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Electron flux models for different energies at geostationary orbit

2016; American Geophysical Union; Volume: 14; Issue: 10 Linguagem: Inglês

10.1002/2016sw001506

1542-7390

Richard Boynton, М. А. Балихин, D. G. Sibeck, S. N. Walker, S.A. Billings, Natalia Ganushkina,

Astro and Planetary Science

Space WeatherVolume 14, Issue 10 p. 846-860 Research ArticleOpen Access Electron flux models for different energies at geostationary orbit R. J. Boynton, Corresponding Author R. J. Boynton rboynton85@gmail.com Department of Automatic Control and Systems Engineering, University of Sheffield, Sheffield, UK Correspondence to: R. J. Boynton, rboynton85@gmail.comSearch for more papers by this authorM. A. Balikhin, M. A. Balikhin Department of Automatic Control and Systems Engineering, University of Sheffield, Sheffield, UKSearch for more papers by this authorD. G. Sibeck, D. G. Sibeck NASA Goddard Space Flight Center, Greenbelt, Maryland, USASearch for more papers by this authorS. N. Walker, S. N. Walker Department of Automatic Control and Systems Engineering, University of Sheffield, Sheffield, UKSearch for more papers by this authorS. A. Billings, S. A. Billings Department of Automatic Control and Systems Engineering, University of Sheffield, Sheffield, UKSearch for more papers by this authorN. Ganushkina, N. Ganushkina Finnish Meteorological Institute, Helsinki, Finland University of Michigan, Ann Arbor, Michigan, USASearch for more papers by this author R. J. Boynton, Corresponding Author R. J. Boynton rboynton85@gmail.com Department of Automatic Control and Systems Engineering, University of Sheffield, Sheffield, UK Correspondence to: R. J. Boynton, rboynton85@gmail.comSearch for more papers by this authorM. A. Balikhin, M. A. Balikhin Department of Automatic Control and Systems Engineering, University of Sheffield, Sheffield, UKSearch for more papers by this authorD. G. Sibeck, D. G. Sibeck NASA Goddard Space Flight Center, Greenbelt, Maryland, USASearch for more papers by this authorS. N. Walker, S. N. Walker Department of Automatic Control and Systems Engineering, University of Sheffield, Sheffield, UKSearch for more papers by this authorS. A. Billings, S. A. Billings Department of Automatic Control and Systems Engineering, University of Sheffield, Sheffield, UKSearch for more papers by this authorN. Ganushkina, N. Ganushkina Finnish Meteorological Institute, Helsinki, Finland University of Michigan, Ann Arbor, Michigan, USASearch for more papers by this author First published: 14 October 2016 https://doi.org/10.1002/2016SW001506Citations: 25 The copyright line for this article was changed on 18 NOV 2016 after original online publication. AboutSectionsPDF ToolsRequest permissionExport citationAdd to favoritesTrack citation ShareShare Give accessShare full text accessShare full-text accessPlease review our Terms and Conditions of Use and check box below to share full-text version of article.I have read and accept the Wiley Online Library Terms and Conditions of UseShareable LinkUse the link below to share a full-text version of this article with your friends and colleagues. Learn more.Copy URL Share a linkShare onFacebookTwitterLinkedInRedditWechat Abstract Forecast models were derived for energetic electrons at all energy ranges sampled by the third-generation Geostationary Operational Environmental Satellites (GOES). These models were based on Multi-Input Single-Output Nonlinear Autoregressive Moving Average with Exogenous inputs methodologies. The model inputs include the solar wind velocity, density and pressure, the fraction of time that the interplanetary magnetic field (IMF) was southward, the IMF contribution of a solar wind-magnetosphere coupling function proposed by Boynton et al. (2011b), and the Dst index. As such, this study has deduced five new 1 h resolution models for the low-energy electrons measured by GOES (30–50 keV, 50–100 keV, 100–200 keV, 200–350 keV, and 350–600 keV) and extended the existing >800 keV and >2 MeV Geostationary Earth Orbit electron fluxes models to forecast at a 1 h resolution. All of these models were shown to provide accurate forecasts, with prediction efficiencies ranging between 66.9% and 82.3%. Key Points A set of electron flux forecast models is deduced for energy ranges sampled by GOES 13 The deduced models forecasting performance statistics are detailed The models will be implemented as a real-time forecasting tool 1 Introduction The radiation belts consist of energetic particles trapped by the terrestrial magnetic field and were discovered from the first in situ space radiation measurements. The outer radiation belt is made up of trapped electrons ranging in energy from keVs to several MeVs. Blake et al. [1992] and Reeves [1998] showed that the electron fluxes can vary by several orders of magnitude in a few hours. The high fluence of these energetic electrons can cause a number of problems on spacecraft depending on the electron energy. For example, low-energy electrons (1 keV to 100 keV) can cause surface charging that interferes with the satellite electronic systems [Olsen, 1983; Mullen et al., 1986], while higher energies (above 1 MeV and above) cause deep dielectric charging that may permanently damage the materials on board the satellite [Baker et al., 1987; Wrenn et al., 2002; Gubby and Evans, 2002; Lohmeyer and Cahoy, 2013; Lohmeyer et al., 2015]. There are still many unanswered questions about the mechanisms involved within the radiation belts, such as the acceleration mechanisms and loss processes of the electrons [Friedel et al., 2002]. Since we do not have a complete understanding of the physics, radiation belt models based on first principals struggle to capture the variable dynamics of the system [Horne et al., 2013b]. As such, these models often exhibit large errors between the forecast and the observed electron population [Horne et al., 2013a]. The system identification approach has also been applied to modeling the radiation belts. In this approach, models are automatically deduced from input-output data by the system identification algorithms. The system identification methodologies include linear prediction filters [Baker et al., 1990], dynamic linear models [Osthus et al., 2014], neural networks [Koons and Gorney, 1991; Freeman et al., 1998; Ling et al., 2010], and Nonlinear Autoregressive Moving Average with Exogenous inputs (NARMAX) [Wei et al., 2011; Boynton et al., 2013a, 2015]. While linear prediction filters and dynamic linear models are suitable for linear systems, the main advantage of NARMAX and neural networks is that they are capable of modeling nonlinear dynamics within the system. NARMAX and neural networks can both provide accurate and reliable models for nonlinear systems such as the radiation belts; however, NARMAX has the advantage of interpretability over neural networks. Neural networks result in the relationship between input and output measurements being described through a maze of multilayered neurones, in which each connection has an associated weight factor and each neurone has an activation function. This makes neural networks extremely difficult to interpret, i.e., to find out how the input variables couple together to produce changes in the output. In contrast, NARMAX models can result in a simple polynomial, from which understanding how the inputs change the output is intuitive. Therefore, this study uses the NARMAX methodologies to model the electron fluxes observed by the Geostationary Operational Environmental Satellites (GOES), situated in Geostationary Earth Orbit (GEO). The main aim of this study is to create reliable forecast models for the electron flux energy ranges observed by the third-generation GOES. The second aim is to increase temporal resolution of the forecast to that which currently operates on the University of Sheffield Space Weather Website (http://www.ssg.group.shef.ac.uk/USSW/UOSSW.html) and was developed by Boynton et al. [2015]. In section 2, we discuss the methodology used to deduce the forecast models. This includes a brief description of the NARMAX algorithm. Section 3 presents an extension of the current 24 h resolution >800 keV and >2 MeV GEO electron flux models, developed by Boynton et al. [2015], to 1 h resolution and a calculation of their performance. In section 4, the methodology and data used to derive the low-energy models and the results of the model performances are shown. The limitations of the models and their performance are discussed in section 5, and the conclusions from this study are presented in section 6. 2 NARMAX Methodology As stated in section 1, NARMAX models provide reliable forecasts and are also easy to interpret. As such, the methodology has been applied to a wide range of scientific fields, from analyzing the adaptive changes in the photoreceptors of Drosophila flies [Friederich et al., 2009] to modeling the tide at the Venice Lagoon [Wei and Billings, 2006]. In the field of space physics, the methodology was first used to model the Dst index using the half-wave rectifier (solar wind velocity multiplied by the southward interplanetary magnetic field (IMF) component) as the input [Balikhin et al., 2001; Boaghe et al., 2001]. More recently, due to lack of knowledge about the inputs to the Dst index system, Boynton et al. [2011b] used the NARMAX model structure detection methodology to identify the main control parameter, or solar wind coupling function, for geomagnetic storms quantified using the Dst index. This coupling function was , where p is the pressure, V is the velocity, is the tangential IMF, and is the clock angle of the IMF in GSM coordinates. Boynton et al. [2011a] used this coupling function to deduce a reliable model for the Dst index. Boynton et al. [2013b] and Balikhin et al. [2011] employed a similar approach to identify the solar wind control parameters for electron fluxes at GEO. In these studies, they found that the solar wind velocity and density were the main control parameters. The interpretability of these results allowed Balikhin et al. [2012] to make a direct comparison with the energy diffusion equation, where they found that acceleration due to local diffusion does not dominate at GEO. Recently, the NARMAX model structure detection methodology has been employed by Beharrell and Honary [2016] to determine the relationship between the solar wind and SYM-H. NARMAX models were first proposed by Leontaritis and Billings [1985a, 1985b] who demonstrated that the models have the potential to represent a wide class of nonlinear systems. A Multi-Input Single-Output NARMAX model, which was used in this study to model the electron fluxes at GEO, is expressed by (1) where y, u, and e represent the output, input, and error terms, respectively, F[·] represents some nonlinear function (a polynomial in the case of this study), m is the number of system inputs, and ny, ,…, , ne are the maximum time lags for the output, each of the m inputs, and the error, respectively. Billings et al. [1988] developed the first Forward Regression Orthogonal Least Squares (FROLS) algorithm that automatically fits a NARMAX model using input-output training data sets. Simply put, the overall algorithm developed by Billings et al. [1988] involved three stages. The first stage is model structure detection, which identifies the variables or combination of variables that control the evolution of the system. In equation 1, the expansion of F[·] in terms of a high-degree polynomial results in a huge number of monomials, especially if there are many possible inputs. The vast majority of the possible monomials will have little influence on the system; i.e., the coefficients of the monomial will be zero. Therefore, only a small number of monomials are required to represent the dynamics of the system. The FROLS procedure identifies the most significant monomials by use of the Error Reduction Ratio (ERR). Once the model structure is detected, the second stage is to estimate the coefficient for each of the monomials detected in the model. These first two stages are referred to as training the model. The final stage is to validate the model. Since its inception, many variants on the FROLS algorithm have been developed [Billings et al., 1989; Mao and Billings, 1997; Wei and Billings, 2008]. This study employs the Iterative Orthogonal Forward Regression (IOFR) algorithm, developed by Guo et al. [2014], which is more likely to detect the optimal model when the data are oversampled. The IOFR is largely based upon the initial FROLS algorithm, where the ERR of each of the monomials is calculated with respect to the output. The monomial with the highest ERR is then selected as the first monomial for the initial model structure. For the next step of the algorithm, all other monomials are orthogonalized relative to the first monomial that has just been selected. This effectively removes the first monomials contribution to the output from the remaining monomials. The ERRs of these orthogonalized monomials are then calculated with respect to the output, and the one with the highest ERR is selected as the second monomial for the initial model. For the third step, the remaining monomials are orthogonalized relative to both the first and second monomials selected for the initial model and the ERR is calculated. Again, the orthogonalized monomial with the highest ERR is selected and this will be the third monomial for the model. This process of orthogonalizing the remaining monomials with respect to all the selected model terms then selecting the orthogonalized monomial with the highest ERR for the model is continued until the model has the optimum number of model monomials. To decide the optimum number of model terms, this study employed the Adjustable Prediction Error Sum of Squares (APRESS) [Billings and Wei, 2008]. After each monomial is selected during every step of the FROLS algorithm, the APRESS is calculated from the ERR (2) where N is the number of data points, k is the number of monomials that has been selected, and λ is an adjustable factor that was between 5 and 10. At each step, i, APRESS(i) is calculated and compared to the previous APRESS(i − 1). APRESS will decrease as each significant monomial is added to the model until a local minima is reached. After this turning point, the addition of more model monomials is less likely to increase the performance of the model and may lead to the model becoming overfit [Billings and Wei, 2008]. Therefore, the turning point in APRESS dictates the optimum number of model monomials and the initial model polynomial structure is obtained. A least squares procedure then identifies the coefficients for each monomial to yield the model. 3 Increasing the Time Resolution of the Existing >800 keV and >2 MeV GEO Electron Flux Models Models for forecasting the fluxes of >800 keV and >2 MeV electrons at GEO were developed by Boynton et al. [2015]. These models were deduced using the NARMAX methodology and provide a 1 day resolution forecast for 1 day ahead. Both of these models were shown to have a high prediction efficiency for estimating the next day's electron flux value [Boynton et al., 2015]. The forecast results can be found online at www.ssg.group.shef.ac.uk/USSW/UOSSW.html. The original model only produces one forecast for the day. This forecast is for the average electron flux between 00:01 UTC 1 day to 00:00 UTC on the next day, calculated at 00:01 UTC. This means that at the start of every UTC day the original model calculates a forecast for the average electron flux over the next 24 h. One of the aims of this study is to increase the temporal resolution of these forecasts. Therefore, the time resolution of the >800 keV and >2 MeV GEO electron flux models was extended to give a forecast of the electron fluxes every hour for the next 24 h in contrast to only one daily forecast per day. This means that every hour the model will calculate a forecast for the average electron flux over the next 24 h, producing 24 forecasts per day. 3.1 Data and Methodology The >800 keV and >2 MeV electron flux models rely on solar wind inputs to forecast the electron flux. The solar wind inputs are the daily average velocity and density and the amount of time the IMF is southward in a 24 h period. The 1 min solar wind velocity, density, and IMF Bz component data were obtained from the OMNI website (http://omniweb.gsfc.nasa.gov/ow_min.html) from 1 January 2011 to 28 February 2015. At every hour, the past 24 h average of the solar wind velocity and density was calculated. For example, the point at 10:00:00 UTC on 5 January 2015 is an average of the 1440 1 min points between 10:01:00 UTC on 4 January 2015 and 10:00:00 UTC on 5 January 2015. In addition, the number of minutes that the IMF was southward during the past 24 h was determined for the final input. The electron flux data used to analyze the performance of the extended temporal resolution >800 keV and >2 MeV GEO electron flux models were from GOES 13. The electron fluxes on board the GOES 13 satellite are measured by the Energetic Proton Electron and Alpha Detector (EPEAD) [Hanser, 2011] and the Magnetospheric Electron Detector (MAGED). [Hanser, 2011]. The data for these instruments can be accessed from http://www.ngdc.noaa.gov/stp/satellite/goes/dataaccess.html, and the MAGED will be discussed in section 4.1. The EPEAD measures the relativistic integral electron fluxes and has two detectors pointing in opposite directions, both tangential to the spacecrafts orbit, named the east and west detectors. Since the EPEAD measures integral flux, the >2 MeV electrons will be measured by the >800 keV channel; however, the >2 MeV electrons account for less than 3% of the electrons detected on average. These data were used to assess the 1 h temporal resolution of the SNB3GEO electron flux models (SN stands for Sheffield NARMAX, and B3 corresponds to the letters of surnames of three model developers and GEO stands for geostationary orbit). The data period used for this part of the study was from 1 January 2011 to 28 February 2015. The study employed the >800 keV and >2 MeV energy channels from both the east and west detectors on board the GOES 13 satellite. The 5 min proton-corrected electron flux values were averaged between the east and west detectors. This was then time averaged resulting in a data set with 1 h resolution, such that each 1 h point was determined by averaging the 5 min data over the past 24 h; e.g., the point at 10:00:00 UTC on 5 January 2015 is average of the 288 5 min points between 10:05:00 UTC on 4 January 2015 and 10:00:00 UTC on 5 January 2015. These data were then compared to the model forecast. The 1 h moving average data will allow for a more continuous forecast of the daily average electron flux, such that every hour the online model will be able to forecast the electron flux value over the next 24 h, compared to only producing one forecast for each UTC day. Therefore, the forecast horizon for both the >800 keV and >2 MeV models will be 24 h. 3.2 Model Performance The >800 keV and >2 MeV GEO electron flux models were run using the 1 h resolution input data, and the results were compared to the EPEAD 1 h electron flux data, for the period from 1 January 2011 to 28 February 2015. The performance of the models during the period could then be analyzed. The performance of the models was assessed statistically by the correlation coefficient (CC), equation 3, and the prediction efficiency (PE), equation 4, which are commonly used to assess models [Temerin and Li, 2006; Li, 2004; Boynton et al., 2011a; Wei et al., 2004; Boynton et al., 2015; Rastatter et al., 2013]. (3) (4) Here EPE is the PE, ρ is the CC, y(t) is the output at time t, is the estimated output from the model, N is the length of the data, and the bar signifies the average. 3.2.1 The >800 keV Model Figure 1a shows the past 24 h average >800 keV electron flux measured by GOES in blue and the model 24 h ahead forecast in orange for the period from 1 January 2011 to 28 February 2015, while Figure 1b depicts the model error ( ). During this period, the PE was 72.1% and the CC was 85.1%. Figure 1Open in figure viewerPowerPoint (a) The past 24 h average >800 keV electron flux measured by GOES in blue and the model 24 h ahead forecast in orange for the period from 1 January 2011 to 28 February 2015. (b) The >800 keV electron flux model error ( ). (c) The past 24 h average >2 MeV electron flux measured by GOES in blue and the model 24 h ahead forecast in orange for the period from 1 January 2011 to 28 February 2015. (d) The >2 MeV electron flux model error ( ). 3.2.2 The >2 MeV Model Figure 1c shows the past 24 h average >2 MeV electron flux measured by GOES in blue and the model 24 h ahead forecast in orange for the period from 1 January 2011 to 28 February 2015, Figure 1d depicts the >2 MeV electron flux model error. The PE for the >2 MeV model was 82.3%, while the CC was 90.9%. Figures 1a and 1c reflect the better statistical performance of the >2 MeV model over the >800 keV model, since it can clearly be seen that the >2 MeV model follows more closely the blue observed GOES electron flux, particularly for the lower electron flux values. 4 Modeling the Low-Energy Electron Fluxes Measured by GOES 13 Models to forecast the low-energy electrons measured by GOES satellites were deduced using the NARMAX IOFR algorithm. This method requires input-output data for training the models. 4.1 Data and Methodology The electron flux data for the training and validation of these models come again from GOES 13. The MAGED has nine telescopes pointing in different directions and measures the lower energy differential electron fluxes in five energy channels: 30–50 keV, 50–100 keV, 100–200 keV, 200–350 keV, and 350–600 keV [Hanser, 2011]. The data period used for this part of the study was from 1 May 2010 to 28 February 2015 and employed all energy channels available from the instrument. This study is concerned mainly with the trapped electrons and therefore should not use a telescope that is directed in the loss cone, which is 800 keV and >2 MeV had minimum time delays of 1 day, and thus, it is possible to forecast 1 day into the future. However, same day values of the solar wind affect the current low-energy electron flux. Therefore, it is not possible to forecast 1 day ahead. To get around this problem, the past 24 h averages were calculated for each hour, as previously described. Therefore, the input time lags in the algorithm, , were shifted hourly not daily. For example, if input U(t − 10 h) is selected by the model, this monomial represents the average of the points between U(t − 10 h) and U(t − 34 h). Initially, a number of window intervals from 1 h averages, past 3 h, past 12 h, and 24 h were investigated. The 12 and 24 h windows gave the better results, but it was decided to use 24 h averaging for convenience because the same inputs could be used for >2 MeV and 800 keV models. This also makes the procedure simpler when implemented online and therefore less chance of bugs. The algorithm was run for the five energy ranges using lagged inputs from 2 to 48 h. These inputs were the solar wind velocity V and density n, the amount of time the IMF is southward in a 24 h period TBs, the Dst index, and the term resulting from the coupling function proposed by Balikhin et al. [2010] and Boynton et al. [2011b], (where is the tangential IMF and is the clock angle of the IMF). Therefore, the NARXAX model of the electron flux J will be (5) where the lags are in hours. When F is expanded to a second degree polynomial, there will be over 10,000 monomials for the FROLS algorithm to search through. For the 30–50 keV electrons, a compromise had to be made between producing a reliable forecast and the forecast horizon, the amount of time the model can forecast into the future. The model detected by the algorithm included input terms I, with a minimum lag of 6 h J(t) = F[I(t − 6),...]. Therefore, employing the inputs at the present time t, it is possible to estimate the electron flux 6 h into the future, J(t + 6) = F[I(t),...]. To increase the forecast horizon, the ≤6 h time lagged monomials were manually removed from the algorithms search to see if the performance of the model, based on PE and the CC, dropped significantly on a period of test data from 1 May 2010 to 28 February 2011. It was found that there was only a negligible drop in performance if the detected model had input terms with a minimum of 7 h time lag. This process of manually removing monomials with larger and larger time lags was continued until there was a significant performance drop in the model output. Figure 3 shows the results of this process with PE having a significant drop at a minimum lag of 11 h. Therefore, the model with a minimum of 10 h lag was selected as the final 30–50 keV model and could forecast the past 24 h average of the flux 10 h in the future. This methodology was repeated for the other four energy channels, and as with the studies by Boynton et al. [2013b] and Balikhin et al. [2012], the time delay of electron fluxes increased with the energy. The forecast horizons for each of the models are shown in Table 1. In each of the NARMAX models, the monomial with the minimum lag is due to a velocity component within the monomial. For example, in the 30–50 keV model, the FROLS algorithm selected as the exogenous monomial with the highest ERR. The exogenous monomial with the highest ERR in each of the models had a component of the velocity at the models minimum lag. For the three lowest energies the velocity was coupled with the IMF factor, while for the two higher energies the FROLS algorithm selected the linear velocity. Figure 3Open in figure viewerPowerPoint The PE of a 30–50 keV model between 1 May 2010 and 28 February 2011 versus the minimum lag included in that model. Table 1. Table Showing the Performance of the Five Low-Energy Electron Flux Models As Well As the Forecast Length Model Forecast Horizon 1 Day PE (%) 1 Day CC (%) 1 h PE (%) 1 h CC (%) 30–50 keV 10 h 72.0 84.9 66.9 82.0 50–100 keV 12 h 70.7 84.2 69.2 83.5 100–200 keV 16 h 71.1 84.4 73.2 85.6 200–350 keV 24 h 69.5 83.7 71.6 84.9 350–600 keV 24 h 69.9 83.8 73.6 85.9 4.3 Final Model Performance The performance of the models was analyzed statistically using the PE and CC. Each of the models were run on the data from 1 March 2013 to 28 February 2015. At first, the models were run on the daily averaged data which resulted in 730 points for the period. Then, the models were extended to 1 h resolution of the past 24 h average, which contains 17,520 points, to assess each of the model's performance with an increased time resolution. Table 1 lists the performance of the five low-energy electron flux models, showing the PE and CC for the 1 day and 1 h resolution data. The table also shows the minimum time lag used in the model and thus how far ahead the model can forecast into the future. This is in agreement with the studies by Boynton et al. [2013b] and Balikhin et al. [2012], since the minimum time lags increase with energy. The PEs of the models are between 66.9% and 73.6%, which means that the mean square error is well within the variance of the fluxes, and the CC 82% and 85.9%. The results of the five models for the 1 h resolution data are illustrated in Figures 4a (30–50 keV model), 4(c) (50–75 keV model), 4e (100–200 keV model), 5a (200–35