# Improving the Arrival Time Estimates of Coronal Mass Ejections by Using
Magnetohydrodynamic Ensemble Modeling, Heliospheric Imager data, and Machine
Learning

Talwinder Singh  Center for Space Plasma and Aeronomic Research, The
University of Alabama in Huntsville, AL 35805, USA  Bernard Benson  McLeod
Software Corporation, Birmingham, AL 35242, USA  Syed A. Z. Raza  Department
of Space Science, The University of Alabama in Huntsville, AL 35805, USA  Tae
K. Kim  Center for Space Plasma and Aeronomic Research, The University of
Alabama in Huntsville, AL 35805, USA  Nikolai V. Pogorelov  Center for Space
Plasma and Aeronomic Research, The University of Alabama in Huntsville, AL
35805, USA  Department of Space Science, The University of Alabama in
Huntsville, AL 35805, USA  William P. Smith  Department of Physics, Brown
University, RI 02912, USA  Charles N. Arge  Solar Physics Lab, NASA/GSFC,
Greenbelt, MD 20771, USA

###### Abstract

The arrival time prediction of Coronal mass ejections (CMEs) is an area of
active research. Many methods with varying levels of complexity have been
developed to predict CME arrival. However, the mean absolute error (MAE) of
predictions remains above 12 hours, even with the increasing complexity of
methods. In this work we develop a new method for CME arrival time prediction
that uses magnetohydrodynamic simulations involving data-constrained flux-
rope-based CMEs, which are introduced in a data-driven solar wind background.
We found that, for 6 CMEs studied in this work, the MAE in arrival time was
$\sim$8 hours. We further improved our arrival time predictions by using
ensemble modeling and comparing the ensemble solutions with STEREO-A&B
heliospheric imager data. This was done by using our simulations to create
synthetic J-maps. A machine learning (ML) method called the lasso regression
was used for this comparison. Using this approach, we could reduce the MAE to
$\sim$4 hours. Another ML method based on the neural networks (NNs) made it
possible to reduce the MAE to $\sim$5 hours for the cases when HI data from
both STEREO-A & B were available. NNs are capable of providing similar MAE
when only the STEREO-A data is used. Our methods also resulted in very
encouraging values of standard deviation (precision) of arrival time. The
methods discussed in this paper demonstrate significant improvements in the
CME arrival time predictions. Our work highlights the importance of using ML
techniques in combination with data-constrained magnetohydrodynamic modeling
to improve space weather predictions.

##  1 Introduction

Coronal Mass Ejections (CMEs) are coherent structures of plasma and magnetic
field that originate from the Sun and propagate in to the interplanetary space
through the ambient solar wind (SW). CMEs are major sources of geomagnetic
storms at Earth, which have several undesirable consequences for many space-
and ground-based technologies. Throughout the years, numerous observatories
and instruments, both on the ground and in space, have assisted us in
comprehending the connection between the Sun and Earth. Many CME propagation
models have been developed over the years using the data from these
observatories to provide CME arrival time forecasts. They include the
empirical models (e.g. Vandas et al., 1996; Brueckner et al., 1998; Gopalswamy
et al., 2001, 2005; Wang et al., 2002; Manoharan et al., 2004), drag based
models (Vršnak, 2001; Vršnak and Gopalswamy, 2002), and also such physics-
based models as Shock Time of Arrival (STOA) (Dryer and Smart, 1984), STOA-2
(Moon et al., 2002), and HAFv.2 (Fry et al., 2001). Newer-generation
prediction models are based on magnetohydrodynamic (MHD) simulations which use
over-pressurized blobs to initiate the CME propagation (e.g.,
Wang–Sheeley–Arge (WSA)-ENLIL-Cone model, Emmons et al., 2013)). All these
models result in arrival time mean absolute errors (MAEs) of more than 10
hours. Riley et al. (2018) compared several CME arrival time prediction models
and found that even the best performers had MAEs of 13 hours and standard
deviation of 15 hours. It was also shown that the arrival time accuracy of the
models is not improving with the model complexity (Riley et al., 2018). This
has been demonstrated by other studies (e.g., Gopalswamy et al., 2013), which
showed that the performance of the ENLIL-cone model is similar to that of the
much simpler empirical shock arrival model (Gopalswamy et al., 2005). In
addition, all such models cannot predict the north-south magnetic field
component ($B_{z}$) in the CME at Earth because they ignore the development of
the associated magnetic flux ropes.

CMEs disturb the ambient SW during their travel through the inner heliosphere
(IH). Reciprocally, the SW can impact the kinetic and magnetic properties of
CMEs. Understanding this CME-SW interaction is important for accurate
forecasting of the arrival time of CMEs. With the availability of Solar
Terrestrial Relations Observatory (STEREO) Heliospheric Imager (HI) data
(Eyles et al., 2009), it became possible to track the kinematics of CMEs up to
large distances from the Sun (Liu et al., 2010; Lugaz et al., 2010) and study
the CME-SW interaction with more detail. HI data have been used to develop
several arrival time forecasting methods. These methods track CME fronts in
the ecliptic plane using the HI data. The pixels corresponding to the ecliptic
plane are identified in the running-difference HI images. These pixels are
arranged in a vertical formation over a specified period of time, such that
time forms the horizontal axis, to create time-elongation maps, also called
J-maps. CME fronts reveal themselves as diagonal bright streaks in J-maps. To
calculate the distance the CME has traveled along the Sun-Earth line using
these J-maps, several models have been developed such as the fixed $\phi$
(Kahler and Webb, 2007), harmonic mean fitting (Lugaz et al., 2009),
triangulation with harmonic mean (HM, Lugaz et al., 2010), and self-similar
expansion (SSE, Davies et al., 2012) models. All of them use very simple
assumptions about the shape of a CME. Both HM and SSE models assume CMEs to
have a spherical shape, with the sphere either connected to the solar center
(in the HM model) or expanding and propagating in the IH (in the SSE model).
Forecasting methods that use these models usually extract CME time-elongation
data from the J-maps to where the bright fronts are traceable and then use
drag-based models to forecast arrival time at Earth. These methods, however,
have not been able to surpass the accuracy of such models as the WSA-ENLIL-
Cone model, which use only the coronagraph data to forecast CMEs. Simplistic
assumptions about initial CME shapes in the IH and ignorance of the CME–SW
interactions, which can distort their shapes during propagation can be a
reason for this (Liu et al., 2006; Riley et al., 1997; Manchester IV et al.,
2004).

The EIEvoHI model, first described by Rollett et al. (2016) assumes a more
flexible CME shape while converting the CME-front elongation angle to the
distance a CME has traveled along the Sun-Earth line. That model assumes CMEs
to have elliptic shape with a flexible ellipse aspect ratio. It also tracks
CME fronts in HI data to a particular distance and uses a drag-based model to
find the CME arrival time at Earth. Ensemble modeling with EIEvoHI has been
shown to be a promising method of CME arrival time predictions (Amerstorfer et
al., 2018). A recent study by Hinterreiter et al. (2021a) reported MAE of 7.5
hours and a standard deviation of 9.5 hours by hind-casting 12 CMEs using
EIEvoHI. EIEvoHI is only suitable for forecasting the arrival time and speed
of CMEs at Earth. It does not consider the associated flux ropes, so it is
unable to estimate the magnetic field carried by a CME towards Earth.

MHD modeling can reproduce realistic SW-CME interaction and eliminate the
necessity of making assumptions about the CME shape in the IH. It provides us
with a promising tool for forecasting the properties of the SW and CME plasma,
and magnetic field at Earth. This requires that the solar wind is driven by
data and CMEs simulated on the basis of flux rope models (Jin et al., 2017;
Scolini et al., 2019; Singh et al., 2020a, 2022). Flux rope models are usually
constrained by CME observations in coronagraphs. Scolini et al. (2019) and
Singh et al. (2019) constrained the poloidal flux in their models using the
reconnected flux in the source active regions. Singh et al. (2022) introduced
a constant turn flux rope model and used it to simulate the 12 July 2012 CME.
This model can be constrained by the CME direction, tilt, speed, half-angle,
aspect ratio, magnetic flux, and helicity sign. Singh et al. (2022) reproduced
the plasma and magnetic field properties of this CME at Earth with good
accuracy. They also described a method to create ensemble members for this
flux-rope-based CME and used them to study the role of observational
uncertainties in the CME properties obtained at Earth with numerical
simulations. HI observations can be used to compare the evolution of different
ensemble members with the actual CME behavior. This type of comparison was
done earlier by Barnard et al. (2020) to improve the arrival time accuracy of
their CME model.

In this work, we use the method described by Singh et al. (2022) to perform an
MHD, ensemble modeling of six CMEs. We propose a novel approach to comparing
our ensemble members with observations in the IH using ML methods. We show
that incorporating HI data in MHD forecasting methods can substantially
improve the arrival time predictions. We describe the data used in this work
in Section 2. In Section 3, we describe the SW model and CME models used in
this study. We show our results in Section 4, which is followed by our
conclusions in Section 5.

##  2 Data Used

In this work we analyze six CMEs listed in Table 1. For each CME, we use the
level 0.5 FITS data from the Sun-Earth Connection Coronal and Heliospheric
Investigation (SECCHI)/Cor2 (Howard et al., 2008) coronagraphs in Solar
Terrestrial Relations Observatory (STEREO) A & B (Kaiser et al., 2008). These
files are processed to level 1 using the secchi_prep program in IDL’s
SolarSoft library which converts the units from data numbers (DN) into Mean
Solar Brightness (MSB). We also use level 1 FITS data from Large Angle
Spectroscopic Coronagraph (LASCO)/C2/C3 (Brueckner et al., 1995) coronagraph
in the Solar and Heliospheric Observatory (SOHO) spacecraft. The coronagraph
data are used to estimate the direction, tilt, half-angle, aspect ratio, and
speed of the CMEs. We use long-term background-subtracted data from HI 1 & 2
(Eyles et al., 2009) onboard STEREO A & B to track the CMEs in the IH. We also
use level 2 HI data which are already corrected for cosmic rays, shutterless
readout, saturation effects, flat fields, and instrument offsets from
spacecraft pointing. We also use the solar extreme ultraviolet (EUV) and line
of sight (LOS) magnetic field observations from the Solar Dynamics Observatory
(SDO) Atmospheric Imaging Assembly (AIA) (Pesnell et al., 2012) and
Helioseismic and Magnetic Imager (HMI) (Schou et al., 2012; Hoeksema et al.,
2014) instruments to study, respectively, the source regions of the six CMEs.
We use aia_prep program in SolarSoft to read in and calibrate AIA level 1 data
to level 1.5. Level 1 data includes bad-pixel removal, de-spiking, and flat-
fielding. Level 1.5 data additionally has roll correction that makes the solar
north in the images vertical, re-scales the images to $0^{\prime\prime}.6$
pixels, and performs translation that puts the solar disk center at the image
center. The hmi_prep program in SolarSoft is utilized to read HMI data. This
program processes the magnetograms to guarantee the proper roll angle and
adjusts their placement so that the center of the solar disk aligns with the
centers of the magnetograms. AIA and HMI data of the source active regions are
used to estimate the magnetic flux and helicity sign of the six CMEs. The
methods used to estimate these values are described in Section 4. One hour
averaged data, provided by NASA/GSFC’s OMNI data through OMNIWeb (King and
Papitashvili, 2005), are used to compare the in situ measurements at Earth
with our simulations. All CMEs we study in this work are Earth-directed and
show magnetic cloud signatures at Earth. The CME speeds near the Sun are in a
wide range between 500 and 1500 km/s.

##  3 Simulation Models

We perform MHD simulations using the Multi-Scale Fluid-Kinetic Simulation
Suite (MS-FLUKSS) in this study. MS-FLUKSS is a collection of modules capable
of performing adaptive mesh refinement (AMR) simulations of the SW in the
presence of neutral atoms, non-thermal ions, turbulence, etc. (Pogorelov et
al., 2014; Fraternale et al., 2021). This code is highly parallel, allowing us
to perform simulations much faster than in real time. We describe the SW and
CME models used in this study in the following subsections.

###  3.1 Solar wind model

The IH model implemented in MS-FLUKSS solves the ideal MHD equations with the
finite-volume, total variation diminishing (TVD) approximations of
conservation laws on non-uniform spherical grids. For this study, we chose the
grid with 150, 256, and 128 cells in the $r$, $\phi$, and $\theta$ directions,
respectively. We set the inner boundary of our domain at 0.1 AU, where the SW
has already attained super-fast magnetosonic speed. The outer boundary is set
at 1.5 AU. The inner boundary conditions are provided as a time-series of WSA
solutions (e.g., Kim et al., 2020). The WSA results at this height are
obtained using the Air Force Data Assimilative Photospheric Flux Transport
(ADAPT) synchronic maps based on the NSO/GONG magnetograms (Arge et al., 2010,
2011, 2013; Hickmann et al., 2015) as input at the solar surface. The WSA
model employs the potential field source surface (PFSS) model to extend the
photospheric magnetic field to a spherical source surface located at 2.5
$R_{\odot}$ and utilizes the Schatten current sheet model (PFCS, Schatten,
1972) to take it even further to the outer boundary of the WSA at 0.1 AU. The
SW speed at the WSA outer boundary is calculated as a function of the flux
expansion factor and distance to the nearest coronal hole boundary (Arge et
al., 2003, 2005; McGregor et al., 2011). SW density and temperature at 0.1 AU
are estimated by using the empirical correlations between the SW speed,
density, and temperature based on the OMNI data (Elliott et al., 2016). The
process of using WSA boundary conditions in the IH model is the same as
described in Singh et al. (2020a) and Singh et al. (2022). ADAPT-WSA provides
12 realizations that can be used as the inner boundary in our IH model. To
perform our CME simulations, we use the best ADAPT-WSA realization based on
their comparison with near-Earth SW data.

Figure 1: The initial structure of the constant turn flux rope used to
simulate the 12 July 2012 CME (case 5). The speed at the apex is 1097 km/s,
and the poloidal and toroidal magnetic fluxes are set to $1.4\times 10^{22}$
Mx and $7.6\times 10^{21}$ Mx respectively. The flux rope is inserted into the
IH at the physical time of 13 July 2012 at 03:33. The apex of the flux rope is
at $70R_{\odot}$, while its latitude, longitude, and tilt are
$\mathbf{-10.1^{\circ}}$, $\mathbf{0.7^{\circ}}$, and $\mathbf{52.0^{\circ}}$
respectively. The semi-translucent slice by the plane containing the flux rope
axis shows the radial velocity. Magnetic field lines are shown with black
lines. The red sphere indicates the inner boundary of the IH model at 0.1 AU.

###  3.2 CME model

In this work, we use a constant turn flux rope model described in Singh et al.
(2022) to simulate CMEs. This model assumes the initial CME structure to be
croissant-like with a circular cross-section and two legs rooted at the center
of the Sun. This shape is based on FRiED model geometry described by Isavnin
(2016). The magnetic field of this flux rope model is based on the analytic
solutions provided by Vandas and Romashets (2017). The constant turn flux rope
can be initialized in any direction defined by the apex latitude and
longitude. The flux rope can have any tilt w.r.t. the solar equatorial plane.
Moreover, this flux rope can have desired half-angle and aspect ratio. The
magnetic field in the flux rope can be initialized to have desired poloidal
and toroidal fluxes, and helicity sign. The helicity sign can take the +1 or
-1 values and describes the direction of the twist of magnetic field lines in
the flux rope, with the +1 (-1) sign representing the direction described with
the right- (left-) hand rule. The velocity of plasma inside the flux rope is
initialized to have desired value at the apex and a self-similar expanding
profile (Singh et al., 2022). We can introduce a uniform-density plasma inside
this flux rope. Singh et al. (2022) showed that the constant turn flux rope
model is suitable for simulating CMEs in the IH. It reproduced the CME
magnetic field, density, and velocity reasonably well at Earth for the 12 July
2012 CME, when this model was constrained with the observed direction, tilt,
half-width, aspect ratio, speed, mass, helicity sign, and magnetic flux. These
properties of the CMEs can be obtained using various observational data as
discussed in Singh et al. (2020b). This CME is also included in the current
work as case 5.

Flux ropes are inserted into the ambient SW in such a way that the former are
initially superimposed with the SW background (Singh et al., 2020a). When a
flux rope is superimposed with the background, we replace the SW magnetic
field and velocity with those in the flux rope. During this superimposition,
The density and internal energy density of the flux rope are added to the
values in the ambient SW density and internal energy. Figure 1 shows the
initialized flux rope for the simulation of 12 July 2012 CME. The properties
of this flux rope are summarised in Table 1. We will describe the method of
constraining this model with observations in Section 4.1.

##  4 Results

###  4.1 Constraining initial CME parameters with data

The CME properties that we need to constrain our flux rope are its direction,
tilt, speed, half-angle, aspect ratio, mass, poloidal and toroidal magnetic
fluxes, and the magnetic helicity sign. These properties for our six cases are
summarized in Table 1. These quantities are estimated using different
observations as discussed below.

We can estimate the CME direction, tilt, speed, half-angle, aspect ratio, and
speed using multi-viewpoint observations of STEREO-A & B COR2 and SOHO C2/C3
coronagraphs. A suitable model for utilizing these multi-viewpoint
observations to find 3D CME evolution is the Graduated Cylindrical Shell (GCS)
model (Thernisien, 2011). This model fits a croissant shape with a curved
front and conical legs over a CME as seen in the coronagraph field of view
(FOV). Table 1 shows the fitted latitude, longitude, tilt angle from the solar
equatorial plane, half-angle, and aspect ratio for the six CMEs we have
considered in this study. Each time series of GCS fitted heights gives us the
CME speed via linear regression (see e.g. Hess and Zhang, 2014). This value
has also been reported in Table 1. It should be noted that the GCS model can
work only when a CME is in the coronagraph FOV. However, we insert our flux
rope into IH when its apex is already at $70R_{\odot}$, by which time the CMEs
have already left the STEREO and SOHO coronagraph FOVs. We assume a self-
similar expansion of CMEs between the time we are last able to fit the GCS
model and when the CME apex should reach $70R_{\odot}$ according to the drag-
based model (DBM) (Vršnak and Zic, 2007). We used the drag parameter of
$10^{-8}$ ${km^{-1}}$ and the asymptotic SW speed equal to 450 km/s for all
our cases. Vršnak et al. (2014) found that this combination of SW speed and
drag parameter ensures roughly the same arrival time accuracy for the DBM and
WSA-ENLIL-Cone models. DBM also estimates the CME speed at $70R_{\odot}$ (see
Table 1). We superimpose each flux rope with the SW background at this
DBM–estimated time and assume the DBM–estimated speed at the apex.

The total mass of a CME can be found using the CME brightness in the white
light coronagraph images. The brightness of a CME in coronagraphs is due to
the Thomson scattering of photospheric light by the CME plasma electrons
(Billings, 1966). We can calculate the true mass of a CME by integrating the
brightness over the CME area in coronagraph images and removing the projection
effects, which is done with the method based on the multiple coronagraph
viewpoints (Colaninno and Vourlidas, 2009). These true masses of the six CMEs
under consideration are given in Table 1. In this study, we uniformly
distribute 0.1% of this mass inside the flux rope when initializing it. The
motivation behind using this factor is based on the investigation of Singh et
al. (2022) who showed that this results in a reasonable reproduction of the
observed plasma density at Earth for 12 July 2012 CME. The physical
interpretation of this factor being so small is that only the flux-rope part
of a CME is inserted into the ambient SW in our simulation. The coronagraph
observations show that flux rope resides in the cavity region of a CME and has
density much lower than those in the sheath and core, which contribute the
majority of mass to CMEs (Riley et al., 2008).

We estimated the poloidal magnetic flux of our CMEs by using the reconnected
flux in post-eruption arcades (PEAs) in the source active regions (Gopalswamy
et al., 2018). This technique necessitates EUV observations when the PEA
structure has reached its full development, usually during the decline stage
of the flare.. Gopalswamy et al. (2017) showed that the poloidal flux of CME
flux ropes is equal to one half of the unsigned flux in the area covered by
the PEAs at this time. In this paper, we estimate the toroidal fluxes of our
CMEs by using an analytic relation between the poloidal and toroidal magnetic
fluxes in CME flux ropes given by Qiu et al. (2007). The poloidal and toroidal
fluxes found by using these methods for our six CMEs are given in Table 1. The
toroidal flux can also be estimated by using the coronal dimming area in the
EUV observations at the time of eruption. However, coronal dimming is not
apparent for many CME eruptions, so the toroidal flux estimates based this
method are possible only for a limited number of CMEs. Therefore, we opted to
use the analytic relationship by Qiu et al. (2007). The constant turn flux
rope model can be initialized with desired poloidal and toroidal fluxes. This
is an advantage of this model compared to the spheromak model where the
poloidal and toroidal fluxes cannot be set independently.

The sign of the helicity in a CME flux rope can be established by examining
the pre-eruptive magnetic field configuration of the active region, as
reported by Bothmer and Schwenn (1998). For instance, Luoni et al. (2011)
illustrate how the magnetic tongues present in active regions can be used to
determine the helicity sign of the flux ropes above them. We use this method
to find the helicity sign for our six cases (see Table 1). The constant turn
flux rope model can be initialized to have either right-handed or left-handed
magnetic field turns based on the helicity sign found by using this method.
The hemispheric helicity rule can also be used to determine the helicity sign
of the ARs, which states that ARs in the solar northern (southern) hemisphere
have negative (positive) helicity. However, Liu et al. (2014) showed that this
rule did not apply to about a quarter of the 151 ARs studied by them. We found
that all six CMEs discussed here follow the hemispheric helicity rule.

Table 1: Properties of the six CMEs discussed in this study. CME # |  1 |  2 |  3 |  4 |  5 |  6
---|---|---|---|---|---|---
Date |  2010/08/01 08:00 |  2011/09/06 23:05 |  2011/09/13 22:10 |  2012/01/19 14:36 |  2012/07/12 16:48 |  2012/09/28 00:00
Latitude |  17.9 |  26.8 |  21.2 |  43 |  -10.1 |  11.7
Longitude |  -37.9 |  40 |  17.6 |  -30.4 |  0.7 |  31.3
Tilt |  75.5 |  16.2 |  -11.2 |  -57 |  52 |  -78.3
Half angle |  35 |  55 |  30 |  43 |  33 |  40
Ratio |  0.5 |  0.5 |  0.5 |  0.5 |  0.5 |  0.5
Linear speed |  1441 |  746 |  547 |  1214 |  1267 |  976
DBM speed |  1218 |  708 |  541 |  1061 |  1097 |  887
Mass ($\times 10^{15}$ g) |  3.8 |  10.3 |  4.5 |  18.1 |  16.5 |  9.6
Poloidal Flux ($\times 10^{21}$ Mx) |  1.7 |  2.9 |  3.6 |  3.7 |  14.0 |  0.9
Toroidal flux ($\times 10^{21}$ Mx) |  0.6 |  1.2 |  1.5 |  1.6 |  7.6 |  0.3
Helicity |  -1 |  -1 |  -1 |  -1 |  1 |  -1

###  4.2 Ensemble modeling

The observations described in the previous subsection are not free from
uncertainties. These uncertainties can be due to measurement errors, model
assumptions, and subjective errors. Singh et al. (2022) focused on the
uncertainties in the GCS fitting parameters and found the subjective
uncertainties in CME latitude, longitude, tilt, and speed by comparing the
fitting results of 56 CMEs reported in multiple studies and catalogs. They
found that the GCS estimates of the CME latitude, longitude, tilt, and speed
have average uncertainties of $5.7^{\circ}$, $11.2^{\circ}$, $24.7^{\circ}$,
and 11.4% respectively. Thernisien et al. (2009) have reported the average
uncertainties in aspect ratio and half-angle GCS parameters as $+0.07/-0.04$
and $+13^{\circ}/-7^{\circ}$ respectively. Using the GCS uncertainties, Singh
et al. (2022) created 77 ensemble members of the 12 July 2012 CME and analyzed
how the GCS uncertainties can create a spread in the simulated CME properties
at Earth.

In this study, we similarly create 77 ensemble members for each of our six
cases. The first member is called the seed ensemble member and the other 76
are created by introducing uncertainties to its GCS parameters in various
combinations. The seed ensemble member with the GCS properties of latitude
$\theta$, longitude $\phi$, tilt $\gamma$, speed $V_{\mathrm{CME}}$, half-
angle $\alpha$, and aspect ratio $\kappa$ can be represented by a set
$\\{\theta,\phi,\gamma,V_{\mathrm{CME}},\alpha,\kappa\\}$. The other 76
ensemble members are created by introducing the average GCS uncertainties to
these properties. These members are represented by the following sets:

  1. 1.

$\\{\theta\pm 5.7^{\circ},\phi\pm 11.2^{\circ},\gamma\pm
24.7^{\circ},V_{\mathrm{CME}}\pm
11.4\%,\alpha_{-7^{\circ}}^{+13^{\circ}},\kappa_{-0.04}^{+0.07}\\}$, (64
members)

  2. 2.

$\\{\theta\pm 5.7^{\circ},\phi,\gamma,V_{\mathrm{CME}},\alpha,\kappa\\}$ (2
members)

  3. 3.

$\\{\theta,\phi\pm 11.2^{\circ},\gamma,V_{\mathrm{CME}},\alpha,\kappa\\}$ (2
members)

  4. 4.

$\\{\theta,\phi,\gamma\pm 24.7^{\circ},V_{\mathrm{CME}},\alpha,\kappa\\}$ (2
members)

  5. 5.

$\\{\theta,\phi,\gamma,V_{\mathrm{CME}}\pm 11.4\%,\alpha,\kappa\\}$ (2
members)

  6. 6.

$\\{\theta,\phi,\gamma,V_{\mathrm{CME}},\alpha_{-7^{\circ}}^{+13^{\circ}},\kappa\\}$
(2 members)

  7. 7.

$\\{\theta,\phi,\gamma,V_{\mathrm{CME}},\alpha,\kappa_{-0.04}^{+0.07}\\}$ (2
members)

Ensembles for each of our six CMEs consist of 12 realizations provided by
ADAPT-WSA. These are used as boundary conditions to obtain the SW properties
at Earth. To perform a CME simulation, we choose a realization that provides
the best performance at Earth for the ambient SW. Currently we use a manual
approach to select the best-performing ADAPT-WSA ensemble member. For each of
our six CMEs, we conduct 77 simulations by initializing the flux ropes using
the CME properties listed in Table 1 for the simulation of the seed ensemble
member and using the method described previously to create the remaining 76
ensemble members. In this work, we simulate ensemble members with apex speeds
at $70R_{\odot}$ as low as 480 km/s and as high as 1357 km/s. The simulated
CMEs had poloidal fluxes ranging between $0.9\times 10^{21}$ Mx and
$14.0\times 10^{21}$ Mx. We simulate ensemble members that have CME half
angles ranging between $23^{\circ}$ and $68^{\circ}$. This demonstrates the
ability of our constant turn flux rope model to simulate CMEs with a wide
range of speeds, magnetic fluxes, and widths.

Figure 2 shows the CME properties at Earth for all six cases. Each panel has
been tagged with the CME number given in Table 1. All panels show the magnetic
field components in the radial-tangential-normal (RTN) coordinates, density,
and radial bulk velocity. The RTN system is based on the position of Earth and
is aligned with the line connecting the Sun and the Earth. The R axis, which
is directed radially away from the Sun and passes through Earth, is the
primary axis. The T axis, which is perpendicular to the R axis and parallel to
the solar equatorial plane, is determined by the cross product of the Sun’s
spin vector and the R axis. The T axis is positive in the direction of the
planet’s rotation around the Sun. The N axis completes the right-handed set of
axes. The blue and black lines in each graph show the OMNI data, and our
simulation results at Earth for the seed ensemble member, respectively. The
green lines are the simulation results for the rest of ensemble members. One
can see that the uncertainties in GCS parameters can introduce quite large
spreads in CME magnetic field, density, and velocity at Earth. The vertical
gray line represents the arrival time of the observed CME. The dotted red
lines show the arrival time of the earliest and latest arriving ensemble
members. The spread of arrival times range from as low as 12 hours in case 5
to as large as 46 hours in case 4. The arrival time of the simulated CME
ensemble members is determined by comparing the simulated density at Earth
between a simulation with an initialized CME and a simulation without it. The
arrival time is defined as the point at which the density in the CME-
initialized simulation surpasses 10% of the density in the no CME-initialized
simulation. The accuracy of this method is confirmed through visual inspection
of the simulated density at Earth.

In Cases 3, 4, and 5, the CME density at Earth is reproduced reasonably well.
The results for cases 1 and 6 considerably overestimate densities at Earth. In
case 2, our simulations were not able to reproduce the observed large density
in the CME sheath region. The reason for disagreements in Cases 1,2, and 6 can
be that we insert only 0.1% of the total observed CME mass into the flux rope.
The effect of this choice on the simulated CME densities at Earth needs to be
further investigated. The plasma speeds in the simulated CMEs show a
reasonable agreement with observations in all considered cases. The ensemble
members show quite a large arrival time window for all six cases. For example,
the arrival time window of the 12 July 2012 CME (case 5) is 12 hours wide. The
rest of the 5 CMEs also had arrival time windows wider than 10 hours. In this
study, we focus on the improvement of the arrival time errors. This is done by
comparing the results obtained using our ensemble members with the STEREO
heliospheric imager data.

Since our CME model involves a flux rope, the modifications our simulations
impose onto the magnetic field at Earth resemble magnetic clouds with smoothly
rotating magnetic field components. A visual inspection shows that our model
underestimates the magnetic field magnitude at Earth for cases 1, 2, 4, and 6.
Magnetic field magnitude was reproduced well in cases 3 and 5. Direction of
the CMEs in both these cases was most aligned with the Sun-Earth line (see
latitude and longitude values in Table 1. This suggests that our model
underestimates magnetic field at its flanks. One of the reasons for this can
be due to the reconnection of the cloud magnetic field with the SW magnetic
field. This effect deserves further investigation which will be done in the
future. Our ensemble modeling shows that the GCS uncertainties can introduce
large changes in the CME magnetic field at Earth. CME magnetic field magnitude
and direction can have a large spread in ensemble simulations. It should be
noted that the range is only due to the uncertainty in GCS parameters that are
used to constrain the constant turn flux rope model. If additional ensemble
members were to be created to represent the uncertainty in poloidal and
toroidal magnetic fluxes that are used to constrain the model, we could expect
to see even a larger spread in the simulated magnetic field at Earth. This
demonstrates significant difficulties in the implementation of the models,
such as ours, for forecasting the CME magnetic field at Earth. However, we can
try to weigh the individual ensemble members using their agreement with the
heliospheric imager data to provide a probabilistic forecasting. This will be
done in the future. Here we decided to concentrate on improvement of the
arrival time predictions.

\begin{overpic}[scale={0.1},angle={0},width=199.16928pt,keepaspectratio]{1AU_data_1.png}
\put(82.0,92.0){\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}{
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\begin{overpic}[scale={0.1},angle={0},width=199.16928pt,keepaspectratio]{1AU_data_2.png}
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3}} \end{overpic}
\begin{overpic}[scale={0.1},angle={0},width=199.16928pt,keepaspectratio]{1AU_data_4.png}
\put(82.0,92.0){\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}{
4}} \end{overpic}
\begin{overpic}[scale={0.1},angle={0},width=199.16928pt,keepaspectratio]{1AU_data_6.png}
\put(82.0,92.0){\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}{
5}} \end{overpic}
\begin{overpic}[scale={0.1},angle={0},width=199.16928pt,keepaspectratio]{1AU_data_7.png}
\put(82.0,92.0){\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}{
6}} \end{overpic}
Figure 2: Plasma density, radial velocity components, and the heliospheric
magnetic field in the 1 hr averaged OMNI data (blue lines) and our simulation
results at Earth (black and green lines) for Cases 1 to 6. The case numbers
are shown beside the panels. The black lines represent the results obtained
when using the seed ensemble members. The green lines show the results for the
76 ensemble members, which were found by introducing uncertainties in the GCS
parameters of the seed ensemble member. The vertical gray line signifies the
arrival of the observed CME, while the dotted red lines indicate the arrival
times of the earliest and latest simulated CME ensemble members.

###  4.3 Comparison with HI data

HI1 imagers on board STEREO-A & B have fixed FOVs that extend between
4${}^{/}circ$ and 24${}^{/}circ$ of elongation angle. HI2s similarly have FOVs
that extend between 18.7 and 88.7 degrees of elongation angles. This
translates to distances of up to 1 AU and beyond for HI1 and HI2 combined.
CMEs expand in the IH and consequently their density decreases as they
propagate. This means that CMEs become fainter as they propagate in the HI
FOVs, so that most of CMEs become untraceable near the outer edge of the HI2
FOV. A common approach to tracking CMEs in HI FOV is to create time-elongation
maps, also known as J-maps. These are created by extracting the pixels
corresponding to the ecliptic plane in running-difference HI images. These
extracted pixels are then stacked vertically for a specified time range to
create an image called a J-map. CME fronts reveal themselves as oblique bright
fronts after this procedure. The top panels of Figure 3 show such J-maps for
the 12 July 2012 CME created using STEREO-A & B data. The 12 July 2012 CME can
be seen clearly in both STEREO-A & B J-maps. We can trace the bright fronts of
a CME in both J-maps to obtain time-elongation data. J-maps can have multiple
CMEs visible in them, but the bright front corresponding to the CME under
consideration can easily be found by considering the eruption time of a CME.

Simulations can be compared with observations by creating synthetic J-maps.
The middle row in Figure 3 shows the synthetic J-maps we have created with the
simulation based on the seed ensemble member for the 12 July 2012 CME (for
STEREO-A on the left and STEREO-B on the right). We create these maps by
integrating the density along the line-of-sight (LOS) towards STEREO in the
ecliptic plane and taking a running difference with time. We then track the
bright front in the synthetic J-maps to get time-elongation data. We have
automated this tracking because of the large number of ensemble members. The
time-elongation data in simulations and observations can be easily compared.
The bottom panel of Figure 3 shows the time-elongation graphs for observations
(black line) and our simulated ensemble members (color lines). The plotlines
corresponding to the different ensemble members are colored according to the
travel time of CME from Sun to Earth for that ensemble member, which is found
by subtracting the eruption time of the 12 July 2012 CME from the arrival time
for that ensemble member. Similar comparison plots for the rest of 5 cases are
shown in Figure 4. We can see that, for all our cases, some ensemble members
agree better than others with the CME observations in the IH. The agreement
can be better visualized by creating plots that show the difference between
simulated and observed elongations as function of time, Figure 6 shows these
plots for the 12 July 2012 CME. In the next subsection, we will describe the
methods to use these comparisons to improve the arrival time estimates.

---
\begin{overpic}[scale={0.1},angle={0},width=170.71652pt,keepaspectratio]{Time_EA_Ensembles_A_2012-07-12-eps-
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5-A}} \end{overpic}
\begin{overpic}[scale={0.1},angle={0},width=170.71652pt,keepaspectratio]{Time_EA_Ensembles_B_2012-07-12-eps-
converted-to.pdf}
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5-B}} \end{overpic}
Figure 3: Top panel: J-maps created using HI1 and HI2 data from STEREO-A
(left) and STEREO-B (right) showing the 12 July 2012 CME (Case 5). The tracked
CME front is shown with red symbols in both J-maps. Middle panel: Synthetic
J-maps created from the simulation of the seed ensemble member for the 12 July
2012 CME from STEREO-A (left) and STEREO-B (right) vantage point. The CME
front is traced with the blue line. Bottom panel: The time-elongation data
extracted from the J-maps by tracing the CME fronts in observed J-map (black
line) and synthetic J-maps for all ensemble members (colored according to the
travel time in hours from eruption to Earth impact). The left panel is for
STEREO-A and the right panel is for STEREO-B. These panels are tagged with
“case number - spacecraft” format tags.
\begin{overpic}[scale={0.1},angle={0},width=142.26378pt,keepaspectratio]{Time_EA_Ensembles_A_2010-08-01-eps-
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1-B}} \end{overpic}
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2-A}} \end{overpic}
\begin{overpic}[scale={0.1},angle={0},width=142.26378pt,keepaspectratio]{Time_EA_Ensembles_B_2011-09-06-eps-
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2-B}} \end{overpic}
\begin{overpic}[scale={0.1},angle={0},width=142.26378pt,keepaspectratio]{Time_EA_Ensembles_A_2011-09-13-eps-
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3-A}} \end{overpic}
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converted-to.pdf}
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3-B}} \end{overpic}
\begin{overpic}[scale={0.1},angle={0},width=142.26378pt,keepaspectratio]{Time_EA_Ensembles_A_2012-01-19-eps-
converted-to.pdf}
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4-A}} \end{overpic}
\begin{overpic}[scale={0.1},angle={0},width=142.26378pt,keepaspectratio]{Time_EA_Ensembles_B_2012-01-19-eps-
converted-to.pdf}
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4-B}} \end{overpic}
\begin{overpic}[scale={0.1},angle={0},width=142.26378pt,keepaspectratio]{Time_EA_Ensembles_A_2012-09-28-eps-
converted-to.pdf}
\put(12.0,65.0){\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}{
6-A}} \end{overpic}
\begin{overpic}[scale={0.1},angle={0},width=142.26378pt,keepaspectratio]{Time_EA_Ensembles_B_2012-09-28-eps-
converted-to.pdf}
\put(12.0,65.0){\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}{
6-B}} \end{overpic}
Figure 4: The time-elongation data obtained by tracing the CME fronts in the
observed (black line) and synthetic J-maps for all ensemble members (colored
according to the travel time in hours from eruption to the Earth impact). The
left panel is for STEREO-A and the right panel is for STEREO-B. The panels are
tagged with the “case number - spacecraft” format tag in the top left corners.
Case 5 has been shown previously in Figure 3

###  4.4 Improving arrival times with ML

We have simulated the seed ensemble members for each of our six CMEs with the
properties listed in Table 1. The errors in arrival times for these
simulations are given in Table 2. Using the errors $\Delta t_{i}$ (in case
$i$), we can calculate the mean error (ME), MAE, and standard deviation (SD)
as follows:

$\textup{ME}=\frac{1}{N}\sum_{i=1}^{N}\Delta t_{i}$

$\textup{MAE}=\frac{1}{N}\sum_{i=1}^{N}|\Delta t_{i}|$

$\textup{SD}=\sqrt{\frac{1}{N}\sum_{i=1}^{N}|\Delta t_{i}-\textup{ME}|^{2}}$

The MAE obtained by simulating only the seed ensemble members, for all six
cases, is $\sim$8 hours. All simulated CMEs in the seed members arrived later
than in the observations. Therefore the ME has the same magnitude as MAE, but
the sign is negative. This shows that our CME simulations tend to have a
negative bias in the arrival time estimates. Cases 2, 3, and 6 have arrival
time errors equal to $-15$ hours, $-15.2$ hours, and $-9.1$ hours and their
contribution to the MAE is therefore dominant. These three cases also have the
launch speeds lower than the other three CMEs. This shows the tendency of our
model to have large arrival time errors for slower CMEs.

MAE is a more meaningful metric than ME to gauge the performance of a model,
since ME calculations are affected by the mutual cancellation of positive and
negative $\Delta t_{i}$ (Riley et al., 2018). We can further reduce MAE by
using the arrival time distribution for all ensemble members and their
corresponding differences between the simulated and observed elongation angles
in the IH. We can achieve this by answering the following question: what will
be the arrival time in a CME simulation of hypothetical ensemble member whose
elongation angle evolution in its synthetic J-map matches perfectly with the
observations in the IH? This hypothetical ensemble member can be represented
by the “elongation difference = 0” line in plots shown with the black line in
Figure 6.

We have used two machine learning-based methods to answer this question: (1)
the lasso regression (LR) (Tibshirani, 1996) and (2) the fully connected
neural networks (NNs). In a machine learning approach, a hypothesis is
formulated to map the input variables onto an output variable with the goal
that the algorithm learns the mapping of a function $h:X-y$ such that $h(x)$
is a good approximation for the target variable $y$. Training of the algorithm
involves the minimization of the cost/loss function for our hypothesis. To
reduce over-fitting, penalties are added to the cost function to avoid large
coefficients which make the model unstable. The method of adding penalties to
the cost function is known as regularization.

LR, also known as L1 Regularization, is a linear regression variant that adds
a penalty to the loss function to reduce the magnitude of the coefficients for
the features or inputs to the model. The penalty term is the absolute value of
the coefficients multiplied by a scalar constant, called the regularization
parameter. The objective function to be minimized in LR is the mean squared
error plus the penalty term, which is defined as follows:

$J(w)=\frac{1}{N}\sum_{i=1}^{N}(y_{i}-\sum_{j=1}^{p}w_{j}x_{ij})^{2}+\lambda\sum_{j=1}^{p}|w_{j}|,$

where $w$ is the vector of coefficients, $N$ is the number of samples, $p$ is
the number of inputs/features, $x_{ij}$ is the $j$th feature of the $i$th
sample, $y_{i}$ is the target variable, and $\lambda$ is the regularization
parameter that controls the amount of shrinkage of the coefficients. The LR
forces some of the coefficients to be exactly zero, resulting in the
elimination of some features. This means that only the most important features
are included in the model, reducing the risk of overfitting.

Fully connected networks are based on the idea of artificial neurons, which
are modeled after biological neurons in the human brain. These artificial
neurons receive inputs from other neurons, process these inputs through a non-
linear activation function, and produce an output that can be used as input
for other neurons in the network. By stacking multiple layers of neurons, a
fully connected network can learn complex representations of data, allowing it
to make accurate predictions and decisions. Neural networks can approximate a
function by calculating the weighted sum of inputs, including the bias term,
computing the activation function response, and passing the output to the
output neuron. This constitutes the forward pass or the forward propagation.
The cost/loss function is calculated by “back propagation” of the cost
function gradients (Rumelhart et al., 1986). Machine learning techniques,
including the NNs, have been successfully applied to prediction and
forecasting problems in heliophysics (Fernandez Borda et al., 2002; Qu et al.,
2003; Li et al., 2007; Qahwaji and Colak, 2007; Wang et al., 2008; Yuan et
al., 2010; Ahmed et al., 2013; Bobra and Couvidat, 2015; Jonas et al., 2018;
Benson et al., 2019, 2020, 2021). Camporeale (2019) provides an extensive
review of the methods and applications of machine learning research in
heliophysics.

Figure 5: Training and evaluation pipeline for machine learning algorithms.
“Ensemble Data” contains independent variables: time since the eruption, and
elongation angle difference between observed and simulated CMEs, and the
dependent variable: travel time of the simulated CME to reach Earth. In
“Holdout Data”, elongation angle difference is set as zero at all times since
CME eruption. “Predictions” are the travel time and it’s standard deviation
for the given “Holdout Data”.

For both methods, the variables, such as time since the eruption, elongation
angle difference from STEREO-A, and elongation angle difference from STEREO-B
(as shown in Figure 6), are considered to be independent, whereas the travel
time for an ensemble member is considered to be the dependent variable. These
independent and dependent variables constitute the “Ensemble Data” in Figure
5, which shows the model training and evaluation pipeline for the machine
learning experiments. The input ensemble data is divided into training and
test sets using a 80%-20% split. The ML model (LR or NN) is trained using the
training data and the errors in predictions are evaluated using the test set.
This process is iterated until we find the best model fit. The final model is
then evaluated using the holdout data to make the final predictions of the CME
travel time.

For the LR method, we frame the arrival time prediction as a simple regression
problem in which we are predicting a continuous variable for a given set of
inputs. The standard algorithm for regression problems is linear regression
(Montgomery et al., 2006) where the input variables are assumed to depend
linearly on the target variable. The experiments using LR are implemented
using the widely available Scikit-Learn library (Pedregosa et al., 2011). We
train our model for each CME separately. The model is provided the arrays of
time since eruption and elongation angle differences as dependent variables.
The travel time of an ensemble member is considered to be a dependent
variable. The model is tested on a holdout data set which has “elongation
difference = 0” for all times since eruption. The model is therefore
determining the travel time which it expects from a hypothetical ensemble
member which matches perfectly with observations in IH.

Similarly, we use a fully connected NN with twelve thousand parameters using
the same input parameters to predict the CME arrival time. The NN consists of
an input layer, four hidden layers with 128, 64, 32, and 16 fully connected
neurons, and a single output layer that outputs a continuous variable. A ReLU
activation layer is used after every layer with batch normalization and a
dropout of 20% is used as means of regularization to avoid over-fitting. All
experiments involving NNs are implemented using Tensorflow 2.0 (Abadi et al.,
2016). The data for each CME are split into training, testing, and validation
sets using a 60%, 20%, and 20% random split. The best model is then evaluated
using a separate holdout set which has “elongation difference = 0” for all
times since eruption.

We have performed two experiments each using both of the ML methods. The first
experiment uses data from both STEREO-A & B both and the second experiment
uses data from the STEREO-A only. The second experiment is motivated by the
current operational availability of STEREO-A data only. Our aim is to
understand the applicability of our approach in this case.

The results reported in Table 2 show the MAE, ME, and standard deviation for
each experiment. The results show that the LR model based on STEREO-A & B data
performs best with a CME arrival time MAE (SD) of 4.1 (3.2) hours, followed by
the NN model using STEREO-A & B data of 5.3 (6.1) hours. However, while using
the STEREO-A data alone, the NN model offers better performance than the LR
model with an arrival time of 5.1 (5.2) hours compared to 6.9 (5.3) hours. We
can see that all our experiments have improved the MAE compared to using just
seed member simulations for the six CMEs. Perhaps the most encouraging result
is the MAE of 5.1 hours achieved by using the NNs and only the STEREO-A data.
It shows that NNs are more capable of handling non-linear relationships in
data. This result is most relevant to the current data availability of STEREO
data since only STEREO-A is operational. All our experiments resulted in a
reduction in arrival time errors for cases 2, 3, and 6, which were
contributing most to MAE. Our work highlights the importance of combining
machine learning and physics-based MHD modeling to improve space weather
predictions. The machine learning models allow us to continuously update the
arrival time estimates of CMEs by using simulation-to-observation comparisons
in the IH. This proof-of-concept study shows that our methods can improve the
arrival time estimates with sufficient lead times.

The MAE we have achieved with our methods is better than those achieved by
such models as HM, SSE, and ElEvoHI, even though those models assume CME
shapes that should perfectly match the HI observations. The advantage of our
model is in a more realistic CME-SW interaction. This allows for CMEs to
acquire complex shapes by interacting with local SW structures, e.g., high-
speed streams. In contrast, the above-mentioned models assume much simpler CME
geometries throughout the IH. A newer version of ElEvoHI, called ElEvoHI 2.0
(Hinterreiter et al., 2021b) allows for a deformable front and takes CME-SW
interaction into account through an empirical drag force. Encouragingly,
Hinterreiter et al. (2021b) showed an improvement in the CME arrival time
estimates when a realistic CME-SW interaction and a deformable front are taken
into account. This study supports our argument that MHD modeling that
reproduces realistic CME-SW interactions can provide better results than the
models using CME shape assumptions throughout the IH.

In this study, we observed that the MAE achieved was significantly lower than
the average MAE of currently operational models. However, it is important to
note that the cases simulated in this study were well observed, while the MAE
of the operational models typically encompasses a wider range of CMEs with
varying degrees of complexity, including those with CME-CME interactions. To
validate our findings, we plan to expand the scope of our study in the future
to include a more diverse set of CMEs. Despite the limited sample size of six
CMEs in this study, the benefit of combining machine learning and MHD CME
modeling to enhance arrival time predictions is quite evident.

\begin{overpic}[scale={0.1},angle={0},width=170.71652pt,keepaspectratio]{CME12JulyA_with_obs-
eps-converted-to.pdf}
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5-A}} \end{overpic}
\begin{overpic}[scale={0.1},angle={0},width=170.71652pt,keepaspectratio]{CME12JulyB_with_obs-
eps-converted-to.pdf}
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5-B}} \end{overpic}
---
Figure 6: The difference of the elongation angles obtained from the synthetic
and observed J-maps for 12 July 2012 CME for STEREO-A (left panel) and
STEREO-B (right panel). The panels are tagged with the “case number -
spacecraft” format tag in the top left corners. Table 2: Errors in arrival
times of the six CMEs discussed in this study, alongside with the mean error
(ME), mean absolute error (MAE), and standard deviation (SD) for each method.

CME 01/08/10 06/09/11 13/09/11 19/01/12 12/07/12 28/09/12 ME MAE SD Seed
member -3.9 -15.0 -15.2 -3.7 -0.9 -9.1 -8.0 8.0 6.1 LR (A & B) -3.3$\pm$0.8
-9.3$\pm$0.2 1.8$\pm$0.3 -6.8$\pm$1.7 1.9$\pm$0.1 -1.8$\pm$0.4 -2.9 4.1 3.2 LR
(A) -7.2$\pm$0.1 -14.8$\pm$1.3 0.8$\pm$0.3 -7.9$\pm$2.1 -2.6$\pm$0.1
-8.0$\pm$1.3 -6.6 6.9 5.3 NNs (A & B) -2.2$\pm$2.6 -12.7$\pm$1.0 5.9$\pm$1.3
-4.8$\pm$1.7 -1.0$\pm$0.3 -5.2$\pm$1.3 -3.3 5.3 6.1 NNs (A) -3.9$\pm$0.2
-10.2$\pm$1.9 5.4$\pm$2.1 -5.3$\pm$1.6 -1.1$\pm$2.2 -4.6$\pm$5.5 -3.3 5.1 5.2

##  5 Conclusion

In this study, we have used a constant turn flux rope model to simulate six
CMEs in a data-driven SW background. The flux rope model was constrained with
the observed CME direction, tilt, speed, half-angle, aspect ratio, mass,
magnetic fluxes, and magnetic helicity sign. The average uncertainties in
observed direction, tilt, speed, half-angle, and aspect ratio allowed us to
create 76 additional ensemble members for each CME from their seed ensemble
members. We have presented the simulation results for all the ensemble members
for our six considered CMEs at Earth. Using synthetic and observed J-maps, we
were able to compare the ensemble members with observed CMEs. These
comparisons were used in the LR and NN models to improve the arrival time
estimates of the CMEs. The results of this work can be summarized as follows:

  1. 1.

Constant turn flux rope model is suitable to simulate CMEs in the IH.

  2. 2.

This model can simulate CMEs with a large range of speeds, magnetic fluxes,
and widths. In this study, we have simulated CMEs that were launched with
speeds ranging between 480 km/s and 1357 km/s, poloidal fluxes ranging between
$0.9\times 10^{21}$ and $14.0\times 10^{21}$ Mx, and half angles ranging
between $23^{\circ}$ and $68^{\circ}$.

  3. 3.

By simulating only the seed ensemble member of each CME, we found the MAE to
be 8 hours and the standard deviation is 6.1 hours. All these simulations
resulted in the simulated CMEs arriving later than the observed ones, showing
a negative bias in our method.

  4. 4.

The uncertainty in GCS parameters can be used to create ensemble members and
the CME arrival time estimates can be improved by comparing simulated ensemble
members with STEREO HI observations.

  5. 5.

We obtained the MAE for our six CMEs as small as 4.1 hours using the LR and
STEREO-A & B data, 6.9 hours using the LR and STEREO-A data only, 5.3 hours
using NNs and STEREO-A & B data, and 5.1 hours using NNs and STEREO-A data.

  6. 6.

The standard deviation for the six CMEs was found to be 3.2 hours using the LR
and STEREO-A & B data, 5.3 hours using LR and STEREO-A data, 6.1 hours using
NNs and STEREO-A & B data, and 5.2 hours using NNs and STEREO-A data.

  7. 7.

These MAE and standard deviation measures are better than the other currently
operational and proposed arrival time forecasting models. This demonstrates
that the combination of HI data and MHD modeling together with machine
learning can substantially improve space weather predictions.

The simulation of approximately 6 days of physical time on a grid of
$150\times 128\times 256$ by MS-FLUKSS requires 2 hours of computation using
128 CPUs. Given the ability to run all ensemble members simultaneously, the
ensemble modeling carried out here is practical for predicting space weather
in real-time. The machine learning methods we have used are quite robust and
do not take more than a few minutes to run. We have used science data from HIs
onboard STEREOs for this analysis. However, these data become available
several days after observations. Future work will test the applicability of
the beacon HI data to our methods, which can make our methods suitable for
real-time forecasting. This work also highlights the importance of having HI
observatories on Lagrangian points 4 and 5 to continuously image the CME
evolution in IH. The European Space Agency’s Vigil spacecraft is slated to be
stationed at the L5 point in the mid-2020s, equipped with HI instruments to
observe CMEs in the interplanetary space. Our current research can be
leveraged to make the most of these data, thereby enabling enhanced space
weather forecasts.

### Acknowledgments

The authors acknowledge support from NSF/NASA SWQU grant 2028154. TKK
acknowledges support from AFOSR grant FA9550-19-1-0027 and NASA grant
80NSSC20K1453. NP was also supported, in part, by NSF-BSF grant 2010450 and
NASA grants 80NSSC19K0075 and 80NSSC21K0004. Supercomputer allocations were
provided on SGI Pleiades by NASA High-End Computing Program award
SMD-21-44038581 and also on TACC Stampede2 and SDSC Expanse by NSF XSEDE
project MCA07S033. We acknowledge the NASA/GSFC Space Physics Data Facility’s
OMNIWeb for the SW and IMF data used in this study. We also used SOHO and
STEREO coronagraph data fromhttps://lasco-www.nrl.navy.mil and stereo-
ssc.nascom.nasa.gov respectively. SDO EUV and magnetogram data have been taken
from http://jsoc.stanford.edu/ajax/exportdata.html. This work utilizes data
produced collaboratively between the Air Force Research Laboratory (AFRL) and
the National Solar Observatory.

## References

  * Abadi et al. [2016] Martín Abadi, Ashish Agarwal, Paul Barham, Eugene Brevdo, Zhifeng Chen, Craig Citro, Greg S. Corrado, Andy Davis, Jeffrey Dean, Matthieu Devin, Sanjay Ghemawat, Ian Goodfellow, Andrew Harp, Geoffrey Irving, Michael Isard, Yangqing Jia, Rafal Jozefowicz, Lukasz Kaiser, Manjunath Kudlur, Josh Levenberg, Dan Mane, Rajat Monga, Sherry Moore, Derek Murray, Chris Olah, Mike Schuster, Jonathon Shlens, Benoit Steiner, Ilya Sutskever, Kunal Talwar, Paul Tucker, Vincent Vanhoucke, Vijay Vasudevan, Fernanda Viegas, Oriol Vinyals, Pete Warden, Martin Wattenberg, Martin Wicke, Yuan Yu, and Xiaoqiang Zheng.  TensorFlow: Large-Scale Machine Learning on Heterogeneous Distributed Systems.  _arXiv e-prints_ , art. arXiv:1603.04467, March 2016.  doi: 10.48550/arXiv.1603.04467.
  * Ahmed et al. [2013] O. W. Ahmed, R. Qahwaji, T. Colak, P. A. Higgins, P. T. Gallagher, and D. S. Bloomfield.  Solar flare prediction using advanced feature extraction, machine learning, and feature selection.  _Solar Phys._ , 283(1):157–175, Mar 2013.  ISSN 1573-093X.  doi: 10.1007/s11207-011-9896-1.
  * Amerstorfer et al. [2018] T. Amerstorfer, C. Möstl, P. Hess, M. Temmer, M. L. Mays, M. A. Reiss, P. Lowrance, and P.-A. Bourdin.  Ensemble prediction of a halo coronal mass ejection using heliospheric imagers.  _Space Weather_ , 16(7):784–801, 2018.  doi: https://doi.org/10.1029/2017SW001786.  URL https://agupubs.onlinelibrary.wiley.com/doi/abs/10.1029/2017SW001786.
  * Arge et al. [2005] C. N. Arge, G. de Toma, and J. G. Luhmann.  Comparison of the Stream Structure and Coronal Sources of the Solar Wind During the April 7 & May 12, 1997 Halo CMEs.  In K. Sankarasubramanian, M. Penn, and A. Pevtsov, editors, _Large-scale Structures and their Role in Solar Activity_ , volume 346 of _Astronomical Society of the Pacific Conference Series_ , page 371, December 2005.  URL https://ui.adsabs.harvard.edu/abs/2005ASPC..346..371A.
  * Arge et al. [2010] C. N. Arge, Carl J. Henney, Josef Koller, C. Rich Compeau, Shawn Young, David MacKenzie, Alex Fay, and John W. Harvey.  Air force data assimilative photospheric flux transport (adapt) model.  _AIP Conference Proceedings_ , 1216(1):343–346, 2010.  doi: 10.1063/1.3395870.
  * Arge et al. [2011] C. N. Arge, C. J. Henney, J. Koller, W. A. Toussaint, J. W. Harvey, and S. Young.  Improving Data Drivers for Coronal and Solar Wind Models.  In N. V. Pogorelov, E. Audit, and G. P. Zank, editors, _5th International Conference of Numerical Modeling of Space Plasma Flows (ASTRONUM 2010)_ , volume 444 of _Astronomical Society of the Pacific Conference Series_ , page 99, October 2011.  URL https://ui.adsabs.harvard.edu/abs/2011ASPC..444...99A.
  * Arge et al. [2013] C. N. Arge, Carl J. Henney, Irene Gonzalez Hernandez, W. Alex Toussaint, Josef Koller, and Humberto C. Godinez.  Modeling the corona and solar wind using adapt maps that include far-side observations.  _AIP Conference Proceedings_ , 1539(1):11–14, 2013.  doi: 10.1063/1.4810977.
  * Arge et al. [2003] Charles N. Arge, Dusan Odstrcil, Victor J. Pizzo, and Leslie R. Mayer.  Improved method for specifying solar wind speed near the sun.  _AIP Conference Proceedings_ , 679(1):190–193, 2003.  doi: 10.1063/1.1618574.  URL https://aip.scitation.org/doi/abs/10.1063/1.1618574.
  * Barnard et al. [2020] L. Barnard, M. J. Owens, C. J. Scott, and C. A. de Koning.  Ensemble cme modeling constrained by heliospheric imager observations.  _AGU Advances_ , 1(3):e2020AV000214, 2020.  doi: https://doi.org/10.1029/2020AV000214.  URL https://agupubs.onlinelibrary.wiley.com/doi/abs/10.1029/2020AV000214.  e2020AV000214 10.1029/2020AV000214.
  * Benson et al. [2019] B. Benson, W. D. Pan, G. A. Gary, Q. Hu, and T. Staudinger.  Determining the parameter for the linear force-free magnetic field model with multi-dipolar configurations using deep neural networks.  _Astron.Comput._ , 26:50 – 60, 2019.  ISSN 2213-1337.  doi: https://doi.org/10.1016/j.ascom.2018.11.002.
  * Benson et al. [2020] B. Benson, W. D. Pan, A. Prasad, G. A. Gary, and Q. Hu.  Forecasting solar cycle 25 using deep neural networks.  _Solar Phys._ , 295(5):65, May 2020.  ISSN 1573-093X.  doi: 10.1007/s11207-020-01634-y.
  * Benson et al. [2021] B. Benson, W. D. Pan, A. Prasad, G. A. Gary, and Q. Hu.  On the estimation of the sharp parameter meanalp from aia images using deep neural networks.  _Solar Physics_ , 296(11):163, Nov 2021.  ISSN 1573-093X.  doi: 10.1007/s11207-021-01912-3.  URL https://doi.org/10.1007/s11207-021-01912-3.
  * Billings [1966] Donald E. Billings.  _A guide to the solar corona_.  1966\.
  * Bobra and Couvidat [2015] M. G. Bobra and S. Couvidat.  Solar flare prediction using SDO/HMI vector magnetic field data with a machine-learning algorithm.  _Astrophys.J._ , 798(2):135, Jan 2015.  doi: 10.1088/0004-637x/798/2/135.
  * Bothmer and Schwenn [1998] V. Bothmer and R. Schwenn.  The structure and origin of magnetic clouds in the solar wind.  _Annales Geophysicae_ , 16(1):1–24, January 1998\.  doi: 10.1007/s00585-997-0001-x.
  * Brueckner et al. [1995] G. E. Brueckner, R. A. Howard, M. J. Koomen, C. M. Korendyke, D. J. Michels, J. D. Moses, D. G. Socker, K. P. Dere, P. L. Lamy, A. Llebaria, M. V. Bout, R. Schwenn, G. M. Simnett, D. K. Bedford, and C. J. Eyles.  The Large Angle Spectroscopic Coronagraph (LASCO).  _Solar Physics_ , 162(1-2):357–402, December 1995\.  doi: 10.1007/BF00733434.
  * Brueckner et al. [1998] G. E. Brueckner, J. P. Delaboudiniere, R. A. Howard, S. E. Paswaters, O. C. St. Cyr, R. Schwenn, P. Lamy, G. M. Simnett, B. Thompson, and D. Wang.  _Geophys Res Lett_ , 25(15):3019–3022, Jan 1998.  doi: 10.1029/98GL00704.
  * Camporeale [2019] E. Camporeale.  The challenge of machine learning in space weather: Nowcasting and forecasting.  _Space Weather_ , 17(8):1166–1207, 2019.  doi: 10.1029/2018SW002061.
  * Colaninno and Vourlidas [2009] Robin C. Colaninno and Angelos Vourlidas.  First Determination of the True Mass of Coronal Mass Ejections: A Novel Approach to Using the Two STEREO Viewpoints.  _The Astrophysical Journal_ , 698(1):852–858, June 2009.  doi: 10.1088/0004-637X/698/1/852.
  * Davies et al. [2012] J. A. Davies, R. A. Harrison, C. H. Perry, C. Möstl, N. Lugaz, T. Rollett, C. J. Davis, S. R. Crothers, M. Temmer, C. J. Eyles, and N. P. Savani.  A SELF-SIMILAR EXPANSION MODEL FOR USE IN SOLAR WIND TRANSIENT PROPAGATION STUDIES.  _The Astrophysical Journal_ , 750(1):23, apr 2012\.  doi: 10.1088/0004-637x/750/1/23.  URL https://doi.org/10.1088/0004-637x/750/1/23.
  * Dryer and Smart [1984] M. Dryer and D.F. Smart.  _Advances in Space Research_ , 4(7):291 – 301, 1984.  ISSN 0273-1177.  doi: https://doi.org/10.1016/0273-1177(84)90200-X.  URL http://www.sciencedirect.com/science/article/pii/027311778490200X.
  * Elliott et al. [2016] Heather A. Elliott, David J. McComas, and Craig E. DeForest.  LONG-TERM TRENDS IN THE SOLAR WIND PROTON MEASUREMENTS.  _The Astrophysical Journal_ , 832(1):66, nov 2016\.  doi: 10.3847/0004-637x/832/1/66.  URL https://doi.org/10.3847%2F0004-637x%2F832%2F1%2F66.
  * Emmons et al. [2013] D. Emmons, A. Acebal, A. Pulkkinen, A. Taktakishvili, P. MacNeice, and D. Odstrcil.  Ensemble forecasting of coronal mass ejections using the wsa-enlil with coned model.  _Space Weather_ , 11(3):95–106, 2013.  doi: https://doi.org/10.1002/swe.20019.  URL https://agupubs.onlinelibrary.wiley.com/doi/abs/10.1002/swe.20019.
  * Eyles et al. [2009] C. J. Eyles, R. A. Harrison, C. J. Davis, N. R. Waltham, B. M. Shaughnessy, H. C. A. Mapson-Menard, D. Bewsher, S. R. Crothers, J. A. Davies, G. M. Simnett, R. A. Howard, J. D. Moses, J. S. Newmark, D. G. Socker, J. P. Halain, J. M. Defise, E. Mazy, and P. Rochus.  The Heliospheric Imagers Onboard the STEREO Mission.  _Sol Phys_ , 254(2):387–445, February 2009.  doi: 10.1007/s11207-008-9299-0.
  * Fernandez Borda et al. [2002] Roberto A. Fernandez Borda, P. D. Mininni, C. H. Mandrini, D. O. Gómez, O. H. Bauer, and M. G. Rovira.  Automatic solar flare detection using neural network techniques.  _Solar Phys._ , 206(2):347–357, Apr 2002.  ISSN 1573-093X.  doi: 10.1023/A:1015043621346.
  * Fraternale et al. [2021] Federico Fraternale, Nikolai V. Pogorelov, and Jacob Heerikhuisen.  Transport of interstellar neutral helium throughout the heliosphere.  _The Astrophysical Journal Letters_ , 921(2):L24, nov 2021.  doi: 10.3847/2041-8213/ac313c.  URL https://dx.doi.org/10.3847/2041-8213/ac313c.
  * Fry et al. [2001] C. D. Fry, W. Sun, C. S. Deehr, M. Dryer, Z. Smith, S. I. Akasofu, M. Tokumaru, and M. Kojima.  _J Geophys Res_ , 106(A10):20985–21002, Oct 2001.  doi: 10.1029/2000JA000220.
  * Gopalswamy et al. [2005] N. Gopalswamy, A. Lara, P. K. Manoharan, and R. A. Howard.  _Advances in Space Research_ , 36(12):2289–2294, Jan 2005.  doi: 10.1016/j.asr.2004.07.014.
  * Gopalswamy et al. [2013] N. Gopalswamy, P. Mäkelä, H. Xie, and S. Yashiro.  _Space Weather_ , 11(11):661–669, Nov 2013.  doi: 10.1002/2013SW000945.
  * Gopalswamy et al. [2017] N. Gopalswamy, S. Yashiro, S. Akiyama, and H. Xie.  Estimation of Reconnection Flux Using Post-eruption Arcades and Its Relevance to Magnetic Clouds at 1 AU.  _Solar Physics_ , 292(4):65, April 2017.  doi: 10.1007/s11207-017-1080-9.
  * Gopalswamy et al. [2018] N. Gopalswamy, S. Akiyama, S. Yashiro, and H. Xie.  Coronal flux ropes and their interplanetary counterparts.  _Journal of Atmospheric and Solar-Terrestrial Physics_ , 180:35 – 45, 2018.  ISSN 1364-6826.  doi: https://doi.org/10.1016/j.jastp.2017.06.004.  URL http://www.sciencedirect.com/science/article/pii/S1364682617303607.  Variability of the Sun and Its Terrestrial Impacts.
  * Gopalswamy et al. [2001] Nat Gopalswamy, Alejandro Lara, Seiji Yashiro, Mike L. Kaiser, and Russell A. Howard.  _J Geophys Res_ , 106(A12):29207–29218, Dec 2001.  doi: 10.1029/2001JA000177.
  * Hess and Zhang [2014] Phillip Hess and Jie Zhang.  Stereoscopic Study of the Kinematic Evolution of a Coronal Mass Ejection and Its Driven Shock from the Sun to the Earth and the Prediction of Their Arrival Times.  _The Astrophysical Journal_ , 792(1):49, September 2014.  doi: 10.1088/0004-637X/792/1/49.
  * Hickmann et al. [2015] Kyle S. Hickmann, Humberto C. Godinez, Carl J. Henney, and C. Nick Arge.  Data Assimilation in the ADAPT Photospheric Flux Transport Model.  _Sol Phys_ , 290(4):1105–1118, April 2015.  doi: 10.1007/s11207-015-0666-3.
  * Hinterreiter et al. [2021a] Jürgen Hinterreiter, Tanja Amerstorfer, Martin A. Reiss, Christian Möstl, Manuela Temmer, Maike Bauer, Ute V. Amerstorfer, Rachel L. Bailey, Andreas J. Weiss, Jackie A. Davies, Luke A. Barnard, and Mathew J. Owens.  Why are elevohi cme arrival predictions different if based on stereo-a or stereo-b heliospheric imager observations?  _Space Weather_ , 19(3):e2020SW002674, 2021a.  doi: https://doi.org/10.1029/2020SW002674.  URL https://agupubs.onlinelibrary.wiley.com/doi/abs/10.1029/2020SW002674.  e2020SW002674 2020SW002674.
  * Hinterreiter et al. [2021b] Jürgen Hinterreiter, Tanja Amerstorfer, Manuela Temmer, Martin A. Reiss, Andreas J. Weiss, Christian Möstl, Luke A. Barnard, Jens Pomoell, Maike Bauer, and Ute V. Amerstorfer.  Drag-based cme modeling with heliospheric images incorporating frontal deformation: Elevohi 2.0.  _Space Weather_ , 19(10):e2021SW002836, 2021b.  doi: https://doi.org/10.1029/2021SW002836.  URL https://agupubs.onlinelibrary.wiley.com/doi/abs/10.1029/2021SW002836.  e2021SW002836 2021SW002836.
  * Hoeksema et al. [2014] J. Todd Hoeksema, Yang Liu, Keiji Hayashi, Xudong Sun, Jesper Schou, Sebastien Couvidat, Aimee Norton, Monica Bobra, Rebecca Centeno, K. D. Leka, Graham Barnes, and Michael Turmon.  The helioseismic and magnetic imager (hmi) vector magnetic field pipeline: Overview and performance.  _Solar Physics_ , 289(9):3483–3530, Sep 2014\.  ISSN 1573-093X.  doi: 10.1007/s11207-014-0516-8.  URL https://doi.org/10.1007/s11207-014-0516-8.
  * Howard et al. [2008] R. A. Howard, J. D. Moses, A. Vourlidas, J. S. Newmark, D. G. Socker, S. P. Plunkett, C. M. Korendyke, J. W. Cook, A. Hurley, J. M. Davila, W. T. Thompson, O. C. St Cyr, E. Mentzell, K. Mehalick, J. R. Lemen, J. P. Wuelser, D. W. Duncan, T. D. Tarbell, C. J. Wolfson, A. Moore, R. A. Harrison, N. R. Waltham, J. Lang, C. J. Davis, C. J. Eyles, H. Mapson-Menard, G. M. Simnett, J. P. Halain, J. M. Defise, E. Mazy, P. Rochus, R. Mercier, M. F. Ravet, F. Delmotte, F. Auchere, J. P. Delaboudiniere, V. Bothmer, W. Deutsch, D. Wang, N. Rich, S. Cooper, V. Stephens, G. Maahs, R. Baugh, D. McMullin, and T. Carter.  Sun Earth Connection Coronal and Heliospheric Investigation (SECCHI).  _Space Science Reviews_ , 136(1-4):67–115, April 2008.  doi: 10.1007/s11214-008-9341-4.
  * Isavnin [2016] A. Isavnin.  FRiED: A Novel Three-dimensional Model of Coronal Mass Ejections.  _Astrophys J_ , 833(2):267, December 2016.  doi: 10.3847/1538-4357/833/2/267.
  * Jin et al. [2017] M. Jin, W. B. Manchester, B. van der Holst, I. Sokolov, G. Tóth, A. Vourlidas, C. A. de Koning, and T. I. Gombosi.  Chromosphere to 1 AU Simulation of the 2011 March 7th Event: A Comprehensive Study of Coronal Mass Ejection Propagation.  _The Astrophysical Journal_ , 834(2):172, January 2017.  doi: 10.3847/1538-4357/834/2/172.
  * Jonas et al. [2018] E. Jonas, M. Bobra, V. Shankar, J. T. Hoeksema, and B. Recht.  Flare prediction using photospheric and coronal image data.  _Solar Phys._ , 293(3):48, Feb 2018.  ISSN 1573-093X.  doi: 10.1007/s11207-018-1258-9.
  * Kahler and Webb [2007] S. W. Kahler and D. F. Webb.  V arc interplanetary coronal mass ejections observed with the solar mass ejection imager.  _Journal of Geophysical Research: Space Physics_ , 112(A9), 2007.  doi: https://doi.org/10.1029/2007JA012358.  URL https://agupubs.onlinelibrary.wiley.com/doi/abs/10.1029/2007JA012358.
  * Kaiser et al. [2008] M. L. Kaiser, T. A. Kucera, J. M. Davila, O. C. St. Cyr, M. Guhathakurta, and E. Christian.  The STEREO Mission: An Introduction.  _Space Science Reviews_ , 136(1-4):5–16, April 2008.  doi: 10.1007/s11214-007-9277-0.
  * Kim et al. [2020] T. K. Kim, N. V. Pogorelov, C. N. Arge, C. J. Henney, S. I. Jones-Mecholsky, W. P. Smith, S. D. Bale, J. W. Bonnell, T. Dudok de Wit, K. Goetz, P. R. Harvey, R. J. MacDowall, D. M. Malaspina, M. Pulupa, J. C. Kasper, K. E. Korreck, M. Stevens, A. W. Case, P. Whittlesey, R. Livi, D. E. Larson, K. G. Klein, and G. P. Zank.  Predicting the Solar Wind at the Parker Solar Probe Using an Empirically Driven MHD Model.  _Astrophys J Suppl_ , 246(2):40, February 2020.  doi: 10.3847/1538-4365/ab58c9.
  * King and Papitashvili [2005] J. H. King and N. E. Papitashvili.  Solar wind spatial scales in and comparisons of hourly wind and ace plasma and magnetic field data.  _Journal of Geophysical Research: Space Physics_ , 110(A2), 2005.  doi: 10.1029/2004JA010649.  URL https://agupubs.onlinelibrary.wiley.com/doi/abs/10.1029/2004JA010649.
  * Li et al. [2007] R. Li, H. Wang, H. He, Y. Cui, and Z. Du.  Support vector machine combined with k-nearest neighbors for solar flare forecasting.  _Chinese.J.Astron.Ast._ , 7(3):441–447, jun 2007\.  doi: 10.1088/1009-9271/7/3/15.
  * Liu et al. [2006] Y. Liu, J. D. Richardson, J. W. Belcher, C. Wang, Q. Hu, and J. C. Kasper.  Constraints on the global structure of magnetic clouds: Transverse size and curvature.  _Journal of Geophysical Research: Space Physics_ , 111(A12), 2006.  doi: 10.1029/2006JA011890.  URL https://agupubs.onlinelibrary.wiley.com/doi/abs/10.1029/2006JA011890.
  * Liu et al. [2014] Y. Liu, J. T. Hoeksema, and X. Sun.  Test of the Hemispheric Rule of Magnetic Helicity in the Sun Using the Helioseismic and Magnetic Imager (HMI) Data.  _Astrophys J Lett_ , 783(1):L1, March 2014.  doi: 10.1088/2041-8205/783/1/L1.
  * Liu et al. [2010] Ying Liu, Jackie A. Davies, Janet G. Luhmann, Angelos Vourlidas, Stuart D. Bale, and Robert P. Lin.  Geometric Triangulation of Imaging Observations to Track Coronal Mass Ejections Continuously Out to 1 AU.  _Astrophys J Lett_ , 710(1):L82–L87, Feb 2010.  doi: 10.1088/2041-8205/710/1/L82.
  * Lugaz et al. [2009] N. Lugaz, A. Vourlidas, and I. I. Roussev.  Deriving the radial distances of wide coronal mass ejections from elongation measurements in the heliosphere - application to CME-CME interaction.  _Annales Geophysicae_ , 27(9):3479–3488, September 2009.  doi: 10.5194/angeo-27-3479-2009.
  * Lugaz et al. [2010] N. Lugaz, J. N. Hernandez-Charpak, I. I. Roussev, C. J. Davis, A. Vourlidas, and J. A. Davies.  DETERMINING THE AZIMUTHAL PROPERTIES OF CORONAL MASS EJECTIONS FROM MULTI-SPACECRAFT REMOTE-SENSING OBSERVATIONS WITHSTEREOSECCHI.  _The Astrophysical Journal_ , 715(1):493–499, apr 2010.  doi: 10.1088/0004-637x/715/1/493.  URL https://doi.org/10.1088%2F0004-637x%2F715%2F1%2F493.
  * Luoni et al. [2011] M. L. Luoni, P. Démoulin, C. H. Mandrini, and L. van Driel-Gesztelyi.  Twisted Flux Tube Emergence Evidenced in Longitudinal Magnetograms: Magnetic Tongues.  _Solar Physics_ , 270(1):45–74, May 2011.  doi: 10.1007/s11207-011-9731-8.
  * Manchester IV et al. [2004] Ward B. Manchester IV, Tamas I. Gombosi, Ilia Roussev, Darren L. De Zeeuw, I. V. Sokolov, Kenneth G. Powell, Gábor Tóth, and Merav Opher.  Three-dimensional mhd simulation of a flux rope driven cme.  _Journal of Geophysical Research: Space Physics_ , 109(A1), 2004.  doi: 10.1029/2002JA009672.  URL https://agupubs.onlinelibrary.wiley.com/doi/abs/10.1029/2002JA009672.
  * Manoharan et al. [2004] P. K. Manoharan, N. Gopalswamy, S. Yashiro, A. Lara, G. Michalek, and R. A. Howard.  _Journal of Geophysical Research (Space Physics)_ , 109(A6):A06109, Jun 2004.  doi: 10.1029/2003JA010300.
  * McGregor et al. [2011] S. L. McGregor, W. J. Hughes, C. N. Arge, M. J. Owens, and D. Odstrcil.  The distribution of solar wind speeds during solar minimum: Calibration for numerical solar wind modeling constraints on the source of the slow solar wind.  _Journal of Geophysical Research (Space Physics)_ , 116(A3):A03101, March 2011.  doi: 10.1029/2010JA015881.
  * Montgomery et al. [2006] Douglas C. Montgomery, Elizabeth A. Peck, and Geoffrey G. Vining.  _Introduction to Linear Regression Analysis (4th ed.)_.  Wiley & Sons, 2006.  ISBN 0471754951.  URL https://www.bibsonomy.org/bibtex/24187b5938a0ffe49b30bb53ef9808edb/beyondhor1zon.
  * Moon et al. [2002] Y. J. Moon, M. Dryer, Z. Smith, Y. D. Park, and K. S. Cho.  _Geophys Res Lett_ , 29(10):1390, May 2002.  doi: 10.1029/2002GL014865.
  * Pedregosa et al. [2011] Fabian Pedregosa, Gaël Varoquaux, Alexandre Gramfort, Vincent Michel, Bertrand Thirion, Olivier Grisel, Mathieu Blondel, Andreas Müller, Joel Nothman, Gilles Louppe, Peter Prettenhofer, Ron Weiss, Vincent Dubourg, Jake Vanderplas, Alexandre Passos, David Cournapeau, Matthieu Brucher, Matthieu Perrot, and Édouard Duchesnay.  Scikit-learn: Machine Learning in Python.  _Journal of Machine Learning Research_ , 12:2825–2830, October 2011.  doi: 10.48550/arXiv.1201.0490.
  * Pesnell et al. [2012] W. Dean Pesnell, B. J. Thompson, and P. C. Chamberlin.  The Solar Dynamics Observatory (SDO).  _Solar Physics_ , 275(1-2):3–15, January 2012\.  doi: 10.1007/s11207-011-9841-3.
  * Pogorelov et al. [2014] Nikolai Pogorelov, Sergey Borovikov, Jacob Heerikhuisen, Tae Kim, Igor Kryukov, and Gary Zank.  Ms-flukss and its application to modeling flows of partially ionized plasma in the heliosphere.  In _Proceedings of the 2014 Annual Conference on Extreme Science and Engineering Discovery Environment_ , XSEDE ’14, pages 22:1–22:8, New York, NY, USA, 2014. ACM.  ISBN 978-1-4503-2893-7.  doi: 10.1145/2616498.2616499.  URL http://doi.acm.org/10.1145/2616498.2616499.
  * Qahwaji and Colak [2007] R. Qahwaji and T. Colak.  Automatic short-term solar flare prediction using machine learning and sunspot associations.  _Solar Phys._ , 241(1):195–211, Mar 2007.  ISSN 1573-093X.  doi: 10.1007/s11207-006-0272-5.
  * Qiu et al. [2007] Jiong Qiu, Qiang Hu, Timothy A. Howard, and Vasyl B. Yurchyshyn.  On the magnetic flux budget in low-corona magnetic reconnection and interplanetary coronal mass ejections.  _The Astrophysical Journal_ , 659(1):758–772, apr 2007.  doi: 10.1086/512060.  URL https://doi.org/10.1086%2F512060.
  * Qu et al. [2003] M. Qu, F. Y. Shih, J. Jing, and H. Wang.  Automatic solar flare detection using mlp, rbf, and svm.  _Solar Phys._ , 217(1):157–172, Oct 2003.  ISSN 1573-093X.  doi: 10.1023/A:1027388729489.
  * Riley et al. [1997] Pete Riley, J. T. Gosling, and V. J. Pizzo.  A two‐dimensional simulation of the radial and latitudinal evolution of a solar wind disturbance driven by a fast, high‐pressure coronal mass ejection.  _Journal of Geophysical Research: Space Physics_ , 102(A7):14677–14685, 1 1997.  ISSN 0148-0227.  doi: 10.1029/97JA01131.  URL https://doi.org/10.1029/97JA01131.
  * Riley et al. [2008] Pete Riley, Roberto Lionello, Zoran Mikić, and Jon Linker.  Using global simulations to relate the three-part structure of coronal mass ejections to in situ signatures.  _The Astrophysical Journal_ , 672(2):1221–1227, jan 2008.  doi: 10.1086/523893.  URL https://doi.org/10.1086/523893.
  * Riley et al. [2018] Pete Riley, M. Leila Mays, Jesse Andries, Tanja Amerstorfer, Douglas Biesecker, Veronique Delouille, Mateja Dumbović, Xueshang Feng, Edmund Henley, Jon A. Linker, Christian Möstl, Marlon Nuñez, Vic Pizzo, Manuela Temmer, W. K. Tobiska, C. Verbeke, Matthew J West, and Xinhua Zhao.  _Space Weather_ , 16(9):1245–1260, 2018.  doi: 10.1029/2018SW001962.  URL https://agupubs.onlinelibrary.wiley.com/doi/abs/10.1029/2018SW001962.
  * Rollett et al. [2016] T. Rollett, C. Möstl, A. Isavnin, J. A. Davies, M. Kubicka, U. V. Amerstorfer, and R. A. Harrison.  ElEvoHI: A NOVEL CME PREDICTION TOOL FOR HELIOSPHERIC IMAGING COMBINING AN ELLIPTICAL FRONT WITH DRAG-BASED MODEL FITTING.  _The Astrophysical Journal_ , 824(2):131, jun 2016\.  doi: 10.3847/0004-637x/824/2/131.  URL https://doi.org/10.3847/0004-637x/824/2/131.
  * Rumelhart et al. [1986] David E. Rumelhart, Geoffrey E. Hinton, and Ronald J. Williams.  Learning Representations by Back-propagating Errors.  _Nature_ , 323(6088):533–536, 1986.  doi: 10.1038/323533a0.  URL http://www.nature.com/articles/323533a0.
  * Schatten [1972] Kenneth H. Schatten.  Current Sheet Magnetic Model for the Solar Corona.  In Charles P. Sonett, Paul Jerome Coleman, and John Marsh Wilcox, editors, _NASA Special Publication_ , volume 308, page 44. 1972\.  URL https://ui.adsabs.harvard.edu/abs/1972NASSP.308...44S.
  * Schou et al. [2012] J. Schou, P. H. Scherrer, R. I. Bush, R. Wachter, S. Couvidat, M. C. Rabello-Soares, R. S. Bogart, J. T. Hoeksema, Y. Liu, T. L. Duvall, D. J. Akin, B. A. Allard, J. W. Miles, R. Rairden, R. A. Shine, T. D. Tarbell, A. M. Title, C. J. Wolfson, D. F. Elmore, A. A. Norton, and S. Tomczyk.  Design and Ground Calibration of the Helioseismic and Magnetic Imager (HMI) Instrument on the Solar Dynamics Observatory (SDO).  _Solar Physics_ , 275(1-2):229–259, January 2012\.  doi: 10.1007/s11207-011-9842-2.
  * Scolini et al. [2019] C. Scolini, L. Rodriguez, M. Mierla, J. Pomoell, and S. Poedts.  Observation-based modelling of magnetised coronal mass ejections with euhforia.  _A &A_, 626:A122, 2019.  doi: 10.1051/0004-6361/201935053.  URL https://doi.org/10.1051/0004-6361/201935053.
  * Singh et al. [2019] T. Singh, M. S. Yalim, N. V. Pogorelov, and N. Gopalswamy.  Simulating Solar Coronal Mass Ejections Constrained by Observations of Their Speed and Poloidal Flux.  _The Astrophysical Journal_ , 875(2):L17, apr 2019\.  doi: 10.3847/2041-8213/ab14e9.  URL https://doi.org/10.3847%2F2041-8213%2Fab14e9.
  * Singh et al. [2020a] Talwinder Singh, Tae K. Kim, Nikolai V. Pogorelov, and Charles N. Arge.  Application of a Modified Spheromak Model to Simulations of Coronal Mass Ejection in the Inner Heliosphere.  _Space Weather_ , 18(5):e02405, May 2020a.  doi: 10.1029/2019SW002405.
  * Singh et al. [2020b] Talwinder Singh, Mehmet S. Yalim, Nikolai V. Pogorelov, and Nat Gopalswamy.  A modified spheromak model suitable for coronal mass ejection simulations.  _The Astrophysical Journal_ , 894(1):49, may 2020b.  doi: 10.3847/1538-4357/ab845f.  URL https://dx.doi.org/10.3847/1538-4357/ab845f.
  * Singh et al. [2022] Talwinder Singh, Tae K. Kim, Nikolai V. Pogorelov, and Charles N. Arge.  Ensemble simulations of the 2012 july 12 coronal mass ejection with the constant-turn flux rope model.  _The Astrophysical Journal_ , 933(2):123, jul 2022\.  doi: 10.3847/1538-4357/ac73f3.  URL https://dx.doi.org/10.3847/1538-4357/ac73f3.
  * Thernisien [2011] A. Thernisien.  Implementation of the Graduated Cylindrical Shell Model for the Three-dimensional Reconstruction of Coronal Mass Ejections.  _apjs_ , 194(2):33, June 2011.  doi: 10.1088/0067-0049/194/2/33.
  * Thernisien et al. [2009] A. Thernisien, A. Vourlidas, and R. A. Howard.  Forward Modeling of Coronal Mass Ejections Using STEREO/SECCHI Data.  _Sol Phys_ , 256(1-2):111–130, May 2009.  doi: 10.1007/s11207-009-9346-5.
  * Tibshirani [1996] Robert Tibshirani.  Regression shrinkage and selection via the lasso.  _Journal of the Royal Statistical Society. Series B (Methodological)_ , 58(1):267–288, 1996.  ISSN 00359246.  URL http://www.jstor.org/stable/2346178.
  * Vandas and Romashets [2017] M. Vandas and E. Romashets.  Magnetic cloud fit by uniform-twist toroidal flux ropes.  _Astro and Astrophys_ , 608:A118, December 2017.  doi: 10.1051/0004-6361/201731412.
  * Vandas et al. [1996] M. Vandas, S. Fischer, M. Dryer, Z. Smith, and T. Detman.  Simulation of magnetic cloud propagation in the inner heliosphere in two dimensions 2. A loop parallel to the ecliptic plane and the role of helicity.  _J Geophys Res_ , 101(A2):2505–2510, Feb 1996.  doi: 10.1029/95JA02446.
  * Vršnak and Zic [2007] B. Vršnak and T. Zic.  Transit times of interplanetary coronal mass ejections and the solar wind speed.  _A &A_, 472(3):937–943, 2007.  doi: 10.1051/0004-6361:20077499.  URL https://doi.org/10.1051/0004-6361:20077499.
  * Vršnak et al. [2014] B. Vršnak, M. Temmer, T. Žic, A. Taktakishvili, M. Dumbović, C. Möstl, A. M. Veronig, M. L. Mays, and D. Odstrčil.  HELIOSPHERIC PROPAGATION OF CORONAL MASS EJECTIONS: COMPARISON OF NUMERICAL WSA-ENLIL$+$CONE MODEL AND ANALYTICAL DRAG-BASED MODEL.  _The Astrophysical Journal Supplement Series_ , 213(2):21, jul 2014.  doi: 10.1088/0067-0049/213/2/21.  URL https://doi.org/10.1088%2F0067-0049%2F213%2F2%2F21.
  * Vršnak [2001] Bojan Vršnak.  _Sol Phys_ , 202(1):173–189, Aug 2001.  doi: 10.1023/A:1011833114104.
  * Vršnak and Gopalswamy [2002] Bojan Vršnak and Nat Gopalswamy.  _Journal of Geophysical Research (Space Physics)_ , 107(A2):1019, Feb 2002.  doi: 10.1029/2001JA000120.
  * Wang et al. [2008] H. N. Wang, Y. M. Cui, R. Li, L. Y. Zhang, and H. Han.  Solar flare forecasting model supported with artificial neural network techniques.  _Adv.Space Res._ , 42(9):1464 – 1468, 2008.  ISSN 0273-1177.  doi: https://doi.org/10.1016/j.asr.2007.06.070.
  * Wang et al. [2002] Y. M. Wang, P. Z. Ye, S. Wang, G. P. Zhou, and J. X. Wang.  _Journal of Geophysical Research: Space Physics_ , 107(A11):SSH 2–1–SSH 2–9, 2002.  doi: 10.1029/2002JA009244.
  * Yuan et al. [2010] Y. Yuan, F. Y. Shih, J. Jing, and H. Wang.  Automated flare forecasting using a statistical learning technique.  _Res.Astron.Astrophys_ , 10(8):785–796, jul 2010\.  doi: 10.1088/1674-4527/10/8/008.