11institutetext: Space Research Institute, Austrian Academy of Sciences,
Schmiedlstraße 6, 8042 Graz, Austria
11email<EMAIL_ADDRESS>22institutetext: Institute of Physics,
University of Graz, Universitätsplatz 5, 8010 Graz, Austria 33institutetext:
Zentralanstalt für Meteorologie und Geodynamik, Hohe Warte 38, 1190 Vienna,
Austria 44institutetext: Department of Physics, Imperial College London,
London, UK 55institutetext: Technical University of Braunschweig,
Braunschweig, Germany

# Multi point analysis of coronal mass ejection flux ropes using combined data
from Solar Orbiter, BepiColombo and Wind

Andreas J. Weiss 11 2 2 Christian Möstl 11 E.E. Davies 44 Tanja Amerstorfer 11
Maike Bauer 11 2 2 Jürgen Hinterreiter 11 2 2 Martin A. Reiss 11 Rachel L.
Bailey 33 T.S. Horbury 44 H. O’Brien 44 V. Evans 44 V. Angelini 44 D. Heyner
55 I. Richter 55 H-U. Auster 55 W. Magnes 11 D. Fischer 11 W. Baumjohann 11

(Received: 29 March 2021 — Revised: - — Accepted: -)

###### Abstract

Context.  The recent launch of Solar Orbiter and BepiColombo opened a brief
window in which these two spacecraft, and existing spacecraft at L1, were
positioned in a constellation that allows for the detailed sampling of any
Earth-directed coronal mass ejections (CMEs). Fortunately, two such events
occurred during this time period with in situ detections of an interplanetary
coronal mass ejection (ICME) by Solar Orbiter on the 19th of April and the
28th of May 2020. These two events were subsequently also observed in situ by
BepiColombo and Wind around a day later.

Aims.  We attempt to reconstruct the observed in situ magnetic field
measurements for all three spacecraft simultaneously using an empirical
magnetic flux rope model. This allows us to test the validity of our flux rope
model on a larger and more global scale. It additionally allows for cross-
validation of the analysis with different spacecraft combinations. Finally, we
can also compare the results from the in situ modeling to remote observations
obtained from the STEREO-A heliospheric imagers which were able to capture the
early evolution of the coronal mass ejections.

Methods.  We make use of the 3D coronal rope ejection model (3DCORE) in order
to simulate the ICME evolution and reconstruct the measured flux rope
signatures at the spacecraft positions. For this purpose, we adapt a
previously developed approximate Bayesian Computation sequential Monte-Carlo
(ABC-SMC) fitting algorithm for the application to multi point scenarios. This
approach not only allows us to find global solutions, within the limits of our
model, but also naturally generate error estimates on the model parameters and
detect potential ambiguities.

Results. We show that we are able to generally reconstruct the flux ropes
signatures at three different spacecraft positions simultaneously using our
model in combination with the flux rope fitting algorithm. For the well-
behaved 19th of April ICME our approach works very well while only showing
minor deficiencies. The 28th of May ICME, on the other hand, shows the
limitations of our approach for lower quality ICME measurements or deformed
shapes. Unfortunately, the usage of multi point observations for these events
does not appear to solve inherent issues, such as the estimation of the
magnetic field twist or flux rope aspect-ratios due to the specific
constellation of the spacecraft positions, which all lie near the ecliptic
plane. As our general approach can be used for any fast forward simulation
based model we give a blueprint for future studies using more advanced ICME
models.

###### Key Words.:

Sun: coronal mass ejections (CMEs) – solar terrestrial relations

##  1 Introduction

Coronal mass ejections (CMEs) are a highly energetic process in which a large
amount of magnetized plasma is ejected from the Sun. Interplanetary CMEs
(ICMEs), which can be observed and identified using heliospheric imagers,
propagate far into the heliosphere and arrive at the inner planets within a
few days (Möstl et al., 2009b; Rouillard, 2011). On Earth, they are
responsible for the most extreme geomagnetic storms that can occur and are
therefore the strongest possible driver for the Earth’s space weather system
(Gosling et al., 1991; Farrugia et al., 1993; Gonzalez et al., 1999; Schwenn,
2006). As such, they represent one of the most extreme events that must be
taken into account when relying on modern applications or equipment that are
affected by the space weather environment (Fuller-Rowell et al., 1994; Bolduc,
2002; Pulkkinen et al., 2017).

While the geomagnetic effectiveness of an ICME arriving at Earth relies on
multiple factors, the two most important indicators are the speed and the
north-south magnetic field component $B_{z}$. Given a strong southward
orientation of the magnetic field, a ICME can facilitate the injection of
energy into the magnetosphere leading to the aforementioned geomagnetic events
(Burton et al., 1975; Gonzalez et al., 1994). The in situ magnetic field
signatures of ICMEs often exhibit similar structure with a frontal shock and
sheath region that is followed by a highly structured rotating magnetic field
(Burlaga et al., 1981; Klein & Burlaga, 1982; Kilpua et al., 2017). This
rotating magnetic field, which is commonly referred to as the magnetic
obstacle, can potentially manifest itself a strongly south orientated magnetic
field over the duration of multiple hours. As such, the understanding of the
structure of these magnetic obstacles is key to linking ICMEs with the
resulting geomagnetic effects (Bothmer & Schwenn, 1998).

The magnetic obstacles are often described as magnetic flux ropes (MFR)
structures (Marubashi, 1986; Burlaga, 1988; Lepping et al., 1990). While
alternative pictures exist, based on spheromak models (Farrugia et al., 1995;
Vandas & Romashets, 2017; Singh et al., 2020), the most common depiction uses
some form of a flux bent flux tube that remains connected to the solar surface
while propagating away from the Sun (Dasso et al., 2005). A variety of
different successful magnetic flux rope models have been developed for the
purpose of describing the in situ signatures of ICMEs which generally assume a
rigid cylindrical or toroidal geometry. They are furthermore divided into two
general classes, those based on uniform-twist fields (Gold & Hoyle, 1960; Hu
et al., 2015; Vandas & Romashets, 2017) or constant $\alpha$ fields
(Lundquist, 1950; Lepping et al., 1990; Hidalgo et al., 2002; Nieves-
Chinchilla et al., 2018).

Along with the wide variety of descriptions for the magnetic field, the global
shape of ICMEs also remains largely hidden. While the initial structure can be
partially observed due to Thomson scattering close to the Sun (Cremades &
Bothmer, 2004; Thernisien et al., 2006), the in situ magnetic field
measurements represent a strong projection of the internal magnetic field that
is intertwined with the global or local geometry. This can make it
exceptionally difficult to draw conclusions on the shape using just single in
situ magnetic field measurements. It is broadly understood that there are
various processes that influence the shape and evolution of ICMEs during their
propagation through the heliosphere (Manchester et al., 2017). This includes
rotations or deflections close to the Sun (Lugaz et al., 2012; Kay et al.,
2015) and interactions with the ambient solar wind (Riley & Crooker, 2004; Liu
et al., 2006; Démoulin & Dasso, 2009). More complex interactions are also
possible such as CME-CME collisions (Möstl et al., 2012; Lugaz et al., 2017).

More recently there has been a focus on the study of multi point events, i.e.
ICMEs that are observed in situ by multiple spacecraft (Möstl et al., 2009a;
Kilpua et al., 2009; Good et al., 2019; Davies et al., 2020; Salman et al.,
2020) in an attempt to gain a better global understanding of ICME structure
and their evolution. Unfortunately, such events are inherently rare due to the
limited number of spacecraft with on board magnetometers in different solar
orbits within similar time frames. The successful modeling of these multi
point events invariably requires global ICME models, that include magnetic
fields and ICME evolution, which goes beyond the capability of analytical
models with rigid cylindrical or toroidal shapes. While it may be tempting to
use large scale MHD models (e.g. Manchester et al., 2008; Török et al., 2018;
Verbeke et al., 2019), they are too computationally expensive and unwieldy for
fitting the magnetic field measurements of events. For this purpose multiple
semi-empirical models have been developed, such as those described in Isavnin
(2016); Möstl et al. (2018); Weiss et al. (2021), that aim to mimic the
general properties that ICMEs are thought to have. These models serve as a
bridge in between the more complex MHD models and the fully rigid analytical
models and can include simple interaction with the solar wind, some form of
deformations, and an internal magnetic field while retaining some of the
computational simplicity of the simpler analytical models.

In our previous paper (Weiss et al., 2021), henceforth referred to as W21, we
showcased the successful application of the semi-empirical 3D coronal rope
ejection (3DCORE) model to a single point event detected by Parker Solar
Probe. The 3DCORE model is an semi-empirical forward simulation model that
describes an ICME as a self-similarly expanding torus-like structure that
remains attached to the Sun. It includes ICME evolution factors such as flux
rope expansion and deceleration due to slower solar wind which are described
using empirical relations. The internal magnetic field structure is based on
an analytical toroidal uniform-twist solution described in Vandas & Romashets
(2017).

In this paper we attempt the novel approach of simultaneously fitting multiple
magnetic field measurements from different spacecraft at different locations.
So far the majority of multi point studies have only compared different single
point fits from different spacecraft from the same underlying event (Möstl et
al., 2012). The results from our analysis will show if it is even possible to
describe multiple observations using the global assumptions within our model
and, if not, give a hint on its deficiencies. The comparison of the single
point fits and multi point fits of the same event also allows for a sort of
cross-validation which can further the understanding on the event and model.

Section 2 describes the two events that we will test our approach on. The
first event, the April 19th ICME, represents very close by high-quality
measurements for which we would expect our approach to have the highest
chances of success. The second event, the May 28th ICME, presents a bigger
challenge and was primarily chosen to test the limits of our methods. In
Section 3 we give a brief recap of the 3DCORE model from W21 and also describe
the two changes that were made to update the model. Section 4 describes the
adapted ABC-SMC fitting algorithm that was used for fitting the multi point
data. Section 5 shows the reconstructed magnetic field measurements for both
events and the inferred model parameters from multiple spacecraft
combinations. Finally, in Section 6 we discuss the results and the conclusions
we can draw.

##  2 Data and Events

Figure 1:  In situ magnetic field measurements in local radial tangential
normal (RTN) coordinates from Solar Orbiter (1a/e), BepiColombo (1b/f) and
Wind (1c/g) for the 20th of April 2020 (left) and 28th of May 2020 (right)
event. The two bottom panels (1d/h) show the in situ proton density number
$N_{p}$ and the associated plasma beta coefficient $\beta_{p}$. The blue
shaded area represents our estimated intervals for the unperturbed magnetic
flux ropes (UMFR) which we will be subsequently using for our fitting
approach.

To demonstrate the applicability of our 3DCORE model and the associated Monte-
Carlo based fitting method for modeling the magnetic field of multi point
ICMEs, we choose two recent events that were both observed in situ by the
Solar Orbiter (SolO), Wind and BepiColombo (Bepi) spacecraft.

The first and primary event that we will make use of is the 2020 April 19 ICME
which we henceforth refer to as the April event. This event was initially
detected in situ by Solar Orbiter at a heliocentric distance of $0.80$ AU on
2020 April 19 05:06 UTC, at about 4 degrees longitude (HEEQ) east of Earth.
The ICME shock subsequently arrived a day later at Wind and BepiColombo on the
April 20 at 01:34 UTC and 03:09 UTC respectively. For an extensive description
of this event we refer the reader to Davies et al. (2021) as we will only
focus on the details that are of interest for our analysis.

The second event that we have chosen is the 2020 May 28 ICME which we
henceforth refer to as the May event. Similar to the April event, this ICME
was again first detected by Solar Orbiter at a significantly smaller
heliocentric distance of $0.55$ AU on May 28 15:00 UTC. The ICME was again
also observed by Wind and BepiColombo at a very large longitudinal separation
on May 29 at 15:15 UTC and May 30 at 01:15 UTC respectively. Due to the lack
of clear indicators of a shock front these arrival times represent rough
estimates.

Figure 1 shows the in situ measurements from the SolO MAG magnetometer
(Horbury et al., 2020), the Wind MFI magnetometer (Lepping et al., 1995), the
Bepi MPO-MAG magnetometer (Glassmeier et al., 2010; Heyner et al., 2021) and
the Wind SWE instrument (Ogilvie et al., 1995) for both events. The magnetic
field measurements are shown in their respective local radial-tangential-
normal (RTN) coordinate systems. For the plasma measurements from the Wind
spacecraft we only show the proton density $N_{p}$ and the associated computed
plasma beta coefficient $\beta_{p}$. The blue shaded area represents our
estimated intervals for the unperturbed magnetic flux ropes (UMFR, see Davies
et al. (2021) for the April event) which will be the intervals that we use for
our fitting analysis. The UMFR intervals were estimated using both the
magnetic field measurements and the plasma measurements and matched with each
other using features in the magnetic field data.

Figure 2:  Spacecraft positions for Solar Orbiter, BepiColombo, Parker Solar
Probe and STEREO-A on 2020 April 20 00:00 UTC (left) and on the 2020 May 29
00:00 UTC (right) in HEEQ coordinates. Furthermore shown are the planets
Mercury (brown), Venus (orange) and Earth (green). Colored trails show orbital
movements of the spacecraft within a time frame of two months around the
specifically chosen position. Two open ended cones show the field of view of
the HI-1 ($20^{\circ}$) and HI-2 ($70^{\circ}$) imaging instruments aboard
STEREO-A.

Figure 2 shows the different spacecraft positions within the heliosphere on
2020 April 20 00:00 UTC and the 2020 May 29 00:00 UTC including other
satellites such as Parker Solar Probe (PSP) and the inner planets. As can be
seen from these two figures the STEREO-A spacecraft, which carries
heliospheric imagers (HI, Eyles et al., 2009), was ideally positioned at both
times to image and track Earth-directed CMEs. The captured images for both
events are briefly discussed in Section 2.1. Unfortunately, PSP was located on
the back side of the Sun and therefore incapable of recording any of these two
events. Table 1 summarizes the positions of all three spacecraft that observed
the events in situ on the same dates in Heliocentric Earth equatorial (HEEQ)
coordinates.

Table 1: Positions of Solar Orbiter, Wind and BepiColombo on the 2020 April 20 00:00 UTC (top) and 2020 May 29 00:00 UTC (bottom) in HEEQ coordinates. |  | Solar Orbiter | Wind | BepiColombo
---|---|---|---|---
April | r [$au$] | $0.80$ | $0.99$ | $1.01$
lon [$deg$] | $-3.80$ | $0.18$ | $-1.23$
lat [$deg$] | $-3.85$ | $-5.15$ | $-5.54$
May | r [$au$] | $0.55$ | $1.00$ | $0.97$
lon [$deg$] | $30.11$ | $-0.14$ | $-5.26$
lat [$deg$] | $4.70$ | $-0.91$ | $-2.70$

The magnetic obstacle of the April event appears as an almost pristine flux
rope in all three in situ measurements with a nicely rotating magnetic field.
The normal component $B_{N}$ is bipolar, the transversal component $B_{T}$
unipolar and the radial component $B_{R}$ negative near zero. According to the
standard classification schemes this corresponds to a low inclination left-
handed South-East-North flux rope (Bothmer & Schwenn, 1998; Davies et al.,
2021). Due to the textbook like structure of this flux rope it represents a
unique opportunity to test the validity of our model with multiple close-by in
situ measurements. The May event shows significantly more complex structure
and was primarily chosen to identify the limits of our modeling approach at
extremely large longitudinal separations. While one can still clearly identify
a right-handed South-West-North (SWN) flux rope in the SolO MAG measurements,
which represents a low inclination flux rope with the opposite axial direction
and handedness compared to the April event, the magnetic field is strongly
perturbed at the center for Bepi and Wind. Nonetheless, taking only the
endpoints of the Bepi and Wind measurement into account the magnetic field
still exhibits a magnetic field rotation that is comparable to Solar Orbiter.
The differences can be explained due to large spatial distance in between the
spacecraft and poses a significant challenge for our modeling efforts. We will
partially alleviate these issues by applying strong smoothing onto the
magnetic field data from the May event to suppress the smaller scale
perturbations and use the less perturbed endpoints for our analysis.

###  2.1 Remote Observations

Figure 3:  White light images, with the stars subtracted using a median
filter, from the STEREO-A HI-1 instrument showing the April event on the 16th
of April 07:29 UTC (left) and the May event on the 26th of May 15:29 UTC
(right). The image of the April event shows a slightly flattened Earth-
directed CME with a faint leading edge, elliptical cavity and a strong density
pile up at the back. The image of the May event shows a structure at a higher
latitude that is hard to identify in a single image (marked with the yellow
arrow) and does not contain an obvious cavity.

Figure 3 shows two white light images taken by the HI-1 instrument on 2020
April 16th 07:29 UTC (left) and 2020 May 26th 15:29 UTC (right). The HI-1
instrument was able to nicely follow the evolution of the April CME over a
time frame of almost 24 hours showing a faint leading edge, an elliptical
cavity and a strong density pile up at the trailing edge. The CME cavity,
which represents the magnetic obstacle of interest, is seen edge-on and
appears to be of elliptical shape with a time-invariant aspect ratio of around
2 (see Davies et al., 2021). These images for the April event appear to be
consistent with the approximation of an elliptical cross-section that is one
of the central assumptions built into our 3DCORE model. The white light images
for the May event on the other hand show a smaller and fainter structure
propagating above the ecliptic. The structure also darkens considerably faster
making it impossible to observe the evolution of the shape for a longer
duration than a few hours.

Both the April111https://www.helcats-
fp7.eu/catalogues/event_page.html?id=HCME_A__20200415_01 and
May222https://www.helcats-
fp7.eu/catalogues/event_page.html?id=HCME_A__20200526_01 event are part of the
HELCATS catalog. Despite only being observed from a single vantage point one
can extract rough estimates for the propagation speed and direction by
tracking the CME evolution. Under the assumption of self-similar expansion the
given values for the April event are an average CME speed of $339\pm
11\,\textrm{kms}^{-1}$ and a propagation direction of $-2\deg$ longitude and
$-6\deg$ latitude in HEEQ coordinates. For the May event one receives an
average speed of $497\pm 64\,\textrm{kms}^{-1}$ and a propagation direction of
$36\deg$ longitude and $11\deg$ latitude again in HEEQ coordinates.

##  3 Model

In this paper we use an updated version of the 3DCORE model to analyze our two
selected events. While we will give a brief general overview we will focus on
the changes to the implementation that we have made in order to facilitate
analyzing multi point events. The two primary changes that are worth
mentioning are the “twist reparametrzitation” where we substituted the
magnetic field twist parameter using a different related parameter which has
beneficial numerical properties and the introduction of a new noise model. For
the interested reader we refer to W21 for a detailed description of the
original model.

###  3.1 3DCORE Overview

The 3DCORE model is an empirically based forward simulation model which can
generate synthetic in situ magnetic field measurements anywhere within the
heliosphere. It assumes a self-similarly expanding torus-like MFR structure
that propagates along a fixed direction radially away from the Sun. The end
points of the torus-like structure stay attached to the Sun. The expansion of
the flux rope structure is mimicked using a simple scaling relation. The
interaction with the background solar wind is described by using a drag-based
model (Vršnak et al., 2013) where the frontal part of the structure is
accelerated or decelerated, depending on the relative speed to the ambient
solar wind speed. The solar wind speed is assumed to be constant everywhere.
The “pancaking” effect, that is described in Riley & Crooker (2004), is
approximated by stretching the cross-section of the torus into an elliptical
shape.

Figure 4:  Example of a 3DCORE simulation showing the torus-like structure at
near $1$ AU with a highly elliptical cross-section and two magnetic field
lines at differing radial distances($\tau=5$).

The internal magnetic field of the MFR is simulated by inserting an analytical
magnetic field model into the structure. We use a model based on the solutions
described in (Vandas & Romashets, 2017) which we have slightly modified in
order to accommodate the elliptical cross-section. We furthermore make the
assumption that the total number of twists along the torus stays constant in
time. This leads to a decreasing twist per unit length as the structure
expands. The decrease of the magnetic field strength as the structure
propagates forward and expands is again described by a simple scaling relation
(Leitner et al., 2007). While the implemented magnetic field is based on
analytically derived equations, it is important to note that it is not fully
physical and violates various conservation laws. We assume that the errors
that are introduced by this approximation are less severe than those arising
from the basic assumptions on the geometry.

One of the key points of our model is that the simulation is extremely fast as
it is only based on empirical relations and does not solve any expensive and
complicated MHD equations. This efficiency makes it significantly easier to
explore the full parameter space of the model and allows the usage Monte-Carlo
based methods which require in the order of thousands or millions of
simulation runs.

In total the model contains 12 controllable parameters: the initialization
time $t_{0}$, propagation direction and orientation $lat,\ lon,\ inc$, cross-
section width $d_{1au}$, cross-section aspect ratio $\delta$, initialization
distance $r_{0}$, initial speed $v_{0}$, magnetic field strength $b_{1au}$ at
1AU, twist number $\tau$, global solar wind speed $v_{sw}$ and drag parameter
$\Gamma$. Both the diameter of the cross-section and magnetic field strength
are modified by scaling relations with $d(t)=d_{1au}r(t)^{1.14}$ and
$b(t)=b_{1au}r(t)^{-1.68}$, where $r(t)$ is the distance of the CME front from
the Sun.

###  3.2 Twist Reparameterization

One of the key findings of W21 was that we were not able to accurately infer
the magnetic field twist $\tau$ and the aspect ratio of the torus cross-
section $\delta$ using single-point observations. We found a strong
$\tau-\delta$ degeneracy meaning that these two parameters are strongly
correlated. The dependency relation roughly follows
$\tau\propto\delta/E(\delta)$ where $E(\delta)$ is the function for the
circumference of an ellipse with a given aspect ratio.

This issue exists as we are not directly capable of measuring the twist of the
magnetic field, and instead only measure the local direction of the magnetic
field vectors. Depending on the geometry or size, in this case the height of
the CME structure given by the aspect-ratio, the total number of twists along
a fixed length can change drastically. On top of the issue of not being able
to accurately infer these model parameters simultaneously we found this
degeneracy to be a numerical issue when using our Monte-Carlo based fitting
algorithm as the degeneracy is strongly non-linear and is inefficiently
sampled using multivariate Gaussian transition kernels. The shape of this
degeneracy, as seen in a two dimensional histogram, closely resembles a
“banana shape” and presents a classical problem for the class of algorithms
that we use (Haario et al., 1999; Filippi et al., 2013).

We therefore attempt to at least remove the non-linearity of this degeneracy
by defining a new twist parameter $T_{\tau}=\tau E(\delta)$. The introduction
of this new parameter significantly reduces the numerical instability of our
approach and increases the overall speed of the fitting algorithm as the
sampling efficiency is increased. The original $\tau$ values remain
recoverable in a post-processing step.

###  3.3 Noise Model

In order to estimate the uncertainties within our model with our fitting
algorithm we need to be capable of simulating the measured noise. In the
context of our model we gather all unwanted small scale components of the
magnetic field under the general term of “noise”. This includes the expected
instrument noise but also other seemingly random phenomena such as MHD
turbulence and waves or fluctuations due to undetermined physical processes.
In the case of instrument noise, we assume that the data sets are thoroughly
cleaned and that the most significant spacecraft influences are correctly
removed.

In W21 we made the assumption of a simple additive Gaussian noise model that
was described by a single parameter $\sigma$ which determined the standard
deviation of the additive noise component. As was briefly mentioned in W21,
this is a very basic approach which is easy to implement but fails to fully
capture the small scale behaviour of the magnetic field that is seen in the
observations. We will therefore introduce a more sophisticated additive noise
model based on the power spectrum distribution (PSD) $P(k)$. Our
implementation for generating the noise is inspired by the analogous process
that is used to generate primordial perturbations for dark matter N-body
simulations using the matter power spectrum (see Hahn & Abel, 2011).

For our purpose, we are interested in using the PSD to reproduce the type of
fluctuations within our model that are seen in the actual observations. As a
first step we require an appropriate PSD from which we generate the noise. The
most adequate choice is the PSD from the underlying event which is being
analysed as the distributions can vary significantly in between different
events. In our case we ignore cross-correlations between the three different
components of the magnetic field which significantly simplifies the process as
we only need to generate three independent noise samples for each component.
Given a random noise field $\mu(t)\sim\mathcal{N}(0,1)$ and its corresponding
Fourier transform $\tilde{\mu}(k)=\mathcal{F}\left(\mu(t)\right)$ one can
generate a random time series:

$\delta(t)=\mathcal{F}^{-1}\left(\sqrt{P(k)}\tilde{\mu}(k)\right)$ (1)

so that $\tilde{\delta}(t)\approx P(k)$. This process allows us to generate
random and unique noise fields which all have the same underlying PSD
statistic. In the final step we add this random field to our model output to
combine the large scale magnetic field rotation with the random small scale
fluctuations.

We compute the PSD for each magnetic field component independently using
Welch’s algorithm (Welch, 1967) from the time interval of interest. In order
to remove the large scale fluctuations that partially describe the magnetic
field rotation that is described by our flux rope model we pre-process the
measurements using a linear de-trending process on an hourly scale.

As the PSD varies only little for each component we generate a single general
distribution from the average over all three components. In the next step we
use the computed PSD to generate a random sequence with the same underlying
PSD that will act as our noise. As white noise (Gaussian noise) has a flat
power spectrum we can generate the desired sequence by first generating a
random Gaussian sequence $\delta\sim\mathcal{N}(0,1)$ and multiplying the
newly generated sequence in Fourier space with $P(k)$ before transforming the
sequence back into real space. This process is performed independently for
each component of the magnetic field on top of which the noisy $\delta$
sequence is then added.

The biggest advantage of this newer noise model is that it eliminates the
necessity of a noise level parameter as the severity of noise is directly
estimated from the measurements. In theory this approach also allows us to fit
profiles with an extremely high density of fitting points as the the point-to-
point fluctuations are more accurately modelled to decrease with increasing
proximity. In practice this is not done as the inaccuracies are dominated by
the assumption of the geometry and magnetic field model.

While this noise model is expected to be a large improvement over the model
employed in W21 it is still important to note one of its deficiencies. The
fluctuations in the magnetic field strength are generally around two
magnitudes weaker than those of its components, this is a strong hint that the
underlying fluctuations would be better described using rotations, rather than
the additions that are performed in our noise model. But as our summary
statistic that we use for our fitting algorithm only makes use of the magnetic
field components and not the total field we expect the difference to be
negligible.

##  4 Multi point fitting algorithm

In this section we focus on the details of the fitting algorithm that we use
to analyze the magnetic field measurements with our 3DCORE model. The
underlying algorithm, an approximate Bayesian Computation sequential Monte-
Carlo (ABC-SMC) fitting algorithm, remains largely unchanged from W21 and
requires only minimal adaptions for the multi point scenario.

The general idea behind the ABC-SMC fitting algorithm is to generate large
ensemble simulations from an initial parameter space and classify the goodness
of fit for each simulation run according to an error metric, called the
summary statistic, and reject all simulations which fall above a certain error
threshold. An important step is also the rejection of all simulations that do
not generate valid results at all due to the underlying CME geometry not
hitting any of the observers or producing magnetic field signatures that are
significantly time shifted due to the simulated CME arriving too late or
early.

By repeating this procedure and using the accepted simulations as a
representation of the initial parameter space for the next iteration one can
sequentially reduce the error threshold, starting from a larger value, and
arrive at a good estimate of the underlying posterior after a certain number
of iterations.

Due to the stochastic nature of the simulations, as we include the noise model
introduced in Section LABEL:sec:noise_model, and the correct implementation of
an ABC algorithm we are able to interpret the generate ensembles as
probability distributions (in the Bayesian view). This naturally gives us not
only best-fit model parameters but also non-Gaussian error estimates or
parameter correlations which can further deepen the understanding of the
results and the underlying model.

We discuss the changes that we have made to the summary statistic and the pre-
filtering process for determining if a simulation run can be considered as
valid or invalid. We also take a brief look at the inherent biases that may
appear in our analysis.

###  4.1 Summary Statistic

For each single observation we use a normalized root mean square error
statistic between model output $x$ and reference observation $y$ which is
defined as followed:

$\rho(x,y)=\sqrt{\frac{\sum_{i=1}^{K}(x(t_{i})-y(t_{i}))^{2}}{\sum_{i=1}^{K}y(t_{i})^{2}},}$
(2)

where $t_{i}$ is one of the $K$ time points at which the magnetic field is to
be compared.

At this point it may appear viable to create a combined statistic that
condenses multiple statistics into a single number such as the mean RMSE over
multiple observations. In practice we have found that this approach runs into
the issue in which the algorithm can make a hidden trade-off in between
improving the fit for one observation at the expense of the others. For this
reason we adapted our method so that it simultaneously optimizes multiple
error metrics. While this does not necessarily guarantee optimal convergence
it does not allow for a scenario where one observation is optimized at the
negative trade-off of another and the end result clearly shows if a certain
observation cannot be properly fitted.

As a consequence, a simulation output for N observations
$x=(x^{1},\ldots,x^{N})$ is accepted if and only if
$\sigma(x^{k},y^{k})<\epsilon^{k}$ for all $k\in[1,N]$. The iterative
threshold values $e^{k}$ are computed in the same way as in W21 with
$e^{k}_{0}=\sigma(0,y^{k})$ and $e^{k}_{j+1}$ is set as a quantile fraction of
$e^{k}_{j}$.

###  4.2 Adaptive Time Markers

An important aspect of the fitting algorithm is to determine if a simulation
run can be considered as valid in the first place. In a large ensemble
simulation a majority of runs will not generate any valid results due to
missing one or many of the defined observers or producing signatures that are
strongly time-shifted when compared with the reference observation that is to
be reproduced. In W21 we used two time markers, $t_{s}$ and $t_{e}$, to
determine valid simulation runs.

The condition that we initially set was that any simulation run is to be
considered valid if and only if it produces a valid magnetic field signature
at any of the time points $t_{i}$ that are part of the summary statistic and
produces an invalid measurement at $t_{s}$ and $t_{e}$. This requires the
observer to be within the CME geometry, as assumed by the 3DCORE model, at any
time point $t_{i}$ and to be outside of it at $t_{s/e}$. By setting $t_{s}$
just before $t_{1}$ and $t_{e}$ just after $t_{K}$ one is able to closely
control in situ duration of any single valid run.

The problem that arises, specifically in the multi point scenario, is that it
is initially very hard to find simulations which produce signatures of just
the right length. As one adds additional constraints by adding observations
the efficiency of the fitting algorithm can suffer significantly. For this
reason we implemented an adaptive time marker scheme, in which the the $t_{s}$
and $t_{e}$ markers are set at a large temporal distance to the the fitting
time points $t_{i}$. This distance is iteratively reduced, as is the error
threshold, until it reaches a final value after a certain number of
iterations. This considerably helps the sampling process in the initial stages
when the explored parameter space is still large. From basic tests we have
concluded that this initial approximation does not have any significant
negative effects on the final results.

###  4.3 Bias

An inherent property of our 3DCORE model is that it is an empirical model and
does not contain any physical calculations. As we will show, this can lead to
some peculiar results which require some additional treatment. While our
fitting algorithm and error metric guarantees that we only select model runs
and their associated model parameters that match the observations well it can
only make use of “valid” runs. Simulation runs that do not generate any
results due to the 3DCORE shape not hitting any of the predefined observers or
runs that generate magnetic fields that are too strongly time shifted are
automatically rejected before computing any error metrics.

While the size of the shapes is limited by using bounds for the model
parameters this nonetheless leads to an extremely strong bias towards the
largest allowed shapes as they do not fall victim to this selection bias. The
two parameters in our model that control the size of our structure are the
$d_{\text{1AU}}$ (torus width) and $\delta$ (torus height) parameters. The
width parameter $d_{\text{1AU}}$ is constrained by the duration of the in situ
measurement and can therefore be more or less estimated by our algorithm but
the height is completely unknown due to all observers being largely located
within the ecliptic plane. Due to the $\delta-\tau$ degeneracy that we mention
previously we know that we cannot constrain the $\delta$ parameter using a
single in situ measurement. This generally leads to linear bias
$p(\delta)\alpha\delta$ as a structure that has a larger aspect ratio $\delta$
has a larger chance of hitting the observer.

Under certain circumstances this bias does not occur, such as a CME that
directly propagates towards the observer, which makes the handling of this
issue hard to handle. In general this must be done at a case by case basis by
looking at the model simulations that are accepted and rejected due to hitting
or missing the observers. The conclusion is that our analysis will normally
generate extremely large estimates for the $\delta$ parameter while the
correct interpretation is that nothing about this parameter is actually known.
One hope is that this bias would disappear when adding multiple observations
which would largely resolve this issue.

##  5 Results

Figure 5:  Reconstructed synthetic magnetic field measurements (dashed) with
$1\,\sigma$ confidence interval (shaded area) using the triple point
constellation for the April event and a dual-point constellation for the May
event. The reconstructed magnetic field is overlayed on top of the real in
situ measurements for Solar Orbiter (a/d), BepiColombo (b/e) and Wind (c/f).
The blue shaded area represents the defined UMFR interval and the dotted
vertical lines show the used fitting points $t_{i}$. All magnetic fields are
shown in local RTN coordinates.

In this section we present the results of our paper in which we fitted the
April and May event using our updated 3DCORE model and ABC-SMC fitting
algorithm. A constellation of the three different satellites, combined with
their in situ measurements, not only allows us to create a single large
analysis encompassing all three satellites but also three dual-point analyses
in which a combination of two of the three available in-situ measurements are
used. Counting the single-point results this then leads to a total of seven
different possible data combinations which can be used. This allows us to
study in detail how the results using our method differs when using either the
full data set or smaller sub sets.

Figure 5 shows the resulting triple-point fit of the 2020 April 19 CME using
all three in situ measurements and a dual-point fit of the 2020 May 29 CME for
SolO and Wind in local RTN coordinates. While we were capable of producing a
triple-point fit for the April event we did not achieve satisfactory results
for the May event. The dashed lines show one single representative fit from
the generated ensemble. The shaded area around this single fit shows the
1-$\sigma$ spread of all ensemble members. The vertical lines show the
selected fitting time points $t_{i}$ and the area in between the final $t_{s}$
and $t_{e}$ markers is shaded in light blue. For the adaptive time marker
scheme we used single hour offsets for the single point results, four hour
offsets for dual-point results and six hours for the triple point results. In
all cases the offset was reduced by one hour per iteration of the sequential
ABC-SMC algorithm.

The model initialization time $t_{0}$ is set to a specific date for both
events, the initialization distance $r_{0}$ is fixed to 35 solar radii, and
the initial speed $v_{0}$ and background solar wind speed $v_{s}w$ are bounded
within a smaller range. All other parameters are largely unbounded within a
sensible range. For the April event we set $t_{0}=$ 2020-04-15T23:00 UTC,
$v_{0}\in[350,750]$ and $v_{sw}\in[275,375]$. For the May event we use
$t_{0}=$ 2020-05-26T14:00 UTC, $v_{0}\in[300,650]$ and $v_{sw}\in[250,450]$.
The initialization times, distances and CME initial speeds are roughly
estimated from the remote observations from STEREO-A. Due to projection
effects from the images we can only estimate the lower bound for the speed.
The solar wind speed is taken from in situ data prior to the CME arrivals.

For the April event we largely used equidistant fitting points with a one hour
separation. While this was also possible for the SolO measurement for the May
event this was no longer the case at Bepi or Wind due to the strong
perturbations at the center. As can be seen in the Figure 5 we used an
extremely low number of points for these two measurements which are located
towards the endpoints of the flux rope signature.

The results for the April event show that we were generally capable of
reproducing all three measurements simultaneously with our model. The largest
discrepancy can be found within the SolO measurement where we are not able to
reproduce the sloped radial component $B_{R}$ and instead generate an almost
constant field as would be expected from a rigid flux rope structure. We also
do not find any issues with fitting the varying arrival times which is a
strong indication that the general assumptions on the propagation are more or
less valid within the vicinity of the three spacecraft. The small jumps that
can be seen in the representative ensemble members are artifacts that occur
due to our magnetic field model when combined with high aspect-ratios.

The results for the May event, despite only being a dual-point fit, are
significantly less accurate as can be seen alone from the spread. Since we
only took the SolO and Wind measurements into account we can see that the
arrival time at Bepi is incorrect with most of the ensemble runs arriving
later than expected. For the magnetic field we are more or less able to
reproduce the positive constant $B_{T}$ component although its strength is
underestimated in both cases, the bipolar $B_{N}$ component is overestimated
and the radial $B_{R}$ is far off from the actual measurement at SolO and more
or less accurate for Wind. On top of not being able to fully fit the magnetic
field we were also not able to find solutions with the correct duration of
passing, most ensemble members are significantly longer in duration than the
pre-defined UMFR interval. This shows that during the fitting the process the
adaptive offset for the $t_{s}$ and $t_{e}$ markers was not reduced to zero as
the iteration was aborted beforehand.

Figure 6 shows the 3DCORE shapes for representative ensemble members, using
the same model parameters as the representative fits in Figure 5, from a
side/top and a frontal viewpoint. The color codes for the plotted inner
planets and spacecraft are the same as in Figure 2. In both cases we show
three different fits, for the April event these are the individual SolO and
Wind fits and the combined triple-point fit. In the case for the May event we
show the individual SolO and Wind fit including the combined SolO + Wind fit.

Figure 6:  Representative 3DCORE geometries for multiple different in situ
combinations including the inner planets and satellites using the same color
codes as in Figure 2. The initial parameters for the representative ensemble
members are the same as for the representative in situ signatures in Figure
5.The two top panels show a view from the top while the bottom panels show a
view from the front.

Table 2 shows the inferred model parameters for the April and May event using
all seven different spacecraft combinations. The final row in each case shows
the final achieved error threshold $e^{k}$ for each observation. From these
tabular results we can see that the general solutions for the April event do
not geometrically change much when using different observations. A larger
difference can be found in the parameters that describe the CME kinematics
with the single point observations showing a large range of possible values
for $v_{0},˛v_{sw}$ and $\Gamma$. While the $\Gamma$ parameter seems to be
largely unconstrained in the multi point results the variations on the initial
CME velocity and solar wind background speed are reduced significantly showing
that these multi point observations, combined with our rigid CME geometry set
strong constraints for the kinematics.

The ensemble solutions for the May event show larger variations, specifically
concerning the orientation of the flux rope axis which is described by the
$inc$ parameter. The uncertainty in the kinematics does not significantly
change in between the single point and dual point scenarios showing that the
model has issues describing these observations. The same conclusion can be
drawn by looking at the $\epsilon_{\text{final}}$ parameters which show that
the achieved errors from the fitting procedure are significantly larger when
using dual point observations.

Table 2: 3DCORE parameter fitting results for the April (top) and May (bottom) event in HEEQ coordinates from Solar Orbiter (S), Wind (W) and BepiColombo (B). The $\epsilon_{\textrm{final}}$ values describes the last $\epsilon^{k}$ values that were achieved during the iterative fitting algorithm. The shown $\tau$ values were directly computed from the fitted $T_{\tau}$ (not shown) and $\delta$. |  | Units | SolO | Bepi | Wind | S + B | S + W | B + W | S + B + W
---|---|---|---|---|---|---|---|---|---
19 April 2020 | lon | [$deg$] | $16^{+2}_{-2}$ | $31^{+3}_{-2}$ | $26^{+2}_{-2}$ | $26^{+2}_{-2}$ | $23^{+2}_{-2}$ | $27^{+2}_{-2}$ | $25^{+2}_{-2}$
lat | [$deg$] | $0^{+6}_{-6}$ | $-11^{+4}_{-4}$ | $-9^{+3}_{-3}$ | $-8^{+5}_{-4}$ | $-8^{+4}_{-3}$ | $-9^{+3}_{-3}$ | $-8^{+4}_{-3}$
inc | [$deg$] | $-15^{+3}_{-3}$ | $-14^{+3}_{-3}$ | $-16^{+3}_{-3}$ | $-13^{+3}_{-2}$ | $-14^{+2}_{-2}$ | $-15^{+2}_{-2}$ | $-14^{+3}_{-3}$
$d_{1au}$ | [$au$] | $0.10^{+0.012}_{-0.011}$ | $0.09^{+0.007}_{-0.008}$ | $0.08^{+0.008}_{-0.008}$ | $0.09^{+0.006}_{-0.006}$ | $0.09^{+0.006}_{-0.006}$ | $0.09^{+0.005}_{-0.005}$ | $0.11^{+0.010}_{-0.010}$
$\delta$ |  | $9.53^{+1.85}_{-3.24}$ | $8.75^{+2.40}_{-3.64}$ | $9.11^{+2.27}_{-3.28}$ | $8.65^{+2.57}_{-5.01}$ | $9.42^{+1.89}_{-3.19}$ | $8.71^{+2.46}_{-4.35}$ | $9.11^{+2.18}_{-4.15}$
$v_{0}$ | [$\textrm{kms}^{-1}$] | $313^{+19}_{-9}$ | $455^{+138}_{-91}$ | $385^{+152}_{-51}$ | $318^{+18}_{-13}$ | $314^{+13}_{-10}$ | $348^{+79}_{-23}$ | $319^{+17}_{-14}$
$\tau$ |  | $-0.53^{+0.16}_{-0.21}$ | $-1.59^{+0.41}_{-1.02}$ | $-1.34^{+0.32}_{-0.72}$ | $-1.05^{+0.29}_{-1.37}$ | $-0.92^{+0.20}_{-0.47}$ | $-1.36^{+0.34}_{-1.26}$ | $-1.04^{+0.24}_{-0.80}$
$b_{1au}$ | [$nT$] | $13.31^{+0.50}_{-0.59}$ | $15.05^{+0.47}_{-0.49}$ | $15.10^{+0.53}_{-0.52}$ | $14.16^{+0.50}_{-0.47}$ | $14.18^{+0.47}_{-0.45}$ | $15.10^{+0.45}_{-0.46}$ | $14.36^{+0.43}_{-0.43}$
$\Gamma$ |  | $1.22^{+0.60}_{-0.72}$ | $1.25^{+0.53}_{-0.61}$ | $1.31^{+0.49}_{-0.59}$ | $1.30^{+0.48}_{-0.56}$ | $1.02^{+0.65}_{-0.51}$ | $1.31^{+0.48}_{-0.70}$ | $1.24^{+0.53}_{-0.54}$
$v_{sw}$ | [$\textrm{kms}^{-1}$] | $292^{+27}_{-13}$ | $315^{+29}_{-24}$ | $303^{+39}_{-17}$ | $365^{+7}_{-9}$ | $357^{+11}_{-11}$ | $331^{+33}_{-30}$ | $362^{+9}_{-11}$
$\epsilon^{k}_{\textrm{final}}$ |  | 0.19 | 0.15 | 0.16 | 0.33 + 0.20 | 0.29 + 0.21 | 0.19 + 0.20 | 0.32 + 0.22 + 0.21
29 May 2020 | lon | [$deg$] | $25^{+4}_{-3}$ | $20^{+9}_{-9}$ | $20^{+6}_{-6}$ | $20^{+5}_{-2}$ | $15^{+10}_{-3}$ | $33^{+4}_{-3}$ | -
lat | [$deg$] | $2^{+4}_{-3}$ | $3^{+12}_{-12}$ | $4^{+11}_{-12}$ | $2^{+21}_{-8}$ | $5^{+19}_{-5}$ | $14^{+14}_{-21}$ | -
inc | [$deg$] | $164^{+2}_{-2}$ | $162^{+14}_{-15}$ | $173^{+6}_{-7}$ | $171^{+8}_{-9}$ | $170^{+8}_{-10}$ | $174^{+8}_{-8}$ | -
$d_{1au}$ | [$au$] | $0.14^{+0.009}_{-0.014}$ | $0.11^{+0.020}_{-0.018}$ | $0.09^{+0.023}_{-0.017}$ | $0.13^{+0.013}_{-0.017}$ | $0.13^{+0.012}_{-0.014}$ | $0.11^{+0.022}_{-0.017}$ | -
$\delta$ |  | $8.36^{+2.56}_{-3.15}$ | $8.54^{+2.51}_{-3.88}$ | $8.18^{+2.98}_{-3.74}$ | $6.41^{+4.06}_{-2.51}$ | $4.89^{+5.39}_{-1.97}$ | $10.05^{+1.45}_{-3.28}$ | -
$v_{0}$ | [$\textrm{kms}^{-1}$] | $364^{+18}_{-11}$ | $621^{+95}_{-105}$ | $538^{+140}_{-105}$ | $588^{+112}_{-121}$ | $445^{+99}_{-69}$ | $682^{+52}_{-91}$ | -
$\tau$ |  | $1.10^{+0.61}_{-0.27}$ | $0.44^{+1.20}_{-0.92}$ | $0.63^{+0.87}_{-0.50}$ | $3.22^{+2.50}_{-1.97}$ | $2.76^{+1.72}_{-2.17}$ | $0.36^{+0.73}_{-0.47}$ | -
$b_{1au}$ | [$nT$] | $9.41^{+0.27}_{-0.28}$ | $10.14^{+2.58}_{-2.28}$ | $11.14^{+1.24}_{-1.15}$ | $7.51^{+2.00}_{-1.28}$ | $8.92^{+1.58}_{-1.35}$ | $9.80^{+1.51}_{-1.48}$ | -
$\Gamma$ |  | $1.82^{+0.14}_{-0.22}$ | $0.93^{+0.73}_{-0.56}$ | $1.05^{+0.69}_{-0.65}$ | $1.03^{+0.61}_{-0.57}$ | $1.16^{+0.63}_{-0.65}$ | $0.50^{+0.64}_{-0.22}$ | -
$v_{sw}$ | [$\textrm{kms}^{-1}$] | $253^{+4}_{-2}$ | $413^{+27}_{-47}$ | $387^{+45}_{-49}$ | $423^{+19}_{-35}$ | $407^{+33}_{-34}$ | $401^{+36}_{-59}$ | -
$\epsilon^{k}_{\textrm{final}}$ |  | 0.13 | 0.38 | 0.17 | 0.72 + 0.68 | 0.64 + 0.48 | 0.62 + 0.48 | -

Figure 7 shows a scatter plot for the twist and aspect-ratio solutions from
the overall ensembles of the April event. In this plot we see that the
$\delta-\tau$ degeneracy persist despite the high bias towards higher $\delta$
values. As previously explained in Section 4.3 this can be explained due to a
purely geometrical effect. The solid lines show the
$\tau\propto\delta/E(\delta)$ trend for each single case. Unfortunately this
means that one of the key issues in W21 remains unresolved. We are not able to
solve the $\delta-\tau$ degeneracy using dual or triple point observations.
Taking a closer look at the spacecraft positions in the April event in Figure
6 we are able to see that all three spacecraft are at a similar distance below
the inferred flux rope axis. This furthermore reduces the amount of
information that can be extracted on the aspect ratio within our magnetic
field model.

Figure 7:  Scatter-plot showing the $\tau$-$\delta$ degeneracy for the April
event with the SolO, Bepi and Solo+Bepi+Wind ensembles. The solid lines show
the $\tau\propto\delta/E(\delta)$ trend.

Lastly we can also compare some of the results to the remote observations
shown in Section 2.1. We find that in general the inferred directions match
the given values from the HELCATS catalog very well. For the April event we
find a HEEQ longitude and latitude of $16$ to $31\deg$ and $0$ to $-10\deg$
where the estimated value is $-2/-6$ from the images. For the May event we
infer a HEEQ logitude and latitude of $15$ to $33\deg$ and $2$ to $14\deg$
where the estimated value is $+36/+11$.

##  6 Conclusion

As a summary, we have shown in this paper that it is possible to reconstruct
multiple in situ flux rope measurements simultaneously using an empirical CME
MFR model. We chose two different scenarios to test our approach, with the
first (the April event) consisting of three separate close by measurements and
the second (the May event) consisting of three measurements at larger
longitudinal separations.

In the first case, with the close by measurements, we do not find any major
issues in our attempt to reconstruct the magnetic field measurements using our
model. The seven different combinations for reconstruction show more or less
the same picture further reinforcing the validity of our description on
smaller distance scales.

The second case illustrates the limits of our modeling approach as we are no
longer able to consistently describe all three observations simultaneously.
This can be attributed to large longitudinal separations of the spacecraft
which allows for a vastly different solar wind environment at the different
locations. The two measurements for Wind and Bepi are strongly perturbed which
may be a hint that they only sample the CME structure at the very edge. This
hypothesis can be further strengthened when looking at the multi point
reconstruction from Figure 6 which shows that with our assumed geometry these
two spacecraft are also on the edges.

The problem of estimating the cross-section aspect ratio and the twist has not
been solved by using multi point observations. Due to the constellation of the
April event, where the model depicts all the spacecraft to lie below the flux
rope axis at similar distances, it is still not entirely clear if this problem
will persist for other multi point events or is inherent. The specific
magnetic field model that is used in our analysis may prove problematic in
this aspect. Unless we are given clear observations of a crossing both below
and above the flux rope axis, i.e. a positive and negative unipolar $B_{r}$
signature we believe that we will not be able to resolve the degeneracy of the
cross-section aspect ratio and the twist number when using our model. Further
studies using different magnetic field models, specifically models that are
more physical and include a proper description elliptical cross-sections (e.g.
Nieves-Chinchilla et al., 2018), may resolve this problem or at least reduce
the uncertainties.

In general we believe that our approach works well as a first order
approximation for multi point observations in close proximity. Due to the
different solar wind environments that can be present over longitudinal
separations larger than ca. 10 $\deg$ our model will be limited due to not
properly being able to describe the arrival times and flux rope orientation at
varying positions. The next obvious step would be the construction of more
complex and computationally expensive models that include deformations due to
the solar wind environment. This will invariably require a proper 3D
description of the solar wind within at least 1 AU and a CME model with a de-
formable front. As our approach is, in principal, valid for any forward based
simulation model the entire fitting pipeline can be re-used for this newer
class of models. Our current model can also be used as a first step in feeding
a more complicated model with initial conditions in order to limit the volume
of the parameter space.

The April event represents one of the best multi point observations so far
recorded. New missions have been successfully launched within the last years,
including BepiColombo, Solar Orbiter and Parker Solar Probe. Expected launches
of other missions such as JUICE combined with the rising solar activity of
cycle 25 should allow us to observe many additional multi point events within
the next decade. These observations will be important for testing our existing
approach and future studies in order to develop models that can better
describe the global ICME structure.

###### Acknowledgements.

A.J.W, C.M., R.L.B, M.A.R., T.A., M.B. and J.H. thank the Austrian Science
Fund (FWF): P31521-N27, P31659-N27, P31265-N27. We have benefited from the
availability of Solar Orbiter, STEREO, Wind, and BepiColombo data, and thus
would like to thank the instrument teams, and the SPDF CDAWeb and Solar
Orbiter (http://soar.esac.esa.int/soar/) data archives for their distribution
of data. The Solar Orbiter magnetometer was funded by the UK Space Agency
(grant ST/T001062/1). This research was supported by funding from the Science
and Technology Facilities Council (STFC) studentship ST/N504336/1 (E.D.) and
STFC grant ST/S000361/1 (T.H.). D.H., I.R., H.-U. A. were supported by the
German Ministerium für Wirtschaft und Energie and the German Zentrum für Luft-
und Raumfahrt under contract 50 QW 1501. The open source code for the 3DCORE
model is available at https://github.com/ajefweiss/py3dcore, and the heliosat
package to download and process spacecraft data is accessible at
https://github.com/ajefweiss/heliosat.

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