Statistical analysis of extreme values in space science

Abstract

The extreme values of many parameters measured by the space science community are important because they have a significant impact on human activities. For example, estimates of extreme radiation belt fluxes, plasma temperatures, and solar particle events must be considered when designing spacecraft, and extreme auroral currents must be considered when designing ground-based electric power systems. The mathematical tools that have been developed to address such problems are known as the statistics of extreme values. Here these tools have been applied to three data sets in order to determine if extreme value parametric models can accurately describe space weather parameters. The three examples chosen are (1) the annual maxima of the magnetic index Ap for the years from 1932 to 1997, (2) the annual maxima of the daily average flux of >60 MeV protons measured by the Interplanetary Monitoring Platform spacecraft from 1973 to 1998, and (3) the exceedances of the flux over a threshold of 104 el cm -2 s -1 sr -1 of >2 MeV electrons at geosynchronous orbit measured by the GOES spacecraft from 1986 to 1999. The parameters for each model were determined using the maximum likelihood estimate. The validity of each model was tested using a quantile-quantile (Q-Q) plot. In each case an excellent fit to the extremes of the sample distribution function is given by an appropriate extreme value distribution function. These extreme value distribution functions were also used to estimate the 50-and 100-year values for each data set. The results show that the extreme values observed to date are not unusual in that they are well fit by extreme value models. They also show that larger values than observed to date can be expected for each of the parameters during any 100-year period. 10,915 10,916

Distribution of extreme solar energetic proton fluxes

“…Thus Xapsos et al (1998) fitted the largest annual ( 410 MeV) solar proton event peak fluxes during the period 1967-1994 to the type II (Fré chet) extreme value distribution with a power-law asymptote for large values. Koons (2001) presented an excellent demonstration of the application of the extreme value analyses techniques to the annual maxima of the space weather variables: Ap index, 460 MeV proton fluxes measured by IMP spacecraft and 42 MeV electron fluxes measured by the GOES satellite. Crosby (2008) argued that the statistical extreme value approach is more useful for understanding the underlying physics than the ''case by case'' studies of extreme events traditionally used.…”

“…Crosby (2008) argued that the statistical extreme value approach is more useful for understanding the underlying physics than the ''case by case'' studies of extreme events traditionally used. The problem with the fitting to a distribution approach, as well emphasized in Koons (2001), is that a fit to the extreme value distribution function critically depends on the length of the input data sets. In principle, if we completely understood the physics underlying the phenomena we might be able to say what form the entire distribution function had and there would be no problem in describing its tail.…”

Reproducible Aspects of the Climate of Space Weather Over the Last Five Solar Cycles

“…Pioneering application to space weather includes the F10.7 index (Vedder & Tabor, 1992), the half-daily aa index (Silbergleit, 1999;Siscoe, 1976), and the D ST index (Silbergleit, 1996;Tsubouchi & Omura, 2007). Another application of EVT in the context of space weather has been to energetic electron fluxes (Koons, 2001;Meredith et al, 2015), which also established an unexpected upper bound to the flux of relativistic killer electrons (O'Brien et al, 2007). Important fine-scale details of geomagnetic ground effects have recently been studied using this framework (Beggan et al, 2013;Thomson et al, 2011).…”

Frequency distributions: from the sun to the earth

“…Xapsos et al (2000) developed a model predicting cumulative solar proton event fluence distributions using the Maximum Entropy approach and results correspond well with the measured solar proton distributions. Koons (2001) applied extreme value analysis to the magnetic index Ap, solar proton daily averaged flux values, as well as to the excess of >2 MeV electron flux values over a threshold value at geosynchronous orbit. Results show that the extreme values observed to date are not unusual in that they are well fit by extreme value models.…”