Using Extreme Value Theory for Determining the Probability of Carrington-Like Solar Flares

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1 arxiv [stat.ap] 12 Apr 2016 Preprint April 12 th 2016 Using Extreme Value Theory for Determining the Probability of Carrington-Like Solar Flares S. Elvidge* & M. J. Angling Space Environment and Radio Engineering Group, University of Birmingham, Birmingham, UK. Space weather events can negatively affect satellites, the electricity grid, satellite navigation systems and human health 1. As a consequence, extreme space weather has been added to the UK and other national risk registers 2. However, by their very nature, extreme events occur rarely and statistical methods are required to determine the probability of occurrence solar storms. Space weather events can be characterised by a number of natural phenomena such as X-ray (solar) flares, solar energetic particle (SEP) fluxes, coronal mass ejections and various geophysical indices (Dst, Kp, F10.7). Here we use extreme value theory 3 (EVT) to investigate the probability of extreme solar flares. Previous work has suggested that the distribution of solar flares follows a power law 4. However such an approach can lead to overly fat-tails in the probability distribution function and thus to an under estimation of the return time of such events. Using EVT and GOES X-ray flux data we find that the expected 150 year return level is an X60 flare ( Wm -2, 1-8 Å X-ray flux). We also show that the EVT results are consistent with flare data from the Kepler space telescope mission 5. In the popular press, solar flares have become synonymous with space weather events even though they do not directly cause most of the major space weather effects 6. However, the magnitude of a flare usually provides an indication of the total amount of energy in a space weather event. Flare peak fluxes are classified by the letters, A, B, C, M and X where each letter indicates a flare one order of magnitude larger than the one preceding. An A1 flare has a peak X-ray flux of Wm -2 (measured in the 1 to 8 Å range) whilst an X1 flare has Wm -2. Extreme space weather events (solar superstorms) are often compared to the Carrington event of The Carrington event is thought to be the largest observed space weather event in the last 200 years. The flare associated with the Carrington event has been estimated to be an X45 ± 5 (i.e. 45 ± Wm -2 ) (8). The solar X-ray flare data used in this study is from the NASA Geostationary Operational Environmental Satellites (GOES) satellite missions. The tail of the distribution of solar flares has long been assumed to follow a power law 4,9, dn de E α where E is the flare energy, N is the number of flares and α is the shape parameter 10. The value of α has been estimated to be between 1.7 and 2 11,12. It should be noted that a power law distribution with shape parameter less than 2 has an undefined mean and standard deviation due to the distribution being heavy-tailed. This means that all values are expected to occur eventually. Previous work on space weather risk has been based on the assumption that the flare distribution follows such a law 9,13. 1

2 Extreme value theory (EVT) provides a more advanced tool for estimating probability distribution functions which avoids the starting assumption that the tail follows a simple power law 3. EVT has a wide range of applications; for example in modelling metal alloy strengths 14, estimating extreme wind speeds 15 and a variety of uses in quantitative finance 16. In the various branches of space weather, EVT has been used to investigate the distribution of the daily Aa index (a measure of the disturbance of the Earth s magnetic field) 17, disturbance storm time (Dst) index (an indication of the strength of the equatorial electrojet) 18, geomagnetic data ( dh dt where H is the horizontal geomagnetic component) 19 and relativistic electron fluxes 20. Here we use the Pickands-Balkema-de Haan (PBdH) theorem 21,22 from EVT. The PBdH theorem (also known as the second theorem of EVT) requires independent data, which is achieved by declustering the data. PBdH provides an approach for modelling the tail of an unknown distribution above a threshold value (u). Specifically, the theorem states that for some shape (ξ), location (μ) and scale (σ) parameter (which define the distribution), then for a large enough value u, the distribution of exceedances of this threshold, given that the random variable is greater than the threshold, is given by a generalised Pareto distribution (GPD) H(y) = 1 (1 + 1 ξy ) ξ. (1) σ+ξ(u μ) Determining the threshold value u is crucial to the success of the PBdH theorem. The value of u should be such that for all u 0 > u the parameters of the GPD associated with the exceedances of u 0 are the same subject to a change of scale. Equivalently, for values u 0 > u, the expectation of the exceedances, given that the value exceeds the threshold, E(X u X > u), should be a linear function of u. The mean exceedance plot for the GOES X-ray flux data is shown in Figure 1. This is approximately linear for values of u greater than (and less than where values of u become unreliable due to the lack of data 3 ). Therefore a value of u = is used as the GPD threshold value. This leads to a set of 171 exceedance values (i.e 171 flares greater than X3.5 in the dataset). Maximum log likelihood can then be used to estimate the GPD parameters, resulting in: σ = 1.08 (±0.02) 10 4 and ξ = 0.48 (±0.12), where the standard errors are found from the covariance matrix. Since ξ > 0 the distribution has no upper limit. Figure 2 compares the GPD fit of the data (above the threshold value) with a power-law using a scale factor of 1.7 (fitted to the full data set). One can see that the power law fit is extremely good for flare intensities from M1 to ~X6. However above these values the data dips below the line. This is often noted and ascribed to a lack of data 11. EVT allows the use of this data to examine the tail more rigorously. It can be seen that the GPD fit contains all data up to X17 flares within its 95% confidence intervals. Further evidence of the validity of the method can be found by considering data which is not included in the GPD fit. For example the X35 (±5) flare associated with the so-called Halloween storm on November 4 th, is included in Figure 2. 2

3 Shibayama et al. 5 looked for superflares on Sun like stars using 500 days of Kepler data. The study found approximately 5000 stars similar to our Sun (rotational period longer than 25 days, surface temperature between 5600 K and 6000 K and surface gravity of log(g) > 4). Amongst these stars they found evidence of four superflares (approximately X300; converting energy to GOES flare classification using the method described in Schrijver et al. 23 ). These observations lead Shibayama et al. to the conclusion that such flares could be expected on the Sun once in approximately 2,000 years. Such a result is within the 95% confidence intervals of our EVT analysis as shown in Figure 2 giving further assurance to the validity of the method. It should, however, be noted that Aulanier et al. 24 suggest that the largest flare possible on our Sun is ~X200. The EVT fit can be used to estimate the largest expected flare in a given time period. For example, what is the largest flare we would expect to see in a 50-year period? To calculate this, the m-observation return level is given by x m = u + σ ξ [(mdn c n )ξ 1], (2) where m is the year return level, u the threshold value, σ and ξ the estimated GPD parameters, d the number of observations in a year, n c the number of clusters greater than u and n the total number of observations. The return levels, and 95% confidence intervals, can then be produced. Figure 3 shows the X-ray flare return level plot. Using these return levels we estimate that a Carrington like flare (X45) is expected once in a ~90 year period, with 95% confidence intervals of 40 to 290 years. The largest possible flare of X200 (24) has a return time of 2,000 years (500 15,000 year 95% confidence intervals). Extreme value theory is used to provide rigorous statistical estimation of rare events. Part of that rigour is to acknowledge the errors in estimation, hence the large confidence intervals where there is no data. However this paper has shown that using EVT on solar flare data is consistent with both GOES X-ray flux data and data from the Kepler mission. There are two ways for a national risk register to consider the worst case scenario : the return time of events of a particular intensity, or the largest expected event in a given time scale. If considering the worst case in terms of intensity of event, e.g. the return period of an X45 (Carrington) flare, then our analysis shows that the worst case should be considered as a one in 40 year event (the lower bound of the 95% confidence interval). A reasonable worst case would be to consider X45 as a one in 90 year event. If one considers the largest event in a given time scale, e.g. 150 years, then the worst case is an X100 flare and a reasonable worst case is an X60 flare. 3

4 References 1. Cannon, P. S. & et al. Extreme space weather: impacts on engineered systems and infrastructure. (2013). doi: /swe UK Cabinet Office. The National Risk Register of Civil Emergencies. (2015). at < 9/ _2015-NRR-WA_Final.pdf> 3. Coles, S. An Introduction to Statistical Modeling of Extreme Values. (Springer, 2001). 4. Lu, E. T. & Hamilton, R. J. Avalanches and the Distribution of Solar Flares. Astrophys. J. 380, L89 L92 (1991). 5. Shibayama, T. et al. Superflares on solar-type stars observed with Kepler. I. Statistical properties of superflares. Astrophys. J. Suppl. Ser. 209, 5 (2013). 6. Hapgood, M. Astrophysics: Prepare for the coming space weather storm. Nature 484, (2012). 7. Carrington, R. C. Description of a Singular Appearance seen in the Sun on September 1, Mon. Not. R. Astron. Soc. 20, (1859). 8. Cliver, E. W. & Dietrich, W. F. The 1859 space weather event revisited: limits of extreme activity. Sp. Weather Sp. Clim. 3, (2013). 9. Riley, P. On the probability of occurrence of extreme space weather events. Sp. Weather 10, (2012). 10. Hudson, H. S. Solar physics: Solar flares add up. Nat. Phys. 6, (2010). 11. Boffetta, G., Carbone, V., Giuliani, P., Veltri, P. & Vulpiani, A. Power Laws in Solar Flares:Self-Organized Criticality or Turbulence? Phys. Rev. Lett. 83, (1999). 12. Aschwanden, M. J. & Freeland, S. L. Automated solar flare statistics in soft X-rays over 37 years of GOES observations: the invariance of self-organized criticality during three solar cycles. Astrophys. J. 754, (2012). 13. Cannon, P. S. Extreme Space Weather-A Report Published by the UK Royal Academy of Engineering. Sp. Weather 11, (2013). 14. Tyron, R. G. & Cruse, T. A. Probabilistic mesomechanics for high cycle fatigue life prediction. J. Eng. Mater. Technol. 122, (2000). 15. Della-Marta, P. M. & et al. The return period of wind storms over Europe. Int. J. Climatol. 29, (2009). 16. Rocco, M. Extreme value theory in finance: a survey. J. Econ. Surv. 28, (2014). 4

5 17. Silbergleit, V. M. The Most Geomagnetically Disturbed 24 Hours. Stud. Geophys. Geod. 43, (1999). 18. Tsubouchi, K. & Omura, Y. Long-term occurrence probabilities of intense geomagnetic storm events. Sp. Weather 5, (2007). 19. Thomson, A. W. P., Dawson, E. B. & Reay, S. J. Quantifying extreme behavior in geomagnetic activity. Sp. Weather 9, (2011). 20. Meredith, N. P., Horne, R. B., Isles, J. D. & Rodriguez, J. V. Extreme relativistic electron fluxes at geosynchronous orbit: Analysis of GOES E > 2 MeV electrons. Sp. Weather 13, (2015). 21. Balkema, A. & de Haan, L. Residual life time at great age. Ann. Probab. 2, (1974). 22. Pickands, J. Statistical inference using extreme order statistics. Ann. Stat. 3, (1975). 23. Schrijver, C. J. et al. Estimating the frequency of extremely energetic solar events, based on solar, stellar, lunar, and terrestrial records. J. Geophys. Res. 117, (2012). 24. Aulanier, G. et al. The standard flare model in three dimensions II. Upper limit on solar flare energy. Astron. Astrophys. 549, (2013). Acknowledgements. The GOES data was collected from the NGDC website at Correspondence. Correspondence and requests for materials should be addressed to S. Elvidge ( s.elvidge@bham.ac.uk). 5

6 Figure 1. Mean exceedance plot for the GOES X-ray flare data. The x-axis represents increasing values of u, whilst the y-axis shows the mean exceedance for the given value. 95% confidence intervals are also plotted. The graph is roughly linear for values of u greater than (X3.5 flares) and less than where values of u become unreliable due to the lack of data 3. 6

7 Figure 2. Shown in red dots are flare intensities against the number of times that intensity (or greater) occurs in the data. In blue is a power law fit of the data, and in black is the generalised Pareto distribution (GPD) fit (solid line) along with the 95% confidence intervals (dotted). In green the largest Halloween storm flare (X35 ±5) and Kepler data is shown. This data was not used in the analysis. The vertical dashed orange lines represent the expected flare intensity of a Carrington flare (X45 ±5). The vertical green dashed line represents the Aulanier et al. 24 estimate of the maximum expected flare our Sun is capable of (X200). 7

8 Figure 3. Return level plot, with 95% confidence intervals, for the GOES X-ray flare data. An X45 flare is expected once in a 90 year period, with 95% confidence intervals of 40 to 290 years. The largest expected flare in a 150 year period is an X60, with 95% confidence intervals of X35 to X100. 8

9 Methods Data The Geostationary Operational Environmental Satellites (GOES) satellite X-ray flux data is recorded by the Space Environment Monitor (SEM) subsystem. The raw data is collected approximately every three seconds. The data used in this work uses one minute averages of this raw data. There have been a number of GOES satellites since GOES-1 launched in October The main change in the data set arises from the switch from spinning satellites (GOES-7 and previous) to three-axis stabilized (GOES-8 onwards). A scaling factor is required to make the data from each GOES satellite consistent. To get the true X-ray flux value from the latest GOES satellites the data needs to be divided by 0.7. This ensures that the classification level (e.g. X5) is consistent across all the GOES satellites 25. The SEM has also shown to saturate during the most extreme events. During the storm of October/November 2003 the instrument saturated at a value of Wm -2 (X17 flare), whereas the largest flare of this period is estimated to be X35 (±5) 8. More recent GOES satellites have not yet experienced such a large flare, but it is expected they will saturate at similar flux levels. Consequently, in this analysis all but one X17 value has been removed from the dataset. One value has been left in since it is has been independently estimated that the flare on October 28 th 2003 was an X17.2 flare 26. Extreme Value Theory Extreme value theory (EVT) is mainly based around two theorems: the Fisher-Tippett- Gnedenko (FTG) theorem 3 and the Pickands-Balkema-de Haan (PBdH) theorem 21,22. FTG states that for a suitably normalized sample, an independent and identically distributed (iid) random variable converges to one of only three possible distributions: the Gumbel distribution 27, the Fréchet distribution 28 or the Weibull distribution 29. The distributions each have a shape ( ξ), location ( μ) and scale ( σ) parameter which defines the distribution. However the requirement that the variable must be iid leads to difficulties in using the FTG theorem in many real world cases. In the case of X-ray flares the data is neither independent nor identically distributed. Flare events are usually recorded multiple times in the GOES data (since the events last longer than one minute). Thus a large flux value is often followed by other large fluxes. Also the number of events is related to the 11-year solar cycle: during solar maximum there is an increased number of flare events, and a corresponding decrease during solar minimum. Thus the underlying distributions vary temporally. Such temporal variations could be removed by, for example, taking the ten year maximum (thereby accounting for the solar cycle). However this would leave only five data points, which is not enough to work with. Instead the PBdH theorem (also known as the second theorem of EVT) is more applicable to the data. The theorem only requires the data be independent, and this can be accomplished by declustering the data. The declustering is achieved by not counting contiguous class X data 9

10 records; i.e. it is assumed that contiguous events are in fact a single event. For a new event to be counted there must have been 15 consecutive values of class M flares or lower between the two X flares. This approach changes the autocorrelation of lag one between the data sets from ~0.98 to ~0.23, providing confidence that the underlying declustered flare dataset is independent. The PBdH theorem defines an approach for modelling the tail of an unknown distribution above a threshold value. Specifically, the theorem states that for some μ, σ > 0 and ξ from the FTG theorem, then for a large enough threshold value (u), the distribution of exceedances of this threshold, given that the random variable is greater than the threshold, is given by generalised Pareto distributions (GPD) H(y) = 1 (1 + 1 ξy ) ξ. (3) σ+ξ(u μ) PBdH further states that the parameters defined by the GPD are directly associated to the corresponding distribution from the FTG theorem. In particular the shape parameter ξ is equal in the two theorems. The parameters of the GPD are estimated using a maximum likelihood method. Assume the values {y 1,, y k } are the k exceedances greater than u (i.e. y i = x i u) then the GPD parameters are estimated by maximising: l(σ, ξ) = k log(σ) (1 + 1 ξ ) k i=1 (4) log (1 + ξy i ). σ References 25. Machol, J. & Viereck, R. GOES X-ray Sensor (XRS) Measurements. (2015). at < 26. Blagoveshchensky, D. V, MacDougall, J. W. & Piatkova, A. V. Ionospheric effects preceding the October 2003 Halloween storm. J. Atmos. Solar-Terrestrial Phys. 68, (2006). 27. Gumbel, E. J. Les valeurs extrêmes des distributions statistiques. Ann. l Institut Henri Poincaré 5, (1935). 28. Fréchet, M. Sur la loi de probabilité de l écart maximum. Ann. Soc. Pol. Math. 6, (1927). 29. Weibull, W. A statistical distribution function of wide applicability. J. Appl. Mech. - ASME 3, (1951). 10

arxiv: v4 [stat.ap] 19 Oct 2017

arxiv: v4 [stat.ap] 19 Oct 2017 arxiv: 1604.03325v4 [stat.ap] 19 Oct 2017 Preprint October 19 th 2017 Using Extreme Value Theory for Determining the Probability of Carrington-Like Solar Flares S. Elvidge and M. J. Angling Space Environment