Abstract
We present an analysis of coronal mass ejections (CMEs) observed by the Heliospheric Imagers (HIs) onboard NASA’s Solar Terrestrial Relations Observatory (STEREO) spacecraft. Between August 2008 and April 2014 we identify 273 CMEs that are observed simultaneously, by the HIs on both spacecraft. For each CME, we track the observed leading edge, as a function of time, from both vantage points, and apply the Stereoscopic SelfSimilar Expansion (SSSE) technique to infer their propagation throughout the inner heliosphere. The technique is unable to accurately locate CMEs when their observed leading edge passes between the spacecraft; however, we are able to successfully apply the technique to 151, most of which occur once the spacecraftseparation angle exceeds \(180^{\circ }\), during solar maximum. We find that using a small halfwidth to fit the CME can result in inferred acceleration to unphysically high velocities and that using a larger halfwidth can fail to accurately locate the CMEs close to the Sun because the method does not account for CME overexpansion in this region. Observed velocities from SSSE are found to agree well with singlespacecraft (SSEF) analysis techniques applied to the same events. CME propagation directions derived from SSSE and SSEF analysis agree poorly because of known limitations present in the latter.
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1 Introduction
In addition to the continuous outflow of the solar wind, coronal mass ejections (CMEs: e.g. Webb and Howard, 2012) are a phenomenon by which the Sun releases large quantities of energy in the form of magnetised plasma. They are known to drive magnetic disturbances at Earth, and they are in fact the purveyors of the most extreme spaceweather effects (e.g. Gosling et al., 1991; Kilpua et al., 2005; Richardson and Cane, 2012; Kilpua et al., 2017). CMEs, and their evolution within the solarwind environment, have been the subject of spacebased observations since they were first discovered in images taken by the Orbiting Solar Observatory 7 (OSO 7, 1971 – 74: Tousey, Howard, and Koomen, 1974). Over the following decades, nearcontinuous coronagraph coverage has been provided by both groundbased instruments, such as the Mauna Loa MK3 Coronameter (Fisher et al., 1981), and their spacebased counterparts, most notably the Large Angle and Spectrometric Coronagraph (LASCO: Brueckner et al., 1995) instruments onboard the Solar and Heliospheric Observatory (SOHO). SOHO was launched in 1995 and, despite a brief loss of communication, the LASCOC2 and C3 coronagraphs have operated nearcontinuously ever since; however, the inner C1 camera was lost in 1998. The launch of the Solar Mass Ejection Imager (SMEI, 2003 – 11: Eyles et al., 2003), onboard the Coriolis spacecraft, extended the coverage of CME observations to far greater solar elongation angles into the heliosphere, by means of wideangle imaging. Since 2006, the STEREO Heliospheric Imagers (HIs: Eyles et al., 2009) have continued to provide wideangle imaging of CMEs. Each STEREO spacecraft possesses two HIs: the HI1 cameras have an angular range in elongation from 4 – 24^{∘} and the HI2 cameras cover 18.7 – 88.7^{∘}, aligned to the Ecliptic. Since the launch of STEREO, HI observations, complemented by the STEREO coronagraphs (COR1 and 2: Howard et al., 2008), have provided a considerable amount of information about CME evolution and propagation through the heliosphere (e.g. Byrne et al., 2010; Davis, Kennedy, and Davies, 2010; Möstl et al., 2010; Savani et al., 2012; Harrison et al., 2012; Rollett et al., 2014; Temmer et al., 2014). Coronagraphs provide coverage close to the Sun, for example a planeofsky (POS) range of 1.1 – 32 R_{⊙} in the LASCO field of view (FOV). The POS is defined as the plane that is perpendicular to the Sun–observer line. Conversely, the inner limit of the FOV of the HIs is \(4^{\circ }\) solar elongation (approximately 15 R_{⊙} in the POS).
Based on observations from LASCO, prior to the loss of C1, and SOHO’s Extreme Ultraviolet Imaging Telescope (EIT) Zhang et al. (2001, 2004) characterise the acceleration of CMEs into three phases: initiation, impulsive acceleration, and propagation. The initiation phase represents the initial acceleration up to approximately 1.3 to 1.5 R_{⊙}. Although the subsequent impulsive acceleration phase of a CME can vary significantly in magnitude and duration, it is typically limited to the inner corona (defined by Zhang et al. (2001) as ≈1 – 3 R_{⊙}). However, the impulsive acceleration phase can extend throughout the entire LASCO FOV (St. Cyr et al., 2000; Zhang et al., 2004). Typically, beyond a few solar radii, the CME enters its socalled propagation phase. This is characterised by a relatively constant speed, although the very fastest events are seen to exhibit a deceleration, well into the LASCO FOV, and the very slowest events an acceleration (Yashiro et al., 2004; Gopalswamy et al., 2009). This is evidence of drag forces acting on CMEs and causing their speeds to tend toward the ambient solarwind speed, which is typically 300 – 500 km s^{−1}. Sachdeva et al. (2017) quantify the contributions from the Lorentz and drag forces for 38 CMEs observed by SOHO and find that the former are most significant between 1.65 – 2.45 R_{⊙}, whereas the latter can begin to dominate beyond 4 R_{⊙}, or up to 50 R_{⊙} for slow CMEs.
CMEs can deviate from radial trajectories due to magnetic forces and interaction with background solarwind plasma (e.g. Kilpua et al., 2009; Byrne et al., 2010; Lugaz et al., 2011; Möstl et al., 2015; Manchester et al., 2017). Cremades and Bothmer (2004) study 276 CMEs observed by LASCO between 1996 and 2002. They find an average latitudinal deflection of \(18.6^{\circ }\) towards the Equator during solar minimum (up to 1998) and a poleward deflection of \(7.1^{\circ }\) during solar maximum (2000), with a period of intermediate behaviour in 1999. Isavnin, Vourlidas, and Kilpua (2014) studied 14 flux ropes associated with CMEs using multiviewpoint coronagraph observations combined with MHD modelling to propagate the structures to 1 AU. Whilst deflection most commonly occurs inside 30 R_{⊙}, they find that significant deflection, particularly in longitude, can occur out to 1 AU. Such longitudinal deflections are due to interactions with the background Parker Spiral solarwind structure. Wang et al. (2004) show that this causes faster CMEs to deflect from West to East and slower CMEs to deflect from East to West. Isavnin, Vourlidas, and Kilpua (2014) showed a maximum longitudinal deflection of 29^{∘} from the Sun to Earth for an average speed CME, and Wang et al. (2014) study an individual interplanetary CME that exhibits a longitudinal deflection of \(20^{\circ }\).
Due to the wideangle nature of heliospheric imaging, it is possible to estimate the threedimensional direction of propagation of a CME using HI data from a single vantage point by assuming that these features are moving at a constant velocity. This is clearly demonstrated in socalled time–elongation maps, or Jmaps (Sheeley et al., 1999, 2008), constructed from HI data, in which this constant linear speed is manifested as an apparent angular acceleration that depends on the propagation direction of the feature with respect to the observing spacecraft. Davies et al. (2009) show that the path of a CME through time–elongation maps may be fitted to retrieve its speed, direction, and launch time. Three geometries commonly used to fit CME parameters using a single vantage point are fixed\(\phi \) (FP: Kahler and Webb, 2007; Rouillard et al., 2008; Sheeley et al., 2008), selfsimilar expansion (SSE: Davies et al., 2012; Möstl and Davies, 2013), and harmonic mean (HM: Lugaz, Vourlidas, and Roussev, 2009; Lugaz, 2010). Each of these models is based on the assumptions that the CME possesses a circular crosssectional front that expands selfsimilarly with a constant halfwidth [\(\lambda \)] as the CME propagates. The FP and HM models use respective halfwidths of \(0^{\circ }\) and \(90^{\circ }\). The SSE geometry is generalised to any halfwidth and, as such, the FP and HM geometries can be considered as special cases of SSE, where FP is a point source and HM a circle anchored to the Sun. When these geometries are applied to fit CME kinematic properties from time–elongation data, the fitting methods are referred to as FPF, SSEF, and HMF. Many studies have shown that CME expansion is indeed close to selfsimilar at interplanetary distances (e.g. Bothmer and Schwenn, 1997; Liu, Richardson, and Belcher, 2005; Savani et al., 2009), however cases of flux ropes that deviate from selfsimilar expansion are presented by Kilpua et al. (2012), Savani et al. (2011, 2012).
In Article 1 (Harrison et al., 2018) we present the HICAT (www.helcatsfp7.eu/catalogues/wp2_cat.html) catalogue, which contains a list of all CMEs that were observed using HI (965 by STEREOA and 936 by STEREOB) during the science phase of the STEREO mission (April 2007 to September 2014). In Article 2 (Barnes et al., 2019), we present the kinematic properties of these CMEs from the FPF, SSEF, and HMF methods, based on singlespacecraft observations from STEREOA and STEREOB, which resulted in the generation of the HIGeoCAT (www.helcatsfp7.eu/catalogues/wp3_cat.html) CME catalogue containing 801 and 654 CMEs for STEREOA and B, respectively. Here, we take a further subset of these events, which we determine to be CMEs observed in HI images from both spacecraft simultaneously. We apply stereoscopic selfsimilar expansion (SSSE) geometrical analysis, presented by Davies et al. (2013), to determine the kinematic properties of these CMEs using observations from both STEREO spacecraft. The SSSE method is based on the SSE geometry. SSSE with \(\lambda =0^{\circ }\) (i.e. a point source) corresponds to the socalled geometric triangulation (GT) technique, first performed by Liu et al. (2010). SSSE with \(\lambda =90^{\circ }\) (i.e. a circle anchored to the Sun) corresponds to the tangent to a sphere (TAS) technique, introduced by Lugaz et al. (2010). The extra information afforded by using a second vantage point allows one to drop the assumption that a CME is travelling in a constant direction and at a constant speed. However, the CME is still assumed to be selfsimilarly expanding at a constant halfwidth. These stereoscopic methods may only be applied to features that propagate in the plane containing both observing spacecraft and the Sun, which, in the case of STEREO, is the Ecliptic.
Lugaz (2010) analysed 12 CMEs that occurred between 2008 and 2009 and were seen in HI on both STEREO spacecraft using singlespacecraft and stereoscopic methods based on the FP (\(\lambda =0^{\circ }\)) and HM (\(\lambda =90^{\circ }\)) geometries. For both methods, they found discrepancies between the propagation direction derived from observations using STEREOA and STEREOB, particularly for CMEs propagating outside of \(60^{\circ }\pm 20^{\circ }\) from the Sun–spacecraft line. However, this discrepancy was larger when using the FPF method than when using the HMF method in all but one case. Within \(60^{\circ }\pm 20^{\circ }\), the two methods result in directions within \(\pm 15^{\circ }\) of each other. They identified three main sources of error: the assumption of constant propagation direction when using observations from a singlespacecraft, the assumption of negligible width when using \(\lambda =0^{\circ }\), and the assumption of constant CME velocity. The authors show that the first and third of these may be addressed by using stereoscopic observations, whilst the second may by addressed by modelling the CMEs with nonzero halfwidth. The singlespacecraft methods, and their ability to predict arrival times, are assessed by Möstl et al. (2011), who find arrival times within ±5 hours can be achieved if CMEs are tracked to at least 30^{∘} elongation. Möstl et al. (2017) study the HiGeoCAT CMEs, whose kinematic properties are derived by the SSEF method (using \(\lambda =30^{\circ }\)), and their ability to predict arrival times at different spacecraft. They find a range of 23 – 35% of predicted arrivals are actually observed in situ at the various spacecraft. The predicted arrival times are early by an average of 2.6 ± 16.6 hours, excluding predicted and insitu signatures that lie outside a oneday time window. More sophisticated geometries for modelling CME morphology include the Graduated Cylindrical Shell (GCS) model (Thernisien, Howard, and Vourlidas, 2006), ElEvoHI (Rollett et al., 2016), which employs an elliptical CME front and includes the effects of drag, and 3DCORE (Möstl et al., 2018), which is used to measure CME rotation and deflection and also to give information about magnetic field orientation based on data from the Solar Dynamics Observatory. Both Volpes and Bothmer (2015) and Palmerio et al. (2019) apply SSSE analysis to HI data, where the halfwidth of the CME is first determined using coronagraph observations. This is rather labourintensive and so is more challenging to apply to large statistical studies, such as that which we present here.
These singlespacecraft (SSEF, including FPF and HMF) and stereoscopic (SSSE, including GT and TAS) analysis techniques are based on assumptions that often fail to include the more complex physical processes that occur during CME propagation, such as rotations (e.g. Möstl et al., 2008; Vourlidas et al., 2011; Möstl et al., 2018), deformations (Savani et al., 2010), and, for singlespacecraft techniques in particular, deflections (e.g. Byrne et al., 2010; Wang et al., 2014). These effects result from CMEs interacting with other structure in the heliosphere including highspeed streams and other CMEs (Lugaz, Vourlidas, and Roussev, 2009; Lugaz et al., 2012; Liu et al., 2014; Lugaz et al., 2017). Whilst CME–CME interactions may be quite rare during solar minimum, a CME rate of 5 –10 day^{−1} is not unusual at solar maximum (e.g. Yashiro et al., 2004; Robbrecht, Berghmans, and der Linden, 2009; Gopalswamy, 2010; Vourlidas et al., 2017; Harrison et al., 2018). Indeed, Zhang et al. (2007) show that of the 88 major geomagnetic storms (defined by minimum disturbance storm time index, Dst ≤ 100 nT) that occurred during Solar Cycle 23, 60% were associated with individual CMEs; however 27% were the result of CME interactions with background structure or other CMEs.
Whilst the STEREO mission comprises two spacecraft, contact with STEREOB was lost in October 2014. The recently launched Parker Solar Probe and Solar Orbiter missions both possess wideangle imagers (WideField Imager for Parker Solar Probe (WISPR): Vourlidas et al., 2016 and SoloHI: Howard et al., 2013, respectively), as will the upcoming Polarimeter to Unify the Corona and Heliosphere (PUNCH) mission in lowEarth orbit and the potential future mission to the Sun–Earth L_{5} point (Kraft, Puschmann, and Luntama, 2017). These newly launched spacecraft are already returning CME images (Hess et al., 2020, using WISPR) and it is therefore important to realise the benefits and the limitations of the methods that we use to analyse the data, particularly the assumptions involved when observing from just a single vantage point.
Section 2 includes a description of the SSSE method and an explanation of how we apply it to time–elongation profiles from HI. Section 3 presents the results of the statistical analysis of the SSSE fitting results, including CME acceleration. Finally, we present a comparison between the kinematic properties derived using stereoscopic analysis methods and those that were determined using observations from just a single spacecraft. For a thorough description of the compilation of HICAT the reader is urged to refer to Article 1 (Harrison et al., 2018); furthermore the compilation of HiGeoCAT is the subject of Article 2 (Barnes et al., 2019).
2 Method
Due to the overlap of the FOVs of the HI cameras on the two STEREO spacecraft, a number of CMEs occur that are imaged from both STEREOA and B vantage points. Prior to 2014, the HIA and HIB cameras were offpointed towards the eastern and western limbs of the Sun, respectively. The amount by which the FOVs of each spacecraft overlap therefore increased from the start of the mission until late 2010, when the spacecraft separation was close to 180^{∘}, after which time it progressively decreased. This means that it is typically easier to identify CMEs that are observed by HI on both spacecraft near the end of 2010. Additionally, the CME rate was very low at the start of the mission because it coincided with solar minimum (Articles 1 and 2). As a result of these two factors, the first event that we identify to be observed in HI on both spacecraft is in August 2008, nearly two years after the launch of STEREO.
We identify joint events by manual inspection of those events contained in HICAT, based upon their time of entry into the respective HI1 FOVs. If a HICAT CME enters the HI1A FOV, we identify if any CMEs have entered the HI1B FOV within a ±twoday window. This time window is chosen to be large enough that no CME that is observed by both spacecraft is likely to lie outside it. We then determine if the two events are the same CME by examination of sequences of simultaneous images from HI1A and HI1B, which have a nominal cadence of 40 minutes. This is achieved by identifying similarities in CME size, morphology, and internal structure. The events that satisfy these conditions are listed in a new, separate catalogue that is available on the HELCATS website (HIJoinCAT: www.helcatsfp7.eu/catalogues/wp2_joincat.html). HIJoinCAT contains just two columns, which list the unique identifier of the CME observed in HI1 data from STEREOA and STEREOB, respectively, as they appear in the singlespacecraft HICAT list (and therefore HIGeoCAT, of which the former is a superset). HiJoinCAT contains a total of 273 CMEs, observed between 31 August 2008 and 02 April 2014. Unlike both HICAT and HIGeoCAT, the HIJoinCAT list is no longer being updated due to the fact that it requires data from both spacecraft, a condition that is no longer satisfied since the loss of STEREOB in 2014.
Two HIJoinCAT CMEs are shown in Figure 1; the HICAT unique identifier for each event is printed at the bottom of each panel. The top two panels show simultaneous images from HI1A (left) and HI1B (right) at 16:09UT on 02 June 2011 when STEREOA and STEREOB were 192^{∘} apart. We are able to determine quite easily that these images are of the same CME because they exhibit very similar structure. The bottom two panels in Figure 1 show simultaneous images from HI1A (left) and HI1B (right), at 20:09 UT on 26 October 2013, when the STEREO spacecraft were 290^{∘} apart. Although it is less obvious than in the previous example, similar structures can still be identified in each image, which leads us to conclude that this is indeed the same CME observed by both spacecraft.
To the CMEs that are contained in the new catalogue, we apply the SSSE analysis method of Davies et al. (2013) to the time–elongation data that were already determined for each CME front for HIGeoCAT as described in Article 2. It should be noted that the time–elongation data of each CME front were determined at a position angle (PA) close to the apex, which is occasionally far from the Ecliptic. The SSSE fitting method must be applied to a CME front in the plane that contains both spacecraft and the Sun; in the case of STEREO this is the ecliptic plane. As such, we must assume that those time–elongation profiles recorded away from the Ecliptic are a reasonable approximation to the time–elongation profile of the CME front in the ecliptic plane. Examples of CME time–elongation maps are shown in Figure 2, where panels a and b correspond to the CME in the top panels of Figure 1. Likewise, panels c and d of Figure 2 correspond to the CME shown in the bottom of Figure 1. The SSSE method makes the same assumptions about CME morphology as does the SSEF method (Davies et al., 2012): that the CME is a selfsimilarly expanding structure, with a specified half width and a circular crosssectional front. However, unlike with the singlespacecraft methods, the extra information afforded by using two spacecraft to track the CME means that we no longer require the assumptions of constant speed and constant propagation direction that were necessary with the singlespacecraft fitting technique. By applying the SSSE method to a CME, using an angular halfwidth [\(\lambda \)] its position is defined by the points where the line of sight from each spacecraft intercepts its leading edge. Assuming a circle of fixed halfwidth, we determine the heliocentric distance of the CME apex [\(R\)] using the following equation from Davies et al. (2013):
where \(d\) is the heliocentric distance of the spacecraft, \(\epsilon \) is the solar elongation angle of the CME leading edge, and \(\phi \) is the spacecraft–Sun–CME apex angle. The subscripts A and B refer to the observing spacecraft. From Equation 1, the time–elongation profiles that were used to compile HIGeoCAT are used to solve for \(R\) and \(\phi \), as a function of time. For a full mathematical derivation, the reader is referred to Davies et al. (2013), but Figure 3 illustrates the concept for our two example CMEs.
Figure 3 shows a schematic representation of the solutions for two CMEs with \(\lambda =40^{\circ }\) that were derived for two timesteps in the time–elongation profiles for the CMEs shown in Figures 1 and 2. The method for solving Equation 1 for \(R\) and \(\phi _{A}\) or \(\phi _{B}\) (Davies et al., 2013) depends on a square root and therefore has two solutions. Typically, however, one of these solutions is unphysical and may be easily discarded. Mathematically, the blue circle in panels a and b of Figure 3 describes a CME propagating away from the Sun with a negative \(R\), of which the trailing edge corresponds to the observed elongation. This is why the blue line does not connect from the Sun to the CME centre. In such cases, we can easily discard this solution as incorrect. Conversely, the red circle represents the leading edge of a CME travelling approximately between the two spacecraft, which is consistent with the observations in Figure 1. In some cases there exist two ambiguous, realistic solutions and the appropriate result must be selected manually, for example the second row in Figure 3. In these cases, we assume the CME to be directed approximately towards the Earth because this is the region where the HI1 FOVs overlap. There exists a further limitation of stereoscopic methods when the observed lines of sight are close to parallel, that is when the CME leading edge passes directly between the two spacecraft. When this happens, \(R\) and \(\phi \) become strongly influenced by small errors in \(\epsilon \) and the resulting solutions give CMEs that vary significantly in apex position between successive observations. For this reason we discard these solutions from the analysis presented in this article; however, these CMEs are still included in HIJoinCAT, which does not contain CME kinematics. This configuration is most common when the spacecraft are separated by approximately 180^{∘}, which is close to solar maximum and when the overlap between the HI FOVs is also maximised, and therefore the time at which the majority of the joint CMEs are detected.
The image cadences of the HI1 and HI2 cameras are 40 minutes and 2 hours, respectively, and so we may use successive sets of observations to track the time–elongation profile of the CME’s leading edge, using Jmaps, as it propagates through the heliosphere, as shown in Figure 2. The SSSE technique requires that the elongation of the CME front as observed from STEREOA and B be simultaneous, so the time–elongation profile from each spacecraft is mapped onto a set of common times using a linear, stepwise interpolation, separated by 30 minutes, limited by the time interval for which data from both vantage points are available. For a given timestep, a value of \(R\) and \(\phi \) is calculated using Equation 1 for ten different halfwidths increasing from 0^{∘} to 90^{∘} in increments of 10^{∘}. Such analysis is performed for each timestep to produce time profiles of \(R\) and \(\phi \) for the CME. To derive the profiles of the CME velocity [\(V(t)\)] and acceleration [\(A\)] we fit a function to the CME apex radial distance [\(R\)] profile. As is the case with many existing CME catalogues (e.g. Yashiro et al., 2004; Vourlidas et al., 2017), we choose to fit a secondorder polynomial.
An example of the analysis of HCME_A__20131026_01 and HCME_B__20131026_01 is shown in Figure 4, where the CME is tracked for just over 24 hours (50 halfhour timesteps). The CME apex longitude, in Heliocentric Earth Ecliptic (HEE) coordinates, as a function of time (Figure 4a) can be seen to shift from approximately +20^{∘} to −15^{∘}, in the most extreme case of \(\lambda =90^{\circ }\), over this period, where positive is westward. A deflection of this magnitude is feasible (Wang et al., 2014; Isavnin, Vourlidas, and Kilpua, 2014), and the westtoeast direction is consistent with the findings of Wang et al. (2004) for fast CMEs. However, we expect that some contribution is likely to result from errors in the fitting method. Liu et al. (2013) find the HM geometry to be an inaccurate approximation for CMEs near the Sun due to the fact that CMEs expand at a rate greater than selfsimilarity in their early propagation phase. Indeed, the deflection shown in Figure 4 is most pronounced for \(\lambda =90^{\circ }\) and least so for \(\lambda =0^{\circ }\). Figure 4b shows the CME apex heliocentric distance, in AU, as a function of time. The first timestep corresponds to a CME apex distance close to 0.2 AU, relatively independent of \(\lambda \), and, depending on the chosen halfwidth, the CME is tracked to just beyond 0.6 AU (red line; \(\lambda =90^{\circ }\)) or well beyond 1.5 AU (black line; \(\lambda =0^{\circ }\)). For each halfwidth, the secondorder polynomial fitted to the \(R\)profile is overplotted as a solid line. The velocity profile, in km s^{−1}, derived from this secondorder fit is shown in Figure 4c, as is the acceleration, in Figure 4d. The fit to the 0^{∘} halfwidth CME suggests an acceleration rate of over 40 m s^{−2}, resulting in a speed that increases from 800 km s^{−1} to well over 2000 km s^{−1} in less than eight hours. Such a speed increase is inconsistent with typical CME behaviour, particularly at these radial distances, which suggests that using this halfwidth to model the CME is a poor approximation. Indeed, for the 90^{∘} halfwidth model we find a CME accelerating at approximately 1 m s^{−2}, maintaining a speed between 800 – 900 km s^{−1}, which, although fast, is certainly more realistic behaviour. This example illustrates an important result that is common to many of the CMEs analysed in this article: using a small halfwidth often results in unphysical CME acceleration to very high velocities. This is unrealistic given that the average speed of ICMEs determined from insitu measurements is found to be approximately 450 km s^{−1}, regardless of heliocentric distance (Richardson, Liu, and Belcher, 2005).
3 CME Statistical Properties
3.1 CME Frequency
Initially we identify CMEs in HICAT that are observed using both STEREOA and B. For each CME observed in HI1A images we identify any CMEs that enter the HI1B FOV within ±twodays of the time that the CME enters the HI1A FOV. For all HICAT CMEs (965 from STEREOA and 936 from B) observed prior to the loss of communication with STEREOB, we produced a preliminary list of 475 potentially common CMEs using this method. We refine this list through examination of HI1 images. This results in a subset of 273 CMEs imaged by both HI1A and HI1B for inclusion in our socalled HIJoinCAT catalogue. It is likely that we erroneously exclude some events that were actually observed by both spacecraft due to nonoptimal viewing geometry. To these 273 CMEs, we apply the aforementioned stereoscopic fitting analysis to the STEREOA and STEREOB time–elongation profiles from the HIGeoCAT catalogue. Figure 5 shows the temporal distribution of the CME count with a bin size of one month. The white bins show the greatest number of CMEs observed by in HI on either STEREOA, or B, from HICAT. The shaded regions show the total number of HIJoinCAT CMEs, which is greatest during 2010 and 2011, corresponding to the time when the spacecraft were close to \(180^{\circ }\) separation; Solar Cycle 24 peaked soon after: in 2012. The lightergrey region of the histogram shows the number of CMEs that were confirmed to be imaged by both HI1A and HI1B but that were excluded from the final analysis for one of two reasons: Firstly, we exclude some CMEs for which time–elongation profiles from both STEREOA and STEREOB view points are not available in the HIGeoCAT catalogue. This is due to data gaps or CMEs that were too difficult to track. Secondly, and more significantly, the SSSE method breaks down when the LOSs of the observed leading edge of the CME from both spacecraft are approximately parallel. This occurs commonly when the boresights of the HI1 cameras are directly opposite, because the majority of CMEs were found not to be tracked far into the HI2 FOV in Article 2 (Barnes et al., 2019). As the HI1 FOVs are centred at \(14^{\circ }\) elongation in the Ecliptic, this alignment occurs around August 2010, when the spacecraft are separated by \(152^{\circ }\). Figure 5 shows the result of this problem, where all 35 dualspacecraft CMEs observed in the months July – November 2010 are excluded from the final analysis. The number of CMEs that were analysed using SSSE is greatest during 2011 and 2012, which is when the spacecraft separation approaches \(270^{\circ }\) and coincides with solar maximum. In total, the stereoscopic analysis was successfully applied to 151 CMEs.
3.2 CME Acceleration and Deflection
Figure 6 shows the distributions of both CME acceleration (panels a, c and e), which is assumed to be constant, and CME longitudinal deflection (panels b, d, and f), determined using SSSE with halfwidths of 0^{∘} (top row), 30^{∘} (middle row), and 90^{∘} (bottom row). The acceleration distributions are peaked near zero, showing that, typically, CMEs do not experience much acceleration in the HI FOV. For \(\lambda =0^{\circ }\), 46% of events have −1\(< A<+\)1 m s^{−2}, for \(\lambda =30^{\circ }\) the corresponding value is 48% and for \(\lambda =90^{\circ }\) it is 51%. Although accelerations tend to be small, their distribution depends quite strongly on the chosen halfwidth. For \(0^{\circ }\) halfwidth, 115 (77%) of CMEs are accelerating and 34 (23%) are decelerating, for \(30^{\circ }\) halfwidth 98 (66%) of CMEs are accelerating and 51 (34%) are decelerating, and for \(90^{\circ }\) halfwidth 79 (53%) of CMEs are accelerating and 70 (47%) are decelerating. That most CMEs are accelerating in the HI FOV is inconsistent with results established previously: whilst St. Cyr et al. (1999) show the majority of CMEs to be accelerating within 2.44 R_{⊙}, Gopalswamy et al. (2009) show that almost all have stopped accelerating by 32 R_{⊙}. The mean acceleration for CMEs analysed using \(\lambda =0^{\circ }\) is 6 m s^{−2}, which is skewed well away from zero by CMEs that have unphysically large accelerations that result from fitting with small halfwidths, as was discussed in the previous section. In Figure 3c, for example, the location of the CME shown in red is derived using a halfwidth of \(40^{\circ }\). A CME fitted with a halfwidth of \(0^{\circ }\) will be further from the Sun than the apex of that 40^{∘} halfwidth CME, at the points where the dashed lines intersect, whereas the apex of a CME with a halfwidth greater than \(40^{\circ }\) will be closer to the Sun. The SSSE method using \(\lambda =0^{\circ }\) can result in increasingly large speeds and large accelerations, as is seen in panels c and d of Figure 4. This effect tends to be less apparent for larger halfwidths; in the cases of \(\lambda =30^{\circ }\), and \(\lambda =90^{\circ }\), the mean accelerations are 1 and 0 m s^{−2}, respectively. Regardless of the halfwidth chosen, we still find, as noted above, that the number of accelerating CMEs is always greater than the number of those decelerating. Even for an intermediate halfwidth of \(30^{\circ }\), almost two thirds of the events appear to experience acceleration. The results do suggest that, in general, CMEs continue to experience acceleration within the HI FOVs; however, this is usually not significant in magnitude.
Figure 6b, d, and f shows the distributions of CME deflections in ecliptic longitude, determined using SSSE analysis with respective halfwidths of 0^{∘}, 30^{∘}, and 90^{∘}. The deflection refers to the difference between the final and initial longitudinal position of the CME apex. For all three halfwidths, significant deflections are often seen. For \(\lambda =0^{\circ }\), 33% of CMEs deflect by more than \(\pm 10^{\circ }\); corresponding values for \(\lambda =30^{\circ }\) and \(90^{\circ }\) are 45% and 54%, respectively. In some cases we appear to observe deflections of up to \(90^{\circ }\), far in excess of the maximum deflection of \(29^{\circ }\) observed by Isavnin, Vourlidas, and Kilpua (2014) and indeed of the \(20^{\circ }\) deflection by Wang et al. (2014). This is in contradiction to any known physical process and is instead the result of inadequacies with the assumptions of the analysis method. As was discussed in the previous section, and is seen in Figure 4, the SSSE method is sometimes poor at determining the orientation of wider CMEs when they are close to the Sun, due to the fact that they expand at a rate greater than selfsimilarity (Liu et al., 2013). Indeed, because of this issue, it is difficult to ascertain how much of the longitudinal deflections determined using SSSE analysis is due to inaccuracies in the method, because we are unable to accurately measure their initial longitude. However, the propagationdirection measurements become more constant as the CME propagates further into the heliosphere and so determining this value further out into the HI FOV is likely to provide a more reliable estimate of the ultimate CME propagation direction.
Figure 7a, d, and e shows a comparison between the initial CME velocities and the CME acceleration using SSSE with respective halfwidths of \(\lambda =0^{\circ }\), 30^{∘}, and \(90^{\circ }\). Figure 7 shows no clear correlation between the initial velocity and the measured acceleration when using \(\lambda =0^{\circ }\) to model the CME and the regression line is strongly skewed by CMEs with high acceleration values exceeding the plot range. Likewise, there is little correlation between initial velocity and acceleration for \(\lambda =30^{\circ }\) in Figure 7c. For \(\lambda =90^{\circ }\), in Figure 7e, there appears to be a slight tendency for the slowest CMEs to experience a positive acceleration and, conversely, for the faster CMEs to experience a deceleration, as expected (Yashiro et al., 2004; Gopalswamy et al., 2009). As noted previously, smaller halfwidths, particularly \(\lambda =0^{\circ }\), can lead to unphysically large accelerations. Figure 7e is consistent with the idea that CMEs experience drag from the ambient solar wind that causes their speed to tend towards the typical solarwind speed, which is not the case for Figures 7a and c. This may suggest that a halfwidth between \(30^{\circ }\) and \(90^{\circ }\) is a better representation of the observed CMEs. Indeed, Yashiro et al. (2004) show that the mean width of CMEs observed using LASCO increases from 46^{∘} in 1996 (solar minimum) to 57^{∘} in 2000 (solar maximum); however, this width is measured in PA and not longitude. The regression line in Figure 7e suggests that the juncture between CMEs that accelerate and those that decelerate corresponds to an initial speed of \(660\pm 346\) km s^{−1}, which, although rather imprecise, does correspond to the typical slow solarwind velocity at 1 AU of around 400 – 500 km s^{−1}.
Figures 7d, d, and f show a comparison between the initial CME speed and the final CME speed derived using halfwidths of \(0^{\circ }\), \(30^{\circ }\), and \(90^{\circ }\), respectively. In the case of \(\lambda =0^{\circ }\) (Figure 7b) there is little correlation between initial and final velocities, which is due to the fact that many CMEs are found to have unphysically high final speeds when using this halfwidth, regardless of their initial speed. The regression line is strongly skewed by CMEs with final velocities exceeding 2000 km s^{−1}. In fact, 22 CMEs (15%) analysed assuming \(\lambda =0^{\circ }\) have final velocities that exceed the 2000 km s^{−1} upper limit of the plot, whereas only three CMEs do so for \(\lambda =30^{\circ }\) and only one for \(\lambda =90^{\circ }\). For \(\lambda =30^{\circ }\) (Figure 7d) and \(90^{\circ }\) (Figure 7f), there is a strong correlation between initial and final velocity, with respective correlation coefficients of 0.75 and 0.74. The CME final velocity is less spread than that of initial velocity in each case, which can be seen in the histograms at the top and right of each panel. This can be explained by the idea that CMEs tend towards the ambient solar, wind speed. For the CMEs analysed using \(\lambda =90^{\circ }\), 60% of slower events, those that have an initial velocity below 500 km s^{−1}, are accelerating and 57% of those with an initial velocity above this value are decelerating. In the case of \(\lambda =30^{\circ }\), the majority of both slower and faster events are accelerating, which is inconsistent with established CME behaviour (e.g. Yashiro et al., 2004; Gopalswamy et al., 2009) and suggests that this may demonstrate inadequacies in the use of this geometry to describe the CMEs.
3.3 A Comparison of SingleSpacecraft and Stereoscopic Techniques
Figure 8 shows a comparison between the velocities determined for each CME from singlespacecraft SSE, and stereoscopic SSSE, analyses, resulting from three assumed geometries corresponding to \(\lambda =0^{\circ }\), \(30^{\circ }\), and \(90^{\circ }\) (equivalent to the singlespacecraft FPF, SSEF30, and HMF techniques); \(v_{\mathrm{A}}\) and \(v_{\mathrm{B}}\) are the velocities resulting from the singlespacecraft fitting methods applied to STEREOA and STEREOB time–elongation profiles, respectively, and \(v_{\mathrm{A+B}}\) is the initial speed from the stereoscopic method. Figures 8a, b, and c (top row) correspond to results using a halfwidth of 0^{∘}, Figures 8d, e, and f (middle row) use 30^{∘} and Figures 8g, h, and i (bottom row) use 90^{∘}. Each column presents panels corresponding to the three combinations of pairs of \(v_{\mathrm{A}}\), \(v_{\mathrm{B}}\), and \(v_{\mathrm{A+B}}\). The histograms at the top and left of each panel show the speed distribution corresponding to the \(x\) and \(y\)parameters plotted in that panel. The correlation coefficient [\(R\)] between each pair of velocity measurement ranges between 0.64 (Figure 8h) and 0.77 (Figure 8c), showing that there is reasonable agreement between speeds derived from all three methods, for each halfwidth. The best agreement is found for \(\lambda =0^{\circ }\) and the worst agreement for \(\lambda =90^{\circ }\). The CME final speeds derived using the stereoscopic analysis method, overplotted in light grey (panels in the second and third columns), are found to show a much poorer correlation with the singlespacecraft speeds (with \(R\) ranging from 0.26 in panel b to 0.50 in Figures 8f and i). This correlation is worst for \(\lambda =0^{\circ }\) (Figures 8b and c) and improves with increasing halfwidth; the best correlation is seen in Figures 8h and i, using \(\lambda =90^{\circ }\). As shown in Figure 4, for example, the final speeds derived using the stereoscopic method with \(\lambda =0^{\circ }\) are often unphysically high. However, even for \(\lambda =90^{\circ }\), the correlation between the final velocity derived from stereoscopic analysis and from singlespacecraft fitting is still far worse than that of the initial velocities. In fact the correlation coefficient has a value of only 0.26 between \(v_{\mathrm{A+B}}\) (final) and \(v_{\mathrm{A}}\) and a value of 0.38 between \(v_{\mathrm{A+B}}\) (final) and \(v_{\mathrm{B}}\). This is consistent with the results of Liu et al. (2013), who find an “apparent late acceleration” for CMEs fitted using FPF (equivalent to SSEF with \(\lambda =0^{\circ }\)). They show that the HMF (\(\lambda =90^{\circ }\)) method can reduce this effect, however, that it can still produce an overestimate of CME speed further out into the heliosphere. Singlespacecraft derived speeds of those CMEs included in the HIGeoCAT catalogue that impact spacecraft throughout the inner heliosphere were compared to insitu signatures by Möstl et al. (2017). For those predicted impacts of HIGeoCAT CMEs that matched with insitu impacts, the predicted arrival times (derived using SSE with \(\lambda =30^{\circ }\)) were found to be 2.4±17.1 hours early for HIA and 2.7±16.0 hours early for HIB, for events within a time window of ±1 day. The HiGeoCAT speeds were on average 191±341 km s^{−1} greater than those measured insitu for HIA CMEs and 245±446 km s^{−1} greater for HIB CMEs. However, a similar study has not been performed using the CMEs in the HIJoinCAT, which are analysed using the SSSE method, and which would provide a measure of ground truth with which to compare all three fitting methods. Without such a ground truth with which to determine whether SSSE or SSEF analysis, and using halfwidth, is most accurate at determining the true CME speed, we are instead able to draw some conclusions from the results by identifying the weaknesses of these models. The good agreement between SSEF speed and SSSE initial speed suggests that both methods are in fact a reliable means with which to measure CME speed. The extra information afforded by using two spacecraft to track the CME would suggest that this is a better method to use, if data are available from two vantage points; however, the apparent late acceleration identified by Liu et al. (2013) means that the results become less accurate when the CME is observed further into the heliosphere, particularly when using small halfwidths.
Figure 9 shows a comparison between the longitudinal propagation angles determined for each CME, from each of the three methods. Here, lon_{A} and lon_{B} are the derived longitude of the CME apex in Heliocentric Earth Equatorial (HEEQ) coordinates from singlespacecraft analysis and lon\(_{\mathrm{A+B}}\) is the final CME apex longitude derived from stereoscopic analysis in the same coordinate system. The top, middle, and bottom row of panels correspond to a halfwidth of 0^{∘}, \(30^{\circ }\), and \(90^{\circ }\), respectively. Each plot shows the difference in longitude between two of the three fitting methods, as a function of spacecraftseparation angle. The histogram at the top of each plot shows the distribution of CMEs as a function of spacecraftseparation angle, which increases as a function of time, and therefore they correspond approximately to the darkgrey distribution in Figure 5. The total range of separation angles over which the CMEs are observed is 73^{∘} to 291^{∘} and the majority of CMEs (78%) occur once the spacecraft separation is greater than 180^{∘} because this equates to solar maximum. The longitudes derived from singlespacecraft analysis are in fairly poor agreement with each other; moreover, neither agrees well with the results from stereoscopic analysis. In addition, the discrepancy between each set of results appears to show a systematic variation with spacecraft separation. For each row (i.e. different halfwidth) the difference between the SSE_{A} and SSE_{B} longitudes (lefthand column) suggests that the methods produce a bias towards a certain range of CME propagation directions relative to the spacecraft. As the spacecraft move apart in longitude, in opposite directions, this bias results in the observed correlation between separation angle and longitude difference. In Sections 3.2 and 3.3 of Article 2 (Barnes et al., 2019), we studied the distribution of CME propagation angle [\(\phi \)] relative to the spacecraft, for all 1455 CMEs in version 5 of the singlespacecraft fitting catalogue HiGeoCAT. We showed that the maximum of each distribution was peaked at around 78^{∘} (\(\lambda =0^{\circ }\)), 72^{∘} (\(\lambda =30^{\circ }\)), and 84^{∘} (\(\lambda =90^{\circ }\)) for STEREOA CMEs and 72^{∘} (\(\lambda =0^{\circ }\)), 69^{∘} (\(\lambda =30^{\circ }\)), and 77^{∘} (\(\lambda =90^{\circ }\)) for STEREOB CMEs. This is believed to be due to two effects: The first is an observational effect, whereby it is somewhat easier to observe CMEs travelling close to the Thomson surface (Tappin and Howard, 2009). The second is an inherent bias in the singlespacecraft fitting models, found by Lugaz (2010) who shows that assuming \(\lambda =0^{\circ }\) for a wide CME (with \(90^{\circ }\)) causes a bias towards propagation directions close to \(60^{\circ }\) from the Sun–spacecraft line, for CMEs that propagate at more than \(\pm 20^{\circ }\) from this direction. Systematic effects are also seen in the second (Figures 9b, e, and h) and third (Figures 9c, f, and i) columns, where we compare the stereoscopic results to each of the singlespacecraft results. However, they are less significant because the stereoscopic method only suffers from the Thomsonsurface effect and not the bias from the singlespacecraft model assumptions. The regression line in Figure 9a crosses zero on the \(y\)axis when the spacecraft separation is close to \(150^{\circ }\), which is the sum of the median propagation directions found from SSEF with \(\lambda =0^{\circ }\) (FPF) for each spacecraft in Article 2 (Barnes et al., 2019). That is, these biases cause the FPF longitudes to coincide when the spacecraft separation is \(150^{\circ }\). As we have no ground truth to determine the best method, and halfwidth, with which to accurately determine CME propagation, we do so based on the observed biases in the results. Clearly, the inherent biases identified by Lugaz (2010) are apparent in the results presented here and we must agree with their findings: using a small halfwidth with SSEF analysis is a poor method to determine CME propagation direction, and the extra information afforded by SSSE analysis, applied using a large halfwidth, is the best method to avoid these biases. However, CME overexpansion means that using SSSE analysis to locate CMEs at low elongation angles is less reliable than when it is applied to observations further out in the heliosphere.
4 Summary
From all CMEs observed by HI on STEREOA and STEREOB, whilst contact still existed with the latter, we identify 273 CMEs, occurring between 31 August 2008 and 02 April 2014 that are observed by both. We apply SSSE analysis techniques to these CMEs in order to determine their kinematic behaviour. During this time the spacecraftlongitude separation increased from 73^{∘} to 291^{∘}. This period spans approximately half a solar cycle, beginning at solar minimum and ending at the peak of Solar Cycle 24.
The main conclusions are summarised as follows:

i)
The SSSE method fails when the CME passes between the observing spacecraft and the linesofsight of the CME leading edge are close to parallel because small errors in elongation translate to large errors in determining CME position. We therefore apply the technique to just 151 CMEs, 78% of which occur close to solar maximum, after the spacecraft separation exceeds 180^{∘}. These data are too few to perform a thorough investigation of the optimal spacecraft configuration with which to analyse CMEs using the SSSE method. However, the results show that two spacecraft situated at L_{4} and L_{5}, with a separation of \(120^{\circ }\), would be a feasible configuration to track an Earthdirected CME, until its front reaches \(60^{\circ }\) elongation, where the LOSs would become parallel.

ii)
Accelerations derived using the SSSE technique are much higher when a smaller halfwidth is chosen. For \(\lambda =0^{\circ }\), 76% of CMEs were found to have positive acceleration and 15% showed a final velocity exceeding 2000 km s^{−1}. For \(\lambda =30^{\circ }\), 66% of all CMEs were found to be accelerating, regardless of their initial velocity. Conversely, using \(90^{\circ }\) results in approximately half of CMEs accelerating and half decelerating (52% versus 48%, respectively), suggesting that this model agrees best with the average CME width of 47 – 60^{∘} (Yashiro et al., 2004). For slower CMEs, with initial speeds below 500 km s^{−1}, 60% are seen to accelerate and 57% of CMEs faster than 500 km s^{−1} are seen to decelerate when using \(\lambda =90^{\circ }\). This is consistent with the wellestablished understanding that drag between CMEs and the background solar wind causes the CME speed to tend towards the ambient solarwind speed. This is an important result, which indicates that CMEs are capable of experiencing acceleration well into the HI FOV. This is contrary to the main assumption used in SSEF analysis, a method that is commonly used to analyse interplanetary CME propagation.

iii)
The inferred longitude of CMEs is found to vary greatly when they are close to the Sun, due to the fact that our SSSE analysis does not account for CME overexpansion. As the CME is tracked to larger elongation angles, the propagation direction is found to approach a constant value. It is therefore difficult to draw meaningful information about CME deflections, because we cannot accurately know their initial longitude. However, the final longitude for \(\lambda =90^{\circ }\) is expected to provide a good estimate of the ultimate CME propagation direction.

iv)
The velocity for each of the 151 CMEs is determined using three different means: SSEF using STEREOA data, SSEF using STEREOB data, and SSSE analysis using data from both spacecraft. Each technique is applied using three different halfwidths to fit the CMEs: 0^{∘}, 30^{∘}, and 90^{∘}. Agreement between initial CME speed is good between all methods; however, the final CME speed derived from SSSE analysis does not agree with that from SSEF. This is in part due to the overestimation of CME acceleration when using a small halfwidth in the SSSE analysis and in agreement with the apparent late acceleration found by Liu et al. (2013). We therefore conclude that it is best to use the SSSE method with \(\lambda =90^{\circ }\) to determine CME speeds; however, the results become less reliable when the CME is tracked further into the heliosphere.

v)
Similarly, we compare the difference in HEEQ longitude of the CME apex between each pair of fitting methods, again using \(\lambda =0^{\circ }\), \(30^{\circ }\), and \(90^{\circ }\). The agreement between the SSEF methods from each spacecraft is poor, with differences close to 180^{∘} in the worst cases. The effect is systematic and is a function of spacecraftseparation angle, which is due to three causes: Firstly, projection effects caused by Thomson scattering; secondly, biases in the CME direction determined when using a small halfwidth, as identified by Lugaz (2010); and, thirdly, the incorrect assumptions employed by the singlespacecraft fitting methods. We therefore find that the SSSE method using \(\lambda =90^{\circ }\) is also the best method to determine CME propagation directions, however, it is less accurate at low elongation angles due to the fact that it fails to account for CME overexpansion nearer the Sun.
Whitelight heliospheric imaging offers a unique way to track CMEs through the inner heliosphere. In Article 1 (Harrison et al., 2018) we presented a catalogue of interplanetary CMEs that, at the time of writing, contains over 2000 events and spans an entire solar cycle. Many of these CMEs have been studied using singlespacecraft analysis techniques in Article 2 (Barnes et al., 2019), and 151 of these events, presented here, have been analysed using stereoscopic observations. If we wish to track CMEs in the heliosphere, for the purposes of both science and space weather, there are, however, many limitations that result from doing so with observations from just one spacecraft, and many still with observations from two. Many of the limitations identified in this article are possible to address by modifying the way that the SSSE analysis is applied to the data. For example, modelling a CME with super selfsimilar expansion would account for CME overexpansion in the early propagation phase. Alternatively, it may be preferable to analyse the CME in coronagraphs separately using models, such as GCS, that can measure CME expansion, before moving to SSSE analysis at greater distances from the Sun. With the upcoming launches of the PUNCH mission in Earth orbit, ESA’s Lagrange mission to L_{5}, as well as the continued coverage from STEREOA and the recent launches of the Parker Solar Probe and Solar Orbiter, we are entering an era of unprecedented coverage from wideangle imagers. It will therefore soon be possible to scrutinise these methods further: for example, the extra information available from three or more vantage points will allow the measurement of noncircular CME fronts and will greatly limit the cases where the SSSE method fails due to parallel LOS observations or when observing CMEs at small elongation angles close to the Sun. Further to this, Solar Orbiter will observe from higher latitudes giving a truly threedimensional view of CMEs when combined with observations from the Ecliptic. The PUNCH mission possesses the unprecedented advantage of measuring polarisation of Thomsonscattered light in heliospheric observations, which provides a further means to constrain the location of observed features along the LOS (e.g. DeForest et al., 2016).
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Acknowledgements
This work was carried out as part of the EU FP7 HELCATS (Heliospheric Cataloguing, Analysis and Techniques Service) project (www.helcatsfp7.eu/). We acknowledge support from the European Union FP7–SPACE–2013–1 programme for the HELCATS project (#606692). The HI instruments on STEREO were developed by a consortium that comprised the Rutherford Appleton Laboratory (UK), the University of Birmingham (UK), Centre Spatial de Liège (CSL, Belgium), and the Naval Research Laboratory (NRL, USA). The STEREO/SECCHI project, of which HI is a part, is an international consortium led by NRL. We recognise the support of the UK Space Agency for funding STEREO/HI operations in the UK. C. Möstl thanks the Austrian Science Fund (FWF): P31521N27, P31659N27.
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Barnes, D., Davies, J.A., Harrison, R.A. et al. CMEs in the Heliosphere: III. A Statistical Analysis of the Kinematic Properties Derived from Stereoscopic Geometrical Modelling Techniques Applied to CMEs Detected in the Heliosphere from 2008 to 2014 by STEREO/HI1. Sol Phys 295, 150 (2020). https://doi.org/10.1007/s1120702001717w
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DOI: https://doi.org/10.1007/s1120702001717w