Skip to main content

Geoeffectiveness of Coronal Mass Ejections in the SOHO Era

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 space-weather 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.

This is a preview of subscription content, access via your institution.

Figure 1
Figure 2
Figure 3
Figure 4
Figure 5
Figure 6
Figure 7
Figure 8
Figure 9
Figure 10
Figure 11
Figure 12

References

  1. Akasofu, S.-I.: 1981, Energy coupling between the solar wind and the magnetosphere. Space Sci. Rev. 28, 121. DOI .

    ADS  Article  Google Scholar 

  2. Cargill, P.J.: 2004, On the aerodynamic drag force acting on interplanetary coronal mass ejections. Solar Phys. 221, 135. DOI .

    ADS  Article  Google Scholar 

  3. Cid, C., Cremades, H., Aran, A., Mandrini, C., Sanahuja, B., Schmieder, B., Menvielle, M., Rodriguez, L., Saiz, E., Cerrato, Y., Dasso, S., Jacobs, C., Lathuillere, C., Zhukov, A.: 2012, Can a halo CME from the limb be geoeffective? J. Geophys. Res. 117, 11102. DOI .

    Article  Google Scholar 

  4. Dungey, J.W.: 1961, Interplanetary magnetic field and the auroral zones. Phys. Rev. Lett. 6, 47. DOI .

    ADS  Article  Google Scholar 

  5. Farrugia, C., Berdichevsky, D.: 2004, Evolutionary signatures in complex ejecta and their driven shocks. Ann. Geophys. 22, 3679. DOI .

    ADS  Article  Google Scholar 

  6. Gopalswamy, N., Yashiro, S., Akiyama, S.: 2007, Geoeffectiveness of halo coronal mass ejections. J. Geophys. Res. 112, 6112. DOI .

    Article  Google Scholar 

  7. Gopalswamy, N., Makela, P., Yashiro, S., Davila, J.M.: 2012, The relationship between the expansion speed and radial speed of CMEs confirmed using quadrature observations of the 2011 February 15 CME. Sun Geosph. 7, 7.

    ADS  Google Scholar 

  8. Gosling, J.T., Bame, S.J., McComas, D.J., Phillips, J.L.: 1990, Coronal mass ejections and large geomagnetic storms. Geophys. Res. Lett. 17, 901. DOI .

    ADS  Article  Google Scholar 

  9. Huttunen, K.E.J., Schwenn, R., Bothmer, V., Koskinen, H.E.J.: 2005, Properties and geoeffectiveness of magnetic clouds in the rising, maximum and early declining phases of solar cycle 23. Ann. Geophys. 23, 625. DOI .

    ADS  Article  Google Scholar 

  10. Kim, R.-S., Cho, K.-S., Moon, Y.-J., Dryer, M., Lee, J., Yi, Y., Kim, K.-H., Wang, H., Park, Y.-D., Kim, Y.H.: 2010, An empirical model for prediction of geomagnetic storms using initially observed CME parameters at the Sun. J. Geophys. Res. 115, 12108. DOI .

    Article  Google Scholar 

  11. Koskinen, H.E.J., Huttunen, K.E.J.: 2006, Geoeffectivity of coronal mass ejections. Space Sci. Rev. 124, 169. DOI .

    ADS  Article  Google Scholar 

  12. Lepping, R.P., Acũna, M.H., Burlaga, L.F., Farrell, W.M., Slavin, J.A., Schatten, K.H., Mariani, F., Ness, N.F., Neubauer, F.M., Whang, Y.C., Byrnes, J.B., Kennon, R.S., Panetta, P.V., Scheifele, J., Worley, E.M.: 1995, The wind magnetic field investigation. Space Sci. Rev. 71, 207. DOI .

    ADS  Article  Google Scholar 

  13. Maričić, D., Vršnak, B., Stanger, A.L., Veronig, A.M., Temmer, M., Roša, D.: 2007, Acceleration phase of coronal mass ejections: II. Synchronization of the energy release in the associated flare. Solar Phys. 241, 99. DOI .

    ADS  Article  Google Scholar 

  14. McComas, D.J., Bame, S.J., Barker, P., Feldman, W.C., Phillips, J.L., Riley, P., Griffee, J.W.: 1998, Solar Wind Electron Proton Alpha Monitor (SWEPAM) for the advanced composition explorer. Space Sci. Rev. 86, 563. DOI .

    ADS  Article  Google Scholar 

  15. Mishra, W., Srivastava, N., Chakrabarty, D.: 2014, Evolution and consequences of interacting CMEs of 2012 November 9 – 10 using STEREO/SECCHI and in situ observations. Solar Phys., in press. arXiv . ADS .

  16. Moon, Y.-J., Choe, G.S., Wang, H., Park, Y.D., Gopalswamy, N., Yang, G., Yashiro, S.: 2002, A statistical study of two classes of coronal mass ejections. Astrophys. J. 581, 694. DOI .

    ADS  Article  Google Scholar 

  17. Moon, Y.-J., Choe, G.S., Wang, H., Park, Y.D., Cheng, C.Z.: 2003, Relationship between CME kinematics and flare strength. J. Korean Astron. Soc. 36, 61.

    ADS  Article  Google Scholar 

  18. Möstl, C., Davies, J.A.: 2013, Speeds and arrival times of solar transients approximated by self-similar expanding circular fronts. Solar Phys. 285, 411. DOI .

    ADS  Article  Google Scholar 

  19. Möstl, C., Farrugia, C.J., Kilpua, E.K.J., Jian, L.K., Liu, Y., Eastwood, J.P., Harrison, R.A., Webb, D.F., Temmer, M., Odstrcil, D., Davies, J.A., Rollett, T., Luhmann, J.G., Nitta, N., Mulligan, T., Jensen, E.A., Forsyth, R., Lavraud, B., de Koning, C.A., Veronig, A.M., Galvin, A.B., Zhang, T.L., Anderson, B.J.: 2012, Multi-point shock and flux rope analysis of multiple interplanetary coronal mass ejections around 2010 August 1 in the inner heliosphere. Astrophys. J. 758, 10. DOI .

    ADS  Article  Google Scholar 

  20. Ogilvie, K.W., Chornay, D.J., Fritzenreiter, R.J., Hunsaker, F., Keller, J., Lobell, J., Miller, G., Scudder, J.D., Sittler, E.C. Jr., Torbert, R.B., Bodet, D., Needell, G., Lazarus, A.J., Steinberg, J.T., Tappan, J.H., Mavretic, A., Gergin, E.: 1995, SWE, a comprehensive plasma instrument for the wind spacecraft. Space Sci. Rev. 71, 55. DOI .

    ADS  Article  Google Scholar 

  21. Pitman, J.: 1993, Probability, Springer, New York.

    Book  MATH  Google Scholar 

  22. Richardson, I.G., Cane, H.V.: 2010, Near-Earth interplanetary coronal mass ejections during solar cycle 23 (1996 – 2009): Catalog and summary of properties. Solar Phys. 264, 189. DOI .

    ADS  Article  Google Scholar 

  23. Richardson, I.G., Cane, H.V.: 2011, Geoeffectiveness (Dst and Kp) of interplanetary coronal mass ejections during 1995 – 2009 and implications for storm forecasting. Space Weather 9, 7005. DOI .

    ADS  Article  Google Scholar 

  24. Richardson, I.G., Webb, D.F., Zhang, J., Berdichevsky, D.B., Biesecker, D.A., Kasper, J.C., Kataoka, R., Steinberg, J.T., Thompson, B.J., Wu, C.-C., Zhukov, A.N.: 2006, Major geomagnetic storms (Dst≤−100 nT) generated by corotating interaction regions. J. Geophys. Res. 111, 7. DOI .

    Google Scholar 

  25. Rodriguez, L., Zhukov, A.N., Cid, C., Cerrato, Y., Saiz, E., Cremades, H., Dasso, S., Menvielle, M., Aran, A., Mandrini, C., Poedts, S., Schmieder, B.: 2009, Three frontside full halo coronal mass ejections with a nontypical geomagnetic response. Space Weather 7, 6003. DOI .

    ADS  Article  Google Scholar 

  26. Russell, C.T., McPherron, R.L., Burton, R.K.: 1974, On the cause of geomagnetic storms. J. Geophys. Res. 79, 1105. DOI .

    ADS  Article  Google Scholar 

  27. Schwenn, R., dal Lago, A., Huttunen, E., Gonzalez, W.D.: 2005, The association of coronal mass ejections with their effects near the Earth. Astrophys. J. 23, 1033.

    Google Scholar 

  28. Smith, C.W., L’Heureux, J., Ness, N.F., Acuña, M.H., Burlaga, L.F., Scheifele, J.: 1998, The ACE magnetic fields experiment. Space Sci. Rev. 86, 613. DOI .

    ADS  Article  Google Scholar 

  29. Srivastava, N.: 2005, A logistic regression model for predicting the occurrence of intense geomagnetic storms. Ann. Geophys. 23(9), 2969. DOI .

    ADS  Article  Google Scholar 

  30. Srivastava, N.: 2006, The challenge of predicting the occurrence of intense storms. J. Astrophys. Astron. 27, 237. DOI .

    ADS  Article  Google Scholar 

  31. Srivastava, N., Venkatakrishnan, P.: 2004, Solar and interplanetary sources of major geomagnetic storms during 1996 – 2002. J. Geophys. Res. 109, 10103. DOI .

    Article  Google Scholar 

  32. Stirzaker, D.: 2003, Elementary Probability, Cambridge University Press, New York.

    Book  MATH  Google Scholar 

  33. Stone, E.C., Frandsen, A.M., Mewaldt, R.A., Christian, E.R., Margolies, D., Ormes, J.F., Snow, F.: 1998, The advanced composition explorer. Space Sci. Rev. 86, 1. DOI .

    ADS  Article  Google Scholar 

  34. Uwamahoro, J., McKinnell, L.A., Habarulema, J.B.: 2012, Estimating the geoeffectiveness of halo CMEs from associated solar and IP parameters using neural networks. Ann. Geophys. 30, 963. DOI .

    ADS  Article  Google Scholar 

  35. Valach, F., Revallo, M., Bochníček, J., Hejda, P.: 2009, Solar energetic particle flux enhancement as a predictor of geomagnetic activity in a neural network-based model. Space Weather 7, 4004. DOI .

    ADS  Article  Google Scholar 

  36. Verbanac, G., Mandea, M., Vršnak, B., Sentic, S.: 2011, Evolution of solar and geomagnetic activity indices, and their relationship: 1960 – 2001. Solar Phys. 271, 183. DOI .

    ADS  Article  Google Scholar 

  37. Verbanac, G., Živković, S., Vršnak, B., Bandić, M., Hojsak, T.: 2013, Comparison of geoeffectiveness of coronal mass ejections and corotating interaction regions. Astron. Astrophys. 558, A85. DOI .

    ADS  Article  Google Scholar 

  38. Vršnak, B., Sudar, D., Ruždjak, D.: 2005, The CME-flare relationship: Are there really two types of CMEs? Astron. Astrophys. 435, 1149. DOI .

    ADS  Article  Google Scholar 

  39. Vršnak, B., Ruždjak, D., Sudar, D., Gopalswamy, N.: 2004, Kinematics of coronal mass ejections between 2 and 30 solar radii. What can be learned about forces governing the eruption? Astron. Astrophys. 423, 717. DOI .

    ADS  Article  Google Scholar 

  40. Vršnak, B., Žic, T., Vrbanec, D., Temmer, M., Rollett, T., Möstl, C., Veronig, A., Čalogović, J., Dumbović, M., Lulić, S., Moon, Y.-J., Shanmugaraju, A.: 2013, Propagation of interplanetary coronal mass ejections: The drag-based model. Solar Phys. 285, 295. DOI .

    ADS  Article  Google Scholar 

  41. Yashiro, S., Gopalswamy, N., Michalek, G., St. Cyr, O.C., Plunkett, S.P., Rich, N.B., Howard, R.A.: 2004, A catalog of white light coronal mass ejections observed by the SOHO spacecraft. J. Geophys. Res. 109, 7105. DOI .

    Article  Google Scholar 

  42. Yermolaev, Y.I., Nikolaeva, N.S., Lodkina, I.G., Yermolaev, M.Y.: 2012, Geoeffectiveness and efficiency of CIR, sheath, and ICME in generation of magnetic storms. J. Geophys. Res. 117. DOI .

  43. Zhang, J., Dere, K.P., Howard, R.A., Bothmer, V.: 2003, Identification of solar sources of major geomagnetic storms between 1996 and 2000. Astrophys. J. 582, 520. DOI .

    ADS  Article  Google Scholar 

  44. Zhang, J., Richardson, I.G., Webb, D.F., Gopalswamy, N., Huttunen, E., Kasper, J.C., Nitta, N.V., Poomvises, W., Thompson, B.J., Wu, C.-C., Yashiro, S., Zhukov, A.N.: 2007, Solar and interplanetary sources of major geomagnetic storms (Dst≤−100 nT) during 1996 – 2005. J. Geophys. Res. 112, 10102. DOI .

    Article  Google Scholar 

Download references

Acknowledgements

This work has received funding from the European Union Seventh Framework Programme (FP7/2007-2013) 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 Solar-Terrestrial Physics (STP) Division of NOAA’s (National Oceanic and Atmospheric Administration) National Geophysical Data Center (NGDC) for providing solar flare data.

Author information

Affiliations

Authors

Corresponding author

Correspondence to M. Dumbović.

Appendix

Appendix

Here we provide a supplement to Section 5 that reports the detailed step-wise 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 example-CME 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.

  • First-order 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 X-ray 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 first-order 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 (non-halo 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.

Figure 13
figure13

The geometric probability distribution, P(X=k), 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):

$$ m_{\mathrm{GD}} = E(X) = \sum_{X \epsilon k}{X \cdot P(X)} = \sum_{k = 1}^{\infty }{k \cdot p \cdot(1-p)^{k-1}} = \frac{1}{p}. $$
(9)

Therefore, the probability of the success in each trial, p, can be calculated if the mean of the geometric distribution, m GD is known,

$$ p = \frac{1}{m_{\mathrm{GD}}}. $$
(10)

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 GDm 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

$$ m_{\mathrm{GD}} = 1 + \frac{m_{\mathrm{DST}}~[\mathrm{nT}]}{100}. $$
(11)

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),

$$ m_{\mathrm{DST}}(v) = a\cdot v + b. $$
(12)

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

$$ P(k) = \frac{P(k)}{\sum_{k=1}^{4}{P(k)}}. $$
(13)

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

$$ \sum_{k=1}^{4}{P_{v}(k)} = \sum _{k=1}^{4}{P(k)} = 1. $$
(14)

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 power-law function (see Figure 9),

$$ m_{\mathrm{DST}}(r) = a\cdot r^b. $$
(15)

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),

$$ m_{\mathrm{DST}}(w) = a\cdot w^2 + b\cdot w + c. $$
(16)

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),

$$ m_{\mathrm{DST}}(f) = a \cdot f^2 + b \cdot f + c. $$
(17)

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 power-law function (see Figure 9),

$$ m_{\mathrm{DST}}(i) = a \cdot i^b. $$
(18)

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).

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

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/s11207-014-0613-8

Download citation

Keywords

  • Coronal mass ejections
  • Solar flares
  • Geomagnetic storms