11institutetext:  Physikalisch-Meteorologisches Observatorium Davos, World
Radiation Center, 7260, Davos Dorf, Switzerland 22institutetext: ETH-Zürich,
Wolfgang-Pauli-Str. 27, 8093 Zürich, Switzerland 33institutetext: Instituto de
Astronomia y Fisica del Espacio, IAFE, UBA-CONICET, Ciudad Universitaria, CC.
67, 1428, Buenos Aires, Argentina 44institutetext: Université Paris-Saclay,
CNRS, Institut d’Astrophysique Spatiale, 91405, Orsay, France 55institutetext:
Solar-Terrestrial Centre of Excellence – SIDC, Royal Observatory of Belgium,
Ringlaan -3- Av. Circulaire, 1180 Brussels, Belgium 66institutetext: Institute
of Geodynamics of the Romanian Academy, 19-21 Jean-Louis Calderon, 020032,
Bucharest, Romania

# Stereoscopic Measurements of Coronal Doppler Velocities

O. Podladchikova, Corresponding author: Olena Podladchikova Stereoscopic
Measurements of Coronal Doppler VelocitiesStereoscopic Measurements of Coronal
Doppler Velocities<EMAIL_ADDRESS>L. Harra Stereoscopic
Measurements of Coronal Doppler VelocitiesStereoscopic Measurements of Coronal
Doppler VelocitiesStereoscopic Measurements of Coronal Doppler
VelocitiesStereoscopic Measurements of Coronal Doppler Velocities K.Barczynski
Stereoscopic Measurements of Coronal Doppler VelocitiesStereoscopic
Measurements of Coronal Doppler VelocitiesStereoscopic Measurements of Coronal
Doppler VelocitiesStereoscopic Measurements of Coronal Doppler Velocities C.H.
Mandrini Stereoscopic Measurements of Coronal Doppler VelocitiesStereoscopic
Measurements of Coronal Doppler Velocities F. Auchère Stereoscopic
Measurements of Coronal Doppler VelocitiesStereoscopic Measurements of Coronal
Doppler Velocities D. Berghmans Stereoscopic Measurements of Coronal Doppler
VelocitiesStereoscopic Measurements of Coronal Doppler Velocities É. Buchlin
Stereoscopic Measurements of Coronal Doppler VelocitiesStereoscopic
Measurements of Coronal Doppler Velocities L. Dolla Stereoscopic Measurements
of Coronal Doppler VelocitiesStereoscopic Measurements of Coronal Doppler
Velocities M. Mierla Stereoscopic Measurements of Coronal Doppler
VelocitiesStereoscopic Measurements of Coronal Doppler VelocitiesStereoscopic
Measurements of Coronal Doppler VelocitiesStereoscopic Measurements of Coronal
Doppler Velocities S. Parenti Stereoscopic Measurements of Coronal Doppler
VelocitiesStereoscopic Measurements of Coronal Doppler Velocities L. Rodriguez
Stereoscopic Measurements of Coronal Doppler VelocitiesStereoscopic
Measurements of Coronal Doppler Velocities

(Received February 02, 2021; accepted July 30, 2021)

###### Abstract

Context. The Solar Orbiter mission, with an orbit outside the Sun–Earth line
and leaving the ecliptic plane, opens up opportunities for the combined
analysis of measurements obtained by solar imagers and spectrometers. For the
first time, different space spectrometers will be located at wide angles to
each other, allowing three-dimensional (3D) spectroscopy of the solar
atmosphere.

Aims. The aim of this work is to prepare the methodology to facilitate the
reconstruction of 3D vector velocities from two stereoscopic LOS Doppler
velocity measurements using the Spectral Imaging of the Coronal Environment
(SPICE) on board the Solar Orbiter and the near-Earth spectrometers, while
widely separated in space.

Methods. We develop the methodology using the libraries designed earlier for
the STEREO mission but applied to spectroscopic data from the Hinode mission
and the Solar Dynamics Observatory. We use well-known methods of static and
dynamic solar rotation stereoscopy and the methods of EUV stereoscopic
triangulation for optically-thin coronal EUV plasma emissions. We develop new
algorithms using analytical geometry in space to determine the 3D velocity in
coronal loops.

Results. We demonstrate our approach with the reconstruction of 3D velocity
vectors in plasma flows along ”open” and ”closed” magnetic loops. This
technique will be applied to an actual situation of two spacecraft at
different separations with spectrometers on board (SPICE versus the Interface
Region Imaging Spectrograph (IRIS) and Hinode imaging spectrometer) during the
Solar Orbiter nominal phase. We summarise how these observations can be
coordinated.

###### Key Words.:

Sun: UV radiation – Sun: corona – Sun: corona – Instrumentation: high angular
resolution – Techniques: spectroscopic

##  1 Introduction

The Solar Orbiter (SO) space mission (Müller 2020), launched in February 2020,
provides both remote sensing and in situ measurements of the solar atmosphere
and heliosphere. The main goal of this mission is to understand how the
heliosphere is formed and sustained. The trajectory of Solar Orbiter takes it
out of the Earth’s orbit and into an orbit around the Sun, reaching within the
orbit of Mercury to 0.28 AU. A spacecraft whose orbit is away from the
Sun–Earth line opens up huge possibilities for the representation of the
three-dimensional (3D) imaging and spectroscopic data together with near-Earth
imaging and spectroscopy. For the first time, spectrometers will be located at
different angles from each other, allowing 3D spectroscopy of the solar
atmosphere.

The presence of persistent high-temperature, high-speed upflows from the edges
of active regions (Harra et al. 2008) is a key discovery from Hinode (Culhane
et al. 2007). Measurements from the Extreme Ultraviolet (EUV) Imaging
Spectrometer (EIS) indicate that the upflows reach velocities of 50 km s-1
with spectral line asymmetries approaching 100 km s-1 and more (see, e.g.,
Dolla & Zhukov 2011). It has been suggested that these upflows may lie on open
magnetic field lines that connect to the heliosphere and may be a significant
source of the low-speed solar wind (Harra et al. 2008; Brooks & Warren 2011;
Mandrini et al. 2014, and references therein). All of active regions observed
by Hinode/EIS show upflows. Different explanations have been given for the
physical mechanism of the upflows, including waves, reconnection in the corona
(Baker et al. 2009; Mandrini et al. 2015), and reconnection in the
chromosphere (De Pontieu et al. 2007) driving energy upwards. The blue shifts
in these regions are ubiquitous, and indicate the presence of upflows. Various
studies have used modelling to determine whether the upflows that are seen at
the edges of the active regions become plasma outflowing into the solar wind
(e.g. Boutry et al. 2012; Edwards et al. 2016).

Attempts have been made to understand how these flows vary with the location
on the disc. Limb-to-limb studies of an active region were carried out by
Démoulin et al. (2013) and Baker et al. (2017). The highest plasma velocities
in the three spectral lines that they explored have similar magnitudes, and
their magnitudes increase with temperature. The authors concluded that their
results are compatible with the active-region upflows originating from
reconnection between active-region loops and neighbouring loops. However,
having two spectroscopic views of an active region will enhance our
understanding of flows.

First work on solar stereoscopy was carried out by assuming temporal stability
of the features under study and allowing the Sun to rotate in order to obtain
more than one viewpoint (e.g. Berton & Sakurai 1985; Koutchmy & Molodenskii
1992; Aschwanden & Bastian 1994; Aschwanden et al. 1995, 1999; Feng et al.
2007b). Such an approach is used also now (see, e.g., Nisticò et al. 2013).
The review by Aschwanden (2011) presents in detail the methods of static and
dynamic solar rotation stereoscopy for coronal loops observed in optically-
thin coronal plasma EUV emissions.

Two simultaneous viewpoints of the Sun were provided by the Solar Terrestrial
Relations Observatory (STEREO, Kaiser et al. 2008). The two STEREO spacecraft
orbit the Sun with increasing separation angles, providing stereoscopic images
of the Sun’s atmosphere. The stereoscopic images obtained by the two nearly
identical EUV broad-band imagers of the Sun Earth Connection Coronal and
Heliospheric (SECCHI) suite (Howard et al. 2008; Wuelser et al. 2004; Eyles et
al. 2009) on board STEREO helped us to understand the 3D geometry of a rich
variety of optically thin solar structures (Liewer et al. 2009; Patsourakos et
al. 2009; Aschwanden 2009a; West et al. 2011; Delannée et al. 2014;
Podladchikova et al. 2019; Mierla et al. 2008; Aschwanden 2009a; Mierla et al.
2009; Temmer et al. 2009; Mierla et al. 2010; Feng et al. 2009; de Patoul et
al. 2013). Methods of stereoscopic triangulation using a stereoscopic pair of
EUV or white light coronal images have been developed. A lot of progress has
been made to improve magnetic field models based on stereoscopic information,
and to beat down the discrepancy between theoretical magnetic field models and
observed stereoscopically triangulated loop 3D coordinates (Aschwanden 2011) .

Several studies have been carried out in parallel to calculate the true
location of coronal structures in 3D space (Pizzo & Biesecker 2004; Inhester
2006; Feng et al. 2007a; Aschwanden et al. 2008; Howard & Tappin 2008). These
are based on the direct geometric triangulation using a series of line-of-
sight (LOS) measurements taken from different spacecraft views towards the
apparent edges of the structures. The true 3D coordinates of the structures
are calculated from the intersections of these LOSs. The theoretical
background for solar stereoscopy based on the direct triangulation of solar
structures is given by Inhester (2006). The direct triangulation technique can
be applied after prior identification and matching of the targeted structures
in two images (see the review by Wiegelmann et al. 2009). The alternative
method of magnetic stereoscopy has been proposed by Wiegelmann & Neukirch
(2002). The success of magnetic stereoscopy depends on the quality of
theoretical magnetic field models. A critical assessment of non-linear force-
free field (NLFFF) models identified a substantial mismatch between
theoretical magnetic field models extrapolated from photospheric magnetograms
and stereoscopically triangulated loops, in the order of a 3D misalignment
angle of $\alpha_{mis}\approx 20^{\circ}$ to $40^{\circ}$ (De Rosa et al.
2009; Sandman et al. 2009; Aschwanden 2011). Rodriguez et al. (2009)
highlighted that every pixel in an image is a result of the LOS integration of
the emission of the optically thin coronal plasma. This problem of LOS
integration persists in the analysis of the EUV data, further complicating the
matching of points between the two images.

The triangulation method has been applied to detailed 3D reconstructions of
coronal loops in active regions obtained from a stereoscopic pair of images
(Inhester 2006; Aschwanden et al. 2008; Aschwanden & Wülser 2011; Aschwanden
et al. 2012a; Nisticò et al. 2013; Aschwanden et al. 2015; Chifu et al. 2017).
An example of such work (Rodriguez et al. 2009) reveals that loops that appear
to be co-spatial in 171 Å and 195 Å images have in fact different heights and
occupy different volumes. These results are key to understanding coronal
heating.

Here, we develop a 3D EUV spectroscopy methodology for active regions that is
used to build velocity vectors from a pair of EUV images and from Doppler
shift maps taken from different perspectives. We do this using the STEREO
triangulation technique of EUV images of active regions and we develop novel
methods of analytical geometry in space to determine 3D velocities in coronal
loops. An advantage of having two different views of an active region is that
one can determine the projection angles of the loops, and hence direction of
plasma flows along them. We use Hinode/EIS data acquired at different times in
order to replicate the situation of two spacecraft with spectrometers on
board, and we choose an active region that did not show significant changes in
time (Section 2). In addition, we describe methods with which to reconstruct
the velocity vectors; these will be released as a suite of 3D spectroscopy
algorithms (Software package DOVES –– DOppler VElocities Stereoscopically) for
velocity vector reconstruction. We describe the spatio-temporal co-alignment
between EUV broad-band and spectroscopic images (Section 3), the
reconstruction of the 3D geometry of coronal loops (Section 4), and
deprojection algorithms of the measured LOS Doppler shifts into velocity
vectors of plasma flows onto straight and curved coronal loop structures
(Section 5). We also perform magnetic field modelling of the active region to
investigate the possibility of using it to derive the 3D loop geometry
(Section 5.2.4).

A description of the instruments that could be used for 3D spectroscopy is
given in Section 6. We provide recommendations on the optimum possible
spacecraft configuration and spatial resolution of the instruments for 3D
spectroscopy within the framework of the Solar Orbiter mission. The proposed
3D spectroscopy method is of particular interest when aiming to understand how
the corona is heated and how the solar wind is formed.

##  2 Data preparation for 3D spectroscopy

###  2.1 Instrumentation

Initial data for 3D velocity reconstruction includes a stereoscopic pair of
EUV images, and simultaneous Doppler shifts measurements from two angularly
separated spectrometers. Now, to simulate the stereo view we get usable
simultaneous data sets with SPICE, we use solar rotation to produce two
different viewpoints, as it was done for STEREO sofware preparation. These
viewpoints are imitated now by imaging data obtained by the Atmospheric
Imaging Assembly (AIA) on board the Solar Dynamic Observatory (SDO, Pesnell et
al. 2012) and spectroscopic data from Hinode/EIS. By applying direct
triangulation methods to SDO/AIA intensity images, we restore the 3D
coordinates of the observed structures. Then, the triangulation of the Doppler
shifts observed by Hinode/EIS allows us to restore the vector velocities in
these structures.

Hinode is a Japanese mission launched in 2006 by the Institute of Space and
Astronautical Science (ISAS) of the Japan Aerospace Exploration Agency (JAXA)
in collaboration with the National Astronomical Observatory of Japan (Kosugi
et al. 2007). Hinode/EIS has two wavelength bands, 170–211 Å and 246–292 Å,
that include spectral lines formed over a wide range of temperatures, from
chromospheric to flare temperatures. Hinode/EIS has an effective spatial
resolution of about 3–4 arcsec. The high spectral resolution (0.06 Å) allows
the determination of Doppler velocity maps with an accuracy of 3 km s-1. The
Hinode/EIS campaign used in this work has a slit size of 1 arcsec, a field of
view (FOV) of 467 arcsec $\times$ 511 arcsec and a raster duration of 67 min.

The SDO/AIA, built by Lockheed Martin Solar and Astrophysics Laboratory,
provides continuous broad-band full-disc images from the chromosphere to the
solar corona in seven EUV wavelengths covering the temperature range $2\times
10^{4}-2\times 10^{7}$ K, with a cadence of 12 s and a spatial resolution of
1.2 arcsec (corresponding to two pixels). SDO is in a circular geosynchronous
orbit at an altitude of 35 800 km.

Active region NOAA AR 2678 was observed simultaneously by SDO/AIA and Hinode
EIS on 19 and 22 November 2017. The separation of the two observing positions
due to solar rotation is 37∘, which is within the angular range
$9^{\circ}-120^{\circ}$ in which stereoscopic vision and 3D spectroscopy
becomes applicable. We discuss this aspect in Section 3.

###  2.2 Observations of NOAA AR 2678

Figure 1: Whole Sun images obtained with SDO/AIA at 193 Å on 19 November 2019
(position 1, left) and 22 November 2017 (position 2, right). The active region
(pointed with arrows) has a simple magnetic configuration ($\alpha$ according
to the Zurich classification). These two views correspond to a separation of
$37^{\circ}$ between position 1 and position 2, allowing the use of
stereoscopy for the 3D reconstruction of the active region. We illustrate the
active region dynamic evolution with four animations: 193 Å from 19 to 22
November  (Movie 1), stabilized 193 Å from 19 to 22 November  ( Movie 2), 2D
SDO/HMI photospheric magnetic field with overlaid 193 Å ( Movie 3), 171 Å  (
Movie 4). Animations demonstrate that some of the observed simple shape loops
can be traced throughout the entire animation.

On each viewpoint, three simultaneous images were obtained: (1) Fe xii
intensity from SDO/AIA, (2) Fe xii intensity from Hinode EIS and (3) Fe xii
Doppler velocity map. The SDO/AIA images are used to establish the 3D
coordinates of coronal points and the Hinode EIS images for velocity vector
construction in 3D space. Figure 1 shows whole Sun images obtained with
SDO/AIA at 193 Å. Because the active region evolves slowly, its apparent
morphological changes are due to the different viewing angles.

##  3 Spatio-temporal data co-alignment for 3D spectroscopy

###  3.1 Hinode/EIS data

We used the eis_prep.pro routine in SolarSoftware to calibrate the Hinode/EIS
data. The dark current and cosmic rays were removed, and the hot pixels were
corrected. The intensity DN (digital number) values were calibrated to the
spectral radiance in units of erg $\cdot$ (cm2 $\cdot$ s $\cdot$ sr $\cdot$
Å)-1. We focused on the strongest emission line observed by Hinode EIS, Fe xii
at 195.12 Å with formation temperatures $\log T=6.1$. For each spectrum, we
fitted a single Gaussian function with the eis_auto_fit routine and obtained
the line peak intensity, Doppler velocity, and line width using
eis_get_fitdata.pro.

###  3.2 SDO/AIA data processing

We have chosen the following conditions for the SDO/AIA 193 Å data, the
observation time corresponds to that of the whole Hinode/EIS raster scan. The
FOV is slightly larger than the region observed by Hinode/EIS and it is
tracked with solar rotation at the Carrington rate. Images were transformed to
a grid with a resolution of 0.6 arc seconds per pixel. The pre-processed data
were aligned with the solar north and divided by the exposure time. The
strongly saturated frames were removed. We selected SDO/AIA 193 Å images with
more than 2 s of exposure time to obtain suitable images for long-lived
coronal structure analysis. We processed the images with an image stacking
technique to increase photon-to-noise statistics and then with a wavelet high-
pass filter to enhance coronal structures (Stenborg et al. 2008).

####  3.2.1 Hinode/EIS and SDO/AIA data co-alignment

Dynamic structures on the solar surface evolve with a significantly shorter
time than the duration of the Hinode/EIS raster observation of 67 min. We
created a pseudo SDO/AIA raster analogous to the Hinode/EIS raster to
compensate for possible dynamic changes. To this end, we extracted the SDO/AIA
data closest in time to each position of the Hinode/EIS slit. These data were
merged into a single SDO/AIA map (hereafter referred to as “artificial SDO/AIA
raster map”), corresponding to Hinode/EIS raster map using the cross-
correlation method described herein. The offset between the artificial SDO/AIA
raster map and the Hinode/EIS raster map was applied together with a pointing
correction. Based on the new Hinode/EIS pointing details, the new artificial
SDO/AIA raster map was created and aligned to the new Hinode/EIS map. In our
analysis, we set the expected accuracy as half the size of the Hinode/EIS
pixel.

Figure 2: Hinode/EIS intensity (middle) and Doppler velocity (right) raster
maps with corrected pointing information, and the spatio-temporally
corresponding artificial SDO/AIA raster map (left) obtained by the instruments
in position 1.

Figure 2 shows the Hinode/EIS intensity and Doppler velocity raster maps and
the corresponding artificial SDO/AIA raster map. A similar procedure was
presented in Barczynski et al. (2018) to co-align SDO/AIA and the Interface
Region Imaging Spectrometer (IRIS) raster data. Here, 3D point triangulation
is performed with SDO/AIA data, and each 3D point is assigned to a pair of
Doppler shifts via spatio-temporal SDO/AIA – Hinode/EIS co-alignment.

##  4 3D triangulation of NOAA AR 2678

###  4.1 Triangulation through epipolar geometry

The 3D coordinates are calculated as (Earth-based) Stonyhurst heliographic
longitude and latitude, along with the radial distance in solar radii. Then,
the coordinates are converted into the Heliocentric Earth Equatorial (HEEQ)
coordinates for a Cartesian representation of the data. A two-dimensional (2D)
solar image taken by a spacecraft is usually identified by the $(i,j)$
coordinates in the image coordinate system or by the latitude and longitude in
the heliographic (or helioprojective Cartesian) coordinate system, when
$(i,j)$ are projected on the solar surface (Thompson 2006). The multi-point
heliospheric observation in the corona requires a complete 3D heliocentric
coordinate system, as explained in Section 4.1.1. Figure 3 shows the HEEQ
coordinate system with the origin in the centre of the Sun and the object
position in the heliosphere described by the Cartesian coordinates $X,Y,Z$.
The Stonyhurst coordinate system describes the 3D position of a feature with
the spherical coordinates $R,\Theta,\varphi$.

Figure 3:  Stereoscopic observation of NOAA AR 2678 with two spacecraft. The
positions of the two spacecraft and any point in the solar corona to be
triangulated define a plane called the epipolar plane (see, e.g., Inhester
2006). Because the two LOSs (LOS1 and LOS2) must lie in the same epipolar
plane, their intersection in this plane is unambiguously defined by the 3D
point. For convenience, the 3D coordinates of the points are reconstructed in
HEEQ heliocentric coordinates (axis X is directed towards the Earth).
Calculation of the 3D-point coordinates is the first step in the velocity
vector construction.

As seen from Figure 3, two observing spacecraft positions together with a
targeted point in the solar corona define a plane called the epipolar plane.
All targeted points have planes in common that contain the two spacecraft
positions. Given that every epipolar plane is seen head-on from both
spacecraft, it is reduced to a line in the respective image projections; this
line is called an epipolar line.

Any targeted corona point identified to be situated on a certain epipolar line
in one image must lie on the same epipolar line in the other image. The
epipolar lines therefore provide a natural coordinate system for stereoscopic
reconstructions. Consequently, finding a correspondence between pixels in the
images taken by two EUV imagers separated in space is reduced to establishing
a correspondence between pixels along the same epipolar lines in the two
images.

Once the correspondence between the pixels is found, the 3D reconstruction is
performed by calculating the LOSs that belong to the respective pixels in the
image and back-tracking them into 3D space. Because the LOSs must lie in the
same epipolar plane, their intersection in this plane is defined
unambiguously. This procedure is often called “tie-pointing” (see, e.g.,
Inhester 2006; Mierla et al. 2009; Liewer et al. 2009; Aschwanden 2011). The
geometrical reconstruction errors are related to both the separation angle
$\gamma$ between the two spacecraft and the spatial resolution $ds$ of an
image (Inhester 2006; Aschwanden et al. 2015); see Section 6.3 for more
information on reconstruction errors in the context of the Solar Orbiter
mission.

####  4.1.1 3D-triangulation software

In this study, we use the HEEQ system. The origin of the HEEQ system is the
intersection of the solar equator with the central meridian as seen from
Earth, and the solar feature location is given either in Cartesian coordinates
X, Y, Z (HEEQ coordinates) or in spherical coordinates (Stonyhurst
heliographic coordinates: latitude $\Theta$, longitude $\varphi$ and
heliospheric radius $R$) (Thompson 2006) (see Figure 3). The locations of the
features were processed with the World Coordinate System routine
scc_measure.pro of the Solar Software (Thompson 2006). The structures were
traced in stereoscopic pairs of images using projections along the epipolar
lines. The routine uses a combination of the information in the header, such
as the pixel-to-degree conversion, and the SPICE (Spacecraft Planet Instrument
C-matrix Events orientation) database of orbital kernels by calling the
routine convert_sunspice_coord.pro, which is based on SPICE orbital ephemeris
kernels containing the spacecraft location and pointing information. The
coordinate system of each spacecraft image plane can be related to a
heliocentric coordinate system, and stereoscopic analysis is performed. The
output of scc_measure.pro is given in the Stonyhurst heliographic coordinates.

####  4.1.2 3D triangulation of AR 2678 points

Direct measurement of the 3D coordinates of a point located high in the solar
corona is not possible. However, if the two LOSs traced back from the LOS
projections on the 2D image intersect higher in the corona at the observed
point, then the 3D coordinates of the point can be evaluated. The procedure is
performed as follows:

  * •

Point selection on a 2D plane of sky (Image 1); see Figure 4a.

  * •

Transition to 3D coordinates.

  * •

LOS drawing from satellite 1 to the observed point (LOS 1 or epipolar lines in
stereoscopy). The LOSs are not visible on the 2D Image 1 (Figure 4a), because
they are projected to one point.

  * •

Image 2 (Figure 4b) shows the same area in the solar corona from Perspective
2\. LOS1 is visible in Image 2 (Figure 4b) due to observation from a different
point.

  * •

The intercept of LOS 1 with coronal loops, the structure of the observation,
determines the projection of the 3D point on Image 2 (crosses in Figure 4b).
Thus, the problem converges, as long as we deal with two LOS projections of
the same point in 3D space.

To check the obtained coordinates the procedure can be then performed in
reverse order (Figure 4c,d). In Figure 4d we start from the obtained points on
Image 2 and describe how we arrive at the initially selected points on Image 1
(Figure 4c) using the LOS projection technique. We repeat the procedure for
points 1-2-3 located on the same loop (same for points 4-5 and 6-7) selected
on the SDO / AIA images, and express their true spatial position in a 3D
heliocentric coordinate system.

Figure 4: (a) Points 1 to 7 on Image 1 selected for triangulation. (b) Points
1 to 7 on Image 2 are found on the corresponding epipolar lines (dashed lines)
at the intersection with the observed structures. Points 1, 2 and 3 belong to
closed loops (we call as such those for which we can identify both of their
bases in AIA EUV images); whereas points 4, 5, 6, and 7 belong to open loops
(i.e. those for which we can identify only one of their bases in AIA EUV
images). To validate the tie-pointing procedure, we performed the reverse
process: first starting from the obtained points on Image 2 (panel d) and then
selecting their stereoscopic pairs on the corresponding epipolar lines on
Image 1 (panel c), obtaining the same points.

####  4.1.3 3D loop parameterisation

Here we study the loops, which are shown in Figure 4 as straight, curved or
semicircular, which is typical for loops that are inclined towards the surface
near the center of the disc (Reale 2010). Aschwanden (2011) presents the
technique of semi-circular loop point positions and loop plane inclination
definitions. The plane in the strict geometric sense has zero thickness;
nevertheless, the terms “loop plane” and “loop plane inclination” are the
established and widely used terms in the analysis of the parameters of coronal
loops (see, e.g., Section ”Stereoscopic Fitting of Circular Loop Geometry” in
Aschwanden (2009b), or Nisticò et al. (2013) and Aschwanden & Wülser (2011)),
even though the real loops are thick (Klimchuk et al. 1992).

From the 3D triangulation, we obtain several points on the same loop. We model
the loop by a circle, as explained in Rodriguez et al. (2009) and Aschwanden
(2011), defining the curvature radius, and we use three loop points
triangulated as described above to fit the loop with the model. The loop can
be defined in 3D space by a set of three points with coordinates $(X,Y,Z)$ in
the HEEQ coordinate system and the two sets of 2D coordinates of the solar
images $(i_{Los1},j_{Los1})$ and $(i_{Los2},j_{Los2})$; however, it is
possible to use more points to fit the circle. We save the projected
coordinates of the 3D points $(i_{Los},j_{Los})$ on the images obtained from
the two perspectives for the subsequent operations with Doppler velocity
measurements. More details on loop parameterisation are given in Aschwanden et
al. (2015), Aschwanden (2011) and Nisticò et al. (2013).

When working with optically thin objects in the solar corona, the most
critical task is to ensure that two LOSs truly intersect on the object under
study in 3D EUV images of the corona. Figure 5 shows the coronal loop under
study, presented in the 3D HEEQ coordinate system and reconstructed from
points 1, 2 and 3. Blue lines outside the loop show tangents to the loop
plotted at the observed points.

Figure 5: Semi-circular curved coronal loop under study observed
stereoscopically in NOAA AR 2678 in November 2017. The loop is shown in the
HEEQ coordinate system. Blue lines outside the loop show tangents to the loop
plotted at the selected points.

####  4.1.4 NOAA AR 2678 3D loops

Table 1:  Latitude, longitude, and height above the photosphere of the seven selected points in Figure 4. The 1st column indicates the point number. The 2nd, 3rd, and 4th columns show the latitude and longitude from Earth, Position 1 and Position 2 viewpoints, respectively. The 5th column lists the height of the points above the photosphere, and the 6th column shows the inclination angles $\alpha$ between the open coronal loop, or closed coronal loop plane, and the vertical to the solar surface. See text for the meaning of closed and open loops.  | Earth | Position 1 | Position 2 | Height above |  Inclination $\alpha$
---|---|---|---|---|---
| (Lat./Long.) | (Lat./Long.) | (Lat./Long.) | Photosphere |  Degrees
Closed loop |  |  |  |  |
Point 1 | $-10.7^{\circ}/-3.04^{\circ}$ | $-13.2^{\circ}/-3.13^{\circ}$ | $-12.7^{\circ}/35.16^{\circ}$ |  $16.0$ Mm |
Point 2 | $-12.9^{\circ}/\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ 5.2^{\circ}$ | $-16.6^{\circ}/-5.9^{\circ}$ | $-16.2^{\circ}/36.40^{\circ}$ |  $76.1$ Mm |  $15.5^{\circ}$
Point 3 | $-12.9^{\circ}/-12.2^{\circ}$ | $-15.6^{\circ}/-12.9^{\circ}$ | $-15.3^{\circ}/26.2^{\circ}$ |  $29.6$ Mm |
Open loop |  |  |  |  |
Point 4 | $-10.9^{\circ}/-16.5^{\circ}$ | $-13.4^{\circ}/-17.24^{\circ}$ | $-13.2^{\circ}/21.3^{\circ}$ |  $22.3$ Mm |
Point 5 | $-11.7^{\circ}/-17.5^{\circ}$ | $-14.8^{\circ}/-19.2^{\circ}$ | $-14.6^{\circ}/21.4^{\circ}$ |  $55.1$ Mm |  $54.1^{\circ}$
Open loop |  |  |  |  |
Point 6 | $-6.9^{\circ}/-18.5^{\circ}$ | $-9.6^{\circ}/-19.9^{\circ}$ | $-9.3^{\circ}/20.02^{\circ}$ |  $48.8$ Mm |
Point 7 | $-3.2^{\circ}/-21.7^{\circ}$ | $-5.90^{\circ}/-24.7^{\circ}$ | $-5.7^{\circ}/17.4^{\circ}$ |  $89.1$ Mm |  $55.5^{\circ}$

Table 1 summarises the results of the 3D reconstructed points in the
Stonyhurst coordinate system from the Earth point of view, as well as from the
Position 1 and Position 2 perspectives. The first entry in the table lists the
results for points selected on a closed loop within the active region core,
where the dominant Doppler velocity indicates a downflow, and the bottom part
of the table shows results for points on open loops at the edges of the active
region, where the Doppler velocities show blue-shifted plasma (upflow). Notice
that we use the terms closed loops for those for which we can identify their
two bases in AIA EUV images, and open loops for those for which we can
identify only one of their bases in AIA EUV images.

##  5 3D spectroscopy methodology

###  5.1 Vector velocity measurements

Figure 6:  Stereoscopic view of NOAA AR 2678, recorded from different
perspectives in November 2017. (a) and (b) show the elevated 3D points
projected on the 2D SDO/AIA intensity maps through the LOS technique. LOS1
from Perspective 1 (panel a) is projected down on Image 2 (panel b). Crosses
denote the projection of the elevated coronal points on 2D images. Cross-
calibration has been applied for the SDO/AIA and Hinode/EIS images (panels c
and d). The Hinode/EIS intensity and Doppler velocity maps match, as long as
they are obtained by the same spectrometer at the same time. The LOS
projections of the elevated points on the 2D Doppler velocity maps (panels e
and f) provide two LOS velocity values attached to their reference point in 3D
space. The majority of the reconstructed points are associated with upflows
(blue) and only two points are associated with downflows (red).

We developed a sequence of algorithms designed to perform triangulation of
Doppler shift maps of the solar corona recorded from different perspectives,
with the ultimate goal of enabling reconstruction of the vector velocity
field.

Solar-rotation stereoscopy was used to reconstruct the 3D coordinates of the
highly elevated points in the solar corona. The same loops were identified in
two 2D EUV images in order to calculate the true loop location in 3D corona
coordinates. We used the coronal loops of NOAA AR 2678 projected in the plane
of the sky observed by SDO/AIA from two different viewing angles (Figure
6a,b). The region was stable in time and did not show any flaring activity as
demonstrated by  ( Animation 1). The crosses on the images show the different
locations of the elevated coronal points when they are projected through the
LOS onto 2D images. The SDO/AIA images and Hinode/EIS intensity images were
co-aligned as described in Section 3 (Figure 6c,d). As long as the Hinode/EIS
intensity and velocity maps are co-aligned, the crosses on the intensity maps
can be directly translated onto the velocity maps. Thus, the crosses on the
two Hinode EIS velocity maps indicate two LOS velocities measured in the
considered elevated coronal points (Figure 6e,f).

Figure 7: Stereoscopic observation of NOAA AR 2678 on 19 November 2017, shown
in the 3D HEEQ coordinate system. The X axis is directed towards the Earth and
the Z axis is aligned with the solar rotation axis passing through the solar
north. The two observing satellites are separated from each other by 37∘. The
blue shifts at point 4 are measured along their respective LOSs and are
represented by the vector magnitudes $v_{LOS1}$ and $v_{LOS2}$. The velocity
vector $\@vec{v_{ep}}$, measured in the epipolar plane, is calculated from
$v_{LOS1}$ and $v_{LOS2}$ following the geometry depicted in the figure. In
this case, the true velocity vector of point 4 is directed along the coronal
loop, showing the plasma flow confined by the magnetic field, and it is
calculated directly from the deprojection of $\@vec{V_{3D}}$ on the coronal
loop. If the plasma flow is perpendicular to the epipolar plane, its velocity
cannot be measured, i.e., $\@vec{V_{3D}}$ cannot be derived when it makes an
angle of 90∘ degrees with the epipolar plane.

###  5.2 Velocity vectors for NOAA AR 2678 coronal loops

In this section, we describe how to obtain a velocity vector at a point
situated high above the photospheric level, using two LOS Doppler measurements
obtained by spectrometers distributed in the heliosphere. Previously, we
restored the true 3D coordinates of points in the high corona and the
coordinates of their projections onto each of the 2D Doppler maps.

####  5.2.1 Deprojection rules of two LOS velocity components in the epipolar
plane

Observing two LOS Doppler velocities at any identifiable point in the corona
enables us, first of all, to find the velocity vector in the epipolar plane
formed in the 3D heliosphere by the position of the two satellites and the
observation point. The gray plane in Figure 7 shows the epipolar plane defined
by the observed point 4 on the open coronal loop and the location of the two
satellites. The two Doppler blue-shifted velocities measured at point 4 along
the LOSs situated between the observed point and the observing satellites
represent the 1D vectors $\@vec{v}_{LOS1}$ and $\@vec{v}_{LOS2}$. Figure 8
shows two LOS Doppler velocities measured at points 4, 5, 6 and 7. Here,
$\@vec{v_{LOS1}}$ and $\@vec{v_{LOS2}}$ are different velocities measured
along the LOSs. The velocity vector $\@vec{v_{ep}}$ is the projection of the
3D velocity in the epipolar plane, and can be reconstructed by calculating the
intersection point of the two perpendiculars deprojected from
$\@vec{v_{LOS1}}$ and $\@vec{v_{LOS2}}$ in the epipolar plane. The velocity
vector $\@vec{v_{ep}}$, defined in the epipolar plane, is an especially useful
estimate when we cannot identify any structure to which this point belongs. In
this case, $\@vec{v_{ep}}$ can be an indicator of the plasma flow behaviour.

Figure 8:  Each plot presents the two LOS velocity components
$\@vec{v_{LOS1}}$ and $\@vec{v_{LOS2}}$ (dashed arrows), measured by each
spectrometer at points 4, 5, 6, and 7 in the corona and the reconstructed
$\@vec{v_{ep}}$ (solid arrows) in the epipolar plane (gray). The epipolar
plane is defined by the locations of the two spacecraft and the point in the
solar corona to be triangulated. $\@vec{v_{ep}}$ represents the projection of
$\@vec{V_{3D}}$ on the epipolar plane. When projected on LOS1 and LOS2,
$\@vec{v_{ep}}$ represents the velocities $\@vec{v_{LOS1}}$ and
$\@vec{v_{LOS2}}$ measured by each spectrometer.

####  5.2.2 Plasma outflows along open loops

In this subsection, we demonstrate the technique of velocity vector
reconstruction in the open coronal loops using the pre-calculated geometry in
3D space. If the pair of points (4, 5) or (6, 7) belongs to the open coronal
loop approximated as a straight line between two points, we can restore the
velocity vector magnitude of the plasma flow by the direct deprojection of
$\@vec{v_{ep}}$ on the loop by using the formula:

$\@vec{V_{3D}}=\frac{\@vec{v_{ep}}}{\cos(\beta)},$ (1)

where $\beta$ is the angle between the $\@vec{v_{ep}}$ velocity and the
coronal loop ( Figure 9). If $\beta=90^{\circ}$ , the deprojection cannot be
made. Correspondingly, small $\beta$ angles lead to higher accuracy of the
$\@vec{V_{3D}}$ deprojection.

Figure 9:  Open loops traced in 3D solar coronal coordinates at points 4–5 (panels a and b) and 6–7 (panels c and d). Gray planes represent epipolar planes crossing the loops at the points under observation. The epipolar planes are formed in the heliosphere by the two spacecraft locations and each of the points 4, 5, 6, and 7; $\@vec{v_{ep}}$ is reconstructed in the plane as shown in Figure 8. The $\@vec{V_{3D}}$ velocity vectors are reconstructed using the measured angle $\beta$ and $\@vec{v_{ep}}$ according to Equation 1. The magnitudes of the vectors are shown in Table 2.  Table 2:  Numerical values of the fully reconstructed velocity vectors $\@vec{V_{3D}}$, which characterise outflows along open loops shown by black arrows in Figure 9. $\beta$ is the angle between the velocity vector $\@vec{v_{ep}}$ measured in the epipolar plane and the loop. $\@vec{V_{3D}}$ can be reconstructed by deprojecting $\@vec{v_{ep}}$ onto the coronal loop. The calculations were carried out for each of the points 4, 5, 6, and 7, which lie on open loops. The minus sign is implied to indicate outflows.  | $v_{LOS1}$ | $v_{LOS2}$ | $v_{ep}$ | $\beta$ | $V_{3D}$
---|---|---|---|---|---
| km s-1 | km s-1 | km s-1 | degrees | km s-1
Open loop |  |  |  |  |
Point 4 | -13.61 km s-1 | -16.71 km s-1 | -16.73 km s-1 | 28.86∘ | -19.11 km s-1
Point 5 | -4.54 km s-1 | -5.67 km s-1 | -5.67 km s-1 | 28.54∘ | -6.45 km s-1
Open loop |  |  |  |  |
Point 6 | -3.37 km s-1 | -5.27 km s-1 | -5.45 km s-1 | 36.29∘ | -6.76 km s-1
Point 7 | -2.03 km s-1 | -3.17 km s-1 | -3.29 km s-1 | 36.32∘ | -4.07 km s-1

We reconstructed the velocity vector $\@vec{V_{3D}}$ for the points in the
open-loop outflows using deprojections:
$\@vec{v_{ep}}\rightarrow\@vec{V_{3D}}$. Figure 9 shows the locations of the
coronal loop in 3D corona coordinates and the epipolar planes crossing loops
at the points under observation, where two LOS Doppler velocities are measured
and $\@vec{v_{ep}}$ is obtained (blue arrows in the planes). After the
measurement of the $\beta$ angle between $\@vec{v_{ep}}$ (blue arrows) and the
loop in 3D space, we obtain $\@vec{V_{3D}}$ (black arrows along the loops)
using Equation 1. In all cases, the resulting vector velocities
$\@vec{V_{3D}}$ are directed outwards. The numerical values of the velocity
vectors are given in Table 2. The small variation in the angle $\beta$ between
the LOS Doppler velocities (same spacecraft position) and the open loop when
measured from points on the same loop reflects the location change of the
observation point.

####  5.2.3 Plasma outflows in closed loops

In this subsection, we demonstrate a technique for determining the velocity
vector for flows following curved paths traceable in the solar corona. When
measuring the LOS Doppler velocities with two spectrometers at points
belonging to a closed, semicircular or simply curved, coronal structure,
$\@vec{V_{3D}}$ can be determined again using Equation 1. However, in this
case, the deprojection should be performed on the tangent to the coronal-loop
direction, as long as the velocity vector is oriented in this direction.
Figure 10 (left panel) shows the deprojected velocity $\@vec{v_{ep}}$ from the
two measured LOS Doppler velocities $\@vec{v_{LOS1}}$, $\@vec{v_{LOS2}}$.
Figure 10 (right panel) shows $\@vec{V_{3D}}$ at points 1, 2, and 3. Detailed
information on the LOS Doppler velocity magnitudes and their angles with the
tangents are listed in Table 3.

Figure 10:  Plasma flow velocity vectors in the 3D closed coronal loop under investigation. The location of the loop points in the high corona and the 3D geometry of the loop were determined using the LOS projection technique. (a) shows the velocity vector $\@vec{v_{ep}}$, deprojected from the two Doppler LOS velocities measured by the two spectrometers at point 2. The epipolar plane (gray) is the plane containing the two satellites and the observation point. The vectors $\@vec{V_{3D}}$ were constructed by the deprojection of $\@vec{v_{ep}}$ on the tangent to the loop direction (panel b). The absolute values of the vectors are shown in Table 3.  Table 3:  Velocity vectors characterising the flows in the investigated closed loops observed in NOAA AR 2678 on November 2017. $\@vec{V_{3D}}$, shown in Figure 10, is reconstructed at points 1, 2, and 3 on the loop. The velocity vectors $V_{3D}$ are found by deprojection from $\@vec{v_{ep}}$ on the tangent to the coronal loop directions. The minus sign is implied to indicate outflows. | $v_{LOS1}$ | $v_{LOS2}$ | $v_{ep}$ | $\beta$ | $V_{3D}$
---|---|---|---|---|---
| km s-1 | km s-1 | km s-1 | degrees | km s-1
Closed loop |  |  |  |  |
Point 1 | 2.38 km s-1 | -0.68 km s-1 | 4.91 km s-1 | 7.51∘ | 4.95 km s-1
Point 2 | 0.71 km s-1 | -1.97 km s-1 | 4.27 km s-1 | 17.98∘ | 4.48 km s-1
Point 3 | -7.35 km s-1 | -9.4 km s-1 | -9.41 km s-1 | 14.11∘ | -9.70 km s-1

Thus, the velocities deprojected from two LOS Doppler velocities measured at
points of a closed coronal loop provide true intrinsic information of the
plasma flow behaviour in the loop. In this case, the deprojection is not
carried out on the structure, as described in the previous subsection, but on
the direction of the tangent to the loop reconstructed at the observed point.
This technique can be applied to any curved structure in the solar atmosphere,
provided that we can define its 3D geometry using, for instance, epipolar
geometry stereoscopy or magnetic modelling of the loops.

####  5.2.4 Comparison of deduced inclinations and linear force-free field
extrapolation of NOAA AR 2678

Magnetic field modelling has been used extensively to interpret the morphology
of coronal magnetic loops (Chifu et al. 2015). An example by Harra et al.
(2008) illustrates how using the angles from magnetic field modelling enables
a more accurate determination of the upflowing plasma at the edge of an active
region.

We compute the coronal magnetic field topology of NOAA AR 2678 during its disc
transit on 11 November 2017 at 08:28 UT and 22 November 2017 at 03:44 UT. The
LOS magnetic field is extrapolated to the corona using a linear force-free
field (LFFF) configuration where $\@vec{J}\times\@vec{B}=0$ and
$\nabla\times\@vec{B}=\alpha_{\rm M}\@vec{B}$, with $\alpha_{\rm M}$ constant
(Mandrini et al. 1996; Démoulin et al. 1997).

Figure 11: LFFF extrapolation for NOAA AR 2678 overlaid on an SDO/AIA
intensity map. The left panel shows the LFFF model results at 08:28 UT on 19
November 2017. The right panel shows the LFFF model extrapolation at 03:44 UT
on 22 November 2017.

Figure 11 (left panel), shows the LFFF model results for 19 November 2017.
Similar results are obtained on 22 November 2017 (right panel). Both panels
show the modelled global coronal structure for which we have used the closest
in time magnetograms to each EIS image as boundary condition. The panels are
constructed superposing field lines computed using different values of
$\alpha_{\rm M}$, which have been selected to better match the shape of the
observed loops in AIA 193 Å. To do this comparison, the model is first
transformed from the local frame to the observed frame as discussed in
Mandrini et al. (2015) (see the transformation equations in the Appendix of
Démoulin et al. 1997). This allows a direct comparison of our computed coronal
field configuration to AIA EUV lopps. Furthermore, in order to determine the
best matching $\alpha_{\rm M}$ values we have followed the procedure discussed
by Green et al. (2002).

Figure 12 shows the definition of the loop inclination angles $\delta_{EUV}$
computed for EUV SDO/AIA loops using triangulation methods, which are compared
with those $\delta_{M}$ derived from our LFFF extrapolation. The inclination
angles $\delta_{M}$ derived from the model are computed as discussed in
Démoulin et al. (2013); they are the angles between the line tangential to the
loop at the coronal points studied here and the vertical to the solar surface
as shown on the Figure 12.

Figure 12:  Closed coronal loop of NOAA AR 2678 traced in 3D corona
coordinates by stereoscopic (red) and LFFF (black) methods. We computed here
the inclination $\delta_{EUV}$ between the vertical to the solar surface
(blue) and the tangential line at the studied points for SDO/AIA loops using
stereoscopic methods and we found a close similarity with the $\delta_{M}$
computed using the LFFF extrapolations.

Both LFFF and stereoscopic methods obtained rather close ranges of inclination
angles to the vertical: $[20^{\circ},30^{\circ}]$ at point 1 and
$[35^{\circ},45^{\circ}]$ at point 3 on the closed loop. For open loops, the
inclination $\delta_{EUV}$ and $\delta_{M}$ to the vertical at point 6 and
point 7 (where the field lines are quite simple) are in the same range
$[10^{\circ},30^{\circ}]$. At point 4 and point 5 the inclination angles
estimated by both methods are in the range between $[15^{\circ},25^{\circ}]$.
Since the projection of an observed coronal loop has a certain thickness, a
set of computed infinitely thin field lines can match the visible shape of a
loop; this is the origin of the range of values for the inclination
$\delta_{M}$ coming from the magnetic field model.

In general, we do not observe discrepancies between the field lines computed
from the LFFF model and stereoscopic reconstruction, although photospheric
footpoints are difficult to identify in EUV images and from the magnetic model
we trace the full line starting at the photospheric level. The latter
demonstrates the future opportunity of pairing stereoscopic loop
reconstruction using Solar Orbiter data with magnetic field modelling, which
is expected to significantly improve the accuracy of stereoscopic vector
velocity measurements (see, e.g., Aschwanden et al. 2015).

##  6 3D Spectroscopy by two distributed imaging spectrometers

We developed an approach using a pair of space-born spectrometers that provide
stereoscopic views of Doppler shift maps in the solar corona. Measurement of
the velocity vector in the corona can be achieved with one extraterrestrial
remote spectrometer (SO/SPICE) and with the orbital spectrometer Hinode/EIS or
IRIS. Dual spectrometers allow measurement of the 2D velocity vector in any
identifiable coronal point, and eventually reconstruction of 3D velocity
vectors, using the morphology of the coronal structures. In this section we
describe how we derive the optimal spacecraft configurations for 3D
spectroscopy with SO. The spectral and spatial resolution of spectrometers
influences directly the velocity vector measurement accuracy.

###  6.1 SPICE, Hinode/EIS and IRIS spectrometers

High-resolution spectroscopy of optically thin plasma allows measurements of
plasma parameters such as temperatures, densities, chemical abundances, and
Doppler and non-thermal motions (Phillips et al. 2008; Del Zanna & Mason
2018).

In the science phase of the Solar Orbiter mission, there will be opportunities
to carry out 3D spectroscopy in the chromosphere, transition region, and
corona for the first time by combining data from Hinode/EIS (Culhane et al.
2007), IRIS (De Pontieu et al. 2014b) and SO/SPICE (Anderson et al. 2020).
Hinode/EIS wavelength range is 1170-210 Å and 250-290 Å, and its angular
resolution is 2 arcsec. There are four slit and slot positions: 1 arcsec slit,
2 arcsec slit, 40 arcsec slot, and 266 arcsec slot. The temporal resolution is
a few seconds in dynamic events, 10 s in active regions and around 1 minute in
coronal holes. The maximum FOV is 360 arcsec $\times$ 512 arcsec.

The SO/SPICE instrument is also a high-resolution imaging spectrometer
operating in EUV/UV wavelengths. Its design is focused on studies that combine
remote sensing and in situ instruments on board Solar Orbiter. The wavelength
range is 704–790 Å and 973–1049 Å and the accuracy of the line shifts is 5
km/s. Table 4 shows the different emission lines used by SPICE, their
wavelengths, temperature of formation and their closest equivalents for
Hinode/EIS. In most cases, the emission lines for the two instruments do not
represent the same ion, but the formation temperature is similar. Some caution
must be used for lines that are optically thick, such as He ii, when comparing
them.

IRIS is an imaging spectrometer with a slit-jaw imaging system that probes the
chromosphere. It explores the solar chromospheric dynamics to determine how
the energy flows through the chromosphere and the transition region. The
spacecraft has a polar sun-synchronous orbit. IRIS has a spatial resolution of
0.33–0.4 arcsec, a temporal resolution of 2 s, and a velocity resolution of 1
km/s over an FOV of up to 175 arcsec $\times$ 175 arcsec (De Pontieu et al.
2007, 2014a).

Table 4: SPICE emission lines, wavelength, temperature of formation (T), and
the closest equivalents in Hinode/EIS.

SPICE | Wavelength (Å) | $\log T$ | EIS | Wavelength (Å) | $\log T$
---|---|---|---|---|---
C iii | 977.03 | 4.5 | He ii | 256.7 | 4.7
O v | 760.43 | 5.4 | O v | 192.9 | 5.4
Ne viii | 770.92 | 5.8 | Mg vii | 278.39, 280.75 | 5.8
|  |  | Si vii | 275.35 | 5.8
|  |  | Fe viii | 185.21, 194.66 | 5.8
Mg ix | 706.02 | 6.0 | Fe x | 184.5, 257.2 | 6.0
Fe x | 1028.04 | 6.0 |  |  |
Mg xi | 997.44 | 6.2 | Fe xiii | 196.54, 202.04, 203.83 | 6.2
Si xii | 520.67 | 6.3 | Fe xiv | 274.20, 264.79 | 6.2
Flare lines
Fe xviii | 974.84 | 6.9 | Ca xvii | 192.82 | 6.7
Fe xx | 721.55 | 7.0 | Fe xxiii | 263.76 | 7.2
|  |  | Fe xxiv | 255.11 | 7.2

IRIS and Hinode/EIS frequently carry out simultaneous observations, pointing
at the same target. This mode of collaboration will be of great interest
alongside the different FOVs of Solar Orbiter. The IRIS wavelength range
covers the chromosphere, and the Hinode/EIS wavelength range covers the
corona. The SPICE wavelength range provides measurements of the chromosphere,
transition region, and corona. Combining the three spectrometers will provide
the first 3D solar spectroscopy. Table 5 shows a comparison of the SPICE and
IRIS wavelengths.

Table 5: SPICE emission lines, wavelength, temperature of formation, and the
closest equivalents in IRIS.

SPICE | Wavelength (Å) | $\log T$ | IRIS | Wavelength (Å) | $\log T$
---|---|---|---|---|---
H i | 1025.72 | 4.0 | Mg ii | 803, 2796 | 3-7–3.9
C ii | 1036.34 | 4.3 | C ii | 1334, 1335 | 4.3
O iv | 787.72 | 5.2 | O iv | 1399, 1401 | 5.2
Mg xi | 997.44 | 6.2 | Fe xii | 1349.4 | 6.2
Flare lines
Fe xx | 721.55 | 7.0 | Fe xxi | 1354.1 | 7.0

###  6.2 SPICE observing with the Extreme Ultraviolet Imager and SDO/AIA

For 3D geometry reconstruction, we used SDO/AIA intensity images and their
coupling with Doppler velocity maps to perform operations with vector
velocities. In this context, Solar Orbiter Doppler maps can be combined with
the intensity images of the Extreme Ultraviolet Imager (EUI) on board the same
satellite. In both cases, the match between Doppler maps and intensity images
provides all the data necessary to compute velocity vectors.

####  6.2.1 The EUI

The Extreme Ultraviolet Imager (EUI, Rochus et al. 2019), on board Solar
Orbiter observes the solar atmosphere from the top of the chromosphere to the
low corona using three imaging telescopes: the Full Sun Imager (FSI) and two
High Resolution Imagers (HRIEUV and HRILya). The FSI provides a 3.8∘ $\times$
3.8∘ FOV in the 174 Å and 304 Å passbands with a typical temporal cadence of
600 s, and a spatial resolution of 9 arcsec. This telescope provides a context
view of the upper solar atmosphere up to $4R_{\sun}$ at perihelion.

Both HRIEUV and HRILya telescopes provide a 17 arcmin $\times$ 17 arcmin FOV
with a plate-scale of 0.5 arcsec and unprecedented imaging cadence of 1 s. The
passband of HRIEUV (174 Å) is comparable to that of FSI. The second high-
resolution telescope, HRILya, observes the Sun in the Lyman $\alpha$ line
(Ly$\alpha$ 1216 Å). The HRI instruments will study the small-scale structures
and highly dynamic events in the upper solar atmosphere. Ly$\alpha$ is the
most intense emission line in the solar spectrum and affects planetary
atmospheres, so it is of great interest; however, so far observations in
Ly$\alpha$ are rare and were carried out mainly aboard sounding rockets
(Korendyke et al. 2001; Vourlidas et al. 2010, 2016; Chua et al. 2013;
Chintzoglou et al. 2018) and the Transition Region and Coronal Explorer space
observatory (Golub et al. 1999). Table 6 shows the closest corresponding SPICE
emission line to the EUI passband. Again, caution must be used for the
temperature of formation with Ly$\alpha$, as it is a complex, optically thick
spectral line.

Table 6: SPICE emission lines, wavelength, and temperature of formation, and
the closest equivalent filter in EUI.

SPICE | Wavelength (Å) | $\log T$ | EUI | Wavelength (Å) | $\log T$
---|---|---|---|---|---
C iii | 977.03 | 4.5 | Ly$\alpha$ | 1216 | 4.7
Mg viii | 772.31 | 5.9 | Fe ix | 174 | 5.9
Ne viii | 772.31 | 5.9 | Fe ix | 174 | 5.9

###  6.3 3D-point triangulation uncertainties

####  6.3.1 Spacecraft separation angle and spatial resolution

An issue of great importance when planning observations with two spacecraft is
understanding the optimum separation between them. In this section, we discuss
how the accuracy of the measurement varies with spacecraft separation.Inhester
(2006) calculated that the errors in 3D coronal point triangulation depend on
both the spatial resolution $ds$ of an image and on the spacecraft separation
angle $\gamma$:

$\varepsilon_{\Delta}=\frac{ds}{\sin(\gamma/2)},$ (2)

$ds$ is the pointing error across the epipolar line, which is linearly
dependent on the spatial resolution of the instrument and can have values from
1 to 10 pixels, depending on with what accuracy one can identify the same
feature in two images. When the spatial resolution is fixed and $\gamma=0$,
the errors are very large. For large $\gamma$ the errors are small, but
additional errors are introduced due to complications in identifying the same
feature in both images. When $\gamma=180^{\circ}$, the errors are very large
for off-limb features because the LOSs of the two spacecraft are parallel. The
low optical thickness of the solar corona enables the observation of the same
off-limb feature when the separation is near 180∘, but makes it impossible to
observe the same on-disk feature (Aschwanden et al. 2012b, 2015).

Figure 13:  Geometric error for triangulating a single-point object as a
function of spacecraft separation angle. The object is observed by two
separated spacecraft, e.g., the Solar Orbiter and one Earth-orbiting
spacecraft (SDO, Hinode, or IRIS). The separation angles between the Earth–Sun
line and the Solar Orbiter-Sun line varies in the all range of angles during
the cruise and nominal phase of the mission. Angles of $\pm 90^{\circ}$
provide the minimal triangulation error of 1.4 pixels for the 3D geometry
reconstruction using Solar Orbiter high resolution imagers HRIEUV and HRILya.
Separation angles greater than 100∘ between spacecraft can complicate pointing
at the same area of the Sun with two spectrometers and is therefore not
recommended.. However, the angle interval of $10^{\circ}$ – $100^{\circ}$
provides a sufficiently small triangulation error and good pointing
possibilities for spectrometers.

The geometric error in Equation 2 is the error for the triangulation of a
single-point object. For a 1D object such as a section of a loop, the error is
more complex, as it additionally increases with decreasing angle between the
loop tangent and the epipolar plane, and theoretically becomes infinite if
this angle becomes zero. The intuitive reason for this behaviour is that a
smaller angle with the epipolar plane results in a smaller visual angle
between the projected tangent and the epipolar lines in both images.
Consequently, the point of intersection of the loop with the epipolar lines
can be less well determined (Inhester 2006).

Aschwanden et al. (2012b) showed that stereoscopy of coronal loops is feasible
with good accuracy from around $6^{\circ}$ to $127^{\circ}$. This range is
also predicted theoretically by taking into account the triangulation errors
due to finite spatial resolution and uncertainty in the identification of
stereoscopic correspondence in image pairs, which is disturbed by projection
effects and foreshortening for viewing angles near the limb. By combining the
accuracy of altitude triangulation with the stereoscopic correspondence
ambiguity, the authors estimated that a spacecraft separation angle of
$22^{\circ}$ – $125^{\circ}$ is most favourable for stereoscopy, when using an
instrument with a spatial resolution of 2.6 arcsec (pixel size of 1.6 arcsec)
such as Extreme Ulltaviolet Imager aboard STEREO.

We used Equation 2 to calculate the triangulation errors for different
spacecraft separation angles $\gamma$ between the Solar Orbiter and the near-
Earth spacecraft Hinode or IRIS. Figure 13 shows the triangulation error as a
function of spacecraft separation angle, where $ds$ is 1 pixel. The physical
measure contained in a 1-pixel unit will vary as (100-350) km2 for HRI and as
(900–3000) km2 for FSI when the Solar Orbiter travels between aphelion and
perihelion. The angle interval of $10^{\circ}$ – $100^{\circ}$ is expected to
provide a sufficiently small triangulation error and good pointing
possibilities for 3D solar spectroscopy.

##  7 Discussion and conclusions

We have developed a technique that reconstructs velocity vectors in coronal
loops from the Doppler maps and EUV images obtained by two spacecraft located
at significant (on a heliospheric scale) distances from each other. The
methodology is built upon the dynamic solar rotation EUV spectroscopy and on
STEREO triangulation methods for coronal loops observed in optically thin
coronal EUV emissions lines (Aschwanden 2011). Novel algorithms use methods of
spatial analytical geometry in order to determine 3D velocities in coronal
loops using different LOSs Doppler shifts measurements. The study is inspired
by the launch of the Solar Orbiter in February 2020, the first solar mission
to travel far beyond the ecliptic plane, which carries the EUV spectrometer.
For the first time, stereoscopic pairs of solar Doppler maps will be available
by combining data from the distant Solar Orbiter and the near-Earth
spectrometers. We, therefore, explore the optimum conditions for future vector
velocity observations with Solar Orbiter. The methodology consists of the
following steps:

  1. 1.

Determination of 3D geometry of structures through the LOS projection
technique with high resolution broad band EUV imagers aboard Solar Orbiter
EUI, Solar Dynamics Observatory and STEREO.

  2. 2.

Cross-calibration and alignment between monochromatic and broad–band EUV
images.

  3. 3.

Vector velocity reconstruction

    1. (a)

Data acquisition: Stereoscopic pair of Doppler shifts at each coronal point.

    2. (b)

Deprojection ”1D$\rightarrow$2D”:
$\@vec{v_{LOS1}}$,$\@vec{v_{LOS1}}\rightarrow$ $\@vec{v_{ep}}$. Two measured
LOS velocity components are deprojected into velocity vector $\@vec{v_{ep}}$
defined in the plane formed by the two spacecrafts and the coronal point we
reconctruct the velocity vector. When orthogonally projected on LOS1 and LOS2,
$\@vec{v_{ep}}$ represents $\@vec{v_{LOS1}}$ and $\@vec{v_{LOS2}}$ velocities
measured by each spectrometer.

    3. (c)

Deprojection ”2D$\rightarrow$3D”: $\@vec{v_{ep}}\rightarrow\@vec{V_{3D}}$. The
velocity vector $\@vec{V_{3D}}$ is reconstructed by the orthogonal
deprojection of the defined in the observational plane $\@vec{v_{ep}}$ onto
the direction of plasma flow in the 3D solar corona.

    4. (d)

Limits of application: If the plasma flow in the solar corona is orthogonal to
the observational plane the velocity vector cannot be estimated.

  4. 4.

Spacecraft separation setup: The 3D solar spectroscopy methodology developed
here can be applied efficiently over a wide range $10^{\circ}$ \-
$100^{\circ}$ of separation angles between Solar Orbiter SPICE and Hinode/EIS
or IRIS for practically the entire duration of the Solar Orbiter mission,
subject to observation of the same area in the Sun. The most favourable for
spacecraft separation is expected to be $90^{\circ}$.

The first attempt of Doppler maps stereoscopic observations in coronal lines
was conducted by the consortium during Solar Orbiter cruise phase on 21-28
April 2021, as part of a quadrature observation campaign. Test data for vector
velocity reconstruction will be available for the community shortly after
calibration. Velocity vectors in plasma flows propagating at small angles to
the observation plane (the epipolar plane, where the LOS Doppler velocities
are located) can be reconstructed with higher accuracy, but the triangulation
error may increase. The high spectral and spatial resolution of the
instruments also contributes to an increase in the accuracy of the 3D velocity
reconstruction, i.e., Solar Orbiter observations in perihelion positions
minimise triangulation errors. The applicability of the methodology is
determined by the correct solution of two problems, stereoscopic tie-pointing
and triangulation of features, and application of analytical geometry in
space. All these facts must be taken into account when planning stereoscopic
observations using Solar Orbiter, and the detailed error assessment will be
provided upon the algorithm test with real stereoscopic Doppler maps data
sets.

###### Acknowledgements.

The authors acknowledge the anonymous referee and the Editor for providing
valuable suggestions and comments. We are grateful to the EUI, SPICE
instrument teams and to ESA Solar Orbiter Remote Sensing working group for
detailed discussions during the consortium meetings, to Hardi Peter and Bart
de Pontieu for discussions about the wavelengths and stereoscopic observations
of Doppler maps. OP is grateful to Conrad Schwanitz for sharing his data
analysis experience, William Thompson for the vector velocity definition
velocity and to Anik de Groof for Solar Orbiter orbits and observations
planning. OP acknowledges funding from Karbacher Fonds. KB acknowledges
funding from SNSF. CM acknowledges grants PICT 2016-0221 (ANPCyT) and UBACyT
20020170100611BA. CM is a member of the Carrera del Investigador Científico of
the Consejo Nacional de Investigaciones Científicas y Técnicas (CONICET). The
authors thank the Belgian Federal Science Policy Office (BELSPO) for the
provision of financial support in the framework of the PRODEX Programme of the
European Space Agency (ESA) under contract numbers PEA 4000112292 and
4000117262.

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