defining targets that are scheduled with a locally fixed patch area and
sequence, i.e. these two regions of the EWS are observed with the same
strategy as the EDFs and EAFs.

The resulting schedule is called the ‘stage-1 schedule’, and its computation
consists of three steps:

  1. 1.

analysis of each target (strategy and visibility);

  2. 2.

placement of each target in longitude and year;

  3. 3.

assignment of timestamps to each placement.

####  6.3.1 Target analysis

In general, a target has four types of constraints: a fixed location on the
sky, a specific observation sequence, a minimum depth, and a cadence (or other
time constraints). From these we compute the target’s ‘window of visibility’,
i.e. the range of ecliptic longitudes (of the Sun) in which the target may be
observed (on the leading or on the trailing side of the orbit). Typically,
each target has two windows of visibility per year.

The computation of each target window of visibility reveals its flexibility in
placement and, more important, possible conflicts (of placement) with other
targets. The procedure is essentially manual (aided by software tools). Three
cases are of particular interest (see [Sc21] for details).

The first case is the EDF-Fornax, which is planned to be observed gradually
along the mission for a total of 52 times, taking into account the larger
local background. It can be observed twice per year. However, the window of
visibility of the EDF-Fornax partially collides with that of the EDF-South,
and with some orientations of the CPC-South. This means that scheduling one of
these targets strongly restricts the placement of the others. To overcome this
conflict, the visits to CPC-South are all scheduled in the first year, leaving
those longitudes free for EDF-Fornax, in the following years (5 times once
every six months, plus a short 2 times visit on the first year, as required).
Likewise, the EDF-South is observed from the second year onward (but offset
from EDF-Fornax).

The second case is the scheduling of the COSMOS and SXDS photo-$z$ calibration
targets. These require an observation of a $2\times 2$ pattern that, if
observed in one go, would take 15 days each. This would pose a great
difficulty for the scheduling of the EWS, by adding a long interruption. In
general, when scheduling the EWS, it is possible to bridge over interruptions
(such as calibrations), if shorter than five days. Long interruptions are not
bridgeable, forcing a interruption of the EWS build-up, something that must be
compensated in the next passage (at least a half-year later). It also reduces
the opportunity to place PSF calibrations (that require a week of EWS
observations prior to calibration). The solution is to observe these fields in
four visits, observing half of it each time, first to an intermediate depth,
and then a second time to the final depth.

The third case are the self-calibration and VIS non-linearity calibrations,
which both require approximately a monthly cadence of observations. In order
to decrease the number of interruptions to the EWS observations and the number
of large slews used, these observations are always scheduled in sequence and
executed at the same sky field, defining a ‘recurring calibration block’

Finally, every four weeks starting on Mondays at noon (UTC) $\pm$ 1.2 h, there
is one block of 12 h that is reserved for spacecraft orbit maintenance
operations (SOP). During SOP time survey data cannot be taken.

####  6.3.2 Target placement

The second part of the computation of the stage-1 schedule consists on placing
each target in a table (named the ‘design schedule’), at a given longitude and
year, striving to avoid collisions with other targets. In that table, choices
have to be made such as what target goes into each year, and at what longitude
to place it. There is some freedom in this procedure. For instance, many
targets do not require to be observed in a specific year. However, some
targets are more stringent than others; i.e., have shorter visibility windows
or have a single annual visibility. Therefore, for simplicity, it is
preferable to place the targets by decreasing order of perceived difficulty,
filling the table year by year. This process is performed manually.

While being manual, the process does not need to be very precise: some overlap
in longitude between targets is allowed. Any overlaps are rectified in the
next step and the longitudes are converted to timestamps, transforming the
design schedule into the stage-1 schedule.

The design schedule is filled according to a strategy that prioritises the
placement of the targets with the most constraining observing conditions. In
the first year the order of placement is as follows.

  1. 1.

Place all ten CPC-South visits at transit longitude. These are the most
demanding targets.

  2. 2.

Place recurring calibration blocks (self-cal+VIS non-linearity) at an
approximate step of $$$/11=$$$. This promotes a synergy with CPC-North.

  3. 3.

Place the ten CPC-North visits. The orientation of these targets has a step of
$None$.

  4. 4.

Place eleven targets of the northern polar cap. The width of these patches and
the angular offset was chosen to match the placement of the CPC-North targets
(i.e. immediately before or after). The remaining two targets are placed in
the second year.

  5. 5.

Place all eight targets of the southern polar cap.

  6. 6.

Place the first two targets of the COSMOS-wide. This covers half the area
needed on this target in the first year.

  7. 7.

Place a $\times 2$ pass visit to EDF-Fornax, slightly offset from its ideal
placement, to avoid collision with one of the CPC-South targets.

  8. 8.

Place the AEGIS target (required for the first year).

  9. 9.

Place the two NISP nonlinearity targets. Because these are freely placeable,
the choice is to place them on the slot of longitudes where there is less area
of EWS within the RoI.

  10. 10.

Place a double visit to EDF-North to avoid a small gap of time between other
targets. Since it can be placed all year round, the visits to EDF-North may be
used as a filler (as in this case).

At this point, the first year is filled. It would be possible to pack more
targets into it, but that would reduce the size and number of EWS windows,
which in turn reduces the opportunity to place VIS PSF calibration fields. The
latter need to be scheduled within EWS windows larger than seven days (see
Sect. 6.1.2).

With all CPC observations placed on the design schedule, the most demanding
target left to be scheduled is EDF-Fornax. Unlike the EDF-South, the EDF-
Fornax cannot be reached far from transit through a large depointing because
of its low ecliptic latitude. The remaining years follow a common strategy,
with most of the targets (EDF-Fornax, EDF-South, recurring calibration blocks,
EDF-North, NISP wavelength dispersion) being scheduled almost exactly the same
way. The exception are the EAF targets that vary from year to year: In the
current schedule, the SXDS is placed in the 3rd and 4th years, and the VVDS
and GOODS-North in the 5th year.

The strategy for the remaining years is as follows.

  1. 1.

Place two $\times 5$ passes to EDF-Fornax at their transits.

  2. 2.

Place two pairs of visits to EDF-South. One pair of one pass visit plus a
$\times 2$ pass visit, separated by $None$, and a second pair of two $\times
2$ pass visits, also separated by $None$ (ensuring survey-windows not smaller
than $5$ days, given an average orbital progression of $None$ per day). Each
group is clustered around each EDF-Fornax pointing.

  3. 3.

Place the 11 recurring calibration blocks. The blocks are slightly offset from
a nominal cadence to avoid conflicts with the EDF-Fornax and EDF-South
observations.

  4. 4.

Place the two remaining northern polar targets (second year only).

  5. 5.

Place the large block of NISP wavelength dispersion target (second year only).

  6. 6.

Place EAF targets (these vary from year to year).

  7. 7.

Place two visits to EDF-North, one $\times 4$ pass and another $\times 2$
pass, near the centre of the range of EWS RoI scarcity, matching surrounding
recurring calibration blocks.

  8. 8.

Place the two NISP non-linearity targets.

Figure 29: An example of the full schedule, including also the stage-1
schedule discussed in Sect. 6.3.3. Time goes left to right for each row, which
spans one year of the survey. Subsequent years go from top to bottom. See text
for a description

####  6.3.3 Timestamp assignment

The last stage of the computation of the intermediate schedule is the
automated conversion of the year-by-year longitudes into timestamps. Given a
starting date to the survey routine phase (currently but not frozen yet, this
is expected to start on 8392.5 Modified Julian Date, or 2022-12-23T11:59:23Z),
it is easy to compute the corresponding Sun longitude ($None$). Then, it is a
simple matter of “reading” the design schedule, year-wise.

The first target following the starting longitude is a recurring calibration
block at $None$, approximately two days after the beginning. This timestamp is
assigned to the first target found. Practically, this involves only computing
the next timestamp when the Sun is at a given longitude. The process
continues, assigning timestamps to targets of the first year, by order of
longitude (wrapping around at longitude $None$). Once the traversing reaches
the initial longitude, the process continues in the second year, starting at
the same longitude where the first year ended. Once the second year is
completed, the process continues on the third year and so on, until the
intermediate schedule is completed. There is a possibility that the initial
longitude for traversing any year coincides with the middle of a target. In
those cases, the traversing continues after that target.

Besides converting longitudes to timestamps, an algorithm disentangles
overlaps between targets and prevents, when possible, the occurrence of too
small EWS windows. The overlap of two or more targets (allowed in the previous
stage), is fixed by offsetting those targets from their initial placement,
minimising the overall offset, within the range allowed by their window of
visibility. In most cases, this process is sufficient to resolve overlaps. In
case of failure, the solution is to go back to the previous stage and fix the
overlap manually. The same process is applied to eliminate the occurrence of
small windows. Except that, now, the offsetting is in the opposite direction,
pushing targets closer to each other.

####  6.3.4 Results

Figure 29 shows an example of the full Euclid schedule. The result of the
stage-1 procedure, in this diagram, is the sequence of coloured boxes. The
time allocated for each observation of a target is represented by a labelled
box of a unique colour. The pink and white boxes represent the periods
available to observe the EWS (at the end of the stage-1 scheduling they are
all still unallocated). During the stage-2 scheduling (Sect. 7) the EWS is
scheduled in part of the available time (shown by the pink boxes), while some
time periods remain unallocated (shown by the white boxes that are
increasingly longer towards the final years of the survey, cf. Sect. 8.2).

The alignment of boxes with the same colour across years indicates that the
respective targets are scheduled at the same time every year. The labels (and
widths) of the boxes indicate when consecutive visits are made to the same
field. The Deep Fields are often scheduled with more than one visit in a row,
in particular EDF-Fornax is usually visited 5 times to efficiently use its
short time visibility that occurs twice per year.

Differently from the other calibrations, PSF calibrations are scheduled within
the pink boxes, i.e., the EWS observation is interrupted to point to a PSF
field (for $\sim 20.5$ h) and then return to the same position. This allows
for a much better stability of the SAA and AA values used (that will match the
ones used on that observation of the EWS), than if they were scheduled in the
recurring calibration block, always together with the other targets of
approximately monthly cadence. The visibility windows of twenty preliminary
PSF fields are shown as horizontal red bars within the pink EWS boxes, while
the vertical red bars show the actual scheduled time of the PSF observations
(the corresponding Sun longitude for each of the fields is indicated by the
numbers 1 to 20 on the axes at the edges of the Fig. 29).

Table 8: Time budget for calibration, EDFs and EAFs observations.

Type | Name | Time | Total
---|---|---|---
|  | [days] | [days]
| VIS PSF calibration | 38 |
Instrument | VIS nonlinearity | 28 |
| NISP wavelength dispersion | 8 |
| NISP nonlinearity | 32 | 106
EAFs | The six fields | 34 |
| Self-calibration | 54 | 88
| EDF-N | 35 |
EDFs | EDF-S | 89 |
| EDF-F | 61 |
| CPC | 48 | 233
| 427

Table 8 summarises the time allocated to make all calibration, EDF and EAF
observations, which is 427 days. Note that the time allocated for the self-cal
field is here included in the EAFs budget and not in the instrument
calibrations, since its repeated observations will make it the deepest of the
Euclid fields. Also note that due to the synergy between CPC and EDFs, 36 of
the 48 days needed for CPC calibrations also contribute for the completion of
the Euclid Deep Survey. Details are given in [Sc21].

The breakdown of the time-allocation by observing type is:

  * •

instrument calibrations, 25$\%$;

  * •

auxiliary fields, $21\%$;

  * •

deep fields, $54\%$.

##  7 Computing the Euclid Wide Survey

The computation of the EWS is a complex optimisation problem, for which we
have developed the scheduling tool ECTile. After the preliminary stage-1
described in Sect. 6.3 that determines the schedule of the various
calibrations, EDFs, EAFs, and polar caps targets, ECTile proceeds with the
computation of the EWS in its stage-2. Before turning to a detailed
description of stage-2 of ECTile from Sect. 7.2 onwards, we give a brief
description of pre-ECTile explorations.

###  7.1 Early explorations

The derivation of the optimal survey is a complex process, and ECTile is one
of many possible solutions. It is, however, worth stressing that it is the
outcome of a lengthy process, in which alternatives have been explored, but
ultimately rejected. For instance, the first solutions of the EWS were
delivered by industry to demonstrate the feasibility of the survey, but
ignored some important additional considerations, such as observing areas with
low zodiacal background first (see Sect. 5.1.1). Other solutions (Amiaux et
al., 2012) were produced using ESA’s Euclid Sky Survey Planning Tool (ESSPT)
(Gómez-Alvarez et al., 2018), which allows the user to manually place patches
on a sky map and fill them with Euclid FoVs. In this section we provide a
brief overview of the prior investigations that led to ECTile as it is today.

Figure 30: An example of scheduling the EWS with a rigid tiling and requiring
—AA— $<1\degr$, covering parts of the northern ecliptic hemisphere. Different
colours indicate different observing epochs. _Left panel_ : The empty dark
regions are caused by interruptions of the EWS. _Right panel_ : The empty
regions are only partially recovered at a later stage, and new gaps are formed
at lower latitudes.

To cover a maximum area with minimum overlap between single observations, a
pre-determined tiling is almost unavoidable. The early approaches (see Tereno
et al., 2014) therefore defined rigid FoVs placed parallel to the ecliptic
meridians. The fields were scheduled starting at high latitudes and moving up
and down along ecliptic meridians, observing in transit, that is with ${\rm
SAA}=$$$ (and ${\rm AA}=$$$) when the local meridian coincides with the
$Y_{\rm SC}$ direction defined in Sect. 2.1 (see also Sect. 2.5). The extent
to which we move across latitudes before moving to the next longitude, defines
a latitude band, to be observed in one year. Due to the convergence towards
the poles, the number of fields per band decreases with latitude, and
conversely the height of the bands increase with latitude.

Whenever there is an interruption in the EWS schedule due to an observation of
calibration or EDF targets, a corresponding gap is left in the band. After one
revolution in the orbit, a latitude band is finished and the scheduling of the
next band, on a lower latitude, starts. The gaps can be recovered in the
following year when in transit again. For this, the height of the next
latitude band needs to be smaller, in order to create a time buffer that
allows one to cover the gaps while not creating new gaps in the lower band.
This way, the missing area can be observed slightly off-transit, tilting the
telescope arouns $X_{\rm SC}$. The tilt must be accompanied by a rotation
around the $Z_{\rm SC}$ axis (hence changing AA) to compensate and keep the
field aligned with the tiling. However, given the very stringent constraints
on AA ($|{\rm AA}|<$$$ at the time of the early explorations) the time buffers
are necessarily small, and it is only possible to partially cover the gaps.
This process is illustrated in Fig. 30. It is clear that gaps in the survey
are unavoidable, and attempts to fill them only leads to more gaps elsewhere.
It was thus concluded that the use of a fixed tesselation and fixed latitude
bands was not viable.

The AA limit was eventually relaxed to $3\degr$ at the Preliminary Design
Review, (PDR) and later to $6\degr$ at the Critical Design Review (CDR), being
currently fixed at $5\degr$. This makes the use of a fixed tessellation a
viable approach, enabling an efficient coverage of the sky. However, the use
of fixed latitude bands remains not viable and more complex strategies needed
to be developed, as described in the next sections.

###  7.2 Introducing ECTile’s stage-2

The computation of the EWS revolves around the concept of a patch, which may
be loosely defined as a compact set of fields with a valid observation
sequence. This is the basic building block for this computation. This section
describes the steps that precede the computation of a patch, summarised in
Fig. 31. The first step is the computation of the tessellated RoI, a set of
tiles covering the designed RoI. In parallel, survey-windows are computed from
the stage-1 schedule, defining the intervals of time left to EWS observations.
These two elements are then combined into patch-sources, compact sub-sets of
each quadrant of the tessellated RoI within reach of a given survey-window. A
patch-source is, in general, further divided into one or more patch-segments,
which are sets of tiles guaranteed to be not only compact but also having
their contour shaped like a lat-long rectangle (an essential property for the
computation that follows it). Patch-segments from the same patch-source are
then transformed into a patch by establishing upon them an ordered sequence of
observations.

Figure 31: Steps and concepts required for the computation of the EWS, linking
the RoI and stage-1 schedule to patches (defined later).

###  7.3 Major constraints for the EWS

In this section we present the main inputs and associated constraints for the
EWS optimisation algorithm, and briefly review their impact on the current
solution.

####  7.3.1 Constraints due to overlap

Figure 32: Example of a few rows of a tile driven tessellation drawn in a
cylindrical projection. Individual tiles are indicated by the red rectangles,
whereas the (distorted) footprints of the FoV for non-zero latitudes are shown
in blue (sizes are exaggerated for clarity). The number of tiles per row
varies with latitude.

We define a tile as the largest rectangle in latitude–longitude that is
completely contained in a single FoV (see Fig. 32). The survey area must then
be observed through geometrically contiguous tiles, with an overlap of
boundaries between adjacent tiles of a $0.5$% wide strip (1% overlap,
overall), to cope with the non-null pointing error. The goal of this
requirement is to enable efficient coverage of the sky, whilst ensuring a
minimum overlap between adjacent fields. This can be achieved with a
tessellation of tiles laid out along parallels of latitude, with adjacent
tiles on the same row and tiles between adjacent rows overlapping by $1\%$.
Due to the convergence towards the poles, the number of tiles per row
decreases with latitude, as shown in Fig. 32. Given the geometric shape of the
FoV, a rectangle on a sphere, the overlap between FoVs also increases with
latitude.

####  7.3.2 Constraints due to SAA and AA

Figure 33: The subset of the tessellation observable at given epoch, with the
Sun pointing at $0\,\text{deg}$ and the transit meridians at $90\,\text{deg}$
(trailing-side) and $270\,\text{deg}$ (leading-side). This shows that the need
to observe fields aligned with meridians strongly restricts visibility (see
Fig. 5)

As described in Sect. 2.5, the limited range of the pointing angles implies
that most of the sky must be observed at, or at least close to transit. As we
motivate later, most of the EWS is observed with fields aligned with the
ecliptic meridians. In general, these fields are almost never observed at
transit, thus requiring a rotation around $Z_{\rm SC}$ to realign the FoV with
the local meridian. However, the constraints on AA and SAA severely limit the
extent to which a field may be observed away from transit. This is highlighted
in Fig. 33 which shows the region of the sky that is observable at a given
transit. It is mostly constrained by the AA range, except at lower latitudes,
where the constraint on SAA dominates.

Figure 34: Example of adjacent FoV slew ability at high latitude (_left
panel_) and low latitude (_right panel_). The plots show the slew from a
central tile (blue) to its neighbours; up to ${1\aas@@fstack{\circ}2}$ in
light-red, and from ${1\aas@@fstack{\circ}2}$ to ${3\aas@@fstack{\circ}6}$ in
dark-red. At high latitudes, slew movements between FoVs of a tessellation are
severely restricted.

####  7.3.3 Constraints due to slews

The limitations associated with the cost of a slew (see Sect. 2.4), imply that
EWS fields observed consecutively in time must also be, as much as possible,
spatially adjacent to each other. In this way, large slews are mostly reserved
for moving between EWS fields and calibration or Deep Fields, or between
patches of the EWS.

Figure 34 shows the reach from a given field, when slewing with a small-slew
within a limit of ${1\aas@@fstack{\circ}2}$ and ${3\aas@@fstack{\circ}6}$, for
a field placed at two different latitudes. The example demonstrates that, at
low latitude, it is possible to slew to all adjacent tiles; if considering
slews up to ${3\aas@@fstack{\circ}6}$, it is possible to slew to tiles two
rows away (recall that the slew can be depicted as the arc separating two
different directions on the sky plus a rotation around the latter). However,
at high latitude, the slew between adjacent tiles is much more limited by the
size of the not small change in AA needed to keep the alignment with the local
meridian of the tessellation. There, adjacent tiles on the same row are
further apart, strongly limiting the field-to-field slewing which, in this
case, is performed by a rotation around the $Z_{\rm SC}$-axis. For example,
given the FoV width of ${0\aas@@fstack{\circ}7}01$, tiles placed at latitude
of ${79\aas@@fstack{\circ}0}8$ are separated exactly by
${3\aas@@fstack{\circ}6}$ of longitude. Thus, above $79\degr$, it is not
possible to slew sideways between two adjacent fields (aligned with the
tessellation).

###  7.4 From EWS tessellation to patches

####  7.4.1 The tessellated RoI

The generation of the tessellated RoI begins with the computation of a global
tessellation on the sphere, covering the sphere without polar caps from
latitudes $\pm 79\degr$ towards the equator, with non-overlapping tiles
aligned with the meridians. This tessellation is then filtered by selecting
tiles that have at least one of their corners inside the RoI. This represents
the tessellated RoI and specifies the FoVs eligible to be observed (see Fig.
35). Figure 36 shows two zoomed views, highlighting the dependence of overlap
with latitude. The EWS solution schedules a large subset of the tessellated
RoI, which then becomes the Euclid ‘footprint’.

Figure 35: Tessellated RoI (in green), covering the four quadrants. The dashed
red line represents the RoI.

Figure 36: _Left panel_ : Zoomed view of the tessellation for the EWS at low
latitude. _Right panel_ : Same, but now for high latitude. The increased
overlap of neighbouring tiles at high latitude is evident. Orange tiles
contain a blinding star (black dots) and are skipped from observation.

####  7.4.2 Covering the polar caps

As discussed in Sect. 7.3.3, above $79\degr$ it is not possible to slew
sideways, when traversing fields aligned with the tessellation. Indeed, at
high latitudes, the centres of adjacent fields on the same row of the
tessellation are separated by an eigenslew larger than
$3\aas@@fstack{\circ}7$. As the amplitude of an eigenslew includes the
rotations needed to align the fields and not only the angular separation
between the fields centres, the part of the EWS above $+79\degr$ and below
$-79\degr$, the polar caps, are not part of the global tessellation.

Instead, each polar cap is covered by a fixed number of patches of fixed area,
as shown in Fig. 37. The northern cap is covered with 13 patches, while there
are six patches covering the southern cap. The southern polar cap is smaller
due to the presence of the LMC, which lies outside the EWS RoI. Each patch of
the polar caps is a target field scheduled during stage-1 with the strategy
used for observing the EDFs and calibration fields.

At high latitudes, overlap between neighbouring fields cannot be avoided,
because there is longer room for a rotation of the FoV. The average FoV
overlap on the polar caps is $\sim$18%, while the average overlap on the
regular wide is under 3%. This leads to a small loss in survey efficiency.

(a)  (b)

Figure 37: Coverage of the northern and southern ecliptic polar-caps (left and
right panels, respectively), with the design of polar-caps drawn in red and
patches drawn in blue. Extra patches below $79\degr$ are also added to polar-
caps (two in the North and one in the South) to observe corners of the RoI
that would be difficult to observe with the regular wide strategy. The
presence of the Galactic plane affects both caps. In addition, the southern
boundary avoids the LMC. Blinding stars are also shows; in particular the
small gap in the southern cap is due to the presence of the extremely bright
star R-Doradus.

####  7.4.3 Survey-windows and patch-sources

The next step in the computation of the EWS is to match the global
tessellation with the survey-windows defined by the stage-1 schedule. A
‘survey-window’ is the span of time between consecutive calibration blocks. It
defines uninterrupted time intervals available for observing the RoI.
Currently, there are approximately $100$ survey-windows. As explained later,
these are processed in chronological order, one at a time. But we first show
how a single survey-window intersects with the RoI and how it is populated
with EWS observations.

A survey-window begins at the end of the last pointing of a calibration block,
and lasts until the first pointing of the following calibration block. Let $a$
and $b$ be the two pointings delimiting a survey-window, and $\Delta t$ a
nominal observation time, defined as ROS time, including the typical field-to-
field slew time. The approximate number of FoVs possible to observe within a
survey-window (i.e., its capacity) is then given by $n=(b-a)/\Delta t$.

During the slot of time defined by a survey-window, spanning from $a$ to $b$,
two transit meridians scan two opposite sectors of the sphere, representing
the areas within reach of the survey-window. In turn, both of these sectors
intersect with two or more of the four quadrants of the RoI, identifying the
eligible FoVs within reach of a given survey-window. The intersection of each
of these sectors with a single quadrant of the RoI defines a ‘patch-source’,
i.e., a contiguous subset of the tessellated RoI within reach of a survey-
window (as exemplified in Fig. 38). The number of patch-sources per survey-
window varies from two, intersecting only the mainlands, to four, intersecting
all quadrants. In rare configurations, a survey-window intersects the same
mainland twice, defining two separate patch-sources.

Figure 38: Example of four patch-sources (in red) generated by a survey-
window. Notice how the patch-source in the northern-island intersects that
quadrant only partially.

A patch-source provides a simplification of the wide survey time window
(‘survey-window’). Given the need to observe fields in sequence, ones near
each other (within the slew constraints), the observations within a wide
survey-window must necessarily form a contiguous compact set. As described
below, these compact sets of tiles defines a patch.

A ‘patch’ is the building block of the survey. It constitutes a unit of
observation of the wide, using some of the time budget for the wide (which may
all or part of a survey-window) and covering some of the RoI. By construction,
different patches do not intersect, neither in time or space. Moreover, the
RoI is covered as much as possible in an orderly fashion, with patches
stacking one on top of each other. In that respect, the EWS may be seen as a
long sequence of patches (scheduled around the calibration blocks).

Patch-sources of the same survey-window compete for the same time, with each
one (potentially) generating a patch. Hence, in general, several candidate
patches are available, with the actual choice of one among the possible ones
determined by several aspects. In the simplest scenario, one or more patches
fill all the time available and, in that case, it suffices to choose one of
them (and discard the rest). In more complex scenarios, a survey-window only
intersects a quadrant partially, producing a patch-source that does not
consume all the available time. In these cases, the solution is to fill the
survey-window with patches from several patch-sources. One patch is selected,
reducing the extent of the survey-window. Then, the process is restarted,
recomputing the patch-sources and generating a new set of patches. The process
is iterated until all available time is exhausted.

Inevitably, due to the cyclic nature of the scanning of the sky, the selection
of a patch reduces the RoI available for the generation of later patches. This
does not cause any problem and it is easily coped with by flagging observed
tiles as they are scheduled, thus avoiding selecting them again in subsequent
compilations of patch-sources. However, cyclic placement of patches around the
sphere creates a “dented” boundary of observed tiles. Over time, this leads
some of the observed regions in the RoI to acquire a boundary shaped like a
polyline in latitude and longitude (see Fig. 39).

Figure 39: Example of a patch-source reduced by previously scheduled patches
(plot of region S-.5pt
IV
only, no star skipping for clarity). Previously scheduled patches are shown in
blue, with the red region depicting the patch-source of some window. Notice
how the bottom side of this quadrant acquired a lat-long shape after the
stacking of some patches. Figure 40: Example of sibling patch-segments from a
fragmented patch-source (of Fig. 39). Patch-segments are displayed in shades
of light blue, with previously observed patches in dark blue. Tiles containing
a blinding star are skipped for observation, here displayed in orange. Tiles
in the vicinity of an extremely bright star are also skipped (even if not
containing it), as is the case of the star marked with a circle at
$\sim(323,-36)$.

####  7.4.4 Patch-source partitioning

In the previous section, the process of extracting a patch from a patch-source
was simplified, for the sake of clarity. Actually, the process is slightly
more complex, requiring the definition of the concept of a patch-segment.
First, tiles are selected on the condition that tiles observed in the same
time slot should also be close to each other. However, because the contour
line of the unscheduled part of the RoI may become irregular, the condition on
proximity might not be feasible to meet. The solution is to partition the
tiles of a patch-source into patch-segments, where a patch-segment is simply a
group of tiles amenable to be visited with a sequence of small-slews.

When creating a patch-segment, tiles must be selected evenly across longitude,
matching the rate of fields observable per unit of time (approximately, 20
fields per day $\equiv$ per degree of longitude). The first time a patch is
extracted from a patch-source, the base of the corresponding RoI is bounded by
a straight line (the side of the mainlands close to the poles), resulting in a
single, possibly large, patch-segment of tiles. But, after a few iterations,
the base becomes a polyline (see Fig. 39). Then, the process of selecting
tiles evenly across longitude may, potentially, produce a fragmented
selection, made of two or more separate patch-segments (Fig. 40 shows an
example of this). This fragmentation is inevitable, but it is not an obstacle.
In general, a patch-source produces several patch-segments in order to fill
its span of time. In the case of multiple patch-segments, the solution is to
schedule them separately, followed by a merge into a continuous single
schedule.

###  7.5 Scheduling patches

This section describes the core scheduling functionality for the computation
of the EWS. We describe two algorithms that we used. Each algorithm takes a
patch-segment as input and produces an ordering of the tiles, i.e., a
sequence, geometrically appropriate for observation (within the constraints of
the spacecraft). We then describe how several patch-segments, properly
ordered, are combined in a single schedule.

We describe first the “look-ahead” algorithm and later on the “diffusion”
algorithm that replaced the former.

####  7.5.1 The look-ahead algorithm

The look-ahead algorithm was the first successful attempt at scheduling a
patch-segment of tiles, within mission constraints, allowing the generation of
a compliant survey. While superseded by the diffusion algorithm (Sect. 7.5.2),
it gave much insight into the key factors at play, and paved the way for the
design of the latest algorithm.

The look-ahead algorithm was designed around the idea that a scheduling
sequence must traverse a patch-segment of tiles following a “natural” zig-zag
scheme, monotonously across ecliptic longitudes. At its core, the algorithm
traverses the patch-segment from right to left (longitude), going up and down
(latitude), with minimal reversing of direction. The natural ordering is
computed iteratively, moving from a given position and given direction of
traversal (going-up or going-down) to the next. If the current direction is
going-up, the next tile in the sequence is the first unvisited tile of the row
above found by scanning the patch-segment from right to left. Should that tile
not exist, then the tile on the same row immediately to the left is selected.
In the latter case, the direction is reversed from going-up to going-down,
setting a flag that the top border of the patch-segment was reached. If the
current direction is going-down, the choices are reversed; the next tile in
the sequence is the first unvisited tile of the row below found by scanning
the patch-segment from right to left or, if this does not exists, it is the
tile on the same row immediately to the left. Likewise, in the later case, the
direction is reversed from going-down to going-up, setting a flag that the
bottom border was reached.

With this algorithm one can then define the full process. At the beginning, a
starting tile is chosen (i.e. a tile on the rightmost side of the patch-
segment), as well as an initial direction, and a starting timestamp.
Typically, there are many tiles close to or at the same longitude as the
rightmost tile; all are suitable as starting tiles. Afterwards, the algorithm
computes a path to traverse the patch-segment, propagating along timestamps
for the observations of each tile.

In general, this algorithm does not cover the patch-segment completely. It may
get stuck in one of three ways:

  * •

a dead-end is reached, with no unvisited tiles to jump to;

  * •

an unvisited adjacent tile is identified, but it cannot be observed within the
slew constraints;

  * •

a non-valid observation is encountered because an observation went outside the
valid range of AA or SAA when assigning timestamps.

This can be understood as follows. Due to convergence towards the ecliptic
poles, patch-segments at high latitude are very asymmetric, having most of
their tiles either at the top or the bottom. In this case, a sequence of
observations should spend more time at latitudes with many tiles, making only
occasional excursions to less populated latitudes. However, the algorithm is
designed for full vertical excursions (whenever possible). Also, some patch-
segments have unique geometric features; or corners odd enough to trap the
single path of the traversing strategy.

The solution is to extend the algorithm in two ways: making it explore more
paths and allowing changes of direction in mid-excursion. The first should
promote sequences that adapt to odd shapes, while the second should be able to
cope with the asymmetry of patch-segments at high latitude. To these ends a
‘probing step’ is implemented. Instead of blindly progressing up and down,
inverting direction only at the boundaries, the algorithm first runs two
probes to determine if, in the next step, the sequence should advance by
going-up or going-down (i.e. whether it should continue or invert direction).

The probing step is simple. Before advancing, the algorithm first computes the
natural sequence for the remaining unvisited tiles for two scenarios: first,
for a natural sequence that continues going-up, and then for a natural
sequence that continues going-down. The direction to take for the next step is
then given by the length of the two natural sequences just computed. If the
lengths are different, it takes the direction of the longest sequence
(possibly inverting direction), otherwise, it just keeps going in the same
direction as before. The resulting sequence is obtained by applying this
probing process iteratively, until all tiles are visited (returning success)
or until the algorithm gets stuck (returning failure). As an optimisation, if
one of the probing sequences traverses all of the remaining tiles, that
sequence is taken and completes the schedule. If the above run fails, the
process is repeated by trying other starting tiles (from the subset of tiles
with longitude close to the rightmost tile). Varying the starting point
explores different configurations, greatly improving the chances of success.
Figure 41 illustrates this process.

Figure 41: Example output of the look-ahead algorithm. The sequence starts at
mid-height, going downwards first. It traverses all fields running up and
down, from right to left. Notice how, sometimes, it shows an irregular
behaviour, zig-zagging to the sides instead of smoothly following the expected
up and down directions.

This algorithm was used to generate the EWS from the period of time between
the mission PDR to the CDR. However, following the CDR it was realised that
the resulting surveys exhibited uncontrolled excursions over the full AA range
(visible in Fig. 41 and the left panel of Fig 43), degrading the thermal
stability of the telescope and thus affecting the PSF estimation. This was
particularly acute for patch-segments at high latitude. Part of this failure
is due to the restrictiveness of small slews, limited (at the time this was
developed) to a maximum of $1\aas@@fstack{\circ}2$. The slew range was then
relaxed, allowing now for a small number of slews up to $None$. However, the
look-ahead algorithm is intrinsically limited by its simplicity and lack of
flexibility (low number of parameters), making it difficult to accommodate new
constraints such as skipping tiles with bright stars. These disadvantages
prompted the development of a more capable algorithm, as explained in the
following section.

(a)  (b)  (c)  (d)  (e)

Figure 42: Example run of the diffusion algorithm. The green and blue squares
represent the fields in a patch. The blue ones mark the base row. The red
line-segments depict the progress of the algorithm as it traverses all the
fields. Panel (a) shows diffusion after fours iterations, extending the
threads upwards. Remaining panels illustrate several points of the diffusion
process: at panel (b) two threads skip across the hole left by a bright star,
while two other threads merge, at panel (c) two threads merge while the
remaining reach the top, at panel (d) the threads are extended two rows
downwards, panel (e) shows the final sequence, after the linking of adjacent
threads. The plots are in elliptic coordinates.

####  7.5.2 The diffusion algorithm

Following analysis of the full satellite structural thermal optical
performance (STOP) made by Euclid’s industrial prime contractor, we analysed
the impact of the spacecraft attitude on the PSF stability. It was found that
that, in addition to the applicable limitations on SAA and AA, a further
minimisation of the field to field variation of these angles was desirable.

The new diffusion algorithm, which is presented here, achieves this goal,
whilst also facilitating the avoidance of bright stars. To minimise the angle
variations, the patch-segment must be traversed in columns of alternating
directions. This strategy avoids lateral slews on the same row that would
cause spikes in $\delta_{\text{AA}}$ and $\delta_{\text{SAA}}$, detrimental to
PSF stability. Moreover, since the slew reach is longer in the vertical
direction (see Sect. 2.5), moving mostly in the vertical direction increases
the chances of successfully skipping across rows. This feature allows the
algorithm to skip adjacent tiles (if already observed), and jump over holes
created by bright stars.

To illustrate the diffusion algorithm, we consider an input patch-segment in
the northern hemisphere close to the polar cap. The proximity to the pole
highlights the patch-segment convergence, which is relevant in this context.
The same strategy is easily adapted to other latitudes (with less convergence)
and the southern hemisphere (by swapping up and down). The algorithm is
divided in two steps. The first step computes parts of the final sequence,
called ‘threads’, which connects tiles along columns. The second step obtains
the final sequence by tying adjacent threads together.

The computation of the threads begins by selecting the widest row of the
patch-segment (not necessarily the one at the bottom), assigning a thread to
each of its tiles. Then, the threads are computed in parallel, sequentially
joining tiles of the current row with tiles of the row immediately above,
until the top row is reached. The algorithm then returns to the starting point
(the widest row), extending the threads downwards and thus completing the
threads.

The double step approach is needed because, in general, the widest row is not
at the bottom or at the top. Typically, a patch is bounded by a lat-long
rectangle, like the examples of Fig. 41 and Fig. 42. However, patches
alongside the RoI boundary may get asymmetrical shapes, acquiring some of the
shape of the adjoining boundary. In those cases, the widest row is usually
some row in the middle.

The computation of threads is based on two parameters of the rows: row length
and row capacity. The length of a row $i$, $\text{len}(i)$, is the number of
tiles of that row. The capacity of a row $i$, $\text{cap}(i)$, is the maximum
length of a row or of any row above it, defining the minimum number of threads
that must cross a given row. If $\text{cap}(i)=\text{len}(i)$, then all
threads visit all tiles in row $i$. If $\text{cap}(i)>\text{len}(i)$, then
$\text{cap}(i)-\text{len}(i)$ threads do not touch row $i$, and must skip it,
but participate in some row above it. Due to the convergence towards the
poles, the patch-segments in the northern hemisphere funnel on the upper part.
Similarly, the rows capacity reduces with increase in latitude.

Let $k$ be the row (or one of the rows) of largest length, being also the
starting row for computing the threads. Each thread is initialised with
exactly one tile, of row $k$. Let $n_{\text{thread}}$ be the initial number of
threads. The process then begins by extending the threads, now at row $k$, to
the tiles of row $k+1$, guided by the following criteria that depend on
$\text{len}(k+1)$ and $\text{cap}(k+1)$:

  1. 1.

If $\text{len}(k+1)=n_{\text{thread}}$, then we have the simplest case with a
one to one correspondence. It suffices to extend each thread to row $k+1$ by
making an ordered assignment, from right to left (ensuring the threads do not
cross);

  2. 2.

If
$\text{cap}(k+1)<n_{\text{thread}}\;\land\;\text{cap}(k+1)>n_{\text{thread}}-2$,
then it is not possible to extend all threads; the number of tiles of the row
above is smaller than the number of threads. In this case, we extend the
threads closely aligned to some tile of the row above it. This is performed by
trying all combinations of ordered (non-crossing) assignments between a subset
of $\text{len}(k+1)$ threads and all of the $\text{len}(k+1)$ tiles of row
$k+1$. For each tried combination that was accepted, we compute the cumulative
variation in longitude (“verticality”) of the thread. The combination with the
lowest variation in longitude is selected, extending $\text{len}(k+1)$ threads
(and leaving the rest unchanged),

  3. 3.

If $\text{cap}(k+2)<n_{\text{thread}}-2$, it is also not possible to extend
all the threads but, unlike the previous case, it is possible to eliminate a
pair of threads (leaving $n_{\text{thread}}-2$ threads). Actually, it is
necessary to eliminate threads to avoid the risk of having threads stalled
(i.e., not reaching the top of the patch). A pair of threads may be eliminated
if they are adjacent in the list of threads. To eliminate them, it suffices to
merge the two threads together, short-circuiting their paths, by connecting
the two adjacent threads to the same tile (of row $k+1$, creating an inverted
“v”). This case is slightly more complex than the previous one. Now, it needs
to try all combinations of ordered (non-crossing) assignments between a subset
of $\text{len}(k+1)+1$ threads and all of the $\text{len}(k+1)$ tiles of row
$k+1$, considering that pairs of consecutive threads extend to the same tile
(merging those threads). Like before, all combinations are scored against
verticality, selecting the one with lowest cumulative variation in longitude.
It extends $\text{len}(k+1)-2$ threads, leaving $n_{\text{thread}}-2$ active
threads, and merges two threads (ending their progress).

Figure 43: Variation of AA when observing the same EWS patch with the two
different algorithms: look-ahead (left panel) and diffusion (right panel). The
latter has comparatively a more monotonic progression with much smaller field
to field AA variations and spikes. The bottom panels are the time derivative
of the top panels.

This process is illustrated in Fig. 42. Figure 42a shows the first step,
extending threads from the initial row (in blue) to the row immediately above.
Given the difference in row length, one thread is held up. After a few
iterations, Fig. 42b shows a case where a thread on the left-side jumps over
four threads (approximately at longitude $None$, latitude 60-$None$), and a
thread in the middle (approximately at longitude $None$, latitude 66-$None$)
is kept straightly vertical by skipping a row. This is an example of the
capability of the diffusion process to adjust to a varying row length, by
“squeezing” more threads than the length of each individual row, promoting
straighter threads. After a few more iterations, Fig. 42c shows two mergers of
two neighbouring threads (rows at latitudes $None$ and $None$). This reduces
the number of threads from seven to five (which is enough to cover the tiles
of the rows above). This mechanism ensures a monotonous decrease of the number
of threads, keeping it close to the row’s capacity. Figure 42d shows the
threads fully extended upwards (with two pair of threads ended being merged in
the process). In this case, there is no need to extend the threads downwards
also, since the starting row is also the bottom row. Lastly, Fig. 42e shows
the final sequence, obtained after connecting adjacent threads (arrows show
the temporal sequence of the covering).

The net effect of the above strategy is to grow threads upwards, striving to
be as vertical as possible, and merging adjacent threads when necessary, to
cope with the reduction of rows capacity (with latitude). In a way, this
growth resembles a diffusion process, hence its name. It should be noticed
that the number of threads crossing each row may be larger than the actual
number of tiles at that row. This is desirable. It is the mechanism that
enables the threads to accommodate to the slight irregularities of the patch-
segment, while still be vertically aligned. In addition, this same mechanism
allows the threads to skip tiles containing blinding stars (to be implemented
in a future release).

The input patch-segment is bounded, by construction, to a lat-long rectangle.
Therefore, in general, all the threads traverse the full extent of the patch-
segment, from top to bottom (the exception being the cases truncated by the
RoI boundary). So, the thread endpoints (top and bottom) are expected to be
close to each other meridian-wise. The final step of the computation is to
pairwise connect the threads from right to left. This last step creates larger
moves in longitude, detrimental for PSF stability. However, these moves are
limited to the number of threads, which is much lower than the length of the
patch-segment).

The diffusion algorithm is by far the most expensive part of the EWS
computation, with the combinatorial exploration taking most of its cost.
However, it has so far proven to be stable and robust, solving a large number
of diverse patch-segment configurations while optimising thermal stability,
thus yielding a much more stable PSF. Figure 43 shows a comparison of the
resulting AA field to field variation in the same patch, when scheduling it
with the the look-ahead algorithm versus diffusion algorithm. The time
behaviour of AA in the latter case is much smoother and with fewer spikes than
the previous results. This improvement translates into a better thermal
stability and an overall decrease in time variations of the PSF, which
therefore can be better modelled.

####  7.5.3 Linking patch-segments into a patch

The diffusion algorithm proposes a sequence of observations for a pattern of
tiles, which is a segment of a patch. This is the building block for computing
a patch. First, the diffusion algorithm is applied to all patch-segments of a
patch-source. Then, if all are successful, the resulting patch-segments are
linked together in a single observation sequence, by assigning timestamps for
observation, generating a single sequence that covers the time slot of a
survey-window; i.e., a patch (see Fig. 44).

Timestamps are assigned adding observation and slew times along the order of
the sequence, checking compliance with constraints. This is performed first
forward in time, starting at the pointing (quaternion and timestamp) that
defines the start of the survey-window. Next, the timestamp of the first
observation is computed by adding the slew time from the start pointing. Then,
the timestamp of the second time observation is computed by adding the ROS
observation time (a fixed value) plus the slew time from the previous
observation (a variable value). The process is repeated throughout the
sequence, assigning timestamps sequentially. In this process, slew-times are
computed according to the slew-time estimator. By construction, slews within a
patch-segment are expected to be small slews. In contrast, jumps between
patch-segments are considered to be large slews, adding to the large-slew
budget. However, this number of large-slews is relatively small. Typically,
survey-windows are filled with patches generated from one or two patch-
sources, with each patch being split in a low number of patch-segments, at
most. Hence, the initial ${\sim}100$ survey-windows produce no more than a few
hundreds of patch-segments in the end.

As explained above, the algorithm first attempts to assign timestamps from the
beginning of the survey-window forward. If this succeeds, it creates a patch
flushed backward in time, leaving some idle time at the end of the window.
This is because in general, an integer number of observations with varying
slew-times does not fit perfectly into the slot of time previously defined by
a survey-window. If this succeeds, this is the preferable solution. If it
fails, then a reverse assignment is attempted, assigning timestamps from the
end of the survey-window backward (and reversing the computations of slew-
time). If this succeeds, it may leave some idle-time at the beginning of the
window.

Usually, there is a sufficient number of tiles distributed along the range of
longitudes (covered by a patch-source) to generate a patch covering all the
time slot of a survey-window. However, in some cases, such as when a patch-
source intersects only a corner of a quadrant, it is not possible to have a
path extending the full width of the respective survey-window. Trying to flush
a patch to both end-sides of a survey-window enables patching those odd cases,
promoting also the generation of patches that adapt to the boundary of the
RoI.

Figure 44: Example of a three sibling patch-segments (cf. Fig. 40). Patch-
segments are displayed in alternating colours. Previously observed tiles in
dark blue. Red arrows represent large-slews (moving from patch-segment to
patch-segment).

In the process of assigning timestamps, an observation of a field with a given
timestamp may fail to comply with the constraints of SAA and/or AA. If, at any
point, a failure of compliance is encountered, the computation is terminated
and all segments are discarded. The rationale for this strategy is simple: if
a sequence, computed as parallel to meridians as possible, fails the timestamp
assignment, then it might not be viable in the first place. Most likely, in
those cases, the patch has some geometry feature that stretches the scheduling
flexibility too far. The result of this stage is a patch that is flushed,
time-wise, to the beginning or to the end of the interval of time allotted to
it. The path is thus a composite structure, made by a string of segments.

Figure 45: An example the ECTile output, showing the build-up of the coverage
of the EWS in RSD_2021A (cf. Sect. 8) over time in ecliptic coordinates. In
the main panel the EWS patches are coloured according to different years, in
correspondence to the time bars below. The part of the RoI which is not
observed is in light green (we remind that the RoI covers a larger area than
the one needed to be observed). Under the main panel, the progression over the
six years is unfolded, with two rows for each year indicating the trailing and
leading directions of the telescope. Different colours represent the
progression in time, from deep blue to light blue, whereas the thin line
segments represent time intervals reserved for calibrations and Deep Fields.
The light grey boxes for the leading or trailing directions mark when the
telescope is actually observing in the opposite direction (for each grey box
there is a corresponding coloured box). Towards the end of the mission some of
the survey-windows are completely or partially light-red, showing that
particular time slot has run out of sky areas to observe.

Figure 46: Reference Survey Definition 2021A (14 514 deg2) chronology shown in
celestial coordinates. RoI boundaries are shown as solid red lines. Blinding
stars cause 809 avoidance areas within the reference survey, with an average
of $0.785\,\text{deg}^{2}$ per avoidance area, totalling $635\,\deg^{2}$
Dashed lines (1300 deg2 in white per Galactic cap) delimit the highest SNR
areas. The ecliptic referential is over-plotted in red. The three EDFs (bright
green) and the six EAFs (red diamonds, not in size) are shown.

###  7.6 Placing patches

The EWS is computed sequentially, covering the RoI by placing one patch at a
time. In some way, it is like solving a jigsaw puzzle on a surface of a
sphere, but where the shapes of the pieces are not fixed from the start,
instead being computed as the scheduling progresses. Two objectives guide the
build-up of the EWS, which can be summarised as: “observe as much as possible,
as early as possible”. More specifically,

  1. 1.

We want a compact footprint (per quadrant), so that the footprint of
observations forms a single, continuous region. This means patches should
match perfectly next to each other, with no holes in between;999The only
planned “holes” are the tiles skipped because of the presence of blinding
stars.

  2. 2.

We want to observe fields as early as possible, which means that all slots in
survey-windows are assigned to observations, provided there is an unobserved
part of the RoI within reach. After a few iterations, the RoI within reach of
some survey-windows is typically scheduled (given the cyclic nature of the
process).

Eventually, either no area is left to schedule (within a range of latitudes of
the RoI) or there is no time left (within the six years of the mission). When
no suitable area can be scheduled for a given period, the schedule just leaves
it unassigned, giving rise to unallocated time. This highlights that presently
it is not possible to assign it to particular observations, but that it can be
done at a later time.

The EWS is computed by filling the RoI one survey-window at the time, in
chronological order. At each step, a survey-window is processed by assigning
observations to its time slot. As explained below, this may require a few
iterations. Only when all possible assignments are handled, the computation
moves on to the next survey-window.

As discussed in Sect. 7.4.3, each survey-window generates one or more patch-
sources, which in turn may generate one patch (made of a single patch-segment
or of a string of several patch-segments), flushed to the beginning or to the
end of its survey-window. All the patches are continuously linked to a
particular stage-1 observation (e.g. a calibration, deep-field, or polar cap
patch). All these patches compete for the same slot of time, and we need to
decide which patch to add to the survey. In general, the choice is to build
the EWS layer by layer of ecliptic latitude, striving to go from high to low
SNR regions. When this criteria is not decisive, the choice is to select the
patch that better matches (horizontally, along ecliptic latitudes) some
previously selected adjacent patch.

The last step in the processing of a survey-window is to place PSF
calibrations and SOPs. At most, one PSF calibrations is inserted per window.
This is achieved by analysing the patches just selected, identifying the
timestamps where a PSF calibration target is within reach. A jump to a PSF
calibration takes place at the end of some wide observation. Typically, a
single patch generates several such candidates. In parallel, candidate
timestamps for the required SOPs are identified. Again, these are timestamps
of the end of some wide observation. There are one to two SOP candidates per
window, at most. Then, the two types of candidates are considered, inserting
them in chronological order; one PSF calibration, if available, and one or two
required SOPs. Among all PSF candidates, preference is given to the ones
occurring between patch-segments or between a preceding SOP and a following
patch-segment, in order to save large-slews. The insertion is performed by
cutting a wide patch at the required timestamp, adding the PSF calibration or
the SOP, and pushing the remaining of the patch forward in time. This process
may require also an adjustment of the following calibration block, pushing it
forward in time (if now overlapped by the preceding patch). The process of SOP
insertion is also applied to the following calibration block before processing
the next survey-window.

The islands are located at lower latitudes, and comprise less area.
Consequently, most of the survey build up is shared between the two mainlands
(S-.5pt
I
and S-.5pt
IV
, see Fig. 17) with the islands (S-.5pt
II
and S-.5pt
III
) becoming relevant in the last year of the survey. This can be seen in Fig.
45, which shows an example of the build-up of EWS patches over the duration of
the mission, coloured from deep blue to light blue as time progresses. This
strategy does not only forces the two mainlands to grow at an equal rate but,
more importantly, it is crucial to guarantee a compact survey footprint.

The height of a patch is dictated by the speed of the orbit, which varies
slightly with latitude. It roughly corresponds to the time the orbit takes to
scan the width of a FoV divided by the nominal observation time. To some
degree, the height (and width) may vary slightly from its natural size
(approximately, $\pm 2$ rows). Hence, in practice the height of a patch is
more or less fixed, making the layer approach optimal. It guarantees a maximum
of free RoI above previously computed patches, giving ample space (i.e.,
height) for the generation of each new patch. The sole exception is when the
top layer of the survey reaches the boundary of the RoI. If the available
height is less then the minimum patch height, the patch generation fails. Due
to the layering approach, this obstacle arises only for a small part of the
RoI, namely the top layer of each quadrant. For the rest of the RoI, the
stacking of patches ensures a compact filling.

##  8 The Euclid reference survey definition

In this section we present the latest version of the “Euclid reference survey
definition” (RSD), RSD_2021A. This is the result of the stage-1 and stage-2
scheduling procedures (cf Sects. 6.3 and 7), and the corresponding ECTile
outputs are the ones shown in Figs. 29 and 45.

We recall that the RSD observations start three months after launch. They are
preceded by a one-month commissioning phase, followed by a two-month
performance verification (PV) phase. During the PV phase the first survey-like
data will be obtained, which are used to verify the data processing, and to
validate and eventually tune or adjust the nominal sequence of operations
(possible minor changes to the RSD can be implemented in less than one week
time, producing a new version of the RSD; more complex changes of course
require more time). Moreover, during PV some survey specific observations will
be carried out. The latter will, for instance, provide direct measurements of
the zodiacal light and stray light to verify and refine our models.

###  8.1 Survey performance

Figure 46 shows the footprint of RSD_2021A. The different colours indicate
different observing epochs of the EWS. The three EDFs and the six EAFs (cf.
Sect. 6.2) are tied into the EWS. In the two mainlands of the EWS, the
observations start from the ecliptic poles and progress towards the equator.
The best sky areas around both Galactic caps are covered within the first
three years of the mission. The observations of the two EWS islands only take
place in the final two years of the mission. Note that some of the worst parts
of the RoI (totalling an area of ${\sim}2696\,\text{deg}^{2}$) are left
unobserved. These correspond to the uncoloured areas in the islands and at low
latitudes in the mainlands. The areas of the sky with longitudes between
$150\degr$ and $225\degr$, and between $330\degr$ and $45\degr$, are
observable at the same time since they are separated by $180\degr$. They
contain much area within the RoI, and moreover the EDF-F and EDF-S are also
located there. This means that there is not enough time for the EWS to observe
all that area in the six years of the mission and hence their worst-quality
regions are not observed.

The RSD contains 44 065 fields (28 080 to build the EWS and 15 985 for EDFs,
EAFs and calibration targets observations). The EWS fields are contained in
256 patches (seen in Figs. 45 and 46). The vast majority of the field slews,
used to point the telescope, are below $1\aas@@fstack{\circ}2$, as shown in
the right panel of Fig. 47. This is the most efficient slew regime in terms of
propellant usage. As shown in Fig. 47, all telescope rotations are done within
the allowed SAA and AA limits. Most of the observations are done close to
transit, with $90\%$ of the SAA values used between $88\degr$ and $94\degr$.
The statistics of AA usage shows that $71\%$ of the telescope rotations are
done with $|AA|<1\degr$. Even though SAA and AA values spread over the full
range allowed, the field-to-field variations (between consecutive
observations) of SAA and AA are very small throughout the survey: smaller than
$1\degr$ in $97.4\%$ (SAA) and $98.6\%$ (AA) of the field-to-field transitions
over the full mission. This feature is extremely important for the thermal
stability, which ensures a stable PSF for WL shape measurements. It was
possible to achieve this performance thanks to the implementation of the
diffusion algorithm described in Sect. 7.5.2.

Figure 47: The distributions of angles and slews. _Left panel:_ AA angle.
_Central panel:_ SAA angle. _Right panel:_ field slews up to $None$. All
quantities do comply with the allowed ranges.

###  8.2 Unallocated time

The existence of a deficit of area on some longitudes (see Sect. 5.2.1),
compared to the available observing time, is evident from Fig. 48. The blue
curve is the area available in the RoI at a given ecliptic longitude (in bins
of 1°). The RoI areas in longitudes separated by $180\degr$ are added, since
that pair of longitudes can be observed at the same time, from the trailing or
the leading direction. Due to this six-month periodicity, the $x$–axis range
only extends to 180°. The red curve denotes the cumulated number of days
during which a given longitude is visible for EWS observations, assuming
transit observations, and converted to equivalent area (1 day corresponding to
$10\,\deg^{2}$). The available time is not uniform, it is determined after the
stage-1 schedule is defined (see Sect. 6.3), which creates a strong variation
along the year (i.e. wiggles in the red curve). For example, the absolute
minimum corresponds to the highly booked longitudes of the EDFF and EDFS,
where less time is left for EWS observations.

In longitudes where the red curve is above the blue curve, there is a deficit
of area for the time available for EWS, leading to unallocated time.
Conversely, in longitudes where the blue curve is above the red curve, there
is an excess of area for the time available for EWS, leading to unobserved
areas in the RoI. In Fig. 46 this corresponds to the areas with no patches,
which clearly are on the areas of the RoI of worst quality.

Figure 48: Area of the RoI as function of (folded) longitude (blue).
Equivalent area visible as function of (folded) longitude (red). Area is the
available RoI area at transit per degree of longitude. The red curve assumes
that the EWS observations are done at transit. Unallocated time arises when
the red curve is above the blue curve. See text for details.

Note that the presence of unallocated time in the EWS schedule does not mean
that there will be any idle time, because some areas of the EWS may be re-
observed or new areas that do not qualify for the EWS, but have scientific
value nonetheless, may be observed instead. In doing so, we can either
consider fields that are observable within the thermal and pointing
constraints enforced for the EWS, or we can operate outside these constraints,
thus with the risk of perturbing the continuation of the EWS afterwards.
Therefore one needs to have the real in-flight characteristics to get a solid
picture of the possibilities and constraints.

Some examples that maintain the strict survey limitations are:

  * •

possible multiple exposures ($\>4$) on areas in the ecliptic plane which do
not qualify for SNR with ROS single visit;

  * •

possible decontamination procedures;

  * •

repeat some suitable but lower-quality EWS regions to boost their SNR, or fill
possible gaps due to unexpected events which might interrupt the basic
scheduling;

  * •

increase the depth of the self-cal field, which could become a reference field
for a dedicated supernova program in a possible extension of the mission;

  * •

build a medium deep field (EMDF) covering $200-300$ deg2 that is one magnitude
deeper than the EWS (this would require five additional visits that would
preferentially be done once every year);

  * •

observe suitable astronomical objects that are located outside the RoI that
would benefit from localised repeats. Examples are observations on the
ecliptic or Galactic plane, Galactic bulge (e.g. microlensing), specific low
surface brightness objects, nearby galaxies, or clusters of galaxies;

  * •

use of the blue grism on targets during new or repeat visits on specific
targets.

How to best use the time that cannot be used for single pass EWS will be
decided at a later stage upon a consolidated scenario by the Euclid Science
Team and ESA.

Figure 49: Growth of the area surveyed by the RSD_2021A EWS with time (upper
curve). Growth of unallocated time with time (lower curve), here in units of
EWS equivalent area (one month corresponds to about 300 deg2). Table 9: EWS
area observed in RSD_2021A per year and cumulatively at the end of each year.
Last column shows the cumulative unallocated time (see text).

RSD_2021A | Area (per year) | Cumulative area | Unallocated time
---|---|---|---
year | [deg2] | [$\%$] | [deg2] | [$\%$] | [days]
1 | 2 656 | 18.3 |  02 656 |  018.3 |  000
2 | 2 762 | 19.0 |  05 418 |  037.3 |  010
3 | 2 708 | 18.7 |  08 126 |  056.0 |  033
4 | 2 400 | 16.5 | 10 526 |  072.5 |  089
5 | 2 147 | 14.8 | 12 673 |  087.3 | 174
6 | 1 840 | 12.7 | 14 514 | 100.0 | 292

###  8.3 Euclid footprint area

The area of the RSD_2021A Euclid footprint is 14 514 $\deg^{2}$. Table 9 lists
the observed EWS area at the end of each year of the mission, while the growth
with time of the area covered by the EWS is shown in Fig. 49.

During the first year many calibration observations are scheduled to support
the first data releases. As a result, the EWS initially progresses slower than
in the second and third year, but it still reaches an area in excess of 2500
deg2. At the end of the third year, more periods occur when standard EWS
observations cannot be made due to the increasing paucity of available
unobserved areas within the RoI; the slope of the growth in time flattens
there, causing the staircase-like pattern seen in Fig. 49. These periods of
‘unallocated time’ (see Sect. 8.2) have a 6-month periodicity due to the
intersection of the Galactic plane with the ecliptic plane, have an increasing
duration and slow down the progression of the EWS.

The time available to implement the EWS, once the other mandatory observations
are carried out and the geometry of the RoI is taken into account, is an
important input for the construction of the EWS and the final covered areas.
In RSD_2021A this time is $2190-427-39-292=1432$ days, where 427 days are used
for calibrations, EDAs and EDFs observations, 39 days are reserved for SOP and
there are 292 unallocated days (Fig. 50 depicts the time allocation
breakdown). Given the length of the ROS, close to 4400 s per field (Sect.
4.1), and the FoV of 0.53 deg2 (Sect. 3.3), the effective EWS area increases
by 10.1 deg2 per day, (considering the effective average field overlap of 3%).
This means that, with the current calibration, EDFs, EAFs, and SOP required
times, the EWS can reach an area of 15 000 deg2 only if the unallocated time
is shorter than 240 days.

Figure 50: The breakdown of the total time of the Euclid mission (2190 days)
in EWS and non-EWS usage, for RSD_2021A, corresponding to the schedule
depicted in Fig. 29.

Figure 51: Evolution of the time available for EWS observations in the RSDs
produced from 2015 to 2021. The changes are be due to increase or decrease
(denoted by the symbols + or -, larger changes have double symbol) of
calibrations, RoI area (directly impacting the unallocated time) and ROS time.
See text for details.

The time available for the EWS has generally decreased over the years, as the
mission matured. Figure 51 depicts this evolution, indicating the driving
changing factor for each transition (increse/decrease of calibrations, RoI or
ROS). In earlier surveys, such as the RSD_2015A prepared for PDR, the time
needed for calibrations was small and the RoI was larger (driven by WL
counts). In early 2018, RSD_2018A used for SPV2 introduced a smaller RoI based
on the more restrictive conditions for GC, which led to a larger distance from
the Galactic plane (because of effects from star density on spectra), with a
smaller area to be observed and an increase of unallocated time (cf. Sect.
8.2). Shortly after, a major redefinition of the calibration plan, increased
its allocated time by 100 days. At the same time the redefinition of the slew
concept (see Sect. 7.4.2) made it no longer possible to schedule the EWS at
high latitudes with the global tessellation, leading to a faster filling of
the EWS and a faster build-up of unallocated time. These two factors led to a
strong decrease of the EWS available time in RSD_2018B for CDR (Laureijs et
al., 2020). In 2019, RSD_2019A introduced a new algorithm to schedule the
high-latitude regions. Given the fact that the throughputs were measured to be
larger than specified, the RoI was reverted to the limits defined in Laureijs
et al. (2011). The larger area of this RoI led to an increase in the available
time for the EWS, and to a solution that covered 15 000 deg2. The year 2020
introduced the new, better high-quality RoI, described in Sect. 5, based on
the latest estimates of SNR. This again decreased the available EWS time, now
causing the RSD_2020A to barely reach 15 000 deg2. In RSD_2020B the diffusion
algorithm was introduced to increase thermal stability. This was achieved
without decreasing the schedule efficiency and the available time for the EWS.
Nevertheless the EWS time decreased due to a further increase of time needed
for the calibrations. Finally, in RSD_2021A we implemented the skipping of
bright stars, together with the latest revised times for both ROS (significant
increase of the dither steps duration) and SOP (halved).

RSD_2021A falls short of covering the desired 15 000 deg2 area for the EWS by
$\sim 3\%$. The missing 486 deg2 are the equivalent of one and a half months
of EWS observing time. In part this can be recovered by making local
tessellations, to allow shifting the position of the centre of the tiles
affected by stars, such as to avoid the blinding stars falling on the
detectors, instead of skipping the whole tile. Moreover, it is expected that
the EWS available time will further decrease sightly with the insertion of the
complete set of PSF calibrations. In fact, the expected cadence is not yet
fully respected in RSD_2021A, because the stringent stability constraints
could not always be fulfilled, especially in the final years of the survey.
Study is underway to tackle this issue and, once this problem will be solved,
the time needed amounts to $\approx$ 10 days (100 deg2) to be taken off from
the EWS present coverage. It must be noticed, however, that a 3% reduction in
survey area translates into a comparable reduction in the final dark energy
FoM (actually even less since the still uncovered areas of the RoI are of
lower quality than the average). An area a few percent lower than originally
expected is therefore of no consequence for achieving the original goal of the
mission, i.e. a FoM larger than 400 from the two main probes for a
$w_{0}-w_{a}$ CDM model with no other prior information (Laureijs et al.,
2011). Indeed, for the present coverage, one preliminarily expects the FoM to
scale down from 500 (Euclid Collaboration: Blanchard et al., 2020) to $\approx
480$.

Nevertheless, the final area of the Euclid footprint is not yet fixed and its
value can change in either direction. On the one hand, possible further
synergies between targets and optimisations will likely increase the area
covered by the EWS. On the other hand, modifications in the ROS, the
calibrations plan, and in the amount of overlap between tiles might further
reduce the final area.

##  9 Summary

We have presented the status of the reference survey for the Euclid mission at
the beginning of year 2021. The reference survey encompasses all the six years
of the mission baseline, fulfilling all the constraints while combining
observations of the EWS, all the slews, the calibrations, the deep and
auxiliary field plus other calibration fields. We also discussed and gave
models of the main backgrounds which affect the space observations. We also
presented the complementary ground-based observations needed for photometric
redshifts.

This is a non-trivial achievement, because of the complex nature of the
mission: the Euclid step-and-stare strategy is severely constrained by
pointing limitations with respect to the Sun, whilst it also has to carry out
a large number of dedicated observations for calibration purposes and sample
characterisation. This leads to a complex interplay of timing observations,
visibilities and manoeuvres. Although the two main instruments are operated in
concert, they do require three separate calibration strategies, thus adding to
the complexity. This makes Euclid not only different from a typical
observatory mission (e.g. _Herschel_), but also from all-sky missions that
have a single scanning strategy (e.g. _Planck_ and Gaia).

Despite these challenges, we have found a highly optimised solution, that
takes into account the main spacecraft and instrument characteristics and
limitations, the current models for the expected background, the various
calibrations, the dithering pattern and the methods to cover the wide area
expected to be observed. The latest version of the Euclid Reference Survey
Definition (RSD_2021A) fulfils practically all requirements, resulting in a
EWS that covers $\approx 14\,500\deg^{2}$ of the extragalactic sky. This is 3%
short of initial target mainly because of the paucity of good sky for Euclid
and of the severe constraints on the pointing (if feasible, 10 months of
presently unallocated time would allow in principle to add 3000 square degrees
to EWS). A companion paper will detail the rationale and results for the
Euclid Deep Fields that cover about 40 deg2, but at much greater depth.

The definition of the reference survey is an ongoing process, and some of the
results presented will continue to evolve because the Euclid Consortium has
developed both the expertise and the specific tools that allow one to probe
significant variations of any among the multiple boundary conditions which
originate from either the knowledge of the spacecraft, the instruments or the
astronomical environment. The current solution is, however, a close proxy for
the final survey. Crucially, it demonstrates the feasibility of the core
mission within the numerous constraints. Future plans include:

  * •

further optimisations to reach the nominal 15 000 deg2 (e.g. double exposures
on the ecliptic plane to compensate the local high background);

  * •

implement refined simulations;

  * •

implement refined background models;

  * •

incorporate possible future changes in parameters (e.g. changes in ROS or in
the calibration plan);

  * •

consider “what if” scenarios for non-recurrent operations (decontamination,
phase diversity calibrations) and possible failures (electronics or other
systems);

  * •

make revisions and updates based on real measurements and in-flight
performance.

These further improvements will present challenges in their own right, but
given the current level of maturity, there is little doubt that Euclid will
dramatically advance our understanding of the nature of dark matter and dark
energy, whilst impacting many aspects of astronomy thanks to the tremendous
legacy value of these unique data.

###### Acknowledgements.

The Euclid Consortium acknowledges the European Space Agency and a number of
agencies and institutes that have supported the development of Euclid, in
particular the Academy of Finland, the Agenzia Spaziale Italiana, the Belgian
Science Policy, the Canadian Euclid Consortium, the Centre National d’Etudes
Spatiales, the Deutsches Zentrum für Luft- und Raumfahrt, the Danish Space
Research Institute, the Fundação para a Ciência e a Tecnologia, the Ministerio
de Economia y Competitividad, the National Aeronautics and Space
Administration, the National Astronomical Observatory of Japan, the
Netherlandse Onderzoekschool Voor Astronomie, the Norwegian Space Agency, the
Romanian Space Agency, the State Secretariat for Education, Research and
Innovation (SERI) at the Swiss Space Office (SSO), and the United Kingdom
Space Agency. A complete and detailed list is available on the Euclid web site
(http://www.euclid-ec.org).

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##  Appendix A Acronyms

Table 10: Acronyms used in the paper. Acronym | Name
---|---
1D |  1-Dimensional
2D |  2-Dimensional
AA |  Alpha Angle
AOCS |  Attitude and Orbit Control System
APE |  Absolute Pointing Error
BGS |  “Blue” Grism
CCD(s) |  Charge-Coupled Device(s)
CDR |  Critical Design Review
CPC |  Completeness Purity Calibration field
CPC-N |  CPC-North
DE |  Dark Energy
DGL |  Diffuse Galactic Light (cirrus)
DM |  Dark Matter
DR# |  Data Release number #
EAFs |  Euclid Auxiliary Fields
EC |  Euclid Consortium
ECTile |  EC Tiling program
EDFs |  Euclid Deep Fields
EDFN |  EDF-North
EDFS |  EDF-South
EDFF |  EDF-Fornax
EDS |  Euclid Deep Survey
EMDF |  Euclid Medium Deep Field
EOL |  End Of Life
ERS |  Euclid Reference Survey
ESA |  European Space Agency
ESSPT |  Euclid Sky Survey Planning Tool
EWS |  Euclid Wide Survey
FGS |  Fine Guidance Sensor
FoM |  Figure of Merit
FoV |  Field of View
FPA |  Focal Plane Array
FWA |  Filter Wheel Assembly
FWHM |  Full Width Half Maximum
GC |  Galaxy Clustering
GWA |  Grism Wheel Assembly
LMC |  Large Magellanic Cloud
LoS |  Line of Sight
LSB |  Low Surface Brightness
$\Lambda$CDM |  $\Lambda$ Cold Dark Matter
M1 |  Main mirror
MC |  Monte Carlo
MCDR |  Mission Critical Design Review
NDI |  Normalised Diffusion Irradiance profile
NEO |  Near Earth Object
NEP |  Northern Ecliptic Pole
NIR |  Near-InfraRed
NISP |  Near infrared Imager and SPectrometer
NISP-S |  NISP Spectroscopy
NISP-P |  NISP Photometry
Q# |  Quick Data Release number #
PDR |  Preliminary Design Review
PLM |  PayLoad Module
PN |  Planetary Nebula
PSF |  Point Spread Function
PV |  Performance Verification
RGS# |  “Red” Grism #
RoI |  Region of Interest
ROS |  Reference Observation Sequence
RSD |  Reference Survey Definition
RSD_# |  Reference Survey Definition _#
SAA |  Solar Aspect Angle
SEDs |  Spectral Energy Distributions
SEL2 |  Sun-Earth Lagrangian point L2
SiC |  Silicon Carbide
SMC |  Small Magellanic Cloud
SNR |  Signal to Noise Ratio
SOP |  Spacecraft Orbit and Platform maintenance
SPSAA |  Solar Panel Solar Aspect Angle
SPV# |  Science Performance Verification #
STOP |  Satellite Structural Thermal Optical Performance
SVM |  SerVice Module
VIS |  VISible instrument
WL |  Weak gravitational Lensing
2MASS |  Two Micron All-Sky Survey
AEGIS |  All-wavelength Extended Groth strip International Survey
ASAP |  Analysis of Stellar Atmospheres and Pulsation
ATLAS |  Asteroid Terrestrial-impact Last Alert System
CANDELS |  Cosmic Assembly Near-IR Deep Legacy Survey
CDFS |  Chandra Deep Field South
CFHT-MegaCam |  Canada-France-Hawaii Telescope Mega Camera
CFHT |  Canada-France-Hawaii Telescope
CFIS |  Canada-France Imaging Survey
COBE |  COsmic Background Explorer
COSMOS |  Cosmic Evolution Survey
DES |  Dark Energy Survey
DETF |  Dark Energy Task Force
DUNE |  Dark UNiverse Explorer
GOODS-N |  Great Observatories Origins Deep Survey - North
HSC |  Subaru Hyper Suprime-Cam
HST |  Hubble Space Telescope
HUDF |  HST Ultra Deep Field
IPAC |  Infrared Processing and Analysis Center
IRAS |  InfraRed Astronomical Satellite
JEDIS-g |  Javalambre-Euclid Deep Imaging Survey in g band
JST |  Javalambre Survey Telescope
JWST |  James Webb Space Telescope
LSST |  Legacy Survey of Space and Time
Pan-STARRS |  Panoramic Survey Telescope & Rapid Response System
SDSS |  Sloan Digital Sky Survey
SMEI |  Solar Mass Ejection Imager
SNAP |  SuperNova / Acceleration Probe
SXDS |  Subaru/XMM-Newton Deep Survey
UDF |  Ultra Deep Field
UNIONS |  Ultraviolet Near-Infrared Optical Northern Survey
VVDS |  VIsible Multi Object Spectrograph (VIMOS) Very Large Telescope (VLT) Deep Survey
VVDS-Deep |  VIsible Multi Object Spectrograph (VIMOS) Very Large Telescope (VLT) Deep Survey – Deep
WISHES |  Wide Imaging with Subaru HSC of the Euclid Sky
AKM# |  Arendt, Kashlinski and Mosley #
Sc21 |  Scaramella et al., in prep. (Euclid Deep Survey)