# Drag-based model (DBM) tools for forecast of coronal mass ejection arrival
time and speed

Mateja Dumbović 1,∗, Jaša Čalogović 1, Karmen Martinić 1, Bojan Vršnak 1,
Davor Sudar 1, Manuela Temmer 2, and Astrid Veronig 2,3

###### Abstract

## 1

Forecasting the arrival time of coronal mass ejections (CMEs) and their
associated shocks is one of the key aspects of space weather research. One of
the commonly used models is, due to its simplicity and calculation speed, the
analytical drag-based model (DBM) for heliospheric propagation of CMEs. DBM
relies on the observational fact that slow CMEs accelerate whereas fast CMEs
decelerate, and is based on the concept of MHD drag, which acts to adjust the
CME speed to the ambient solar wind. Although physically DBM is applicable
only to the CME magnetic structure, it is often used as a proxy for the shock
arrival. In recent years, the DBM equation has been used in many studies to
describe the propagation of CMEs and shocks with different geometries and
assumptions. Here we give an overview of the five DBM versions currently
available and their respective tools, developed at Hvar Observatory and
frequently used by researchers and forecasters. These include: 1) basic 1D
DBM, a 1D model describing the propagation of a single point (i.e. the apex of
the CME) or concentric arc (where all points propagate identically); 2)
advanced 2D self-similar cone DBM, a 2D model which combines basic DBM and
cone geometry describing the propagation of the CME leading edge which evolves
self-similarly; 3) 2D flattening cone DBM, a 2D model which combines basic DBM
and cone geometry describing the propagation of the CME leading edge which
does not evolve self-similarly; 4) DBEM, an ensemble version of the 2D
flattening cone DBM which uses CME ensembles as an input and 5) DBEMv3, an
ensemble version of the 2D flattening cone DBM which creates CME ensembles
based on the input uncertainties. All five versions have been tested and
published in recent years and are available online or upon request. We provide
an overview of these five tools, of their similarities and differences, as
well as discuss and demonstrate their application.

##  2 Keywords:

coronal mass ejections, solar wind, interplanetary shocks,
magnetohydrodynamical drag, space weather forecast

##  3 Introduction

Coronal mass ejections (CMEs) are one of most prominent drivers of space
weather in the heliosphere. They are the causes of largest geomagnetic storms
[e.g. 1] as they may carry enhanced and specifically oriented magnetic fields
[see e.g. 2, 3]. Forecasting the arrival time of CMEs and their associated
shocks is therefore one of the key aspects of space weather research.
Therefore, there is a diversity of CME models available today, some focusing
on the arrival time forecast only and other, more complex models include
forecast of other CME properties as well [see e.g. 4, 5, 6, 7, and references
therein].

Propagation of CMEs in the heliosphere with the purpose of obtaining the CME
time of arrival (ToA) and speed of arrival (SoA), can be modelled by empirical
models [e.g. 8, 9], kinematic shock propagation models [e.g. 10, 11, 12],
machine-learning models [e.g. 13, 14], numerical 3D MHD models (e.g. H3DMHD
model by Wu et al. [15], WSA-ENLIL+Cone model by Odstrcil et al. [16],
EUHFORIA model by Pomoell and Poedts [17], CORHEL model by Mikić et al. [18]
or AWSoM model by van der Holst et al. [19]) and drag-based models (see
below). All CME propagation models need CME input as well as input of
characteristics of the background solar wind, where both may have large
uncertainties. Therefore, it is not surprising that despite their differences,
ToA errors of different propagation models revolve around 10 hours [20, 6].

One of the most popular CME propagation setups used in forecast models today
is the drag-based propagation. In this concept, the CME which is initially
under the influence of Lorentz force, gravity and drag-force due to
interaction with the ambient medium, at a certain distance from the Sun is
influenced dominantly by the drag force [see e.g. 21, 22, and references
therein]. This concept is supported by the observational fact that slow CMEs
accelerate whereas fast CMEs decelerate [23, 24, 25]. The drag force can be
represented by the aerodynamic drag equation describing the kinetic drag
effect in a fluid [26, 27], however it should be noted that in the IP space,
i.e. collisionless solar wind environment, the drag is caused primarily by the
emission of MHD waves and not particle collisions [28].

Drag-based models typically use the same form of the basic drag equation
applied to various geometries representing the CME structure of different
dimensionality, e.g. 1D Drag-Based Model [DBM, 29, 30] and Enhanced DBM [31,
32], 2D Drag-Based Model [33], the 2D Ellipse Evolution Model [ElEvo, 34] and
a version of ElEvo using data from Heliospheric Imagers [ElEvoHi, 35], and 3D
flux rope models such as ANother Type of Ensemble Arrival Time Results
[ANTEATR, 36] or 3-Dimensional Coronal ROpe Ejection [3DCORE, 37]. Since drag-
based models use an analytical equation to describe the time-dependent
evolution of the CME, they are computationally efficient and thus widely used
in probabilistic/ensemble modelling approaches [e.g. 38, 39, 40, 36, 41, 42].
The advantage of ensemble modelling is that it gives the probability of
arrival, as well as the range of possible arrival times and speeds.

Starting with a basic 1D DBM [29], five versions of the drag-based model
versions have been developed by the Hvar Observatory solar and heliospheric
group in close collaboration with the solar and heliospheric group at the
University of Graz. These five versions include three different geometries, as
discussed in Section 4.2, and two different ensemble versions, as discussed in
Section 4.3. We give an overview of these five DBM versions and their
respective tools in Section 4 and demonstrate their application on a real
event in Section 5.

##  4 Overview of Drag-based model tools

###  4.1 The basic description of the model

The Drag-Based Model (DBM) tools are all based on the equation of motion
analogous to the aerodynamic drag:

$a(t)=-\gamma(v(t)-w)|v(t)-w|\,,$ (1)

where $a(t)=\mathrm{d}^{2}R(t)/\mathrm{d}t^{2}$ is the CME acceleration,
$v(t)=\mathrm{d}R(t)/\mathrm{d}t$ is the CME speed, $R(t)$ is the heliospheric
distance, $\gamma$ is the drag parameter which describes CME speed change-rate
and is assumed to be constant, and $w$ is the solar wind speed, also assumed
to be constant. Along with the initial properties of the CME, which can be
obtained from the coronagraphic observation, $\gamma$ and $w$ have to be
specified to obtain analytical solutions of Equation 1 for a specific CME,
$R(t)$ and $v(t)$, given by Vršnak et al. [29]:

$\begin{split}R(t)=\frac{S}{\gamma}ln[1+S\gamma(v_{0}-w)t]+wt+R_{0}\\\
v(t)=\frac{v_{0}-w}{1+S\gamma(v_{0}-w)t}+w\,,\end{split}$ (2)

where $v_{0}=v(t=0)$ is the initial CME speed, $R_{0}=R(t=0)$ the
corresponding starting radial distance, and S is a sign function ($S=1$ for
$v_{0}>w$, $S=-1$ for $v_{0}<w$). These solutions describe the time-dependent
part of the drag-based CME propagation and are thus the same in all tools,
regardless of different geometries.

We note that generally speaking $\gamma$ and $w$ are not constant in time.
However, it can be shown that at a sufficient distance from the Sun $\gamma$
and $w$ become approximetely constant and may be represented by their
asymptotic values, which are approximately equal to the values at 1 AU [see
27, 29, 33, 43, for details]. Theoretically, the distance at which
$\gamma=const.$ and $w=const.$ assumptions should hold is beyond $\approx
15\mathrm{R_{\odot}}$ [33]. On the other hand, the distance at which the drag
force becomes dominant varies case-to-case, and is farther away for slower
CMEs [44, 45, 25]. Therefore, $R_{0}\leq 15\mathrm{R_{\odot}}$ might not be an
optimal choice for the model. The Lorentz force was found to generally peak
between 1.65 and 2.45 $\mathrm{R_{\odot}}$ and becomes negligible for faster
CMEs at $3.5-4\mathrm{R_{\odot}}$, for slower at $12-50\mathrm{R_{\odot}}$
[46]. As the optimal value for the starting radial distance the DBM tools
recommendation is $\geq 20\mathrm{R_{\odot}}$, as this assumption was shown to
be valid for a number of cases [47, 29] and should hold unless the CME is very
slow and/or has a prolonged acceleration phase [44, 45, 25, 46]. Nevertheless,
it is recommended to always check whether the early CME kinematics indicates
that the starting radial distance $\geq 20\mathrm{R_{\odot}}$ is suitable for
the observed CME.

####  4.1.1 Using empirical $w$ and $\gamma$ values in DBM

An important DBM issue is how to determine the input for $w$ and $\gamma$.
There are several options to determine the input for $w$: 1) using an
empirical value determined from statistical analysis; 2) using a solar wind
model (e.g. numerical heliospheric model, an empirical model based on CH
observation or persistence model); 3) using the solar-wind speed based on the
in-situ measurements at 1 AU at the time of the ICME take-off. Using $w$ based
on the in-situ measurements at the time of the ICME take-off was shown to be
the same or even worse than using empirically obtained values [29].
Statistical analysis has shown that the most appropriate values for $w$ should
be in the range $300-600\,\mathrm{km\,s}^{-1}$, with
$w=500\,\mathrm{km\,s}^{-1}$ as the optimal value [i.e. applicable to the
broadest subset of CMEs 29]. However, the optimal empirically derived value is
sample-dependent and was found to be lower for a different sample [30].
Therefore, as an optimal empirically-based value $w=450\,\mathrm{km\,s}^{-1}$
is set for all tools. Recent analysis has shown that this value seems optimal
even during the conditions of low solar activity [48]. It should be noted that
this value might not be valid if there is an equatorial coronal hole in the
vicinity of the CME source region, where one should apply a higher value to
take into account CME propagation through the high speed stream. For that
purpose one can use a model of the solar wind speed, where empirical solar
wind models are especially suitable, due to their simplicity and speed. DBM
tools available at the European Space Agency (ESA) Space Situational Awareness
(SSA) portal can be coupled with the Empirical Solar Wind Forecast tool, which
is based on empirical modelling of the high-speed stream (HSS) arrival derived
from coronal hole area observations [see 49, 50, 51, 52].

The $\gamma$ parameter is given by the expression [e.g. 29]:

$\gamma=\frac{c_{\mathrm{d}}A\rho_{\mathrm{w}}}{M+M_{\mathrm{v}}}=\frac{c_{\mathrm{d}}}{L(\frac{\rho}{\rho_{\mathrm{w}}}+\frac{1}{2})}\,,$
(3)

where $A$ is the CME cross-sectional area, $\rho_{\mathrm{w}}$ is the solar-
wind density, $M$ is the CME mass, $M_{\mathrm{v}}$ is the so-called virtual
mass (i.e. the mass of the material piled-up in front of the CME, $L$ is the
CME thickness in the radial direction, $\rho$ is the CME density and
$c_{\mathrm{d}}$ is the dimensionless drag coefficient, which in the DBM tools
is taken to be 1 according to [26]. Theoretically, it is possible to estimate
relative CME mass density and radial size to determine $\gamma$ based on
coronagraphic measurements. However, the errors corresponding to these
estimations [$\approx 15\%$ for the mass 53] can yield $\gamma$ with a very
large uncertainty. For a CME several times denser than the surrounding corona
(e.g. $\rho/\rho_{\mathrm{w}}\approx 5$), and of radial size
$1-10\mathrm{R}_{\odot}$ one finds approximate range of $\gamma=0.2-2\cdot
10^{-7}\,\,\mathrm{km}^{-1}$, which roughly corresponds to the range obtained
from statistical analysis [29]. The distribution of the $\gamma$ obtained from
statistical analysis is highly asymmetrical and weighted towards the lower
values [29], where $\gamma=0.2\cdot 10^{-7}\,\,\mathrm{km}^{-1}$ was found as
an optimal value in combination with $w=450\,\mathrm{km\,s}^{-1}$ [29, 30].
Therefore, this value has been chosen as optimal empirically-based value for
DBM tools (of course customized values are allowed).

In addition, some of the DBM tools offer $\gamma$ options for slower and
faster CMEs. Observationally, CME peak speed is related to the peak soft X-ray
flux [54, 55], and the flare fluence to the CME mass [56, 57]. This is
interpreted in the context of a feedback relationship between the CME dynamics
and the reconnection process in the wake of the CME [58]. Consequently, we
would expect faster CMEs to be more massive and thus expect lower $\gamma$ for
faster CMEs and higher $\gamma$ for slower CMEs. Additional empirical-based
fine-tuning of the $\gamma$ parameter may be performed by the user according
to the relative CME brightness in the coronagraphic images, which is generally
related to the CME mass [see e.g. 59, and references therein]. Massive CMEs
are generally observed as brighter objects in the coronagraphic images,
therefore one may use a lower value of $\gamma$ in case of very bright CMEs or
increase it for very faint CMEs. However, one needs to keep in mind that the
observed intensity of a CME (and thus mass calculation) depends on the angle
between the line-of-sight of the observer and the plane-of-sight, i.e. CME
direction with respect to the Thomson surface [for details see 60, 59].
Finally, fine-tuning of the $\gamma$ parameter may be performed to account for
the pre-conditioning of the interplanetary space due to preceding CME(s).
Namely, preceding CME(s) may ”deplete“ the heliospheric sector before the CME
in question, resulting in lower density and thus lower drag forces [61, 62,
63]. This effect can be taken into account by using a lower value for the
$\gamma$ parameter [see e.g. 61, 64]. However, when ”customising“ $\gamma$ one
needs to be careful not to underestimate or overestimate it, as this can lead
to underestimation or overestimation of the transit time, respectively. It was
recently shown by Paouris et al. [65] that underestimated $\gamma$ can lead to
significant underestimation of the transit time, even if $w$ is
underestimated, especially for the fast CMEs.

####  4.1.2 Running DBM for shock propagation

While propagating in the interplanetary space CMEs may or may not drive
shocks, however if they do, the arrival of the CME magnetic structure (i.e.
ejected twisted magnetic structure) is preceded by the shock arrival [for ICME
overview see e.g. 66, 67]. Physically, the DBM equation of motion describes
the propagation of the CME magnetic structure and not of the associated shock.
However, the comparison of the DBM with the heliospheric model ENLIL [16], in
which the CME is initiated as a pressure pulse and thus more suitable to track
the shock front, has shown that there is in general a good agreement between
the two when a lower value of the $\gamma$ parameter is applied [30].
Moreover, Hess and Zhang [32] have found that both the shock front and the CME
leading edge can be modelled in the heliosphere with a drag model, where the
CME ejecta front undergoes a more rapid deceleration than the shock front and
the propagation of the two fronts is not completely coupled in the
heliosphere. Indeed, some drag-based models such as the ElEvo [34] and ElEvoHi
[35] standardly follow the shock front. Dumbović et al. [39] have also used a
lower $\gamma$ value ($\gamma=0.1\cdot 10^{-7}\,\mathrm{\,}{km}^{-1}$) to
apply the DBM ensemble version to simulate CME shock propagation, whereas
Temmer and Nitta [61] and Guo et al. [68] have used DBM to model both shock
and CME propagation separately, using both different input and a lower
$\gamma$ value for the shock propagation. Therefore, we note that DBM tools
can be used to simulate both CME and shock propagation, however it is
important to keep in mind that: 1) the shock propagation is not necessarily
coupled to CME propagation; 2) proper CME/shock input is used and 3) lower
$\gamma$ values should be applied to the shock as compared to the CME
propagation.

###  4.2 DBM tools with different geometries

The basic form of the DBM was formulated by Vršnak and Žic [27], Vršnak et al.
[47] and analysed in detail by [29], where the basic 1D DBM tool was first
presented. The basic 1D version of DBM is available as an online tool at Hvar
Observatory webpage111http://oh.geof.unizg.hr/DBM/dbm.php and relies on
solutions given in Equation 2. Since it is a 1D equation it considers
propagation of a single point, i.e. CME apex. The tool is also applicable to
determine the propagation of an arbitrary, non-apex point of the CME leading
edge, assuming that the CME leading edge evolves self-similarly as a circular
arc concentric with the solar surface (i.e. all elements of the ICME front
have the same heliocentric distance). As can be seen in Figures 1 and 2, this
concentric geometry results in a self-similarly evolving CME leading edge.
However, since the tool does not consider CME angular extent or its direction,
it does not provide information whether or not this point hits a specific
target. The basic assumptions, input, output and tool specifications are given
in the second column of Table 1.

Figure 1: The differences between the fronts at 6 arbitrarily chosen time-
steps for the three DBM tools: basic 1D DBM (the concentric leading edge,
blue), advanced 2D DBM with self-similar cone geometry (green) and the
advanced 2D DBM with flattening cone geometry (red). The subplots show the
position of the leading edge in the XY coordinate system (i.e. the solar
equatorial plane). The heliospheric distance of the apex, $R_{a}$ is
highlighted in each time-step. The following DBM parameters were used to
create the plots: initial CME speed of $1000\mathrm{\,}{km\,s}^{-1}$, initial
distance of $20\mathrm{R}_{\odot}$, $\gamma$ of $0.2\cdot
10^{-7}\mathrm{\,}{km}^{-1}$ and solar wind speed $w$ of
$450\mathrm{\,}{km\,s}^{-1}$.

The advanced form of the DBM was formulated by [33], who applied a 2D cone
geometry to the basic 1D DBM solutions given in Equation 2. The cone geometry
was selected as it is a standard geometry used in heliospheric models such as
ENLIL [16] or EUHFORIA [17] and therefore their input would be suitable for
use in DBM as well. The cone angular dependence is introduced in DBM in the
following form:

$\begin{split}R(\alpha)=R_{0}\frac{\cos\alpha+\sqrt{\tan^{2}\omega-\sin^{2}\alpha}}{1+\tan\omega}\\\
v(\alpha)=v_{0}\frac{\cos\alpha+\sqrt{\tan^{2}\omega-\sin^{2}\alpha}}{1+\tan\omega}\,,\end{split}$
(4)

where $R_{0}$ and $v_{0}$ are distance and speed of the plasma element at the
CME apex, $\omega$ is the half-width of the cone (i.e. of the CME opening
angle) and $\alpha$ is the opening angle corresponding to the plasma element
in question. Depending on the application of the cone-geometry given by
Equation 4 to the basic 1D DBM solutions given in Equation 2, two different
evolutions of the CME leading edge are possible, self-similar cone evolution
and the flattening cone evolution.

The self-similar evolution of the cone leading edge is obtained assuming that
the CME front does not change its shape, i.e. when the DBM solutions for a
plasma element after time $t$ at the angular distance $\alpha$ from the apex
at the leading edge is given by:

$\begin{split}R(\alpha,t)=R_{0}(t)\frac{\cos\alpha+\sqrt{\tan^{2}\omega-\sin^{2}\alpha}}{1+\tan\omega}\\\
v(\alpha,t)=v_{0}(t)\frac{\cos\alpha+\sqrt{\tan^{2}\omega-\sin^{2}\alpha}}{1+\tan\omega}\,,\end{split}$
(5)

where $R_{0}(t)$ and $v_{0}(t)$ are given by Equation 2. The self-similar cone
leading edge is compared to the concentric geometry as well as the flattening
cone leading edge in Figures 1 and 2. It has been adopted in the online DBM
tool that runs on the Hvar Observatory
webpage222http://oh.geof.unizg.hr/DBM/dbm.php, as well as the Community
Coordinated Modelling Centre (CCMC)333https://ccmc.gsfc.nasa.gov. Since the
tool does implement information on the CME angular extent and its direction,
it also provides information of whether or not the CME hits the target.The
basic assumptions, input, output and tool specifications are given in the
third column of Table 1.

The flattening cone leading edge evolution is obtained by propagating each
plasma element of the CME leading edge independently, using the CME 2D cone
geometry given by Equation 4 as the initial leading edge. The DBM solutions
for a plasma element after time $t$ at the angular distance $\alpha$ from the
apex at the leading edge is given by:

$\begin{split}R(\alpha,t)=\frac{S}{\gamma}ln[1+S\gamma(v_{0}(\alpha)-w)t]+wt+R_{0}(\alpha)\\\
v(\alpha,t)=\frac{v_{0}(\alpha)-w}{1+S\gamma(v_{0}(\alpha)-w)t}+w\,,\end{split}$
(6)

where $R_{0}(\alpha)$ and $v_{0}(\alpha)$ are given by Equation 5. The
flattening cone leading edge is also shown in Figures 1 and 2 and similarly as
2D self-similar DBM provides information of whether or not the CME hits the
target. The basic assumptions, input, output and tool specifications are given
in the fourth column of Table 1.

To summarize, three different geometries of the CME leading edge (CME front)
are considered in DBM tools: concentric arc, self-similarly evolving cone, and
flattening cone. The differences between the fronts and their evolution for
the three tools described above are shown in Figure 1 for halfwidth
$<90^{\circ}$ at several arbitrarily chosen time-steps. The 6 subplots (a-f)
show the position of the leading edge in the XY coordinate system for 6
different time-steps. It can be seen that initially (at $t=0$) we differ only
2 geometries, the concentric arc and cone geometry leading edge. Although the
initial shape of the flattening cone leading edge is that of a 2D cone, at
$t>0$ the shape of the leading edge starts to deviate from the initial cone
shape more and more. This is because each plasma element of the leading edge
is propagated independently using different initial parameters. A plasma
element at the flank will have a lower value of the initial speed than e.g. a
plasma element at the apex and will therefore experience less drag if the CME
is faster than the solar wind and more drag if the CME is slower than the
solar wind. Since the drag will not act equally on each plasma element across
the leading edge, the evolution of the leading edge will not be self-similar.
Instead, as can be seen in Figure 1, during the evolution the leading edge
will gradually change from the initial cone shape towards a flatter shape.

This can be seen more prominently in Figure 2, which shows the time-evolution
of the curvature of the CME leading edge with respect to the center of the
Sun, calculated as $K=\mathrm{\Delta}\Theta/\mathrm{\Delta}L$, where
$\mathrm{\Delta}\Theta=|\Theta_{1}-\Theta_{2}|$ is the angular distance and
$\mathrm{\Delta}L=\int_{\Theta_{1}}^{\Theta_{2}}\sqrt{r^{2}+(\mathrm{d}r/\mathrm{d}\Theta)^{2}}\,\mathrm{d}\Theta$
is the corresponding arc length of the curve in polar coordinates. Note that
thus defined $K$ does not correspond to the standard mathematical term
curvature, which is defined with respect to the center of the circle and thus
remains always constant across the circular arc. Instead, we define quantity
$K$ to differ between the concentric arc and self-similar cone in the polar
coordinates with origin at the center of the Sun. We can see that $K$ of the
concentric arc is constant across the leading edge, whereas $K$ of the 2D cone
at the apex is identical to that of the concentric arc, but then increases
towards the flanks. However, the difference in $K$ between flank and apex
remains constant in time for a self-similarly evolving cone front, whereas it
reduces for the flattening cone front.

The last two subplots of Figure 2 show the ratio of the flank distance to the
apex distance, $R_{f}/R_{a}$ and the ratio of the curvature at the flank and
at the apex, $K_{f}/K_{a}$. For self-similarly evolving fronts $R_{f}/R_{a}$
and $K_{f}/K_{a}$ are constant and in the specific case of a concentric
leading edge both equal 1 (values at the flank are equal to the values at the
apex). We see that the apex evolves identically in all three cases. For a
self-similarly evolving cone $R_{f}/R_{a}$ and $K_{f}/K_{a}$ remain constant.
For the flattening cone $R_{f}/R_{a}$ and $K_{f}/K_{a}$ are not constant, as
the $R_{f}/R_{a}$ increases and $K_{f}/K_{a}$ decreases in time, both
approaching the values for the concentric leading edge. It should be noted
however, that they never actually reach the values for the concentric leading
edge. This is because although the flank experiences different drag than the
apex, it is slower than the apex. The difference between $R_{f}$ and $R_{a}$
is increasing, converging to a certain value, as the drag eventually adjusts
the speed of both the apex and the flank to the ambient solar wind speed. As
the distance from the Sun increases, the difference between $R_{f}$ and
$R_{a}$ becomes very small compared to values of $R_{f}$ and $R_{a}$,
therefore $R_{f}/R_{a}$ seems to converge to 1, although mathematically it
will never reach it and the flattening cone will never truly become a
concentric arc. It is also important to note that for halfwidth of
$90^{\circ}$ all three fronts are semi-circles with the origin at the Sun
center and thus evolve identically, as the concentric arc.

Figure 2: The differences between the curvatures of the fronts at different
time-steps for the three DBM tools: basic 1D DBM (the concentric leading edge,
blue), advanced 2D DBM with self-similar cone geometry (green) and the
advanced 2D DBM with flattening cone geometry (red). The time-steps and DBM
parameters correspond to those used in Figure 1. The subplots (a-f) show how
the curvature $K$ along the leading edge behaves in time. The curvature at the
apex, $K_{a}$ is highlighted in each time-step. The subplot g) shows how the
ratio of the flank heliospheric distance and apex heliospheric distance
($R_{f}/R_{a}$) evolves in time, whereas subplot h) shows how the ratio of the
curvature at the flank and at the apex ($K_{f}/K_{a}$) evolves in time.

###  4.3 The ensemble versions of the DBM

As noted in Section 3, drag-based models are computationally efficient and
thus widely used in probabilistic/ensemble modelling approaches. Ensemble
forecasting takes into account the errors and uncertainties of the input to
quantify the resulting uncertainties in the model predictions. The variability
of an observational input is introduced by making an ensemble, i.e. sets of
CME observations to calculate a distribution of predictions and forecast the
confidence in the likelihood of the prediction. This can be achieved in two
ways: 1) by taking independently built sets of CME observations (e.g. as
provided by different observers) or 2) by creating sets of CME observations
(by e.g. using measurements and error estimations provided by a single
observer). These two ensemble options were adopted in the Drag-based ensemble
model (DBEM) [39] and DBEMv3 web tool [48], respectively, which both use 2D
DBM with flattening cone as a background physical model. The output of both
tools is the probability of arrival, which is calculated as the ratio of the
number of runs that predict a hit and total number of runs. Based on the runs
that predict a hit, distributions of arrival time and speed are generated,
where the calculated medians represent the likeliest arrival time and speed,
and the uncertainty range is given by the $95\%$ confidence interval. We note
that initially there was a DBEMv2, which was replaced by a more advanced
DBEMv3.

For a single CME, DBEM uses an ensemble of $n$ measurements of the same CME,
which may not be mutually related in any way (e.g. it might be obtained by
different observers, different methods or even measurements from different
instruments). Each ensemble member has the same weight. Moreover, there is no
assumption that the CME measurements of a particular CME, i.e. ensemble
member, are independent of each other or that their spread in values follow a
certain distribution. The variability of solar wind speed, $w$, and drag
parameter, $\gamma$, is taken into account by producing $m$ of their synthetic
values. These synthetic values are combined with an ensemble of $n$ CME
measurements, to give a final ensemble of $n\cdot m^{2}$ members as an input,
which, after $n\cdot m^{2}$ runs, produces a distribution of $n\cdot m^{2}$
calculated CME transit times and arrival speeds. The synthetic values for $w$
and $\gamma$ are produced assuming that their real measurements follow a
normal distribution with a mean value and standard deviation serving as the
model input. A cumulative standard normal distribution is then generated,
defined on an interval $[0,m-1]$, where $m$ is the number of synthetic
measurements, also needed as the model input. The $m$ values which correspond
to the integer values of the cumulative standard normal distribution are
selected as synthetic measurements. This way, for identical distribution and
$m$, the selection always results in an identical set of synthetic values,
which include the tips of the distribution tail. Therefore, for small $m$ the
distribution of chosen synthetic measurements is too heavily weighted to the
tail compared to the normal distribution, and larger $m$ is needed for
synthetic measurements to be weighted properly, $m>15$ [39]. The basic
assumptions, input, output and tool specifications of DBEM are given in the
fifth column of Table 1.

In DBEMv3 the CME ensemble is not produced by the observer, but the tool.
Observational input values and uncertainties are provided for CME input as
well as $w$ and $\gamma$, from which the tool generates $m$ ensemble members.
Each ensemble member is produced by randomly picking one value for each input
parameter, assuming it follows a normal distribution with observational input
value as mean and standard deviation derived from uncertainty
(uncertainty$=3\sigma$). Due to this randomness (which cannot be controlled),
the ensemble is not likely to be identical each time identical input is used,
which produces small differences in the output of the model for different runs
using identical input. However, for large ensembles, $m>10000$, the
differences of the output are negligible [see 48]. The basic assumptions,
input, output and tool specifications of DBEMv3 are given in the last column
of Table 1.

We note that in the DBEMv3 the CME input parameters are considered to be
independent of each other and therefore the procedure is somewhat similar to
error propagation. In DBEM the CME parameters within one measurement set are
not necessarily independent of each other, CME sets are independent of each
other. This is important due to the nature of the model input used, i.e.
obtaining CME input from coronagraphic measurements. Coronagraphs only display
a projection of a 3D structure. Therefore, in order to derive parameters of a
3D CME, some assumptions need to be made on the CME geometry. These
assumptions as well as their application can vary from observer-to-observer
and result in CME measurement sets where the distribution of single parameter
variability may differ substantially from the normal distribution. A single
observer on the other hand, is more likely to provide CME measurements with
errors that follow a normal distribution. While a single observer is more
likely to bias the mean of the normal distribution of an input parameter and
thus introduce errors, we note that in the near-real-time forecasting, where a
quick estimation of the CME input is needed, DBEMv3 is more applicable, since
it uses input provided by only one observer/method.

##  5 Running the DBM tools: example event

Figure 3: April 3rd 2010 CME observed in running difference images in
STEREO/COR2 and SOHO/LASCO coronagraphs and the GCS reconstruction of the CME
(green mesh).

We demonstrate the performance of DBM tools described in Section 4 by running
all the tools using the same example event. As the example event we choose a
previously studied CME that erupted on April 3rd 2010 and hit Earth on April
5th 2010 [e.g. 69, 70, 71]. This event can be found in the SOHO/LASCO CME
catalog 444https://cdaw.gsfc.nasa.gov/CME_list/, where it is listed as a halo
with first appearance in LASCO-C2 April 3rd 2010 at 10:33 UT. In order to
reconstruct the flux rope structure of the CME we use the Graduated
cylindrical shell (GCS) model [72, 73, 74].

GCS is a geometrical model used to represent the flux rope structure of the
CME to study its three-dimensional morphology, position, and kinematics. The
flux rope is approximated with a self-similarly expanding hollow croissant
originating from the center of the Sun, where the legs are conical, cross
section circular, and the front pseudo-circular. The croissant is fully
defined by six GCS parameters, which are: 1) longitude, 2) latitude, 3) height
corresponding to the apex of the croissant, 4) the tilt of the croissant axis
to the solar equatorial plane, 5) the croissant half-angle measured between
the apex and the central axis of its leg, and 6) the ”aspect ratio“ (i.e. the
sine of the angle defining the ”thickness“ of the croissant leg). The GCS
parameters are obtained by fitting its 2D projections to the respective
coronagraphic images, where at least 2 different vantage points are needed to
constrain the geometry [for further details on GCS see 72, 73, 74].

To perform the reconstruction we use coronagraph images taken by SECCHI/COR2
[75] onboard STEREO-A and B, as well as LASCO-C3 [76] onboard SOHO spacecraft
at April 3rd 2010 10:54 UT. Furthermore, the GCS reconstruction was done in 4
consecutive time-steps, following the CME leading edge while changing only the
height parameter, i.e. assuming self-similar expansion. Based on these 4
measurements, a CME linear speed of $920\,\mathrm{km\,s}^{-1}$ was estimated.
The GCS best fit parameters and linear speed are given in the second column of
Table 2 and the reconstruction is shown in Figure 3.

Figure 4: Left: The projection of the GCS croissant of the April 3rd 2010 CME
in the XY plane of the Heliocentric Earth Equatorial (HEEQ) coordinate system
(i.e. plane of the solar equator). The direction of the apex is marked by a
solid black line. Right: The direction and angular extent of the April 3rd
2010 CME (black arrow and black arc, respectively) and the positions of the
three spacecraft in the HEEQ coordinate system at the CME liftoff time:
STEREO-B (blue), SOHO (green), and STEREO-A (red).

Based on the results of the GCS reconstruction we derive input for the DBM
tools (see column 4 in Table 2). Using the CME linear speed, we extrapolate
the CME apex to $R_{0}=20\mathrm{R}_{\odot}$ assuming constant speed, which is
taken as the initial CME speed, $v_{0}$. The CME angular extent (i.e. CME
angular half width, $\lambda$) in the solar equatorial plane was estimated
based on the GCS-derived tilt, as well as GCS face-on and edge-on widths, as
described by Dumbović et al. [64] and adopted in DBEMv3. The projection of the
GCS reconstructed CME in the solar equatorial plane is shown in Figure 4, as
well as the calculated CME angular extent and the positions of the spacecraft.
We can see that due to relatively small tilt and large half-angle, the angular
extent of the CME is quite large. In addition, we can see that the direction
of the apex (given by the longitude of the CME source region,
$\phi_{\mathrm{CME}}$) is very close to the Sun-Earth line. We next run DBM
tools for the input obtained from the GCS reconstruction (bottom rows of Table
2). In order to run DBEM, for which different sets of CME measurements are
needed as input, we utilize measurements from previous studies on this event
[77, 71] and from online catalogs provided by the Advanced Forecast For
Ensuring Communications Through Space (AFFECTS)555http://www.affects-
fp7.eu/home/ catalogs. We use CME input provided by the AFFECTS-GCS database
and AFFECTS-CAT database, where the latter is obtained with the CME Analysis
Tool (CAT) modeling technique developed by Millward et al. [78]. We also run
all other DBM tools for these various CME inputs (as given in columns (5-8).
The DBM input for these CME measurements was derived the same way as for the
GCS reconstruction performed here. The CME speed for AFFECTS-GCS catalog was
assumed to be the same as in other two GCS reconstructions. CME measurements
provided by five different observers using three different measurement
methods, as well as the corresponding five DBM inputs is given in Table 2.

The inner boundary, the drag parameter, solar wind speed and target distance
are the same for each of the five DBM inputs and are
$R_{0}=20\,\mathrm{R}_{\odot}$, $\gamma=0.2\cdot 10^{-7}\,\,\mathrm{km}^{-1}$,
$w=450\,\mathrm{km\,s}^{-1}$, and $R_{\mathrm{target}}=1$ au, respectively. In
DBEM, 15 synthetic values of $\gamma$ and $w$ were used within the uncertainty
ranges $\pm 0.1\cdot 10^{-7}\,\mathrm{km}^{-1}$ and $\pm
50\,\mathrm{km\,s}^{-1}$, respectively. In DBEMv3 default uncertainty ranges
of the tool were used: $\pm 30$ min, $\pm 0.1\cdot 10^{-7}\,\mathrm{km}^{-1}$,
$\pm 50\,\mathrm{km\,s}^{-1}$, $\pm 200\,\mathrm{km\,s}^{-1}$, $\pm
15^{\circ}$, and $\pm 30^{\circ}$ for lift-off time, $\gamma$, $w$, $v_{0}$,
$\lambda$, and $\phi_{\mathrm{CME}}$, respectively, and 10000 runs were
performed to obtain the results. It can be seen in Table 2 that the difference
in the output of different DBM tools is very similar for all DBM tools which
use a single input set (basic 1D DBM, 2D self-similar DBM, 2D flattening cone
DBM, and DBEMv3). This is because the direction of the apex is very close to
the direction of the target, i.e. the CME is likely to hit the target close to
the apex, where all geometries evolve similarly. This is also the reason for
very high arrival probability, given that the most of the input sets consider
a relatively wide CME. The input set derived based on measurements by Wood et
al. [77] is the only one yielding a DBEMv3 arrival probability $<100\%$,
because the estimated CME half width is low compared to the uncertainty. The
difference of the output is more prominent between different input sets than
between different tools, with the exception of DBEM, which shows slightly
different results compared to other tools. This is because, unlike other DBM
tools, DBEM does not use a single input but takes into account the variability
of different input sets.

The last two rows of Table 2 show the observational results for the CME
arrival, which are based on the CME-ICME association made by Möstl et al. [69]
and arrival time and speed values provided by the ICME catalog of Richardson
and Cane [79]666http://www.srl.caltech.edu/ACE/ASC/DATA/level3/icmetable2.htm,
where the mean ICME speed is taken as arrival speed and ICME start (not the
start of the disturbance) is taken as the arrival time. Measured ICME peak
speed is also given for reference. We can see that all DBM outputs in Table 2
overestimate arrival time by couple of hours and underestimate arrival speed
by couple of tens of $\,\mathrm{km\,s}^{-1}$, indicated that the drag was
overestimated. Indeed, as this is a fast event, 2D flattening cone DBM, DBEM
and DBEMv3 would suggest the $\gamma=0.1\cdot 10^{-7}\,\mathrm{km}^{-1}$
option. Running DBEMv3 and DBEM with $\gamma=0.1\cdot
10^{-7}\,\mathrm{km}^{-1}$ and keeping other input identical as given in Table
2 yields arrival times 2010-04-05 13:20 and 2010-04-05 11:50 UT, respectively,
i.e. very close to the observed arrival time. The DBEMv3 and DBEM output for
arrival speed is $705\,\mathrm{km\,s}^{-1}$ for both tools, which is closer to
the observed ICME peak speed instead of the ICME mean speed.

##  6 Discussion and conclusion

The basic DBM equations (Equations 1 and 2) describe CME propagation in a
simple, physics-based and analytical way. Therefore, even when CME geometry is
included (2D DBM) and in ensemble mode, the model runs very quickly (Table 1).
With the development of different tools and their performance analysis,
optimized DBM parameters (e.g. initial distance, solar wind speed and drag
parameter) have been established, which are offered as default parameters in
the DBM tools (Table 1). This makes the tools very easy to use even for the
unexperienced users. On the other hand, the tools allow customized input for
more experienced users (as described in Section 4.1.1). Therefore, DBM tools
are simple to use and computationally efficient, which is their main advantage
compared to numerical MHD models.

DBM tools offer three different geometries, which identically describe the
propagation of the CME apex, but differ in the description of the CME flanks,
and might therefore differ in the applicability. For instance, both 2D DBM
tools assume initial cone geometry, however one propagates it self-similarly,
whereas the other does not. Therefore, it is reasonable to assume that 2D
self-similar DBM might be more suitable for CMEs with (near) self-similar
expansion. This might be the case with slower CMEs which show more symmetric
in situ profiles [80]. Since they propagate with speeds closer to the solar
wind speed, they experience less drag. On the other hand, 2D flattening cone
DBM might be more suitable for very fast CMEs which show quite asymmetric in
situ profiles [80], indicating non-self similar expansion. They propagate with
speeds much larger than the solar wind speed, and thus experience more drag.
The non-self similar vs. self-similar evolution might become even more
important for considerations of 3D geometries. It is important to note that 2D
flattening cone DBM does not consider change of the front in a manner that the
flanks catch-up or overtake the apex. It is a purely geometrical effect, a
change from a highly-curved cone geometry towards a less-curved concentric-
arc-like geometry, and not related to e.g. non-homogeneous drag or internal
forces that might cause the ’pancaking effect’ [e.g. 81]. The cone geometry is
also suitable to describe the propagation of a CME driven shock, which is
typically faster and stronger at the nose compared to flanks [e.g. 82] and
thus the flanks are ’delayed’ with respect to the nose. On the other hand, for
a freely propagating shock, assuming it propagates in homogeneous medium, a
concentric arc geometry might be more suitable. As demonstrated on an example
event, for CME propagation near the apex, all geometries and therefore all DBM
tools show similar results (provided that the CME input is the same, see Table
2).

The ensemble options of the DBM provide a more comprehensive prediction
compared to the other three tools, as they additionally calculate arrival
probability and confidence interval of the arrival time and speed. In
addition, although they rely on a large number of DBM runs ($>1000$), they are
still computationally inexpensive (Table 1). Therefore, they are quite useful
from the aspect of space weather forecast and its evaluation. Since there is a
difference in implementation of the CME input in DBEM and DBEMv3, their
applicability may also differ. DBEMv3 tool is much faster and only needs input
from one observer, thus it is easy to use in near-real time forecasts. On the
other hand, DBEM may use various CME input sets, provided by different
observers, different methods or even different instruments, and may thus be
more suitable for evaluation purposes.

To summarize, this paper provides an overview of the assumptions, application
and performance of the five DBM tools developed at Hvar Observatory. It is
important to note that although these tools were developed sequentially and
therefore each more recent tool contains improvements compared to the older
version, the older versions still have their applicability.

## Acknowledgments

M.D. acknowledges support by the Croatian Science Foundation under the project
IP-2020-02-9893 (ICOHOSS) and International Space Science Institute (ISSI)
team “Understanding Our Capabilities in Observing and Modeling Coronal Mass
Ejections”, led by C. Verbeke and M. Mierla. J.C., B. V. and D.S. acknowledge
support by the Croatian Science Foundation under the project 7549 (MSOC). K.M.
acknowledges support by the Croatian Science Foundation in the scope of the
Young Researchers Career Development Project Training New Doctoral Students.
This research has received financial support from the European Union’s Horizon
2020 research and innovation program under grant agreement No. 824135
(SOLARNET).

Table 1: Comparison of DBM tools |  |  | 1D DBM | 2D DBM | DBEM | DBEMv3
---|---|---|---|---|---|---
|  |  | self-similar cone | flattening cone
BASIC |  ASSUMPTIONS | drag parameter | $\gamma=const$ | $\gamma=const$ | $\gamma=const$ | $\gamma=const$ | $\gamma=const$
solar wind speed | $w=const$ | $w=const$ | $w=const$ | $w=const$ | $w=const$
self-similarity | Y | Y | N | N | N
2D geometry | concentric circular arc | ice cream cone | ice cream cone | ice cream cone | ice cream cone
INPUT777not including basic CME input which is the same for all tools: CME take-off date & time, CME initial speed | optimal $R_{0}[\mathrm{R_{\odot}}]$ | $R_{0}=20$ | $R_{0}=20$ | $R_{0}=20$ | $R_{0}=20$ | $R_{0}=20$
CME width input | N | Y | Y | Y | Y
source position input | N | Y | Y | Y | Y
GCS input | N | N | N | N | Y
optimal $w[\,\mathrm{km\,s}^{-1}]$ | $w=450$ | $w=450$ | $w=450$ | $w=450$ | $w=450$
modelled $w$ option888using the ESWF tool, see Section4.1.1 | N | N | Y | N | Y
optimal $\gamma[10^{-7}\,\mathrm{km}^{-1}]$ | 0.2 | 0.2 | 0.1 (fast CME) | 0.1 (fast CME) | 0.1 (fast CME)
0.2 (normal CME) | 0.2 (normal CME) | 0.2 (normal CME)
0.5 (slow CME) | 0.5 (slow CME) | 0.5 (slow CME)
OUTPUT | CME arrival time | Y | Y | Y | Y | Y
CME arrival speed | Y | Y | Y | Y | Y
CME arrival probability | N | N | N | Y | Y
TOOL |  SPECS | typical runtime | 0.5 sec | 2 sec | 2 sec |  $<2$ min (average PC) | 6 sec
source | OH999http://oh.geof.unizg.hr/DBM/dbm.php | OH, CCMC101010https://ccmc.gsfc.nasa.gov | ESA SSA111111http://swe.ssa.esa.int | run-on-request121212runs are available upon request to<EMAIL_ADDRESS>| ESA SSA
reference | [29] | [33] | [33] | [39] | [48]
Table 2: CME measurements and the corresponding DBM input and output for the
April 3rd 2010 CME for 5 different observers using 3 different methods.

| observer 1 (GCS) | Rodari+2018 (GCS) | Wood+2017 | AFFECTS-GCS | AFFECTS-CAT
---|---|---|---|---|---
INPUT |  | obs time | 04/03 10:54 | 04/03 11:24 | 04/03 10:00 | 04/03 12:08 | 04/03 12:08
| height [$\mathrm{R}_{\odot}$] | 8.1 | 15.6 | 7.3 | 13.6 | 21.5
CME | lon [deg] | 5 | 3 | 3 | 4 | 0
measurements | lat [deg] | -26 | -28.5 | -16 | -26 | 24.6
| tilt [deg] | 11 | 1.7 | -80 | -1 | –
| kappa | 0.35 | 0.3 | 0.21 | 0.42 | –
| halfangle [deg] | 37 | 24.3 | 60 | 16 | –
DBM input | liftoff time | 04/03 13:24 | 04/03 12:19 | 04/03 12:30 | 04/03 13:29 | 04/03 11:47
v0 [$\,\mathrm{km\,s}^{-1}$] | 920 | 920 | 960 | 920 | 812
lon [deg] | 5 | 3 | 3 | 4 | 0
half width [deg] | 53 | 41 | 19 | 41 | 30
ICME | ToA* | 04/05 12:00 | 04/05 12:00 | 04/05 12:00 | 04/05 12:00 | 04/05 12:00
OBSERVATION | SoA** [$\,\mathrm{km\,s}^{-1}$] | 640 (790) | 640 (790) | 640 (790) | 640 (790) | 640 (790)
OUTPUT |  | ToA | 04/05 17:39 | 04/05 16:34 | 04/05 16:45 | 04/05 17:44 | 04/05 19:43
basic | ToA $|O-C|$ [h]*** | 5.7 | 4.6 | 4.7 | 5.7 | 7.7
1D DBM | SoA [$\,\mathrm{km\,s}^{-1}$] | 620 | 620 | 620 | 620 | 597
| SoA $|O-C|$ [$\,\mathrm{km\,s}^{-1}$] | 20 (170) | 20 (170) | 20 (170) | 20 (170) | 43 (193)
| ToA | 04/05 17:51 | 04/05 16:41 | 04/05 17:01 | 04/05 17:56 | 04/05 19:43
2D self-similar | ToA $|O-C|$ [h] | 5.9 | 4.7 | 5.0 | 5.9 | 7.7
cone DBM | SoA [$\,\mathrm{km\,s}^{-1}$] | 618 | 619 | 617 | 618 | 597
| SoA $|O-C|$ [$\,\mathrm{km\,s}^{-1}$] | 22 (172) | 21 (171) | 23 (173) | 22 (172) | 43 (193)
| ToA | 04/05 17:47 | 04/05 16:36 | 04/05 15:37 | 04/05 17:46 | 04/05 19:37
2D flattening | ToA $|O-C|$ [h] | 5.8 | 4.6 | 3.6 | 5.8 | 7.6
cone DBM | SoA [$\,\mathrm{km\,s}^{-1}$] | 619 | 620 | 627 | 620 | 598
| SoA $|O-C|$ [$\,\mathrm{km\,s}^{-1}$] | 21 (171) | 20 (170) | 13 (163) | 20 (170) | 42 (192)
DBEMv3 | arrival probability [%] | 100% | 100% | 91.5% | 100% | 99.2%
ToA | 04/05 18:06 | 04/05 17:14 | 04/05 18:04 | 04/05 18:28 | 04/05 20:51
ToA CI [h]**** | +5.6/-5.1 | +5.1/-5.8 | +5.6/-6.9 | +5.3/-5.8 | +5.9/-7.7
ToA $|O-C|$ [h] | 6.1 | 5.2 | 6.1 | 6.5 | 8.9
SoA [$\,\mathrm{km\,s}^{-1}$] | 618 | 616 | 612 | 615 | 591
SoA CI [$\,\mathrm{km\,s}^{-1}$] | +64/-52 | +70/-63 | +54/-65 | +51/-68 | +51/-59
SoA $|O-C|$ [$\,\mathrm{km\,s}^{-1}$] | 22 (172) | 24 (174) | 28 (178) | 25 (175) | 49 (199)
DBEM | arrival probability [%] | 100%
ToA | 04/05 16:44
ToA CI [h] | +5.8/-5.4
| ToA $|O-C|$ [h] | 4.7
| SoA [$\,\mathrm{km\,s}^{-1}$] | 617
|  | SoA CI [$\,\mathrm{km\,s}^{-1}$] | +97/-65
|  | SoA $|O-C|$ [$\,\mathrm{km\,s}^{-1}$] | 23 (173)

  * *

*ToA=time of arrival

  * **

**SoA=speed of arrival; observed mean ICME speed (peak speed is given in
brackets)

  * ***

***$|O-C|$=absolute value of the difference between observed and calculated
values

  * ****

****CI=confidence interval (95%)

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