Conference PaperPDF Available

Improve GNSS Orbit Determination by using Estimated Tropospheric and Ionospheric Models

Authors:

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.
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
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... During the computation, drag coefficient is also estimated. In all three cases, ionospheric and tropospheric pseudo range biases are also estimated [14]. The analytical method computations return two new TLEs, that can be immediately propagated by usual SGP4/SDP4 algorithms. ...
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Ionospheric conditions at low latitudes are extremely harsh due to the frequent occurrence of scintillation and the presence of strong TEC gradients. For this study, the São Paulo state region in Brazil is chosen as a test area. This study presents a strategy to mitigate the ionospheric impact on RTK positioning with an experimental result. The proposed strategy explores two approaches that can be applied simultaneously: a) to mitigate the scintillation effect on the GNSS signals by refining the stochastic model of the corresponding observations and b) to precisely estimate the residual double difference ionospheric delay by exploiting an accurate TEC map. The strategy was tested on a long baseline kinematic processing under strong scintillation conditions (DOY21 in 2014). Significant improvements were observed when the combined use of the two mitigation approaches described above was compared with the use of conventional state-of-the-art approaches.
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system. That's right. The automatic differentiation system never formulates a symbolic expression for the derivative. Automatically calling on something like Mathematica to produce a symbolic derivative, and then plugging in a value for x is the wrong image entirely. Automatic differentiation is something completely different. Well OK, but so what? Symbolic algebra systems are so prevalent and powerful today, why should we be concerned with avoiding symbolic methods? There are two answers. The first is practical. Symbolic generation of derivatives can lead to exponential growth in the length of expressions. That causes computational problems in real applications. Accordingly, there is a practical applied side to the subject of automatic differentiation, as witnessed by the serious attention of computer scientists and numerical analysts [3, 4]. The second answer is more mathematical. It is a relatively easy task to create a single variable automatic differentiation system capable of evaluating first derivatives. In fact, writing in this MAGAZINE in 1986, Rall [10] gives a beautiful presentation of just such a system. What is mathematically interesting is an amazingly elegant extension of the one-variable/one-derivative system that handles essentially any number of variables and derivatives. The extension is recursively defined, employing an induction on both the number of variables and the number of derivatives, and using fundamental definitions that are virtually identical to the ones used in Rall's system. The purpose of this paper is to present the recursive automatic differentiation system. To set the stage, we will begin with a brief review of Rall's one-variable/onederivative system, followed by an example of the recursive system in action. Then the mathematical formulation of the recursive system will be presented. The paper will end with a brief discussion of practical issues related to the recursive system.