Specification of the Earth’s Plasmasphere with Data Assimilation

A. M. Jorgensen

New Mexico Institute of Mining and Technology, 801 Leroy Place, Socorro, NM, USA

D. Ober

Air Force Research Laboratory, Hanscom, MA, USA

J. Koller and R. H. W. Friedel

Los Alamos National Laboratory, Los Alamos, NM, USA

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 radiationbelts, an accurate assimilative model of the plasmasphere is crucial for optimizing the accurate predictionof the radiation environments encountered by satellites. One longer time-scales the plasmasphere exhibitssignificant dynamics. Although these dynamics are modeled well by existing models, they require detailedglobal knowledge of magnetospheric configuration which is not always readily available. For that reason dataassimilation 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 simpledata assimilation scheme, inspired by the Ensemble Kalman filtering data assimilation technique does agood job of reproducing the overall structure of the plasmasphere including plume development. This raiseshopes that data assimilation will be an effective method for accurately representing the configuration of theplasmasphere for space weather applications.

Key words:

1. Introduction

Data assimilation techniques are widely used inweather forecasting and that is perhaps the fieldin which they are most well know. However, dataassimilation techniques are used in one form or an-other in a wide variety of data estimation prob-lems. Other examples include radar tracking prob-lems. Data assimilation works by merging, by anymeans, a model which is a physical description ofa system with measurements which constrain thestate or evolution of the system in some relevantway. The free model parameters are then adjustedto maximize the agreement between the model andthe measurements.

One of the most effective data assimilation meth-ods is the Kalman filter (Kalman, 1960), with earlyapplications to radar tracking problems. The origi-nal approach developed by Kalman required all the

derivatives of the model with respect to adjustableparameters, such that for very large problems orcomplex non-linear models this became cumber-some. Several alternatives, some based on statisti-cal approximations, were developed. Among themthe Ensemble Kalman filter (Evensen, 2003) is nowwidely used in weather prediction, and does not re-quire derivatives. Instead it requires the model tobe run many times with different parameters in or-der to sample parameter space statistically.

In recent years Kalman filtering techniques havebeen applied to space physics space weather predic-tion problems, particular to the prediction of the ra-diation belts, with good success (Koller and Fridel,2005; Koller et al., 2007; Maget et al., 2007; Kon-drashov et al., 2007). These projects aim to providea complete specification of the radiation belts basedon satellite measurements and a good but imperfect

Preprint submitted to Advances in Space Research April 4, 2009

Figure 1: KP for December 2006. This interval is used forthe simulations because of the very quiet period at the begin-ning of the interval (days 1-5), the very active period (days14-16), and the period of variable intermediate KP (days6-13).

Figure 2: Orbital positions of the eight satellites at 6-hourintervals for the first two days of December 2006.

physics-based model. The work by Koller uses theRasmussen and Schunk (1990); Rasmussen et al.(1993) plasmasphere model, but does not close thedata assimilation loop around the model, relying in-stead on solar wind parameters to drive the model.The plasmasphere is a significant driving force onthe radiation belts as it is the region which hosts thewaves responsible for acceleration and loss of radi-ation belt particles (e.g. Friedel et al., 2002; Horneand Thorne, 1998).

In this paper we report on initial work to de-velop a data assimilative approach to modeling theplasmasphere. We use the plasmasphere model byOber et al. (1997) and a ensemble data assimilationapproach inspired by Ensemble Kalman filtering.We use a real interval of KP to simulate the plas-masphere and generate simulated data, which wethen input to the data assimilation method in anattempt to recover the plasmasphere configurationand the input KP .

2. Methodology

In this paper we employ a simpler data assimila-tion approach than Ensemble Kalman filtering be-cause of the ease with which it can be implemented.

Modelstate

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mas

pher

e m

odel

Parameter

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control

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stepping

Measurementsimulation

Load stateAdvance model

Data?

ensembleCreate new

Identify bestmodel state

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and compareSimulate data

For each modelin ensemble

StateParameters

StepState

Simulatedmeasurements

Figure 3: Implementation of the simulation. We interfacedthe Ober et al. (1997) plasmasphere model (written in For-tran) with a C-language wrapper which provided access toreading and writing the plasma density and other model pa-rameters. This model is then included in the data assimila-tion loop, computing the plasma density for multiple modelsin parallel.

We use the Ober et al. (1997) plasmasphere model,which is written in the Fortran language. We wrotea C-language wrapper which allows us the neces-sary access to the model internals. This includesthe ability to to read and write the plasma densitymap between the model and a storage array, theability to simulate satellite density measurementsfrom the plasma density map, and the ability toset the external parameter (in this case KP ) andrun the model for a fixed time interval as a subrou-tine. This is illustrated in the left-hand portion ofFigure 3.

The data assimilation approach which we use in-volves an ensemble of models similar to EnsembleKalman filtering. However, for the purpose of sim-plicity we run each model in the ensemble at a fixed

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Figure 4: Demonstration of the data assimilation method fora short interval on December 2, 2006.

KP . At each data assimilation time, where dataand models are compared, the best model is se-lected and its density map is copied to all the otherrunning models.

The assimilation procedure is thus as follows.Several models are run in parallel from the same ini-tial condition with different values of KP for a fixedinterval of time (In reality the different models arerun serially on a single processor, and the plasmadensity maps and model parameters are copied inand out of the model for each). At the end of the in-terval satellite density measurements are simulatedfrom each model and compared with the input data.The cost function for this comparison is the sum ofsquares of fractional errors. The plasma densityfrom the best model is then copied to each of therunning models, and the models are then run againfor another fixed interval of time. This is illustratedin the right-hand section of Figure 3.

In this paper we work with simulated data, whichare generated from a period of real KP values in or-der to have realistic plasma density variations. Weuse the month of December 2006, whose KP valuesare plotted in Figure 1. We simulate data for eightsatellite orbits, including four elliptical orbit satel-lites and four geostationary satellites space evenlyin local time as show in Figure 2. We call thesesimulated data the input data. We do not sim-ulate noise or any systematic effects on the data,but those are factors which must be considered inthe future.

Figure 4 shows a close look at several consecutiveassimilation steps for a short interval of four hourswith data assimilation taking place every hour. Inthe figure the red curve is the input data, the bluecurves, the blue curves each of the models run withdifferent values of KP (in this case the 11 valuesfrom 0 to 10), and the green curve represents thebest model as determined by best agreement be-tween the model and all satellites at the assimila-

tion times, marked by the dotted lines. In this casethe assimilation interval is 1 hour. Notice that atthe beginning of each hour all 11 models begin atthe same point, and then diverge as time progressesbecause of the differing values of KP . Although itappears that at 21 UT the assimilation did not pickthe best-fitting model we should remember that thisfigure shows only one satellite out of eight total.

Throughout this paper we will use the 1-hour as-similation interval, and run either 11 or 31 modelsin parallel, with KP values evenly distributed inthe [0; 10] interval. These are not the same valuesas used to generate the simulation from which theinput data are derived. Those follow the usual en-coding of KP -values, 0, 0.3, 0.7, 1, 1.3, etc.

3. Simulation results

We will report on four different simulations. Thefirst simulation covers the first 16 days of December2006 using 11 parallel models and all eight satel-lites. In the second simulation we use 31 modelsand all eight satellites, whereas in the last two sim-ulation we use 11 models and either the ellipticalor geostationary orbit satellites.

3.1. 16-day simulation

The results of this assimilation run are shown inFigures 5 and 6. Figure 5 shows the input plasmadensity actually measured by the satellites (in red)as well as the plasma density simulated by each ofthe parallel model runs (in blue). The green curveshows the plasma for the model which fit the databest at the hourly assimilation times. This is thesame formatting scheme as is shown in figure 4. Thetop four panels shows the elliptical satellites, thenext four panels the geostationary satellites. Thelast panel shows KP , which is the only free param-eter in the assimilation. The red curve shows theKP used to generate the input data. The greencurve shows the KP of the best model for eachhour interval, and the blue curve shows a double5-hour smooth of the green curve (double in orderto produce a continuous second derivative and anicer look).

Figure 6 shows the density maps at 8-hour in-tervals beginning at 8 UT on December 1, 2006.The layout of those images is explained in the Fig-ure caption: every two rows belong together, withthe upper row showing the recovered plasma map,and the lower row showing the input plasma densitymaps.

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Figure 5: Data assimilation on the plasmasphere for the first 16 days of December 2006. The top four panels of each plot arethe plasma density measurements by the elliptical orbit satellites. The next four panels of each plot are the plasma densitymeasurements by the four geostationary orbit satellites. The bottom panel in each plot is the KP index. In the top eight panelsof each plot the red curve is the input plasma density data, the blue curves are the plasma density measurements correspondingto each of the 11 parallel models, and the green curve is the plasma density corresponding to the best model selected for each1-hour interval. In the bottom panel of each plot the red curve is the input KP value, from Figure 1, the green curve the KP

value corresponding tot he best model for each 1-hour interval, and the blue curve is the green curve smoothed twice with a5-hour boxcar window.

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Figure 6: Images of plasma density as a function of time. In order to maximize the size of the images the scales have been leftout. Each image is of a 16 by 16 RE region centered on Earth with the sun at the left and dusk at the bottom. The colorwhich represents plasma density increases from blue through green to yellow. Images are shown for every 8 hours beginningat 8 UT on December 1, and both the recovered image and the input image are shown in alternating rows with the recoveredimage at the top. Times 8, 16, and 24 are thus represented in the top two rows for December 1, 2, and 3.

In Figure 5 there is generally good agreement be-tween the input plasma density measurements and

the best assimilated plasma density measurements.The first two days (December 1-2), and last three

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Figure 7: Comparison of the effect of increasing the number parallel assimilation models, modeling the first four days ofDecember 2006. The top two panels show plasma density measured by one satellite in the case 11 assimilation states (toppanel) and 31 assimilation states (second panel). The corresponding KP values are shown in the third and fourth panelsrespectively.

Figure 8: Comparison of the effect of using all eight satellites (top panel) versus using only the elliptical orbit satellites (centerpanel) and only the geostationary orbit satellites (bottom panel).

days (December 14-17) of the simulation fit par-ticularly well, with even quite complex structurebetween 14 UT and 20 UT on December 2 beingreproduced well. By contrast the interval from De-cember 3 through December 6 and even part of De-cember 7 is not very well reproduced at all. The restof the time the plasma density is reproduced quitewell although there is a slight tendency for the as-similation step to place the plasmapause closer tothe Earth than indicated by the input data.

In the KP data (the last panels in Figure 5) wesee a similar pattern. Excellent agreement betweeninput and recovered KP on December 1 and 2, 14,15, and 16, very poor agreement on December 3through 6 or 7, and good agreement the rest of thetime, with a slight tendency towards a higher KP

than prescribed in the input data.

In the images (Figure 6) the situation is againsimilar. On December 1 and 2 (First 6 columnsof images of top two rows) the agreement betweenthe recovered plasma density (top row) and the in-put density (second row) is excellent, with evenfine plume structure being reproduced well. OnDecember 3, 4, 5, and 6, the recovered and in-put plasma density are wildly different, with therecovered showing a much smaller plasmapause inagreement with the satellite density measurementsin Figure 5, and in agreement with the larger KP

that was selected during that time interval. OnDecember 7, the recovered and input plasma den-sity maps begin to look more similar, and generallyagree well after that time except for a few differ-

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ences, for example 0 UT on December 12.It is also interesting to note that during most

of the simulation the recovered value of KP (greencurve in the last panels of Figure 5), varies widelyfrom hour to hour while its average (blue curve)is in much better agreement with the input KP

value. A likely explanation for this is that becausenone of KP values available to the assimilation ex-actly match the input KP , the assimilation com-pensates for this by selecting KP values which aregreater than and smaller than the the input KP tomatch the plasma density. It is however surprisingthat much of the time the swings are so great eventhough the average agrees well with the input KP .

3.2. Increasing the number of assimilation states

Next we ask the question whether increasing thenumber of parallel simulations (and thus the num-ber of parameter values) explore. In Figure 7 weplot the value of KP for the first 7 days of December2006, in the same format as earlier. In the interestof brevity we show only the plasma density datafor one elliptical satellite. The first and third panelare for 11 assimilation states, whereas the secondand fourth are for 31 assimilation states. It is clearfrom this that increasing the number of assimilationstates, which is equivalent to increasing the searchspace for best solutions, has the effect of increasingthe accuracy of the recovered plasma density andKP values.

3.3. Geostationary or elliptical satellites?

In Figure 8 we compare the effect of using only theelliptical orbit satellites and only the geostation-ary satellites to using all eight satellites. The toppanel is the result of using all eight satellites, thecenter using only the four elliptical orbit satellites,and the bottom panel the result of using the fourgeostationary satellites. It is clear that using onlythe four elliptical satellites results in a less accu-rate recovery of KP (and also of plasma densities,not shown). In this particular case it also appearsthat using only the geostationary satellites outper-forms using all eight satellites although that is lessclear. One possible explanation for the better per-formance of the geostationary satellites in this caseis the fact that KP is small and therefore the plas-masphere extends to near geostationary orbit andexhibits significant structure there from which thedata assimilation can be constrained. Under thosesame circumstances the elliptical orbit satellites will

spend less time near the plasmapause and thereforeprovide much less constraint on the plasma densitywhich is also seen from the less accurate recoveredKP values. The result which indicates that usingonly the geostationary satellites might produce abetter model recovery than using all eight satel-lites is curious and may be a function of the specificchoice of cost function.

4. Discussion

Overall the method succeeded in recovering theplasmasphere configuration and gross structure aswell as the KP index over most of the simulationinterval. This is despite the simplicity of the dataassimilation approach, and the fact that the modelstates available to the data assimilation did notmatch the states used to generate the input data.

It was also clear that increasing the number ofsimultaneous simulations, and thus the exhaustive-ness of the search, resulted in better recovery of theplasma density and the KP index values. This sug-gests that improving the data assimilation methodwill also result in improving the accuracy of theplasma density maps produced.

It is interesting to observe that despite the verywide swings observed in Figure 5 in hourly KP val-ues, the plasma density is well modeled, and the av-erage KP still agrees well with the input KP . Thiseffect is particularly pronounced when we only al-low the assimilation to use 11 states, whereas it isless pronounced when we allow the assimilation touse 31 states which more closely match the inputKP values, as is seen in Figure 7. That figure showsthat increasing the number of available assimilationstates reduces the swings in the recovered KP val-ues and also results in a more accurately recoveredplasma density. The reason for these wide swingsare likely that the assimilation alternates betweenmore eroding and less eroding plasmasphere mod-els in order to, on the average, approximate a statewhich is not available to the assimilation.

The poor agreement between input and recov-ered plasma density and KP values on December3-6 is somewhat puzzling. A likely reason for thisis a “unlucky” situation with a sub-optimal place-ment of the satellites combined with the selection ofthe wrong state from the small number available tothe assimilation, which then takes the state furtherfrom the true state to the point where the assimila-tion cannot easily recover since it is forced to evolveaccording to the physics imposed by the Ober et al.

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(1997) model. This can also be seen clearly in Fig-ure 5: in the December 3-6 interval there are no bluetraces which approach the red traces, thus indicat-ing that the assimilation has accidentally entereda state from which there is no simple path backto agreeing with the data. The simple assimilationscheme we used in this paper does not include provi-sions for the data to force the plasma density whenthe model deviates too far from the data. Such pro-vision are however contained in the Kalman filteringapproach in which uncertainties in both the modeland the data are accounted for. Of course this re-sults in a model which does not evolve exactly ac-cording to the physical laws of the model. Howevergiven that the model is almost always an approx-imation to reality and that the primary goal is toaccurately recover the configuration of the plasmas-phere this should not be seen as a great loss. It caneven be seen as an advantage in that data forcingof the model is an indicator of deficiencies in themodel which can then be analyzed with the intentto improve the model (e.g Koller et al., 2007).

More data sources are available than the ones wesimulated. Satellite measurements may not be themost powerful data sources for this work becauseof their single-point nature and the small num-ber available. Exception might be the Los AlamosNational Laboratory (LANL) geostationary satel-lites and their plasma instruments (Bame et al.,1993), as well as the GOES satellites. The LANLplasma density measurements may be used to iden-tify plasmapause crossings, and possibly the dis-tance to the plamasphere at other times. They mayalso be used to measure the magnetic field (Thom-sen et al., 1996) to supplement the GOES magneticfield measurements. Over the next several yearsthe Themis mission can also be expected to be aninteresting source of constraining data. In generalany constraining data source can be included whenit is available, and the decision as to its inclusionrests on the amount of effort involved in includ-ing it versus the benefits derived in constrainingthe model. Ground-based instruments are possi-bly more interesting sources of data because theyoften cover much larger regions of space than sin-gle satellites, because they are far less expensive toinstall and maintain, and because they usually per-sist for longer periods of time. These types of mea-surements include Field Line Resonance measure-ments (e.g. Schulz, 1996; Denton and Gallagher,2000), which are now routinely carried out on datafrom the SAMBA, MEASURE, and McMac mag-

netometer chains (Boudouridis and Zesta, 2007;Berube et al., 2005). Other information which canbe included in the assimilation include total elec-tron content measurements based on GPS measure-ments, whistler wave measurements, and improvedmodels of the global magnetic field and global elec-tric field, possibly also obtained using data assimi-lation approaches.

5. Conclusion

We have demonstrated a simple approach to theassimilative modeling of the Earth’s plasmasphere.Despite the simplicity of the data assimilation itperforms well. We also demonstrated and discussedhow the modeling scheme can be further improvedboth by improving the assimilation algorithm andby incorporating other data sources. Overall, thiswork demonstrates that assimilative modeling ofthe plasmasphere can be achieved with relativelysimple tools and data sources.

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