figure table

# Mitigating the effects of particle background on the Athena Wide-Field
Imager

Eric D. Miller  Kavli Institute for Astrophysics and Space Research, Massachusetts Institute of Technology, 77 Massachusetts Avenue, Cambridge, MA 02139, USA  0000-0002-3031-2326 Catherine E. Grant  Kavli Institute for Astrophysics and Space Research, Massachusetts Institute of Technology, 77 Massachusetts Avenue, Cambridge, MA 02139, USA  0000-0002-4737-1373 Marshall W. Bautz  Kavli Institute for Astrophysics and Space Research, Massachusetts Institute of Technology, 77 Massachusetts Avenue, Cambridge, MA 02139, USA  0000-0002-1379-4482 Silvano Molendi  INAF/IASF-Milano, Via Bassini 15, I-20133 Milano, Italy  0000-0002-2483-278X Ralph Kraft  Center for Astrophysics | Harvard & Smithsonian, 60 Garden Street, Cambridge, MA 02138, USA  0000-0002-0765-0511 Paul Nulsen  Center for Astrophysics | Harvard & Smithsonian, 60 Garden Street, Cambridge, MA 02138, USA  ICRAR, University of Western Australia, 35 Stirling Highway, Crawley, WA 6009, Australia  0000-0003-0297-4493 Esra Bulbul  Center for Astrophysics | Harvard & Smithsonian, 60 Garden Street, Cambridge, MA 02138, USA  Max Planck Institute for Extraterrestrial Physics, Giessenbachstr. 1 D-85748, Garching, Germany  0000-0002-7619-5399 Steven Allen  Department of Physics, Stanford University, 382 Via Pueblo Mall, Stanford, CA 94305-4060, USA  David N. Burrows  Pennsylvania State University, Department of Astronomy and Astrophysics, 525 Davey Lab, University Park, PA 16802, USA  0000-0003-0729-1632 Tanja Eraerds  Max Planck Institute for Extraterrestrial Physics, Giessenbachstr. 1 D-85748, Garching, Germany  Valentina Fioretti  INAF Osservatorio di Astrofisica e Scienza dello Spazio di Bologna, via Gobetti 93/3, I-40129 Bologna, Italy  0000-0002-6082-5384 Fabio Gastaldello  INAF/IASF-Milano, Via Bassini 15, I-20133 Milano, Italy  /0000-0002-9112-0184 David Hall  Centre for Electronic Imaging, The Open University, Walton Hall, Milton Keynes, MK7 6AA, UK  Michael W.J. Hubbard  Centre for Electronic Imaging, The Open University, Walton Hall, Milton Keynes, MK7 6AA, UK  0000-0002-4378-826X Jonathan Keelan  Centre for Electronic Imaging, The Open University, Walton Hall, Milton Keynes, MK7 6AA, UK  Norbert Meidinger  Max Planck Institute for Extraterrestrial Physics, Giessenbachstr. 1 D-85748, Garching, Germany  0000-0002-6756-413X Emanuele Perinati  Institut für Astronomie und Astrophysik, Universität Tübingen, Sand 1, 72076 Tübingen, Germany  Arne Rau  Max Planck Institute for Extraterrestrial Physics, Giessenbachstr. 1 D-85748, Garching, Germany  0000-0002-6756-413X Dan Wilkins  Department of Physics, Stanford University, 382 Via Pueblo Mall, Stanford, CA 94305-4060, USA  0000-0002-4794-5998

###### Abstract

The Wide Field Imager (WFI) flying on Athena will usher in the next era of
studying the hot and energetic Universe. Among Athena’s ambitious science
programs are observations of faint, diffuse sources limited by statistical and
systematic uncertainty in the background produced by high-energy cosmic ray
particles. These particles produce easily identified “cosmic-ray tracks” along
with less easily identified signals produced by secondary photons or X-rays
generated by particle interactions with the instrument. Such secondaries
produce identical signals to the X-rays focused by the optics, and cannot be
filtered without also eliminating these precious photons. As part of a larger
effort to estimate the level of unrejected background and mitigate its
effects, we here present results from a study of background-reduction
techniques that exploit the spatial correlation between cosmic-ray particle
tracks and secondary events. We use Geant4 simulations to generate a realistic
particle background signal, sort this into simulated WFI frames, and process
those frames in a similar way to the expected flight and ground software to
produce a realistic WFI observation containing only particle background. The
technique under study, Self Anti-Coincidence or SAC, then selectively filters
regions of the detector around particle tracks, turning the WFI into its own
anti-coincidence detector. We show that SAC is effective at improving the
systematic uncertainty for observations of faint, diffuse sources, but at the
cost of statistical uncertainty due to a reduction in signal. If sufficient
pixel pulse-height information is telemetered to the ground for each frame,
then this technique can be applied selectively based on the science goals,
providing flexibility without affecting the data quality for other science.
The results presented here are relevant for any future silicon-based pixelated
X-ray imaging detector, and could allow the WFI and similar instruments to
probe to truly faint X-ray surface brightness.

###### keywords:

X-rays, background, particles, data processing, Athena

*Send all correspondence to Eric D. Miller<EMAIL_ADDRESS>
##  1 Introduction

Silicon-based X-ray imaging instruments typically characterize detected
photons by reconstructing the energy from the spatial pattern of electrons
liberated by the photon interaction with the detector substrate. This
technique, while allowing the detector to be used as an imaging spectrometer,
is complicated by the fact that highly energetic charged particles undergo
similar interactions in such detectors, producing signals that can be
difficult to separate from the photon signal produced by a celestial source
and properly focused by the optics. In attempts to detect extended, very low
surface brightness sources such as galaxy cluster outskirts and the Warm Hot
Intergalactic Medium, this cosmic-ray-induced background is the dominant
source of both statistical and systematic error, the latter arising from our
incomplete knowledge of the time and spectral variability of the underlying
particle flux.

Understanding and minimizing this particle background is vital for future
advanced X-ray imagers, which will attempt to detect this faint extended
emission in long exposures dominated by signals from cosmic-ray protons, alpha
particles, and electrons, as well as photons from the Galactic foreground and
extragalactic background. The Wide Field Imager (WFI)[1] to fly on Athena[2],
ESA’s next large X-ray observatory, is one such instrument. It will fly a 40′
field-of-view array of DEPFET (depleted p-channel field-effect transistor)
active pixel sensors, fully depleted to 450 $\mu$m with a pixel size of
130$\times$130 $\mu$m, and operating in the 0.2–15 keV band with a full-frame
readout time of 5 ms. The Athena science requirements for the non-X-ray
background are a count rate less than $5.5\times 10^{-3}$ counts s-1 cm-2
keV-1 in the 2–7 keV band and knowledge of the background to within a few
percent[3], both challenging goals for a silicon detector in orbit at either
L1 or L2. These requirements, based on the ambitious faint-source science
goals, require careful pre-launch work to both predict the level of background
and develop algorithms to reduce and characterize it once in orbit.

Previous generations of X-ray detectors have generally used one of two methods
to reduce background from cosmic-ray particles: (1) identifying and
eliminating events with pixel activation patterns more likely to be associated
with particle tracks than with X-ray photons; or (2) the use of anti-
coincidence detectors positioned close to the science detector enabling
simultaneous detection of particle tracks and dropping of events when a signal
appears in both detectors. Strategy (1) is useful in eliminating events
produced by the primary particle itself, but such particles can produce
secondaries when interacting with the instrument structure. Secondaries that
are low-energy photons or electrons have indistinguishable pixel patterns from
the cosmic X-rays constituting the signal, and thus there is an irreducible
limit to how well the background can be rejected by simply considering the
event shape. Strategy (2) overcomes this obstacle by eliminating all signal
recorded during the primary particle interaction, including secondaries.
However, for non-triggered detectors, if the integration time is comparable to
the expected arrival interval of cosmic-ray primaries, then most of the frames
will be rejected and much of the real signal will be lost.

Due to its particular characteristics of detector size, pixel size, and
especially its 5-ms frame time, the WFI inhabits a realm where both of these
methods have some strength, and in the end the choice made depends sensitively
on the science goals of an observation. Since the pattern-based background
rejection technique has been employed on several previous and operating
missions, including XMM-Newton EPIC, Chandra ACIS, Swift XRT, and Suzaku XIS,
it is useful to analyze this real-world data. These instruments have the
benefits that we understand their design and function well, and for some we
have a large amount of full-frame data which contains information from all
pixels, including particle tracks. However, the detectors are different in
design and operation from the WFI DEPFETs, especially ACIS and XIS, and Swift
and Suzaku are additionally in low-Earth orbit, a very different particle
environment from Chandra and XMM-Newton in high-Earth orbit and the expected
L1 or L2 orbit of Athena. This analysis is nevertheless illuminating, as we
found strong spatial and temporal correlations between particle tracks
produced by high-energy cosmic rays and events that would be interpreted as
source X-rays. [4, 5, 6].

A large effort has been underway for several years to predict and model the
expected WFI particle background using Geant4[7, 8] simulations, and to use
these simulations to inform the design of both the camera shielding and on-
board event filtering.[9, 10, 11, 12] In this work, we use a set of these
Geant4 simulations of cosmic rays interacting with the WFI camera body to
model the expected unrejected particle background and explore techniques to
separate this signal from the desired X-ray signal. In particular, we study
correlations between those unrejected events and cosmic ray tracks produced by
the same primary particle interaction; these latter signals have historically
been eliminated from telemetered data due to bandwidth constraints. As we
show, there is a direct spatial correlation between particle tracks and
apparently valid events that can be exploited to, in effect, use the WFI as
its own anti-coincidence detector and reduce the unrejected particle
background in a statistical sense. This “Self Anti-coincidence” (SAC) method
exploits both the spatial correlation between particle tracks and valid
events, and the particular frame time of the WFI, during which we expect an
average of a few cosmic ray interactions that produce signal in the detector.
We present results from this analysis along with a description of how SAC can
be tuned depending on the science goals of a particular observation. This
technique is applicable to any future astronomical X-ray imaging instrument
with a fast frame rate, provided sufficient information is telemetered for
each frame.

This paper is organized as follows. In Section 2, we describe the Geant4
simulation output and how this was converted into simulated WFI frames and
event lists, along with characteristics of the simulated background signal and
validation based on existing XMM-Newton data. In Section 3, we present the
results of an analysis of the spatial correlation of particle tracks and
unrejected, X-ray-like events, along with an application and exploration of
the SAC technique. In Section 4 we summarize our findings. An explanation of
SAC and its various metrics of background reduction as developed by the WFI
Background Working Group (BWG) are presented in Appendix A.

##  2 Data & analysis

###  2.1 Geant4 simulations and sorting of data

The Geant4 simulations were performed at The Open University and consisted of
133 runs of $10^{6}$ Galactic cosmic ray (GCR) proton primaries per run, drawn
from the CREME 96 standard spectral model for solar minimum [13] and generated
on a 70-cm radius sphere surrounding the WFI instrument. These simulations
used a simplified WFI mass model designated E0015261, which includes the
camera, proton shield, filter wheel, and baffle, but excludes a graded-Z
shield under later study by the WFI BWG to reduce the impact of energetic
cosmic X-ray background photons and of secondary electrons produced by GCR
interactions in the proton shield. This is the same mass model used to obtain
results previously presented,[9] and we refer the reader there for more
detailed information about the Geant4 simulation setup and operation. For each
GCR primary that generated signal charge in the WFI detector, the data include
the deposited energy in keV in each pixel and information about the particle
(primary or secondary) responsible for the deposition. The vast majority of
simulated primaries do not interact with the WFI detector; indeed, only
936,934 of 133,000,000 (0.7%) produce signal in any pixels.

The Geant4 output was structured into two different formats for further
analysis. The first dataset was structured on a primary-by-primary basis,
hereafter referred to as “single-primary” frames, and this was used to explore
fundamental properties of the signal produced by individual cosmic rays and
search for useful correlations between particle tracks and events that look
like X-rays that could be exploited to flag the latter. The second type of
dataset has primary GCRs randomly sorted into frames of a finite exposure time
to simulate a real-world observation of the WFI background. While the WFI is
expected to operate at 5 ms per frame[1], we simulated a range of frame times
from 0.2 ms to 5 ms, and focus here specifically on 5 ms and 2 ms, to compare
the effects of readout rate on SAC background reduction. Considering different
frame times also serves as a proxy for sampling solar cycle variability, since
a 2-ms frame will have 40% of the particle fluence of a 5-ms frame, similar to
the factor of $\sim$2 difference in GCR flux observed between solar maximum
and minimum[4]. To construct the datasets, we sorted primaries into frames
using the effective total exposure time given by Eq. 4 of Fioretti et al.
(2012)[14],

$t_{exp}=\frac{N_{p}}{\Phi\times 4\pi^{2}R^{2}}=\frac{N_{p}}{\phi\,\pi
R^{2}},$ (1)

where $N_{p}$ is the number of simulated primary protons, $\Phi$ is the cosmic
ray proton intensity in units of cm-2 s-1 sr-1 at the assumed Athena L1 or L2
orbit, $\phi=4\pi\Phi$ is the cosmic ray proton flux in units of cm-2 s-1, and
$R=70$ cm is the radius of the simulation boundary sphere. The conversion from
intensity to flux assumes an isotropic cosmic ray intensity, and like Fioretti
et al. (2012)[14], we have drawn simulated protons from a cosine law angular
distribution, although without restricting the flux to a small cone.

We assume $\phi$ = 4.1 cm-2 s-1 for GCR protons, based on SPENVIS[15]
simulations of the CREME 96 spectral model for solar minimum [13], yielding
$t_{exp}=15.8$ s for a single Geant4 run of $N_{p}=10^{6}$ primaries. As we
show below, this proton flux produces an average 2–7 keV unrejected count rate
consistent with that derived previously by the WFI BWG for protons only,
$5\times 10^{-3}$ cm-2 s-1 keV-1 [9]. However, since the real particle
background environment includes other species such as GCR alpha particles,
electrons, and gamma rays, we increased the proton flux by 40% to account for
these primaries missing from the Geant4 simulations. This produced a total
average 2–7 keV unrejected count rate consistent with that found by previous
Geant4 analysis amongst the BWG[9], $\sim 7\times 10^{-3}$ cm-2 s-1 keV-1. We
note that the details of the secondary interactions are likely different
between protons and these other species, but to first order this is a
reasonable approximation. We also note that this is a reasonable upper limit
to the GCR flux, as it is based on recent solar minimum observations and in an
extended mission Athena could observe during all parts of one or more solar
cycles.

The scaled GCR primary flux yields a total effective exposure time of 1505 s
for the 133 million primaries, a rate of $8.84\times 10^{4}$ s-1, or 441.9 per
5-ms frame (176.8 per 2-ms frame). Using this as the mean rate, each of the
133 million primaries was assigned a random arrival time drawn from an
exponential distribution, appropriate for modeling arrival intervals of this
Poisson process. Primaries were then assigned into each frame according to
these arrival times. We determine a mean rate of 3.11 interacting primaries
per frame in the 300,967 5-ms frames that were simulated. Of these frames,
95.5% have signal in them, consistent with the expectation from the assumed
Poisson distribution. The simulated 2-ms frames are similarly consistent, with
an average rate of 1.25 interacting primaries per frame, and 71.2% of the
752,331 total frames containing signal.

For each case (single-primary, 5-ms, and 2-ms frames), each frame with signal
was turned into an image of pixel values using the pixel X, Y, and deposited
energy information provided by Geant4. These simulations recorded signal
deposited in a $1181\times 1181$ pixel grid, using 130-$\mu$m pixels and
including a 3-pixel (0.42-$\mu$m) gap between the quadrants. This is larger
than the full WFI Large Detector Array (LDA) field of view, with $512\times
512$ pixel quadrants, or a $1027\times 1027$ pixel full field including the
same gaps. While assembling frames, we simply excised the outer pixels. Any
primaries that have signal only in the excised region were treated as though
they had not interacted with the detector. Any primaries that had signal in
both the outer (excised) and inner regions had their outer signal removed and
inner signal retained. We note that this chip gap is significantly smaller
than the likely WFI design gap, $\sim 4$ mm.

###  2.2 Identifying valid events and particle tracks

Each image was searched for events using a local-maximum method similar to
that employed on-board many X-ray imaging instruments like XMM-Newton EPIC pn
and Chandra ACIS. First an event threshold of 0.1 keV was applied, and pixels
at or above this level were flagged as event candidates. Each candidate pixel
was compared to the other pixels in its 3$\times$3 pixel neighborhood, and if
it was a local maximum it was flagged as an event center. The 5$\times$5
neighborhood around each event center was then searched for pixels at or above
the neighbor (or split) threshold, also set at 0.1 keV. The event pattern was
assigned using EPIC pn rules[16], including single-pixel events (PATTERN=0),
doubles (PATTERN=1–4), triples (PATTERN=5–8), quadruples (PATTERN=9–12), and
everything else (PATTERN=13). In particular, for all non-single-pixel events
which have a 3$\times$3 neighbor above the neighbor threshold, the outer
5$\times$5 was also searched for pixels above the neighbor threshold. Double,
triple, and quad patterns with at least one outer 5x5 pixel above the neighbor
threshold were assigned PATTERN=13. In the remainder of this work, “valid”
events (used interchangeably with “counts”) are those with PATTERN$<$13, as
these are indistinguishable from events produced by X-ray photons. The energy
of the event is the summed energy of all pixels in the inner 3$\times$3 island
that are above the neighbor threshold. Because of the 5$\times$5 pattern
assignment, events with centers within 2 pixels of the edge of a quadrant were
excluded. This reduces the sensitive detector area by 1.6%. Figure 1 shows the
spectra of valid, invalid, and all events.

Figure 1: Spectra of events produced by the Geant4 GCR proton primary
simulations. (left) The spectrum over a wide energy band, showing pattern-
based valid and invalid events separately. Valid events dominate by several
orders of magnitude in the 2–7 keV band, while invalid events dominate above
100 keV, well outside the WFI sensitive band. (right) Spectrum in the 2–7 keV
region, with the WFI unrejected background requirement of $5.5\times 10^{-3}$
s-1 cm-2 keV-1 plotted as a dashed line. Colored lines indicate what types of
particles produce the detected signal for these events–primarily secondary
electrons and photons produced in primary proton interactions with the WFI.
The strong line near 1.5 keV is Al K$\alpha$, and the weaker line near 1.7 keV
is Si K$\alpha$.

We identified particle tracks using image segmentation in each frame.
Hereafter, a “particle track” is defined as a pattern which is either (1) a
spatially contiguous set of five or more pixels above the neighbor threshold,
0.1 keV; or (2) any contiguous set of pixels above 0.1 keV that includes at
least one pixel over 15 keV. This latter energy is called the “MIP threshold”,
an energy above which the Athena mirrors have effectively zero efficiency, and
thus all signal is assumed to be produced by cosmic ray minimum ionizing
particles, or “MIPs”. Detached diagonals are considered contiguous in this
image segmentation, and we did not apply the spatial edge filtering to
particle tracks as we did to events, since these regions contain useful
knowledge about their presence. Note that our definition of “particle track”
differs slightly from that used for the EPIC pn analysis[6] due to option (2).
Each particle track was assigned an ID number to uniquely identify it in the
full dataset. Examples of particle tracks are shown as postage stamps in
Figure 2. A single primary can produce multiple detached particle tracks.

Finally, in each frame, the distance between the central pixel of each event
and the nearest pixel in a particle track was calculated. Many events fall on
particle tracks and so have a distance of zero. Valid events are by definition
unable to fall on a particle track pixel. Thus valid events and particle
tracks are a mutually exclusive set of entities, despite the different methods
used to identify them. A schematic diagram of this distance finding technique
is shown in Figure 3.

To aid our analysis of the correlations between particle tracks and valid
events, we assigned frames to “cases” in the same way as the XMM-Newton EPIC
pn analysis[6], namely:

  * •

Case A: frame contains only particle tracks.

  * •

Case B: frame contains only valid events.

  * •

Case C: frame contains both particle tracks and valid events.

  * •

Case D: frame contains neither particle tracks nor valid events (empty frame).

This sorting was done for the single-primary frames as well as the 2-ms and
5-ms frames. Summary information about the fraction of frames and rates of
particle tracks and valid events in each case is given in Table 1 and explored
in more detailed in the following sections.

Figure 2: Images of a small sample of individual particle tracks, with the color scale in keV. Pixels shown are equivalent to 130 $\mu$m WFI LDA pixels, so the image sizes are not the same and scale with the size of the tracks. Figure 3: Schematic of a frame containing particle tracks (white pixels) and valid events (green pixels). The image segmentation would identify three particle tracks in this frame. Orange arrows indicate the distance between each valid event, defined by the maximum pixel of the 3$\times$3 island, and the nearest particle track or MIP pixel. Table 1: Summary information for frame-by-frame analysis. Type of Frame | Single primary | 5 msec | 2 msec
---|---|---|---
no. frames | 133,000,000 | 300,967 | 752,331
no. frames with signal | 936,934 (0.7%) | 287,424 (95.5%) | 535,360 (71.2%)
no. frames with particle track | 918,662 (0.7%) | 286,580 (95.2%) | 530,114 (70.5%)
no. particle tracks per frame | 0.0078 | 3.45 | 1.38
Case A (frame with only particle tracks)
fraction of all frames | 0.68% | 87.3% | 67.8%
fraction of frames with signal | 97.3% | 91.4% | 95.3%
no. particle tracks per frame | 1.12 | 3.57 | 1.94
fraction of valid events |  $\cdots$ |  $\cdots$ |  $\cdots$
Case B (frame with only valid events)
fraction of all frames | 0.013% | 0.3% | 0.6%
fraction of frames with signal | 1.8% | 0.3% | 0.9%
no. particle tracks per frame |  $\cdots$ |  $\cdots$ |  $\cdots$
fraction of valid events | 64.9% | 3.1% | 18.8%
Case C (frame with both particle tracks and valid events)
fraction of all frames | 0.007% | 7.9% | 2.7%
fraction of frames with signal | 0.9% | 8.3% | 3.8%
no. particle tracks per frame | 1.89 | 4.20 | 2.51
fraction of valid events | 35.1% | 96.9% | 81.2%
Case D (frame with neither particle tracks nor valid events)
fraction of all frames | 99.3% | 4.5% | 28.9%
fraction of frames with signal |  $\cdots$ |  $\cdots$ |  $\cdots$
no. particle tracks per frame |  $\cdots$ |  $\cdots$ |  $\cdots$
fraction of valid events |  $\cdots$ |  $\cdots$ |  $\cdots$

###  2.3 Validating the Geant4 Simulations

The XMM-Newton EPIC pn data[6] were used to validate the Geant4 simulations to
give us confidence that the latter represent a reasonable simulation of the
expected WFI particle environment and background. To first order, the relative
fractions of Case A, B, and C frames should be similar between the two,
although there are differences in the instruments. These include frame time
(5.7 ms for EPIC pn vs. 5 ms for WFI), pixel size (150 $\mu$m for EPIC pn vs.
130 $\mu$m for WFI), and depletion depth (280 $\mu$m for EPIC pn vs. 450
$\mu$m for WFI), along with the absence of detector effects like charge
splitting in the WFI simulations, differences in the instrument structure and
shielding, and differences in the particle environment in the XMM-Newton high-
Earth orbit and the Athena orbit at L1 or L2.

To mimic the Small Window Mode (SWM) used in the EPIC pn study, we used the 5
ms frames from Geant4 and, in every frame, drew a 64$\times$64 pixel square
that included a pixel randomly chosen from those pixels with signal above the
lower threshold. The center of the square was randomly assigned as long as it
contained that pixel and fell within the limits of the full LDA field of view.
Event finding and particle track image segmentation were performed using only
the pixels within this square, using the methods described in Section 2.2. In
this way we performed a similar processing to the EPIC pn SWM mode data, but
only including frames with signal. These frames were sorted into Cases A, B,
and C, and their relative fractions are shown in Table 2. Of the frames with
signal, half as many (1% vs. 2%) contain valid events in the Geant4 data
compared to the EPIC pn SWM data. Of these frames with valid events (Case B
and C), we find that 86% also have a particle track in the Geant4 analysis,
virtually identical to the 87% value for the EPIC pn SWM frames. That there is
a higher fraction of frames with valid events in the XMM data could reflect
differences in instrument design and operation, or an underestimation of the
background rate from Geant4 similar to what has been seen on eROSITA [17]. The
similarity of the Case B and C fractions is remarkable, however, and we
conclude that the Geant4 simulations produce a valid representation of the
expected WFI background for our purposes of exploring correlations between
particles tracks and valid events.

Table 2: Relative frequency of different frame cases in EPIC pn data and Geant4 simulations. Frame Type | Fraction of Frames with | Fraction of Frames with
---|---|---
| Signal (Case A+B+C) | Valid Events (Case B+C)
| EPIC-pn | Geant4 | EPIC-pn | Geant4
Case A (particle track only) | 98.00% | 99.04% |  $\cdots$ |  $\cdots$
Case B (valid event only) | 1.75% | 0.83% | 87% | 86%
Case C (particle track + valid event) | 0.25% | 0.13% | 13% | 14%

###  2.4 Spectral Properties of Particle Tracks in Geant4 Simulations

With all particle tracks identified, we explored whether the small but
systematic differences seen in the EPIC pn SWM Case A and Case C particle
track spectra data[6] were also present in the Geant4 simulation data. For
each particle track, we calculated two versions of the total energy, first
using the full range of pixel energies, and second clipping each pixel at 22.5
keV to mimic the dynamic range of EPIC pn, similar to that expected for the
WFI. In both case, the pixels in a particle track are then summed to get the
total energy, the distribution of which is shown in Figure 4. Interestingly,
we see a flattening or excess of Case C particle tracks at high energy
compared to Case A, similar to what is seen in the EPIC pn SWM data and
providing further validation that the Geant4 results produce a reasonable
simulation of the background. This may indicate different secondary particle
production mechanisms for the Case C particle tracks, which are accompanied by
valid events, compared to the Case A particle tracks, which are not. The fact
that this difference is also seen in the clipped pixel data suggests a
possible method of using the particle track energies to statistically identify
frames which are likely to contain unrejected background. The level of
background improvement and feasibility of this method are left for future
work.

Figure 4: Spectra of summed particle track energy. The left plot sums full
pixel values, while the right plot clips pixels at 22.5 keV to mimic the
dynamic range of EPIC pn, similar to that expected for the WFI. The top plots
show both Case A and Case C, while the bottom plots normalize by Case A. A
high-energy excess is seen in Case C particle tracks, similar to what is seen
in EPIC pn[6]. It is seen in both unclipped and clipped spectra.

##  3 Results

###  3.1 Self-Anti-Coincidence (SAC)

That valid events are spatially correlated with primary or secondary particle
tracks from the same interacting cosmic ray was recognized early on in Geant4
simulations by the WFI Background Working Group[9] and in the analysis of in-
orbit Chandra and Swift data[4, 5]. This correlation can be exploited by
masking around particle tracks and flagging valid events within a certain
distance; such events can later be filtered in ground processing depending on
the science goals of the observation. However, this masking also reduces the
signal and thus the efficiency of the instrument. This optional, partial-veto
method has been termed “Self-Anti-Coincidence” (SAC), since under this scheme
the WFI detector acts as its own anti-coincidence detector. Throughout the
remainder of this work, we analyze the effects of SAC on different background
reduction metrics, and explore the background improvement possible with
enhanced, SAC-enabled post-processing algorithms.

Figure 5: Radial distribution of valid events (left) and particle tracks
(right), normalized to the detector area. The lower panel of the right plot is
a zoom-in to more clearly show the Case C points. The valid events overall
have a flat distribution, however those valid events that accompany a particle
track (Case C) are concentrated toward the center, and those that have no
particle track (Case B) are more likely to be found near the edge. The
particle tracks for those cases follow similar trends. This is expected, as a
valid event detected near the edge is more likely to lose an accompanying
particle track outside the field of view.

###  3.2 The Empirical Correlation Between Particle Tracks and Valid Events

Frames containing single cosmic ray primary particles are key to understanding
the spatial correlation between particle tracks and valid events. The area-
normalized radial distributions of valid events and particle tracks derived
from these single-primary frames are shown in Figure 5. While the valid events
have a flat distribution overall, those that accompany particle tracks (Case
C) are more likely to be found toward the center of the frame, and those that
lack a particle track (Case B) are more likely near the edge. The particle
tracks for those cases follow similar trends. This is expected, since a valid
event detected near the edge is more likely to lose an accompanying particle
track off the edge.

A useful metric to quantify this spatial correlation is the cumulative
probability that a valid event falls within a certain radius of a particle
track resulting from the same cosmic ray interaction. We define this
probability as $P_{cor}$$(<r_{e})$, where $r_{e}$ is the “exclusion radius” to
indicate its use in filtering unrejected background. A detailed analytic
derivation of $P_{cor}$ is presented in Appendix B, based on results from a
previously published WFI Geant4 study[9]. We determine $P_{cor}$ empirically
from our Geant4 results as the cumulative distribution of radius in pixels
between all Case C valid events and the nearest pixel in a particle track (the
orange vectors in Figure 3). To normalize $P_{cor}$ to the full LDA field of
view, we assume that Case B valid events have a corresponding particle track
somewhere outside of the field. Thus we divide the distribution by the total
number of valid events in Cases B and C. The resulting distribution is shown
in Figure 6, plotted with the analytic $P_{cor}$ curves from Figure 26 in
Appendix B, with lines for an infinite plane (black), a full LDA field (blue),
and an LDA quadrant (red). Our orange curve is consistent with the model for
the full LDA field, despite the very different methods used to derive the two.
At the largest $r_{e}$, the correlation probability reaches 35%. This is the
maximum amount of effective background improvement we can achieve by using
SAC; the other 65% of valid events are produced in Case B primary interactions
that do not also produce a particle track in the LDA field (see Table 1).

Figure 6: Cumulative probability that a valid event falls within an exclusion
radius $r_{e}$ of a particle track produced by the same primary. The orange
line is derived from our single primary Geant4 simulation data. The other
lines are from Figure 26 in Appendix B. Figure 7: Distribution of the number
of particle tracks produced by primaries that do not also produce a valid
event (Case A) and those that do (Case C). Valid events are more likely to be
accompanied by a plurality of particle tracks. This can be used to identify
frames that are likely to include valid events. Note that the histograms are
shifted slightly along the X axis to improve clarity.

In addition to a spatial correlation between particle tracks and valid events,
we have found that proton primaries that produce valid events are much more
likely to produce multiple particle tracks. This can also be seen from Table
1, which shows that, among primaries that produce signal in the detector, Case
A primaries produce on average 1.1 particle tracks, while Case C primaries
produce 1.9 particle tracks. To further explore this, we plot in Figure 7 the
distribution of particle track number for Case A and Case C primaries. Only 6%
of Case A primaries produce multiple particle tracks, whereas 30% of Case C
primaries do. Qualitatively, this makes sense; a primary interaction in the
WFI structure can produce a shower of secondaries striking the detector, and
these secondaries include both high-energy particles that produce tracks and
lower energy photons and electrons that produce valid events. The number of
independent particle tracks in a WFI frame contains some information about the
likelihood of a valid event being present, and thus counting them could be a
useful method to reduce the background. However, since this plurality occurs
in 30% of Case C primaries, and such primaries account for only 35% of the
valid events, no more than 10% of the 2–7 keV background may be eliminated
with this method. The potential gain is further reduced by the expectation of
$\sim$3.5 particle tracks per 5-ms frame (see Table 1). Nevertheless, we
continue to explore “multi-track” selective SAC, whereby SAC is applied only
on frames with a certain number of particle tracks, in the remainder of this
work.

###  3.3 Applying SAC to Geant4 Frame Data

We identify three metrics to represent improvement in the particle background.
One is the simple level of the background which is used to define the WFI
requirement. The other metrics, more relevant for certain Athena science
cases, are the signal-to-background ratio, which is an estimator of systematic
uncertainty; and the signal-to-noise ratio, an estimator of statistical
uncertainty. Both are important in the background-dominated regime, although
the level of importance depends on the details of the science goals being
pursued. These metrics are derived in analytical terms in Appendix A.

####  3.3.1 Background Reduction and Lost Signal

The fractional reduced background is $b=B/B_{o}$, where $B_{o}$ is the
original background before SAC is applied, and $B$ is the background after
applying SAC masking, both measured in counts of valid events. Likewise, the
fractional reduced signal is defined as $s$ = $S/S_{o}$, where $S_{o}$ is the
original source signal (in counts) before SAC is applied, and $S$ is the
signal after applying SAC masking. Unless noted otherwise, all of the metrics
below using $s$ and $b$ are independent of the actual source or background
flux, as shown in Appendix A. In some cases this assumes that the observations
are background dominated. We also assume the source is uniform spatially and
temporally, so that $S/S_{o}$ goes as the fraction of area remaining after
SAC, or $(1-A_{R}/A_{T})$ in the notation of Appendix A, where $A_{R}$ is the
rejected area and $A_{T}$ is the total area.

We calculate $s=S/S_{o}$ for a particular SAC exclusion radius $r_{e}$
directly from the Geant4-derived frame data. We create a mask for each frame
and draw a circle of radius $r_{e}$ around each pixel in a particle track (or
MIP pixel). Pixels inside these circles are set to zero, and those outside are
set to one. The remaining fractional area $a_{good}=[1-A_{R}(r_{e})/A_{T}]$,
and thus $S(r_{e})/S_{o}$, is simply the ratio of the amount of masked area to
total area, summed over all frames. This method is shown schematically in
Figure 8. Unlike the Appendix A method, which uses a Monte Carlo simulation to
calculate this value, our method is subject to statistical limitation. For
long frame time and large $r_{e}$, very little area is retained, so the
uncertainty on $A_{R}$ becomes large.

Figure 8:  Example of calculating the fractional remaining area after SAC
masking. The left image shows a single, full LDA frame with several particle
tracks. The center image shows masking with a radius of 10 pixels around each
particle track pixel, and the right frame shows masking with radius 200
pixels. Overlapping masking circles require us to empirically determine the
remaining area.

We calculate $b=B/B_{o}$ for a particular SAC exclusion radius $r_{e}$ in a
similar way from Geant4 results. In this case, we simply eliminate all valid
events within $r_{e}$ of a MIP pixel, using the distance calculated in Section
2.2. $B(r_{e})/B_{o}$ is then the number of remaining valid events divided by
the original number. We note that $B/B_{o}$ is identical whether we restrict
the valid events to 2–7 keV or use all valid events below 15 keV. Since the
latter contains three times as many events as the former, we use all events
below 15 keV to increase the statistics. Nevertheless, as for the masked area,
for long frame time and large $r_{e}$ there are few valid events remaining and
the uncertainty on $B/B_{o}$ becomes large. Where possible, we include these
uncertainties in the following analysis.

In this notation, the first metric, the fractional reduction in background
surface brightness $f_{BG}$, can be written as

$f_{BG}=(1-F_{BG}/F_{o,BG})\,,$ (2)

where $F_{BG}$ and $F_{o,BG}$ are the reduced and original background surface
brightness, respectively. Since

$F_{BG}=B/A_{good}$ (3)

and

$F_{o,BG}=B_{o}/A_{T}\,,$ (4)

where the fractional remaining area is

$a_{good}\equiv A_{good}/A_{T}=(1-A_{R}/A_{T})=s\,,$ (5)

some math tells us that

$f_{BG}=(1-b/s)\,.$ (6)

Since this is a surface brightness, it depends on both the remaining
fractional area and the number of remaining background valid events. We plot
this value as a function of $r_{e}$ in Figure 9, along with $s$ and $b$. We
further note that $f_{BG}$ cannot exceed $P_{cor}$ as shown in Figure 6, since
only background events correlated with the masked particle track contribute to
the background surface brightness reduction. Other background events are
removed at a rate simply proportional to the lost area (denoted $P_{ran}$ in
Section A.2), so there is no reduction in surface brightness. However, this
assumes the random background events are also uniformly distributed; as we
showed in Figure 5, this is not exactly the case.

Figure 9: Remaining signal $s$ and background $b$ as a function of SAC
exclusion radius, for 5-ms (left) and 2-ms (right) frame times. The background
reduction exceeds the signal loss at all masking radii. Also shown for
reference is the fractional reduction in background surface brightness,
$f_{BG}$, as a function of masking radius. This cannot exceed $P_{cor}$, shown
in Figure 6, since only correlated background events contribute to this
improvement. Figure 10: Signal-to-background ratio for 5-ms (left) and 2-ms
(right) frame time. Dashed lines correspond to lines of the same color in
Figure 21, and are calculated from an analytic treatment of $s/b$ for an
infinite plane (black), full LDA field (blue), and a single LDA quadrant (red)
(see Appendix A.3). The solid blue line shows Eq.7 using $P_{cor}$ from Figure
6, and is similar to the full-field analytic relation. The green curves show
$s/b$ for different multi-track settings, with SAC enforced only on frames
that contain at least the number of particle tracks shown. A 1 $\sigma$ error
region is shown for the “any tracks” curve for reference; this curve is
measured directly from the simulation data, and the fact that it is largely
consistent with both the dashed blue analytic curve and solid blue
$P_{cor}$-derived curve provides an important cross-check of the methodology
and different Geant4 simulations that informed each analysis.

####  3.3.2 The Signal-to-Background Ratio

The signal-to-background ratio, $s/b$, is an indicator of the systematic error
in the measurement due to the irreducible limit of knowledge of the
background. We plot this as a function of SAC exclusion radius in Figure 10,
along with curves derived in Appendix A.3 and Figure 21, for frame times of 5
and 2 ms. The different green curves labelled “any tracks”, “$>$1 track”,
etc., indicate the results from selective application of SAC only in frames
that contain at least that many particle tracks. The “any tracks” curve
corresponds to standard SAC, which masks around any particle track pixel in
all frames. This curve is similar to the dashed blue curve calculated from

$\frac{s}{b}=\frac{(S/S_{o})}{(B/B_{o})}=\frac{1}{(1-P_{cor})}\,,$ (7)

which is Eq.24 derived in Appendix A.3. We calculate the empirical version of
$s/b$ using the orange $P_{cor}$ curve in Figure 6 and show that as the thin
blue line in Figure 10. This is consistent with the relation derived
analytically in the Appendix, and also consistent with the directly determined
“any tracks” curve, an important cross-check of the methodology and different
Geant4 simulations that informed each analysis.

Since $s/b$ depends only on $P_{cor}$, there should be no change with frame
time. The differences between 5 and 2 ms in the “any tracks” line (standard
SAC) are due to statistical limitations in calculating the lost area and
reduction in valid events from the background. The differences in multi-track
selective SAC are real; for the shorter frame time, which is also a proxy for
lower GCR flux, there are fewer particle tracks per frame, and thus fewer of
the frames are participating in the background reduction.

It is clear that applying SAC to the full frame has a substantial benefit in
this metric compared to applying it to a quadrant. This remains true when
applying selective SAC only to frames with two or more particle tracks (“$>$ 1
track” curve), at large exclusion radius.

####  3.3.3 The Signal-to-Noise Ratio

In the background-limited regime where $B\gg S$, and assuming counting
statistics dominate any systematic errors, the signal-to-noise ratio is
$SNR=(S/B)^{1/2}$. We follow Eq.29 in Appendix A.4 and define a normalized
signal-to-noise ratio as

${snr}\equiv\frac{S/S_{o}}{(B/B_{o})^{1/2}}\,,$ (8)

Our derived $snr$ curves are shown in Figure 11 as a function of exclusion
radius for 5 and 2 ms frame time, again with multiple curves for multi-track
selective SAC, and including dashed lines derived in Appendix A.4. The solid
blue line shows the solution for Eq.30 using our empirical $P_{cor}$, and this
is fully consistent with the measured “any track” $snr$, indicating that the
two independent Geant4 simulations produce compatible descriptions of
$P_{cor}$. The derived $snr$ is similar to the dashed blue analytically
derived curve, although they deviate at large exclusion radius.

Figure 11: Signal-to-noise ratio for 5-ms (left) and 2-ms (right) frame time.
Dashed lines correspond to lines in Figure 22 of Appendix A.4, and the blue
line with circles shows our result of Equation 30 also from Appendix A.4.
Other notations are the same as in Figure 10. The multi-track method produces
higher snr than standard SAC, especially at large exclusion radius. That the
analytically derived relation is consistent with our empirically derived “any
tracks” relation again provides a valuable cross-check for our methodology.
Figure 12: Signal-to-noise ratio vs. signal-to-background ratio. Applying SAC
in 5-ms frames with at least two particle tracks provides an improvement over
standard SAC.

Applying SAC to the full frame degrades the $snr$, especially at large
exclusion radius, as more signal is lost. To explore the trade-off between
$s/b$ and $snr$, in Figure 12 we plot $s/b$ vs. $snr$ for different settings
of selective SAC. For 5-ms frame time, applying SAC in frames with at least
two particle tracks provides a slight improvement over standard SAC.

####  3.3.4 Effects of Rolling Shutter

We have made a simplifying assumption in the above that the full frame is read
out instantaneously. In practice, the WFI LDA will implement a rolling shutter
whereby each detector row is read out in sequence from top to bottom over the
course of the 5-ms frame time. This means that any arriving cosmic ray may
produce secondary particles that land on the other side of the current rolling
shutter row, and since this happens virtually instantaneously compared to the
speed of the rolling shutter, it results in the primary particle track and
secondary events appearing in different frames. As we show analytically in
Appendix A.5, this degrades the effectiveness of SAC, since it alters the
spatial correlation between particle tracks and valid events by introducing a
temporal dependence.

Figure 13: Schematic showing the effects of the WFI LDA rolling shutter on SAC
from the point of view of a MIP track that accompanies four valid events. The
left two panels show the two major cases, where the MIP track is below (Case
1) or above (Case 2) the current readout row, which moves downward in these
depictions. The right-most panel shows a depiction of the full LDA focal
plane. We assume the rolling shutters are synchronized between the four DEPFET
sensors and move in the directions shown.

We approach this from the perspective of the particle track, since that is a
natural way for the SAC masking to be defined. We define frame $n$ as the
frame in which the particle track is recorded by the WFI, and we assume that
any pixels activated by the particle are done so instantaneously, in a time
much shorter than the row readout time ($\ll 10$ $\mu$s). This includes pixels
that are activated directly by the primary particle or by any secondaries
produced by interaction with the WFI structure. A schematic of a particle
interaction with a simplified WFI LDA is shown in Figure 13. The particle
track of MIP pixels is shown in orange. In this example, the particle produced
four secondary valid events, shown as blue pixels, which here cover all the
possible configurations of the particle track, valid events, and the current
readout row. There are two general cases: the current readout row is above the
particle track (Case 1), or it is below the particle track (Case 2), where
“above” and “below” are defined for the rolling shutter moving downward. In
Case 1, valid events B, C, and D are read into frame $n$ along with the
particle track. Valid event A is above the rolling shutter, so it will be read
into frame $n+1$. In Case 2, valid events A, B, and C will be read into frame
$n$ along with the particle track. Valid event D is below the rolling shutter
and so is read into the frame currently being read, $n-1$. We ignore the case
where a MIP track lands on the current readout row, which should occur for
$<1$% of MIP tracks. We finally assume that the rolling shutters on all four
DEPFET sensors are synchronized, and that they operate as shown in the right
panel of Figure 13.

We first determine the effects of rolling shutter on $P_{cor}$, the cumulative
correlation between particle tracks and valid events they produce, using the
single primary data set. For each primary, we randomly assign a rolling
shutter row, and then for each particle track produced by that primary, we
eliminate valid events on the other side of the shutter row before
accumulating the probability distribution. The resulting $P_{cor}$ is shown in
Figure 14 as a dashed orange line; comparison to the non-rolling-shutter
$P_{cor}$ (solid orange line) shows little difference at small exclusion
radius and a $\sim$20% reduction at large exclusion radius. This makes
qualitative sense. At the smallest particle track/valid event separations, it
is very unlikely the rolling shutter will happen to fall between a particle
track and its nearby secondary events. At intermediate separations, $r_{e}=$
400–600 pixels, this becomes much more likely, and we see a large deviation of
$\sim$ 20% from $P_{cor}$ with no rolling shutter. At the largest separations,
there are very few particle track/valid event pairs contributing to the
cumulative correlation, and so the rolling shutter effect is diluted and
$P_{cor}$ remains about 20% below the non-rolling-shutter value. This 20%
effect is less than the factor of two (or 50%) estimated by the analytic
treatment in Appendix A.5; the latter is really an upper limit, since it
assumes the distribution of secondary valid events on the detector is random,
rather than spatially correlated with the primary particle track as we have
shown.

Figure 14: Cumulative probability that a valid event falls within an exclusion
radius $r_{e}$ of a particle track produced by the same primary, comparing the
results without rolling shutter (from Figure 6) and those including rolling
shutter. There is very little difference at low $r_{e}$, since there is a low
probability of the shutter row interloping between a particle track and its
secondary events at these small distances. At larger $r_{e}$, the correlation
degrades by about 20%. Figure 15: (Left) Signal-to-background ratio for 5-ms
frame time with rolling shutter included. Dashed lines are as in Figure 10.
The solid blue line shows Eg.7 using $P_{cor}$ from Figure 14, and
demonstrates that the rolling shutter effect degrades this SAC background
reduction metric at large exclusion radius ($s/b\approx 1.4$ with rolling
shutter included compared to $s/b\approx 1.5$ without). The green curve and 1
$\sigma$ error region is measured directly from the simulated rolling shutter
data, and it is largely consistent with the solid blue $P_{cor}$-derived
curve. (Right) Signal-to-noise ratio for 5-ms frame time with rolling shutter
included. Dashed lines are as in Figure 11. The thin blue line shows our
result of Equation 30, and is consistent with the green curve derived from
rolling shutter simulations.

To quantitatively measure the effects of the rolling shutter on our SAC
background reduction metrics, we adopt the “minimal exclusion” scheme
described in Appendix A.5, whereby we only exclude valid events in the same
recorded frame as a particle track, instead of also treating the preceding and
trailing frames. Since we are including the effects of rolling shutter in the
simulations but essentially ignoring them in the data analysis, this is a
conservative approach to estimate the impact. We determine the signal-to-
background ratio $s/b$ and signal-to-noise ratio $snr$ as described in
Sections 3.3.2 and 3.3.3, and show the results in Figure 15. We don’t show
results for the “multi-track” analysis here, but rather enforce SAC on frames
with any number of MIP tracks. Once again, the empirically derived relations
(green curves) are very similar to those calculated from the $P_{cor}$-based
relations derived in Appendix A.3 and A.4 (blue curves). The $s/b$ relation
departs from what is shown in Figure 10 without rolling shutter; the
improvement in this metric at large exclusion radius is about 25% lower with
rolling shutter included in the simulated observations and the “minimal
exclusion” SAC analysis scheme implemented. This is fully driven by the
difference in $P_{cor}$. The $snr$ is not greatly different from the non-
rolling-shutter version, and in any event the improvement in $snr$ is
restricted to small $r_{e}$, where the rolling shutter has minimal impact.

In the remaining analysis, unless otherwise noted, we focus on the simplified
simulations that exclude rolling shutter.

###  3.4 Practical Mitigation of the Background Using SAC

The preceding analysis shows that, by employing SAC, we are able to reduce the
background as measured by any of these three metrics:

  * •

Number of unrejected (valid) background events, $b=B/B_{0}$

  * •

Signal-to-background ratio, $s/b=(S/S_{0})/(B/B_{0})$

  * •

Signal-to-noise ratio, $snr=(S/S_{0})/(B/B_{0})^{1/2}$.

Here again, $B_{0}$ and $S_{0}$ are original background and signal counts, and
$B$ and $S$ are background and signal counts that remain after SAC masking.
Regardless of metric, the SAC background reduction is always accompanied by a
loss of signal at the combination of expected frame rate and GCR flux for the
WFI. We show this in Figure 16, which plots the three metrics against the
fractional reduction in signal counts. We also impose notional but somewhat
arbitrary requirements on the SAC technique: it must improve the background by
at least 25% while reducing the signal by no more than 5%. Here “improvement”
depends on the metric; it can be a fractional reduction in the background
count rate, or a fractional increase in $s/b$ or $snr$. The upper left
quadrant satisfies these requirements, and at no point for either 5 ms or 2 ms
frame time, or for enforcing different multi-track SAC thresholds, does the
line pass through this quadrant. Only with short frame times of $<$0.5 ms do
any of the metrics pass through the necessary quadrant. For these frame times,
SAC remains useful, since the exclusion radius is smaller than the LDA field
size. For very short times, normal anti-coincidence can be used, and the full
frame excluded when a MIP pixel is detected.

Figure 16: Background “improvement” as a function of signal loss plotted for
three different background reduction metrics, with different panels showing
results for different frame times. Results from rolling shutter simulations
using the “minimal exclusion” SAC scheme are shown by dashed curves in the
5-ms panel. The shaded quadrants show notional but arbitrary requirements that
might be imposed for a background reduction technique: at least a 25%
improvement in the background metric (above the horizontal dashed line)
accompanied by no more than 5% signal loss (left of the vertical dashed line).
The SAC technique cannot reach such requirements for any metric except in the
shortest frame times.

On first glance, a simple reduction of background counts would appear to
afford the best performance, but it is a specious metric since removal of
relevant secondary background events is always accompanied by a similar
removal of signal. Putting this metric aside, we see that in all instances the
improvement in $s/b$ is greater than that in $snr$. This reveals that SAC is
more effective in reducing systematic errors than statistical ones. Indeed,
for large fractional signal losses, improvement on $s/b$ is reached at the
expense of a loss on $snr$. Even though we cannot meet the notional
requirements, any improvement in $s/b$ can be useful as long as $snr$ does not
suffer, and SAC can be thought of as a way of turning irreducible systematic
errors into statistical errors that are reducible via an increase in exposure
time. The analysis in the previous sections assumes that uncertainties are all
statistical in nature, and that the noise term in $snr$ in the background-
dominated regime is simply the square root of the number of valid events.
Experience with deep observations of low surface brightness emission in XMM-
Newton (e.g., cluster outskirts and galaxy halos) has shown that the limiting
factor in these observations is never statistical uncertainty but always
systematic uncertainty in the level of the background. For XMM-Newton EPIC pn
observations, there is typically a 5% irreducible uncertainty in the
background that dominates the detection and characterization of faint diffuse
emission; this is largely driven by the residual soft proton background[18],
which should be avoided on Athena through orbit selection and use of a
magnetic diverter. However, for investigations of such low surface brightness
sources, any reduction in the absolute level of the background via SAC could
significantly improve the scientific return even if a significant number of
source photons were discarded.

Systematic uncertainty can arise from a number of sources depending on the
strategy of the observations. For field-filling diffuse sources, often a non-
contemporaneous blank-sky pointing is used to constrain both the focused X-ray
and unfocused particle background, introducing systematic effects due to
background time variability and changes in instrumental performance or
calibration. Although quantifying these effects is complicated, in a simple
model we can treat systematic uncertainty as a variance that adds linearly in
the error budget, rather than in quadrature, and is thus not reducible by
increasing the exposure time. The SNR of a diffuse source observed by the WFI
in such a case can be given by

${SNR}=\frac{S_{o}}{(S_{o}+B_{o}+\sigma^{2}B_{o}^{2})^{1/2}}\,,$ (9)

where again $S_{o}$ is the number of source counts and $B_{o}$ is the number
of background counts, where counts refer to valid events. These counts are
related to the source and background surface brightness, $S_{o}^{\prime}$ and
$B_{o}^{\prime}$, both in units of cts s-1 arcmin-2 integrated over some
energy band by the relations

$S_{o}=S_{o}^{\prime}\Omega t_{obs}\,,B_{o}=B_{o}^{\prime}\Omega t_{obs}\,,$
(10)

where $\Omega$ is the solid angle of the region and $t_{obs}$ is the observing
time. The value of $\sigma$ defines the systematic uncertainty expressed as a
fraction of the background level, with $\sigma$ = 0.05 for a typical deep XMM-
Newton observation[18] and $\sigma=0.02$ as a current best estimate for Athena
WFI, based on the requirement for knowledge of the non-focused particle
background above 1 keV[19]. For observations which are both background-
dominated ($B_{o}\gg S_{o}$) and of small regions or short exposure times
($B_{o}\ll 1/\sigma^{2}$), Eq. 9 reduces to the standard
$SNR=S_{o}/B_{o}^{1/2}$. However, in the case where the systematic error of
the background begins to dominate, $B_{o}\gg 1/\sigma^{2}$ and
$SNR=S_{o}/(\sigma B_{o})$. Increasing the exposure time in this case does
nothing to increase the sensitivity because the uncertainty is dominated by
uncertainty in the background level. This is the idea behind SAC; we remove
background at the cost of signal, because that lost signal can always be
recovered by increasing the exposure time.

Although loss of signal is usually undesirable, for some important WFI
observations such as deep surveys, SAC can provide significant improvements in
surface brightness sensitivity that yield important science. An example is
shown in Figure 17, where we show the change in SNR as a function of exposure
time for a number of source sizes by applying SAC aggressively with exclusion
radius $r_{e}=600$ pixels. We use Eq.9 to calculate SNR in the presence of
systematic error, using $\sigma=0.02$ (2%) as our current best estimate from
the Athena WFI particle background knowledge requirement[19]. Since we plot
fractional change in SNR compared to not using SAC, the source flux cancels
out in the assumed highly background-dominated limit (see also the derivation
in Appendix A.4). Shaded regions show results for sources of of 1, 10, and 100
arcmin2 in extent, sampling typical sizes of low-surface-brightness features
that might yield interesting science. The shading spans the expected variation
of the particle background over an extended Athena mission: solid lines are
from our best estimate of the maximum GCR flux at solar min, upper limits
correspond to predicted minimum GCR flux at solar max, and lower limits are if
the pre-launch estimates are low by a factor of two, similar to what is seen
on eROSITA[17]. SAC offers a substantial improvement on large scales
regardless of exposure time, and still significant improvement on smaller
scales, especially during times of low background.

Figure 17: Fractional change in SNR as a function of exposure time, comparing
“aggressive” SAC with $r_{e}=600$ pixels to not using SAC. This assumes a 2%
systematic error and a highly background-dominated observation. Shaded regions
are shown for different sources sizes, and span the expected variation of the
particle background; solid lines are from our best estimate of the maximum GCR
flux at solar min, and lower limits are if these pre-launch estimates are low
by a factor of two, similar to what is seen on eROSITA[17]. Even in this worst
case scenario, SAC is still valuable at improving the SNR for the deepest
exposures of the most extended faint sources. Figure 18: Fractional change in
SNR by SAC as a function of exclusion radius for times of high (left) and low
(right) GCR flux expected over the life of an extended Athena mission. Here
“change” is compared to the SNR achieved by eliminating a full quadrant,
simulated here as an exclusion radius of 300 pixels. A value of zero means
there is no SNR change. The model, described in the text, is a diffuse, faint
emission source observed for 100 ks (orange) and 1.5 Ms (blue), and searched
for extended features of 1 (solid), 10 (dashed), and 100 (dotted) arcmin2 in
size. This includes a systematic background uncertainty of 2%. SAC over the
full FOV provides flexible, selective masking out to large exclusion radius,
inaccessible to single-quadrant SAC, that greatly enhances deep exposures of
extended sources.

The “selective” SAC masking described here works best if the full LDA field is
utilized, rather than a single quadrant. The optimal radius to mask depends
sensitively on the exposure time and the size of the region, which both depend
on the science under study. For an illustration of the power of SAC, we
simulated a diffuse source of 2 keV thermal emission matching the Athena WFI
surface brightness requirement of $6.2\times 10^{-16}$ erg cm-2 s-1 arcmin-2
in the 5–7 keV band[3]. We simulated two levels of accompanying particle
background surface brightness, one at $5.5\times 10^{-3}$ cts cm-2 s-1 keV-1
in the 2–7 keV band, from the WFI requirements to represent solar min (left
panel), and one at half this flux to represent solar max (right panel). From
this we estimated source and background counts in observations of 100 ks and
1.5 Ms, bracketing the depths of the WFI shallow and deep survey, and in
regions of 1, 10, and 100 arcmin2, as in Figure 17. Using Eq. 9 and our
empirical relations of signal loss $s$ and background loss $b$ (see Figure 9),
we then determined the SNR as a function of SAC exclusion radius $r_{e}$,
assuming a systematic background uncertainty of 2% ($\sigma=0.02$), the Athena
WFI requirement. We plot this in Figure 18 as the fractional SNR change over
the maximum background reduction possible with single-quadrant SAC, equivalent
to dropping a quadrant containing a MIP pixel. Here we simulate this as the
SNR at $r_{e}=300$ pixels, where $s/b$ for single-quadrant SAC starts to turn
over and become less effective (red line in Figure 10).

For shorter exposures sampling small regions, there is little improvement over
single-quadrant SAC. However, for the deepest WFI exposures, and for science
that requires detecting coherent structures on large scales, applying SAC to
the full FOV can improve the SNR by up to $\sim$20% over single-quadrant SAC,
with no increase in exposure time. This is especially true for times of lower
GCR flux. To obtain this improvement with single-quadrant SAC would require an
additional 600 ks observation of this field.

It is of course clear that many science cases will not benefit from masking
regions around particle tracks. But allowing the application of SAC by the
science observer maximizes the science return in a way that simply dropping
quadrants or full frames would not. By telemetering every MIP pixel location
in each frame, or at the very least the distance to the closest MIP pixel for
each in-band event, WFI data would enable flexible application of SAC masking.
This method has no impact on science investigations that do not benefit from
it, as the science observer could decide whether to use it or not, or even
experiment with different values of masking radius.

##  4 Summary

We have presented an analysis of Geant4 simulations of the Athena WFI particle
background in an effort to mitigate its effects. The majority of simulated
5-ms frames (87%) contain only particle tracks that cannot be confused with
focused X-rays due to their morphology or total energy; an additional 8% of
frames contain both particle tracks and X-ray-like events. This means that
true anti-coincidence techniques cannot be used to drop frames, as it would
remove $\sim$95% of the source signal. We have developed and presented a
partial veto scheme called Self-Anti-Coincidence, or SAC, which exploits a
spatial correlation between particle tracks and secondary valid events, a
correlation that we have identified and validated with independent Geant4
simulations and in-flight XMM-Newton EPIC pn data. By masking smaller regions
of the FOV around particle tracks, this technique can greatly reduce the
systematic effects of particle background in certain science cases, most
notably observations of very faint, highly extended sources. With sufficient
information included in WFI telemetry, this filtering can be applied
selectively on the ground by the user, enabling detection of very low surface
brightness objects without sacrificing other science.

We stress that the work presented here is not restricted to the Athena WFI,
but is relevant for any future silicon-based pixelated X-ray imaging detector.
In addition to providing a novel background mitigation technique for the WFI,
the results and methodology can be used to generate requirements on elements
such as frame rate, detector size, and particle environment for future
missions. Such considerations will maximize the science return from otherwise
challenging observations of very faint, extended X-ray sources.

##  5 Acknowledgements

This work was done under the auspices of the Athena WFI Background Working
Group, a consortium including MPE, INAF/IASF-Milano, IAAT, Open University,
MIT, SAO, and Stanford. We thank the entire Working Group for valuable
discussions that contributed greatly to this paper. We also thank the
anonymous referee for helpful comments that significantly improved the
manuscript. The US-based co-authors gratefully acknowledge support from NASA
grant NNX17AB07G. The studies undertaken at the Open University were funded by
the UK Space Agency, for which the team are grateful for their ongoing
support.

This paper made use of simulations from Geant4 software[7, 8] and utilized the
following software libraries for data analysis and presentation: the Perl Data
Language (PDL, pdl.perl.org) developed by K. Glazebrook, J. Brinchmann, J.
Cerney, C. DeForest, D. Hunt, T. Jenness, T. Lukka, R. Schwebel, and C.
Soeller; NumPy[20] (numpy.org); Astropy[21, 22] (http://www.astropy.org), a
community-developed core Python package for Astronomy; and Matplotlib[23]
(https://matplotlib.org), a Python library for publication quality graphics.

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Eric D. Miller is a research scientist at the MIT Kavli Institute for
Astrophysics and Space Research. He received his BA degree in physics from
Oberlin College in 1996, and his MS and PhD degrees in astronomy and
astrophysics from the University of Michigan in 1998 and 2003, respectively.
He leads the XRISM In-Flight Calibration Planning Team, develops X-ray imaging
detectors for future missions, and studies galaxy clusters and the diffuse
intergalactic medium.

Biographies and photographs of the other authors are not available.

##  Appendix A Self-Anti-Coincidence (SAC) Estimators

In this Appendix we introduce several quantities which may be used to assess
the effectiveness of the SAC technique and explore their relationship with key
parameters such as the frame time, $t_{f}$, and the exclusion radius, $r_{e}$.

###  A.1 Signal

Let us start with the signal, $S$. We define $S$ as the source valid event
counts accumulated over the region of interest, such as an LDA quadrant or the
full LDA, per unit frame, averaged over many frames. Here we shall assume the
signal to be distributed uniformly over the region of interest. The
probability that, in a given frame, a valid event is lost due to SAC is
$P_{sl}\equiv l_{s}/t_{s}$, where $l_{s}$ is the number of lost events and
$t_{s}$ is the total number of events. Under the assumption of spatial
uniformity of the signal, we have:

$P_{sl}=A_{R}/A_{T}\,,$ (11)

where $A_{R}$ is the area masked or rejected by SAC and $A_{T}$ is the total
area. From this we derive the expression for the signal:

$S=(1-A_{R}/A_{T})\cdot S_{o}\,,$ (12)

where $S_{o}$ is the signal when no SAC is applied. In our calculations we
will make use of the fractional signal loss $(S_{o}-S)/S_{o}$, which can be
expressed as:

$(S_{o}-S)/S_{o}=A_{R}/A_{T}\,.$ (13)

By comparing Eq.11 with Eq.13 we see that the fractional signal loss and the
rejection probability are actually the same thing:

$P_{sl}=(S_{o}-S)/S_{o}\,.$ (14)

For small signal losses, i.e., $(S_{o}-S)/S_{o}\ll 1$, exclusion regions do
not overlap and we can derive a simple formula explicitly relating the signal
loss to the frame time and the exclusion radius. Indeed:

$(S_{o}-S)/S_{o}=\frac{\pi r_{e}^{2}\cdot N_{p}}{A_{T}}\,,$ (15)

where $\pi r_{e}^{2}$ is the area of a single exclusion region and $N_{p}$ is
the number of particle tracks falling in a given frame. This assumes that the
masking regions are circular and that particle tracks are small compared to
the exclusion radius. By rewriting $N_{p}$ as the rate of cosmic ray particle
tracks $cr_{p}$ over the region of interest (a quadrant or the full LDA) times
the frame time $t_{f}$ we find:

$(S_{o}-S)/S_{o}=\frac{\pi r_{e}^{2}\cdot cr_{p}\cdot t_{f}}{A_{T}}\,.$ (16)

Eq.16 shows that the signal loss scales quadratically with the exclusion
radius and linearly with the frame time. As already pointed out, this
derivation is strictly correct in the linear regime, i.e., $(S_{o}-S)/S_{o}\ll
1$; as the signal loss increases, the probability that different exclusion
circles overlap must be accounted for. A simple algebraic expression can also
be worked out for very large exclusion radii. When the exclusion circles
encompass the total area, the surviving signal will be associated with the
fraction of frames in which no particle track appears on the detector. This
fraction is $\exp(-N_{p})$, assuming a Poisson distribution of arriving
primaries. From this we derive:

$(S_{o}-S)/S_{o}=1-\exp(-cr_{p}\cdot t_{f})\,.$ (17)

Note that in the linear regime, the fractional signal loss does not depend
upon the specific size of the region under consideration, quadrant or full
FOV; indeed in Eq.16 the area dependence is found both in the numerator
($cr_{p}$) and in the denominator ($A_{T}$) and cancels out. Conversely, when
we approach saturation, area does matter, as shown in Eq.17, where the term in
the exponent scales linearly with the total area through $cr_{p}$. This is
quite intuitive: the larger the area under consideration, the larger the
number of particle tracks and the smaller the likelihood that, for a given
frame time, a frame is track-free.

In the intermediate regime of signal loss between Eqs.16 and 17, masked areas
overlap and there is no simple formula to estimate $(S_{o}-S)/S_{o}$. Thus we
have resorted to Monte Carlo simulations. We assumed a primary frame-rate
consistent with that reported in Ref. 9 and a time resolution of 1 ms. We drew
primary events in each bin following Poisson statistics and assigned random
positions over 510$\times$510 and 1020$\times$1020 grids representing
respectively a single LDA quadrant and the full LDA FOV. Finally, we re-binned
the time-series to the desired frame time and computed the average rejected
area, $A_{R}$, over a large number of frames. In Fig.19 we plot the simulated
fractional signal loss as a function of exclusion radius for four different
values of the frame time: 1 ms, 2 ms, 5 ms, and 10 ms. For a given value of
the frame time, signal loss increases with increasing $r_{e}$, quadratically
in the linear regime (see Eq.16) and saturating at large $r_{e}$ (see Eq.17).
As frame time goes down, the saturation regime shifts to larger exclusion
radii. In other words, larger exclusion radii can be accepted for smaller
frame times. In the limiting case of $t_{f}\rightarrow 0$ the exclusion circle
can encompass the whole detector with no signal loss.

Figure 19: Fractional signal loss as a function of exclusion radius for four
different values of the frame time, as indicated by different colors. The left
panel, for a 510$\times$510 grid, represents a single WFI LDA quadrant; the
right panel, for a 1020$\times$1020 grid, represents the full LDA FOV.

Let us focus on the left panel of Fig.19, which refers to a single quadrant,
and assume we are willing to accept a certain fractional signal loss, say 20%;
we can distinguish three different regimes.

  1. 1.

$t_{f}>10$ ms: Fractional signal loss rapidly saturates to unity, only small
exclusion radii can be accepted; in this regime SAC is of little or no use.

  2. 2.

$t_{f}<1$ ms: The exclusion circle can encompass the whole quadrant with an
acceptable signal loss; this is the standard anti-coincidence regime.

  3. 3.

1 ms $<t_{f}<$ 10 ms: The exclusion circle is not restricted to very small
values, however it cannot extend to the whole area. This in an intermediate
regime where partial vetoing of the quadrant can be of use.

Since the values of $t_{f}$ marking the transitions between the three regimes
depend mostly on the value of the signal loss at saturation, analogous values
for the full detector case depicted in the right panel of Fig.19 can be
obtained by dividing all frame times by a factor of four (see Eq.17). Thus,
for the full detector case we have:

  1. 1.

$t_{f}>2.5$ ms: SAC not practical;

  2. 2.

$t_{f}<0.25$ ms: standard anti-coincidence regime;

  3. 3.

0.25 ms $<t_{f}<$ 2.5 ms: partial vetoing regime.

These cases are again for an acceptable signal loss of 20%, and the acceptable
level depends strongly on the particular sources and science under study. More
importantly, the signal $S$ does not provide a full description of SAC, as it
does not contain any information on the improvements afforded by this partial
vetoing technique. For this we must turn to other indicators.

###  A.2 Rejected background

We define $B$ as the background valid event counts accumulated over the region
of interest, quadrant or full detector, per unit frame, averaged over many
frames. We shall assume the background to be distributed uniformly over the
region of interest. We define $P_{rb}$ as the probability that, in a given
frame, a valid event produced by a cosmic ray is rejected by SAC, i.e.,
$P_{rb}\equiv r_{b}/t_{b}$, where $r_{b}$ is the number of rejected background
events and $t_{b}$ the total number of background events in the frame. Two
distinct terms contribute to $P_{rb}$:

  1. 1.

$P_{ran}$, the probability that the valid background event fell within the
exclusion circle(s) of cosmic ray tracks from one or more unrelated primaries;
and

  2. 2.

$P_{cor}$, the probability that the valid background event fell within the
exclusion circle of a particle track associated with the primary that
generated it.

The first term has already been introduced when discussing signal loss (see
Eq.11, $P_{ran}=P_{sl}=A_{R}/A_{T}$), and it depends both on the exclusion
circle and the frame time. The second term depends on the exclusion circle,
but does not depend on the frame time; it may be thought of in a simple way as
a sort of cumulative “secondary spread function”, $P_{cor}\equiv
P_{cor}(<r_{e})$, encapsulating the spatial spread of secondary particles.
Like the fractional signal loss, $P_{cor}$ depends on the specific region that
is being considered, and it will differ when considering a single quadrant or
the full detector. A derivation of $P_{cor}$ for these two cases and for the
limiting case of an infinite plane is provided in Appendix B.

Note that $P_{rb}$ cannot be simply written down as the sum of $P_{cor}$ and
$P_{sl}$. Indeed, for long frame times, a secondary event may end up falling
simultaneously within the exclusion circle of the primary that generated it
and in that of one or more unrelated primaries. This can be accounted for by
including in the sum a correction term that accounts for the double counting
of events that belong to both categories, i.e.:

$P_{rb}=P_{cor}+P_{sl}-P_{cor}\cdot P_{sl}\,,$ (18)

where $P_{cor}\cdot P_{sl}$ is the probability that an event ends up falling
simultaneously within the exclusion circle of the primary that generated it
and that of one or more unrelated primaries. By rearranging some of the terms
and using Eq.11 we can rewrite the above equation in the form:

$P_{rb}=(1-P_{cor})\cdot A_{R}/A_{T}+P_{cor}\,.$ (19)

Once $P_{rb}$ is known, the background can be computed from the equation:

$B=(1-P_{rb})\cdot B_{o}\,,$ (20)

where $B_{o}$ is the background when no SAC is applied. Substituting Eq.19
into Eq.20 we find

$B=\\{1-[(1-P_{cor})\cdot A_{R}/A_{T}+P_{cor}]\\}\cdot B_{o}\,.$ (21)

Figure 20: Fractional background rejection as a function of exclusion radius
for a 5 ms frame time and a 510$\times$510 grid, representing a single
quadrant. We show in black the total fractional background; in red the
fractional background if only secondaries that fall in the exclusion circle of
generating primaries are removed; in blue the fractional background if only
secondaries that fall in the exclusion circle of unrelated primaries are
removed.

In Fig.20 we use Eq.21 to plot the fractional background, $B/B_{o}$, as a
function of the exclusion radius, for a frame time of 5 ms. $P_{cor}$ has been
derived from WFI simulations [9] as described in detail in Appendix B, and
$A_{R}/A_{T}$ has been derived from Monte Carlo simulations as described in
Section A.1.

Fig.20 illustrates why $B$ is not a good SAC estimator. Application of self
anti-coincidence results in two different kinds of background reduction: a
favorable one, associated with the removal of secondaries that fall in the
exclusion circle of the primaries that generated them (Fig.20 red curve); and
an unfavorable one, associated with the removal of unrelated secondaries
(Fig.20 blue curve). Using only estimator $B$, we do not have a way of
discriminating between the two.

###  A.3 Signal-to-background ratio

The signal-to-background ratio, $S/B$, can be easily worked out from the
equations for the signal and background, Eqs.12 and 21. With a little algebra
we find:

$\frac{S}{B}=\frac{S_{o}}{B_{o}}\cdot\frac{1}{(1-P_{cor})}\,.$ (22)

Interestingly, unlike $S$ and $B$, $S/B$ does not depend upon frame time but
only on the exclusion radius through $P_{cor}$. Moreover, if we divide both
sides of the equation by $S_{o}/B_{o}$, and define a re-normalized signal to
background ratio

$\frac{s}{b}\equiv\frac{S/S_{o}}{B/B_{o}}\,,$ (23)

we derive a very general formula,

$\frac{s}{b}=\frac{1}{(1-P_{cor})}\,,$ (24)

which does not depend on the specific values of $S_{o}$ or $B_{o}$.

Figure 21: Renormalized signal to background ratio. Different colors refer to
different regions: black refers to an infinite plane, blue to a
1020$\times$1020 grid representing the full detector and red to a
510$\times$510 grid representing a single quadrant.

In Fig.21 we plot the renormalized signal to background ratio, where $P_{cor}$
has been derived from WFI simulations [9] as described in detail in Appendix
B. As already pointed out, $S/B$ does not depend on area loss, and therefore
on frame time; it depends only on $P_{cor}(<r_{e})$, i.e., the cumulative
probability distribution that a secondary fall within a radius $r_{e}$ of its
primary. $S/B$ improves steadily, reaching a maximum values of $\sim 20\%$ for
rejection of a entire quadrant and $\sim 50\%$ for rejection of the full
detector area. $S/B$ is a good indicator of the improvement afforded by SAC,
however it does not provide a full description of its effects because it
contains no information of the area loss.

###  A.4 Signal-to-noise ratio

The signal-to-noise ratio

$SNR\equiv\frac{S}{{(S+B)}^{1/2}}\cdot t^{1/2}\,,$ (25)

where $t$ is the exposure time, can also be easily worked out from the
equations for the signal and background, Eqs.12 and 21. With a little algebra
we find:

${SNR}=\frac{(1-A_{R}/A_{T})^{1/2}\cdot S_{o}}{[S_{o}+(1-P_{cor})\cdot
B_{o}]^{1/2}}\cdot t^{1/2}\,.$ (26)

Interestingly, like $S$ and $B$, and unlike $S/B$, $SNR$ does depend upon
frame time through $A_{R}$ as well as on the exclusion radius through
$P_{cor}$. In the background dominated regime, $B\gg S$, where

${SNR}=\frac{S}{B^{1/2}}\cdot t^{1/2}\,,$ (27)

Eq.26 reduces to:

${SNR}=\frac{S_{o}}{B_{o}^{1/2}}\cdot\frac{(1-A_{R}/A_{T})^{1/2}}{(1-P_{cor})^{1/2}}\cdot
t^{1/2}\,.$ (28)

If we divide both sides of this equation by $S_{o}/\sqrt{B_{o}}\cdot t^{1/2}$,
and define a re-normalized signal to noise ratio:

${snr}\equiv\frac{S/S_{o}}{(B/B_{o})^{1/2}}\,,$ (29)

we derive a very general formula,

${snr}=\frac{(1-A_{R}/A_{T})^{1/2}}{(1-P_{cor})^{1/2}}\,,$ (30)

which does not depend on the specific values of $S_{o}$ or $B_{o}$.

As already pointed out, like $B$, $snr$ depends on area loss and on $P_{cor}$,
and so it is a mixed estimator. However, unlike $B$, it can be of use by
informing us about the exclusion radius that maximizes the signal-to-noise
ratio. By looking at Fig.22, left or right panel, we see that for a given
choice of frame time, $snr$ peaks at specific values of the exclusion radius.
For $t_{f}=10$ ms, maximum $snr$ is reached at $r_{e}=30$ pixels; as we reduce
the frame time, the peak moves to larger exclusion radius. For $t_{f}=1$ ms
the $snr$ attains its peak value at $r_{e}\sim 200$ pixel. However, in all
instances the maximum improvement on the $snr$ is less than 5% with respect to
the no-SAC case, so this is not a particularly significant improvement. In
simpler words, application of Self Anti-Coincidence, be it to a single
quadrant (Fig.22 left panel) or the full detector (Fig.22 right panel), does
not improve the statistical quality of our data in a significant way.

It is worth pointing out that, although the present result has been derived in
the background dominated regime, $B\gg S$, it applies to all regimes. Indeed,
as we can see in Eq.26, when the signal $S$ is larger or comparable to the
background $B$, the noise term becomes less sensitive to the value of the
background and signal-to-noise improvements associated with background
reduction become even less significant than in the background dominated
regime.

Figure 22: Renormalized signal-to-noise ratio in the background-dominated
regime as a function of exclusion radius for four different values of the
frame time, as indicated by different colors. The left panel, for a
510$\times$510 grid, represents a single WFI LDA quadrant; the right panel,
for a 1020$\times$1020 grid, represents the full detector.

###  A.5 Effects of a Rolling Shutter

Figure 23: Schematic representation of how primary and secondary events are
read out, note that the rolling shutter moves from top to bottom. Top left
panel: the primary (black square) hits on row $i$, indicated in light blue,
and the secondary (blue square) impacts on row $j$ highlighted in orange. Top
right panel: primary and secondary are placed as in the top left panel, also
shown in gray is row $k$, which is being read out when the events hit the
sensor. In this case, row $k$ is read out before rows $i$ and $j$ and both
events are read out in the same frame. Bottom left panel: same as top right
panel, however in this case row $k$ is placed between rows $i$ and $j$, this
leads to the primary being read out one frame after the secondary. Bottom
right panel: same as top right however, in this case, the rolling shutter goes
through row $k$ after rows $i$ and $j$, this leads to both events being read
out in the frame after the one depicted here.

The above calculations have been performed tacitly assuming that all rows in a
frame are read out simultaneously; this is true for many detectors, but not
for the WFI LDA, which is operated in a ‘rolling shutter’ mode. This is a read
out mode where rows are continuously read out one after the other, frame after
frame, and it is driven by technical limitations of the spacecraft power
supply and thermal control[1]. It also means that different rows are read out
at different times and that, in some instances, a primary event and its
secondary may end up recorded in different frames. In Figure 23, we provide a
schematic representation of how primary and secondary events are read out,
with the rolling shutter moving from top to bottom. In the top left panel we
show the primary (black square) on row $i$, indicated in light blue, and the
secondary (blue square) on row $j$, highlighted in orange. Rows are numbered
from top to bottom in accordance with the way the rolling shutter is operated,
thus in the current example $i<j$. In the other three panels we also show row
$k$ which is being read out when primary and secondary hit the sensor. In the
top right panel, $k<i<j$, and thus row $k$ is read out before rows $i$ and $j$
and both events end up in the same frame. In the bottom left panel, $i<k<j$,
and the primary is read out one frame after the secondary. In the bottom right
panel, $i<j<k$, and both events are read out in the next frame. Note that the
velocity at which particles propagate in and around the detector is much
larger than that at which the shutter is operated. Thus, within the current
assessment, we can safely assume that primary and secondary impact the
detector at the same time.

Through a representation similar to the one presented in Figure 23, it is easy
to show that in the case where the primary lands on a higher row than the
secondary, $j<i$, we can distinguish between three possible cases: 1) $k<j<i$,
both events are read out in the same frame; 2) $j<k<i$, the secondary is read
out one frame after the primary; and 3) $j<i<k$, both events are read out in
the same frame. Finally, if primary and secondary hit on the same row, $i=j$,
the two will be read out in the same frame.

In summary, for any given value of $i$ the secondary is read out during one of
two frames; which of the two depends on the row $k$ that is being read out
when primary and secondary impinge on the detector. (There is one minor
exception to this rule: when $i=j$, the two events are read out in the same
frame for any value of $k$.) If ($k<i$ and $k<j$) or ($k>i$ and $k>j$) the
secondary will be read out in the same frame as the primary; if ($i<k<j$) or
($j<k<i$) the secondary and primary will be read out in different frames.
Probabilities for the above cases can be easily computed. We make use of the
following definitions: $P_{=}$ is the probability of primary and secondary
being read out in same frame; $P_{\neq}$ is the probability of primary and
secondary being read out in different frames; and $n_{r}$ is the number of
rows. By requiring that the sum of all probabilities be unity we derive

$P_{=}+P_{\neq}=1\,,$ (31)

and by noting that the probability of secondary and primary to be read out in
different frames must be proportional to the number of rows between $i$ and
$j$,

$P_{\neq}=\frac{|i-j|}{n_{r}}\,.$ (32)

Finally, by combining Eq.31 and 32 we derive:

$P_{=}=\frac{n_{r}-|i-j|}{n_{r}}\,.$ (33)

From our analysis, we have determined that, except for the case where the
secondary falls in the same row as the primary, the secondary can always be
found in one of two frames. The question then is how to incorporate this
information into our SAC calculations. We can consider two limiting
approaches: 1) “minimal exclusion”, throwing away only rows from the frame
where the primary is detected; in this case the area loss term $A_{R}/A_{T}$
is unchanged, but $P_{cor}$, the probability that the secondary falls within
the exclusion circle of the primary that generated it, will be significantly
diminished, by up to a factor of two; and 2) “maximal exclusion”, removing
rows from both frames; in this case it is $P_{cor}$ that remains unchanged
while the $A_{R}/A_{T}$ increases, again by a factor of about two. A wide
range of intermediate solutions could be also considered. One could exclude
rows from one frame only for rows that are close to the row in which the
primary is located (rows with high $P_{=}$), and exclude rows from both frames
for other rows (with low $P_{=}$). In the current work we shall use “maximal
exclusion” as the most conservative background reduction case. In Figure 24,
we show the fractional signal loss as a function of exclusion radius, as shown
in Figure 19 but with the inclusion of the rolling shutter effect.

In the linear regime, i.e., $(S_{0}-S)/S_{0}\ll 1$, the increase in signal
loss is about a factor of two. However, as we move to larger exclusion radii,
overlaps between excluded regions become more frequent and the increase in
signal loss becomes smaller. By adopting the “maximal exclusion” option, the
signal-to-background ratio $s/b$ remains unchanged because $P_{cor}$ remains
unchanged. Conversely, since the signal-to-noise ratio depends on the area
loss, it will be affected by the rolling shutter. This is shown in Figure 25,
where we show the same plots as in Figure 22 with the rolling shutter effect
included.

Figure 24: Fractional signal loss as a function of exclusion radius for four
different values of the frame time, as indicated by different colors. The left
panel, for a 510$\times$510 grid, represents a single WFI LDA quadrant; the
right panel, for a 1020$\times$1020 grid, represents the full LDA FOV. The
effect of the rolling shutter has been included following the “maximal
exclusion” option. Figure 25: Renormalized signal-to-noise ratio in the
background-dominated regime as a function of exclusion radius for four
different values of the frame time, as indicated by different colors. The left
panel, for a 510$\times$510 grid, represents a single WFI LDA quadrant; the
right panel, for a 1020$\times$1020 grid, represents the full detector. The
effect of the rolling shutter has been included following the “maximal
exclusion” option.

##  Appendix B Computing the secondary distribution function for WFI

As pointed out in Section A.2, the probability that a secondary is detected
within a certain radius of the primary generating it, $P_{cor}(<r_{e})$,
depends on the specific region that is being considered, in other words it
will differ when considering a single quadrant or the full detector. To derive
$P_{cor}(<r_{e})$ for the WFI we use the following procedure. We define the
probability, $P^{\infty}_{cor}(<r_{e})$, for the ideal case of an infinite
plane as a parametric function of the form:

$P^{\infty}_{cor}(<r_{e})=2/\pi\arctan[(r_{e}/r_{\ast})^{\alpha}]\,,$ (34)

where the free parameters are the scale radius $r_{\ast}$ and the slope of the
power law $\alpha$. We insert trial values for $r_{\ast}$ and $\alpha$ and
perform Monte Carlo simulations to compute from $P^{\infty}_{cor}(<r_{e})$ the
probabilities for a single quadrant, $P^{q}_{cor}(<r_{e})$, and the full
detector, $P^{d}_{cor}(<r_{e})$. We then use Eq.24 to compute the normalized
signal-to-background ratio for a single quadrant, $(s/b)_{q}$, and the full
detector, $(s/b)_{d}$ and compare these with estimates based on detailed
Geant4 simulation of the WFI. For this exercise we use previously published
data from a different set of Geant4 simulations [9] for the single quadrant
and a value provided to the Athena WFI Consortium (T. Eraerds, private
communication) for the full detector. We then iterate the procedure until
$(s/b)_{q}$ and $(s/b)_{d}$ adequately reproduce the estimates based on Geant4
simulations.

Figure 26: Probability that a secondary fall within $r_{e}$ of its primary.
The black, blue, and red curves refer respectively to: an infinite plane, a
1020$\times$1020 grid (detector), and a 510$\times$510 grid (quadrant).

In Fig.26 we show the probability distributions $P^{\infty}_{cor}(<r_{e})$,
$P^{q}_{cor}(<r_{e})$ and $P^{d}_{cor}(<r_{e})$ that have resulted from the
procedure we have just described. Note how, for large radii, the three curves
converge to different values: 0.21 for $P^{q}_{cor}(<r_{e})$, 0.32 for
$P^{d}_{cor}(<r_{e})$ and 1, by construction, for $P^{\infty}_{cor}(<r_{e})$.

In Fig.27 we show the normalized signal-to-background ratio for a single
quadrant, $(s/b)_{q}$, and the full detector, $(s/b)_{d}$ and compare these
with estimates based on Geant4 simulations of the WFI. The careful reader may
note that, while in the case of $(s/b)_{q}$, left panel, the model fits the
data points reasonably well, for $(s/b)_{d}$, right panel, we have a point and
a star which are respectively well above and in agreement with the model. The
point comes from a presentation at a WFI consortium meeting where results for
both a single quadrant and the full detector were shown; the star comes from a
rescaling of the point with the ratio of the result for the quadrant presented
at the same meeting divided by the more recent estimate shown in the left
panel of Fig.27.

Figure 27: Renormalized signal-to-background ratio for 510$\times$510 grid
(quadrant, left panel) and 1020$\times$1020 grid (detector, right panel) vs.
exclusion radius. Symbols come from Geant4 simulations, curves are computed
through Eq.24 from $P^{q}_{cor}(<r_{e})$ for the left panel and
$P^{d}_{cor}(<r_{e})$ for the right panel.