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Specification of the Earth\'s Plasmasphere with Data Assimilation

Specification of the Earth\'s Plasmasphere with Data Assimilation






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    Specification of the Earth\'s Plasmasphere with Data Assimilation Specification of the Earth\'s Plasmasphere with Data Assimilation Document Transcript

    • 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 radiation belts, an accurate assimilative model of the plasmasphere is crucial for optimizing the accurate prediction of the radiation environments encountered by satellites. One longer time-scales the plasmasphere exhibits significant dynamics. Although these dynamics are modeled well by existing models, they require detailed global knowledge 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. Key words: derivatives of the model with respect to adjustable 1. Introduction parameters, such that for very large problems or Data assimilation techniques are widely used in complex non-linear models this became cumber- weather forecasting and that is perhaps the field some. Several alternatives, some based on statisti- in which they are most well know. However, data cal approximations, were developed. Among them assimilation techniques are used in one form or an- the Ensemble Kalman filter (Evensen, 2003) is now other in a wide variety of data estimation prob- widely used in weather prediction, and does not re- lems. Other examples include radar tracking prob- quire derivatives. Instead it requires the model to lems. Data assimilation works by merging, by any be run many times with different parameters in or- means, a model which is a physical description of der to sample parameter space statistically. a system with measurements which constrain the state or evolution of the system in some relevant In recent years Kalman filtering techniques have way. The free model parameters are then adjusted been applied to space physics space weather predic- to maximize the agreement between the model and tion problems, particular to the prediction of the ra- the measurements. diation belts, with good success (Koller and Fridel, One of the most effective data assimilation meth- 2005; Koller et al., 2007; Maget et al., 2007; Kon- ods is the Kalman filter (Kalman, 1960), with early drashov et al., 2007). These projects aim to provide applications to radar tracking problems. The origi- a complete specification of the radiation belts based nal approach developed by Kalman required all the on satellite measurements and a good but imperfect Preprint submitted to Advances in Space Research April 4, 2009
    • Model Interface Measurement simulation Plasmasphere model Model state Parameter Figure 1: KP for December 2006. This interval is used for control the simulations because of the very quiet period at the begin- ning of the interval (days 1-5), the very active period (days Time 14-16), and the period of variable intermediate KP (days stepping 6-13). For each model in ensemble State Load state Parameters Advance model Step Save state State No Data? Yes Create new ensemble Figure 2: Orbital positions of the eight satellites at 6-hour Simulate data Simulated measurements intervals for the first two days of December 2006. and compare Identify best model state physics-based model. The work by Koller uses the Rasmussen and Schunk (1990); Rasmussen et al. (1993) plasmasphere model, but does not close the Figure 3: Implementation of the simulation. We interfaced data assimilation loop around the model, relying in- the Ober et al. (1997) plasmasphere model (written in For- tran) with a C-language wrapper which provided access to stead on solar wind parameters to drive the model. reading and writing the plasma density and other model pa- The plasmasphere is a significant driving force on rameters. This model is then included in the data assimila- the radiation belts as it is the region which hosts the tion loop, computing the plasma density for multiple models waves responsible for acceleration and loss of radi- in parallel. ation belt particles (e.g. Friedel et al., 2002; Horne and Thorne, 1998). We use the Ober et al. (1997) plasmasphere model, In this paper we report on initial work to de- which is written in the Fortran language. We wrote velop a data assimilative approach to modeling the a C-language wrapper which allows us the neces- plasmasphere. We use the plasmasphere model by sary access to the model internals. This includes Ober et al. (1997) and a ensemble data assimilation the ability to to read and write the plasma density approach inspired by Ensemble Kalman filtering. map between the model and a storage array, the We use a real interval of KP to simulate the plas- ability to simulate satellite density measurements masphere and generate simulated data, which we from the plasma density map, and the ability to then input to the data assimilation method in an set the external parameter (in this case KP ) and attempt to recover the plasmasphere configuration run the model for a fixed time interval as a subrou- and the input KP . tine. This is illustrated in the left-hand portion of Figure 3. 2. Methodology The data assimilation approach which we use in- In this paper we employ a simpler data assimila- volves an ensemble of models similar to Ensemble tion approach than Ensemble Kalman filtering be- Kalman filtering. However, for the purpose of sim- cause of the ease with which it can be implemented. plicity we run each model in the ensemble at a fixed 2
    • tion times, marked by the dotted lines. In this case the assimilation interval is 1 hour. 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 satellite out of eight total. Throughout this paper we will use the 1-hour as- Figure 4: Demonstration of the data assimilation method for a short interval on December 2, 2006. similation interval, and run either 11 or 31 models in parallel, with KP values evenly distributed in the [0; 10] interval. These are not the same values KP . At each data assimilation time, where data as used to generate the simulation from which the and models are compared, the best model is se- input data are derived. Those follow the usual en- lected and its density map is copied to all the other coding of KP -values, 0, 0.3, 0.7, 1, 1.3, etc. running models. The assimilation procedure is thus as follows. 3. Simulation results Several models are run in parallel from the same ini- tial condition with different values of KP for a fixed We will report on four different simulations. The interval of time (In reality the different models are first simulation covers the first 16 days of December run serially on a single processor, and the plasma 2006 using 11 parallel models and all eight satel- density maps and model parameters are copied in lites. In the second simulation we use 31 models and out of the model for each). At the end of the in- and all eight satellites, whereas in the last two sim- terval satellite density measurements are simulated ulation we use 11 models and either the elliptical from each model and compared with the input data. or geostationary orbit satellites. The cost function for this comparison is the sum of squares of fractional errors. The plasma density 3.1. 16-day simulation from the best model is then copied to each of the The results of this assimilation run are shown in running models, and the models are then run again Figures 5 and 6. Figure 5 shows the input plasma for another fixed interval of time. This is illustrated density actually measured by the satellites (in red) in the right-hand section of Figure 3. as well as the plasma density simulated by each of In this paper we work with simulated data, which the parallel model runs (in blue). The green curve are generated from a period of real KP values in or- shows the plasma for the model which fit the data der to have realistic plasma density variations. We best at the hourly assimilation times. This is the use the month of December 2006, whose KP values same formatting scheme as is shown in figure 4. The are plotted in Figure 1. We simulate data for eight top four panels shows the elliptical satellites, the satellite orbits, including four elliptical orbit satel- next four panels the geostationary satellites. The lites and four geostationary satellites space evenly last panel shows KP , which is the only free param- in local time as show in Figure 2. We call these eter in the assimilation. The red curve shows the simulated data the input data. We do not sim- KP used to generate the input data. The green ulate noise or any systematic effects on the data, curve shows the KP of the best model for each but those are factors which must be considered in hour interval, and the blue curve shows a double the future. 5-hour smooth of the green curve (double in order Figure 4 shows a close look at several consecutive to produce a continuous second derivative and a assimilation steps for a short interval of four hours nicer look). with data assimilation taking place every hour. In Figure 6 shows the density maps at 8-hour in- the figure the red curve is the input data, the blue tervals beginning at 8 UT on December 1, 2006. curves, the blue curves each of the models run with The layout of those images is explained in the Fig- different values of KP (in this case the 11 values ure caption: every two rows belong together, with from 0 to 10), and the green curve represents the the upper row showing the recovered plasma map, best model as determined by best agreement be- and the lower row showing the input plasma density tween the model and all satellites at the assimila- maps. 3
    • Figure 5: Data assimilation on the plasmasphere for the first 16 days of December 2006. The top four panels of each plot are the plasma density measurements by the elliptical orbit satellites. The next four panels of each plot are the plasma density measurements by the four geostationary orbit satellites. The bottom panel in each plot is the KP index. In the top eight panels of each plot the red curve is the input plasma density data, the blue curves are the plasma density measurements corresponding to each of the 11 parallel models, and the green curve is the plasma density corresponding to the best model selected for each 1-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 a 5-hour boxcar window. 4
    • Figure 6: Images of plasma density as a function of time. In order to maximize the size of the images the scales have been left out. 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 color which represents plasma density increases from blue through green to yellow. Images are shown for every 8 hours beginning at 8 UT on December 1, and both the recovered image and the input image are shown in alternating rows with the recovered image 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- the best assimilated plasma density measurements. tween the input plasma density measurements and The first two days (December 1-2), and last three 5
    • Figure 7: Comparison of the effect of increasing the number parallel assimilation models, modeling the first four days of December 2006. The top two panels show plasma density measured by one satellite in the case 11 assimilation states (top panel) and 31 assimilation states (second panel). The corresponding KP values are shown in the third and fourth panels respectively. Figure 8: Comparison of the effect of using all eight satellites (top panel) versus using only the elliptical orbit satellites (center panel) and only the geostationary orbit satellites (bottom panel). days (December 14-17) of the simulation fit par- than prescribed in the input data. ticularly well, with even quite complex structure In the images (Figure 6) the situation is again between 14 UT and 20 UT on December 2 being similar. On December 1 and 2 (First 6 columns reproduced well. By contrast the interval from De- of images of top two rows) the agreement between cember 3 through December 6 and even part of De- the recovered plasma density (top row) and the in- cember 7 is not very well reproduced at all. The rest put density (second row) is excellent, with even of the time the plasma density is reproduced quite fine plume structure being reproduced well. On well although there is a slight tendency for the as- December 3, 4, 5, and 6, the recovered and in- similation step to place the plasmapause closer to put plasma density are wildly different, with the the Earth than indicated by the input data. recovered showing a much smaller plasmapause in In the KP data (the last panels in Figure 5) we agreement with the satellite density measurements see a similar pattern. Excellent agreement between in Figure 5, and in agreement with the larger KP input and recovered KP on December 1 and 2, 14, that was selected during that time interval. On 15, and 16, very poor agreement on December 3 December 7, the recovered and input plasma den- through 6 or 7, and good agreement the rest of the sity maps begin to look more similar, and generally time, with a slight tendency towards a higher KP agree well after that time except for a few differ- 6
    • ences, for example 0 UT on December 12. spend less time near the plasmapause and therefore provide much less constraint on the plasma density It is also interesting to note that during most which is also seen from the less accurate recovered of the simulation the recovered value of KP (green KP values. The result which indicates that using curve in the last panels of Figure 5), varies widely only the geostationary satellites might produce a from hour to hour while its average (blue curve) better model recovery than using all eight satel- is in much better agreement with the input KP lites is curious and may be a function of the specific value. A likely explanation for this is that because choice of cost function. none of KP values available to the assimilation ex- actly match the input KP , the assimilation com- pensates for this by selecting KP values which are 4. Discussion greater than and smaller than the the input KP to Overall the method succeeded in recovering the match the plasma density. It is however surprising plasmasphere configuration and gross structure as that much of the time the swings are so great even well as the KP index over most of the simulation though the average agrees well with the input KP . interval. This is despite the simplicity of the data assimilation approach, and the fact that the model 3.2. Increasing the number of assimilation states states available to the data assimilation did not Next we ask the question whether increasing the match the states used to generate the input data. number of parallel simulations (and thus the num- It was also clear that increasing the number of ber of parameter values) explore. In Figure 7 we simultaneous simulations, and thus the exhaustive- plot the value of KP for the first 7 days of December ness of the search, resulted in better recovery of the 2006, in the same format as earlier. In the interest plasma density and the KP index values. This sug- of brevity we show only the plasma density data gests that improving the data assimilation method for one elliptical satellite. The first and third panel will also result in improving the accuracy of the are for 11 assimilation states, whereas the second plasma density maps produced. and fourth are for 31 assimilation states. It is clear It is interesting to observe that despite the very from this that increasing the number of assimilation wide swings observed in Figure 5 in hourly KP val- states, which is equivalent to increasing the search ues, the plasma density is well modeled, and the av- space for best solutions, has the effect of increasing erage KP still agrees well with the input KP . This the accuracy of the recovered plasma density and effect is particularly pronounced when we only al- KP values. low the assimilation to use 11 states, whereas it is less pronounced when we allow the assimilation to 3.3. Geostationary or elliptical satellites? use 31 states which more closely match the input In Figure 8 we compare the effect of using only the KP values, as is seen in Figure 7. That figure shows elliptical orbit satellites and only the geostation- that increasing the number of available assimilation ary satellites to using all eight satellites. The top states reduces the swings in the recovered KP val- panel is the result of using all eight satellites, the ues and also results in a more accurately recovered center using only the four elliptical orbit satellites, plasma density. The reason for these wide swings and the bottom panel the result of using the four are likely that the assimilation alternates between geostationary satellites. It is clear that using only more eroding and less eroding plasmasphere mod- the four elliptical satellites results in a less accu- els in order to, on the average, approximate a state rate recovery of KP (and also of plasma densities, which is not available to the assimilation. not shown). In this particular case it also appears The poor agreement between input and recov- that using only the geostationary satellites outper- ered plasma density and KP values on December forms using all eight satellites although that is less 3-6 is somewhat puzzling. A likely reason for this clear. One possible explanation for the better per- is a “unlucky” situation with a sub-optimal place- formance of the geostationary satellites in this case ment of the satellites combined with the selection of is the fact that KP is small and therefore the plas- the wrong state from the small number available to masphere extends to near geostationary orbit and the assimilation, which then takes the state further exhibits significant structure there from which the from the true state to the point where the assimila- data assimilation can be constrained. Under those tion cannot easily recover since it is forced to evolve same circumstances the elliptical orbit satellites will according to the physics imposed by the Ober et al. 7
    • (1997) model. This can also be seen clearly in Fig- netometer chains (Boudouridis and Zesta, 2007; ure 5: in the December 3-6 interval there are no blue Berube et al., 2005). Other information which can traces which approach the red traces, thus indicat- be included in the assimilation include total elec- ing that the assimilation has accidentally entered tron content measurements based on GPS measure- a state from which there is no simple path back ments, whistler wave measurements, and improved to agreeing with the data. The simple assimilation models of the global magnetic field and global elec- scheme we used in this paper does not include provi- tric field, possibly also obtained using data assimi- sions for the data to force the plasma density when lation approaches. the model deviates too far from the data. Such pro- vision are however contained in the Kalman filtering 5. Conclusion approach in which uncertainties in both the model and the data are accounted for. Of course this re- We have demonstrated a simple approach to the sults in a model which does not evolve exactly ac- assimilative modeling of the Earth’s plasmasphere. cording to the physical laws of the model. However Despite the simplicity of the data assimilation it given that the model is almost always an approx- performs well. We also demonstrated and discussed imation to reality and that the primary goal is to how the modeling scheme can be further improved accurately recover the configuration of the plasmas- both by improving the assimilation algorithm and phere this should not be seen as a great loss. It can by incorporating other data sources. Overall, this even be seen as an advantage in that data forcing work demonstrates that assimilative modeling of of the model is an indicator of deficiencies in the the plasmasphere can be achieved with relatively model which can then be analyzed with the intent simple tools and data sources. to improve the model (e.g Koller et al., 2007). More data sources are available than the ones we simulated. Satellite measurements may not be the References most powerful data sources for this work because of their single-point nature and the small num- Bame, S. J., McComas, D. J., Thomsen, M. F., Barraclough, ber available. Exception might be the Los Alamos B. L., Elphic, R. C., Glore, J. P., Gosling, J. T., Chavez, National Laboratory (LANL) geostationary satel- J. C., Evans, E. P., Wymer, F. J., 1993. Magnetospheric lites and their plasma instruments (Bame et al., plasma analyzer for spacecraft with constrained resources. Rev. Sci. Instr. 64, 1026. 1993), as well as the GOES satellites. 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