real-time thermospheric density estimation via two-line...
TRANSCRIPT
manuscript submitted to Space Weather
Real-Time Thermospheric Density Estimation Via1
Two-Line-Element Data Assimilation2
David J. Gondelach1, and Richard Linares13
1Department of Aeronautics and Astronautics, Massachusetts Institute of Technology, Cambridge, MA,4
USA5
Key Points:6
• Thermospheric density is estimated using a dynamic reduced-order thermosphere7
model and two-line element data.8
• The two-line element data is assimilated using a Kalman filter to provide both den-9
sity and uncertainty estimates.10
• The estimated densities are validated against CHAMP and GRACE accelerometer-11
derived density data.12
Corresponding author: David Gondelach, http://orcid.org/0000-0002-8511-9523,
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manuscript submitted to Space Weather
Abstract13
Inaccurate estimates of the thermospheric density are a major source of error in low Earth14
orbit prediction. Therefore, real-time density estimation is required to improve orbit pre-15
diction. In this work, we develop a dynamic reduced-order model for the thermospheric16
density that enables real-time density estimation using two-line element (TLE) data. For17
this, the global thermospheric density is represented by the main spatial modes of the18
atmosphere and a time-varying low-dimensional state and a linear model is derived for19
the dynamics. Three different models are developed based on density data from the TIE-20
GCM, NRLMSISE-00 and JB2008 thermosphere models and are valid from 100 to max-21
imum 800 km altitude. Using the models and TLE data the global density is estimated22
by simultaneously estimating the density and the orbits and ballistic coefficients of sev-23
eral objects using a Kalman filter. The sequential estimation provides both estimates24
of the density and corresponding uncertainty. Accurate density estimation using the TLEs25
of 17 objects is demonstrated and validated against CHAMP and GRACE accelerometer-26
derived densities. The estimated densities are shown to be significantly more accurate27
and less biased than NRLMSISE-00 and JB2008 modelled densities. The uncertainty in28
the density estimates is quantified and shown to be dependent on the geographical lo-29
cation, solar activity and objects used for estimation. In addition, the data assimilation30
capability of the model is highlighted by assimilating CHAMP accelerometer-derived den-31
sity data together with TLE data to obtain more accurate global density estimates. Fi-32
nally, the dynamic thermosphere model is used to forecast the density.33
Plain Language Summary34
The trajectory of satellites that orbit the Earth at low altitudes (below 1000 km)35
is affected by drag caused by the Earth’s upper atmosphere. To accurately compute the36
effect of the drag on the orbit, the mass density of the atmosphere needs to be know. This37
density can however change quickly due to changing solar activity. To model the changes38
in the upper atmosphere, we developed a computationally-inexpensive numerical model.39
Using the model and observations of satellite orbits, we derive information about the at-40
mospheric density. These estimated densities can be used for improving atmospheric and41
orbit predictions.42
1 Introduction43
Accurate knowledge of the thermospheric density is essential for orbit prediction44
in low Earth orbit and in particular for conjunction assessments. The models of the ther-45
mosphere with most potential for good forecast capabilities, in particular during storm46
conditions, are physics-based models (Sutton, 2018), such as the Global Ionosphere-Thermosphere47
Model (GITM) (Ridley et al., 2006) and the Thermosphere-Ionosphere-Electrodynamics48
General Circulation Model (TIE-GCM) (Qian et al., 2014). These models solve the con-49
tinuity, momentum and energy equations for a number of neutral and charged compo-50
nents. Their modeling and prediction performance comes, however, at a high computa-51
tional cost. The models are very high-dimensional, solving Navier-Stokes equations over52
a discretized spatial grid involving 104-106 state variables and 12-20 inputs and inter-53
nal parameters. In addition, to fully exploit the forecasting potential of physics-based54
models the schemes employed for data assimilation need to be improved (Sutton, 2018).55
Empirical models (Jacchia, 1970; Hedin, 1987; Picone et al., 2002; Bowman et al.,56
2008; Bruinsma, 2015), on the other hand, capture the average behaviour of the atmo-57
sphere using low-order, parameterized mathematical formulations based on historical ob-58
servations. A major advantage of empirical models is that they are fast to evaluate, mak-59
ing them ideal for drag and orbit computations. The accuracy of these empirical mod-60
els is however limited (He et al., 2018), especially during space weather events. Improved61
densities can be obtained by calibrating empirical density models using satellite data.62
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The current Air Force standard is the High Accuracy Satellite Drag Model (HASDM)63
(Storz et al., 2005), which is an empirical model that is calibrated using observations of64
calibration satellites. These satellite observations are used to determine atmospheric model65
parameters based on their orbit determination solutions. Due to the lack of access to space66
surveillance observations, publicly available two-line element (TLE) data have been used67
in the past to estimate the thermospheric density (Picone et al., 2005; Emmert et al.,68
2006) and calibrate empirical models (Cefola et al., 2004; Yurasov et al., 2005; Doorn-69
bos et al., 2008; Chen et al., 2019). Cefola et al. (2004) and Yurasov et al. (2005) cal-70
ibrated the GOST and NRLMSISE-00 density models using two scaling parameters based71
on the fitted ballistic coefficient (BC) values. Doornbos et al. (2008) estimated spher-72
ical harmonics coefficients to calibrate NRLMSISE-00 model using TLE-derived density73
estimates and Sang et al. (2011) and Chen et al. (2019) adjusted the 187 coefficients of74
the DTM87 density model directly during orbit determination of multiple objects. On75
the other hand, Crowley and Pilinski (2017) used TLE data for data assimilation in the76
physics-based Dragster model using Ensemble Kalman filtering. Except for the Drag-77
ster model, the calibrated empirical models have limited forecasting capability which re-78
duces their effectiveness for orbit prediction.79
Recently, a new methodology for modelling and estimating the thermosphere us-80
ing reduced-order modeling was developed by Mehta and Linares (2017) to overcome the81
high-dimensionality problem of physics-based models. The technique combines the pre-82
dictive abilities of physics-based models with the computational speed of empirical mod-83
els by developing a Reduced-Order Model (ROM) that represents the original high-dimensional84
system using a smaller number of parameters. The order-reduction is achieved using proper85
orthogonal decomposition (POD) (Golub & Reinsch, 1970; Rowley et al., 2004), also known86
as principal component analysis (PCA) (Jolliffe, 2011), empirical orthogonal functions87
(EOF) or Karhunen-Loeve expansion (Loeve, 1977). The POD approach uses singular88
value decomposition (SVD) to compute spatial modes that are orthogonal with respect89
to each other. Using the POD modes, a ROM can be constructed from experimental or90
numerical data. In addition, POD modes have been used by Matsuo and Forbes (2010)91
and Sutton et al. (2012) to study and model variations of the thermospheric density. To92
model the dynamic thermosphere, a dynamic ROM was developed by Mehta et al. (2018)93
by determining the best fit linear dynamical system from density data using the recently94
developed Dynamic Mode Decomposition (DMD) technique (Schmid, 2010). The DMD95
approach assumes a linear system xk+1 = Axk, where xk is the kth data sampled from96
a sequential dataset and A is the unknown system matrix. In particular, Dynamic Mode97
Decomposition with control (DMDc) (Proctor et al., 2016) can include the effect of con-98
trol and extends the DMD approach to systems with the form xk+1 = Axk+Buk where99
uk is the system input. The DMDc method was used by Mehta et al. (2018) to develop100
a quasi-physical dynamic ROM for the thermosphere. The application of the ROM ap-101
proach for atmospheric density estimation was demonstrated by data assimilation of ac-102
celerometer derived mass density (Mehta & Linares, 2018b) and simulated GPS mea-103
surements (Mehta & Linares, 2018a) using Kalman filters. This technique enables both104
the accurate estimation of thermospheric density and forecasting of the future density105
through the ROM dynamic model.106
The benefits of this approach are that: 1) the modes provide an optimal low-order107
representation and therefore capture most variance of the system; 2) the global density108
and uncertainty can be estimated in real-time thanks to the dynamic model, which is109
not possible with a static model; 3) the dynamic model can be used for forecasting (for110
static models only the fitted coefficients can be extrapolated in time).111
In this work, the reduced-order modelling technique for density estimation is fur-112
ther developed and TLE data is used to estimate the thermospheric density. The avail-113
ability of TLE data for thousands of objects make them attractive for density estima-114
tion; however, the use of TLEs is challenging due to the limited accuracy of the orbital115
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data. The density estimation using TLE data is achieved by simultaneously estimating116
the orbits and BCs of several objects and the reduced-order density state using an un-117
scented Kalman filter. The main contributions of the paper are:118
1. Nonlinear space weather inputs are introduced to improve ROM prediction.119
2. Two new ROM models based on the NRLMSISE-00 and JB2008 models are de-120
veloped to extend the maximum altitude to 800 km.121
3. Modified equinoctial elements are employed to express the orbit and measurements122
to retain “Gaussianity” of the state probability density function.123
4. Accurate thermospheric density estimates are computed by assimilating TLE data124
in reduced-order density models.125
5. Accurate density estimation over extended periods of time using a limited amount126
of TLE data during both high and low solar activity is demonstrated.127
6. The estimated densities are validated against CHAMP and GRACE accelerometer-128
derived density data.129
7. Improved global density estimates are obtained by assimilating CHAMP accelerometer-130
derived density data together with TLE data.131
The paper is structured as follows. First the development of a dynamic reduced-132
order density model described. After that, the estimation of the density via TLE data133
assimilation using an unscented Kalman is discussed. Then the performance of the ROM134
density estimation is assessed using simulated and real TLE data, and the uncertainty135
quantification and prediction capability of the model are demonstrated. Finally, the re-136
sults are discussed and conclusions are drawn.137
2 Methodology138
The neutral density estimation approach consists of two main components: 1) the139
development of a dynamic reduced-order model (ROM) for the thermosphere and 2) the140
calibration of the ROM through assimilation of TLE data.141
2.1 Reduced-order modeling142
The main idea of reduced-order modeling is to reduce the dimensionality of the state143
space while retaining maximum information. In our case, the full state space consists144
of the neutral mass density values on a dense uniform grid in latitude, local solar time145
and altitude. The goal is to develop a model for the density evolution over time. First,146
to make the problem tractable, the state space dimension is reduced using POD. Sec-147
ond, a linear dynamic model is derived by applying DMDc.148
2.1.1 Proper orthogonal decomposition149
The concept of order reduction using POD is to project the high-dimensional sys-150
tem and its solution onto a set of low-dimensional basis functions or spatial modes, while151
capturing the dominant characteristics of the system. Consider the variation x of the152
neutral mass density x with respect to the mean value x:153
x(s, t) = x(s, t)− x(s) (1)154
where s is the spatial grid. A significant fraction of the variance x can be captured by155
the first r principal spatial modes:156
x(s, t) ≈r∑
i=1
ci(t)Φi(s) (2)157
where Φi are the spatial modes and ci are the corresponding time-dependent coefficients.158
The spatial modes Φ are computed using a SVD of the snapshot matrix X that contains159
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x for different times:160
X =
| | |x1 x2 · · · xm
| | |
= UΣV> (3)161
where m is the number of snapshots. The spatial modes Φ are given by the left singu-162
lar vectors (the columns of U). The state reduction is achieved using a similarity trans-163
form:164
z = U−1r x = U>r x (4)165
where Ur is a matrix with the first r POD modes and z is our reduced-order state. Pro-166
jecting z back to the full space gives approximately x that allows us to compute the den-167
sity:168
x(s, t) ≈ Ur(s) z(t) + x(s) (5)169
More details on POD can be found in Mehta and Linares (2017).170
2.1.2 Dynamic Mode Decomposition with control171
To enable prediction of the atmospheric density, we develop a linear dynamic model172
for the reduced-order state z. First, let’s consider the full-dimensional case. Since the173
atmosphere is highly sensitive to the solar activity, we derive a linear system that con-174
siders exogenous inputs:175
xk+1 = Axk + Buk (6)176
where uk is the system input, which in our case are the space weather inputs. The dy-177
namic matrix A and input matrix B can be estimated from output data using the DMDc178
algorithm. For this, the outputs of the dynamical system (6) or snapshots, xk, are re-179
arranged into time-shifted data matrices. Let X1 and X2 be the time-shifted matrix of180
snapshots such that:181
X1 =
| | |x1 x2 · · · xm−1| | |
, X2 =
| | |x2 x3 · · · xm
| | |
, Υ =
| | |u1 u2 · · · um−1| | |
(7)182
where m is the number of snapshots and Υ contains the corresponding inputs. The data183
matrices X1 and X2 are related (X2 is the time evolution of X1) through the model in184
Eq. (6) such that:185
X2 = AX1 + BΥ (8)186
The goal now is to estimate A and B. However, because X1 and X2 can be extremely187
large, it is more efficient to perform the DMDc in the reduced-order space:188
Z2 = ArZ1 + BrΥ (9)189
where Z1 = U>r X1 and Z2 = U>r X2 are the reduced-order snapshot matrices, and190
Ar = U>r AUr and Br = U>r B are the reduced-order dynamic and input matrices.191
The above equation is modified such that:192
Z2 = ΞΨ (10)193
where Ξ and Ψ are the augmented operator and data matrices, respectively:194
Ξ ,[Ar Br
]and Ψ ,
[Z1
Υ
](11)195
We now estimate the dynamic and input matrices by minimizing ‖Z2−ΞΨ‖. The aug-196
mented operator matrix is then solved for by computing the pseudoinverse of Ψ:197
Ξ = Z2Ψ+ (12)198
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where the + subscript indicates the pseudoinverse. In MATLAB, this can be easily com-199
puted using the backslash operator: Z2Ψ+ =
((Ψ+)>Z>2
)>=(Ψ>\Z>2
)>. Calculat-200
ing Ar and Br without first computing A and B is an improvement with respect to pre-201
vious work (Mehta et al., 2018) and is also numerically more stable because the num-202
ber of entries that needs to be determined for matrix Ar is much smaller than for A.203
The matrices Ar and Br are discrete-time matrices for the discrete reduced-order204
dynamical system:205
zk+1 = Arzk + Bruk (13)206
that corresponds to the fixed timestep T used for the snapshots. However, for estima-207
tion we require continuous information about the density and therefore need a contin-208
uous dynamical model:209
z = Acz + Bcu (14)210
where Ac and Bc are the continuous-time dynamic and input matrices, respectively. The211
continuous-time matrices are obtained by converting the discrete-time matrices using the212
following relation (DeCarlo, 1989):213 [Ac Bc
0 0
]= log
([Ar Br
0 I
])/T (15)214
where T is the sample time, i.e. the snapshot resolution.215
Using a linear system to approximate the nonlinear thermosphere dynamics results216
in approximation errors. The linear approximation is valid due to the small time step217
used for the DMDc algorithm. In addition, the same small time step (one hour) is used218
during estimation such that the state is corrected hourly by the TLE measurements and219
the error remains small. We refer the reader to the original papers by Mehta and Linares220
(2017) and Mehta et al. (2018) for an analysis on the accuracy and validity of the lin-221
ear model. Furthermore, with respect to Mehta et al. (2018), we have improved the pre-222
diction performance of the linear model by including nonlinear space weather inputs, see223
Sections 2.1.3 and 3.1.224
2.1.3 ROM density model development225
Density training data In this work, we have developed three different ROM den-226
sity models using three different atmospheric models to obtain the snapshot matrices,227
namely the empirical NRLMSISE-00 (Picone et al., 2002) and Jacchia-Bowman 2008 (JB2008)228
models (Bowman et al., 2008) and the physics-based TIE-GCM model (Qian et al., 2014).229
We first defined a spatial grid s in local solar time, geographic latitude and altitude (we230
use local solar time instead of longitude as azimuthal coordinate, because the diurnal bulge231
is stationary in solar local time) and computed the density on this grid for every hour232
over 12 years (one solar cycle), resulting in over 105,000 snapshots. These snapshots were233
then used to compute a dynamic ROM model, as described in the previous section, us-234
ing a reduced order of r = 10. It should be noted that we computed the variation of235
the density x by first taking the log base 10 of the density and then subtracting the mean:236
x = log10 x − log10 x, where x and x are the density and mean density on the spatial237
grid.238
Details on the spatial grid and 12 year periods applied for generating the ROM mod-239
els can be found in Table 1. Note that the JB2008 ROM model was computed over the240
years 1999-2010 instead of 1997-2008, because no continuous space weather data was avail-241
able in the year 1998. The TIE-GCM density data was obtained via simulation using242
the TIE-GCM model. The TIE-GCM model has a different grid partitioning than used243
for the ROM model. Therefore, we interpolated the output of the TIE-GCM simulation244
data to obtain the partitioning used in this work, see Table 1. In addition, the upper bound-245
ary of the TIE-GCM model can vary from ∼450 to 700 depending on solar activity, be-246
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cause it uses a log-pressure scale for the vertical coordinate. Therefore, over a full so-247
lar cycle, the maximum altitude for TIE-GCM-based ROM model is limited to 450 km.248
For the lower boundary of the model around 97 km, the Global Scale Wave Model was249
used to specify migrating diurnal and semidiurnal tidal fields (Hagan et al., 2001) with250
eddy diffusivity according to (Qian et al., 2009). More details on the TIE-GCM simu-251
lation settings and inputs can be found in Mehta et al. (2018). An improvement with252
respect to previous work is the development of ROM models that are valid above 450253
km altitude, which is the limiting altitude for TIE-GCM. The new ROM models based254
on NRLMSISE-00 and JB2008 extend up to 800 km altitude, see Table 1.255
Table 1. ROM characteristics: spatial grid and time period.
Base model Local solar time [hr] Latitude [deg] Altitude [km] YearsDomain Resolution Domain Resolution Domain Resolution
TIEGCM [0, 24] 1 [-87.5, 87.5] 9.2 [100, 450] 17.5 1997-2008NRLMSISE-00 [0, 24] 1.04 [-87.5, 87.5] 9.2 [100, 800] 20 1997-2008JB2008 [0, 24] 1.04 [-87.5, 87.5] 9.2 [100, 800] 20 1999-2010
Space weather inputs The space weather inputs uk used in the dynamical model256
are taken from the inputs required by the original density models, see second column in257
Table 2. In addition to these default inputs, we added the next-hour values for key space258
weather indices to improve the DMDc prediction, see third column in Table 2. Finally,259
a new innovation in this work is the addition of nonlinear space weather terms, such as260
the square of an index, e.g. ap2, or the multiplication of two different indices, e.g. ap·261
F10.7, see nonlinear inputs in Table 2. The improvement of the DMDc model due to adding262
nonlinear terms will be discussed in the results section.263
Table 2. ROM space weather inputs: doy=day of year, hr=hour in UTC, GMST=Greenwich
mean sidereal time, overbars indicate the 81-day average.
Base model Standard inputs Future inputs Nonlinear inputs
TIE-GCMa doy, hr, F10.7, F10.7,Kp F10.7,Kp Kp2,Kp · F10.7
NRLMSISE-00b doy, hr, F10.7, F10.7,ap† F10.7, F10.7,ap
† ap2, apnow · F10.7
JB2008c doy, hr, F10.7, F10.7, S10, S10,M10, M10, F10.7, S10,M10, DSTDTC2,Y10, Y10, DSTDTC,GMST, αSUN , δSUN Y10, DSTDTC DSTDTC · F10.7
† ap indices for the NRLMSISE-00 model consist of 8 ap values for up to 57 hours prior to current timea see Qian et al. (2014)b see Picone et al. (2002)c see Bowman et al. (2008)
2.2 Density estimation264
The neutral mass density is estimated through the assimilation of two-line element265
orbital data in the dynamic ROM model. This is achieved by simultaneously estimat-266
ing the ROM state and the orbit and BC of objects using an unscented Kalman filter.267
2.2.1 Two-line element data268
The US Air Force Space Command publicly distributes the orbital data of thou-269
sands of Earth-orbiting objects in the form of two-line element sets. From this TLE data,270
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2.458131 2.458132 2.458133 2.458134 2.458135 2.458136 2.458137 2.458138 2.458139
Julian date 106
-2
-1
0
1
2
Positio
n e
rror
[km
] Radial Along-track Cross-track TLE
2.458131 2.458132 2.458133 2.458134 2.458135 2.458136 2.458137 2.458138 2.458139
Julian date 106
-4
-2
0
2
Positio
n e
rror
[km
]
Radial Along-track Cross-track TLE
Figure 1. TLE position error with respect to GPS data for a satellite at 494 km altitude
using a single TLE (top) or multiple TLEs (bottom) in 8-day window in Jan 2018.
the state of an object (position and velocity) at any epoch can be extracted using the271
SGP4/SDP4 models (Hoots & Roehrich, 1980; Vallado et al., 2006). Hence, the effect272
of drag can be observed in TLE orbital data if the drag perturbation is strong enough.273
A general concern when using TLE data is the accuracy of the orbital data. In the274
SGP4/SDP4 models only the largest perturbations are included, while higher-order and275
short-periodic effects, a dynamic atmosphere and orbital maneuvers are not accounted276
for (Vallado & Cefola, 2012). As a result, there can be large errors in the orbital data277
(Kelso, 2007). In addition, since 2013, TLEs are fitted to a higher-order orbital solution278
that includes a future orbit prediction (Hejduk et al., 2013). This generally improve the279
TLE accuracy at epoch, but may deteriorate the quality if inaccurate future space weather280
is used for the orbit prediction. To gain understanding about errors in TLE data, we com-281
pared the position according to TLE data against GPS data, see Figure 1. For this, we282
used the GPS data of a Planet Labs satellite at 494 km altitude (more information on283
the Planet Labs ephemeris can be found in Foster et al. (2015) and http://ephemerides284
.planet-labs.com). The position error is largest in the along-track direction and varies285
with a 12-hour period, which is thought to be due to missing tesseral m-daily terms in286
the SGP4 theory (Herriges, 1988). From the figure, it is clear that we can expect sig-287
nificant errors in the orbital data. On the other hand, the errors are expected to be lim-288
ited when computing the state at epochs prior to the TLE epoch (see top plot in Fig-289
ure 1), which corresponds with the period of tracking data used to generate the TLE.290
The objects used for density estimation need to be selected carefully. First of all,291
the objects preferably have a strong drag signal. In particular, the effect of drag should292
be strong with respect to other non-conservative force effects, else errors in non-conservative293
force modelling, such as solar radiation pressure, can result in inaccurate density esti-294
mates. Second, it is important to have an accurate estimate of the true BC of the ob-295
ject to minimize errors due to inaccuracies in the BC. Therefore, the variation of the BC296
over time should preferably be very small or else must be modelled accurately (Bowman,297
2002). An overview of the objects used in this work is shown in Table 3. The BC val-298
ues were taken from Bowman et al. (2004), Emmert et al. (2006) and Lu and Hu (2017).299
A 1-σ uncertainty of 1% was assumed for the BC values. This is smaller than the 5-10%300
error often assumed for the drag coefficient (Emmert et al., 2006), but close to the 2-3%301
accuracy found for 30-year averaged BC estimates (Bowman et al., 2004). Finally, any302
orbit maneuvers and significant outliers in the TLE data must be detected and excluded303
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Table 3. Objects used for density estimation from 2002 to 2008. The BC values were taken
from Bowman et al. (2004), Emmert et al. (2006) and Lu and Hu (2017).
NORAD Object BC Perigee Apogee InclinationCatalog ID [m2/kg] height [km] height [km] [deg]
63 Tiros 2 0.01486 509 - 460 555 - 493 48.5165 DELTA 1 R/B 0.05326 598 - 532 620 - 550 47.9614 Hitchhiker 1 0.01463 319 - 312 2061 - 1707 82.02153 THOR AGENA B 0.03329 501 - 497 2636 - 2598 79.72622 OV1-9 R/B 0.02240 475 - 471 4500 - 4468 99.14221 Azur 0.02201 368 - 362 1817 - 1586 102.76073 Cosmos 482 Debris 0.00378 213 - 206 4985 - 3984 52.17337 Vektor 0.01120 380 - 376 1392 - 1285 83.08744 Vektor 0.01117 380 - 372 1429 - 1328 82.912138 Vektor 0.01115 394 - 390 1596 - 1519 83.012388 Vektor 0.01121 384 - 379 1555 - 1460 83.014483 Vektor 0.01130 390 - 385 1658 - 1581 82.920774 Vektor 0.01168 391 - 387 1764 - 1684 83.023278 Vektor 0.01168 398 - 394 1851 - 1783 83.026405 CHAMP 0.00477† 400 - 314 434 - 318 87.227391 GRACE 1 0.00697 480 - 447 505 - 470 89.027392 GRACE 2 0.00693 480 - 447 506 - 470 89.0
† Average of values reported by Emmert et al. (2006) and Lu and Hu (2017)
from the data before estimation. The TLE time series used in this work do not contain304
any orbit maneuvers and contain only few outliers. These outliers are small and were305
found to have little effect on the density estimation results; therefore, the unfiltered TLE306
data was used for estimation (the largest outlier was found in the TLE data of object307
2153 causing a several-km deviation in the orbit for a couple of days; removing the out-308
liers can result in larger orbit errors due to a gap in the TLE time series).309
2.2.2 Unscented Kalman filter310
To fuse the model and TLE data, we use the square-root unscented Kalman filter311
(UKF). The UKF uses an unscented transform (UT) to avoid large errors in the true pos-312
terior mean and covariance of the variables. Here, the true posterior mean and covari-313
ance are computed to the 3rd order by propagating a carefully selected set of sample points,314
called sigma points, through the true nonlinear dynamics. The UKF is a popular algo-315
rithm that is well documented in literature; therefore, for details about the square-root316
UKF the reader is referred to Wan and Van Der Merwe (2001). The state, dynamics,317
measurements and noise used to estimate the density with the UKF are described in the318
following.319
2.2.2.1 State The state x that is estimated in the UKF consists of the osculat-320
ing orbital states (expressed in modified equinoctial elements) and the BCs of the ob-321
jects plus the reduced-order density state z:322
x =[p1, f1, g1, h1, k1, L1, BC1, ... , pn, fn, gn, hn, kn, Ln, BCn, z>
]>(16)323
where n is the number of objects and the modified equinoctial elements (MEE) are de-324
fined as (Walker et al., 1985):325
p = a(1− e2), f = e cos (ω + Ω), g = e sin (ω + Ω),h = tan (i/2) cos Ω, k = tan (i/2) sin Ω, L = Ω + ω + ν.
(17)326
where a, e, i,Ω, ω and ν are the classical Keplerian orbital elements. The MEE are non-327
singular and tend to behave less nonlinear than the Cartesian coordinates (used in pre-328
vious work), which benefits the Kalman filter estimation by mitigating departure from329
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“Gaussianity” of the propagated state probability density function (PDF) (Poore, 2015,330
Section 7.2). To initialize the state, we use the objects’ orbital states according to TLE331
data and take the BC values from Table 3. The ROM state is initialized using densities332
from the JB2008 model.333
2.2.2.2 Dynamic model The dynamic model fff(x, t) for evolving the state x con-334
sists of propagating the orbital states using orbital dynamics and evolving the ROM state335
using the continuous-time DMDc model:336
x = fff(x, t) =
xyzvxvyvz˙BCz
=
vxvyvz
agrav,x + adrag,xagrav,y + adrag,yagrav,z + adrag,z
0Acz + Bcu
(18)337
where [x, y, z] and [vx, vy, vz] are the inertial position and velocity, respectively, that are338
used for orbit propagation and BC = CDAm is the ballistic coefficient. The ROM state339
z is used to compute the atmospheric density by converting z to the full space (see Eq. 5)340
and interpolating the density grid. The orbital dynamics are expressed in inertial Carte-341
sian coordinates whereas the orbital states estimated in the UKF are expressed in MEE342
to retain “Gaussianity” of the state PDF. For sigma point propagation in the UKF, the343
sigma points and weights are first selected in MEE space. Each sigma points is then trans-344
formed to Cartesian space and propagated to the next epoch. Finally, each sigma point345
is transformed back to MEE space and the state and covariance are computed based on346
the MEE sigma points and weights.347
2.2.2.3 Orbital dynamics The orbital dynamics used in this work considers:348
• Geopotential acceleration computed using the EGM2008 model, up to degree and349
order 48 for the harmonics;350
• Solar radiation pressure with dual-cone shadow model;351
• Third body perturbations from Sun and Moon;352
• Atmospheric drag considering a rotating atmosphere for computing the velocity353
relative to the atmosphere. The atmospheric density is computed using the ROM354
density model.355
According to Bowman et al. (2004), a 48x48 geopotential field provides acceptable ac-356
curacy for density estimation. For solar radiation pressure, we assumed a reflectivity co-357
efficient of 1.2 and extracted the area-to-mass ratio from the object’s BC assuming a drag358
coefficient of 2.2. The orbit propagation is carried out in the inertial J2000 reference frame359
using Cartesian position and velocity while the geopotential and drag accelerations are360
computed in the Earth-fixed ITRF93 frame. NASA’s SPICE toolbox is used both for361
Moon and Sun ephemerides (DE405 kernels) and for reference frame and time transfor-362
mations (ITRF93 and J2000 reference frames and leap-seconds kernel).363
2.2.2.4 Measurements The measurements used for estimation are the osculat-364
ing orbital states extracted from TLE data. At one hour intervals the osculating state365
of each object is computed using the nearest newer TLE by propagating the TLE back-366
ward to the measurement epoch using SGP4. These states are then converted to MEE367
and used as measurements. The 17 objects used in this work for density estimation are368
shown in Table 3. Note that for calibrating the ROM-TIEGCM model only 11 objects369
are used, because 6 of the 17 objects have their perigee above the ROM-TIEGCM max-370
imum altitude of 450 km.371
–10–
manuscript submitted to Space Weather
2.2.2.5 Measurement and process noise The measurements noise R was deter-372
mined empirically by comparing the post-fit orbits with the measurements. The mea-373
surement noise covariance R was chosen such that the post-fit residuals are tightly within374
the 3-σ range. The post-fit residuals in p, f and g were found to increase with increas-375
ing eccentricity; therefore, the variance for the measurements of p, f and g was scaled376
by the orbit’s eccentricity e to obtain the following eccentricity-dependent measurement377
noise:378
[Rp, Rf , Rg, Rh, Rk, RL] = [c1 · 10−8, c2 · 10−10, c2 · 10−10, 10−9, 10−9, 10−8] (19)379
where c1 = 1.5 ·max(4e, 0.0023) and c2 = 3 ·max(e/0.004, 1).380
The process noise variance Q for the state and BC was set to:381
[Qp, Qf , Qg, Qh, Qk, QL, QBC ] = [1.5 · 10−8, 2 · 10−14, 2 · 10−14, 10−14, 10−14, 10−12, 10−16](20)382
The process noise for the ROM state Qz was computed using the 1-hour ROM predic-383
tion error on the training data:384
diag(Qz) = diag (Cov[Z2 − (ArZ1 + BrΥ)]) (21)385
As a result of this approach, the Kalman filter will give more confidence to the model386
prediction with respect to measurements when the ROM prediction on the training data387
is more accurate. These values for the measurements and process noise were found to388
give good estimated density results (i.e. low RMS error of the estimated density) and389
good estimates of the uncertainty in the estimated density. If the measurement noise is390
set too low and the process noise too high, the density estimates will soak up errors in391
the TLE data. On the other hand, if the measurement noise is set too high and the pro-392
cess noise too low, the filter will give too much confidence to the model prediction with393
respect to measurements. Finally, the initial covariance for the state was set to:394
[Pp, Pf , Pg, Ph, Pk, PL, PBC , Pz1 , Pzn ] = [Rp, Rf , Rg, Rh, Rk, RL, (0.01 ·BC)2, 20, 5] (22)395
where Pz1 refers to the covariance for the first reduced-order mode z1 and Pzn to the co-396
variance for all other modes. An overview of the reduced-order model density estima-397
tion technique is shown below.398
Algorithm 1 ROM density estimation
Reduced-order modeling1. Generate density training data X (hourly density on grid) using physics-based or em-pirical density model2. Select reduced order r3. Compute reduced-order model using a SVD of the snapshots X (Eqs. 1-4)4. Compute DMDc for reduced-order training data (Eqs. 7-15)Density estimation5. Download TLE data and estimate true BC6. Select objects with accurate TLEs (check self-consistency) and stable BC (not maneu-vering)7. Generate measurements (orbital states in MEE) every hour from TLEs using SGP4using nearest newer TLE8. Estimate ROM modes z using unscented Kalman filter by simultaneously estimatingthe modes and the state and BC of objects
3 Results399
In this section, the performance of the ROM model forecasting and density esti-400
mation using TLE data is assessed.401
–11–
manuscript submitted to Space Weather
171 172 173 174 175 1760
2
4
6
8
10
12
14
Pre
dic
tion R
MS
err
or
[%]
NRLMSISE - linear
TIEGCM - linear
JB2008 - linear
NRLMSISE - nonlinear
TIEGCM - nonlinear
JB2008 - nonlinear
141 142 143 144 145 1460
10
20
30
40
50
60
Pre
dic
tion R
MS
err
or
[%]
NRLMSISE - linearTIEGCM - linearJB2008 - linearNRLMSISE - nonlinearTIEGCM - nonlinearJB2008 - nonlinear
171 172 173 174 175 176
Day of year
178
180
182
184
186
188
190
192
F10.7
[sfu
]
0.5
1
1.5
2
2.5
3
Kp
141 142 143 144 145 146Day of year
180
185
190
195
200
205
210
F10.7 [sfu
]
0
2
4
6
8
10
Kp
Figure 2. Forecast error (including truncation error due to order-reduction) in reproducing
the 2002 snapshots using different ROM models and using linear or nonlinear space weather
inputs during quiet (left) and storm (right) conditions (compare with Figure 5 in Mehta et al.
(2018)).
3.1 ROM density prediction402
The performance of the dynamic ROM models is tested by comparing density fore-403
casts with training data. Using the three different ROM density models the density was404
predicted for 5 days during quiet space weather conditions and during a geomagnetic storm405
in 2002. The resulting density forecast errors (the root mean square (RMS) percentage406
error on the three-dimensional spatial grid) and space weather conditions are shown in407
Figure 2. The predictions using the ROM model based on JB2008 are most accurate.408
This good performance can be explained by the superior space weather proxies used by409
the ROM-JB2008 model. Table 4 shows the average 1-hour, 1-day and 3-days predic-410
tion accuracy of the ROM models with respect to the training data. The table shows411
that the addition of nonlinear space weather terms improves the prediction accuracy for412
all models. The ROM-JB2008 provides the best predictions with respect to training data413
with an error of only 3.6% after 3 days. The nonlinear terms especially improve the pre-414
diction during a geomagnetic storm, see Figure 2. Finally, the average truncation error415
due to reducing the dimension to r = 10 is between 0.5 and 4%, see Table 4. The small416
truncation error for the JB2008-based ROM model can be explained by the fact that the417
JB2008 model uses a simple density distribution (see also Figure 7).418
3.2 Simulated TLE test case419
To assess whether accurate density estimation using a ROM model and TLE data420
is feasible, we first tested the technique using simulated TLE data. Mehta and Linares421
(2018a) already demonstrated the feasibility of accurate density estimation using sim-422
ulated GPS measurements with a 5-minute resolution. For the TLE scenario, the ‘true’423
orbits and densities are first computed by numerically propagating eight objects using424
–12–
manuscript submitted to Space Weather
Table 4. Average prediction error in density using linear or nonlinear space weather inputs and
average truncation error due to order-reduction with r = 10 with respect to training data. The
errors are computed as the root mean square (RMS) error in density on the three-dimensional
spatial grid. The average error over all 105,000 epochs of the training data are shown. The pre-
diction errors include truncation errors due to order-reduction.
ROM model Space weather Mean prediction error [%] Mean truncationinputs 1 hour 1 day 3 days error [%]
NRLMSISE Linear 3.81 8.26 11.28 3.61Nonlinear 3.77 6.80 8.94
TIEGCM Linear 2.23 4.80 6.52 2.06Nonlinear 2.21 4.30 6.63
JB2008Linear 1.32 4.78 5.95
0.60Nonlinear 0.90 2.61 3.65
Table 5. Initial orbital parameters for the eight simulated true orbits.
a [km] e [-] i [deg] Ω [deg] ω [deg] M [deg] BC [m2/kg]
Object 1 6811.031 3.011E-3 81.208 157.262 106.464 52.070 0.0142Object 2 6777.764 1.300E-3 81.225 184.489 329.642 122.045 0.0170Object 3 6810.172 1.293E-3 81.215 187.594 112.894 78.318 0.0168Object 4 6808.532 5.124E-4 53.014 185.496 118.205 79.004 0.0127Object 5 6794.771 2.901E-3 82.094 76.779 354.982 127.117 0.0560Object 6 6785.760 4.594E-4 97.435 67.678 86.303 88.988 0.0220Object 7 6729.365 1.619E-3 87.251 169.664 52.108 83.135 0.0052Object 8 6828.232 1.135E-3 30.411 270.733 29.570 295.859 0.0536
the full dynamics described in Section 2.2.2.3 and the ROM-TIEGCM density model.425
The initial orbital parameters for the true orbits are shown in Table 5. Based on these426
‘true’ orbits, TLE measurement data is simulated assuming Gaussian noise and a 1-hour427
resolution. The 1-σ errors used for simulated TLE measurements are:428
[σp, σf , σg, σh, σk, σL] = [0.045, 2.0 · 10−5, 2.0 · 10−5, 2.0 · 10−5, 2.0 · 10−5, 1.25 · 10−4] (23)429
These errors were established empirically based on TLE data of near-circular orbits around430
400 km altitude in the years 2017 and 2018. These errors convert to RMS position and431
velocity errors of 0.88 km and 1.0 m/s, respectively. The initial guesses for estimating432
the orbits, BCs and ROM-state also include these errors.433
Figure 3 shows the errors in the estimated BCs and densities along the orbits dur-434
ing the estimation period. The errors in density and BC remain below 2% after 12 days435
estimation. If the assumed TLE 1-σ errors are twice as large, then the results are sim-436
ilar but the errors in density and BC after 12 days increase by 0.5−1%, which indicates437
that accurate density estimation is feasible even when measurement errors are large. In438
Figure 4 the true and estimated values and errors of the first four ROM modes are shown.439
All modes converge to their true values, while a small bias in the first mode remains. The440
convergence of the modes is also correctly displayed by the 3-σ error bounds. These re-441
sults demonstrate that accurate density estimation using the TLE data of multiple ob-442
jects is feasible and that both the BC and density are observable. Nevertheless, in re-443
ality, less accurate density estimates can be expected, because new TLE measurements444
are not available every hour and errors in TLE data are not random nor Gaussian.445
–13–
manuscript submitted to Space Weather
0 3 6 9 12 15
Time [days]
0
10
20
30
40
50
Density e
rror
[%]
Object 1
Object 2
Object 3
Object 4
Object 5
Object 6
Object 7
Object 8
0 3 6 9 12 15Time [days]
0
5
10
15
20
25
30
35
BC
err
or
[%]
Object 1Object 2Object 3Object 4Object 5Object 6Object 7Object 8
Figure 3. Error in estimated density (left) and BC (right) for each object in simulated TLE
test case since 00:00 UTC on day 191 of year 2005.
0 5 10 15
Time [days]
-15
-10
-5
0
5
10
z [
-]
Mode 1
True
Estimated
0 5 10 15
Time [days]
-15
-10
-5
0
5
10
15
Err
or
[-]
Mode 1
Error
3
0 5 10 15
Time [days]
0
1
2
3
4
z [
-]
Mode 2
True
Estimated
0 5 10 15
Time [days]
-10
-5
0
5
10
Err
or
[-]
Mode 2
Error
3
0 5 10 15
Time [days]
-4
-2
0
2
4
z [
-]
Mode 3
True
Estimated
0 5 10 15
Time [days]
-6
-3
0
3
6
Err
or
[-]
Mode 3
Error
3
0 5 10 15
Time [days]
-2
-1
0
1
z [
-]
Mode 4
True
Estimated
0 5 10 15
Time [days]
-6
-3
0
3
6
Err
or
[-]
Mode 4
Error
3
Figure 4. True and estimated ROM modes and corresponding error and 3σ error bounds for
simulated TLE test case since 00:00 UTC on day 191 of year 2005.
–14–
manuscript submitted to Space Weather
Satellite Model Density error [%]Orbit-averaged Daily-averagedµ σ RMS µ σ RMS
CHAMP
NRLMSISE-00 24.2 11.4 26.7 24.0 10.4 26.0JB2008 -22.4 7.7 23.7 -22.4 6.3 23.2ROM-TIEGCM 4.6 10.1 11.1 4.4 7.5 8.6ROM-NRLMSISE -2.6 7.3 7.7 -2.7 5.1 5.7ROM-JB2008 -2.0 6.9 7.2 -2.1 5.4 5.7
GRACE-A
NRLMSISE-00 39.8 16.5 43.1 39.6 14.9 42.2JB2008 -20.8 10.8 23.4 -20.8 9.0 22.6ROM-TIEGCM 12.1 15.3 19.5 12.0 11.3 16.4ROM-NRLMSISE 4.7 10.0 11.1 4.8 7.4 8.8ROM-JB2008 8.4 9.3 12.5 7.9 6.6 10.2
Table 6. Accuracy of estimated and modelled densities along CHAMP’s and GRACE-A’s orbit
during August 2002. The numbers show the mean µ, standard deviation σ and root-mean-square
(RMS) of the error in orbit and daily-averaged density as percentage of true densities.
3.3 Real TLE446
In the following, the density is estimated using real TLE data and compared with447
CHAMP and GRACE accelerometer-derived density data. Figure 5 shows the orbit-averaged448
estimated density along CHAMP’s orbit as well as the density according to CHAMP data449
and the NRLMSISE-00 and JB2008 density models during August 2002 (the first month450
for which we had both CHAMP and GRACE data). All three ROM models perform very451
well. Especially, the ROM-NRLMSISE and ROM-JB2008 models are very well able to452
estimate density variations due to changes in solar activity. The RMS errors in the daily-453
averaged CHAMP density are less than 6% for the ROM-NRLMSISE and ROM-JB2008454
models and less than 9% for the ROM-TIEGCM model, see Table 6. The wiggles in the455
estimated orbit-averaged density (particularly visible for ROM-TIEGCM and ROM-NRLMSISE)456
have a period of 12 hours, which suggests that these are due to errors in the TLEs due457
to missing m-dailies. Similar 12-hour variations can be found in the estimated BCs (not458
shown here). Further tuning of the measurement and process noise can reduce the am-459
plitude of these variations due to TLE errors.460
Overall, the ROM model based on JB2008 performs best with a standard devia-461
tion in orbit-averaged density of only 6.9% and 9.3% along CHAMP’s and GRACE-A’s462
orbit, respectively. This shows the high accuracy and temporal resolution that can be463
achieved by the ROM approach using only TLE data. The error in GRACE-A density464
is possibly higher because less objects around GRACE’s 480 km altitude were used than465
around CHAMP’s altitude of 400 km and because the drag signal is weaker at higher466
altitudes. Only 17 objects were used for calibrating the ROM-JB2008 and ROM-NRLMSISE467
models and only 11 for the ROM-TIEGCM model. This is significantly lower than the468
36 and 48 objects used by Doornbos et al. (2008) and Shi et al. (2015), respectively, but469
comparable to the 16 objects used by Yurasov et al. (2005). It should also be noted that470
the errors presented in this paper are with respect to accelerometer-derived density data471
and not with respect to TLE-derived density data as in some other works (Doornbos et472
al., 2008; Shi et al., 2015).473
A close-up of the density along CHAMP’s orbit on August 14, 2002 is shown in Fig-474
ure 6. In this time window, the ROM-estimated densities are very close to the true den-475
sity (both the mean and variation are estimated well). It is probably not feasible to ex-476
–15–
manuscript submitted to Space Weather
1
2
3
4
5
6
7
Orb
it-a
vera
ged
density [kg/m
2]
10-12
CHAMP
JB2008
NRLMSISE-00
ROM-TIEGCM
1
2
3
4
5
6
7
Orb
it-a
vera
ged
density [kg/m
2]
10-12
CHAMP
JB2008
NRLMSISE-00
ROM-NRLMSISE
213 218 223 228 233 238 243
Day of year
1
2
3
4
5
6
7
Orb
it-a
vera
ged d
ensity [kg/m
2]
10-12
CHAMP
JB2008
NRLMSISE-00
ROM-JB2008
213 218 223 228 233 238 243
Day of year
120
140
160
180
200
220
240
260
F10.7
0
1
2
3
4
5
6
7
Kp
Figure 5. Orbit-averaged density along CHAMP orbit according to ROM estimation,
CHAMP data, and JB2008 and NRLMSISE-00 models from August 1 to 31, 2002.
225.8 225.9 226 226.1 226.2 226.3 226.4
Day of year
1
2
3
4
5
6
7
[kg/m
3]
10-12
CHAMP ROM-TIEGCM ROM-NRLMSISE ROM-JB2008 JB2008 NRLMSISE
Figure 6. Density along CHAMP orbit according to ROM estimation, CHAMP data, and
JB2008 and NRLMSISE-00 models (day 226 of year 2002 is August 14, 2002).
–16–
manuscript submitted to Space Weather
NRLMSISE-00
-90
0
90
La
titu
de
ROM-NRLMSISE
JB2008
0 6 12 18 24
Local solar time
-90
0
90
La
titu
de
0.5 1 1.5 2 2.5 3 3.5
Density [kg/m3] 10-12
ROM-JB2008
ROM-TIEGCM
0 6 12 18 24
Local solar time
Figure 7. Maps of modeled density at 450 km latitude on August 8, 2002 at 0:00:00 UTC.
actly match the true density using TLE data due to the highly-dynamic character of the477
atmosphere and low-temporal resolution of TLE data. Besides the good match, one can478
see differences between the different ROM models in the density variation over one or-479
bit. These differences stem from the underlying base models which result in different spa-480
tial modes for each model. The way that the ROM models mimic their base model can481
also be seen in Figure 7 that shows maps of the modelled and estimated density on Au-482
gust 8, 2002 at 450 km altitude. For example, the simple density distribution in the JB2008483
model is also visible in the ROM-JB2008 density, whereas the NRLMSISE-00 and TIEGCM484
based ROM models show more complex density distributions. On the other hand, in-485
dependent of the base model, the ROM-estimated densities have a similar magnitude,486
which indicates successful calibration.487
3.3.1 Uncertainty488
Figures 8 and 9 show the uncertainty in the estimated density for different altitudes,489
latitudes and local solar time. The uncertainty in the estimated density is smaller for490
lower altitude and inside the diurnal bulge. This indicates that the density estimation491
is more accurate when the drag signal is stronger. In addition, Figure 9 shows that the492
uncertainty grows little at altitudes where measurements are available. The estimated493
1-σ uncertainties at 400 and 500 km altitude of about 6.5% and 7.5%, respectively, are494
close to the actual standard deviation of the estimated density at 416 km (CHAMP) and495
502 km (GRACE-A) altitude of 6.9% and 9.3%, respectively, see Table 6. Furthermore,496
between 100 and 800 km altitude the 1-σ error varies between 3 and 11%, while Mehta497
and Linares (2018b) found a 1-σ error of 5% along CHAMP’s orbit and up to 25% er-498
ror at higher altitudes after assimilating CHAMP accelerometer-derived density data in499
a TIEGCM-based ROM model. This indicates that the use of data from multiple ob-500
jects improves the global density estimates.501
–17–
manuscript submitted to Space Weather
Altitude = 300 km
-90
0
90
Latitu
de [deg]
6
6.5
7
Altitude = 400 km
6
6.5
7
7.5
1 e
rro
r [%
]
Altitude = 500 km
0 6 12 18 24
Local solar time
-90
0
90
Latitu
de [deg]
6.5
7
7.5
8
8.5
Altitude = 600 km
0 6 12 18 24
Local solar time
7
7.5
8
1 e
rro
r [%
]
Figure 8. 1-σ uncertainty in density estimated using ROM-JB2008 at 300 to 600 km altitude
after 30 days estimation on August 31, 2002.
100 200 300 400 500 600 700 800
Altitude [km]
2
4
6
8
10
12
1 e
rror
[%]
Perigee altitude of objects
Uncertainty at local solar time = 2.5h, latitude = -20° (opposite of diurnal bulge)
Uncertainty at local solar time = 14.5h, latitude = 20° (in diurnal bulge)
Figure 9. 1-σ uncertainty in density estimated using ROM-JB2008 at two locations for vary-
ing altitude after 30 days estimation on August 31, 2002.
–18–
manuscript submitted to Space Weather
0
20
40
60
80
100JB2008 ROM-JB2008
0
20
40
60
80
100
Orb
it-a
ve
rag
ed
de
nsity e
rro
r [%
]
NRLMSISE-00 ROM-JB2008
0 50 100 150 200 250 300 350Day of year
100
200
300
F1
0.7
0
3
6
9
Kp
Figure 10. Error in orbit-averaged density with respect to CHAMP data for ROM-JB2008
estimated density, and JB2008 and NRLMSISE-00 models for the year 2007.
Satellite Model Density error [%]2003 2007
µ σ RMS µ σ RMS
CHAMPNRLMSISE-00 4.4 19.7 20.2 26.4 15.4 30.5JB2008 -1.9 12.5 12.7 9.5 13.4 16.5ROM-JB2008 -5.1 11.4 12.5 -6.9 9.0 11.4
GRACE-ANRLMSISE-00 20.7 28.6 35.3 44.1 28.4 52.5JB2008 10.0 18.3 20.9 16.0 28.4 32.7ROM-JB2008 6.5 16.9 18.1 -2.5 21.9 22.1
Table 7. Accuracy of estimated and modelled densities along CHAMP’s and GRACE-A’s or-
bit over the year 2003 and 2007. The numbers show the mean µ, standard deviation σ and root
mean square (RMS) of the error in orbit-averaged density as percentage of true densities.
3.3.2 Full years 2003 and 2007502
The neutral mass density was estimated using the ROM-JB2008 model over the503
entire years 2003 (high solar activity) and 2007 (low solar activity). The difference in504
the estimated and true densities along CHAMP’s and GRACE-A’s orbits are shown in505
Table 7 and Figure 10. Both the ROM estimation and JB2008 model perform very well506
in 2003, whereas the ROM estimates are much more accurate than the JB2008 and NRLMSISE-507
00 models in 2007. Table 7 shows that the ROM-estimated densities are less biased and508
have a smaller variance than the empirical models. In particular in 2007, when the em-509
pirical models are least accurate, the ROM densities have a significantly smaller bias and510
standard deviation. Furthermore, the estimated density along GRACE-A’s orbit are less511
accurate. This agrees with the increasing uncertainty in estimated density (see Figure 9)512
and weaker drag signal at higher altitudes. The accuracy can be further improved by us-513
ing more accurate orbital data and by improving the spatial and temporal coverage of514
the measurements, see Section 3.4.515
3.3.3 Geomagnetic storm516
Figure 11 shows the density along CHAMP’s orbit estimated using the ROM-JB2008517
model during a major geomagnetic storm in 2003. Both the ROM and empirical mod-518
els provide good density estimates during the storm. However, after storm on day 151519
the ROM estimates are much more accurate than the empirical models, which overes-520
timate the density, due to calibration. This example shows that the linear ROM model521
is able to deal with space weather events even though these events are strongly nonlin-522
ear.523
–19–
manuscript submitted to Space Weather
146 147 148 149 150 151 152 153 154
Day of year
1.5
2
2.5
3
3.5
4
4.5
5
5.5
6
Orb
it-a
ve
rag
ed
[
kg
/m2]
10-12
CHAMP
JB2008
NRLMSISE-00
ROM-JB2008
146 147 148 149 150 151 152 153 154
Day of year
100
105
110
115
120
125
130
135
140
145
F10
.7
0
1
2
3
4
5
6
7
8
9
Kp
Figure 11. Orbit-averaged density along CHAMP’s orbit according to ROM estimation,
CHAMP data, and JB2008 and NRLMSISE-00 models during major geomagnetic storm on May
30, 2003 (Kp up to 8.3).
47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69
Day of year
0.5
1
1.5
2
2.5
3
Orb
it-a
vera
ged
[kg/m
2]
10-13
GRACE-A
ROM-JB2008 -
TLE data onlyROM-JB2008 -
TLE + CHAMP data
Figure 12. Orbit-averaged density along GRACE-A’s orbit estimated using only TLE data or
using TLE data and CHAMP accelerometer-derived density data (Feb 16 to Mar 10, 2007).
3.4 Including density data524
For some periods of time, one may have access to highly-accurate density data, such525
as CHAMP, GRACE or GOCE accelerometer-derived densities. This data can be included526
in the data assimilation to improve the global density estimates. Figure 3.4 shows the527
estimated orbit-averaged density along GRACE-A’s orbit after assimilating CHAMP den-528
sity data together with TLE data (here CHAMP was at 360 km and GRACE-A at 480529
km altitude; we assumed a 1-σ error in the CHAMP density data of 5%). This period530
coincides with the period of reduced estimation accuracy shown in Figure 10 around day531
50. One density measurement is included every hour at the same time as the TLE or-532
bit measurements. The inclusion of the CHAMP densities significantly improves the GRACE533
density estimates; the error in daily-averaged density reduced from 16.4% to 11.6%. There-534
fore, global density estimates can be improved by including accurate density data. This535
is especially useful in case the drag signal is weak and consequently the density is dif-536
ficult to observe from orbital data such as during periods of low solar activity.537
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manuscript submitted to Space Weather
243 244 245 246 247 248 249 250 251 252 253 254Day of year
2
3
4
5
6
7
8
Orb
it-a
vera
ged [k
g/m
2]
10-12
CHAMPJB2008NRLMSISEROM-NRLMSISE prediction
243 244 245 246 247 248 249 250 251 252 253 254
Day of year
165
170
175
180
185
190
195
200
205
210
F10.7
0
1
2
3
4
5
6
7
8
Kp
Figure 13. Orbit-averaged density along CHAMP orbit according to ROM prediction,
CHAMP data, and JB2008 and NRLMSISE-00 models and F10.7 and Kp indices (Aug 31 to
Sep 10, 2002).
3.5 Density forecast538
The density along CHAMP’s orbit was predicted for 11 days using the ROM-NRLMSISE539
model, see Figure 13. The initial ROM state used for the prediction was obtained after540
30 days calibration in August 2002, see Figure 5. The ROM does very well in predict-541
ing the density, even during two geomagnetic storms. This example shows that ROM542
models are able to accurately forecast the future density if the initial state of the ther-543
mosphere and the future space weather are accurately known. In future work, predic-544
tion of the future space weather will also be considered.545
4 Discussion546
The key motivation for the use of TLE data is the widespread availability of this547
data source. However, TLE orbital data can have significant errors. The presented re-548
sults indicate that these errors do not limit the ability to use TLE data for density cal-549
ibration. In addition, Figure 3.4 showed that the use of more accurate measurements can550
further improve the estimation of the global density. Therefore, in the future other data551
sources, such as radar and GPS orbital data, could be used to obtain improved density552
estimates.553
To improve the predictive performance of the linear ROM models and therefore of554
the density estimates, nonlinear space weather inputs were used. This enhancement im-555
proved the modelling of the relation between space weather and thermospheric density,556
which is important because the thermosphere is strongly forced by solar activity. In par-557
ticular during space weather events, such as geomagnetic storms, the predictive capa-558
bility of the ROM model is important because the temporal resolution of TLE data is559
limited. In addition, in future work, the reduced-order models can be further improved560
by using more accurate density training data for deriving the models using POD and DMDc.561
In addition to estimating the global density, we obtained estimates of the uncer-562
tainty in the density, see Figures 8 and 9. This information is very valuable for uncer-563
tainty quantification, which is needed for, e.g., conjunction assessments. These uncer-564
tainty estimates are in particular important for short-term predictions when errors in565
the density prediction are dominated by errors in the now-cast, whereas after several days566
errors due to inaccurate space weather forecasting become dominant.567
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manuscript submitted to Space Weather
Finally, the ballistic coefficients in Table 3 may have biases that originate from the568
atmospheric models used in the BC estimation. Such biases may result in inaccuracies569
in the density estimates. The presented methodology counteracts this by simultaneously570
estimating the thermospheric density and ballistic coefficients. Still, some error in BC571
may be soaked up by the density estimates. These errors can be reduced by using the572
geometry models of the satellites and by modeling the gas-surface interactions and phys-573
ical drag coefficient to compute the BC (Mehta et al., 2017) instead of estimating the574
BC from orbital data. This approach can be applied in future work to further debias the575
density and BC estimates.576
5 Conclusions577
In this paper we presented a new approach for thermospheric density estimation578
using TLE data. To obtain accurate global density estimates we assimilated the TLE579
data of spatially-spread objects in a dynamic reduced-order density model by simulta-580
neously estimating the orbits, BCs and density. The estimated densities were compared581
with NRLMSISE-00 and JB2008 modelled densities and CHAMP and GRACE-A accelerometer-582
derived densities. Even though TLE orbital data have significant errors and low tem-583
poral resolution, the density estimates are accurate and have a significantly smaller bias584
and smaller standard deviation than densities computed using the empirical models both585
during periods of high and low solar activity.586
To accurately model the density, we developed three different ROM models based587
on TIE-GCM, NRLMSISE-00 and JB2008 density data with an upper altitude up to 800588
km and improved the prediction performance of the models by including nonlinear space589
weather terms as inputs for the ROM dynamics. In addition, the estimation using an590
UKF was improved by expressing the measurements in modified equinoctial elements and591
by calculating the process noise for the ROM model based on training data performance.592
We obtained improved global density estimates by including accurate density measure-593
ments in addition to TLE data. This indicates that more accurate observations, such as594
radar and GPS orbital data, could be used in the future to obtain improved density es-595
timates.596
Furthermore, the ROM model was shown to be able to provide accurate density597
forecasts. Therefore, the density estimates and predictions can be used for both improved598
orbit determination and orbit prediction. Moreover, the technique provides uncertainty599
estimates for the density, which can be used for uncertainty quantification for, e.g., con-600
junction assessment.601
Future work will focus on further improving the ROM dynamic models to deal with602
nonlinearities by improving the choice of space weather inputs and by applying Koop-603
man operator theory. In addition, we can estimate the global thermospheric neutral den-604
sity using historic TLE data from the 1960’s up to present time to generate a density605
database.606
Acknowledgments607
The authors wish to acknowledge support of this work by the Air Force’s Office of Sci-608
entific Research under Contract Number FA9550-18-1-0149 issued by Erik Blasch. The609
authors would also like to thank Planet Labs, Inc. and Vivek Vittaldev for providing the610
GPS ephemerides data used in this work. In addition, the authors are grateful to Eric611
Sutton for providing the TIE-GCM simulation data. The NRLMSISE-00 and JB2008612
models used in this work can be found on https://www.brodo.de/space/nrlmsise and613
http://sol.spacenvironment.net/jb2008/code.html, respectively. The space weather614
proxies were obtained from http://celestrak.com/SpaceData/ and http://sol.spacenvironment615
.net/jb2008/indices.html. The CHAMP and GRACE densities used in this work were616
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manuscript submitted to Space Weather
derived by Sutton (2008) and can be found, e.g., on http://tinyurl.com/densitysets.617
The MATLAB code and ROM models used in this work for density estimation with TLE618
data will be published after acceptance on https://zenodo.org including a Digital Ob-619
ject Identifier (DOI).620
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