Three-Dimensional Parameters of the Earth-Impacting CMEs Based on the GCS Model

: When a CME arrives at the Earth, it will interact with the magnetosphere, sometimes causing hazardous space weather events. Thus, the study of CMEs which arrived at Earth (here-inafter, Earth-impacting CMEs) has attracted much attention in the space weather and space physics communities. Previous results have suggested that the three-dimensional parameters of CMEs play a crucial role in deciding whether and when they reach Earth. In this work, we use observations from the Solar TErrestrial RElations Observatory (STEREO) to study the three-dimensional parameters of 71 Earth-impacting CMEs from the middle of 2008 to the end of 2012. We ﬁnd that the majority Earth-impacting CMEs originate from the region of [30S,30N] × [40E,40W] on the solar disk; Earth-impacting CMEs are more likely to have a central propagation angle (CPA) no larger than half-angular width, a negative correlation between velocity and acceleration, and propagation time is inversely related to velocity. Based on our ﬁndings, we develop an empirical statistical model to forecast the arrival time of the Earth-impacting CME. Also included is a comparison between our model and the aerodynamic drag model.


Introduction
A Coronal Mass Ejection (CME) is a powerful solar explosion that contains a huge amount of plasma with energy of 10 31 ∼10 32 erg and mass of 10 14 ∼10 16 g [1 -3]. The interplanetary counterpart of the CME is called the Interplanetary Coronal Mass Ejection (ICME).
The properties of ICMEs can be analyzed by in-situ observations using spacecraft such as WIND [4]. Previous studies have shown that ICMEs often contain some or all of the following observational features: (1) enhanced magnetic field strength [5]; (2) large and smooth magnetic field rotation [5]; (3) gradually decreasing solar wind speed [5]; (4) lower proton temperature [6]; (5) lower plasma β [7]; (6) obvious bidirectional electron streaming [8]; (7) abnormal charge state of ions [9,10]; and (8) abnormal particle abundance [11,12]. According to whether the characteristics (1), (2), (5) are strictly met, ICMEs can be divided into magnetic clouds (MCs) and non-MCs [13]. Based on these criteria, some ICME catalogs have been established, and various statistical studies have been carried out for a more comprehensive understanding of ICMEs [14][15][16]. The ICMEs may interact with the Earth's magnetosphere and cause some hazardous space weather events, such as intense geomagnetic storms [17,18]. Hence, it's of great significance to study such events. frontside halo CMEs are obviously of interest. Many studies on Earth-impacting CMEs or geoeffective CMEs have focused on such CMEs [21,23,25,26,28,30]; however, sometimes an Earth-impacting CME is not necessarily a halo. Therefore, there are some studies [24] that analyze the characteristics of Earth-impacting or geoeffective CMEs and obtain some necessary but not sufficient conditions for Earth-impacting CME forecasting.
In this work, the three-dimensional parameters of all Earth-impacting CMEs recorded by STEREO andSOHO (2008-2012) are obtained through the GCS model. Then, we conduct some statistical analysis, trying to find the common features of these Earth-impacting CMEs, and compare our findings to previous studies to obtain a better and fuller picture of Earthimpacting CMEs. This paper is organized as follows: in Section 2, we present a simple explanation of the events we studied, as well as a brief introduction to the GCS model; the fitting results are shown in Section 3; in Section 4, we show how the results are processed and analyzed; finally, we give a conclusion and discussion in Section 5.

Data Collection & Method of Fitting
In recent work, Chi et al. (2016) [16] established a new online ICME catalog (http://space.ustc.edu.cn/dreams/wind_icmes/ (accessed on 23 July 2021)), mainly based on the WIND magnetic field and plasma observations from 1996 to 2014. In our work, to obtain Earth-impacting CMEs, as with a similar method illustrated in Chi et al. (2018) [51], we first check in-situ observations from this catalog to obtain information about ICMEs including their start time and average speed. According to this information about ICMEs, we deduce the possible eruption time of the progenitor solar CMEs. Then we search for a corresponding CME observation using J-map [52] with a 10 day long continuous window. Next, we check the STEREO-COR2 (https://stereo-ssc.nascom.nasa.gov/data.shtml (accessed on 23 July 2021)) observed images to find the corresponding CMEs. If the corresponding CMEs are found, we confirm the information about these CMEs in the observation of SOHO by combining the CME list of CDAW (Coordinated Data Analysis Workshop: http://cdaw.gsfc.nasa.gov/CME_list/ (accessed on 23 July 2021)). Note that, if there are multiple alternative CMEs, especially during times of higher solar activity, we use the GCS model to obtain the velocities and directions of all these CMEs and pick out the most likely progenitor CME. Finally, we obtain an ICME-CME list (see Appendix A.1 Table A1) which contains a series of Earth-impacting CMEs from 21 July 2008 to 11 December 2012. A total of 91 ICMEs have been observed during this period, with the solar source of 77 of those ICMEs being more certain. As a result, these 77 ICMEs will be analyzed in this paper.
The data we use are the white-light coronographic observations from the LASCO instrument onboard SOHO and the Sun-Earth-Connection Coronal and Heliospheric Investigation (SECCHI) [53] onboard STEREO [37].
The model we use to obtain the three-dimensional parameters of the Earth-impacting CMEs is the GCS model. This model is a kind of empirical model meant to reproduce the large-scale structure of flux-rope-like CMEs [38][39][40], which consist of a tubular section forming the main body of the structure attached to two cones that correspond to the "legs" of the CME. It expands in a self-similar way. According to the mathematical deduction from Thernisien (2011) [40], the GCS model has six free geometric parameters, including the longitude (φ), latitude (θ), tilt angle (γ), the height of the CME's leading edge (h f ront (see Figure 1 [39,40]), the half-angle of one cone (δ) and the half-angle between the axis of two cones (α) (see the Schematic of GCS model in Figure 1 [39,40]).
After selecting the events, we use the GCS model based on Interactive Data Language (IDL) and Solar So f tware (SSW) to obtain these six geometric parameters of the CMEs mentioned above. To be specific, we need to obtain the maximum similarity between the projections of the GCS model (the green wireframe in Figure 1d-f) and the actual observations of the white-light coronagraph of CMEs (images like Figure 1a-c) by adjusting the six parameters of the flux rope manually, that is, the forward fitting method. After fitting, we can obtain the GCS model's six geometric parameters of each CME event in a time series during its evolution in the field-of-view of three coronagraphs. Before using these parameters to do any analysis, we have performed some simple processing of them. First, the actual half-angular width is 1 2 ω FO (see Figure 1 [39,40]; hereinafter, we use ω to represent the half-angular width), which means the actual halfangular width equals the half-angular width between two cone axis plus half-angular width of one cone (ω = α + δ). Then, we do the linear fitting for the time series of h f ront to get the average initial velocity (V GCS ) of CMEs. At last, we do the quadratic polynomial fitting for the time series of h f ront again and got the initial acceleration (a GCS ) of CMEs. An example of these two kinds of fitting results is presented in Figure 2. This CME event occurred on 6 June 2012.
In Figure 2, the red cross symbols (x) mark the actual measure of h f ront , the green dash line is the linear fitting result and the texts with the same color are the linear velocity we obtain, the blue line and the acceleration information in texts of the same color are the results from quadratic polynomial fitting.

Results
In Section 2, we mention that there are 77 Earth-impacting CMEs that can be used for the analysis. After finishing GCS fitting and some early simple data processing, we find that eight CMEs among the 77 events are not suitable to be modeled by the GCS model and should be excluded. We suspect that two special events (ICME around 07:30:00 UT, 21 June 2010 and 05:33:00 UT, 28 May 2011 [51]) are the results of two CMEs' interaction, thus we perform GCS fitting for both of the two source CMEs. That means that we finally obtain 71 sets of fitting results. These three-dimensional parameters and some other useful information about the Earth-impacting CMEs are listed in Table 1. Columns from left to right are the sequence number of the CME (No.), initial fitting time (Time_fit), longitude (φ, positive for west), latitude (θ, positive for north), tilt angle (γ), half-angular width (ω), the average initial velocity (V GCS ), the initial acceleration (a GCS ) and the time for the start of the ejecta (Time_arrive) observed by WIND, respectively. Note that we only list the initial value of φ, θ and γ for each event due to the possible deflection by Parker spiral magnetic field [27] after that time.
The uncertainties come from the errors of the original six parameters in the GCS model fitting process. A study by Thernisien et al. (2009) [39] shows the mean uncertainties in the GCS model are about ±4.3 • , ±1.8 • , ±22 • , +13 • −7 • , +0.07 • −0.04 • and ±0.48 R sun (solar radii) for φ, θ, γ, α, δ and h f ront , respectively. Our analysis will use these uncertainties as a reference. So, the uncertainty of ω can be derived, which is about +13.1 • −7.1 • . By taking the uncertainty of h f ront into the linear fitting and the quadratic polynomial fitting process, the uncertainties of V GCS and a GCS for each event can also be calculated. The 1 − σ uncertainties of V GCS and a GCS are shown in Table 1.  [28] (no γ, a GCS and the uncertainty of V GCS ). −−− No corresponding data. # Lower confidence level about ICME-CME causal chain.
Now we have the parameters of CME's longitude, latitude, tilt angle, half-angular width, velocity and acceleration. Next, we will analyze the characteristics of these Earthimpacting CMEs, as shown in the following section.

Location
The first concern in predicting Earth-impacting CME is whether there is a CME, specifically we need to know the explosion time and locations of CMEs. However, due to the obscure physical mechanism of CMEs, it is hard to know these two pieces of information in advance by theoretical derivation. So what we can do now is just giving an early warning based on the observation of ongoing CMEs. In this subsection, the locations and other location-related properties of Earth-impacting CMEs are discussed using data from Table 1.
To get a general view of the locations of these Earth-impacting CMEs, we mark them on solar disk like Figure 3. On the top panel, red diamond symbols stand for CMEs appeared on the solar disk (observed from the Earth, the same goes for later descriptions). The bottom panel of Figure 3 shows a scatter graph of θ and φ with red diamond symbols. The two horizontal dashed blue lines mark the maximum and minimum values of θ, which are 43.00 • and −30.18 • , respectively. The enclosed region between two vertical blue dashed lines stands for the front-side solar disk (φ between −90 and 90 degrees).
If we focus on the latitudinal information, Figure 3 reveals that Earth-impacting CMEs are concentrated in the region between 30 • north and 30 • south, with specific value 64 71 (about 90%). As for longitude, Cane et al. [54] suggested that the location of typical Earthimpacting solar events are in longitude ≤40 • east and west. In this work, 60 71 (about 85%) of CMEs locate at longitude of −40 • ∼40 • , and more than half ( 39 71 , ∼55%) of CMEs come from −20 • ∼20 • . The easternmost and westernmost longitudinal locations are −60.56 • and 62.41 • , respectively, which shows the possibility that some Earth-impacting CMEs originate far from the central region.
According to the characteristic of latitudinal and longitudinal distribution of these Earth-impacting CMEs, we suggest that all the Earth-impacting CMEs are distributed at the front of the solar disk, and are concentrated near the central region except fairly few of them are scattered toward to the solar limb. Given that all these CMEs are Earth-impacting, it seems that deflections of these CMEs beyond the field of view of STEREO/COR2 and LASCO/C2 are not significant. Based on this conclusion, we assume that most Earth-impacting CMEs move outward almost in a radial direction for further analysis in Sections 4.2 and 4.3.

Central Propagation Angle & Half-Angular Width
After obtaining the locations of Earth-impacting CMEs, that is, which region of the solar disk these Earth-impacting CMEs come from, we then need to know what kind of CMEs are theoretically Earth-impacting. So the first thing we consider is what propagation direction conditions need to be met for a CME to reach the Earth.
The central propagation angle (CPA) means the angle between CME's central propagation axis and Sun-Earth line, which usually represents the deviation angle of CME under the assumption that the CME's initial orientation is perpendicular to the solar surface. It can be derived by a simple geometric relationship in which the initial CPA (all following mentions of CPA are the initial CPA unless otherwise stated) or deviation angle satisfies Equation (1). cos = cos θ cos φ. (1) With the help of Equation (1), using data of φ and θ, we can obtain the value of CPA ( ). Its distribution is shown in Figure 4b.
The histogram indicates that ranges from 0 • to 70 • , and is clustered in the region between 0 • and 50 • . Specifically, 93% ( 66 71 ) of are in this range. As expected, the result also suggests that most of the Earth-impacting CMEs are centralized at the solar disk and propagate towards the Earth.
In addition to CPA, the other geometrical parameters of CMEs from GCS also have important effects on whether they could arrive at Earth or not. Note that the parameters we talk about here consist of face-on angular width (ω FO , equals double ω in Table 1; see Figure 1 [39,40]), edge-on width (ω EO ; see Figure 1 [39,40] and γ. For simplification, this work does not explore the influence of ω EO and γ, but only discusses the effect of ω. We assume that a CME moves as a self-similar expansion ball [28], as illustrated in Figure 4a, which is a reasonable and useful assumption in CME propagation studies [55]. Map of CME's Self-Similar Expansion Model. E means the Earth, S stands for the Sun, Front (F) shows the front of CME, D act equals SF which means the actual propagation distance of CME, ω is the half-angular width, is the deviation angle and R CME is the half of face-on width (or radius) of CME [28]. The geometric relationships in this sketch map show that the CME can arrive at the Earth without deviating when its half-angular width is not less than the deviation angle, that is, ω ≥ . Figure 4c shows the scatter-plot of and ω with diamond symbols. The orange dashed line indicates ω = . Based on the previous analysis, CMEs distributed in the region above the dashed line should be able to arrive at the Earth while those in the lower region (ω < ) should not. Here, about 69% of the CMEs have ω no less than . Considering that all these CMEs are Earth-impacting, this proportion derived from the above criterion looks fine, but is not so perfect. There are two possible reasons why the other 31% also arrive at Earth. The first reason we consider is that these CMEs have undergone a large deflection in the process of interplanetary space propagation [55,56]. When a CME or flux rope structure that does not satisfy ω ≥ arrives at Earth after deflection, the parts through which the spacecraft pass is their flanks but not the central flux rope structures. So, the in-situ observation may present a non-MC appearance [55]. On the contrary, if a CME arrives at the Earth without deflection (ω ≥ ), the in-situ observation will present an MC structure. To test this idea, we divide the events into two groups. One with red diamond symbols represent the CME with MCs observed in-situ, the other with blue diamond symbols represent those with Non-MCs observed in-situ, as shown in Figure 4c. Among those CMEs which satisfy ω ≥ , 24 are MCs and the other 25 are non-MCs. For those CMEs that do not meet ω ≥ , 41% ( 9 22 ) are MCs, and 59% ( 13 22 ) are non-MCs. The presence of MC or non-MC appears almost equivalently in both groups. Another reason is that the effects of the error of ω and need to be taken into account. As mentioned in Section 3, the mean uncertainty of θ, φ and ω is about ±1.8 • , ±4.3 • and +13.1 • −7.1 • , respectively. The mean uncertainty of has been calculated to be 3.3 • . Given that the uncertainties differ from one another, although their mean values are not remarkable, it still suggests that there should be more than 69% of Earth-impacting CMEs that satisfy the condition ω ≥ .
In total, ω ≥ is a rough but significant condition for a CME to reach the Earth. Therefore, this criterion should be considered an important part of the space weather forecasting model. On the other hand, there are still some events that do not satisfy our criterion but can be Earth-impacting. It means that this criterion is not a necessary condition. A possible explanation is that they had some deflections during propagation, which cannot be verified by this work because we haven't derived any large changes for latitude and longitude during the investigated period. It is a subject worthy of further study.

Simple Kinematics Analysis
Once we know there might be an Earth-impacting CME, it is a matter of cardinal significance to learn when the CME would arrive. So the kinematics of Earth-impacting CMEs is also another important subject for space weather forecasting studies, for which we need to focus on the velocity and acceleration of CMEs at the beginning.
To get a general view of the distribution of the velocity and acceleration of these Earthimpacting CMEs, we plot the histograms for velocity and acceleration as in Figure 5a,b. The average of V GCS is 563 ± 26 km·s −1 and the median is 496 ± 26 km·s −1 . The distribution of the velocity of CMEs demonstrates that there are not too many high-speed Earth-impacting CMEs during the period we study. For the accelerations, as shown in Figure 5b, they are centralized at 0 m·s −2 , more concretely, about 65% ( 46 71 ) of CMEs have accelerations of −20∼20 m·s −2 . In addition, the average and median values of acceleration are −8.7 ± 8.5 m·s −2 and −5.3 ± 8.5 m·s −2 , respectively. CMEs are decelerating on average, as seen by these two values. It is worth noting that, as shown in Table 1, the uncertainty of a GCS is roughly the same as its values, so it can be ignored in most cases. These general views of a GCS provide a significant piece of information that most of these CMEs maintain uniform speed or have little change on velocity from 3 to 20 R sun (LASCO C2: 2.2-6 solar radii; STEREO-COR2: 3-20 solar radii). To reveal the nature of the characteristics of the acceleration, we analyze the correlation between the velocity and the acceleration. As Figure 5c shows, there is an obvious negative correlation between a GCS and V GCS , with a correlation coefficient (CC) of −0.709. The green dash line in Figure 5c is the linear fitting result. When a GCS equals 0 m·s −2 , the corresponding value of V GCS is 341 ± 114 km·s −1 , which represents the average solar wind speed from 3 to 20 R sun during the investigation period [19]. Our statistical results are consistent with the previous understanding [3,19,24,27,57] that fast CMEs will be hindered and decelerated; conversely, slow CMEs will be dragged and accelerated under the effects of background solar wind.
After analyzing the general characteristics of the velocity and acceleration of Earthimpacting CMEs, we then study the actual propagation time T act (the actual propagation time mentioned here equals Time_arrive from column 9 in Table 1 minus Time_fit from column 2 in Table 1) of CMEs for a further understanding of their kinematic characteristics and also for establishing the propagation model of CMEs to realize Earth-impacting CMEs forecasting in the future.
As shown in Figure 6, CMEs' propagation time range from about 1.5 days to about 6 days, which are inversely related to the velocity of CMEs. The green dash dot line shown in Figure 6 is the hyperbolic fitted curve, and the fitting curve equation is The correlation coefficient (CC) of the hyperbolic fitting is 0.581. The coefficient of V GCS in Equation (2) is 1.4 × 10 −4 , which is comparable to the value in Equation (1) in Wang et al. (2002) [26] which is about 1.14 × 10 −4 after unit conversion. The determined value of propagation time using this equation has a maximum deviation of about 51.6 h, a minimum deviation of about 0.1 h, and an average deviation of about 15.2 h. This average value is comparable with previous results [31,[58][59][60]. Therefore, it is reasonable to establish a simple propagation model of Earth-impacting CME based on Equation (2). Although the empirical model mentioned above can obtain a value comparable with previous results, it just represents the general relationship between propagation time and the CMEs' initial velocities. The details of the CME's motion are still needed to improve the accuracy of the Earth-impacting CME predictions, which are outside the scope of this empirical statistical model. Besides, it is important to remember that the actual propagation distance for a CME to reach the Earth is not always 1 AU. Sometimes this value could be a bit larger than 1 AU.
As Figure 4a has shown, it is obvious that SF > SE, which means that CMEs' fronts have gone beyond 1 AU when they hit the Earth [28]. Using the geometric relationship shown in Figure 4a, we get Equation (3): Then we can solve the SF (hereinafter, we use D act , which means the actual propagation distance to represent SF). Note that we remove 22 invalid events (ω < ) because the simultaneous equation of Equation (3) does not have real roots unless ω ≥ .
Using the aerodynamic drag model, we can calculate the propagation time of these Earth-impacting CMEs by numerical calculation (see Appendix A.2 for detail). The calculation uses the values of initial velocity (V GCS ) computed by the GCS model as the initial value of v CME in Equation (4) and actual propagation distance (D act ) computed using Equation (3) as the final value of r. In the process of calculation, we can also obtain the change in velocity and the change in distance. Figure 7 is an example of the velocity change profile and distance change profile for a CME that occurred on 23 May 2010. The black curve in panel (a) shows how the CME's velocity (v CME ) changes versus the heliocentric distance of the CME's front (r) and the black vertical dashed line represents that the actual propagation distance for this CME is about 1.09 (au). In panel (b), the black line shows how the heliocentric distance of CME's front changes as the propagation time (T pro ) increases, the red vertical dashed line represents that the calculated total propagation time for this CME is about 119.76 h and the blue vertical dashed line shows that the actual propagation time is about 121.62 h. The deviation between these two values is 1.86 h.
Then, we compare the actual propagation time (T act ) and the calculated results (T cal ) of both ADM and the Empirical Statistic Model (ESM, described as Equation (2)) to test their accuracy. Figure 8 shows the comparison results.
In Figure 8, red circles represent the calculation by ADM, blue diamonds are the calculation by ESM, the green dashed line shows the relation T cal = T act . A rough comparison of the two models shows that propagation time calculated by ESM seems to have better accordance with the actual value, that is, those diamonds are closer to the dashed line than those circles. Moreover, we use the least square method to calculate the variance between the T cal and T act , that is, we use the following formula: In Equation (5), N is the number of these events. The result is quite similar to the rough glimpse. ∆ ESM is about 333.3 while ∆ ADM is about 3387.5; the latter is much bigger than the former due to the six events with large T cal , and we can obtain a smaller value ∆ ADM ≈ 594.2 if we remove the six events temporarily. The value of ∆ is much more affected by outliers; a smaller ∆ stands for less fluctuation or less deviation of one model. Considering this aspect, we hold the opinion that ESM is more accurate or more stable than ADM. The expected outperformance [31,61] of ADM over ESM does not occur.
For further analysis, we check the velocity of the six events with large T cal of ADM, which included 148.03 km·s −1 , 155.92 km·s −1 , 187.28 km·s −1 , 193.16 km·s −1 , 206.11 km·s −1 and 221.32 km·s −1 . These are the six minimum velocities among all the events involved in the calculation, which have velocities no larger than 230 km·s −1 . The calculation by ADM suggests that these small velocity CMEs should be quite slow to arrive at the Earth while the actual propagation times are much shorter, indicating that low speed CMEs may not be suitable for using Equation (4) to describe their motion, or their ADM equations probably need different C, β, δ and even v SW . This phenomenon shows that most suitable constants for Equation (4) may be affected by the velocity; previous studies have tried to analyze the optimal coefficients under different velocities [61]. Other parameters of a CME are also likely to influence the amount of drag from the solar wind [3]. Another possible reason is that there may be other mechanisms by which they accelerate such as two CMEs' interactions [57,64]. These are follow-up works and will not be discussed here.

Conclusions & Discussion
This work is a systematic study and analysis of 71 Earth-impacting CMEs from 21 July 2008 to 11 December 2012. The three-dimensional parameters of these CMEs are obtained using the GCS model. They have been used to analyze the geometric and kinematics characteristics of Earth-impacting CMEs, and an attempt has been made to find a simple propagation model of them. We conclude the following: Assuming a radial propagation, the central propagation angle (CPA, or ) of CMEs could represent their deviations. Based on this, we suggest that CMEs with a CPA less than 50 • may well be Earth-impacting, which verifies the previous results, such as 86% of FHCMES have a CPA smaller than 50 • [29]. If taking CMEs' other geometric parameters into account, such as the half-angular width (ω), and considering the effect of the errors, the proportion of Earth-impacting CMEs that satisfy ω ≥ is no less than 69%, which suggests that this inequality should be an important part of Earth-impacting CME forecasting.

3.
The velocity and acceleration of Earth-impacting CMEs follow each other with a negative correlation owing to the drag from the environment of which the background solar wind is a major part, so we use the v-axis intercept 341 km·s −1 as the average solar wind. This conclusion is reasonable and consistent with other studies [3,19,24,27]. The propagation time is inversely related to velocity, based upon which an empirical formula of CMEs' propagation can be obtained by hyperbolic fitting. This formula is T act = 1 0.19+1.4×10 −4 ×V GCS (km·s −1 ) days, which can give, with a good approximation, the propagation time of a CME to determine when it will reach Earth. However, the empirical statistical model only represents the general relationship between propagation time and CMEs' initial velocities; it can not describe the detail of the motion of CMEs, so we introduce an aerodynamic drag model to conduct more analysis. A rough comparison of these two models shows that the deviations between the calculated results and actual values of our empirical statistical model are smaller than those of the aerodynamic drag model, and the equation of the latter we use has low performance in low speed events. The matter needs further investigation.
In general, this work builds up a database based on GCS model fitting results for the geometric parameters of Earth-impacting CMEs, which is useful for further studies. This work also conducts preliminary statistical work to analyze the parameter we obtain and also achieves an empirical model for Earth-impacting CME forecasting.
For discussion, a manual, systematic study of many CMEs free from bias is a challenge due to the lack of a uniform and precise definition of a CME, and the subjective nature of the measurement process when performed by a human operator [65]. The main errors in this work are the artificial fitting process in the GCS model fitting; this problem should be overcome by a repeated fitting process and averaging of the results. Another unavoidable error is missing data in the coronagraphs, which makes the fitting less accurate. In addition, owing to the assumption that the CME is an ideal flux rope in the GCS model, the geometric description of CMEs becomes better. However, this assumption also limits a richer description of CMEs, leading to the inapplicability of the GCS model for more complex shaped CMEs. Therefore, it is also unavoidable that some error may exist in the fitting results, especially for those parameters that involve CMEs' shapes. For example, above 30 R sun (solar radii), the front of flux rope CMEs becomes distorted because of interactions with the ambient solar wind [3,66]-the GCS model is not capable of reproducing such behavior [40]. In total, the GCS model itself also contributes to errors. Recently, some new methods have been developed that aim to eliminate model defects or manual errors, which can be used to estimate a CME's propagation (e.g., CORSET3D [65], iCAF [67] and ELEvoHI [68]).
Moreover, the study of Patsourakos et al. (2010) [69] shows that, by not restraining the foot point O (see Figure 1 [39,40]) to be at the center of the Sun, it allows better GCS model fitting for the CMEs with propagation directions not perpendicular to the solar surface (e.g., the case on 21 June 2011 from Heinemann et al. 2019 [70], the case on 23 July 2017 from Dumbovic et al. 2019 [71]). This finding can be helpful to our further study of a more specific and more vision-broad analysis of CMEs' propagation, considering that our present work only used the data of SOHO/LASCO/C2 and STEREO/SECCHI/COR2. A more complete propagation process of a CME from its eruption to the interplanetary space can be analyzed by combining the data of EUVI, COR1, COR2, HI1, and HI2, in which case the more complex process of a CME's motion needs to be considered [72].
In addition to various geometric models or methods, a numerical simulation based on magnetohydrodynamics is also a particularly popular method for studying CMEs in recent years. For example, Shen et al. investigated the acceleration and deceleration of CMEs on 28-31 March 2001 with a 3D COIN-TVD MHD model [57], and they also investigated the influence of the CME's initial parameters on the simulation results [73,74]. These MHD numerical simulations can study CMEs in more detail, especially CMEs with complex structures that are difficult to analyze with most geometric methods, which is conducive to our understanding of the initial parameters and propagation process of CMEs.
Recently, some new solar probe spacecraft have been launched (e.g., Parker Solar Probe [75] and Solar Orbiter [76]), and other concepts for monitoring the Sun and the inner heliosphere have been proposed by Wang et al. (2020) [77] (Solar Ring mission). With the help of the observation of PSP, Solar Orbiter, and possible Solar Ring, we can observe the generation, propagation and interaction of CMEs from more perspectives (e.g., Braga and Vourlidas 2021 [78]), and some more accurate forecast models of Earth-impacting CMEs can be established.

Data Availability Statement:
The processed data and program underlying this article will be shared on request to the corresponding author.

Conflicts of Interest:
The authors declare no conflict of interest.

Appendix A
Appendix A.1. ICME-CME List The ICME-CME list we got is shown in Table A1. Columns from left to right are the sequence number of the event (No.), initial time observed by STEREO-COR2, initial time observed by SOHO-LASCO-C2, position angle measured from Solar North in degrees (counter-clockwise; PA), the sky-plane width of CMEs (Width) in degrees, linear speed of CMEs (V L ) in km·s −1 , acceleration of CMEs (Acc) in m·s −2 , start of the ejecta observed by WIND, and mean velocity in the ejecta (V E ) in km·s −1 . The parameters 'Initial time by C2', PA, Width, V L , and Acc are all from the CDAW SOHO/LASCO CME catalog (https://cdaw.gsfc.nasa.gov/CME_list/index.html (accessed on 23 July 2021)) [20]. Symbols '− − −' represent that there are no corresponding data.
It is worthwhile to note that 'Initial time by COR2' or 'Initial time by C2' are not always the same as 'Time_fit' in Table 1 because we start GCS fitting when all three spacecraft can observe the CME clearly. And, the width, V L , and Acc in Table A1 are different from the corresponding parameter we got from GCS fitting since the former are the results from only C2 observation but the latter is the three-dimensional parameters.