Ionosphere Magnetic Storm Occurrence Probability

M. Chassan, J-M. Azaïs, G. Buscarlet, N. Suard

Abstract:	Severe Ionosphere magnetic storms are feared events for integrity and continuity of navigation systems such as EGNOS, the European SBAS Complementing GPS and an accurate modelling of the probability of these events is necessary so our aim for the work presented in this paper is to give an estimation of the frequency of such extreme magnetic storm per time unit (year) throughout a solar cycle. Intensive storms being scarce, classical statistical methods for distribution estimation are not precise enough. In many domains, the Extreme Value Theory (EVT) enables to describe the behavior of scarce extreme events probability distribution. But, because of, among other things, the discrete form of our data and the obvious non stationary behavior of the frequency of extreme magnetic storms, the EVT cannot be applied. Thus, we develop an innovative method based on a Proportional hazard model, as the Cox model, with time dependent covariates. This method enables to model the non stationary behavior of the frequency of severe storms. Our observations consist basically of 80 years of registration of the 3 hours ap-index. The ap-index quantifies the intensity of the planetary geomagnetic activity. The main advantage of this data set is its large amount and the absence of gap, contrary to raw geomagnetic data (with gap length varying from one month to several years according to the observatory). The chosen data set contains seven complete solar cycles, from the seventeenth to the twenty-third on the general list. We also use the monthly Smoothed Sunspot Number (SSN) to take into account the solar activity. Before all, we have data pretreatments. Indeed, durations of magnetic storms are very variable, from 3 or 6 hours for a very strong storm until 30 hours for a very low level storm. Here, we assume that storms are one time events. Since we focus on strong storms, which are brief compared to lower storms, it is not unrealistic to make this assumption. We introduce a declustering process of the data in order to consider only one event with the highest intensity even if there are different periods of intensity separated by less active ones (lower indices). This declustering is performed along with a time rescaling. The seven solar cycles have different lengths and we have to warp them to the same time scale. Each storm will correspond to a time of occurrence in [-0.5 ; 0.5] where: - -0.5 corresponds to the beginning of the cycle - 0 is the solar activity maximum date - 0.5 corresponds to the end of the cycle. Nevertheless, extreme storms of level 400 are too scarce (in the range over 20 occurrences in the data set) to make inference about their frequency. Thus, we will consider two types of storms: intermediate level storms (with a maximal level of 179, 207 or 236) and severe (or extreme) storms (level 300 or 400). We will use all the storms with a level greater than a fixed intermediate level to estimate the influence of each covariate and then make an extrapolation to high level 400. We introduce the parameter p400, the probability that an intermediate level storm grows into a storm of level 400. Utilization of this parameter assumes that the level reached by a storm (with a level greater or equal to 179) does not depend of the instant of appearance. A chi-square independence test showed that this assumption is quite acceptable. As said before, we estimate the frequency of extreme magnetic storm per time unit throughout a solar cycle using a model based on a proportional hazard model. Widely used in epidemiology, the Cox model allows expressing the risk of death, for example, as a function of some risk factors (covariates) and proportionally to a reference risk. We model the number of occurrences of storms (with a level greater or equal to 179) during the cycle j by a random variable Nj which is supposed to follow a non-homogeneous Poisson process. Its intensity is expressed as: Lambda0(t) Dj exp (Beta Xj), t in [-0.5 ; 0.5] with: - Lambda0(t), the basic intensity that models the fact that storms occur principally during the second half of the cycle - Dj , the length of the cycle j, ensuring an estimation of the frequency by time unit (and not by cycle) - Xj , the max of the monthly SSN, representing the solar activity of the cycle j - Beta modeling the influence of the solar activity. Our model slightly differs from the Cox model in the following way: - events of interest might occur several times, - we estimate the basic risk, instead of consider it as a nuisance parameter. - we introduce the cycle length (Dj) as a covariate to keep a frequency per time unit - we will extrapolate the obtained results using intermediate levels to the extreme level 400 Estimation is performed as follow: - as p400 is independent of the position in the cycle, the empirical frequency was used - Beta is estimated by is maximum likelihood estimator in a Poisson generalized linear model. - Lambda0(t) is estimated by a kernel estimator: a Gaussian kernel was used and the selection of the bandwidth parameter was done by cross-validation. To avoid edge effects, we also periodize the data. Confidence intervals, impact of our declustering process, influence of the solar cycle activity have been also analyzed and will be presented. The same process is currently applied on the ap gradient obtained by a simple derivation from two consecutive ap values: if interesting results are obtained, they will be added. To check the robustness of our estimation, we compare the results obtained with different intermediate storm levels (207 or 236). The stability of the intensity curve is very promising, enabling us to make predictions about the current solar cycle using our model.
Published in:	Proceedings of the 26th International Technical Meeting of the Satellite Division of The Institute of Navigation (ION GNSS+ 2013) September 16 - 20, 2013 Nashville Convention Center, Nashville, Tennessee Nashville, TN
Pages:	1856 - 1868
Cite this article:	Chassan, M., Azaïs, J-M., Buscarlet, G., Suard, N., "Ionosphere Magnetic Storm Occurrence Probability," Proceedings of the 26th International Technical Meeting of the Satellite Division of The Institute of Navigation (ION GNSS+ 2013), Nashville, TN, September 2013, pp. 1856-1868.
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