Abstract
The main objective of the study is to determine the probability distributions of the geomagnetic Dst index as a function of the coronal mass ejection (CME) and solar flare parameters for the purpose of establishing a probabilistic forecast tool for the intensity of geomagnetic storms. We examined several CME and flare parameters as well as the effect of successive CME occurrence in changing the probability for a certain range of Dst index values. The results confirm some previously known relationships between remotely observed properties of solar eruptive events and geomagnetic storms: the importance of the initial CME speed, apparent width, source position, and the class of the associated solar flare. We quantify these relationships in a form that can be used for future spaceweather forecasting. The results of the statistical study are employed to construct an empirical statistical model for predicting the probability of the geomagnetic storm intensity based on remote solar observations of CMEs and flares.
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Acknowledgements
This work has received funding from the European Union Seventh Framework Programme (FP7/20072013) under grant agreement no. 263252 [COMESEP]. This work has been supported in part by Croatian Science Foundation under the project 6212 “Solar and Stellar Variability”. This research has been funded by the Interuniversity Attraction Poles Programme initiated by the Belgian Science Policy Office (IAP P7/08 CHARM). L. Rodriguez acknowledges support from the Belgian Federal Science Policy Office through the ESA – PRODEX program. We are grateful to the SOHO LASCO CME catalog team for providing the CME data. This 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. SOHO is a project of international cooperation between ESA and NASA. We are also grateful to the SolarTerrestrial Physics (STP) Division of NOAA’s (National Oceanic and Atmospheric Administration) National Geophysical Data Center (NGDC) for providing solar flare data.
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Appendix
Appendix
Here we provide a supplement to Section 5 that reports the detailed stepwise mathematical formulations and procedures used for estimating the probability distribution of the geomagnetic storm level based on the remote solar observation of a CME and the associated solar flare. For this purpose, an exampleCME was used with the following characteristics:

First LASCO C2 appearance: 10 April 2001 05:30 UT.

Associated solar flare GOES peak time: 10 April 2001 05:26 UT.

Firstorder LASCO catalog CME speed: 2411 km s^{−1}.

CME angular width: halo.

CME or flare source region location (distance from the center of the solar disk, r): S23W09 (r=0.4067).

Flare Xray class: X2.3.

CME–CME interaction level: train, T (very likely interacts with a halo CME that first appeared in the C2 field of view 9 April 2001 15:54 UT).
Based on the CME or flare characteristics, the key parameters are defined in the following way:

v is a continuous parameter that is equal to the firstorder CME speed, measured in the LASCO field of view, expressed in km s^{−1} and defined in a range v>400. Therefore, v=2411.

r is a continuous parameter that is equal to the distance of a CME or flare position from the center of the solar disk expressed in solar radii and defined in a range 0<r≤1. Therefore, r=0.4067.

w is a discrete parameter with possible values 1 (nonhalo CMEs), 2 (partial halo CMEs), and 3 (halo CMEs). Therefore, w=3.

f is a discrete parameter with possible values 1 (B or C flare), 2 (M flare), and 3 (X flare). Therefore, f=3.

i is a discrete parameter with possible values 1 (S, no interaction), 2 (S?, interaction unlikely), 3 (T?, probable interaction), and 4 (T, interaction highly probable). Therefore, i=4.
A.1 The Geometric Probability Distribution, P(X=k)
The geometric probability distribution is a probability distribution of a random variable X, where X is the number of Bernoulli trials needed to succeed. There is an equal probability of success of each trial, p, and X is defined on an endless set of discrete values k=1,2,3,… (see, e.g., Pitman 1993, and Stirzaker 2003). It is a discrete analog of the exponential distribution. The probability density function for the geometric distribution is given by Equation (4) in Section 5, and an example is given in Figure 13 for different probabilities of success in each trial, p.
It is easily found that the expected value of the geometrically distributed random variable X, that is, the mean of the geometric distribution, is given by the following expression (for details see Stirzaker 2003):
Therefore, the probability of the success in each trial, p, can be calculated if the mean of the geometric distribution, m _{GD} is known,
We use the formalism for geometric distribution to construct Dst distributions observed throughout Section 4. For that purpose, different Dst levels have to be associated with different numbers of trials (k⟷Dst) and the mean of Dst distribution has to be associated with the mean of the geometric distribution (m _{GD}⟷m _{DST}). We associate different Dst levels with different numbers of trials, k, in the geometric distribution in the following way:

k=1⟷Dst<100 nT;

k=2⟷100 nT<Dst<200 nT;

k=3⟷200 nT<Dst<300 nT;

k=4⟷Dst>300 nT.
Note that the value of k is exactly 100 times lower than the upper boundary for the associated Dst level, expressed in nT. It is reasonable to assume that the mean of the geometric distributions relates in a similar fashion to the mean of the Dst distribution (i.e., would be 100 times smaller). The mean of the Dst distribution is in the first bin for all of the distributions throughout Section 4, i.e. m _{DST}<100 nT. However, because the geometric distribution is defined on a set k=1,2,3,…, the mean is always larger than 1. This is also seen from Equation (9) (p<1). Dividing m _{DST} by 100 would not give a mathematically correct m _{GD}, but adding 1 to this relation solves this problem. Therefore
For simplicity, we refer to P(X=k) as P(k). Note that for constructed distributions we use the set of k values k=1,2,3,4, based on the defined associations k⟷Dst.
A.2 Probability Distribution for the CME Speed (v), P _{ v }(k)
The change of the Dst distribution mean with the CME speed, v, can be described with a linear function (see Figure 5),
Here a=0.04, b=10.45, and Dst is expressed in nT. For a given v=2411, Equation (12) gives the distribution mean m _{DST}(v)=106.89. The geometric distribution mean is then calculated using Equation (11), which in our example gives m _{GD}=2.07. The probability of the success in each trial, p, can be calculated using Equation (10) and in our example equals p=0.48.
For each k=1,2,3,4, a probability that the kth trial is the first success, P(k), can be calculated using Equation (4) in Section 5. In the example the results are as follows:

P(k=1)=0.4833;

P(k=2)=0.2497;

P(k=3)=0.1290;

P(k=4)=0.0667.
Because the geometric distribution does not stop at k=4, this distribution is not normalized, i.e. \(\sum_{k=1}^{4}{P(k)} \ne 1\). Therefore, to define this distribution for k=1,2,3,4, it is necessary to renormalize the distribution
In the example the results are as follows:

P(k=1)=0.5204;

P(k=2)=0.2689;

P(k=3)=0.1389;

P(k=4)=0.0718.
Note that the ratio between different P(k) values does not change.
Finally, we construct an adjusted probability distribution by adding the constants shown in the second column of Table 6, as explained in Section 5. More specifically:

P _{ v }(k=1)=P(k=1)+0.13=0.6504;

P _{ v }(k=2)=P(k=2)−0.10=0.1689;

P _{ v }(k=3)=P(k=3)−0.03=0.1089;

P _{ v }(k=4)=P(k=4)=0.0718.
Note that the adjusted probability distribution is normalized because
P _{ v }(k) represents an empirically obtained probability distribution of Dst level for a specific CME speed (v=2411 km s^{−1}).
A.3 Probability Distribution for CME or Flare Position Distance from the Center of the Solar Disk (r), P _{ r }(k)
The change of the Dst distribution mean with CME or flare position distance from the center of the solar disk, r, can be described with a powerlaw function (see Figure 9),
Here a=30.95, b=−0.83, and Dst is expressed in nT. For a given r=0.4067 Equation (15) gives the distribution mean m _{DST}(r)=65.31.
The geometric distribution mean is calculated using Equation (11). In the example m _{GD}=1.65. The success probability in each trial, calculated using Equation (10), is p=0.60.
Probabilities calculated using Equation (4) in Section 5, P(k) are then:

P(k=1)=0.6049;

P(k=2)=0.2390;

P(k=3)=0.0944;

P(k=4)=0.0373.
The renormalized distribution, calculated using Equation (13) is:

P(k=1)=0.6200;

P(k=2)=0.2450;

P(k=3)=0.0968;

P(k=4)=0.0382.
Finally, the adjusted probability distribution (adding the constants shown in the third column of Table 6, as explained in Section 5) is:

P _{ r }(k=1)=P(k=1)+0.12=0.7400;

P _{ r }(k=2)=P(k=2)−0.12=0.1250;

P _{ r }(k=3)=P(k=3)=0.0968;

P _{ r }(k=4)=P(k=4)=0.0382.
P _{ r }(k) represents an empirically obtained probability distribution of Dst level for a specific CME or flare position distance from the center of the solar disk (r=0.4067 solar radius).
A.4 Probability Distribution for the CME Width (w), P _{ w }(k)
The change of the Dst distribution mean with CME width, w, can be described with a quadratic function (see Figure 9),
Here a=15.06, b=−34.60, c=42.25, and Dst is expressed in nT. For a given w=3, this gives the distribution mean m _{DST}(w)=73.99.
The geometric distribution mean is calculated using Equation (11). In the example m _{GD}=1.74. The success probability in each trial, calculated using Equation (10), is p=0.57.
Probabilities calculated using Equation (4) in Section 5, P(k) are then:

P(k=1)=0.5747;

P(k=2)=0.2444;

P(k=3)=0.1039;

P(k=4)=0.0442.
The renormalized distribution, calculated using Equation (13), is:

P(k=1)=0.5942;

P(k=2)=0.2527;

P(k=3)=0.1075;

P(k=4)=0.0457.
Finally, the adjusted probability distribution (adding the constants shown in the fourth column of Table 6, as explained in Section 5) is:

P _{ w }(k=1)=P(k=1)+0.14=0.7342;

P _{ w }(k=2)=P(k=2)−0.12=0.1327;

P _{ w }(k=3)=P(k=3)−0.02=0.0875;

P _{ w }(k=4)=P(k=4)=0.0457.
P _{ w }(k) represents an empirically obtained probability distribution of Dst level for a specific CME width (w=360^{∘}, halo CME).
A.5 Probability Distribution for the Flare Class (f), P _{ f }(k)
The change of the Dst distribution mean with flare class, f, can be described with a quadratic function (see Figure 9),
Here a=10.41, b=−17.90, c=46.93, and Dst is expressed in nT. For a given f=3, this gives the distribution mean m _{DST}(f)=86.92.
The geometric distribution mean is calculated using Equation (11). In the example m _{GD}=1.87. The success probability in each trial, calculated using Equation (10), is p=0.53.
Probabilities calculated using Equation (4) in Section 5, P(k) are then:

P(k=1)=0.5350;

P(k=2)=0.2488;

P(k=3)=0.1157;

P(k=4)=0.0538.
The renormalized distribution, calculated using Equation (13), is:

P(k=1)=0.5612;

P(k=2)=0.2610;

P(k=3)=0.1214;

P(k=4)=0.0564.
Finally, the adjusted probability distribution (adding the constants shown in the fifth column of Table 6, as explained in Section 5) is:

P _{ f }(k=1)=P(k=1)+0.15=0.7112;

P _{ f }(k=2)=P(k=2)−0.12=0.1410;

P _{ f }(k=3)=P(k=3)−0.02=0.1014;

P _{ f }(k=4)=P(k=4)−0.01=0.0464.
P _{ f }(k) represents an empirically obtained probability distribution of Dst level for a specific flare class (f=3, X class flare).
A.6 Probability Distribution for the Interaction Level (i), P _{ i }(k)
The change of the Dst distribution mean with interaction level, i, can be described with a powerlaw function (see Figure 9),
Here a=38.39, b=0.49, and Dst is expressed in nT. For a given i=4, this gives the distribution mean m _{DST}(i)=65.77.
The geometric distribution mean is calculated using Equation (11). In the example m _{GD}=1.66. The probability of the success in each trial, calculated using Equation (10), is p=0.60.
Probabilities calculated using Equation (4) in Section 5, P(k) are then:

P(k=1)=0.5691;

P(k=2)=0.2452;

P(k=3)=0.1057;

P(k=4)=0.0455.
The renormalized distribution, calculated using Equation (13), is:

P(k=1)=0.5894;

P(k=2)=0.2540;

P(k=3)=0.1094;

P(k=4)=0.0472.
Finally, the adjusted probability distribution (adding the constants shown in the fifth column of Table 6, as explained in Section 5) is:

P _{ i }(k=1)=P(k=1)+0.15=0.7394;

P _{ i }(k=2)=P(k=2)−0.13=0.1240;

P _{ i }(k=3)=P(k=3)−0.01=0.0994;

P _{ i }(k=4)=P(k=4)−0.01=0.0372.
P _{ i }(k) represents an empirically obtained probability distribution of Dst level for a specific interaction level (i=4, interaction highly probable).
A.7 Combined Probability Distribution for a Set of Key Parameters (v,r,w,f,i), P(Dst)
After we obtain the probability distribution of Dst level for each of the key solar parameters (v, r, w, f, and i), their combined probability P(k)=P(Dst) is calculated using Equation (8) in Section 5. For our example this gives:

P(k=1)=P(Dst<100 nT)=0.9982;

P(k=2)=P(100 nT<Dst<200 nT)=0.5253;

P(k=3)=P(200 nT<Dst<300 nT)=0.4056;

P(k=4)=P(Dst>300 nT)=0.2178.
Because the set of parameters v, r, w, f, and i is an incomplete set of independent variables for this distribution, this distribution is not normalized, i.e. ∑P(Dst)≠1. Therefore, it is necessary to renormalize the distribution (similarly to Equation (13)):

P(k=1)=P(Dst<100 nT)=0.4649;

P(k=2)=P(100 nT<Dst<200 nT)=0.2447;

P(k=3)=P(200 nT<Dst<300 nT)=0.1889;

P(k=4)=P(Dst>300 nT)=0.1014.
Note that the ratio between different P(Dst) values does not change. P(Dst) represents an empirically obtained probability distribution of Dst level for a specific set of key parameters (v=2411, r=0.4067, w=3, f=3, i=4).
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Dumbović, M., Devos, A., Vršnak, B. et al. Geoeffectiveness of Coronal Mass Ejections in the SOHO Era. Sol Phys 290, 579–612 (2015). https://doi.org/10.1007/s1120701406138
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Keywords
 Coronal mass ejections
 Solar flares
 Geomagnetic storms