Specification of the Earth\'s Plasmasphere with Data Assimilation
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
Air Force Research Laboratory, Hanscom, MA, USA
J. Koller and R. H. W. Friedel
Los Alamos National Laboratory, Los Alamos, NM, USA
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.
derivatives of the model with respect to adjustable
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
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
14-16), and the period of variable intermediate KP (days
For each model
State Load state
Yes Create new
Figure 2: Orbital positions of the eight satellites at 6-hour Simulate data
intervals for the ﬁrst two days of December 2006. and compare
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
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
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.
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.
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.
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
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
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-
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-
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.
(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
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
most powerful data sources for this work because
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