indeed, for
some pixels along the Galactic plane 90 per cent completeness is
never achieved. This is to be expected given the increased intensity
of diffuse emission along the Galactic plane, which is not fully
removed by the SMHW2 filter. Over regions of the sky left un-
masked by the CG30 mask, we find a mean 90 per cent catalogue
completeness of approximately 500 mJy, corresponding to a mean
reliability of approximately 91 per cent.
From the validation simulations, we calculate the absolute posi-
tional offsets of detected sources from input sources. These cluster
below 10 arcmin, peaking around zero, with a median offset of
approximately 3
.
2 arcmin. Considering the 45 arcmin resolution of
the maps used for source detection, and the significant confusion
effects at this resolution, this is a small positional uncertainty for our
source detection algorithm. Indeed, for the purposes of producing
a point-source mask for the actual C-BASS maps this is more than
sufficient. The histogram of absolute positional offsets of detected
sources, applying the CG30 mask, is shown in Fig.
7
.
We are able to recover input source flux-densities using aperture
photometry, with a plot of detected flux densities against input flux
densities being shown in Fig.
8
. At flux densities greater than 1 Jy,
we recover reasonable flux-density estimates for the majority of
sources. A small fraction of sources are recovered with flux densities
biased slightly high. This is to be expected given the 45 arcmin
resolution of the C-BASS map, meaning that multiple input sources
are associated with each detected source. The detected source flux
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C-BASS northern sources
1949
Figure 6.
Panel (a): A map of the completeness estimates obtained from the validation simulations. Panel (b): A map of the corresponding reliability estimates
.
These estimates were calculated on NSIDE
=
4 sky pixels and are shown in Galactic coordinates. In pixels outside of the CG30 mask, the mean 90 per cent
completeness level is approximately 500 mJy, with a corresponding mean reliability of 91 per cent. It can be seen that along the Galactic plane the comp
leteness
and reliability is significantly lower than at higher Galactic latitudes. This is to be expected given the stronger diffuse emission that is not remove
dbythe
SMHW2 filter. Grey pixels in the completeness map denote pixels where 90 per cent completeness was never achieved.
Figure 7.
The histogram of measured absolute positional offsets,
,
applying the CG30 mask, obtained for the 100 Monte Carlo simulations.
Over the un-masked sky area we find the detected positional offsets peak
around zero, with a median positional offset of 3
.
2 arcmin, which is small
compared to the C-BASS beam of 45 arcmin. Larger positional offsets are
predominantly driven by fainter detected sources below
∼
1 Jy, and sources
detected in regions of brighter diffuse emission. The positional offsets
of all the detected sources will consist of samples drawn from Rayleigh
distributions, which results in a tail of larger offsets beyond the peak of the
positional offset distribution.
density will be the summed contribution from associated input
sources. For flux densities below
∼
1 Jy, the scatter in the recovered
flux densities reaches the same magnitude as the input flux densities.
This is again to be expected; below this level sources become much
more heavily obscured by surrounding diffuse emission, making
photometric estimates of source flux densities significantly more
unreliable. We note here that there is a small cluster of recovered
sources at these low flux densities where the recovered flux-density
estimate is spuriously high. This is largely caused by sources being
detected in regions of bright diffuse and highly extended emission,
leading to errors in the background estimation.
Our validation simulations have demonstrated that our source-
detection algorithm is able to accurately recover an underlying
source population, from a sky consisting of additional contributions
from diffuse emission and noise.
4 THE C-BASS NORTHERN SKY CATALOGUE
In this section, we present the C-BASS northern sky, total-intensity
point-source catalogue. The point-source catalogue was produced
using data obtained from only night-time observations, combining
data from the various elevation scans of the telescope. C-BASS data
were calibrated against Tau A as a primary calibrator and Cas A as a
secondary calibrator, with calibrator flux densities being calculated
from the spectral forms in Weiland et al. (
2011
). For Tau A the
spectral model takes the form
log
S
ν
(Jy)
=
2
.
506
−
0
.
302 log
ν
40 GHz
,
(17)
and for Cas A it takes the form
log
S
ν
(Jy)
=
2
.
204
−
0
.
682 log
(
ν
40 GHz
)
+
0
.
038 log
2
(
ν
40 GHz
)
.
(18)
These spectral models give the flux density of Tau A at 4
.
76 GHz as
S
ν
=
609
.
8
±
4
.
2 Jy in epoch 2005, and the flux density of Cas A as
S
ν
=
736
.
1
±
3
.
4 Jy in epoch 2000. The secular decreases adopted
for Tau A and Cas A were the same as those used for the model
fits in Weiland et al. (
2011
). For Tau A this was
−
0.167 per cent
per year, which was taken from Mac
́
ıas-P
́
erez et al. (
2010
), and for
Cas A the secular decrease was
−
0.53 per cent per year. The C-
BASS calibration is accurate to
∼
5 per cent. The flux-temperature
conversion is defined through
T
source
=
S
ν
(Jy)
π
2
k
B
×
10
26
6
.
1
2
2
α
eff
,
(19)
where
T
source
is the source temperature,
k
B
is the Boltzmann
constant, and
α
eff
=
0.55 is the theoretical aperture efficiency for
C-BASS. Detailed discussions of the C-BASS data reduction and
calibration will be given in the C-BASS survey and commissioning
papers (Pearson et al. in preparation; Taylor et al., in prepara-
tion). We perform the source detection on the C-BASS map at
NSIDE
=
1024, using the high NSIDE map to allow for the precise
determination of source positions.
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