# Quantitative Comparison of Methods for Predicting the Arrival of Coronal
Mass Ejections at Earth based on multi-view imaging

R. C. Colaninno, A. Vourlidas, C.-C. Wu
Space Science Division, Naval Research Laboratory, Washington, District of
Columbia, USA.
<EMAIL_ADDRESS>
###### Abstract

We investigate the performance of six methods for predicting the CME time of
arrival (ToA) and velocity at Earth using a sample of nine Earth-impacting
CMEs between May 2010 and June 2011. The CMEs were tracked continuously from
the Sun to near Earth in multi-viewpoint imaging data from STEREO SECCHI and
SOHO LASCO. We use the Graduate Cylindrical Shell (GCS) model to estimate the
three-dimensional direction and height of the CMEs in every image out to
$\sim$200 R⊙. We fit the derived three-dimensional (deprojected) height and
time data with six different methods to extrapolate the CME ToA and velocity
at Earth. We compare the fitting results with the in situ data from the WIND
spacecraft. We find that a simple linear fit after a height of 50$R_{\odot}$
gives the best ToA with a total error $\pm$13 hours. For seven (78%) of the
CMEs, we are able to predict the ToA to within $\pm$6 hours. These results are
a full day improvement over past CME arrival time methods that only used SOHO
LASCO data. We conclude that heliographic measurements, beyond the
coronagraphic field of view, of the CME front made away from the Sun-Earth
line are essential for accurate predictions of their time of arrival.

## 1\. Introduction

In the last few decades, we have discovered that the space environment around
our planet is as dynamic as terrestrial weather. The source of this space
weather is the Sun which produces winds and storms that affect modern
infrastructure. The most geo-effective aspects of space weather are coronal
mass ejections (CMEs) which are analogous to terrestrial hurricanes. These
powerful storms, comprised of plasma and magnetic fields ejected from the
solar corona, can significantly disrupt Earth’s magnetosphere and cause a
range of terrestrial effects from the aurora to ground induced currents. CMEs
were first observed in visible-light coronagraphs (Tousey and Koomen, 1972) as
bright large-scale density enhancements propagating outwards from the Sun.
Signatures of CMEs are also seen _in situ_ plasma and magnetic field data
(Cane et al., 2000). _In situ_ measurements at Earth give us the real-time
arrival and physical properties of geo-effective CMEs. Throughout the paper,
we use the term CME to describe both the imaging and _in situ_ observations.

Since mid-2007, we are able to continuously monitor the propagation of CMEs
from the Sun to Earth using the observations from the Sun-Earth Connection
Coronal and Heliospheric Investigation (SECCHI; Howard et al. 2008) instrument
suite aboard the _Solar TErrestrial RElations Observatory_ mission (_STEREO_ ;
Kaiser et al. 2008). However, even with complete coverage of the Sun-Earth
line with visible-light imaging data, it remains difficult to accurately
predict the arrival of CMEs at Earth. CMEs are detectable in visible-light due
to Thomson scattering of photospheric sunlight by the electrons within the
CME. This emission is optically thin, thus, the observations are integrations
along the line of sight (LOS) making it difficult to identify individual
features of the diffuse CME structure. The emission from the CME electrons
drops off quickly as the CME expands away from the Sun causing a decrease in
density and scattering efficiency with the viewing angle. By the time CMEs
reach Earth, they are large ($\sim$ 0.5 AU width) with correspondingly long
LOS integration paths. All these elements of CME observations at large
heliocentric distances make it difficult to know the time of arrival (ToA)
even when both the Earth and the CME are visible simultaneously in the same
field of view (FOV).

With careful treatment of the visible-light image data, the position of the
CME front can be measured and the resulting height and time (HT) data points
can be fitted by a curve (e.g. polynomial or spline) to model its motion. If
the height of the CME is not measured up to 1 AU, the fit can then be
extrapolated to predict the ToA of the CME at Earth. This is one of the most
basic space weather prediction techniques that can be applied to imaging
observations.

Even prior to the availability of heliographic data from SECCHI, several
methods had been proposed in the literature to predict the ToA using
coronagraphic data from the Large Angle and Spectroscopic Coronagraph (LASCO;
Brueckner et al. 1995) aboard the _SOlar and Heliospheric Observatory_ mission
(_SOHO_ ; Domingo et al. 1995). Empirical CME propagation models (Gopalswamy
et al., 2000, 2001; Schwenn et al., 2005) were developed using the LASCO data
with limited success. These models use the projected velocity of Earth-
directed CMEs observed in the LASCO FOV. Owens and Cargill (2004) evaluated
the predicted ToA for three empirical models: Gopalswamy et al. (2000, 2001)
and Vršnak and Gopalswamy (2002). They found little difference between the
three models with an average ToA error of 0.46 days. Schwenn et al. (2005)
reported similar results for their method which used the expansion speed of
the CME as a proxy for the CME radial speed. All these models can predict the
arrival of the CME at Earth within a $\pm$24 hours window with a 95% error
margin.

At the time, the inaccuracy of these models was attributed to the inability to
measure the true (deprojected) speed of Earth-directed CMEs since LASCO is
located along the Sun-Earth line offering a head-on view of the propagation.
Lindsay et al. (1999) compiled a data set of CMEs observed in quadrature using
Solar Maximum Mission and Solwind coronagraphs, and _Helios 1_ and _Pioneer
Venus Orbiter_ _in situ_ magnetic field and plasma measurements. Owens and
Cargill (2004) found that using these data where the projection effects are
minimized did not improve the results of the three studied ToA models.
Physics-based shock propagation models, using the speeds from the metric type-
II radio bursts which are not affected by projection effects, do not fare any
better with ToA predictions (Cho et al., 2003).

Kilpua et al. (2012) fitted the stereoscopic data from SECCHI and LASCO with
the Graduated Cylindrical Shell (GCS; Thernisien et al. 2009) model to derive
the three-dimensional (3D) position, direction and speed of 30 CMEs within the
LASCO FOV ($<$ 30 R⊙) between 2008 and 2010. Kilpua et al. (2012) did not
extend their analysis into the SECCHI HI FOV. They applied these deprojected
CME speeds in the models of Gopalswamy et al. (2000, 2001). They compared the
predicted ToA to the _in situ_ measurements and found an error of $\pm$30
hours with a 95% error margin. This result is actually worse than the
predictions obtained using the same model with projected CME speeds. It is
unclear what caused this unexpected result. It may be due to the low average
speeds of the Kilpua et al. (2012) sample, taken during the most unusual
minimum of the space age, compared to samples used in the past. For all
empirical models, the error in ToA prediction is largest for slower CMEs.
Given these uncertainties and apparent insensitivity of empirical models to
deprojected speeds, we will not consider them further.

Instead, we focus on models that make certain assumptions about the CME shape
and/or direction to estimate the 3D speed from heliographic image data. Models
using single-viewpoint heliographic imaging data in this way are Fixed-$\phi$
(Sheeley et al., 1999; Rouillard, 2011), harmonic mean (Lugaz, 2010) and self-
similar expansion (Davies et al., 2012). These models use geometric arguments
to derive the longitude and speed of the CME assuming that these values are
constant throughout the range of the observations. The Fixed-$\phi$ method
simplifies the CME to a single point (or rather a very narrow LOS extension).
The harmonic mean method simplifies the CME to a circle which intersects the
Sun and CME front. The self-similar expansion method is an extension of the
harmonic-mean method in that the CME is no longer anchored at the Sun but is
expanding with a constant angular extent. Lugaz et al. (2012) compared the
predicted ToA from the Fixed-$\phi$ and harmonic mean methods for 20 CMEs
which impacted _STEREO_. They found ToA errors of $\pm$33 hours for the
Fixed-$\phi$ and $\pm$20 hours for the harmonic mean method both with 95%
error margin. The self-similar expansion method has not been applied to CME
data, at this point.

Howard et al. (2006) used Solar Mass Ejection Imager (SMEI) heliospheric
imaging data to predict the ToA of 15 CMEs in 2003 and 2004. Their results are
within a range of -24 to 20 hours for all CMEs. Howard and Tappin (2010) used
a 3D model to predict the ToA of three CMEs also using SMEI data at many
different elongations of the front. Their best predictions were within an hour
of the CME ToA.

Methods such as triangulation (Liu et al., 2010; Liewer et al., 2011; Liu et
al., 2013) and constrained harmonic mean (Lugaz et al., 2010) have been
developed to take advantage of the stereoscopic data from _STEREO_ -SECCHI.
Liu et al. (2010) was able to predict the arrival of the front within 12 hours
for their single CME. Liu et al. (2013) used both triangulation and the
constrained harmonic mean method to study the kinematics of three Earth-
impacting CMEs. The constrained harmonic mean method gave the best results
with an error between -2.3 to 8.4 hours in the ToA of the CME driven shock.

Another approach is to use the _in situ_ detection of the CME to constrain the
imaging observations (Wood et al., 2009a; Rollett et al., 2012; Temmer et al.,
2011). Wood et al. (2009a) used a multiple-function fit to the HT data to
describe the kinematics of a CME at different heliocentric distances. Rollett
et al. (2012) and Temmer et al. (2011) have used spline fits to derive the
velocity and acceleration profiles of the CMEs studied in these papers. This
approach may provide some insight into the CME kinematics but cannot be used
for operational space weather predictions because the ToA is no longer a free
variable.

In this paper, we attempt a more operational approach based on a sample of
nine Earth-impacting CMEs. We use the GCS model to fit the multi-viewpoint
observations from SECCHI and LASCO similar to Kilpua et al. (2012). However,
unlike Kilpua et al. (2012), we extend our fits into the heliospheric
observations as far as $\sim$1 AU, in some cases. We then fit the derived 3D
positions using a variety of models, such as constant speed or accelerating
profiles, restricting the fits to certain HT ranges, taking into account the
geometry of the impact and finally comparing with the _in situ_ measurements.
Our aim is to find the simplest and most reliable model for a set of HT
observations than can lead to better operational ToA predictions.

## 2\. Observations of Earth-impacting CMEs

Our primary data comes from the coronagraphic and heliospheric imaging
observations of _STEREO_ -SECCHI and _SOHO_ -LASCO from March 2010 to June
2011. This data set allows us to continuously track Earth-impacting CMEs from
2 or 3 viewpoints at all times. _STEREO_ is comprised of two spacecraft with
nearly identical instrumentation; the _STEREO_ -Ahead (STA) spacecraft orbits
slightly faster than the Earth and the _STEREO_ -Behind (STB) spacecraft
slightly slower. The two spacecraft separate from Earth at a rate of 22.5o per
year since their launch on 25 October 2006. In this study, we use _STEREO_
-SECCHI observations from the outer coronagraph, COR2, which has a FOV from
2.5 to 15 $R_{\odot}$ (Howard et al., 2008). SECCHI also includes two
heliospheric imagers (HI-1, HI-2) which are similar to coronagraphs but have
no occulter and a FOV off-pointed from the center of the Sun. The heliospheric
imagers view the Sun-Earth line from opposite sides of the heliosphere. HI-1
and HI-2 have square FOVs centered on the elliptic plane from 15 to 84
$R_{\odot}$ (20o) and 66 $R_{\odot}$ to 1 AU (70o), respectively (Howard et
al., 2008). We also use the data from the _SOHO_ -LASCO C2 (FOV 2.2–6
$R_{\odot}$) and C3 (FOV 3.8–32 $R_{\odot}$) coronagraphs (Brueckner et al.,
1995).

When studying CMEs, especially Earth-impacting CMEs, it is advantageous to
combine the data from LASCO and SECCHI since LASCO has a head-on view of the
CME and provides a view of the extent of the CME while SECCHI has a side view
and provides information on the location of the CME front. During the time
period of our study, March 2010 - June 2011, the _STEREO_ spacecraft were
separated from each other by 132o to 190o. On 1 March 2010, STB and STA were
-71∘ and 66o, respectively, from Earth. The spacecraft reached opposition on 6
February 2011 and began moving closer to each other on the far side of the
Sun. On 30 June 2011, STB and STA were -93o and 97o from Earth, respectively.
In this configuration, an Earth-directed CME appears on the West limb in STB
and on the East limb in STA.

Table 1: Studied CME | LASCO Detection | Halo or | Lon |  _Wind_ Detection | Velocity | Detection
---|---|---|---|---|---|---
CME | Date | Time (UT) | Partial | (deg) | Date | Time (UT) | ($kms^{-1}$) | Type
1 | 19-Mar-2010 | 10:30 |  | 27 | 23-Mar-2010 | 23:02 | 284 | CME
2 | 03-Apr-2010 | 10:33 | H | 6 | 05-Apr-2010 | 06:43 | 755 | MC
3 | 08-Apr-2010 | 01:31 | PH | -2 | 11-Apr-2010 | 11:59 | 430 | MC
4 | 16-Jun-2010 | 14:54 | PH | -18 | 20-Jun-2010 | 23:59 | 400 | CME
5 | 11-Sep-2010 | 02:00 | PH | -21 | 14-Sep-2010 | 14:24 | 368 | MC
6 | 26-Oct-2010 | 01:36 |  | 22 | 31-Oct-2010 | 04:48 | 365 | MC
7 | 15-Feb-2011 | 02:24 | H | 2 | 18-Feb-2011 | 00:00 | 510 | MC
8 | 25-Mar-2011aaThe CME was listed as two events in the _SOHO_ LASCO CME Catalog. | 14:36 | PHbbSecond detection. | -27 | 29-Mar-2011 | 14:38 | 378 | MC
9 | 2-Jun-2011 | 8:12 | H | -22 | 04-Jun-2011 | 00:00 | 482 | CME

To determine the ToA and speed of the CME at Earth, we use the _in situ_
plasma data from the _Wind_ spacecraft. The _Wind_ spacecraft, like _SOHO_ ,
orbits the L-1 Lagrange point and is ideally situated for monitoring near-
Earth space weather. In this study, we will use data from the _Wind_ Magnetic
Field Investigation (MFI; Lepping et al. (1995)) and Solar Wind Experiment
(SWE; Ogilvie et al. (1995)). The MFI instrument is a triaxial magnetometer
which provides the magnitude and direction of the solar wind’s magnetic field.
The SWE instrument provides the density, velocity and temperature of the ions
of the solar wind. We use the magnetic field data to confirm the passage of a
CME-like magnetic structure, an increase in magnetic field and smooth rotation
in one of the field components (Cane et al., 2000). With the data from plasma
instrument, we can determine the ToA and velocity of a CME to compare with our
results derived from the imaging data.

### 2.1. Description of CME Event Sample

We analyze nine Earth-impacting CMEs observed between March 2010 and June 2011
in both imaging and _in situ_ data. This time range corresponds to the rising
phase of Solar Cycle 24 and is quite advantageous. CMEs during this period are
more energetic but not so numerous as to result in many CME-CME interaction
which confuse measurements of individual features. The CMEs are identified in
Table 1 by the date and time of their first appearance in the LASCO C2
coronagraph taken from the _SOHO_ LASCO CME Catalog
(http://cdaw.gsfc.nasa.gov/CME_list, Yashiro et al. 2004). We denote each CME
with a number in chronological order in Table 1. We will refer to the CMEs by
these numbers, throughout this paper. In Table 1, we also list whether the CME
was identified as a halo (H) or partial halo (PH) in the catalog. Despite all
nine CMEs being Earth-directed, only three were identified as halos and four
were identified as partial halos. Therefore, a CME can impact the Earth even
if it is not identified as a partial halo in the LASCO catalog. With complete
imaging coverage of the Sun-Earth line, we are able to show that all the
studied CMEs are detected at Earth in the _Wind_ spacecraft data. The
Heliocentric Earth Ecliptic (HEE) longitude of the CME derived from GSC model
fitting is listed in column 5 of Table 1.

We search the SECCHI data set beginning in January 2009, when the spacecraft
were separated by $\sim$88.5o and ending in June 2011. To be included in our
sample, the CMEs must be observed in all the imaging data (SECCHI, LASCO,
eight instruments in total) without a significant period ($<$ 1 hour) of
missing data. The CME must be easily tracked between instruments. Thus the
structure of the CME had to be visible out to nearly the edge of the FOV of
each instrument (with the exception of SECCHI HI-2). Due to the effects of
Thomson scattering, the CME emission is dimmest from the LASCO viewpoint for
Earth-impacting CMEs. Thus the visibility of the CME in the LASCO coronagraphs
is usually the limiting factor for selection. To ensure we properly fit the
CME envelope, we rejected any CMEs that expanded outside the upper or lower
edges of the HI-1 FOV. These restrictions are severe and eliminate many CMEs
from study but are required for robust fitting of the GCS model.

To identify the CME region in the _Wind_ data, we used the criteria of Lepping
et al. (2005) automatic detection technique. The technique was developed to
detect potential magnetic clouds (MC) in the data based on the definition from
Burlaga et al. (1981). The technique can also identify possible CMEs in the
_in situ_ data. The detection requirements for a potential MC are higher than
for a CME detection. The minimum requirements for potential CME detection are;
the proton plasma beta must be $<\beta_{p}>\leq 0.3$, the field directions
must change smoothly, and these two conditions must persist continuously for a
minimum of eight hours. For possible MC detection, a period of data must meet
the minimum criteria above and have (i) a high average magnetic field strength
(B $>$ 7 nT), (ii) a low proton thermal velocity (vth=30 $kms^{-1}$) and,
(iii) a minimum change in the magnetic field latitude ($\Delta\theta=35^{o}$).
All nine CMEs meet the minimum detection criteria of Lepping et al. (2005);
seven of them also met the criteria for MC detection. In Table 1, we list the
_in situ_ detection type for each CME. The detection type only indicates if
the _in situ_ data met the outlined criteria. To determine the presence of a
MC or a MC-like structure in the data further analysis would be needed (Wu and
Lepping, 2007).

In Table 1, we list the time when the CME is detected at the _Wind_
spacecraft. There is no consensus in the literature as to which parameter of
the _in situ_ data marks the arrival of the CME (see the discussion in
Gopalswamy et al. 2003 and Cane and Richardson 2003). Since CMEs are large
structures which can persist in the _Wind_ data for days, the selection of the
CME arrival criteria can affect the ToA by several hours. Two parameters
commonly used for the _in situ_ ToA of a CME are the time of the peak magnetic
field intensity of the shock or the beginning of the MC. To properly compare
the imaging to _in situ_ data, we determine the ToA of the CME at _Wind_ based
on the density since it is the common physical parameter between _in situ_ and
imaging measurements. We do not use the peak of the shock magnetic field, if
present, since it arrives before the CME density front. Similarly, we do not
use the arrival of MC because it occurs after the CME density front.
Therefore, we propose that the ToA of the density increase is the most
appropriate for comparison to the imaging data. In Table 2, we list the
duration of the density front, the time between the density increase and the
region of low plasma beta, and the mean of the velocity detected by _Wind_
during the passage of the density increase.

## 3\. Graduated Cylindrical Shell (GCS) Model

To locate the CME front in 3D space from the imaging data, we use the GCS
model. The graduated cylindrical shell model (GCS) was developed by Thernisien
et al. (2006, 2009). It is a forward modeling method for estimating the 3D
properties and position of CMEs in white-light observations. Unlike the
methods discussed earlier (Sheeley et al., 1999; Rouillard, 2011; Liu et al.,
2010; Liewer et al., 2011; Lugaz, 2010; Möstl and Davies, 2012) that use only
the front of the CME, the GCS model is a complete 3D reconstruction of the CME
envelope. Other such 3D reconstruction models as well as the GCS model are
reviewed in Mierla et al. (2010).

The GCS modeling software allows the user to fit a geometric representation of
the CME envelope to all simultaneous imaging observations. The shape of the
GCS model is designed to mimic that of a cylindrical magnetic flux rope. The
CME is described by a curved hallow body with a circular cross-section
connected by two conical legs anchored at the Sun’s centers. It is important
to note that the GCS model is purely geometric and does not provide any
information about the magnetic field. Complete details of the model geometry
can be found in Thernisien (2011).

The model is fit by overplotting the projection of the cylindrical shell
structure onto each image. The observer then adjusts six parameters of the
model until a best visual fit with the data is achieved. The model is
positioned using the longitude, latitude and the rotation parameters. The
origin of the model remains fixed at the center of the Sun. The size of the
flux rope model is controlled using three parameters which define the apex
height, footpoint separation and the radius of the shell. Figure 1 shows
simultaneous images from three viewpoints, STA and STB HI-1 and LASCO C3, as
well as the GCS model fit to the data. In each image, the model is projected
onto the plane of the image using a grid of points (green) that represent the
surface of the model.

### 3.1. Application of the GCS Model to Remote Sensing Data

We fit the GCS model to all available images from all nine CMEs starting at
the CME’s first appearance in the SECCHI COR2 and LASCO C2 FOVs until the
SECCHI HI-2 FOV. When the CME is visible in the LASCO data, we use all three
viewpoints to make the GCS fit. The LASCO viewpoint is essential for a robust
fit because the projection of an Earth-directed CME is usually quite symmetric
between STA and STB. The LASCO viewpoint can give essential information about
the orientation and dimensions of the CME that is ambiguous in the SECCHI data
for Earth-directed CMEs (Vourlidas et al., 2011).

Once the front of the CME is no longer visible in the LASCO FOV, we must make
some assumptions about its evolution. We assume that it expands self-
similarly. This assumption is implemented by keeping constant all parameters
of the GCS model except height. We believe self-similar expansion is a good
assumption, since for most CMEs the model parameters vary only slightly when
fitted using the LASCO view. A notable exception, CME 4 has a rapid change in
its rotation angle in the LASCO FOV (Vourlidas et al., 2011). The effects of
the rotation on the GCS model fit to this CME are discussed in Nieves-
Chinchilla et al. (2012).

Figure 1: A sample of the remote sensing data used in the study. The panels
are data from STA HI-1, LASCO C3 and STB HI-1 from left to right. The data in
the top and bottom panels are the same. The images in the bottom panel have
been over plotted with the GCS model. The GCS model is represented by a grid
of points on the surface of the model.

The GCS fitting provides measurements of various physical aspects of the CME,
such as size, direction, orientation, etc. In this paper, we concentrate our
analysis only on the measurements of the CME 3D height versus time (HT). In
Figures 2-4, the HT measurements and _in situ_ data are plotted on the same
time axis for each of the nine CMEs. The HT data are plotted in the top panel
for each CME with plus signs. We fit the GCS model at a maximum height of 211
R⊙ (0.98 AU) for CME 2. The average maximum height for all the studied CMEs is
179 R⊙ (0.83 AU). The bottom three panels for each CME in Figures 2-4 show the
magnetic field magnitude, proton density and proton velocity measured _in
situ_ from the _Wind_ spacecraft. The first vertical green dashed line marks
the ToA of the density increase. The second green dashed line is the backend
of the density front and the beginning of the low beta plasma and smooth
magnetic field rotation. The mean of the plasma velocity is also plotted as a
horizontal green solid line in each bottom panel. We will discuss the fits to
the HT data in section 4.

### 3.2. Error Estimation in Stereoscopic Localization

To properly assess the various HT fitting methods for deriving the CME
velocity and extrapolating the ToA, we need to assign an error to our height
measurements. Thernisien et al. (2009) estimated the error associated with the
six GCS model parameters when applied to a CME in the SECCHI COR2 views only.
They found an error of $\pm 0.48$ R⊙ in the height. Since we are using LASCO
data in addition to the SECCHI COR2 data, we consider the errors from
Thernisien et al. (2009) as an upper limit

Figure 2:  HT measurements and _in situ_ data plotted on the same temporal
axis. The HT data are plotted in the top panel with plus signs. The bottom
three panels show the magnetic field magnitude, proton density and proton
velocity _in situ_ data from the _Wind_ spacecraft. The vertical green dashed
line marks the width of the density increase (ToA). Fit 1 and 5 and the their
velocities are plotted with blue and orange solid lines, respectively.

Figure 3:  Same as Figure 2 for CMEs 4-6.

Figure 4: Same as Figure 2 for CMEs 7-9.

for the height measurements in these FOV. Thus we need to estimate the error
for heights measured in the HI images without the LASCO images. As mentioned
in the previous section, once the CME in no longer visible in the LASCO FOV,
we fit the GCS model to the data by only adjusting the height parameter. Thus
in the HI-1 and HI-2 images, the accuracy of the GCS model fit is primarily
driven by the proper localization of the CME front from the two viewpoints.

To estimate this error, we consider the simplified problem of stereo
triangulation (Hartley and Zisserman, 2004). In a digital image, there is
always an error associated with the localization of a feature in the image.
The error can be represented by a cone of uncertainty around the line-of-sight
(LOS) from each viewpoint. In Figure 5, we represent the triangulation
geometry between two points, $P$ and $P^{\prime}$, near the Sun-Earth line
with the _STEREO_ spacecraft. The LOS from each spacecraft is drawn with
dashed lines and the cone of uncertainty is drawn with solid lines. The
intersection of the uncertainty in the LOS from STA and STB creates a region
of uncertainty around the feature. This region is a trapezoid defined by the
angle between the two LOS, $\alpha$, and the uncertainty in locating the
feature in the image. Thus $\alpha$ is given by

$\alpha=2\pi-(\theta_{A}+\theta_{B}+\varepsilon_{A}+\varepsilon_{B})$ (1)

where $\theta_{A}$ and $\theta_{B}$ are the longitudes of the spacecraft
relative to the Sun-Earth line and $\varepsilon_{A}$ and $\varepsilon_{B}$ are
the solar elongation of the feature in each instrument. The insert in Figure 5
shows a close up of the geometry for the region of uncertainty for
$P^{\prime}$. Since the LOS are large and the error in locating the feature is
small for the _STEREO_ case, we can assume that the sides of the trapezoid are
separated by a constant distance $w$. The length of the trapezoid axes, $dx$
and $dy$, are given by the equations,

$dx=\frac{w}{2\cos{\frac{\alpha}{2}}},\quad
dy=\frac{w}{2\sin{\frac{\alpha}{2}}}$ (2)

where $dx$ and $dy$ are parallel and perpendicular to the longitude of the
feature, respectively. Based on our experience, we estimate the error in
locating the CME front in HI-1 to be $\pm$ 5 pixels and in HI-2 is $\pm$ 10
pixels, thus, $w$ is 0.2

Figure 5: The error in fitting of the GCS model in the HI-1 and HI-2 images
can be simplified to the error in triangulating a feature in stereoscopic
images. The LOS from each spacecraft is drawn with dashed lines and the cone
of uncertainty is drawn with solid lines. The intersection of the uncertainty
in the LOS from STA and STB creates a region of uncertainty around the
feature. The insert shows a close up of the geometry for the region of
uncertainty for point $P^{\prime}$ assuming a long LOS.

$R_{\odot}$ and 1.4 $R_{\odot}$ for HI-1 and HI1-2, respectively.

Equations 2 require careful consideration despite their simplicity. For
example, the error $dx$ goes to infinity for $\alpha=\pi$. We can see in
Figure 5 that as the CME front moves between point $P$ and $P^{\prime}$, that
the longitude of the CME will be unconstrained. From equation 1, before the
spacecraft reach opposition ($\theta_{A}+\theta_{B}>2\pi$) the range of values
of angle $\alpha$ includes $\alpha=\pi$. This uncertainty in the CME longitude
is part of the reason why once the CME is no longer visible in the LASCO FOV,
we keep the longitude of the model fixed. Since we can fit the GCS model for
all the HI-1 and HI-2 images without changing the longitude, the error in the
longitude must be within the minimum value of $dx$ for all measurements. If
the error in the longitude is bounded by the minimum of $dx$, then the error
in the height is simply $dy$ for each measurement. The maximum error in the
height measurements for each CME varies between 7.4 and 12.9 $R_{\odot}$ in
the HI-2 FOV. In Figures 2-4, the error in the height is too small to be
visible in the plot. The error bars for the HT measurements are shown in
Figures 7 for the case of CME 9.

## 4\. Height and Time Data Fitting Methods

To find the best HT fitting procedure for predicting the ToA and velocity of
the CME at Earth, we explore six methods that assume various kinematic
profiles for the CME front. It is not possible to measure the front height all
the way to Earth for all the CMEs in our sample. The six fitting methods are
described below in approximately the order of their complexity. We assign a
color to each fit type which is used throughout.

#### Fit 1 - Linear (blue)

We fit a first-order polynomial to the HT measurements above a height of
50$R_{\odot}$ (0.23 AU). We selected the lower cutoff of 50$R_{\odot}$ because
for most of the CMEs the HT measurements appear to be approximately linear
after this height. Also 50$R_{\odot}$ is approximately the mean height at
which the CME front is no longer visible in the LASCO data. Although the LASCO
C3 FOV is 32$R_{\odot}$, the 3D front height for Earth-directed CME usually
reaches 50$R_{\odot}$ within the image. Also we remind the reader that we
assume self-similar expansion of the CMEs after the CME front is no longer
visible in LASCO. Thus the longitude of the GSC model is fixed after
50$R_{\odot}$. We extrapolate the linear fit to 1 AU to find the ToA and
velocity at Earth.

#### Fit 2 - Quadratic (purple)

We fit a second-order polynomial to the HT measurements above a height of
50$R_{\odot}$. While most of the CMEs appear to be well described by Fit 1,
some of the CMEs, notably CMEs 2 and 9, have an obviously curved HT profile.
This fit assumes that the CME continues to Earth with a constant acceleration.
We extrapolate the function and take the first derivative at 1 AU to find the
ToA and velocity at Earth.

#### Fit 3 - Multiple Polynomials (red)

We fit all available HT measurements for a given event with multiple first-
and second-order polynomial functions for different time ranges. The HT
measurements are fit by an initial first-order polynomial and then two second-
order polynomials. The boundaries of the three functions are determined by the
best fit while keeping the function and its first derivative continuous. We
extrapolate the ToA by assuming a constant velocity after the final data
point, again keeping the velocity continuous. This multi-function polynomial
fit method is similar to that used by Wood et al. (2009a, b) to fit the
kinematics of two CMEs observed in _STEREO_. However, Wood et al. (2009a, b)
used the ToA of the CME as a final data point for their fit.

#### Fit 4 - Spline (magenta)

We fit all HT measurements with a ridged spline. Again, we extrapolate the ToA
by assuming the CME continues with a constant velocity after the final data
point. The shape of the ridged spline fit is similar to Fit 3. These two
methods provide similar velocity profiles. The spline fit velocity is,
however, a smoothly varying curve throughout the CME trajectory which seems
more physical than the velocity profiles from Fit 3 which are piecewise
continuous with a discontinuous acceleration. This fit is similar to the
method used by Rollett et al. (2012) and Temmer et al. (2011).

#### Fit 5 - LASCO FOV (orange)

With this fit we try to compare coronagraphic analyses of the past against the
heliospheric data available with _STEREO_. We fit only those data points where
the CME was visible in the LASCO images which is the opposite approach of Fits
1 and 2 where we use HT measurements after the CME front leaves the LASCO FOV.
We fit the LASCO measurements with a second-order polynomial. We then
extrapolate the ToA using a first-order polynomial with the velocity derived
from the final LASCO data point. This method is similar to Kilpua et al.
(2012). However, we use a simple linear extrapolation instead of the empirical
propagration models of Gopalswamy et al. (2000, 2001).

#### Fit 6 - Geometric Correction (light blue)

With this fit, we attempt to take into account the effect of the curvature of
the CME front on the ToA and velocity. So we use the height of the GCS model
along the Sun-Earth line instead of the apex height. These heights take into
account the curvature of the GCS model front and the longitude of the CME
propagation. As an example, Figure 6 shows an ecliptic cut through all GCS
model fits for CME 8 where the central line of the plot is the Sun-Earth line
and the dashed line is the longitudinal direction of the model. In Table 1, we

Figure 6: Ecliptic cut through the GCS model fits for CME 8 where the central
line of the plot is the Sun-Earth line and the dashed line is the longitudinal
direction of the model.

list the HEE longitude from each fit GCS. Obviously, the front height along
the Sun-Earth line is less then the apex height. Thus the curvature of the CME
front delays the arrival of the CME and reduces the velocity. We fit these
curvature corrected HT data in the same way as Fit 1.

In Figure 7, we have plotted all the fit methods for the HT measurements of
CME 9. The CME 9 HT measurements of the apex are plotted with black crosses.
The error for each measurement is plotted in gray. Fits 1 (blue), 2 (purple),
3 (red), 4 (magenta), 5 (orange), and 6 (light blue) and their ToA are plotted
with solid and vertical dashed lines, respectively. The green dashed lines
mark the time of the _in situ_ density front. The light blue squares represent
the HT measurements corrected for the front curvature (Fit 6) as derived from
the GCS fit. In Figures 2-4, we have plotted in the top panel Fit 1 and Fit 5
for each of the CMEs. Again the solid and vertical dashed lines represent the
fits and ToA, respectively.

## 5\. Results

To quantify the accuracy of the various HT fits in predicting the ToA and CME
velocity at 1 AU, we calculate the difference $\Delta$T = ToApredicted-
ToA${}_{\emph{Wind}}$. A negative $\Delta$T implies an early arrival and
conversely, a positive $\Delta$T implies a late arrival. The $\Delta$T in
hours are listed in Table 2 for each fitting method. In the first column, we
list the duration of the _in situ_ density front in hours. In Figures 2-4, the
boundaries of the _in situ_ density front are marked with vertical dashed
green lines. The velocity listed in Table 1, is the mean of the measured
proton velocity during the passage of the _in situ_ density front. The _Wind_
proton velocity is plotted in the bottom panels of Figures 2-4. The mean
velocity, listed in Table 1, is plotted over the _Wind_ measurements with a
horizontal solid green line between the dashed lines of the density front. We
calculate the velocity error by finding the difference between the predicted
velocity and the mean of the plasma velocity within the _in situ_ density
front ($\Delta$V = Vpredicted-$\overline{V}_{\emph{Wind}}$). In Table 2, we
list the range of the measured _in situ_ velocities. We have included the
duration and velocity variability of the _in situ_ density front in our
discussion because they may provide a sense of scale for the prediction
errors.

We visually represent the results from Table 2 in Figure 8. In the left panel,
the $\Delta$T for each fit method is plotted with plus signs by CME number on
the vertical axis. The results for the various fits follow the color code in
section 4. The green line represents the duration of the CME _in situ_ density
front. In the right panel, we plot $\Delta$V using the same scheme. The green
lines in the right panel represent the range of velocities measured within the
_in situ_ density front.

For Fit 1, the $\Delta$T is within $\pm$ 6 hours, for seven of the CMEs. For 6
out of 9 events, the predicted ToA are either 2 hours before or within the
density front. The two events (CMEs 2, 4) with $\Delta$T$\pm$13 hour are
possibly violating the self-similar expansion assumption (Nieves-Chinchilla et
al., 2012; Wood et al., 2011). It is unclear how the violation of this
assumption could affect the ToA, furthermore, the $\Delta$T error is in the
opposite sense for these two events. The CME 2 ToA is late while CME 4

Figure 7: Comparison of the six HT fitting methods for CME 9. The green dashed
lines mark the time of _in situ_ density front. Fits 1 (blue), 2 (purple), 3
(red), 4 (magenta), 5 (orange), and 6 (light blue) and their ToA are plotted
with solid and vertical dashed lines, respectively. Black crosses represent
the deprojected HT measurements and light blue squares represent the same
points corrected for the front curvature as derived from the GCS fit. See
Section 4 for details.

is early. The predicted velocities from Fit 1 do not compare as well as the
ToA. For only two CMEs (6 and 7), $\Delta$V is within $\pm$ 50 $kms^{-1}$ of
the mean _in situ_ velocity. For four of the CMEs (1, 2, 5 and 9) the
$\Delta$V is greater than $\pm$ 100 $kms^{-1}$. Almost all the predicted
velocities are too fast with the exception of CME 2. Clearly, all CMEs in our
sample decelerate on the way to 1 AU.

The increased complexity of Fit 2 (quadratic), improves the $\Delta$T for CMEs
4, 5, and 7. Yet, the improvements to the ToA of CMEs 5 and 7 are trivial and
only vary the $\Delta$T of the CME within the density front. The $\Delta$T of
CME 4 is improved significantly from -12.74 to -2.94 hours. We cannot predict
the ToA for CME 6 because the quadratic fit fails to intersect with 1 AU,
_i.e_ , the CME does not make it to the Earth. The ToA for the remainder of
the events is not improved with Fit 2. This is true even for CMEs 2 and 9
which are not fit well with a constant velocity and hence Fit 2 was expected
to improve $\Delta$T. Overall, $\Delta$V is also not improved with Fit 2. Only
two of the CMEs are within $\pm$ 100 $kms^{-1}$ of the mean _in situ_
velocity. While the ToA of CME 4 is significantly improved with Fit 2, the
predicted velocity is worse. Clearly, the quadratic fit overestimates the CME
deceleration to 1 AU.

Table 2: Error in Predicted Arrival and Velocity at Earth | CME | Duration11The duration of the CME density front. | Fit 1 | Fit 2 | Fit 3 | Fit 4 | Fit 5 | Fit 6
---|---|---|---|---|---|---|---|---
$\Delta$T (hrs) | 1 | 6.00 | -0.94 | -6.17 | -2.47 | -4.07 | -28.93 | 56.42
2 | 6.42 | 12.41 | 15.90 | 9.52 | 15.28 | 3.59 | 13.00
3 | 13.17 | -1.58 | -3.41 | -4.03 | -2.86 | 6.21 | 8.09
4 | 9.83 | -12.74 | -2.94 | -13.97 | -9.39 | -27.45 | 6.83
5 | 11.67 | 2.47 | -0.70 | 9.97 | 0.29 | -25.69 | 30.76
6 | 20.17 | 2.18 |  | 9.30 | 11.48 | -29.82 | 37.32
7 | 10.03 | 3.97 | 1.87 | 0.83 | 1.90 | 8.66 | 4.23
8 | 9.33 | -5.69 | -5.81 | -4.81 | -5.64 | -19.68 | 0.04
9 | 5.95 | -0.10 | 5.60 | 2.50 | 3.53 | -36.92 | 8.34
$\Delta$V ($kms^{-1}$) |  | Velocity Range22The absolute range of in situ speeds detected within the CME density front. |  |  |  |  |  |
1 | 10 | 129 | 243 | 153 | 166 | 308 | -13
2 | 78 | -137 | -273 | -131 | -326 | 55 | -138
3 | 37 | 84 | 174 | 107 | 120 | 23 | 18
4 | 25 | 83 | -136 | 143 | 34 | 190 | -8
5 | 40 | 102 | 183 | 32 | 135 | 390 | -13
6 | 31 | 35 |  | -13 | -27 | 296 | -65
7 | 55 | 6 | 89 | 39 | 65 | -31 | 4
8 | 25 | 76 | 82 | 38 | 74 | 187 | 50
| 9 | 61 | 115 | -169 | -138 | -52 | 1426 | 49

Fit 3 does not improve the ToA predictions despite having more free parameters
than the pervious fits. Only the ToA for CME 7 is improved over Fits 1 and 2.
For only four CMEs, $\Delta$V is within $\pm$ 100 $kms^{-1}$ of the mean _in
situ_ velocity. Similarly, Fit 4 with the most free parameters fails to
provide an overall improvement in the predictions.

The most important finding from this exercise may be the disappointing
performance of Fit 5. Similar to Kilpua et al. (2012), we are investigating
whether accurate 3D HT measurements in coronagraphic FOVs can be used to
reliably predict the ToA of CMEs. Our results and Kilpua et al. (2012) suggest
that restricting the measurements to these heights dramatically increases the
ToA error compared to using the inner heliospheric measurements. Our fit uses
the fewest HT measurements but these measurements are based on images from
three viewpoints and are thus the most constrained. The $\Delta$T for only
three CMEs is within $\pm$12 hour. These results should be of interest to the
operational Space Weather community since most CME ToA prediction methods use
measurements from LASCO coronagraph along the Sun-Earth line. For this reason,
we explore this fit and the influence of the final height in the ToA accuracy
in the next Section.

Interestingly, this fit has the best prediction for the ToA of CME 2 (3.59
hours error) of all methods and leads us to two conclusions: 1) CME 2
underwent most of its kinematic evolution before $\sim 50R_{\odot}$, and 2)
the heliospheric measurements for this event are likely inaccurate. As we
mentioned earlier, this is a peculiar event with an undetermined orientation
which may not conform to the GCS model fitting. The six remaining CMEs are
predicted to arrive $>14$ hours early and the predicted velocities are $>$100
$kms^{-1}$ higher than the _in situ_ velocities. The results from Fits 1-5
confirm

Figure 8: A visual representation of the results in Table 2. The $\Delta$T for
each fit method has been plotted with plus signs in the left panel by CME
number. The green line represents the duration of the CME density front. In
the right panel, we have plotted $\Delta$V. The green lines represent the
range of velocities detected _in situ_ within the density front. The results
for the various fits (described in Section 4) are plotted in the following
color scheme: Fit 1 (blue), Fit 2 (purple), Fit 3(red), Fit 4 (magenta), Fit 5
(orange), Fit 6 (light blue).

past findings that CMEs undergo significant deceleration above 50 R⊙, on
average. Our $\Delta$T results are similar to those of Kilpua et al. (2012).

With Fit 6, we investigate the effect of the CME geometry predicted by the GCS
model on the ToA. Since the front the GCS model, and presumably of the actual
CME, is curved, the intersection of the CME with Earth will be delayed
relative to the 1 AU arrival of the CME apex. Möstl and Davies (2012) found
that for a hypothetical circular CME front, the flank can be delayed by up to
2 days compared to the apex arrival at 1 AU. Our model is a bit more realistic
since the front of the GCS model is not circular but slightly oblate depending
on the footpoint separation. Since all ToAs are based on the CME apex height,
the geometric correction of Fit 6 can only delay the ToA. Hence, only the ToA
errors for CMEs 1, 3, 4, and 8 can be improved by considering the CME front
geometry. With Fit 6, the ToA of CMEs 1 and 3 are "overcorrected"; the ToA is
too late. The correction lowers the $\Delta$T for CMEs 4 and 8 by 6.79 and
5.65 hours, respectively. We discuss the implications in the next section.
Interesting, the geometric correction improves $\Delta$V compared to Fit 1 for
all CMEs, even for CMEs 1, 3, and 9, where the correction increases $\Delta$T.
The $\Delta$V error is within $\pm$100 $kms^{-1}$ for eight CMEs. For CME 2,
the velocity is unchanged.

## 6\. The Effect of Final Height in Fit 5 on the ToA Accuracy

We repeat Fit 5 but instead of using the last LASCO FOV measurement as the
limit for the quadratic fit, we include measurements at larger heights within
the HI FOV. In Figure 9, we plot the resulting $\Delta$T versus the final
height of the second-order fit. The curves trace the errors for a given CME
and the best prediction for each event is highlighted with a red square. The
CME number is given on the right of the plot. For most of the CMEs the fit is
nearly linear and as more points are added, the function become more and more
linear.

It is clear, and generally expected, that the

Figure 9: Time of arrival error, $\Delta$T$=ToA_{fit5}-ToA_{\emph{Wind}}$, for
Fit 5 as a function of the final height used for the fit. The curves trace the
error for a given CME and the best result is shown by the red square. The
event number is shown on the right end of the corresponding curve.

addition of HT measurements beyond 50 R⊙ improve the ToA accuracy, sometimes
considerably (ie., by 40 hours for CME 9). Interestingly, it seems that most
of the gain lies in just extending the measurements to 60 R⊙. Additional
heights do not improve the ToA or can even make it worse (e.g., CME 4).
However, this improvement does not occur for the events with the best ToAs in
Fit 5. In the case of CME 7, the additional HT measurements decrease the ToA
accuracy threefold. If we ignore CMEs 4 and 8 for the moment, we see that the
addition of higher HT points tends to result in later arrival times; namely,
it gives slower velocities at the final point used for the quadratic fit. This
is another indication that CMEs decelerate above 50 R⊙. However, the lower
velocity bias strongly affects the events that have already undergone the
majority of their deceleration (events with $\Delta T>0$, CMEs 3, 5, 7). We do
not have an obvious explanation for this at the moment. Larger sample studies
are needed.

However, we can reach a couple of interesting conclusions from this exercise:
(1) ToA predictions can be improved considerably with a few HT measurements in
the HI FOV ($>50R_{\odot}$) especially for events without strong deceleration
within the LASCO FOV range. (2) ToA predictions for strongly decelerating
events may be better if based on HT measurements below $50R_{\odot}$. (3)
There does not seem to be a “standard” distance range where CMEs undergo most
of their deceleration, as may be suggested by the multi-polynomial plots in
Wood et al. (2009a), for example. CMEs 3, 5, 7 seem to have decelerated by
50$R_{\odot}$ and to have picked up speed after this height; CMEs 1, 6, 9 seem
to decelerate in the $50-60R_{\odot}$ range while CME 2 or 5 seem to propagate
more or less with a constant speed.

## 7\. Discussion

In this paper, we investigate methods for predicting the ToA of Earth-
impacting CMEs based on de-projected HT measurements from multi-viewpoint
coronagraphic and heliospheric images. From the comparison of six methods, we
conclude that a simple linear fit of the HT measurements above 50 R⊙ can
significantly reduce the ToA error. The predicted ToA from the linear fit (Fit
1) is within $\pm$6 hours of the arrival of the density front at the _Wind_
spacecraft for 78% of CMEs. This result is a 9 hour improvement over the
results of Gopalswamy et al. (2001) that reports an accuracy of $\pm$15 hour
for 72% of CMEs studied. If we include all events in our study, we can predict
the arrival of CMEs at Earth with $\pm$13 hours which is almost a half day
improvement over the $\pm$24 hour window with a 95% error margin previously
reported in Schwenn et al. (2005). Our results are also an improvement over
the Fixed-$\phi$ and harmonic mean methods, $\pm$33 and $\pm$20 hours,
respectively, which use heliospheric data without taking advantage of the two
_STEREO_ viewpoints (Lugaz et al., 2012). Even our worst case results are a
significant improvement in predicting CME ToA. Therefore, deprojected HT
measurements using images of the CME front obtained from outside the Sun-Earth
line can improve the accuracy of the ToA prediction of Earth-impacting CMEs by
a half day compared to single-view coronagraphic observations obtained along
the Sun-Earth line.

The CMEs with the poorest ToA results (2 and 4) are peculiar. They may violate
the self-similar expansion assumption used to fit the GCS model to the HI-1
and HI-2 images. Nieves-Chinchilla et al. (2012) found that CME 4 is rotating
between 0.5 AU and 1 AU and that its appearance is subject to considerable
projection effects. The CME 2 orientation is ambiguous despite being the
subject of several studies. Wood et al. (2011), for example, found that the
cross section of the CME is significantly elliptical irrespective of the
actual orientation. An elliptical cross-section may indicate that the
expansion of the CME was not self-similar; rotation is also likely. In any
case, the heliospheric HT measurements for this CME are suspect as it is the
only event with an improved ToA from Fit 5. Given the small sample of CMEs,
and the even smaller number of discrepancies, we cannot reach a firm
conclusion on whether CME rotation or other projection effects may be
responsible for the poor ToA predictions.

We are not aware of any previous studies of the CME ToA that report the
predicted velocity at 1 AU as well. We think that this is a serious omission,
since a reliable prediction of the CME velocity at Earth can, in turn, provide
reliable estimates of the CME ram pressure and hence help predict one more
geo-effective parameter. We also use the predicted velocity as a diagnostic of
our fit methods. Since the distance traveled by the CMEs is fixed, we would
assume a correlation with $\Delta$T and $\Delta$V. In other words, if the
fitted velocity is too fast, we would expect the CME to arrive early and vice
versa. In Figure 10, we have plotted $\Delta$T versus $\Delta$V where the
results are plotted using the CME number and the color scheme from section 4.
It is clear that while $\Delta$T is evenly distributed around zero (with the
exception of Fit 5), $\Delta$V is largely positive. More precisely,
$\overline{\Delta T}=1.1$ hours and $\overline{\Delta V}$ = 53 $kms^{-2}$.
There is no obvious trend or correlation, between $\Delta$T and $\Delta$V,
within a particular fit or among the fitting methods with the exception of Fit
5. For Fit 5, the faster velocities are somewhat correlated with early ToA, as
one would expect. Fit 6 has the smallest velocity error but it has the three
largest ToA errors. The geometric correction of Fit 6 systematically decreases
the predicted velocity, as expected, but it does not increase the ToA
accuracy. We conclude that _a linear fit to the HT measurements above 50
$R_{\odot}$ is sufficient for predicting the ToA but fails to capture the true
kinematics of the CME. _

This result would seem to suggest that the CMEs are traveling at a constant
speed between 50 $R_{\odot}$ and 1 AU. However, closer analysis of our of
results does not support this claim. First, if the CME reached a constant
velocity by 50$R_{\odot}$, we would expect the results from Fit 5 (LASCO FOV
only) to be as accurate as Fit 1. But Fit 5 results in early arrivals which
implies that the velocity derived at 50$R_{\odot}$ with the quadratic fit is
too high and hence the velocity of the CME must decrease after 50 $R_{\odot}$.
This deceleration, however, must occur very gradually otherwise Fit 2
(quadratic) would perform better than Fit 1 (linear). It is well known that
the velocities measured in the LASCO FOV have a broader range compared the
velocities at Earth which converge around the average solar wind speed
(Gopalswamy et al., 2000). However, it is not known at what heights CMEs reach
a constant velocity. We assert a CME should reach a constant velocity only
after its velocity matches the ambient solar wind velocity. If there is a
difference in the velocity of the CME and the ambient solar wind, the CME will
be effected by a drag force (Vršnak and Gopalswamy, 2002). Six of our CMEs (2,
3, 4, 6, 7, 8, 9) exhibit an abrupt increase in the _in situ_ velocity
coincident with the density front. Thus, they are still traveling faster than
the solar wind and are still decelerating. For the two CMEs that are traveling
with the solar wind velocity (1 and 5), the $\Delta$V from Fit 5 is 308
$kms^{-2}$ and 390 $kms^{-2}$, respectively. Therefore, these CME decelerated
sometime after 50$R_{\odot}$ but before reaching 1 AU and did so smoothly
since the linear fit gives the best ToA for those events. We conclude that
_all CMEs in our sample are decelerated between 50 $R_{\odot}$ and 1 AU._

Figure 10: The error in the arrival time, $\Delta$T versus the error in the 1
AU velocity, $\Delta$V. The color scheme is the same as in Figure 8.
Counterintuitively, there is no obvious correlation between the two variables,
with the exception of Fit 5 (orange). There is a slight tendency to
overestimate the arrival velocity by about 50 $kms^{-1}$.

We assume that the primary cause of the CME deceleration is the drag force due
to the interaction with the ambient solar wind. While the drag force could
also accelerate a CME between between 50$R_{\odot}$ and 1 AU, we did not
measure such a CME in our sample. While the drag force varies as
$|V_{CME}-V_{SW}|^{2}$, the effect on the velocity would not be quadratic. The
drag force is degenerate; as the velocity of the CME decreases so does the
drag force. Thus the deceleration due to the drag would be very gradual and
occur smoothly as we see in our HT measurements. Also the transition of the
CME into equilibrium with the solar wind would also occur smoothly. This would
explain why the HT profiles are not well fit by Fit 2 (quadratic) and Fit 3
(multiple polynomials) but are better represented by Fit 1 (linear). We
believe that Fit 4 (rigid spline) failed because there is too much error in
the HT measurements.

But we have to reconcile our two conclusions: (1) A linear fit to the HT data
is the best method for predicting the ToA; (2) All measured CME are
decelerating. The obvious suggestion is that the linear fit provides the mean
velocity of the gradually decelerating CME front between 50 $R_{\odot}$ and 1
AU. This also explains the systematic overestimation of the CME velocity with
Fit 1. The mean velocity of a gradually decelerating function will always be
higher than the final velocity. Thus we somewhat alter our original
conclusion. _The mean velocity of a CME between 50 $R_{\odot}$ and 1 AU is the
best parameter for predicting the ToA._ The linear fit is a simple method for
calculating the average velocity from the HT data.

We compare the results from Fit 1 and 6 since the HT measurements were fitted
in the same way. For Fit 6 we used the height of the GCS model along the Sun-
Earth line as opposed to the apex height in Fit 1 (see Figure 6). The
corrected height is less than the apex height depending on the width and
longitude of the GCS model. We find that the apex height is a better predictor
of the CME arrival. We interpret this result as evidence for flattening of the
CME fronts during Earth propagation. The flattening of the CME front has been
theorized in the past (Riley and Crooker, 2004, and references therein) and
seems to occur in the HI-1 and HI-2 images perpendicular to the ecliptic (but
see discussion in Nieves-Chinchilla et al. (2012)). In the heliographic
images, we do not have reliable information about the extent or curvature of
the CME in the ecliptic plane. However, if the curvature of the CME was the
dominant factor in the CME ToA error, we would expect the results of Fit 1 to
be systematically early ($-\Delta$T). We do not see this. Only CMEs 1, 3, 6,
and 8 have early predicted ToA and, therefore, could benefit from the
correction. However, all four "corrected" arrivals result in much later ToA,
i.e., they are overcorrected. Thus we have to assume that the front of these
CMEs is not as curved the GCS model predicts. Therefore, we have indications
of flattening of the CME front in the ecliptic beyond 50 R⊙, for some events.
Further investigations on the role of projection effects (e.g, Nieves-
Chinchilla et al., 2012) and on the proper identification of the CME
substructures (e.g., Vourlidas et al., 2013) is needed.

## 8\. Conclusions

With Fits 1 to 4, we add complexity with each fit by increasing the number of
free parameters in an attempt to capture the kinematics of the CME in the
heliosphere. We assume that the increased number of free parameters would
result in better fits to the HT measurements and that the ToA and velocity
prediction would correspondingly improve. Surprisingly, Fit 1 while having the
fewest free parameters, gives the best results. We find that the best results
are obtained by ignoring complex fitting functions to the full data range,
even discarding the coronal observations, and fitting a simple straight line
to the HT measurements above 50 R⊙ only. We show that measurements close to
the Sun, as those provided by coronagraphs, are not sufficiently robust for
ToA predictions even if those HT measurements are deprojected somehow.
Furthermore, we find that being able to follow a CME front all the way to
Earth (e.g., CMEs 2 and 8 but see CME 9 for a counterexample) does not
actually improve the ToA. Correcting for the CME curvature does not improve
the ToA. Imaging observations integrate along a long LOS, which becomes longer
with increasing heliocentric distance. Therefore, the location of a CME
feature can be subject to considerable uncertainty, including a bias towards
the location of the Thompson sphere (Vourlidas and Howard, 2006), if the CME
is undergoing rotation or other interaction with the ambient environment. Such
evolution is likely to affect the derived CME longitude and its curvature.

These results have important implications for Space Weather and CME
propagation studies:

  1. 1.

A simple linear fit to deprojected HT measurements of the CME front only above
50 $R_{\odot}$ is sufficient to predict the ToA within $\pm 6$ hours (for 7/9
events) and the 1 AU velocity within $\pm$ 140 $kms^{-1}$.

  2. 2.

Deprojected HT measurements of CMEs made using imaging from outside the Sun-
Earth line can improve the Earth ToA prediction of CMEs by a half day compared
to single-view coronagraphic observations along the Sun-Earth line.

  3. 3.

CMEs decelerate slowly and smoothly between 50$R_{\odot}$ and 1 AU.

  4. 4.

HT measurements within coronagraphs FOVs (30 $R_{\odot}$) even if they are
deprojected, are insufficient for accurate Earth ToA or CME velocity
predictions.

  5. 5.

Despite the improvements in CME size and direction, achieved using _STEREO_
data, there remain several open issues in the interpretation of the images
such as the precise localization of the Earth-impacting part of the CME.

#### Acknowledgments

R.C and A.V are supported by NASA contract S-136361-Y to the Naval Research
Laboratory. C.W. is supported by Navy ONR 6.1 program. The SECCHI data are
produced by an international consortium of the NRL, LMSAL and NASA GSFC (USA),
RAL and Univ. Bham (UK), MPS (Germany), CSL (Belgium), IOTA and IAS (France).
LASCO was constructed by a consortium of institutions: NRL, MPIA (Germany),
LAM (France), and Univ. of Birmingham (UK). The LASCO CME catalog is generated
and maintained at the CDAW Data Center by NASA and The Catholic University of
America in cooperation with the Naval Research Laboratory.

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