# Consequences of constant elevation scans for instrumental systematics in
Cosmic Microwave Background Experiments

Daniel B. Thomas<EMAIL_ADDRESS>School of Physics and Astronomy,
Queen Mary University of London, London, E1 4NS, UK  Nialh McCallum  Michael
L. Brown  Jodrell Bank Centre for Astrophysics, School of Physics & Astronomy,
The University of Manchester, Manchester M13 9PL, UK

###### Abstract

Instrumental systematics need to be controlled to high precision for upcoming
Cosmic Microwave Background (CMB) experiments. The level of contamination
caused by these systematics is often linked to the scan strategy, and scan
strategies for satellite experiments can significantly mitigate these
systematics. However, no detailed study has been performed for ground-based
experiments. Here we show that under the assumption of constant elevation
scans (CESs), the ability of the scan strategy to mitigate these systematics
is strongly limited, irrespective of the detailed structure of the scan
strategy. We calculate typical values and maps of the quantities coupling the
scan to the systematics, and show how these quantities vary with the choice of
observing elevations. These values and maps can be used to calculate and
forecast the magnitude of different instrumental systematics without requiring
detailed scan strategy simulations. As a reference point, we show that
inclusion of even a single boresight rotation angle significantly improves
over sky rotation alone for mitigating these systematics. A standard metric
for evaluating cross-linking is related to one of the parameters studied in
this work, so a corollary of our work is that the cross-linking will suffer
from the same CES limitations and therefore upcoming CMB surveys will
unavoidably have poorly cross-linked regions if they use CESs, regardless of
detailed scheduling choices. Our results are also relevant for non-CMB surveys
that perform constant elevation scans and may have scan-coupled systematics,
such as intensity mapping surveys.

##  1 Introduction

Ground-based Cosmic Microwave Background (CMB) surveys are a crucial part of
ongoing (e.g. [Ade et al. (2015a), Adachi et al. (2019), Sayre et al. (2020),
Choi et al. (2020)]) and future (e.g. [Ade et al. (2019), Abazajian et al.
(2019)]) attempts to map the CMB temperature and polarisation in exquisite
detail.

Systematics, including instrumental systematics, need to be controlled to high
precision in order to achieve the science goals of these surveys (see e.g. [Hu
et al. (2003)]), and various techniques are being investigated for mitigating
and controlling different systematics (see e.g. [Ade et al. (2015b), Wallis et
al. (2014), Williams et al. (2021)]). Some systematics, particularly
instrumental systematics such as differential gain and pointing errors between
differenced detectors, couple to the scan strategy (see e.g. [Hu et al.
(2003), O’Dea et al. (2007), Shimon et al. (2008), Miller et al. (2009), Bock
et al. (2009), Wallis et al. (2016), Hivon et al. (2017), McCallum et al.
(2021a)]). This coupling means that knowledge of the scan strategy is required
to compute the expected magnitude of the contamination from these systematics,
and also that a careful choice of scan strategy may be able to mitigate these
systematics. For so-called “ideal111Scans where each sky pixel is visited at a
large number of regularly spaced angles.” scans, the contamination from these
systematics is reduced to zero. [Wallis et al. (2016)] showed that satellite
scans can be constructed to reduce the contamination from these systematics by
2 to 3 orders of magnitude. However, no such general study has been performed
for ground based experiments. In particular, it is not clear how to optimise a
ground-based scan strategy to get close to an ideal scan, and how close to an
ideal scan can be reached in principle.

The scan strategies of ground based surveys are complicated and hard to
parameterise. As a result, detailed simulations of these scan strategies
require many involved choices and considerations, such as: how and when to
scan different parts of the sky; how long to scan each patch for before moving
to a different patch of sky; when and how to avoid the Sun and Moon; what
range of azimuth “throw” to use and at what speed to scan in azimuth; which
elevations to scan at; how to equalise depth over different parts of the sky;
and which times of the year to scan. The simulations themselves can be
computationally expensive, depending on the level of realism and detail in the
simulations. Combined with the high number of parameters and choices, this
expense can make searches over scan strategy parameter space prohibitive. In
this paper, we will refer to the full set of choices, parameters, times and
durations that are required to completely specify a ground-based scan
strategy, as the “detailed scheduling choices”.

The majority of ground based surveys are composed of constant elevation scans
(CESs), which involves choosing an observing elevation, and holding the
elevation fixed at this value whilst sweeping the telescope in azimuth. CESs
are used because they are good for instrument stability, and because varying
the elevation during a scan affects the column length of atmosphere through
which the telescope is looking, resulting in more complicated noise and
systematic properties. As a result, the elevation is typically held fixed
whilst the instrument is scanning, even if multiple different elevations are
used for scanning at different times. The use of CESs puts strong constraints
on the possible properties of the survey, particularly in regards to how the
scans couple to contamination from instrumental systematics.

In this work we show that the assumption of CESs allows one to put bounds on
the parameters that couple the scans to the instrumental systematics, and
calculate typical values and maps for these parameters, independently of
detailed scheduling choices. In particular, we show that there is a
fundamental limit to how close any scan strategy composed of CESs can get to
an ideal scan for the purposes of these parameters. These bounds and typical
values allow forecasts, estimations and calculations to be carried out without
making detailed scheduling choices, and these bounds can also be used as a
target when optimising the detailed scheduling choices according to the
science goals of a particular experiment. These results also allow the
relevant features of detailed scans to be approximated with simple
simulations.

The parameters describing the coupling of instrumental systematics to the scan
strategy are closely related to the standard metric for the cross-linking
(Choi et al., 2020). Cross-linking refers to the different paths that the
instrument takes across the sky as it scans, which can be important for
mitigating the effects of 1/f atmospheric noise during the map-making step
(see e.g. Sutton et al. (2009); Poletti et al. (2017)). Whilst we focus on
scan-coupled systematics, we also comment on the relevance of our findings for
any surveys whose science goals require good cross-linking, which plan to use
CESs. We reiterate that these findings are independent of detailed scheduling
choices.

We describe our approach and code implementation in section 2 and we present
the consequences of CESs in section 3. In section 4 we use the limitations of
CESs to examine the crossing-angle coverage achievable for surveys based in
the Atacama Desert, and the resulting bounds on the degree to which scan-
coupled systematics can be mitigated. We present our conclusions in section 5.

##  2 Scan-coupled systematics and ground-based scanning

Many instrumental systematics have a fixed orientation in the focal plane of
the telescope, particularly for polarisation sensitive detectors. These
systematics include pointing errors, anisotropic beam effects, and gain or
calibration effects for polarisation detectors. Since the orientation of the
focal plane with respect to the sky changes according to how the instrument
scans the sky, the contamination caused by these systematics depends on the
scan strategy (Wallis et al., 2016; McCallum et al., 2021a).

The dependence of the contamination on the scan strategy can be computed
within each pixel using the parameters $h_{k}$, defined as

$h_{k}=\frac{1}{N_{\text{hits}}}\sum_{j}e^{ik\psi_{j}}=\frac{1}{N_{\text{hits}}}\sum_{j}\left(\cos(k\psi_{j})+i\sin(k\psi_{j})\right)\text{,}$
(1)

where $k$ takes integer values. The scan information being used here is the
set of crossing angles $\psi_{j}$, which give the relative orientations of the
focal plane to the sky for each measurement within the pixel in question. We
show how to calculate these angles below. Note that similar parameters for
examining crossing angle coverage are used in Hu et al. (2003); O’Dea et al.
(2007); Shimon et al. (2008); Miller et al. (2009); Bock et al. (2009); Hivon
et al. (2017).

The $|h_{k}|^{2}$ parameters measure the variation within a set of crossing
angles222Note that one can add a constant value to every angle in a set
without changing the value of $|h_{k}|^{2}$ for that set.. The value of
$|h_{k}|^{2}$ (for any $k$) ranges from $0$ to $1$, where $0$ is the value for
an ideal scan and $1$ represents the worst possible scan (every crossing angle
in the pixel is the same). For satellite scan strategies, values as small as
$|h_{k}|^{2}\sim 10^{-3}$ can be reached by careful optimisation (Wallis et
al., 2016). There has previously been no detailed investigation for ground
based surveys, in particular one that examines: which decisions and choices in
the complicated ground-based scan strategies are important for determining the
$h_{k}$ values in each pixel; what typical values of $h_{k}$ are for ground-
based surveys; how well ground-based surveys can be optimised in this regard;
and how close ground-based surveys can get to an ideal scan for the purposes
of these parameters.

These $h_{k}$ values are important for several reasons. Different $h_{k}$ are
required to calculate the contamination due to different systematics. For
example, temperature to polarisation leakage due to differential pointing
errors within differenced detector pairs couples to $h_{1}$ and $h_{3}$
(Wallis et al., 2016; McCallum et al., 2021a), and leakage caused by
differential ellipticity effects couples to $h_{2}$ and $h_{4}$. We refer the
reader to section 2 of McCallum et al. (2021a) for general expressions showing
how signals of different spin couple to different $h_{k}$, and to equations
(26)-(28) of Wallis et al. (2016) for some explicit examples of how the
$h_{k}$ parameters manifest in the power spectrum contamination for specific
systematics. We note that these parameters are also relevant when detector-
differencing is not used (McCallum et al., 2021a). In addition, insufficiently
accurate incorporation of the scan strategy (i.e. $|h_{k}|^{2}$ maps) into the
contamination caused by systematics can result in different conclusions for
quadratic estimators that are used to find and clean systematics from the maps
(Williams et al., 2021). Computing $h_{k}$ maps is also critical for
constructing simple and fast map-based simulations that incorporate the
effects of systematics, notably systematics that can usually only be studied
with full time ordered data (TOD) simulations (McCallum et al., 2021a).

In this work we focus on $k=1$ to $4$ as the most relevant for intensity to
polarisation leakage (see e.g. Wallis et al. (2016)), although we note that
there do exist systematics that couple to $h_{k}$ with $k>4$ (McCallum et al.,
2021a). The same approach taken here can be used to study these additional
parameters if required.

Since the $h_{k}$ parameters capture general information about the crossing
angles, they relate to other aspects of scan strategies beyond the coupling of
particular systematics. For example, the $h_{2}$ and $h_{4}$ quantities appear
directly in simple binning polarisation map-making. More interestingly, in the
recent ACT release (Choi et al., 2020) a metric was used to quantify cross-
linking, which corresponds to $\sqrt{|h_{2}|^{2}}$ (in the absence of
boresight rotation; see below and section 4.4). Due to this correspondence, a
corollary to our results is that the constraints on $|h_{2}|^{2}$ constitute a
limitation on the achievable levels of cross-linking (if using CESs) that is
independent of detailed scheduling choices. Such a limitation is likely to
impact the use of maximum likelihood map makers. This is a complicated issue
beyond the scope of this work so we do not explore it here, other than noting
that the maps and values of $|h_{2}|^{2}$ later in this paper facilitate such
a study without requiring detailed scheduling choices.

The connection between $|h_{2}|^{2}$ and cross-linking breaks down in more
complicated scenarios where different types of crossing angles need to be
tracked. For example, when there is boresight rotation (Ade et al., 2015b) or
rotating half wave plates that modulate the incoming photon polarisation (see
e.g. Maino et al. (2002); Brown et al. (2009); Johnson et al. (2007);
MacTavish et al. (2008); Bryan et al. (2016); Buzzelli et al. (2018)), neither
of which improves cross-linking. Here we primarily focus on the simple case
with only sky rotation, and briefly comment on boresight rotation in section
4.4. Half wave plates require tracking the crossing angles and half wave plate
angles separately, resulting in a more complicated set of $h_{k,l}$ parameters
(see Wallis et al. (2015) for details), so we do not consider them further in
this work.

There are ways to remove these systematics such as during map-making. This
requires knowledge of the spins of the systematics and which $h_{k}$ they
couple to (see e.g. Wallis et al. (2014); McCallum et al. (2021a, b). In this
work we focus on understanding the range of possible values and map structures
of the $h_{k}$ for ground-based scan strategies, and which aspects of the scan
strategies these parameters depend on.

###  2.1 Calculating crossing angles in ground-based scans

To calculate the possible $h_{k}$ values and maps for ground based surveys, we
need to calculate the possible crossing angles $\psi_{j}$ that can be achieved
within each pixel. To do this calculation, we make extensive use of the python
pyEphem (Rhodes, 2011) and Healpy333http://healpix.sourceforge.net (Zonca et
al., 2019; Górski et al., 2005) packages.

We define the observatory using an ephem.Observer() object, where, for these
results, the key property of the observatory is its Latitude. We take the
right ascension (RA), and declination (Dec), of the centre of a healpix pixel
as an ephem.FixedBody() object. The FixedBody Compute method is used to
compute the elevation and azimuth444For the purposes of this work we will
define azimuth as increasing clockwise from North, such that a “rising” field
has an azimuth ($\theta$) of $0^{\circ}<\theta<180^{\circ}$. required to see
the healpix pixel from the given observatory at a particular time. This chain
of computational steps is the basic building block of the code.

We then want to calculate the orientation of the instrument focal plane on the
sky. Since we are restricting to CESs, we know that the next observation will
occur at the same elevation, at a slightly increased azimuth value
$\delta\theta$.555Note that we are interested in the angle of the focal plane
with respect to the sky, which means that we cannot apply equation 4 for both
increasing and decreasing azimuth scans without applying an additional
correction to one of the two cases, as the opposite edges of the focal plane
are the leading edges in the two situations. Correctly accounting for this
effect results in the same crossing angle at the instant that a measurement is
taken, regardless of whether azimuth is increasing or decreasing. For
simplicity in our code, and without loss of generality, we always treat
$\delta\theta$ as positive. The RA and Dec corresponding to the new azimuth
can then be calculated using the radec_of method in pyEphem. Given the RA and
Dec of the pixel, and the RA and Dec of the next observation, the orientation
of the focal plane with respect to the sky can be calculated using the two-
argument arctangent as

$\displaystyle\delta\text{Dec}=\text{Dec}_{2}-\text{Dec}_{1}$ (2)
$\displaystyle\delta\text{RA}=\text{RA}_{2}-\text{RA}_{1}$ (3)
$\displaystyle\psi=\arctan
2\left(\delta\text{Dec},\delta\text{RA}\cos(\text{Dec}_{1})\right)\text{.}$
(4)

Appropriate choices of $\delta\theta$ depend on the scan speed and sampling
frequency of the telescope. For this work we use $\delta\theta=0.8^{\prime}$,
corresponding to for example a scan speed of $0.4$ degrees per second and a
sampling frequency of $30$Hz. However, our results are robust to an order of
magnitude change either side of our chosen value.666This means that the
crossing angle results derived here also apply for scan strategies with
varying azimuth scan speed, including during the turnaround periods where the
telescope switches from an “East to West” scan to a “West to East” scan, as
long as the (varying) azimuth scan speed is within this range.

Combining the above, we have a routine that computes the crossing angle $\psi$
(and the required elevation and azimuth for this observation) as a function of
the observatory Latitude, the time, and the RA and Dec of the targeted pixel.
We use this routine in several ways to generate the results in the rest of the
paper. In particular, we use this routine to sample the possible crossing
angles of a given pixel over a fixed period of time (from a single day to 5
years) given a range of possible observing elevations. We also use the routine
to fix the elevation, and then generate all possible crossing angles within
each pixel that can be achieved using that elevation.

By setting up the problem as above, we have made the implicit assumption that
every hit in a scan will be at the centre of the corresponding map pixel. This
assumption introduces a small error, but for the pixel sizes typical of
current surveys, the declination variation within a pixel is small enough for
this to be negligible (we will see in the next section that the declination
variation within each pixel is the only relevant aspect of this assumption).

In this work we are interested in understanding the limits of what can be
achieved in principle, so the setup we use assumes that each element of the
focal plane is free to scan the sky independently. In practice, different
focal plane elements do not have this freedom and effectively scan at slightly
different elevations to the nominal (boresight) elevation. This constraint is
one of the practical reasons that the bounds we derive are unlikely to be
reachable in practice, even when the bounding value is relatively poor.

We note that balloons performing constant elevation scans during circumpolar
flights at an approximately constant latitude will also be well modelled by
our approach, however if there is significant latitude variation amongst the
parts of the balloon flight during which data is taken, then the approach
taken here would need to be generalised in order to describe this situation.

##  3 Limitations of CESs and consequences for scan-coupled systematics

In this section we use the codes described earlier to examine what the
assumption of CESs means for ground-based surveys. The first subsection is not
new, but it is a simple and self-contained summary of the relevant aspects of
crossing angle coverage, which is all easily obtained from our codes. This
summary is required to understand how we construct the limits on $h_{k}$
variation that we present in later subsections.

###  3.1 CES implications for crossing angles

Using the simple simulation tools developed above, we explore the parameter
space that could affect the crossing angle of a given observation. The
parameters we varied were RA; Dec; observing elevation; time, year and season
of observation; rising vs setting (azimuth less than or greater than
$180^{\circ}$); and telescope latitude and longitude. We summarise the key
findings here.

  1. 1.

During a 24-hour period, for a given observing elevation, the point on the sky
in question can be observed at two times, each with a different azimuth; one
rising and one setting.

  2. 2.

Each of these gives a single distinct crossing angle, so for $N$ distinct
observing elevations, there is a maximum of $2N$ distinct crossing angles per
sky pixel.

  3. 3.

The rising and setting crossing angles have the same magnitude but opposite
signs.

  4. 4.

For fixed declination, the largest and smallest (rising) crossing angles
correspond to the largest and smallest observing elevations. The variation
with elevation is monotonic.

  5. 5.

For fixed elevation, there is a monotonic dependence of (rising) crossing
angle on declination.

  6. 6.

The variation of crossing angle with elevation is such that, for fixed
latitude, fields with Dec$\sim$latitude (fields which pass close to overhead),
have the smallest range of possible crossing angles.

  7. 7.

The variation in crossing angle from day to day, over the course of a year, or
from year to year, is negligible.

  8. 8.

The RA and telescope longitude are both unimportant.

We illustrate these results in figure 1, where we show the range of crossing
angles available as a function of declination, for a telescope located at
Latitude $-22^{\circ}56.396^{\prime}$. This latitude corresponds to CMB
experiments in the Atacama Desert, such as the Simons Observatory (SO) site.
The blue lines correspond to the crossing angles obtained from nominal minimum
and maximum observing elevations of $30^{\circ}$ and $65^{\circ}$
respectively. The range of crossing angles possible for each declination
corresponds to the space between the two lines at each point. The narrowing of
the range of crossing angles for fields with Dec$\sim$latitude is clearly
visible. The hatched region corresponds to the approximate range of
declination that will be targeted by the SO Wide survey (Goeckner-Wald et al.,
2018a), and the two darker shaded areas correspond to the approximate ranges
of declination that will be targeted by the SO North and South Deep surveys
(Goeckner-Wald et al., 2018a). We note that the deep scans avoid the range of
declinations where the crossing angle coverage is worst. At the same time, it
is clear from the figure that their placement in declination is not optimal
from the point of view of maximising the crossing angle coverage. The Wide
survey has an unavoidable region of poor crossing angle coverage, that cannot
be mitigated by detailed scheduling choices without removing the assumption of
CESs.

Figure 1: The crossing angles as a function of Declination, for SO location
and elevation bounds $30^{\circ}$ and $65^{\circ}$, are shown by the two blue
lines. The range of crossing angles possible for each declination corresponds
to the space between the two lines at each point. The large hatched region
corresponds to the approximate range of declinations that will be targeted by
the SO Wide survey, and the two darker shaded areas correspond to the
approximate ranges of declinations that will be targeted by the SO Deep
surveys.

###  3.2 Optimising $|h_{k}|^{2}$ in each pixel with CESs

From the previous section, we see that for a fixed observatory location and
targeted declination, the only relevant variables for calculating the possible
values of $h_{k}$ within a pixel at a given sky location are the bounding two
elevations at which observations are taken, and whether we wish to include
observations that are rising, setting or both. Given these minimal pieces of
information, we can calculate $h_{k}$ in that pixel for every possible scan
strategy composed of CESs by sampling the appropriate number of values from
the range of crossing angles bounded by the observing elevations, with
appropriate sign choices according to whether they are rising or setting. The
set of possible pixel values calculated according to this procedure shows the
range of possible pixel values that could be achieved by any ground-based
experiment. In particular, as long as the assumptions of the previous section
hold, most notably that pixels are small enough, no detailed scheduling
choices can deliver a result outside of this range within that pixel.

We know that the worst possible value of $h_{k}$ in any pixel is $1$,
corresponding to only observing that pixel with a single crossing angle. This
is possible within the context of CESs, so to establish the range of possible
$h_{k}$ values within each pixel, we only need to find the optimal (lowest)
value of $h_{k}$ within each pixel. This amounts to finding the best set of
$\psi_{j}$ between the two endpoints (corresponding to the observing elevation
bounds), such that the magnitudes of
$\frac{1}{N}\sum^{N}_{j}\cos\left(k\psi_{j}\right)$ and
$\frac{1}{N}\sum^{N}_{j}\sin\left(k\psi_{j}\right)$ are minimised.

For $k=1$ to $4$, the elevation range $30^{\circ}-65^{\circ}$, and the SO
latitude, the $h_{k}$ metrics are optimised (minimised) for nearly all pixels
through the simple choice of observing only at the two extremal elevations,
and observing the pixel when it is both rising and setting at each of these
two elevations 777The rising and setting aspect of this result is unsurprising
as it is standard practice for ground-based surveys to observe fields when
they are both rising and setting.. We provide some details and explanation for
why this is the case in appendix A.

In other words, for a given pixel, telescope latitude and minimum and maximum
elevation bounds, this setup sets the optimal $h_{k}$ values that can be
achieved in that pixel, for any detailed choice of scheduling; using any
additional intermediate elevations will not improve the crossing-angle
coverage. As a result, for each pixel, observatory latitude, and pair of
bounding observing elevations, we can use the $|h_{k}|^{2}$ values calculated
from the four crossing angles (two bounding elevations, rising and setting for
each of them) as a bound on what can possibly be achieved in that pixel for
any scan strategy composed of CESs, regardless of detailed scheduling choices.
We call this approach for creating per-pixel lower bounds the two elevation
rising and setting (2ERS) setup. We use this 2ERS setup in the rest of this
work to create bounds on what is theoretically possible assuming CESs,
independently of detailed scheduling choices. Note that 2ERS is not a scan
strategy per se, as it is not a complete specification of the detailed
scheduling choices.

The procedure used to arrive at the per pixel optimality of the 2ERS setup
assumes that each pixel is scanned with at least two elevations. The case of
only a single observing elevation is implicitly included in the 2ERS setup in
the limit that the two elevations are arbitrarily close to each other, however
for completeness, we explicitly consider the single elevation case in appendix
B. Here we just note that even allowing for only a single elevation to be
used, 2ERS is optimal when considering the whole visible field, in the sense
that it minimises both the mean and the maximum of the distribution of
$|h_{k}|^{2}$ values over the visible field.

For simplicity in the 2ERS setup we assume the elevations are equally weighted
over the whole field. Generalising the analysis to include the possibility of
weighting makes no difference to the utility of intermediate elevations.
Whilst the two endpoints can be weighted differently, in practice doing so
makes little difference unless one considers the extreme case, where it
approaches the limiting case of observing with a single elevation. This is the
case covered in appendix B.

##  4 $h_{k}$ values from ground based surveys

Having established that the 2ERS setup gives the optimal values of the
$|h_{k}|^{2}$ parameters in each pixel given the bounding elevations at which
observations are taken (for any scan strategy composed of CESs with at least
two observing elevations), we can now make maps of these optimal values, and
investigate their declination and elevation dependence. We do this for
$|h_{k}|^{2}$ with $k=1$ to $4$.

We work within the elevation range $30-65^{\circ}$, and we set the Latitude to
$-22^{\circ}56.396^{\prime}$ corresponding to the Atacama Desert, a common
site for CMB experiments. However, note that due to the symmetry of the
situation, Northern Latitudes ($+\phi^{\circ}$) behave the same as Southern
Latitudes ($-\phi^{\circ}$).

###  4.1 Maps and declination dependence

We show maps of $|h_{2}|^{2}$ in figure 2, computed using the 2ERS setup for
three different pairs of elevations. There are several main features of these
maps. The first is that the range of pixels seen by both elevations reduces as
the upper elevation is increased, as expected. The second is that there is a
clear pattern to the $h_{2}$ maps as a function of Dec, most notably a band of
declinations within the visible area where the parameter is worst (largest).
These features persist for all pairs of elevation bounds in the range
$30-65^{\circ}$. Finally, the peak $|h_{2}|^{2}$ value for each map is lowest
(i.e. best) for the map with the widest range of observing elevations: the
peak values across the field are $0.99$, $0.91$ and $1.0$ for the elevation
pairs $(30^{\circ},35^{\circ})$, $(30^{\circ},65^{\circ})$, and
$(60^{\circ},65^{\circ})$ respectively. The peak values for various bounding
elevations are shown in appendix C.

A key consequence of these maps is that $|h_{2}|^{2}$ from any scan strategy
composed of CESs is unlikely to be reasonably homogeneous over a large field.
Moreover, since the 2ERS map shows the optimal value that can be achieved,
making the field homogeneous would in practice simply amount to making the
value of $|h_{2}|^{2}$ worse over most of the field. For any scan strategy
composed of CESs, there are parts of the sky where $|h_{2}|^{2}$ will always
be close to one and cannot be significantly reduced, in contrast with
satellite scan strategies. We reiterate that, due to the correspondence with
the cross-linking metric used in Choi et al. (2020), the achievable cross-
linking is limited in the same way as these limitations on $|h_{2}|^{2}$.

We examine $|h_{2}|^{2}$ in more detail in figure 3, where we have taken 1D
slices through the maps at RA$=180^{\circ}$ (this choice is arbitrary and a
different choice does not affect the results). The top plot shows the
variation of $|h_{2}|^{2}$ with declination as the lower elevation bound is
varied, and the bottom plot shows the variation of $|h_{2}|^{2}$ as the upper
elevation bound is varied. The lower plot shows the decrease in visible area,
and slight shift of the peak location, as the upper elevation increases. The
decrease in the peak (worst) values with increasing difference between the
elevation bounds is clear in both plots. This peak decrease is not large,
meaning that it will not result in a significant reduction in the effect of
instrumental systematics. However, the connection between $|h_{2}|^{2}$ and
cross-linking means that this peak reduction may be important for any map-
making methods that have problems as $|h_{2}|^{2}$ approaches unity. Precisely
quantifying the required level of cross-linking for these methods is
complicated. Since our focus is on instrumental systematics, we leave a
detailed investigation of this to future work. However, we reiterate that the
bounds calculated in this section are independent of detailed scheduling
choices, so if a detailed study finds these bounds insufficient for a
particular map making method, no scan strategy composed of CESs can deliver
sufficient cross-linking.

Figure 2: $|h_{2}|^{2}$ maps for a telescope at Latitude
$-22^{\circ}56.396^{\prime}$ for the 2ERS setup with elevation bounds (from
top to bottom) $(30,35)$, $(30,65)$, and $(60,65)$. In all cases, there is a
band of declinations within the visible area where the crossing angle coverage
is at its worst. This band moves with the increase in visible area due to the
upper elevation bound, and the worst values improve as the difference between
the two elevation bounds increases. The maps are in equatorial coordinates.

Figure 3: $|h_{2}|^{2}$ as a function of declination as the elevation bounds
are changed for the 2ERS setup and Latitude $-22^{\circ}56.396^{\prime}$. Top:
the upper elevation is fixed to $65^{\circ}$ and the lower bound is varied as
$30^{\circ}$ (blue, solid), $40^{\circ}$ (purple, dashed), $50^{\circ}$ (red,
dot-dashed), $60^{\circ}$(oranged, dotted). Bottom: the lower elevation is
fixed to $30^{\circ}$ and the upper bound is varied as $65^{\circ}$ (blue,
solid), $60^{\circ}$ (purple, dashed), $50^{\circ}$ (red, dot-dashed),
$40^{\circ}$(orange, dotted). In both cases, increasing the difference between
the elevations reduces the peak. The lower plot also shows the decrease of
visible area, and movement of the peak, as the upper elevation is increased.

Figure 4 is the same as the lower panel of figure 3, except that we are now
plotting $|h_{1}|^{2}$, $|h_{3}|^{2}$ and $|h_{4}|^{2}$. Again, the 2ERS setup
is used, with a lower elevation bound of $30^{\circ}$ and varying upper
elevation bound, and we use the same latitude as before. Although the shape of
each function is different to $|h_{2}|^{2}$, $|h_{3}|^{2}$ and $|h_{4}|^{2}$
also possess peak regions where the instrumental systematics cannot be
significantly mitigated. Also similarly to $|h_{2}|^{2}$, the peak (worst)
values are reduced when there is a greater difference between the elevation
bounds. In addition, as for $|h_{2}|^{2}$, there is a small shift in the shape
of the curve and the location of its peak as the upper elevation bound is
changed.

The behaviour of $|h_{1}|^{2}$ is different. The best (minimum) values across
the field are lower for a greater difference between the elevation bounds. As
per the other parameters, $|h_{1}|^{2}$ has regions where the values are
unavoidably poor, however for $|h_{1}|^{2}$ these occur for the declinations
at the edges of the visible field and are reduced by using lower elevations.
Lower elevations are also better for minimising $|h_{1}|^{2}$ when looking
generally across the whole visible field.

The different shapes of the different curves mean that a declination range
chosen to minimise one of these parameters, will not necessarily be minimal
for the others.

We now briefly recap the results of this subsection. Using the 2ERS approach,
we have shown that there is a strong declination dependence to the optimal
values of these parameters in each pixel when CESs are used; this horizontal
stripe structure will persist for any detailed scheduling choices. In
particular, there are declinations for all four parameters for which the
optimal values are poor, no matter which observing elevations one chooses
(although for $|h_{2}|^{2}$, $|h_{3}|^{2}$ and $|h_{4}|^{2}$, a wider range of
observing elevations slightly reduces these worst values, and for
$|h_{1}|^{2}$ choosing lower elevations does the same). These poor values
apply irrespective of detailed scheduling choices.

Figure 4: $|h_{1}|^{2}$ (top), $|h_{3}|^{2}$ (middle) and $|h_{4}|^{2}$
(bottom) as a function of declination as the upper elevation bound is changed
for the 2ERS setup and Latitude $-22^{\circ}56.396^{\prime}$. The lower
elevation is fixed to $30^{\circ}$ and the upper bound is varied as
$65^{\circ}$ (blue, solid), $60^{\circ}$ (purple, dashed), $50^{\circ}$ (red,
dot-dashed), $40^{\circ}$(orange, dotted). As with $|h_{2}|^{2}$, a greater
difference between the elevation bounds lowers the peaks, and increasing the
upper elevation decreases the visible area and moves the position of the peak.

###  4.2 $h_{k}$ tables for systematics

The variation of the $|h_{k}|^{2}$ parameters with declination in the previous
subsection is useful for understanding the possible structures that scan
strategies can cause in maps due to the different systematics, the optimal
values that can be achieved in each pixel, and how field selection can affect
the contamination caused by these systematics. We now consider the power
spectrum contamination caused by the scan coupled systematics; this depends on
the average of the respective $|h_{k}|^{2}$ over all of the observed pixels
(Wallis et al., 2016). We can use the 2ERS setup defined above to calculate
the optimal values of these parameters averaged over a large observed field
consisting of many pixels, and present these in tables. We assume the same
elevation bounds are used in every pixel in the observed field.

We further validated the optimality of our results and the assumptions in the
2ERS setup by repeating the simulations for these tables whilst adding a third
observing elevation. This third elevation was optimally chosen for each pixel
to give a crossing angle midway between the two rising crossing angles from
the elevation bounds. In all cases the average $|h_{k}|^{2}$ values across the
fields were the same or worse than for the 2ERS case, although they typically
do not degrade by much (up to 0.06).

Given a pair of common bounding elevations (and the observable field
corresponding to the table), the value in the table is the lower bound on what
can be achieved by any detailed scheduling choices. There are several uses for
the tables we present. Firstly, for a given field, the values within the table
corresponding to that field can be compared to determine the better elevations
to use for observations, and how much of an effect it has to choose different
observing elevations. The scan strategies from detailed scheduling choices can
be compared to these values to determine how close they are to the limit of
what can be achieved. Secondly, tables for different fields can be compared to
assist in field selection for surveys that have freedom in their choice of
which declinations to observe. Thirdly, the values in each table can be
compared to $1$ to determine how much value there is in trying to optimise the
detailed scheduling choices, and how close to an ideal scan can be reached for
the field represented by that table. Finally, the values (and maps from
earlier) can be used to forecast the best case and worst case scenarios for
particular instrumental systematics, and thus the level of contamination that
can be expected in future surveys, and the possible gain from reducing the
instrumental imperfections.

In tables 1-4 we show the $|h_{k}|^{2}$ values (for $k=1$ to $4$) calculated
using the 2ERS setup at latitude $-22^{\circ}56.396^{\prime}$, and averaged
over the entire observed field. In this case the observed field is the
declination range $-46.96^{\circ}$ to $1.94^{\circ}$, which is the declination
range visible to all elevation pairs in the tables. For completeness and for
reference we include single elevation scans in the tables.

There are two key points from these tables. Firstly, the improvement compared
to the worst possible scan ($|h_{k}|^{2}=1$) in even the most optimal cases is
typically only a factor of a few (a factor of $2.13$ for $h_{2}$, $3.0$ for
$h_{3}$ and $4.35$ for $h_{4}$). The improvement is better for $h_{1}$, where
it can be up to a factor of $12.5$ compared to $|h_{1}|^{2}=1$. These
improvement factors are worse for less optimal elevation choices. Given that
the values in these tables are approximate bounds on what can be achieved for
any choice of detailed scheduling, and may not be reachable in practice, these
values (even for $h_{1}$) are very poor compared to satellite scans, which can
reduce these parameters by $2-3$ orders of magnitude.

Secondly, the variation amongst all of the entries within a particular table
is relatively small, less than a factor of 4 in all cases (a factor of $3.75$
for $h_{1}$, $1.53$ for $h_{2}$, $1.67$ for $h_{3}$ and $2.26$ for $h_{4}$).
This is relatively small given the wide range of elevation values being
considered, and when compared to the variation in values amongst satellite
surveys.

Generally, $|h_{1}|^{2}$ is the easiest to optimise and can be optimised the
most. However, given that the 2ERS setup is a bound that ignores many of the
practicalities of detailed scan strategy choices and scheduling, and the
relatively modest reductions that are achieved in $h_{k}$ even for this ideal
case, from these results we expect that it is not advisable for CMB
experiments using CESs to prioritise optimising any of these parameters.

For $|h_{2}|^{2}$, $|h_{3}|^{2}$ and $|h_{4}|^{2}$ these tables show a similar
general trend as the previous section that increasing the difference between
the elevation bounds improves the quality of the survey in terms of its
inherent mitigation of systematics. For $|h_{1}|^{2}$, the trend is as
expected from the earlier maps and plots: lower elevations are superior.

The diagonal entries of each table show that restricting to a single elevation
is never better (averaged over the whole visible area) than the best 2ERS
setup.

There are some possible complications here that could affect these results.
For a field where the edges cannot be seen at the observing elevations used
for the central part of the field, the central part of the field will behave
as described above. The edge regions can then be treated as one or more
subsets in declination, with a maximum elevation that is reduced in steps from
that used in the central field. Each step between the central region and the
edge of the full field will behave similarly to the central region, just with
a different set of elevation bounds. I.e. one can in effect stitch together
several regions. For simplicity here we have focused on exploring generally
how the individual regions behave. A related point is that we have not applied
any masks; a mask that systematically and significantly affects some
declinations more than others could change the results in these tables,
however most masks and apodisation schemes contain a substantial central
region that is treated uniformly, so it would require a pathological case to
substantially change the results. Due to the assumption that the same
elevation bounds are used in every pixel, the numbers in the tables will not
be strict lower bounds; in principle, small improvements could be made by
making bespoke per pixel elevation choices. We have checked numerically that
such per pixel choices do not have a significant effect on the optimal values
compared to the best case where the pixels use the same elevations, and this
effect does not change our conclusions. Generally, we note that the caveats to
the optimality discussed here are primarily a formality: the bounds we present
here are close to optimal and it is unlikely that these bounds can be achieved
in practice. Moreover, given the other constraints that exist on scan
strategies, attempting to design a scan strategy that achieves the small gains
that could be achieved in principle from exploiting these caveats is likely to
result in something that is unrealistic and poor in other respects. As a
result, these caveats do not affect the conclusions that will be drawn in this
work.

The approach and codes used to make these tables are general, and can be used
to create tables for any observatory targeting any declination range. For
example, we have re-run our analysis restricting to different subsets of the
full declination range in the tables presented here. The results are broadly
similar to the results presented here, with each $|h_{k}|^{2}$ being better in
one half of the full declination range and worse in the other half, as
expected from figures 3 and 4. The variation of the values between different
subsets of declination is typically larger than the variation due to differing
elevations for a fixed declination range.

Since the chosen latitude for our results is the Atacama Desert, the bounds in
this section represent the best possible (smallest) $|h_{k}|^{2}$ (averaged
over the visible sky area) that can be achieved from sky rotation by
experiments in the Atacama Desert (such as ACT or Simons Observatory) using
CESs. In the same fashion, bounds can be constructed for any ground-based
observatory at a different latitude using our approach.

Table 1: $|h_{1}|^{2}$ values for different elevation bounds using 2ERS setup, averaged over the entire sky between declinations $-46.96^{\circ}$ and $1.94^{\circ}$. | $30^{\circ}$ | $35^{\circ}$ | $40^{\circ}$ | $45^{\circ}$ | $50^{\circ}$ | $55^{\circ}$ | $60^{\circ}$ | $65^{\circ}$
---|---|---|---|---|---|---|---|---
$30^{\circ}$ | $0.08$ | $0.08$ | $0.08$ | $0.08$ | $0.09$ | $0.10$ | $0.12$ | $0.15$
$35^{\circ}$ | - | $0.08$ | $0.08$ | $0.09$ | $0.09$ | $0.10$ | $0.12$ | $0.15$
$40^{\circ}$ | - | - | $0.08$ | $0.09$ | $0.10$ | $0.11$ | $0.13$ | $0.16$
$45^{\circ}$ | - | - | - | $0.09$ | $0.10$ | $0.12$ | $0.14$ | $0.17$
$50^{\circ}$ | - | - | - | - | $0.11$ | $0.13$ | $0.15$ | $0.19$
$55^{\circ}$ | - | - | - | - | - | $0.15$ | $0.17$ | $0.21$
$60^{\circ}$ | - | - | - | - | - | - | $0.20$ | $0.25$
$65^{\circ}$ | - | - | - | - | - | - | - | $0.30$
Table 2: $|h_{2}|^{2}$ values for different elevation bounds using 2ERS setup, averaged over the entire sky between declinations $-46.96^{\circ}$ and $1.94^{\circ}$. | $30^{\circ}$ | $35^{\circ}$ | $40^{\circ}$ | $45^{\circ}$ | $50^{\circ}$ | $55^{\circ}$ | $60^{\circ}$ | $65^{\circ}$
---|---|---|---|---|---|---|---|---
$30^{\circ}$ | $0.71$ | $0.72$ | $0.71$ | $0.70$ | $0.68$ | $0.63$ | $0.57$ | $0.48$
$35^{\circ}$ | - | $0.72$ | $0.72$ | $0.71$ | $0.68$ | $0.64$ | $0.58$ | $0.49$
$40^{\circ}$ | - | - | $0.72$ | $0.71$ | $0.68$ | $0.64$ | $0.58$ | $0.50$
$45^{\circ}$ | - | - | - | $0.70$ | $0.67$ | $0.63$ | $0.57$ | $0.50$
$50^{\circ}$ | - | - | - | - | $0.65$ | $0.61$ | $0.56$ | $0.49$
$55^{\circ}$ | - | - | - | - | - | $0.58$ | $0.54$ | $0.48$
$60^{\circ}$ | - | - | - | - | - | - | $0.50$ | $0.47$
$65^{\circ}$ | - | - | - | - | - | - | - | $0.47$
Table 3: $|h_{3}|^{2}$ values for different elevation bounds using 2ERS setup, averaged over the entire sky between declinations $-46.96^{\circ}$ and $1.94^{\circ}$. | $30^{\circ}$ | $35^{\circ}$ | $40^{\circ}$ | $45^{\circ}$ | $50^{\circ}$ | $55^{\circ}$ | $60^{\circ}$ | $65^{\circ}$
---|---|---|---|---|---|---|---|---
$30^{\circ}$ | $0.52$ | $0.49$ | $0.48$ | $0.46$ | $0.46$ | $0.44$ | $0.41$ | $0.33$
$35^{\circ}$ | - | $0.48$ | $0.47$ | $0.46$ | $0.46$ | $0.45$ | $0.42$ | $0.34$
$40^{\circ}$ | - | - | $0.46$ | $0.47$ | $0.47$ | $0.47$ | $0.44$ | $0.36$
$45^{\circ}$ | - | - | - | $0.48$ | $0.49$ | $0.50$ | $0.47$ | $0.39$
$50^{\circ}$ | - | - | - | - | $0.51$ | $0.53$ | $0.51$ | $0.42$
$55^{\circ}$ | - | - | - | - | - | $0.55$ | $0.54$ | $0.46$
$60^{\circ}$ | - | - | - | - | - | - | $0.54$ | $0.48$
$65^{\circ}$ | - | - | - | - | - | - | - | $0.48$
Table 4: $|h_{4}|^{2}$ values for different elevation bounds using 2ERS setup, averaged over the entire sky between declinations $-46.96^{\circ}$ and $1.94^{\circ}$. | $30^{\circ}$ | $35^{\circ}$ | $40^{\circ}$ | $45^{\circ}$ | $50^{\circ}$ | $55^{\circ}$ | $60^{\circ}$ | $65^{\circ}$
---|---|---|---|---|---|---|---|---
$30^{\circ}$ | $0.34$ | $0.38$ | $0.40$ | $0.39$ | $0.37$ | $0.34$ | $0.29$ | $0.23$
$35^{\circ}$ | - | $0.42$ | $0.44$ | $0.44$ | $0.42$ | $0.38$ | $0.33$ | $0.26$
$40^{\circ}$ | - | - | $0.46$ | $0.47$ | $0.45$ | $0.42$ | $0.37$ | $0.29$
$45^{\circ}$ | - | - | - | $0.47$ | $0.46$ | $0.44$ | $0.41$ | $0.32$
$50^{\circ}$ | - | - | - | - | $0.47$ | $0.46$ | $0.44$ | $0.35$
$55^{\circ}$ | - | - | - | - | - | $0.48$ | $0.49$ | $0.40$
$60^{\circ}$ | - | - | - | - | - | - | $0.52$ | $0.47$
$65^{\circ}$ | - | - | - | - | - | - | - | $0.49$

###  4.3 Discussion

From the previous two subsections, we can now answer some of the questions
posed earlier in the paper. The key quantities determining the $h_{k}$ values
in each pixel are the declination and the number of different observing
elevations used888And any detailed scheduling choices that result in non-
uniform distribution of the hits within a pixel between the different
elevations and between rising and setting. Typical values within a pixel for
$|h_{k}|^{2}$ for ground-based surveys using CESs are between $0.1$ and $1$.

When considering the average of $|h_{k}|^{2}$ over the entire observable
field, the possible improvement due to the scan strategy is not significant.
Moreover, the variation with scan strategy of $|h_{k}|^{2}$ averaged over an
observable field is relatively slight: there is a fundamental limit to how
well these quantities can be optimised, and this limit is orders of magnitude
worse than the values that can be achieved by satellite scans. In particular
an ideal scan cannot be approached for ground-based surveys, so care must be
taken if this is assumed whilst forecasting or modelling ground based surveys.

For a particular instrumental systematic, the values in the tables can be used
to evaluate how much further optimisation might be possible for the relevant
$h_{k}$ for a given detailed choice of scheduling. The bounds in tables 1-4,
and the maps in figure 2 (and the equivalent maps for $h_{1}$, $h_{3}$ and
$h_{4}$ with a mask applied if required) can be used in combination with the
expressions in Wallis et al. (2016) & McCallum et al. (2021a) to predict the
magnitude of the distortion due to scan-coupled systematics on both CMB power
spectra and CMB maps.

The relatively narrow range of possible values, and closeness to 1, means that
the effect of these systematics can be calculated and forecasted relatively
accurately without requiring any detailed information about how the instrument
in question will scan the sky. This ability is particularly useful earlier in
the forecasting and design process to get approximate values and best case
scenarios, that can be used to inform field selection and the other design
aspects of the surveys. These results can also be used to calculate the
threshold or tolerance at which the various instrumental systematics will
cause leakage that will affect the science goals.

For example, a differential pointing systematic causes a temperature to
polarisation leakage modulated by $h_{1}$ and $h_{3}$, with a magnitude $\rho$
(for certain focal plane setups) (McCallum et al., 2021a; Wallis et al.,
2016),

$\Delta
C^{B}_{\ell}=\frac{1}{8}\ell^{2}C^{T}_{\ell}\rho^{2}\left(\langle|h_{1}|^{2}\rangle+\langle|h_{3}|^{2}\rangle\right)\text{,}$
(5)

where the angle brackets denote the average value over the observed pixels. To
determine the magnitude of $\rho$ at which the contamination will reach a
certain level thus requires knowledge of the scan strategy. One can assume the
worst case for the scan strategy (i.e. $|h_{1}|^{2}+|h_{3}|^{2}=2$), but for
satellite scan strategies this would give the wrong requirements on $\rho^{2}$
by 3 orders of magnitude. From the tables we have shown here, a “best case”
scenario can now be computed for ground-based experiments. For example, it can
be seen that over the whole field, the absolute best case scenario for the
scan strategy ($|h_{1}|^{2}+|h_{3}|^{2}=0.48$) only reduces the requirement on
$\rho^{2}$ by a factor of $2.1$, showing a stark difference to the satellite
case. This range of best to worst improvement of $2.1$ to $1$ can be used to
forecast the effect of this systematic for Atacama-based CMB-S4 surveys (or
any other Atacama-based surveys), without requiring any detailed scheduling
choices.

###  4.4 Combined sky rotation and boresight rotation

Many CMB experiments use boresight rotation to mitigate systematics and
provide null tests (see e.g. Ade et al. (2015b)). We can use the 2ERS setup to
get an estimate of how sky rotation interacts with boresight rotation, and
estimate the relative efficacy of the two approaches independent of detailed
scheduling choices. We consider the case of a single boresight rotation angle:
whilst many (typically smaller aperture) CMB experiments include multiple
rotation angles, there are cases where only a single angle is possible, such
as the SO large aperture telescope (Thomas et al., 2019; Goeckner-Wald et al.,
2018b). In principle we could extend this to many rotation angles, for example
the set of eight angles separated by $45^{\circ}$ as used by BICEP (Ade et
al., 2015a). We will see that we only need to consider a single angle to show
the superiority of boresight rotation over sky rotation, irrespective of
detailed scheduling choices. Note that in this section $h_{2}$ no longer
relates to the cross-linking, because boresight rotation changes the
orientation of the focal plane with respect to the sky without changing the
path of the instrument across the sky.

We consider the case where the instrument has two values for the boresight
rotation, taken to be $0^{\circ}$ and the boresight rotation value. For each
crossing angle in the 2ERS setup, we now include a pair of angles, where the
second is adjusted according to the rotated value. In some cases, the results
follow trivially from the symmetry of the $\cos$ and $\sin$ functions, as
taken advantage of in e.g. Thomas et al. (2019). For example, $180^{\circ}$
rotation has no effect on $h_{2}$, but reduces $h_{1}$ to zero. We consider
three angles: $180^{\circ}$, $90^{\circ}$ and $45^{\circ}$, and present their
effects on the different metrics in table 5. Note that the ranges in this
table are showing the approximate range of improvement achieved over the 28
elevation combinations considered in the tables earlier, rather than showing
the result for a single choice of bounding elevations. The key result here is
that even the inclusion of a single boresight rotation angle can result in a
strong improvement on the mitigation of scan-coupled systematics compared to
solely using sky rotation. As with our earlier results, this is independent of
the detailed scheduling choices, showing that the gain to $|h_{k}|^{2}$ from
detailed optimisation of the scan strategy is of limited utility.

Table 5: Effects on $h_{k}$ for different choices of boresight rotation angle using 2ERS setup. | $45^{\circ}$ | $90^{\circ}$ | $180^{\circ}$
---|---|---|---
$h_{1}$ |  $1.5-2\times$ worse |  $1.5-2\times$ worse | reduced to zero
$h_{2}$ |  $2-3\times$ better | reduced to zero | no effect
$h_{3}$ |  $5-7\times$ better |  $2\times$ better | reduced to zero
$h_{4}$ | reduced to zero | no effect | no effect

##  5 Conclusions

We have investigated the consequences of scan strategies composed of CESs for
scan-coupled systematics. We used a method that does not require detailed
scheduling choices or simulations, making the conclusions very general. This
method allows us to answer the questions posed in the introduction. The key
results and conclusions are

  1. 1.

We have created a setup that we call 2ERS, which we have used to calculate per
pixel lower bounds on the parameters that couple instrumental systematics to
the scan strategy. We have shown that CESs put strong constraints on the
possible values of these parameters that can be achieved, independently of
detailed scheduling choices.

  2. 2.

Typical per pixel values of $|h_{k}|^{2}$ for $k=1$ to $4$ are between $0.1$
and $1$, regardless of detailed scheduling choices. There is a fundamental
declination dependence of these scan coupling parameters caused by the use of
CESs. This horizontal stripe structure will persist in the maps for any
realistic choice of detailed scheduling. In particular, for all four
parameters, there are declination ranges with unavoidably poor values.

  3. 3.

The optimal (lowest) values of these parameters are driven by the choices of
observing elevations and the declination of the targeted field, however these
optimal values cannot approach an ideal scan, unlike the case of satellite
surveys. The variation with elevation of the optimal achievable parameters is
relatively small when averaged across the visible field, less than a factor of
$4$ for all parameters, and less than a factor of $2$ for $|h_{2}|^{2}$ and
$|h_{3}|^{2}$. In addition, the improvement compared to the worst possible
case for $|h_{2}|^{2}$, $|h_{3}|^{2}$ and $|h_{4}|^{2}$ is at best a factor of
$4.35$; $|h_{1}|^{2}$ can be improved by up to a factor of $12.5$.

  4. 4.

Combined, these results suggest that attempting to strongly optimise detailed
scheduling choices to minimise these parameters is not advisable, particularly
given the other constraints on realistic scan strategies. We have also shown
that inclusion of even a single boresight rotation angle can result in a
greater reduction in the contamination than any scan strategy built from CESs.

  5. 5.

The maps and tables constructed using our approach in section 4 can be used to
calculate and forecast the approximate level of contamination of these
systematics at both the map and power spectrum levels for any CMB experiment
based in the Atacama Desert, without requiring knowledge of detailed scan
strategy choices. These values provide a guide to the relative importance of
different systematics. These values can also be used to assess how far from
optimal a particular detailed choice of scan strategy and scheduling is.
Whilst we have focused on the Atacama Desert as the most important example,
the codes and approach in this work can be easily applied to create such maps
and tables for an observatory at any latitude.

  6. 6.

Generally, $|h_{2}|^{2}$, $|h_{3}|^{2}$ and $|h_{4}|^{2}$ are improved at a
pixel or field level by considering more widely spaced elevations. However,
$|h_{1}|^{2}$ behaves quite differently: it prefers lower elevations more than
widely spaced elevations, and it can be optimised further than the other 3
parameters.

  7. 7.

We also note that the standard metric for cross-linking (Choi et al., 2020) is
closely related to one of the parameters used in this work ($|h_{2}|^{2}$). As
such, a corollary of our results is that there are fundamental limits to the
cross-linking that can be achieved using CESs, regardless of any detailed
scheduling choices. This also means that if homogeneous cross-linking is
required, this can only be achieved by making the better areas worse, and not
by improving the regions where cross-linking is bounded to be poor.

We expect these results to be useful for understanding the limits of CESs for
upcoming CMB surveys, particularly SO and CMB-S4 (Abazajian et al., 2019), and
also for allowing forecasts and simulations for different systematics to be
made, all without requiring detailed choices of scheduling and expensive
simulations. In addition, these results apply to any non-CMB ground based
surveys that perform CESs, where the spins of different quantities are
important, including intensity mapping.

###### Acknowledgements.

DBT acknowledges support from Science and Technology Facilities Council (STFC)
grants ST/P000649/1, ST/T000414/1 and ST/T000341/1. NM is supported by an STFC
studentship. MLB acknowledges support from STFC grant ST/T007222/1. The codes
used to generate these results are available from the authors on request.

##  Appendix A A. Justification and explanation of 2ERS optimality

In the main text we comment that for a given range of observing elevations, as
long as at least two observing elevations are used, the $|h_{n}|^{2}$
parameters are minimised in the majority of pixels by only observing at the
two bounding elevations, and doing this for both rising and setting
observations. In this appendix we explain why this is the case.

###  A.1 Rising and setting

The generic expression for $|h_{n}|^{2}$ in a pixel is given by

$|h_{n}|^{2}_{\text{rise}}=\frac{1}{N^{2}}\left(\sum^{N}_{i}\cos(n\psi_{i})\right)^{2}+\frac{1}{N^{2}}\left(\sum^{N}_{i}\sin(n\psi_{i})\right)^{2}\text{.}$
(6)

Let us assume without loss of generality that the set of $N$ crossing angles
in this sum are all rising. Given the relationship between rising and setting
crossing angles (a sign change) and the fact that $\sin$ is an odd function
and $\cos$ is an even function, we can consider how this expression changes
when a second set of $N$ crossing angles are included, consisting of the
setting counterparts of the initial set of angles,

$\displaystyle|h_{n}|^{2}=\frac{1}{4N^{2}}\left(\sum^{2N}_{i}\cos(n\psi_{i})\right)^{2}+\frac{1}{4N^{2}}\left(\sum^{2N}_{i}\sin(n\psi_{i})\right)^{2}$
(7)
$\displaystyle=\frac{1}{4N^{2}}\left(\sum^{N}_{i}\cos(n\psi_{i})+\sum^{N}_{i}\cos(-n\psi_{i})\right)^{2}+\frac{1}{4N^{2}}\left(\sum^{N}_{i}\sin(n\psi_{i})+\sum^{N}_{i}\sin(-n\psi_{i})\right)^{2}$
(8)
$\displaystyle=\frac{1}{4N^{2}}\left(\sum^{N}_{i}\cos(n\psi_{i})^{2}+\sum^{N}_{i}\cos(n\psi_{i})\right)^{2}=\frac{1}{N^{2}}\left(\sum^{N}_{i}\cos(n\psi_{i})\right)^{2}\leq|h_{n}|^{2}_{\text{rise}}$
(9)

Therefore it will never be advantageous for the $|h_{n}|^{2}$ parameters to
not include the setting counterpart to any rising observations, and vice
versa.

###  A.2 Systematically spaced elevations

Having established that we will always work with rising and setting crossing
angle pairs, the next thing to establish is why additional elevations between
the endpoints do not improve the $|h_{n}|^{2}$ parameters. We start with the
expression for combined rising and setting angles, and we place equally spaced
crossing angles between the endpoints (the endpoints are denoted $\psi_{A}$
and $\psi_{B}$)

$|h_{n}|^{2}=\frac{1}{N^{2}}\left(\sum^{N}_{i}\cos(n\psi_{i})\right)^{2}=\frac{1}{N^{2}}\left(\sum^{N}_{i}\cos\left(n\left[\psi_{A}+\frac{(i-1)}{(N-1)}\left(\psi_{B}-\psi_{A})\right)\right]\right)\right)^{2}$
(10)

We can now use the result that

$\sum^{N-1}_{i=0}\cos\left(a+ib\right)=\cos\left(a+\frac{(N-1)}{2}b\right)\frac{\sin\left(\frac{Nb}{2}\right)}{\sin\left(\frac{b}{2}\right)}\text{,}$
(11)

so for equally spaced angles we can express $|h_{n}|^{2}$ as a function of $N$
as

$|h_{n}|^{2}=\frac{1}{N^{2}}\left(\sum^{N}_{i}\cos(n\psi_{i})\right)^{2}\propto\left(\frac{1}{N}\frac{\sin\left(n\frac{N}{N-1}\frac{\psi_{B}-\psi_{A}}{2}\right)}{\sin\left(n\frac{1}{N-1}\frac{\psi_{B}-\psi_{A}}{2}\right)}\right)^{2}\text{.}$
(12)

For $N\geq 2$, this function is monotonically increasing with $N$, provided
that $\psi_{A}-\psi_{B}<\frac{203.1}{n}$. This condition is satisfied for all
pixels for $n=1,2$ and is satisfied for the overwhelming majority of pixels
for $n=3,4$, with the exception of a small range of declinations at the edges
of the visible field. As a result, the non-optimally of 2ERS for our results
happens only for small declination ranges at the edge of the visible field,
and even for most of these declinations 2ERS is close to optimal. Given the
large sky areas targeted by upcoming CMB experiments, this will not
significantly affect the results in the main text.

##  Appendix B B. Single elevation scans

The optimality of the 2ERS setup as discussed in appendix A assumes at least
two elevations are used. The case of a single elevation is implicitly included
in the limit that the two bounding elevations are equal, or the limit when one
of the bounding elevations is weighted significantly more than the other. For
completeness, in this appendix we explicitly consider the case of a single
elevation and see in which circumstances, and by how much, this could generate
a more optimal bound than the 2ERS setup. Note the optimality of the 2ERS
setup in the sense that intermediate elevations do not improve the
$|h_{k}|^{2}$ parameters is unaffected by the discussion in this appendix.

Note there are clear limitations to scans with a single elevation: a maximum
of 2 crossing angles within each pixel is achievable, and it limits the amount
of time that the target field can be scanned. As such, it is only realistic
for surveys targeting large patches of the sky, and we expect these to cover a
larger declination range than the band in which a single elevation is the best
choice. Of course one could observe a large field by observing different
strips in declination with their respective optimal single elevation, but this
would require much more complicated scheduling and will not result in
significant reductions in $|h_{k}|^{2}$ compared to a 2ERS setup. As a result,
the cases we will discuss below where there are possible small improvements
over the case with two distinct elevations are not necessarily realistic.

In general, 2ERS with two distinct bounding elevations has the best average
over the whole range of visible declinations, even when allowing for single
elevation scans (see tables 1-4).

In addition, the peak (worst) values over the entire field will always be
worse for the case of a single elevation compared to 2ERS with two distinct
bounding elevations.

However, a single elevation can deliver smaller values of $|h_{k}|^{2}$ than
2ERS in specific pixels, or small declination ranges. This situation occurs
when the targeted declination range (dependent on the latitude of the
telescope) is carefully chosen and small enough; in this case the optimal
$|h_{k}|^{2}$ values come from observing at only one of the two limiting
elevations, and adding any other elevations will degrade the quality of the
crossing-angle coverage. Consequently, if using two observing elevations, the
bounds from the 2ERS setup will not quite be strict bounds. However, in
practice, the average, peak, and per pixel values from considering a two
distinct elevations are close to those from a careful choice of single
elevation. As such, none of the conclusions in the main text are altered by
considering this case.

We illustrate this in figure 5. Here we compare the 2ERS setup to setups where
only one each of the two limiting elevations are used (but rising and setting
are both still used). These plots show that 2ERS has the lowest peak over the
whole visible field, is never the worst value over any range, and gives a
competitive bound when averaged over the whole visible area. However, specific
declination ranges can be chosen such that a single elevation setup is optimal
for the entire targeted declination range. For these specific declination
ranges, 2ERS is not a strict lower bound on the $h_{k}$ metrics (but it is
close), and the respective single elevation scan provides the true bound.

In more detail, a single elevation scan using the lower elevation from a 2ERS
scan typically cuts through the 2ERS peak at a slightly lower value (but is
worse for all other declinations). Conversely, a single elevation scan using
the upper elevation is typically better for the range of declinations either
side of the peak. For example, considering the left hand panel of figure 5, if
one is targeting a declination range of $A^{\circ}$ to $B^{\circ}$, then 2ERS
would not be a strict lower bound for the average value over the targeted
field, and perhaps not a strict lower bound for the peak value, although it
does approximate the correct value. If greater precision of the scheduling-
independent bound is required, the single elevation scan at the upper limit of
$65^{\circ}$ does provide this bound, and can be used for simple scheduling-
independent investigations, which is why we have included single elevations in
our results tables.

Figure 5: $|h_{2}|^{2}$ as a function of declination, comparing 2ERS to single
elevation scans for the upper and lower elevations. In each case the black
(solid) line is the 2ERS setup, the blue (dashed) line is the single elevation
setup for the lower bound and the red (dotted) line is the single elevation
setup for the upper bound. The left plot has elevation bounds $(30,65)$ and
the right plot has elevation bounds $(40,55)$. The same properties are visible
in both: 2ERS has the lowest peak, is better when averaged over the whole
visible area, and it is always close to the best per pixel value from a single
elevation. However, specific declination ranges can be chosen such that one of
the single elevation setups is optimal for the entire declination range. For
these specific declination ranges, 2ERS is not a strict lower bound (but it is
close), and the respective single elevation scan provides the true bound.

##  Appendix C C. Peak values

We show the peak value for different pairs of elevation bounds in table 6.

Table 6: $|h_{2}|^{2}$ worst pixel values for different elevation bounds using 2ERS setup, averaged over the entire sky between declinations $-46.96^{\circ}$ and $1.94^{\circ}$). | $30^{\circ}$ | $35^{\circ}$ | $40^{\circ}$ | $45^{\circ}$ | $50^{\circ}$ | $55^{\circ}$ | $60^{\circ}$ | $65^{\circ}$
---|---|---|---|---|---|---|---|---
$30^{\circ}$ | $1.0$ | $0.99$ | $0.98$ | $0.96$ | $0.94$ | $0.93$ | $0.92$ | $0.91$
$35^{\circ}$ | - | $1.0$ | $0.99$ | $0.98$ | $0.97$ | $0.96$ | $0.94$ | $0.93$
$40^{\circ}$ | - | - | $1.0$ | $1.0$ | $1.0$ | $1.0$ | $1.0$ | $1.0$
$45^{\circ}$ | - | - | - | $1.0$ | $1.0$ | $1.0$ | $1.0$ | $1.0$
$50^{\circ}$ | - | - | - | - | $1.0$ | $1.0$ | $1.0$ | $1.0$
$55^{\circ}$ | - | - | - | - | - | $1.0$ | $1.0$ | $1.0$
$60^{\circ}$ | - | - | - | - | - | - | $1.0$ | $1.0$
$65^{\circ}$ | - | - | - | - | - | - | - | $1.0$

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