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belt data assimilation using THEMIS-SST data and the Salammbo-EnKF code
V. Maget, A. Sicard-Piet, S. Bourdarie, D. Lazaro, D.L. Turner, I.A. Daglis, I.
Sandberg
To cite this version:
V. Maget, A. Sicard-Piet, S. Bourdarie, D. Lazaro, D.L. Turner, et al.. Improved outer boundary conditions for outer radiation belt data assimilation using THEMIS-SST data and the Salammbo- EnKF code. Journal of Geophysical Research Space Physics, American Geophysical Union/Wiley, 2015, 120 (7), p. 5608-5622. �10.1002/2015JA021001�. �hal-01232535�
Improved outer boundary conditions for outer radiation belt data assimilation using THEMIS-SST data
and the Salammbo-EnKF code
V. Maget1, A. Sicard-Piet1, S. Bourdarie1, D. Lazaro1, D. L. Turner2, I. A. Daglis3,4, and I. Sandberg3,4
1ONERA, The French Aerospace Lab, Toulouse, France,2Space Sciences Department, The Aerospace Corporation, El Segundo, California, USA,3IAASARS, National Observatory of Athens, Penteli, Greece,4Now at Department of Physics, University of Athens, Athens, Greece
AbstractOver the last decade, efforts have been made in the radiation belt community to develop data assimilation tools in order to improve the accuracy of radiation belts models. In this paper we present a new method to correctly take into account the outer boundary conditions atL* = 8 in such an enhanced model of the radiation belts. To do that we based our work on the Time History of Events and Macroscale Interactions during Substorms/Solid State Telescope data set. Statistics are developed to define a consistent electron distribution atL* = 8 (in both equatorial pitch angle and energy), and a variance-covariance matrix is estimated in order to more realistically drive the Monte Carlo sampling required by the Ensemble Kalman Filter (EnKF).
Data processing isfirst described as well as caveats avoided, and then the use of these information in a machinery such as the EnKF is described. It is shown that the way the Monte Carlo simulations are performed is of great importance to realistically reproduced outer boundary distribution needed by the physic-based Salammbô model. Finally, EnKF simulations are performed and compared during September 2011 in order to analyze the improvements gained using this new method of defining outer boundary conditions. In particular, we highlight in this study that such a method provides great improvement in the reconstruction of the dynamics observed at geosynchronous orbit, both during quiet and active magnetic conditions.
1. Introduction
Modeling Earth’s radiation belts constitutes an activefield of research due to its direct contributions, both to space weather and standard space environment specification used for spacecraft. Preliminary attempts have been conducted to forecast the current state of Earth’s radiation belts based on pure physic-based models in the FP7 project named SPACECAST [Horne et al., 2013] and to better exploit data and modeling using data assimilation in the FP7 project named MAARBLE (Monitoring, Analyzing and Assessing Radiation Belt Loss and Energization) [Daglis et al., 2012]. Besides, the new AE9-AP9 models [Ginet et al., 2013] are based on more physics than previous AE8-AP8 models [Vette, 1991] in order to improve temporal and spatial interpolation and guesses between data. It has been shown in the recent years that a data assimilation scheme based on an Ensemble Kalman Filter (EnKF) [Evensen, 2003] may lead to great improvements in (1) the accuracy of modeling the different regions of Earth’s radiation belts, (2) the possibility to accurately predict the state of the radiation belts, and (3) in accurately reanalyzing a long time period as a basis for specification model and climatology [Maget et al., 2007; Bourdarie and Maget, 2012; Reeves et al., 2012;
Shprits et al., 2013]. However, to be fully useful, such a framework has to be optimized in terms of physics- based models and the data accuracy it relies on.
To achieve such a challenge, many efforts have been undertaken to improve data reliability in the recent years. In particular, one of the MAARBLE project objectives was to enhance data exploitation and to incorporate multispacecraft particle measurements into data assimilation tools. In the same way, the foundation of the new AE9-AP9 model relies on evolved techniques developed to optimize the exploitation of data [O’Brien, 2005; O’Brien and Guild, 2010]. In parallel to these data-oriented improvements,Bourdarie and Maget[2012] also showed that improving the uncertainties in estimations of physics-based processes, which drive the physical model in a data assimilation scheme (the Salammbô code in that case), was necessary. Distributions of radial diffusion and wave-particle interaction coefficients based respectively on uncertainties analyzed in the studies of Lejosne et al.
[2013] andSicard-Piet et al.[2014]] have been made in order to better drive the Salammbô-EnKF code.
Journal of Geophysical Research: Space Physics
RESEARCH ARTICLE
10.1002/2015JA021001
Key Points:
•Radiation belt modeling can be improved using better boundary conditions
•Data assimilation aims at improving radiation belt modeling
•Monte Carlo sampling is used to enhance boundary condition modeling
Correspondence to:
V. Maget,
vincent.maget@onera.fr
Citation:
Maget, V., A. Sicard-Piet, S. Bourdarie, D. Lazaro, D. L. Turner, I. A. Daglis, and I. Sandberg (2015), Improved outer boundary conditions for outer radiation belt data assimilation using THEMIS-SST data and the Salammbo-EnKF code, J. Geophys. Res. Space Physics,120, doi:10.1002/2015JA021001.
Received 11 JAN 2015 Accepted 1 JUL 2015
Accepted article online 4 JUL 2015
©2015. American Geophysical Union.
All Rights Reserved.
In consequence, a consistent set of parameterizations of the Salammbô-EnKF code is now well established.
Up to now, the outer boundary conditions (along the trapping boundary of the electron radiation belts) were kept free, i.e., automatically determined by the analysis phase of the data assimilation scheme. However, this may lead to discrepancies atL* (the RoedererLparameter as defined inRoederer [1970]) values corresponding to altitudes higher than the geosynchronous orbit, since this corresponds to a region where all the physic-based processes taken into account in the Salammbô code become less accurate.
Indeed, the Salammbô code resolves the dynamics of the radiation belts using a Fokker-Planck equation [Bourdarie and Maget, 2012]. It is based on a finite difference resolution method in terms of energy, equatorial pitch angles, andL*. All the major processes acting on the trapped electrons are taken into account: wave-particle interactions, atmosphere-particle interactions, plasmasphere-particle interactions, and radial diffusion. The outer boundary isfixed toL* = 8 in this model, which does not resolve magnetic local time. Besides, the EnKF is a mathematical tool [Evensen, 2003] that performs a multidimensional least squares minimization between data and modeling estimations (based on Salammbô physical prediction) in order to rescale these last ones. In a simplified view, one can consider that the EnKF aims at interpolating between ingested data using a physics-based model (the Salammbô code herein) both spatially and temporally. However, it also performed extrapolation outside the region of ingested data (for example, above geosynchronous orbit) and outside temporal coverage (forecast).
Correctly taking into account the accuracy of the boundary condition in the EnKF is of great importance.Daae et al.[2011] have been thefirst ones to consider this question. They performed a reanalysis of the radiation belts using a 1-D diffusion code and a standard Kalmanfilter. They showed that physical processes acting in the radiation belts are localized principally forL* values between 4 and 6 and that the location of the outer boundary does not depend onL* above 7. This may be effectively true in that particular case when (1) a 1-D diffusion code is used, consequently not resolving pitch angle distribution, and (2) the outer boundary is forced to afixed distribution (no uncertainty supposed on). In the present study, we aims at analyzing further the influence of the outer boundary distribution on the reanalysis in a more general case, when (1) it is used with a 3-D model, consequently moreflexible and capable to restitute the whole distribution of trapped particles in the radiation belts, and (2) sufficient statistical information is available regarding its dynamics, so that the assimilation tool may base its optimization on the most accurate pieces of information.Kondrashov et al.[2011] proposed for the 3-D case to linearize the Fokker-Planck equation in order to directly feed a standard Kalman Filter with the logarithmic value of the phase space density. The authors showed that their method improves the assimilation and prediction skills of their tool when model errors induced by the linearization of the equation become smaller than the observational ones. In our case, our assimilative tool is based on an EnKF assimilating the logarithmic value of the phase space density [Bourdarie and Maget, 2012], while the time interpolation is performed using the natural phase space density. As a consequence, the 3-D Fokker-Planck equation does not required being linearized, the EnKF providing the optimal framework to solve such kinds of nonlinear functions, with a dual benefit:
computation time and all-cases validity.
The general method to take into account uncertainties in the EnKF is to make a Monte Carlo sampling around the expected value of whether data, physics-based processes, or boundary condition spectra, by using the a priori estimated standard deviation, or variance, of the considered distribution. Boundary conditions are generally derived from analyticfits based on plasmasheet characteristics [Christon et al., 1988, 1991] and magnetic activity dependences [Bourdarie and Maget, 2012], or if a sufficiently reliable data set is available, a distribution can be directly obtained [Ni et al., 2011]. In this last case, it is then straightforward to estimate at least the standard deviation of the corresponding distribution. However, for the purpose of data assimilation, not only the standard deviation is required, for example, for each energy channel, but also all the covariance between each of them. Indeed, if only standard deviations are used to sample the distribution around its average, unrealistic spectra may be generated. As a consequence, the EnKF could be inconsistently driven and its accuracy reduced. The large amount of data of the SST instrument onboard Time History of Events and Macroscale Interactions during Substorms (THEMIS) satellites [Turner et al., 2013] allowed us to estimate the whole covariance matrix of the distribution of electronflux at L* = 8, in order to define an accurate outer boundary condition for radiation belt models as well as for data assimilation tools.
In section 2, we first present the THEMIS/SST data as well as the way they have been analyzed. In section 3, complete distributions of electron fluxes, in terms of equatorial pitch angle and energies, are described according to geomagnetic activity (Kp index). Then in section 4, a Monte Carlo sampling method taking into account the variance as well as the covariance of the obtained distributions is explained and the resulting improvements are highlighted. Finally, data assimilation results with these new boundary conditions are presented in section 5. Results and perspectives arefinally discussed in section 6.
2. Data Description and Analysis
The data used in this study are provided by NASA’sfive THEMIS spacecraft. We used energetic electronflux observations measured by the THEMIS Solid State Telescope (SST) instruments. Due to early mission commissioning, we used data after December 2007 only [Turner et al., 2013]. Table 1 summarizes the time period used for each spacecraft, as two of the spacecraft were sent to lunar orbit in 2010. The SST instruments provide omnidirectional electronflux in 11 energy channels ranging from 31 keV to 720 keV, as well as unidirectional ones resolving eight pitch angles between 0° and 180°.
In this statistical study we use both differential omnidirectional and unidirectionalfluxes. TheL*/magnetic local time (MLT) coverage of the data provided by thefive spacecraft as mentioned above is represented on Figure 1 for all magnetic activities (all Kp index mixed), quiet activity (Kp<2), medium activity (2≤Kp<4), and high magnetic activity (Kp≥4). TheL* values were computed using IGRF and Olson-Pfitzer Quiet models. These cartographies show that there is a good MLT coverage of the region betweenL* = 1 andL* = 9. However, we can note that there are poorer statistics at high magnetic activity (Kp≥4) due to the very calm solar cycle during the full period of THEMIS data.
Table 1. Synthesis Table of the Time Periods Used for Statistics for Each THEMIS Spacecraft
Spacecraft Start date Stop date
THEMIS A 01/01/2008 31/12/2013
THEMIS B 01/01/2008 31/12/2009
THEMIS C 01/01/2008 31/03/2010
THEMIS D 01/01/2008 31/12/2013
THEMIS E 01/01/2008 31/12/2013
Figure 1.THEMIS-A, B, C, D, and E MLT coverage between 01/01/2008 and 31/12/2013 for all magnetic activity (allKp), quiet activity (Kp<2), medium activity (2≤Kp<4), and high magnetic activity (Kp≥4).
Aflag of saturation and contamination already exists in the data provided by TDAS (THEMIS Data Analysis Software) and has been taken into account in this study. Concerning the background of the data due to cosmic ray and shield-penetrating contamination as well as the noise floor of the instrument, no flag is provided via TDAS. Thus, an analysis of the data has been done tofilter them according to these contamination sources. A scatterplot offlux level versusL* value is used for each energy channel to determine the threshold above which background is no longer predominant.
Background levels have consequently been obtained for all energy channels, from 31 keV to 720 keV. Data havefinally beenflagged in order not to use bad data in the construction of the boundary condition.
3. Boundary Conditions atL* = 8 for the Salammbô Electron Model
In the Salammbô code, theL* grid ranges from 1 to 8 in order to span a sufficiently large region so that all the internal processes acting on the trapped population (in particular, wave-particle interactions) interfere only slightly with the border of the modeling box. Nonetheless, as the Salammbô code does not resolve magnetic local times, extending itsL*-grid above 8 would induce strong discrepancies since physics, such as electric field influence, magnetic stretching, and drift-orbit bifurcations [e.g., Ukhorskiy et al., 2011], would not be correctly taken into account. Thus, defining boundary conditions for the Salammbô code or the EnKF relying on this model means estimating statistically the phase space density (as a function of both equatorial pitch angle and energy) atL* = 8 using data with the best accuracy as possible. To make that estimate, THEMIS/SST electron data (from allfive spacecraft), only in the nightside for 21 h<MLT<3 h (in order to focus on the injection region and avoid caveats close to the dawn and dusk sectors), are used to characterize the electron distribution atL* = 8 (±0.1).
For thefirst estimate, we limited our analysis to statistics from omnidirectionalfluxes. Figure 2 presents the average values obtained for three classes of magnetic activity (Kp<2 in black, 2<Kp<4 in green, and Kp>4 in red). Thisfigure shows that the mean electron flux at thisL* value seems to depend on the magnetic activity but the behavior is different according to the energy of electron. At low energy, up to 150 keV, electron flux increases with magnetic activity (i.e., with Kp index), while at higher energy (>150 keV) electronflux decreases whenKpincreases. The behavior at low energy is due to the fact that the higher the geomagnetic activity is, the more probable are injections of low-energy particles from the plasmasheet.
The behavior at energy greater than 150 keV is more difficult to explain. At these energies, the combination of both radial diffusion and chorus wave-particle interactions shapes the electron spectra in the outer belt.
During enhanced magnetic activity, high-energy electrons are rapidly pushed inward by radial diffusion at the same time that chorus waves are rising and begin to energize fresh plasmasheet electrons. In the contrary, for quiet and medium magnetic activity, radial diffusion time scale is slowed, thus allowing chorus waves to energize at a given location the same population of high-energy electrons longer. In such a configuration, acceleration of the trapped electrons may be observed as far asL* = 8 [Ni et al., 2011]. As a consequence, the result between these processes acting on the outer electron radiation belt is a net decrease in theflux level when magnetic activity is increasing (main phase of geomagnetic storms).
Besides, one needs to keep in mind that the accuracy of the estimation can be biased due to the poorness of statistics during such enhanced magnetic activity during the THEMIS mission (only a few events have reached Kp≥4 between 2008 and 2013). Figure 3 presents the distribution of the electronflux provided by the Figure 2.Omnidirectional electronflux atL* = 8 versus energy for three
classes of magnetic activity (Kp<2 in black, 2<Kp<4 in green, and Kp>4 in red) in the nightside sector (21 h<MLT<3 h).
five THEMIS/SST instruments (named THEMIS A, B, C, D, and E in the following) between 01 January 2008 and 31 December 2013 for all energy channels and three classes of magnetic activity (Kp<2 on the top, 2<Kp<4 on the middle, and Kp>4 on the bottom). We can note that the number of points decreases when the magnetic activity increases, which lead to a very noisy distribution at high magnetic activity (Kp>4). This confirms the fact that statistics are poor for high magnetic activity, while for the quiet and medium ones, the distribution is not as scattered as for Kp>4.
In order to implement boundary conditions at L* = 8, we see that not only do we need the average distribution value to characterize the seed population according to magnetic activity but also its entire distribution in order to weight its representativeness.
This is most important for when we deal with data assimilation scheme as discussed in the introduction. In particular, the Salammbô-EnKF code requires an ensemble of realistic distributions (in energy, in pitch angle, or both) around the averaged one to drive the assimilation. In consequence, the same analysis and statistics presented in this subsection have been made for the entire distribution at L* = 8, according to equatorial pitch angle (named αeq in the following), energy channels (named Ech in the following) and magnetic activity levels.
For ease, the unidirectionalflux statistics are not presented in this paper since they highlight the same conclusions as for the more easy to understand omnidirectional ones. Characterizing the entire electron distribution atL* = 8 using THEMIS/SST data set does not only imply the estimation of the average and the variance of the distribution for each triplet (αeq, Ech, and Kp). Indeed, the distribution being obviously continuous in pitch angle and energy, the estimation of the correlation or the covariance is of great importance to define realistic distributions to be used in the EnKF.
Based on the same data, we have computed the variance-covariance matrixes between each pair (Ech,αeq) for any levels of magnetic activity. The results are presented in Figure 4 for threeKpvalues (Kp= 0 on the left, Kp= 2 in the middle andKp= 4 on the right). In the mappings, each pixel represents a pair (Ech,αeq). E0 to E10 correspond to the 11 energy channels of the THEMIS/SST instrument: 31 keV, 41 keV, 52 keV, 65.5 keV, 92 keV, 139 keV, 203.5 keV, 293 keV, 408 keV, 565.5 keV and 719.5 keV, andα0 toα3 correspond to the four equatorial pitch angles available in the data: 0–22.5°, 22.5–45°, 45–77.5°, 77.5–90°. The variance of the distribution for each pair (Ech,αeq) corresponds to each element of the diagonal. The other elements of the matrixes correspond to the covariance between corresponding pairs. Such matrixes are not easy to Figure 3.Distributions of electron flux from SST instrument onboard
THEMIS A, B, C, D, and E (all data gathered) between 01/01/2008 and 12/31/2013 for all energy channels and three classes of magnetic activity (Kp<2 on the top, 2<Kp<4 on the middle, andKp>4 on the bottom).
analyze because each line (or column) is not normalized compared to the other ones. To better analyze the spread of the distribution, we have plotted in Figure 5 the correlation matrixes which normalize the previous variance-covariance matrixes. The relationship between them is simple: each correlation element between two pairsC1= (Ech1,αeq1) andC2= (Ech2,αeq2) can be expressed as
corrðC1;C2Þ ¼ covðC1;C2Þ σ1σ2
(1) whereσ1andσ2are respectively the standard deviation of pairC1andC2and cov(C1,C2), the covariance betweenC1andC2. In the correlation matrix, the diagonal is now always equal to one, and it can easily be seen that the correlation between several equatorial pitch angles for a given energy is strong: the pitch angle distribution tends to keep always the same form (forfixedKpandEchvalues). In the contrary, the correlation between energy channels is more restrictive and only a few numbers of neighboring channels are strongly correlated together. This tends to show that for afixedKpvalue, the energy spectra can experience a various number of shapes, highlighting the fact thatKpmay not be the best proxy to classify the trapped population atL* = 8 from tens of keV to MeV energy values.Christon et al.[1988] showed that a large set of spectral shapes can be observed at largeL* values, highlighting the strong dynamics of this more or less trapped population. AlthoughKpmay be a good but sometimes too rough indicator of global magnetic disturbance in the inner magnetosphere, the dynamics of 30 keV injected electrons have clearly different drivers than 700 keV ones. In particular, [Korth and Thomsen, 2001] showed that below a few hundreds of keV the access of the plasmasheet to the inner magnetosphere depends not only on the magnetic activity (that can be approximated usingKp) but also on the electricfields topology, which modifies the global shape of the injected electron distribution too.
Figure 5.Correlation coefficients between the different energies for unidirectional electronflux and for threeKpvalues (Kp= 0 on the left,Kp= 2 in the middle, andKp= 4 on the right). E0 to E10 correspond to the 11 energy channels of THEMIS/SST: 31 keV, 41 keV, 52 keV, 65.5 keV, 92 keV, 139 keV, 203.5 keV, 293 keV, 408 keV, 565.5 keV, and 719.5 keV, andα0 toα3 correspond to the four equatorial pitch angles available in the data: 0–22.5°, 22.5–45°, 45–77.5°, 77.5–90°.
Figure 4.Covariance between the different energies for unidirectional electronflux and for threeKpvalues (Kp= 0 on the left,Kp= 2 in the middle, andKp= 4 on the right). E0 to E10 correspond to the 11 energy channels of THEMIS/SST: 31 keV, 41 keV, 52 keV, 65.5 keV, 92 keV, 139 keV, 203.5 keV, 293 keV, 408 keV, 565.5 keV, and 719.5 keV, andα0 toα3 correspond to the four equatorial pitch angles available in the data: 0–22.5°, 22.5–45°, 45–77.5°, 77.5–90°.
4. Defining a Realistic Ensemble of Outer Boundary Distributions for Salammbô EnKF Code
Three possibilities mainly exist to drive the outer boundary distribution of an EnKF (or Kalman Filter) code as long as these edge points are taken into account during the assimilative processing: (1) one keeps them free (no constrains on and the analysis phase extrapolate inner points dynamics there), (2) one fixes this distribution and thus constrains the outer part of the simulation box (which may lead to discrepancies with assimilated geosynchronous data if the outer edge is not far from that altitude) or at least apply pure variances estimations, (3) one gives all the available and physics-based information to the EnKF in order to constrain its optimization close to the outer edge of the simulation box. Based on the electron distribution statistical properties atL* = 8 detailed in the previous section, we can now generate a realistic set of outer boundary conditions to drive the Salammbô EnKF code using Monte Carlo sampling to test this third possibility, which seems to be the most physics-based solution to us.
Figure 6.Examples of Monte Carlo samplings (10 spectra shown) of the boundary condition from omnidirectionalflux statistics:
on the upper plot, when considering only the variance of each energy channel (no correlation taken into account), and on the bottom plot, when considering the whole variance-covariance matrix of the distribution. The black line represents the median spectrum forKp= 3, while the dashed orange line represents the median + 3 times the standard deviation.
In order to take into account, not only the average and the variance for each pair (Ech,αeq) as it is usually done, we base our sampling on a Cholesky factorization of the covariance matrix [Gill, 1981;Gilli, 2011;Kreyszig, 2011]. It has been shown that multicorrelated samplings can be generated using this kind of factorization [Genz, 1992;Johnson, 1994]. Numerically, we want to define a vector of random variableZ(our distribution at L* = 8 in both equatorial pitch angle and energy dimensions), which is distributed according to a multinormal lawN(μ,P
), where μis the vector of means (the usual averaged distribution), and P the variance-covariance matrix.
E. Schumann (see http://comisef.wikidot.com/tutorial:correlation), referring Gill [1981] and Gilli [2011], explains that since any variance-covariance matrix is symmetric and has real values, P
can be decomposed into
X¼LDL′ (2)
Lis a lower triangular matrix andDis a diagonal matrix with strictly positive elements. One can then write X¼L ffiffiffiffi
pD ffiffiffiffi pD
L′¼CC’ (3)
Cis named the Cholesky factor ofP
, which is a lower triangular matrix. As all elements on the diagonal ofD are strictly positive, we can always calculi their square root, allowing us to define the so-called ffiffiffiffi
pD
matrix.
Based on these mathematical considerations, it is straightforward to derive a sampling of correlated random variables from a vectorXdistributed according to a Normal lawN(0,I), withIthe identity matrix. It is shown that premultiplying such a vector by a matrixMand adding a vectorAconducts the vectorMX+Ato follow a multinormal lawN(A,MIM′) [Genz, 1992]. In consequence,CX+μis distributed according to a multinormal lawN(μ,P
) sinceCIC′=P
, thus corresponding to the desiredZvector.
As afirst implementation, Figure 6 presents the improvement obtained in sampling energy spectra atL* = 8 with or without taking into account the entire variance-covariance matrix, forKp= 3. The upper panel shows 20 samples generated using only the mean values and variance in each energy channel of the SST-based Figure 7.Energy spectra enhancements obtained for unidirectionalflux. This upper (respectively, bottom) plot shows 20 Monte Carlo samplings without (respectively, with) taking into account the correlation between pairs (αeq,Ech). The energy spectra are presented for two interval of equatorial pitch angles.
statistics (the omnidirectionalflux are used here), while the bottom plot shows 20 samples generated using the variance-covariance matrix. On both panels of Figure 6, the median spectra are drawn in black and a rough estimation of the upper envelopes of the distribution (3 times the standard deviation) is drawn in orange (dashed curve). It is interesting to note that such a method constitutes a great improvement to drive data assimilation scheme since the boundary conditions become much more realistic. It avoids oscillation artifacts in energy, improving drastically the form of the spectra. Moreover, one can note that the different samples in the bottom plot have different slope, indicating the possible shapes that the spectra can take. This can be considered as a physical degree of freedom for the data assimilation tool which will be able to choose between different allowed shapes and amplitudes of spectra to better restitute the radiation belt dynamics.
The median and the variance-covariance matrixes based on unidirectionalflux statistics have then been used to sample the boundary condition atL* = 8 according toαeqandEch. This time, instead of sampling only 11 energy channels, we now have to deal with 44 pairs (αeq,Ech). Twenty Monte Carlo sampled distributions have been again generated and are shown in Figures 7 and 8, forKp= 3. As for the omnidirectional case, thesefigures highlight the improvements obtained when using this sampling method, focusing on the energy spectra in Figure 7, and on the pitch angle distribution in Figure 8. On bothfigures, each vertical line corresponds to a sample. Whether the vertical line highlights the energy spectrum of the sample at for a given pitch angle (Figure 7), whether it highlights its pitch angle distribution for a given energy channel (Figure 8). Again, one can note the physics-based smoothing obtained.
5. Improvements Achieved in the Data Assimilation Process
In Figures 6–8, only 20 samples have been computed to highlight the sampling process. In the EnKF framework, 200 members are at least required to span enough the entire domain of allowed states for the radiation belts (the span depends on the distributions of each diffusion process, seeBourdarie and Maget Figure 8.Pitch angle distribution enhancements obtained for unidirectionalflux. This upper (respectively, bottom) plot shows 20 Monte Carlo samplings without (respectively, with) taking into account the correlation between pairs (αeq,Ech).
The pitch angle distributions are presented for two energy channels.
[2012] for more details). As a consequence, at least 200 spectra have to be generated in order to drive each member of a data assimilation simulation correctly at the outer edge of the radiation belts. In order to show the benefits of using a statistically distributed outer boundary condition, we have performed two data assimilation simulations from 7 to 25 September 2011. Both simulations are based on the Salammbô code parameterized as detailed in Bourdarie and Maget [2012], excepted concerning the outer boundary condition, now relying on our new distribution which resolves both pitch angle and energy atL* = 8. We will name in the following discussion thefirst one SIMUno_covaras it uses the new boundary conditions using only the estimated variances to feed the EnKF members. The second one’s is named SIMUcovas it uses this time both the estimated variances and covariances to generate realistic distributions at the outer edge of the simulation box.
In the previous study fromBourdarie and Maget[2012], only a Kappa distribution in energy (no pitch angle resolution) was used for the initialization and then the EnKF scheme had to drive itself the outer boundary at each analysis phase. This solution was chosen in order to limit inaccuracies introduced by forcing unphysically the pitch angle distribution atL* = 8, which is now no more the case. Data from GIOVE-B/
SREM (Standard Radiation Environment Monitor) instrument (all channels, see Taylor et al. [2009]
for details) and NOAA-GOES13 Magnetospheric Electron Detector/Proton Detectors and scanning electron microscope (SEM) instruments (all channels too, see http://ngdc.noaa.gov/stp/satellite/goes/
documentation.html for details) have been used to feed the assimilation scheme. This period of simulation has been chosen because it shows an enhanced magnetic activity period (Kpindex peaks up to 5.3, see Figure 10) followed by a very quiet one (Kpindex below a value of 2, see Figure 9). In particular, whenKp reaches a high value, radial diffusion tends to smooth the distribution in L*, while at lowerKp values, wave-particle interactions enhance the production of MeV electrons in the outer radiation belt. As a consequence, the members of the EnKF do not spread in the same way when radial diffusion or wave- Figure 9.Comparison of the EnKF simulations performed (left column) without or (right column) with the correlations esti- mated between energies and pitch angles atL*=8 during 9–10 September 2011. (a) Comparisons are made against GOES 13/SEM data for>800 keV electronsflux along the satellite orbit. The black curves correspond to observations, while the green ones correspond to the EnKF median value of>800 keVflux. The red curves correspond to pure Salammbô simulations using the average distribution estimated from THEMIS data as a reference. (b) The evolutions of 40 members from the ensembles along GOES 13 orbit are shown. (c) A synthesis of these examples in the form of the evolution of the standard deviation of the ensemble is presented. (d) The evolution ofKpindex during this time period of simulation is also shown.
particle interactions dominate in the outer radiation belt. This is of interest to look at the improvement obtained when defining this new way the outer boundary condition in both configurations. For clarity, we decided to focus our results for the time period from 9 September, 9:00 A.M. to 10 September, 6:00 P.M., 2011. Indeed, this corresponds to a disturbed time period of the simulation during which the improvements are more significant. It is to note that during calm periods of the simulated period, both simulations give very close results when compared to GOES data.
In Figure 9a, we compare our simulation results along GOES 13 orbit for electrons of energies greater than 800 keV. Figure 9a (left) corresponds to SIMUno_cov, while Figure 9a (right) corresponds to SIMUcov. In each plot, the black curves correspond to GOES 13 observations, the green ones to the EnKF outputs, and the red ones to the“pure”Salammbô simulation (e.g., no data is assimilated in this simulation) driven by the average distribution of THEMIS data at the outer edge of the simulation box. For both simulations, the Monte Carlo sampling is performed as explained in the previous section in order to define a realistic distribution (in both energy and equatorial pitch angle) for each member of the EnKF at L* = 8. The difference only resides in the use or not of the estimated correlations to drive the sampling. At about 12:00 A.M. on 9 September, the pure Salammbô simulation presents a strong and rapid decrease in the flux, while only a small decrease is observed at that time in the GOES 13 measurements. At that time, the Kp index is rising up to 5.33 (see the evolution ofKp index plotted in Figure 9c). This deviation from Salammbô is mainly due to the modeling of the magnetopause shadowing effect, which may be too strong in this case compared to the reality observed by GOES 13. However, this is a good example of the value of data assimilation. While theflux drops in the pure Salammbô simulation, both EnKF simulations avoid this. They better reproduce the evolution of theflux observed at GOES altitude, even if during disturbed times around midnight on 10 September, all the rapid decreases and increases observed in the flux are not reproduced correctly. While the GOES 13 satellite is orbiting around Earth during high level of magnetic activity, the geosynchronous orbit may cross the magnetopause from time to time, thus impacting the observed flux [Onsager et al., 2002]. Furthermore, Salammbô does not resolve magnetic local time. Consequently, it assumes that the trapped electron population is homogeneous at given L*
values and can only reproduce averaged distribution values.
Nonetheless, the main purpose of the discussion here is to highlight the improvements obtained with this new way of defining the outer boundary condition in a data assimilation scheme. In Figure 9a, the major deviation from GOES 13 data between 4 P.M. and 5 P.M. on 9 September occurring in SIMUno_cov(Figure 9a, left) is smoothed in SIMUcov(Figure 9a, right). Moreover, the global accuracy is improved during this time period when using the covariances to improve the definition of the outer boundary distribution of our EnKF scheme. Only small spikes are still noticeable, mainly due to the rough hourly Monte Carlo sampling performed to drive the outer boundary conditions. Indeed, each hour a new Monte Carlo sampling is performed to take into account the evolution of Kp index. However, no temporal smoothing is used.
Consequently, the outer boundary distribution may drastically change from an hour to the next one, inducing discontinuous behaviors in the EnKF scheme. This is highlighted in Figure 9b, in whichflux along GOES 13 orbit from 40 members are plotted for both EnKF simulations. Large spikes occur repeatedly in SIMUno_covthat are mainly no more present in SIMUcov. The use of the covariances in the Monte Carlo sampling of the outer boundary condition tends to smooth these discontinuities. This is also noticeable when plotting the standard deviation of the ensembles for both simulations as it is done in Figure 9c.
Spikes are clearly visible too in SIMUno_covand mainly no more in SIMUcov. They correspond to thefirst analysis phase after each resampling of the outer boundary. The EnKF scheme tries to take into account these unrealistic distribution changes as significative dynamics which generate a temporary instability during the analysis phase solutions exist to get completely rid of such artefacts, in particularly a temporal smoothing in the Monte Carlo sampling (see, for example, Evensen[2003] and Evensen [2004] for more details) and will constitute the next step in the optimization of our data assimilation tool.
Thisfirst validation of the improvements obtained due to the use of the whole variance-covariance matrix of the outer boundary distribution has been conducted by comparing simulations outputs to GOES 13 data, which have also been used as driving data for the data assimilation scheme. Although it already constitutes a meaningful comparison, it does not assure that these new boundary conditions have a global positive influence in the restitution of the whole outer radiation belt. As a consequence, we have compared simulation outputs to an independent set of data in Figure 10. The INTEGRAL spacecraft is a
highly elliptical satellite (650 km × 150 000 km, with an inclination of 52°) launched in October 2002, which crosses the outer radiation belt for different band of equatorial pitch angles. Roughly, it crossesL* = 4 for equatorial pitch angles greater than 70°, L* = 5 for pitch angles of between 50 and 70°, andL* = 6 for equatorial pitch angles between 20 and 40°. The SREM/INTEGRAL Radiation Environment Monitor instrument suite on board the satellite provides 15 channels. In particular, the TC1 channel (which corresponds to approximately electrons greater than 1.6 MeV in the outer radiation belt region) has been Figure 10.Comparisons of the simulations with independent INTEGRAL data (TC1 channel which roughly measures
>1.6 MeV electrons in the outer belt) each time it crosses (first panel)L* = 6, (second panel)L* = 5, and (third panel) L* = 4. The black crosses correspond to the observation, the red squares correspond to the pure Salammbô simulation, the blue plus signs correspond to SIMUno_cov, and the green triangles correspond to SIMUcov. The fourth panel shows the evolution ofKpindex during this time period of simulation.
used in Figure 10 to compare our simulation results to in situ data. For this instrument we dispose of both its raw count rates and the response functions of the instrument. In order to avoid convolution uncertainties, we have compared observed count rates to simulated ones, since it is possible in our EnKF tool to both ingest and output data of any types [Bourdarie and Maget, 2012]. Figure 10 presents four panels. The bottom one corresponds to the evolution of theKpindex from 8 to 25 September 2011, while the three other panels show comparisons between data and simulations at (from bottom to top)L* = 4, 5, and 6. The black crosses correspond to the count rates observed in the TC1 channel, while the red squares correspond to the pure Salammbô simulation output, the blue “plus” signs correspond to SIMUno_cov, and the green triangles correspond to SIMUcov. At L* = 6 improvements are noticeable compared to both the pure Salammbô Figure 11.Statistics accuracy depending onKpvalues. The top plot shows the evolution of the median spectra (omnidir- ectionalflux statistics) according toKpvalue. The bottom plot shows the variance-covariance matrix for unidirectionalflux statistics atKp= 6.
simulation and SIMUno_cov. At thisL* value, the result comparison is performed with equatorial pitch angles smaller than 40°. This corresponds to a region where we know that Salammbô is not optimized since the wave-particle interaction model used is imperfect due to the lack of electromagnetic wave measurements in that region. This explains why Salammbô output over INTEGRAL data may reach almost three decades. Both EnKF simulations clearlyfit better INTEGRAL data mainly thanks to the outer boundary distribution resolving pitch angles. Even if GOES 13 data are ingested in the EnKF scheme betweenL* = 5 and 6, these data mostly improve the accuracy in the equatorial region. SIMUno_cov is slightly less accurate than SIMUcov. An improvement factor of 2 to 3 is noticeable depending on the considered crossing time. At lowerL* values, the same conclusions are applicable, even if the improvements are less noticeable due to the fact that the comparisons are performed closer to the equator. The conclusion of this comparison is that the care taken to improve the outer boundary distribution in the EnKF scheme has a noticeable influence especially in the region above L* = 5 and primarily for low equatorial pitch angles which is the major gap to fill in the modeling for that region. This extends the study ofDaae et al.[2011] due to the fact that more accuracy is required and can be obtained for a 3-D code.
6. Discussion
The SST instruments on board the THEMIS spacecraft allowed us to statistically approximate the entire distribution of the electron population atL* = 8, which had been a major lack in the radiation belt models up to now. Of course, the accuracy of this enhanced boundary condition depends on the completeness of the statistics used. For a pure physic-based model such as the Salammbô code, the median distribution obtained is sufficient. Only the poor statistics at high values ofKpdegrade its accuracy. In the case of data assimilation tool such as the EnKF, more information concerning the distribution is needed. As can be seen in Figures 6–8, the use of the variance-covariance matrix provides much more realistic samplings than if we had only drew independent Gaussian variables for each pair (αeq,Ech). In particular, the smoothing in pitch angle is a considerable improvement which both avoids many caveats and improve the restitution of the radiation in belt outside the equator aboveL* = 5, where it is known that modeling is inaccurate.
As highlighted in the previous section, the fact that the Monte Carlo sampling of the distribution is more realistic helps the EnKF to improve its performance. To go further, it is interesting to keep on working in this direction, whether when using data to make empirical models, as it is the case in the new AE9 and AP9 models or when defining physics-based models such as for wave-particle interactions or radial diffusion models. In the framework of data assimilation, it is a necessity to have models that can be sampled using Monte Carlo methods, representing the most realistic dynamics.
Nonetheless, a last caveat has to be kept in mind. When statistics are poor, as is the case for highKpvalues, the information contained in the variance-covariance matrix may be biased. As an example, the THEMIS period has poor statistics forKp= 6. Figure 11 (bottom) shows the variance-covariance matrix forKp= 6.
We can note that the correlation between energies atKp= 6 is strongly reduced compared to the one at Kp= 3. Moreover, when we look at the evolution of the median spectrum as a function ofKp(Figure 11, top), we can see that for all classes of Kpbelow 5, the spectral shapes are similar while above 5, the spectra shapes drastically change. These are artifacts, mainly due to the poorness of the statistics at these Kp values. If such biases are used with no care, this may drastically affect the accuracy of the data assimilation results.
In conclusion, a new numerical method has been implemented in order to optimize information on the outer boundary condition to a data assimilation tool. It is one of thefirst times that data are considered in such a complete manner. By using the variance-covariance matrix, the sampling that we generate to drive the EnKF from L* = 8 is much more realistic and contains much more information than a standard kappa distribution. In particular, the allowed level of spreading is now physically managed. This constitutes a great improvement, both on the data exploitation and data assimilation sides.
References
Bourdarie, S. A., and V. F. Maget (2012), Electron radiation belt data assimilation with an ensemble Kalmanfilter relying on the Salammbô code,Ann. Geophys.,30(6), 929.
Christon, S. P., D. G. Mitchell, D. J. Williams, L. A. Frank, C. Y. Huang, and T. E. Eastman (1988), Energy spectra of plasma sheet ions and electrons from∼50 eV/e to∼1 MeV during plasma temperature transitions,J. Geophys. Res.,93(A4), 2562–2572, doi:10.1029/JA093iA04p02562.
Acknowledgments
MAARBLE has received fundings from the European Community’s Seventh Framework Programme (FP7-SPACE- .2010-1, SP1 Cooperation, Collaborative project) under grant agreement n284520.
This paper reflects only the authors’
views, and the European Union is not liable for any use that may be made of the information contained therein. We also acknowledge NASA contract NAS5-02099 and V. Angelopoulos for the use of data from the THEMIS mission.
Specifically, D. Larson and R.P. Lin are also thanked for the use of SST data. THEMIS data are free for scientific publications and can be accessed here: http://themis.
ssl.berkeley.edu/data_products/.
M. Balikhin thanks Jichun Zhang and another reviewer for their assistance in evaluating this paper.
Christon, S. P., D. J. Williams, D. G. Mitchell, C. Y. Huang, and L. A. Frank (1991), Spectral characteristics of plasma sheet ion and electron populations during disturbed geomagnetic conditions,J. Geophys. Res.,96(A1), 1–22, doi:10.1029/90JA01633.
Daae, M., Y. Y. Shprits, B. Ni, J. Koller, D. Kondrashov, and Y. Chen (2011), Reanalysis of radiation belt electron phase space density using various boundary conditions and loss models,Adv. Space Res.,48(8), 1327–1334, doi:10.1016/j.asr.2011.07.001.
Daglis, I. A., S. Bourdarie, Y. Khotyaintsev, O. Santolik, R. Horne, I. Mann, D. Turner, A. Anastasiadis, V. Angelopoulos, and G. Balasis (2012), Monitoring, analyzing and assessing radiation belt loss and energization, inEGU General Assembly Conference Abstracts, vol. 14, p. 9014.
Evensen, G. (2003), The Ensemble Kalman Filter: Theoretical formulation and practical implementation,Ocean Dyn.,53(4), 343–367, doi:10.1007/s10236-003-0036-9.
Evensen, G. (2004), Sampling strategies and square root analysis schemes for the EnKF,Ocean Dyn.,54, 539–560, doi:10.1007/
s10236-004-0099-2.
Genz, A. (1992), Numerical computation of multivariate normal probabilities,J. Comput. Graph. Stat.,1(2), 141–149, doi:10.1080/
10618600.1992.10477010.
Gill, P. E. (1981),Practical Optimization, Academic Press, London; New York.
Gilli, M. (2011),Numerical Methods and Optimization in Finance, Elsevier/Academic Press, Amsterdam, Boston, Mass.
Ginet, G. P., et al. (2013), AE9, AP9 and SPM: New models for specifying the trapped energetic particle and space plasma environment,Space Sci. Rev., doi:10.1007/s11214-013-9964-y.
Horne, R. B., et al. (2013), Forecasting the Earth’s radiation belts and modelling solar energetic particle events: Recent results from SPACECAST,J. Space Weather Space Clim.,3, A20, doi:10.1051/swsc/2013042.
Johnson, G. E. (1994), Constructions of particular random processes,Proc. IEEE,82(2), 270–285, doi:10.1109/5.265353.
Kondrashov, D., M. Ghil, and Y. Shprits (2011), Lognormal Kalmanfilter for assimilating phase space density data in the radiation belts,Space Weather,9, S11006, doi:10.1029/2011SW000726.
Korth, H., and M. F. Thomsen (2001), Plasma sheet access to geosynchronous orbit: Generalization to numerical globalfield models, J. Geophys. Res.,106(A12), 29,655–29,667, doi:10.1029/2000JA000373.
Kreyszig, E. (2011),Advanced Engineering Mathematics, 10th ed., Wiley, Hoboken, N. J.
Lejosne, S., D. Boscher, V. Maget, and G. Rolland (2013), Deriving electromagnetic radial diffusion coefficients of radiation belt equatorial particles for different levels of magnetic activity based on magneticfield measurements at geostationary orbit,J. Geophys. Res. Space Physics,118, 3147–3156, doi:10.1002/jgra.50361.
Maget, V., S. Bourdarie, D. Boscher, and R. H. W. Friedel (2007), Data assimilation of LANL satellite data into the Salammbô electron code over a complete solar cycle by direct insertion,Space Weather,5, S10003, doi:10.1029/2007SW000322
Ni, B., Y. Shprits, M. Hartinger, V. Angelopoulos, X. Gu, and D. Larson (2011), Analysis of radiation belt energetic electron phase space density using THEMIS SST measurements: Cross-satellite calibration and a case study,J. Geophys. Res.,116, A03208, doi:10.1029/2010JA016104.
O’Brien, P., and T. B. Guild (2010), Trapped Electron Model 2 (TEM-2).
O’Brien, T. P. (2005), A framework for next-generation radiation belt models,Space Weather,3, S07B02, doi:10.1029/2005SW000151.
Onsager, T. G., G. Rostoker, H.-J. Kim, G. D. Reeves, T. Obara, H. J. Singer, and C. Smithtro (2002), Radiation belt electronflux dropouts: Local time, radial, and particle-energy dependence,J. Geophys. Res.,107(A11), 1382, doi:10.1029/2001JA000187.
Reeves, G. D., Y. Chen, G. S. Cunningham, R. W. H. Friedel, M. G. Henderson, V. K. Jordanova, J. Koller, S. K. Morley, M. F. Thomsen, and S. Zaharia (2012), Dynamic Radiation Environment Assimilation Model: DREAM,Space Weather Int. J. Res. Appl.,10, S03006.
Roederer, J. G. (1970),Dynamics of Geomagnetically Trapped Radiation, Springer.
Shprits, Y., A. Kellerman, D. Kondrashov, and D. Subbotin (2013), Application of a new data operator-splitting data assimilation technique to the 3-D VERB diffusion code and CRRES measurements,Geophys. Res. Lett.,40, 4998–5002, doi:10.1002/grl.50969.
Sicard-Piet, A., D. Boscher, R. B. Horne, N. P. Meredith, and V. Maget (2014), Effect of plasma density on diffusion rates due to wave particle interactions with chorus and plasmaspheric hiss: Extreme event analysis,Ann. Geophys.,32(8), 1059–1071, doi:10.5194/angeo-32-1059-2014.
Taylor, B., C. I. Underwood, K. A. Ryden, and P. A. Morris (2009), A GIOVE Derived Galileo Electron Spectrum and Comparison to Models, IEEE Transactions on Nuclear Science,56(6), 3423–3428, doi:10.1109/TNS.2009.2034149.
Turner, D. L., V. Angelopoulos, W. Li, M. D. Hartinger, M. Usanova, I. R. Mann, J. Bortnik, and Y. Shprits (2013), On the storm-time evolution of relativistic electron phase space density in Earth’s outer radiation belt,J. Geophys. Res. Space Physics,118, 2196–2212, doi:10.1002/jgra.50151.
Ukhorskiy, A. Y., M. I. Sitnov, R. M. Millan, and B. T. Kress (2011), The role of drift orbit bifurcations in energization and loss of electrons in the outer radiation belt,J. Geophys. Res.,116, A09208, doi:10.1029/2011JA016623.
Vette, J. I. (1991), The NASA/National Space Science Data Center Trapped Radiation Environment Model Program (1964–1991).
