Specification of the Earth���s plasmasphere with data assimilation A.M. Jorgensen a,*, D. Ober b, J. Koller c, R.H.W. Friedel c a New Mexico Institute of Mining and Technology, 801 Leroy Place, Socorro, NM 87801, USA b AFRL/RVBXP, 29 Randolph Road, Hanscom AFB, MA 01731, USA c Space Science and Applications, ISR-1, MS D466, Los Alamos National Laboratory, Los Alamos, NM 87545, USA Received 6 April 2009 received in revised form 30 May 2010 accepted 6 June 2010 Available online 12 June 2010 Abstract In this paper we report on initial work toward data assimilative modeling of the Earth���s plasmasphere. As the medium of propagation for waves which are responsible for acceleration and decay of the radiation belts, an accurate assimilative model of the plasmasphere is crucial for optimizing the accurate prediction of the radiation environments encountered by satellites. On longer time-scales the plasma- sphere exhibits significant dynamics. Although these dynamics are modeled well by existing models, they require detailed global knowl- edge of magnetospheric configuration which is not always readily available. For that reason data assimilation can be expected to be an effective tool in improving the modeling accuracy of the plasmasphere. In this paper we demonstrate that a relatively modest number of measurements, combined with a simple data assimilation scheme, inspired by the ensemble Kalman filtering data assimilation technique does a good job of reproducing the overall structure of the plasmasphere including plume development. This raises hopes that data assimilation will be an effective method for accurately representing the configuration of the plasmasphere for space weather applications. �� 2010 COSPAR. Published by Elsevier Ltd. All rights reserved. Keywords: Kalman filtering Plasmasphere modeling Space weather Data assimilation 1. Introduction Data assimilation techniques are widely used in weather forecasting and that is perhaps the field in which they are most well known (e.g. Kalnay et al., 1998). However, data assimilation techniques are used in one form or another in a wide variety of data estimation problems. Other examples include radar tracking problems (e.g. Ramachandra, 2000). Data assimilation works by merging, by any means, a model which is a physical description of a system with mea- surements which constrain the state or evolution of the sys- tem in some relevant way. The free model parameters are then adjusted to maximize the agreement between the model and the measurements. One of the most effective data assimilation methods is the Kalman filter (Kalman, 1960), with early applications to radar tracking problems. The original approach developed by Kalman required all the derivatives of the model with respect to adjustable parameters, such that for very large problems or complex non-linear models this became cum- bersome. Several alternatives, some based on statistical approximations, were developed. Among them the ensem- ble Kalman filter (Evensen, 2003) is now widely used in weather prediction, and does not require derivatives. Instead it requires the model to be run many times with dif- ferent parameters in order to sample parameter space statistically. In recent years Kalman filtering techniques have been applied to space weather prediction problems, particularly to the prediction of the radiation belts, with good success (Koller and Friedel, 2005 Koller et al., 2007 Maget et al., 2007 Kondrashov et al., 2007 Naehr and Toffoletto, 2005 Rigler et al., 2004). These projects aim to provide a complete specification of the radiation belts based on satel- lite measurements and a good but imperfect physics-based model. The work by Koller uses the Rasmussen and Schunk (1990), Rasmussen et al. (1993) plasmasphere model, but 0273-1177/$36.00 �� 2010 COSPAR. Published by Elsevier Ltd. All rights reserved. doi:10.1016/j.asr.2010.06.013 * Corresponding author. Tel.: +1 575 835 5450. E-mail address: anders@nmt.edu (A.M. Jorgensen). www.elsevier.com/locate/asr Available online at www.sciencedirect.com Advances in Space Research 47 (2011) 2152���2161

does not close the data assimilation loop around the model, relying instead on solar wind parameters to drive the model. The plasmasphere is a region of dense plasma trapped in the co-rotating portion of the inner magnetosphere. Plasma- sphere dynamics is driven by ionospheric sources and sinks, and by the changing magnetospheric electric field, which with the magnetic field combine to create density gradients and plasma plumes (e.g. Lemaire and Gringauz, 1998 Dar- rouzet et al., 2009). The plasmasphere is a significant driv- ing force on the radiation belts as regions of large density gradients host the waves responsible for acceleration and loss of radiation belt particles (e.g. Friedel et al., 2002 Horne and Thorne, 1998). In this paper we report on initial work to develop a data assimilative approach to modeling the plasmasphere. We use the plasmasphere model by Ober et al. (1997) and a ensemble data assimilation approach inspired by ensemble Kalman filtering. We use a real interval of KP to simulate the plasmasphere and generate simulated data, which we then input to the data assimilation method in an attempt to recover the plasmasphere configuration and the input KP. The Ober et al. (1997) model is a physics-based model of the plasmasphere. The overall operation of the model can be described in three simple equations: one equation which describes the dayside outflow from the ionosphere, contain- ing two free parameters, the plasmasphere saturation den- sity and the maximum outflow rate one equation which describes the nightside inflow to the ionosphere, containing one free parameter, the decay rate and one equation which describes convection and continuity. In addition, the model takes as inputs arbitrary electric and magnetic fields. For the purpose of this paper we use a dipole magnetic field and an electric field which is parametrized by KP (Sojka et al., 1986). 2. Methodology In this paper we employ a simpler data assimilation approach than ensemble Kalman filtering because of the ease with which it can be implemented. We use the Ober et al. (1997) plasmasphere model, which is written in the Fortran language. We wrote a C-language wrapper which allows us the necessary access to the model internals. This includes the ability to read and write the plasma density map between the model and a storage array, the ability to simulate satellite density measurements from the plasma density map, and the ability to set the external parameter (in this case KP) and run the model for a fixed time interval as a subroutine. This is illustrated in Fig. 1(a). The data assimilation approach which we use involves an ensemble of models similar to ensemble Kalman filter- ing. However, for the purpose of simplicity we run each model in the ensemble at a fixed KP. At each data assimi- lation time, where data and models are compared, the best model is selected and its density map is copied to all the other running models. The assimilation procedure is thus as follows. Several models are run in parallel from the same initial condition with different values of KP for a fixed interval of time (in reality the different models are run serially on a single pro- cessor, and the plasma density maps and model parameters are copied in and out of the model for each). At the end of the interval satellite density measurements are simulated from each model and compared with the input data. The cost function for this comparison is the sum of squares of fractional errors. The plasma density from the best model is then copied to each of the running models, and the mod- els are then run again for another fixed interval of time. This is illustrated in Fig. 1(b). This approach is significantly simplified compared to for example ensemble Kalman filtering (e.g. Evensen, 2003). In ensemble Kalman filtering the full probability distribution is approximated by an ensemble, and formal errors are derived. In our approach, which is an initial test, for the sake of simplicity, we only track the best-fitting model amongst an ensemble of fixed drivers. The success of this simple approach will inform whether more sophisticated modeling approaches are warranted. In this paper we work with simulated data, which are generated from a period of real KP values in order to have realistic plasma density variations. We use the first half of December 2006, whose KP values are plotted in Fig. 2. We simulate data for eight satellite orbits, including four ellip- tical orbit satellites and four geostationary satellites space evenly in local time as show in Fig. 3. We call these simu- lated data the input data. We do not simulate noise or any systematic effects on the data, but those are factors which must be considered in the future. Fig. 4 shows a close look at several consecutive assimi- lation steps for a short interval of 4 h with data assimila- tion taking place every hour. In the figure the red curve is the input data, the blue curves each of the models run with different values of KP (in this case the 11 values from 0 to 10), and the green curve represents the best model as determined by best agreement between the model and all satellites at the assimilation times, marked by the dotted lines. In this case the assimilation interval is 1 h. Notice that at the beginning of each hour all 11 models begin at the same point, and then diverge as time progresses because of the differing values of KP. Although it appears that at 21 UT the assimilation did not pick the best-fitting model we should remember that this figure shows only one satel- lite out of eight. Throughout this paper we use the 1-h assimilation inter- val, and run either 11 or 31 models in parallel, with KP val- ues evenly distributed in the [0 10] interval. These are not the same values as used to generate the simulation from which the input data are derived. Those follow the encod- ing of KP-values, 0, 0.3, 0.7, 1, 1.3, etc. We should note that although real Kp only covers a discrete set of values in the [0 9] interval, the model is defined on a continuous variable and still produces a reasonable convection pattern outside of these discrete values, as well as in the [9 10] interval. We do not expect the assimilation to make much use of values above 9, but give it access to this range of parameter A.M. Jorgensen et al. / Advances in Space Research 47 (2011) 2152���2161 2153