real-time thermospheric density estimation via two-line...

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,

[email protected]

–1–


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

–2–


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

–3–


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

–4–


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

–5–


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

–6–


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

–7–


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

–8–


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

–9–


“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–


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–


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–


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–


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–


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–


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–


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–


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–


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–


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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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

–21–


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

–22–


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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