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Improve GNSS Orbit Determination by using

Estimated Tropospheric and Ionospheric Models

Cazabonne Bryan and Maisonobe Luc

CS Group

Toulouse, France

bryan.cazabonne@c-s.fr, luc.maisonobe@c-s.fr

Abstract— Orbit Determination is a technique used to

estimate the position of a satellite from its observable

measurements. Missing or incorrect modeling of troposphere and

ionosphere delays is one of the major error source in space

geodetic techniques such as Global Navigation Satellite Systems

(GNSS). Accurate computation of these two delays is a

mandatory step to cope with accuracy needs which are close to

centimeter or millimeter levels. This paper presents the different

steps of development of estimated tropospheric and ionospheric

models. All these models are included in the Orekit open-source

space flight dynamics library. Adding estimated tropospheric

and ionospheric models into an orbit determination process can

be a difficult procedure. Computing and validating measurement

derivatives with respect to troposphere and ionosphere

parameters are critical steps. To cope with this constraint, we

used the Automatic Differentiation technique to avoid the

calculation of the derivatives of long equations. Automatic

Differentiation is equivalent to calculating the derivatives by

applying chain rule without expressing the analytical formulas.

Therefore, Automatic Differentiation allows a simpler

computation of the derivatives and a simpler validation. This

paper presents how the Jacobian measurement matrix is

computed by Automatic Differentiation. It also describes the

impact of using estimated tropospheric and ionospheric models.

Finally, a study of different model configurations is performed in

order to highlight the relevant tropospheric and ionospheric

parameters to estimate. The performance of the different models

is demonstrated under GPS orbit determination conditions. Both

satellite state vector estimation and measurement residuals

quality are used as indicator to quantify the orbit determination

performance. This paper addresses that estimated tropospheric

and ionospheric models are actually more accurate than

empirical models to estimate satellite state vector in GNSS orbit

determination. A gain of about 60% is obtained on the

estimation of the satellite position when estimated models are

used, without altering the computation time.

Keywords— GNSS; Orbit Determination; Open-Source; Orekit;

Ionospheric delay; Tropospheric delay

I.

I

NTRODUCTION

Population needs are always evolving. Services provided

by satellites are increasingly present in our everyday life.

Among these services, Global Navigation Satellite Systems

(GNSS) applications (e.g., agriculture, vehicle navigation,

precise time reference, and climate change) are omnipresent for

both military and civilian purposes. For this type of application,

accuracy requirements are close to centimeter level [1], [2].

Orbit determination is a technique used to estimate the

position of a satellite from observable measurements. In the

case of GNSS satellites, the process shall be both fast and

accurate to cope with the accuracy requirements of GNSS

applications. Before reaching the receiver, GNSS satellites

signal is slowed down by the atmosphere, mainly because of

two distinct layers: the ionosphere and the troposphere.

Missing or incorrect modeling of these two effects is one of the

key sources of error in GNSS orbit determination [3].

Ionosphere is a high altitude layer of the Earth’s

atmosphere composed of an ionized medium of charged

particles which slow down the signal emitted by the GNSS

satellites. This medium is dispersive. As a consequence, the

ionospheric path delay for the GNSS signal depends on the

frequency of the signal [4]. For this reason, it is possible to

mathematically remove the ionospheric delay by using an

ionosphere-free combination of measurements. However, this

technique has some drawbacks. First, a single-frequency

ionosphere-free combination of measurements, such as

GRAPHIC, imposes to solve the unknown ambiguity of the

carrier phase measurement. Moreover, a dual-frequency

ionosphere-free combination of measurements can be used

only for multi-frequency receivers, limiting its use. The

troposphere is the lowest layer of the Earth’s atmosphere. It is a

non-dispersive medium. The path delay on the navigation

signal is caused both by the water vapor and the dry gases

(mainly N

2

and O

2

)

in the troposphere [5]. Empirical models

are usually used to calculate the ionospheric delay (e.g.,

Klobuchar [6], NeQuick [7], and Global Ionosphere Maps [8])

and the tropospheric delay (e.g., Saastamoinen [9]) because

they are easy to use and to implement. However, for a better

accuracy and to take into account the strong temporal and

spatial variations of the ionosphere, finding new approaches to

calculate the ionospheric and tropospheric delays becomes

essential [10], [11].

The lack of precision in the empirical ionospheric and

tropospheric delay models is due to the rapid fluctuations and

the unpredictability of some effects in the ionosphere and the

troposphere. These effects are characterized by the total

electron content (TEC) for the ionosphere and by the wet

zenith delay for the troposphere. Estimating the TEC parameter

and the wet zenith delay provides highest performance in the

orbit determination process. This paper presents the different

steps of development of the estimated ionospheric and

tropospheric models. Ionospheric and tropospheric parameters

are estimated simultaneously with the position and the velocity

of the satellite during the orbit determination process. The

estimation process is based on a batch least squares estimator.

In order to compute the derivatives with respect to the

satellite’s orbital parameters and the ionospheric and

tropospheric parameters during the orbit determination process,

we use the automatic differentiation technique [12]. GPS

satellite orbit determination highlights the performance of the

estimated delay models. Only pseudorange measurements are

used. Satellite’s orbit estimation and measurement residuals are

used as indicator to quantify the orbit determination

performance.

The structure of this paper is as follows. Section II presents

the Orekit open-source space flight dynamics library. Section

III provides the theoretical elements of the estimated

ionospheric and tropospheric delay models. Section IV

describes the orbit determination process and the integration of

the estimated delay models. Section V provides experimental

results demonstrating the impact of the estimated models.

Section VI ends the paper with the concluding remarks.

II. O

REKIT

Orekit is an open-source space flight dynamics library [13].

It is written in Java and provides low level elements for the

development of flight dynamics applications. Orekit project

started in 2002 as an in-house closed project developed by CS

Group. Since 2008 Orekit is distributed under the open-source

Apache License version 2.0 [14]. Fig. 1 illustrates the packages

organization in Orekit.

Current version of Orekit is 10.1. It was released early

2020. Many features are implemented in Orekit in order to

enable the users to build their own flight dynamics application.

The main features are summarized in Table I.

TABLE I. O

REKIT MAIN FEATURES

Feature Subfeature

Predefined orbit frames

Inertial frames

ITRF from 1998 to 2014

Local Orbital Frames

Time scales

UTC, TAI, UT1, and TT

Navigation satellites time scales

Orbit Keplerian, Equinoctial, Cartesian, Circular

File formats

Rinex observation files

Antex fi les

CCSDS Orbit and Tracking Data Messages

SEM and YUMA files

SP3 orbit files

Models

Tropospheric and ionospheric delays

Atmosphere

Geomagnetic field

Tidal displacements and ocean loading

Tessellation

Attitude modelling

Predefined laws

Navigation satellites attitude

Orbit propagation

Two-Lines Element

Numerical propagation with customizable

force models

Semi-analytic propagation with customizable

force models

Navigation satellites propagation

Event handling during propagation

Taylor algebra

Orbit determination

Extended Kalman filter

Batch least squares filtering

Single satellite orbit determination

Multi-satellite orbit determination

Range, Phase, Position/Velocity and Doppler

measurements

Measurement biases estimation

Propagation parameters estimation

Satellite and ground station clock estimation

III. T

ROPOSPHERIC AND

I

ONOSPHERIC

M

ODELS

In this section, the theoretical elements of the estimated

tropospheric and ionospheric delay models are developed.

A. Modelling the GNSS Measurements

GNSS pseudorange and carrier phase measurements are

used to obtain the apparent distance between the receiver and

the satellite. In addition to the geometric range between the

receiver and the satellite, pseudorange and carrier phase

measurements are affected by atmospheric propagation delays

Fig. 1. Orekit packages organization. Orekit’s organization is based on

fourteen main packages providing, inter alia, data parsing, orbit

propagation, and orbit determinantion features.

and clock biases. The carrier phase measurement is more

precise than the pseudorange measurement. However, it is

ambiguous by an unknown integer number of cycles.

Therefore, pseudorange and carrier phase measurements are

given by [15]

R = ρ + c(dt

rec

−dt

sat

) + T

+

I

−K

R,sat

+ ε

R

Φ = ρ + c(dt

rec

−dt

sat

) + T −I−K

Φ,sat

+ λN + λω + ε

Φ

(1)

where R and Φ are the pseudorange and carrier phase

measurements, respectively; ρ is the geometric distance

between the antenna phase centers of the receiver and the

satellite; c is the speed of light (i.e., c = 299792458 meters per

second); dt

rec

and dt

sat

are the receiver’s and satellite’s clock

biases, respectively; T is the tropospheric delay; I is the

ionospheric delay; K

R,sat

and K

Φ,sat

are the instrumental delays

of the pseudorange and carrier phase measurements,

respectively; λ is the wavelength of the carrier signal; N is the

carrier phase integer ambiguity; ω represents the wind-up

effect; and ε is the receiver noise.

B. Tropospheric Delay

Tropospheric delay is one of the main propagation delays

for GNSS signals. It is due to the refractive index of the

Earth’s troposphere [3]. For an accurate computation of the

tropospheric delay, it is necessary to consider that the Earth’s

troposphere is composed of a dry and a wet part. Following

the nomenclature in (1), the tropospheric delay can be

expressed as

T = T

dry

+ T

wet

(2)

where T

dry

is the dry tropospheric delay and T

wet

is the wet

tropospheric delay. Moreover, both tropospheric delays in (2)

can be expressed as the product of a zenith delay and a

mapping function

T

dry/wet

(E) = T

dry/wet

z

. m

dry/wet

(E) (3)

where T

dry/wet

z

is the zenith delay (dry or wet); E is the

elevation angle of the satellite observed from the ground

station; and m

dry/wet

(E) is the mapping function (dry or wet).

The mapping function is used for the modeling of the

elevation dependence of the tropospheric delay. Finally, using

the expression (3) into (2), the delay is given by [16]

T(E) = T

dry

z

. m

dry

(E) + T

wet

z

. m

wet

(E) (4)

In (4), the dry zenith delay can be accurately computed

using the Saastamoinen model [3], [9], [17]. Empirical models

can also be used to compute the dry and wet mapping

functions (e.g., Niell Mapping Function [18], Global Mapping

Function [19], and Vienna Mapping Function [20], [21]).

The wet zenith delay is caused by the water inside the

troposphere’s clouds. Therefore, this parameter depends on the

weather conditions, making it difficult to model. Moreover,

the wet zenith delay varies faster than the dry zenith delay.

Nevertheless, it is possible to estimate the wet zenith delay

during the orbit determination process to cope with these

constraints. It can be estimated simultaneously with the

satellite’s position and velocity. In this case, observable

measurements are used instead of empirical models to

quantify the contribution of the wet component of the

tropospheric delay. Furthermore, the wet part of the delay

accounts for about 10% of the tropospheric delay [17], [22].

Because of the one order of magnitude difference between the

dry and wet components of the tropospheric delay, numerical

stability problems can occur during the orbit determination.

Therefore, estimating the total zenith delay instead of the wet

zenith delay improves the accuracy and the computation time

of the estimation process. The tropospheric delay can be

expressed as

T(E) = T

dry

z

. m

dry

(E) + (T

total

z

−T

dry

z

) . m

wet

(E) (5)

where T

total

z

is the total zenith delay estimated during the

orbit determination process. In Section V, a comparison

between the estimation of the total zenith delay and the wet

zenith delay is performed.

C. Ionospheric Delay

Ionospheric delay is the second propagation delay for

GNSS signals. It is due to the dispersive nature of the

ionosphere. Following the nomenclature in (1), the

ionospheric delay on GNSS signal can be expressed, at first-

order, as [17]

I =

40.3

f

2

. sTEC (6)

where f is the signal’s frequency; and sTEC is the slant

TEC in electrons per square meter. It is possible to express the

sTEC in TEC units (TECU), where 1 TECU = 10

16

electrons

per square meter. Higher order terms can be used in (6) to

express the ionospheric delay. However, they account for

about 0.1% of the total ionospheric delay [3], [17]. Therefore,

they can be neglected. Moreover, the sTEC can be expressed

as the product of a vertical TEC and an ionospheric mapping

function

sTEC(E) = vTEC . m

I

(E) (7)

where vTEC is the vertical TEC in electrons per square

meter; E is the elevation angle; and m

I

(E) is the ionospheric

mapping function. Finally, using the expression (7) into (6),

the ionospheric delay is given by

I(E) =

40.3

f

2

.

vTEC . m

I

(E) (8)

In (8), the ionospheric mapping function can be computed

using the Single Layer Model mapping function [23]

m

I

(E) = 1

1

sin

2

z

*

,sinz

*

= R

E

R

E

+ h . sinz (9)

where R

E

is the Earth’s radius; h is the height of the

ionospheric single layer; and z = (π / 2) – E is zenith angle.

Typical values for R

E

and h are 6371 kilometers and 450

kilometers, respectively. However, the single layer assumption

with a constant height introduces significant errors in the

ionospheric delay computation. To improve the computation of

the ionospheric mapping function recent models propose to

define multi-layers mapping functions. Moreover, other models

propose to define a variable ionospheric height instead of the

constant height of 450 kilometers [24]-[26]. The study of the

different mapping functions is not performed in this paper.

Only the Single Layer Model mapping function is used.

In (8), the vTEC can be obtained from the global vTEC

maps provided by the International GNSS Service (IGS) with a

geographic resolution of 2.5° in latitude and 5.0° in longitude

and a temporal resolution of one hour. Therefore, both a

temporal interpolation and a geographic interpolation are

needed to obtain the local value of the vTEC at a given date

[8]. However, performing two interpolations can reduce the

accuracy of computations. In this case, observable

measurements can be used instead of empirical models to

quantify the vTEC. For this reason, estimating the vTEC

simultaneously with the position and the velocity of the

satellite and the total zenith delay can improve the accuracy of

the GNSS orbit determination.

IV. O

RBIT

D

ETERMINATION

In this section, the orbit determination process is presented.

The influence of the estimated delay models on the orbit

determination is also developed.

A. The Batch Least Squares Orbit Determination

Orekit provides two orbit determination algorithms to

estimate the state vector of a satellite and model parameters.

The first one is a batch least squares estimator and the other

one is an extended Kalman filter. This paper focuses only on

the batch least squares orbit determination.

Let Y be the state vector of the satellite. The state vector

contains all the parameters estimated during the orbit

determination process. It can be defined as

Y=X

sat

Y

sat

Z

sat

X

sat

Y

sat

⋯

⋯Z

sat

C

r

T

total

z

vTEC dt

rec

(10)

where X

sat

Y

sat

and Z

sat

are the three coordinates of the

satellite’s position; X

sat

Y

sat

and Z

sat

are the three coordinates

of the satellite’s velocity; C

r

is the solar radiation coefficient

(it must be estimated for a better accuracy); T

total

z

and vTEC

are the estimated tropospheric and ionospheric parameters,

respectively; and dt

rec

is the receiver’s clock bias that must be

also estimated during the orbit determination process. Of

course, other parameters could be estimated as well, the

parameters above correspond to the ones needed for this study.

Let Y

t

0

be the user’s initial guess for the satellite’s state vector

at the epoch t

0

. The batch least squares estimator provides the

best estimate of the satellite’s state vector as

Y

t

0

= Y

t

0

+δy (11)

where Y

t

0

is the satellite’s state vector estimated by the

batch least square estimator at the epoch t

0

; and δy is the

differential-correction. In order to obtain the differential-

correction, the batch least squares estimator has to solve the

non-linear equation given by [27], [28]

δy = A

T

WA

-1

A

T

Wb (12)

where A is the partial derivative matrix; A

T

is the A

transpose; W is the weighting matrix; and b is the residual

vector. The partial derivative matrix contains the partial

derivatives of the observable measurements with respect to the

parameters estimated during the orbit determination process.

Let N be the number of observable measurements and S be the

number of estimated parameters (i.e., S = 10). The dimensions

of the A, W and b elements are N × S, N × N and N,

respectively.

The weighting matrix, W, is built according to the

measurement weight value initialized by the user [28]. This

value is initialized at the beginning of the orbit determination

process for all the observable measurements. Moreover, the

residual vector, b, is computed, for each measurement, by the

difference between the observed and the estimated values of

the measurement. Finally, the partial derivative matrix, A, is

given by [28]

= ∂ρ

i

∂Y

t0

, i=1, …,N (13)

where ρ

i

is an observable measurement at epoch t; and i is

the index of the measurement. The batch least squares

estimator performs the computation of the differential-

correction until convergence.

Let k be the iteration number, the converge criterion is

defined by

RMS

k-1

−

RMS

k

RMS

k-1

≤ ε (14)

where ε is the convergence threshold; and RMS =

b

T

b

is

the root mean square of the residuals.

B. Computation of the Partial Derivative Matrix

In this subsection, the partial derivative matrix is studied.

Let Q be the vector containing the six satellite’s orbital

elements (i.e., position and velocity); P the vector containing

the estimated force model parameter (i.e., solar radiation

coefficient); and M be the vector containing the three

measurement parameters (i.e., total zenith delay, vTEC, and

the receiver’s clock bias).

The estimated satellite state vector, Y

t

0

, can be rewritten as

Y

t

0

= Q

t

0

P

t

0

M

t

0

(15)

Following (15), the partial derivative matrix can be divided

into three submatrices

= (A

A

A

) (16)

where

A

1

=∂ρ

i

∂Q

t

0

=∂ρ

i

∂Q

t

. ∂Q

t

∂Q

t

0

A

2

=∂ρ

i

∂P

t

0

=∂ρ

i

∂Q

t

. ∂Q

t

∂P

t

0

A

3

=∂ρ

i

∂M

t

0

=∂ρ

i

∂M

t

. ∂M

t

∂M

t

0

∂Q

t

∂Q

t

0

and ∂Q

t

∂P

t

0

are called the state transition

matrices; ∂ρ

i

∂Q

t

⁄ and ∂ρ

i

∂M

t

⁄ are the observation matrices;

and ∂M

t

∂M

t

0

is a 3 × 3 identity matrix.

During an orbit determination process, the batch least

squares estimator performs two important steps. The first step

is the computation of the estimated value of the observed

measurement following (1). During this step, the estimator

computes both the estimated value and the partial derivatives

of the observed measurement with respect to the orbital

parameters and the measurement parameters. In that respect,

observation matrices are computed during the measurement

estimation. The second step consists in introducing the

dynamic effects in the orbit determination by performing the

numerical integration of the equations of motion. During this

step, the state transition matrices are computed.

C. Automatic Differentiation

In this subsection, theoretical elements about the automatic

differentiation technique are presented. Depending on the

number of observable measurements and the number of

estimated parameters, (13) and (16) show that a significant

quantity of partial derivatives can be computed during an orbit

determination process. Computing and validating partial

derivatives is a critical step to establish an accurate orbit

determination process. However, it is possible, without altering

the complex structure of the algorithms, to perform

computations on extensions of the real numbers in order to add

additional information to the scalar values of the satellite’s

state components, such as partial derivatives. This is possible

thanks to the automatic differentiation technique [29].

Automatic differentiation provides both the partial

derivatives of the observed measurement and of the satellite’s

state vector components without having to find their analytical

expressions. This is the equivalent of calculating the

derivatives by applying chain rule. Let X

sat

be X coordinate of

the satellite’s position. Automatic differentiation provides all

its partial with respect to the components of the satellite’s state

vector.

X

sat

∂X

sat

∂X

sat

⁄∂X

sat

∂Y

sat

⁄⋯

∂X

sat

∂Z

sat

⁄∂X

sat

∂C

r

⁄⋯∂X

sat

∂dt

rec

⁄ (17)

Therefore, the automatic differentiation technique is used to

compute the state transition matrices and the observation

matrices.

D. The State Transition Matrices

The state transition matrices are used to compute the partial

derivative matrix of the batch least squares estimator. The

computation of these matrices is a significant step that should

not be neglected. The state transition matrices are computed

using the variational equations [30]

d

∂Q

t

∂Q

t0

dt =∂Q

t

∂Q

t

.

∂Q

t

∂Q

t0

d

∂Q

t

∂P

t0

dt =∂Q

t

∂Q

t

.

∂Q

t

∂P

t0

+

∂Q

t

∂P

t

(18)

In (18), Q

t

is the derivative of the Q

t

vector with respect to

the time. In other words, Q

t

contains the coordinates of the

satellite’s velocity and acceleration. The satellite’s

acceleration is computed for all force models used in the orbit

determination process. Thanks to the automatic differentiation

technique, the partial derivatives of the satellite’s acceleration

with respect to the satellite’s position; the satellite’s velocity

and the force model parameters are computed at the same time

as the satellite’s acceleration. In this respect, the automatic

differentiation technique affords a simpler computation and

validation of the ∂Q

t

∂Q

t

and ∂Q

t

∂P

t

matrices. Moreover,

measurement parameters do not affect the computation of the

orbital perturbations. Therefore, the partial derivatives with

respect to the measurement parameters have no effect on the

computation of the state transition matrices.

Finally, the variational equations are added to the set of

equations of motions. They are integrated simultaneously by

the numerical integrator to access the state transition matrices.

V. E

XPERIMENTAL RESULTS

In this section, experimental results are presented

demonstrating the benefits of using estimated ionospheric and

tropospheric models during an orbit determination process.

A. Computer characteristics

The tests were performed on a 3.20 GHz Intel Core i5-3470

laptop with 8 GB RAM.

B. GPS Orbit Determination

A GPS orbit determination is used to highlight the

performance of the estimated ionospheric and tropospheric

models. The estimation process uses a total of 8192

pseudorange measurements from a network of ten stations.

Moreover, the pseudorange measurements were taken between

00H00 and 05H00 (Coordinated Universal Time) the 13

th

February 2016. Carrier phase measurements are not used

during the orbit determination process. Fig. 2 illustrates the

location of the stations.

Several criteria are used to draw conclusions about the

relevance of the estimated ionospheric and tropospheric

models. First, it is important to verify the correct estimation of

the satellite’s orbit. Hence, the estimated satellite’s orbit is

compared to the final orbit products of the IGS. Moreover, it is

interesting to analyze the statistical results of the orbit

determination. Hence, a study of the measurement residuals is

also performed.

The setting used for the orbit determination experiments is

presented in Table II.

TABLE II. O

RBIT

D

ETERMINATION

S

ETTING

Parameter Value

Satellite GPS 07

Epoch (ISO 8601) 2016-02-13T02:31:30.000Z

Inertial frame EME2000

Position (m)

X

sat

=

-2747606.681

Y

sat

= 22572091.306

Z

sat

= 13522761.402

Velocity (m/s)

X

sat

= -2729.515

Y

sat

= 1142.663

Z

sat

= -2523.906

Mass (kg) 1630

Satellite’s clock bias (μs) 469.873238

Ellipsoid WGS84 with IERS 2010 Convention

Earth’s potential (degree × order) 20 × 20

Atmospheric drag No

Solar radiation pressure Yes

Relativity force model Yes

Parameter Value

Third body with solid tides Moon and Sun

Attitude model Center body with yaw steering

Numerical integrator Dormand-Prince 853

Optimization engine Levenberg-Marquardt

Station displacement corrections Yes

Antenna phase center position Yes

Tropospheric mapping function Niell mapping function

Pressure, Temperature, humidity Global Pressure Temperature [31]

C. GPS Orbit Determination with Saastamoinen and

Klobuchar models

In this subsection, an orbit determination using empirical

ionospheric and tropospheric models is performed. The

purpose of this experiment is to have reference results in order

to highlight the performance of the estimated delay models.

Therefore, the Saastamoinen and the Klobuchar models are

used to compute the tropospheric delay and the ionospheric

delay, respectively. The statistical results are presented in

Table III. Fig. 3 illustrates the evolution of the measurement

residuals during the orbit determination process

TABLE III. O

RBIT

D

ETERMINATION USING

S

AASTAMOINEN AND

K

LOBUCHAR MODELS

Residuals Orbit Estimation

RMS

(cm)

Min

value

(m)

Max

value

(m)

Mean

value

(m)

Position

error

(m)

Velocity

error

(m/s)

Time

a

(s)

90.0 -3.91 8.38 1.2e-5 4.7 0.0029 32.9

a.

Duration of the orbit determination

The standard deviation of the measurements is 0.90

meters. Moreover, the difference between the estimated

position and the reference position of the satellite is 4.7

meters. The results show that a reasonable orbit determination

is obtained with the empirical ionospheric and tropospheric

models. However, the purpose is to improve these results with

the estimated delay models. Note that this is by no means

intended to be a geodesy level orbit determination. There is

only one satellite, the time range is less than one half orbit,

carrier phase measurements are not used, etc. It should only

be considered as the reference for the next cases with

tropospheric and ionospheric effects

D. GPS Orbit Determination with Estimated Total Zenith

Delay and Klobuchar Ionospheric Model

In this subsection, the total zenith delay, T

total

z

, is estimated

during the orbit determination process. Moreover, the

tropospheric delay is computed according to (5) with a Niell

mapping function. In order to highlight the impact of the

estimation of the total zenith delay, the ionospheric delay is

still computed using the Klobuchar model. In this respect, the

only difference with the previous experiment is the estimation

of the total zenith delay instead of the use of the Saastamoinen

model.

Fig. 2. Stations used for the GPS orbit determination test. A total of 8192

pseudorange measurements, from ten different stations, is used to

determine the state vector of the GPS satellite. Reference stations are

mainly located in the United States because they correspond to the

satellite visibility during the orbit determination time span.

The experimental results of the second experiment are

summarized in Table IV.

TABLE IV. O

RBIT

D

ETERMINATION USING

E

STIMATED TOTAL ZENITH

DELAY AND

K

LOBUCHAR MODEL

Residuals Orbit Estimation

RMS

(cm)

Min

value

(m)

Max

value

(m)

Mean

value

(m)

Position

error

(m)

Velocity

error

(m/s)

Time

(s)

88.8 -3.92 8.46 2.2e-4 2.8 0.0026 33.6

The second experiment highlights the performance of the

estimation of the total zenith delay. First, a small improvement

is observed on the standard deviation of the residuals.

Moreover, a gain of 1.9 meters is obtained on the estimation of

the satellite’s position (about 40%). A small gain is also

obtained on the estimation of the satellite’s velocity. Finally,

the computation time is on the same order of magnitude as the

previous experiment. Therefore, estimating the total zenith

delay improves the estimation of the satellite’s orbit.

E. GPS Orbit Determination with Estimated Wet Zenith

Delay and Klobuchar Ionospheric Model

In this subsection, the wet zenith delay, T

wet

z

, is estimated

during the orbit determination process. Moreover, the

tropospheric delay is computed according to (4) with a Niell

mapping function. The ionospheric delay is still computed

using the Klobuchar model. The purpose of this experiment is

to compare the estimation of the wet zenith delay with the

estimation of the total zenith delay under orbit determination

conditions.

The experimental results of the third experiment are

summarized in Table V.

TABLE V. O

RBIT

D

ETERMINATION USING

E

STIMATED

W

ET ZENITH

DELAY AND

K

LOBUCHAR MODEL

Residuals Orbit Estimation

RMS

(cm)

Min

value

(m)

Max

value

(m)

Mean

value

(m)

Position

error

(m)

Velocity

error

(m/s)

Time

(s)

88.8 -3.82 8.47 8.9e-4 2.8 0.0026 34.1

The results obtained with the third experiment are close to

those obtained with the second one. First, the position error; the

velocity error; and the RMS are identical. Moreover, the

statistics on the residual values and the computation time are

on the same order of magnitude as the second experiment.

Therefore, estimating the unknown tropospheric parameters

improves the accuracy of the orbit determination process.

However, there is no significant improvement depending on

whether the estimated parameter is the total zenith delay or the

wet zenith delay. For the two last experiments, the tropospheric

delay is computed using (5) with a Niell mapping function.

F. GPS Orbit Determination with Estimated Total Zenith

Delay and Estimated vTEC

In this subsection, the vertical total electron content,

vTEC, is estimated during the orbit determination process.

Therefore, the ionospheric delay is computed using (8).

Moreover, a Single Layer Model mapping function is used.

Finally, the tropospheric delay is computed using (5) with an

estimated total zenith delay. The experimental results are

Fig. 3. Residuals of the pseudorange measurements for each station during the orbit determination process. Ionospheric and tropospheric delays are computed

according to empirical models. The figure shows that the main part of the residuals is between 3.0 meters and -3.0 meters.

summarized in Table VI. Fig. 4 illustrates the evolution of the

measurement residuals during the orbit determination process.

TABLE VI. O

RBIT

D

ETERMINATION USING

E

STIMATED

W

ET ZENITH

DELAY AND

E

STIMATE D VERT ICAL

TEC

Residuals Orbit Estimation

RMS

(cm)

Min

value

(m)

Max

value

(m)

Mean

value

(m)

Position

error

(m)

Velocity

error

(m/s)

Time

(s)

88.1 -3.92 8.40 4.0e-4 1.9 0.0026 34.9

The fourth experiment highlights the performance of the

estimation of the total zenith delay and the vTEC. Compared to

the second and third experiments, a small improvement is

demonstrated on the standard deviation of the residuals.

Moreover, the temporal variation of the residuals in Fig. 4 is

very close to the one on Fig. 3. Finally, the main benefit is on

the estimation of the satellite’s orbit. Although the velocity

error is similar to the two previous experiments, the position

error decreases from 2.8 meters to 1.9 meters. Compared to the

first experiment, the position error decreases from 4.7 meters to

1.9 meters, which represents a gain of about 60% on the

estimation of the satellite’s position. Therefore, estimating the

total zenith delay and the vTEC with the satellite’s orbit during

an orbit determination process can improve significantly the

final results.

G. GPS Orbit Determination with Estimated Total Zenith

Delay and Estimated sTEC

In this subsection, the slant total electron content, sTEC, is

estimated during the orbit determination process. The purpose

of this experiment is to highlight the best configuration

between the estimation of the vTEC parameter and the

estimation of the sTEC parameter. Therefore, the ionospheric

delay is computed using (6). The experimental results are

summarized in Table VII.

TABLE VII. O

RBIT

D

ETERMINATION USING

E

STIMATED

W

ET ZENITH

DELAY AND

E

STIMATED SLANT

TEC

Residuals Orbit Estimation

RMS

(cm)

Min

value

(m)

Max

value

(m)

Mean

value

(m)

Position

error

(m)

Velocity

error

(m/s)

Time

(s)

93.9 -3.84 8.21 -2.4e-3 2.6 0.0028 36.1

The fifth experiment shows a gain of 0.2 meters on the

estimation of the satellite’s position compared to the second

and third experiments. However, there are some drawbacks to

the estimation of the sTEC. First, the value of the standard

deviation increases compared to the previous experiments. An

increase of 5.9 centimeters is obtained. Moreover, the position

error is bigger than the fourth experiment. The same remark

applies to the velocity error. Therefore, higher accuracy is

obtained with the estimation of the vTEC instead of the

estimation of the sTEC.

H. Discussion

In this subsection, the experimental results are discussed.

The orbit determination outputs are all summarized in Table

VIII.

Fig. 4. Residuals of the pseudorange measurements for each station during the orbit determination process. Ionospheric and tropospheric delays are

estimated. The figure shows that the main part of the residuals is between 3.0 meters and -3.0 meters. The temporal variation of the residuals is close to the

one on Fig. 3.

TABLE VIII. O

RBIT

D

ETERMINATION USING

E

STIMATED

W

ET ZENITH

DELAY AND

E

STIMATE D VERT ICAL

TEC

Residuals Orbit Estimation

EX RMS

(cm)

Min

value

(m)

Max

value

(m)

Mean

value

(m)

Position

error

(m)

Velocity

error

(m/s)

Time

(s)

1 90.0 -3.91 8.38 1.2e-5 4.7 0.0029 32.9

2 88.8 -3.92 8.46 2.2e-4 2.8 0.0026 33.6

3 88.8 -3.82 8.47 8.9e-4 2.8 0.0026 34.1

4 88.1 -3.92 8.40 4.0e-4 1.9 0.0026 34.9

5 93.9 -3.84 8.21 -2.4e-3 2.6 0.0028 36.1

First, the five experiments have shown that the estimation

of the total zenith delay and the vTEC improves the accuracy

of the orbit determination process. The benefits are both on the

standard deviation of the residuals and on the estimation of the

satellite’s orbit compared to the final orbit products of the IGS.

Second, it can be noted, thanks to experiments 2 and 3, that

there is no significant difference between the estimation of the

wet zenith delay and the estimation of the total zenith delay.

Indeed, the statistical results are very close between the two

experiments.

Third, experiments 4 and 5 have demonstrated that higher

accuracy is obtained with the estimation of the vTEC instead of

the estimation of the sTEC. The reason is mathematical. In the

case of the fifth experiment, the ionospheric mapping function

is not computed. Therefore, the sTEC parameter is estimated

without the information of the satellite’s orbit. In other words,

the partial derivatives of the sTEC with respect to the satellite’s

position and velocity are equals to zero, degrading the

estimation.

VI. C

ONCLUSION

This paper presents two estimated models to quantify the

ionospheric and tropospheric delays during an orbit

determination process. The tropospheric delay is estimated by

the total zenith delay whereas the ionospheric delay is

estimated by the vertical TEC. The estimated models

demonstrates better experimental results than the empirical

models in terms of standard deviation of the residuals and

satellite’s orbit estimation. The estimation of the ionospheric

and tropospheric delays is currently one of the best solutions

for an accurate and fast single-frequency GNSS orbit

determination. However, the experiments demonstrate that

additional works are still needed to achieve the accuracy level

of the GNSS applications.

In order to improve the accuracy of the orbit determination,

future works will focus on the addition of the carrier phase

measurement on the orbit determination process. Moreover,

future works will also focus on multi-satellites orbit

determination in order to be able to estimate the state vector of

a whole constellation of GNSS satellites during the same orbit

determination process. Finally, there are other improvements

that can be performed on the computation of the ionospheric

mapping function.

A

CKNOWLEDGMENT

The authors would like to thank the Orekit user community

and all the persons who participate to improve this project.

R

EFERENCES

[1] G.M. Someswar, T.P. Surya Chandra Rao, and D.R. Chigurukota,

“Global navigation satellite systems and their applications,” in

International Journal of Software and Web Sciences, vol. 3, no. 1, pp.

17-23, 2013.

[2] J.J. Wang, “Antenna for global navigation satellite system (GNSS),” in

Proceedings of the IEEE, vol. 100, no. 7, pp. 2349-2355, 2012.

[3] A. Martellucci and R.P. Cerdeira, “Review of tropospheric, ionospheric

and multipath data and models for global navigation satellite systems,”

in European Conference on Antennas and Propagations, IEEE, pp.

3697-3702, 2009.

[4] F. Meyer, R. Bamler, N. Jakowski and T. Fritz, “The potential of low-

frequency SAR systems for mapping ionospheric TEC distributions,” in

IEEE Geoscience and Remote Sensing Letters, vol. 4, no. 3, pp. 560-

564, 2006.

[5] J. Askne and H. Nordius, “Estimation of tropospheric delay for

microwaves from surface weather data,” in Radio Science, vol. 22, no. 3,

pp. 379-386, 1987.

[6] J.A. Klobuchar, “Ionospheric time-delay algorithm for single-frequency

GPS user,” IEEE Transactions on aerospace and electronic systems, no.

3, pp. 325-331, 1987.

[7] S.M. Radicella, “The NeQuick model genesis, uses and evolution,” in

Annals of geophysics, vol. 52, no. 3-4, pp. 417-422, 2009.

[8] S. Schaer, W. Gurtner and J. Feltens, “IONEX: The ionosphere map

exchange format version 1,” in Proceedings of the IGS AC workshop,

vol. 9, no. 11, 1998.

[9] J. Saastamoinen, “Atmospheric correction for the troposphere and

stratosphere in radio ranging satellites,” The use of artificial satellites for

geodesy, pp. 247-251, 1972.

[10] J. Park, S.V. Veettil, M. Aquino, L. Yang and C. Cesaroni, “Mitigation

of ionospheric effects on GNSS positioning at low latitudes,” in

NAVIGATION, Journal of The Institut of Navigation, vol. 64, no. 1, pp.

67-74, Spring 2017.

[11] C. Shi, S. Gu, Y. Lou and M. Ge, “An improved approach to model

ionospheric delays for single-frequency precise point positioning,” in

Advances in Space Research, vol. 49, no. 12, pp. 1698-1708, 2012.

[12] D. Kalman, “Doubly recursive multivariate automatic differentiation,” in

Mathematics magazine, vol. 75, no. 3, pp. 187-202, 2002.

[13] L. Maisonobe, V. Pommier and P. Parraud, “Orekit: an open-source

library for operational flight dynamics applications,” in International

conference on astrodynamics tool and techniques, pp. 3-6, 2010.

[14] L. Maisonobe, P.J. Cefola, N. Frouvelle, S. Herbinière, F.X. Laffont, S.

Lizy-Destrez and T. Neidhart, “Open governance of the Orekit space

flight dynamics library,” in International conference on astrodynamics

tool and techniques, 2012.

[15] J. Sanz Subirana, J.M. Juan Zornoza and H. Hernánder-Pajares, GNSS

data processing fundamentals and algorithms, vol. 1, 2013.

[16] J.L. Davis, T.A. Herring, I.I. Shapiro, A.E.E. Rogers and G. Elgered,

“Geodesy for radio interferometry: effects of atmospheric modeling

errors of estimates baseline length,” in Radio science, vol. 20, no. 6, pp.

1593-1607, 1985.

[17] G. Petit and B. Luzum, “IERS convention 2010, models for atmospheric

propagation delays,” no.36, 2010.

[18] A.E. Niell, “Global mapping functions for the atmosphere delay at radio

wavelengths,” in Journal of Geophysical Research: Solid Earth, vol.

101, no. B2, pp. 3227-3246, 1996.

[19] J. Böhm, A.E. Niell, P. Tregoning and H. Schuh, “Global mapping

function (GMF): a new empirical mapping function based on numerical

weather model data,” in Geophysical Research Letters, vol. 33, no. 7,

2006.

[20] J. Böhm, B. Werl and H. Schuh, “Troposphere mapping functions for

GPS and very long baseline interferometry from european center for

medium-range weather forecasts operational analysis data,” in Journal

of Geophysical Research: Solid Earth, vol. 111, no. B2, 2006.

[21] D. Landskron and J. Böhm, “WMF3/GPT3: refined discrete and

empirical troposphere mapping functions,” in Journal of Geodesy, vol.

92, no. 4, pp. 349-360, 2018.

[22] A. Leick, L. Rapoport and D. Tatarnikov, GPS satellite surveying, 4

th

ed.

John Wiley & Sons, 2015.

[23] N. Ya’acob, M. Abdullah and M. Ismail, “Determination of the GPS

total electron content using single layer model (SLM) ionospheric

mapping function,” in International Journal of Computer Science and

Network Security, vol. 8, no. 9, pp. 154-160, 2008.

[24] T. Sakai, T. Yoshihara, S. Saito, K. Matsunaga, K. Hoshinoo and T.

Walter, “Modeling vertical structure of the ionosphere for SBAS,” in

Proceedings of the 22

nd

International Technical Meeting of The Satellite

Division of the Institute of Navigation (ION GNSS), pp. 1257-1267,

2009.

[25] M.M. Hoque and N. Jakowski, “Mitigation of ionospheric mapping

function error,” in Proceedings of the 26

th

International Technical

Meeting of the Satellite Division of The Institute of Navigation (ION

GNSS+ 2013), pp. 1848-1855, 2013.

[26] H. Lyu, M. Hernández-Pajares, M. Nohutcu, A. Garcia-Rigo, H. Zhang

and J. Liu, “The Barcelona ionospheric mapping function (BIMF) and

its application to northern mid-latitudes,” in Gps solutions, vol. 22, no.

3, pp. 67, 2018.

[27] O. Montenbruck and E. Gill, Satellite orbits: models, methodes, and

applications, Springer, Berlin, 2000.

[28] D.A. Vallado, Fundamentals of astrodynamics and applications, 3

rd

ed.

Springer, 2007.

[29] A. Antolino and L. Maisonobe, “Automatic differentiation for

propagation of orbit uncertainties,” in Stardust final conference, 2016.

[30] E. Hairer, S.P. Norsett, G. Wanner, Solving ordinarydifferential

equation (Nonstiff problems), vol. 1, Springer-Verlag, 1993.

[31] J. Böhm, R. Heinkelmann and H. Schuh, “Short note: a global model of

pressure and temperature for geodetic applications,” in Journal of

Geodesy, vol. 81, no. 10, pp. 679-683, 2007.