Methods of precise aircraft positioning in the GPS system with an application of the troposphere correction

Abstract

Journal homepage: http://sjsutst.polsl.pl Article citation information: Kirschenstein, M., Krasuski, K., Goś, A. Methods of precise aircraft positioning in the GPS system with an application of the troposphere correction. Summary. This article presents the results of studies concerning the designation of accuracy in aircraft navigation positioning by means of the SPP and the SBAS code methods. The examination of the aircraft positioning accuracy was made in the aspect of the use of tropospheric correction in observation equations of the SPP and the SBAS positioning methods. The accuracy of the coordinates of the aircraft in the SPP and the SBAS solutions was referenced to the DGPS reference solution. The investigations were conducted on raw observation data and GPS navigation data in an air test in Dęblin. Based on the conducted calculations, it was proved that the lack of use of tropospheric correction in the SPP method causes an error in an aircraft position up to 18.5 m, and in the SBAS method up to 23.2 m. In addition, the statistical measure of RMS accuracy in the absence of applying the tropospheric correction in the SPP method results in an accuracy decrease to 8.6 m, and in the SBAS method to 12.2 m, accordingly.

Full text

Scientific Journal of Silesian University of Technology. Series Transport
Zeszyty Naukowe Politechniki Śląskiej. Seria Transport
Volume 109 2020
p-ISSN: 0209-3324
e-ISSN: 2450-1549
DOI: https://doi.org/10.20858/sjsutst.2020.109.7
Journal homepage: http://sjsutst.polsl.pl
Article citation information:
Kirschenstein, M., Krasuski, K., Goś, A. Methods of precise aircraft positioning in the GPS
system with an application of the troposphere correction. Scientific Journal of Silesian
University of Technology. Series Transport. 2020, 109, 73-84. ISSN: 0209-3324.
DOI: https://doi.org/10.20858/sjsutst.2020.109.7.
Małgorzata KIRSCHENSTEIN
1
, Kamil KRASUSKI
2
, Artur GOŚ3
METHODS OF PRECISE AIRCRAFT POSITIONING IN THE GPS
SYSTEM WITH AN APPLICATION OF THE TROPOSPHERE
CORRECTION
Summary. This article presents the results of studies concerning the designation
of accuracy in aircraft navigation positioning by means of the SPP and the SBAS
code methods. The examination of the aircraft positioning accuracy was made in
the aspect of the use of tropospheric correction in observation equations of the SPP
and the SBAS positioning methods. The accuracy of the coordinates of the aircraft
in the SPP and the SBAS solutions was referenced to the DGPS reference solution.
The investigations were conducted on raw observation data and GPS navigation
data in an air test in Dęblin. Based on the conducted calculations, it was proved that
the lack of use of tropospheric correction in the SPP method causes an error in an
aircraft position up to 18.5 m, and in the SBAS method up to 23.2 m. In addition,
the statistical measure of RMS accuracy in the absence of applying the tropospheric
correction in the SPP method results in an accuracy decrease to 8.6 m, and in the
SBAS method to 12.2 m, accordingly.
Keywords: GPS system, troposphere correction, SPP method, SBAS method,
DGPS method, accuracy
1
Military University of Aviation, Institute of Navigation, Dywizjonu 303 nr 35 Street, 08-521 Dęblin, Poland.
Email: m.kirschenstein@law.mil.pl. ORCID: https://orcid.org/0000-0002-4817-083X
2
Military University of Aviation, Institute of Navigation, Dywizjonu 303 nr 35 Street, 08-521 Dęblin, Poland.
Email: k.krasuski@law.mil.pl. ORCID: https://orcid.org/0000-0001-9821-4450
3 Military University of Aviation, Institute of Navigation, Dywizjonu 303 nr 35 Street, 08-521 Dęblin, Poland.
Email: a.gos@law.mil.pl. ORCID: https://orcid.org/ 0000-0002-4268-8830

74 M. Kirschenstein, K. Krasuski, A. Goś
1. INTRODUCTION
Along with the use of GPS satellite technology in aviation, there has been a rapid increase
in the methods of aircraft precise positioning in the area of air navigation. The methods of
aircraft positioning for GPS satellite technology can be divided into absolute methods (non-
differential) and differential methods (relative) [16]. Moreover, among the absolute and
differential methods, both code and phase observations of GPS measurements are exploited [3].
In addition, the absolute and differential positioning methods may be single- frequency, dual-
frequency or multi-frequency [8].
The basic method of GPS positioning in aviation is a construction of observation equations,
elimination or modelling of systematic errors as well as determining unknown parameters. The
most common methods of GPS positioning in aviation are as follows: SPP code method [7],
SBAS positioning method [4] and the DGPS differential method [5]. In the SPP code method,
the determined parameters are the aircraft position and the GPS receiver on-board clock. The
modelled parameters are satellite-receiver geometric distance, satellite clock error, ionospheric
and tropospheric correction, relativistic correction, TGD hardware delay and a multipath effect
[15]. The method of positioning designated SBAS parameters are also the position of the
aircraft and the GPS receiver clock error. Furthermore, the modelled parameters are as well:
the geometric distance satellite-receiver, satellite clock error, tropospheric and ionospheric
correction, relativistic correction, TGD hardware delay and a multipath effect. It should be
emphasised that the coordinates of the GPS satellite, the GPS satellite clock error, tropospheric
and ionospheric correction are modelled by means of the SBAS correction algorithms [4]. In
the DGPS positioning method, the determined parameters are finally the aircraft position and
the clock difference of a GPS on-board receiver as well as a GPS receiver mounted on the
reference station. Then, the modelled parameters are similarly: the satellite-receiver geometric
distance, tropospheric and ionospheric correction and a multipath effect. However, the satellite
clock error, relativistic correction and TGD hardware delay are eliminated from the observation
equations by applying the differentiation operator [1].
Within this work, the authors intend to present the research results of aircraft positioning
accuracy in the aspect of using the tropospheric correction in the navigation calculations. The
position of the aircraft will be determined based on the SPP code method in the GPS system,
the SBAS method for EGNOS corrections, and the DGPS differential method. This work
explains how a systematic error in the form of the tropospheric correction affects the accuracy
of aircraft positioning for the abovementioned three methods of satellite positioning in air
navigation. The examination exploits real GPS data from an on-board receiver and a ground
reference station. The recorded data were used for numerical calculations in the RTKLIB
v.2.4.2. programme and to develop the results in the Scilab v.6.0.0. programme.
2. RESEARCH METHODOLOGY
A mathematical model to determine the position of the aircraft in the SPP code method, in the
GPS system, can be described as below [12]:
 
· Rel c dtr dts Ion Trop l TGD Mp

       
(1)
where:
l
- code measurement (pseudorange) registered by the airborne receiver in the GPS system,

- the geometric distance satellite and the airborne receiver in the GPS system,

Methods of precise aircraft positioning in the GPS system with… 75.
     
2 2 2
GPS GPS GPS
x X y Y z Z

     
,
 
, ,
GPS GPS GPS
X Y Z
- satellites coordinates in the GPS system,
 
, , x y z
- aircraft coordinates in the geocentric XYZ frame,
c
- speed of light,
dtr
- receiver clock bias,
dts
- satellite clock bias,
Ion
- ionosphere correction,
Trop
- troposphere correction,
Rel
- relativistic correction,
TGD
- timing group delay,
Mp
- multipath effect.
The mathematical model of the aircraft position determination in the SBAS method for
EGNOS corrections can be described as below [10]:
 
* * * *
· Rel c dtr dts Ion Trop l TGD Mp PRC

        
(2)
where:
l
- code measurement (pseudorange) registered by the airborne receiver in the GPS system,
*

- the geometric distance satellite and the airborne receiver in the GPS system, after long-
term EGNOS correction,
     
2 2 2
* * *
GPS GPS GPS
x X y Y z Z

     
,
 
* * *
, ,
GPS GPS GPS
X Y Z
- satellites coordinates in the GPS system, after long-term EGNOS
correction,
 
, , x y z
- aircraft coordinates in the geocentric XYZ frame,
c
- speed of light,
dtr
- receiver clock bias,
*
dts
- satellite clock bias, after long-term EGNOS correction,
*
Ion
- ionosphere correction, based on GRID SBAS model,
*
Trop
- troposphere correction, based on RTCA-MOPS SBAS troposphere model,
Rel
- relativistic correction,
TGD
- timing group delay,
Mp
- multipath effect,
PRC
- fast EGNOS correction.
The mathematical model of determining the aircraft position in the DGPS differential
method in the post-processing mode can be described as below [2]:
·l c dtr Ion Trop Mp

          
(3)
where:
l
- difference between pseudorange registered by the airborne receiver and reference station
in the GPS system,


- difference between geometric distance: satellite-airborne receiver, and satellite-reference
station in the GPS system,

76 M. Kirschenstein, K. Krasuski, A. Goś
c
- speed of light,
dtr
- difference between airborne receiver clock bias and reference station receiver clock bias,
Ion
- difference between ionosphere correction for airborne receiver and reference station,
Trop
- difference between troposphere correction for airborne receiver and reference station,
Mp
- difference between multipath effect for airborne receiver and reference station.
3. THE RESEARCH EXPERIMENT
In the examination test, scientific examinations were conducted to determine the accuracy
of aircraft positioning in the aspect of exploiting the tropospheric correction in navigation
computations. The aircraft position was determined given the SPP code method, the SBAS
positioning method, and the DGPS differential method in post-processing. In the calculations,
to determine the accuracy of aircraft positioning, a comparison between the designated
coordinates in the SPP vs. DGPS and SBAS vs. DGPS solutions was made. Within the SPP and
the SBAS method, the authors obtained two SBAS positioning solutions: the former included
the tropospheric correction, whereas the latter disregarded the tropospheric correction. In the
DGPS solution, the authors considered the tropospheric correction. Furthermore, the designated
aircraft position in the DGPS solution is a reference position for the performed calculations.
The SPP method model had the tropospheric correction model used as Saastamoinen model [9].
In the SBAS solution, the authors used a model of the tropospheric correction as the RTCA
MOPS-SBAS model [11]. Then, in the DGPS solution, the authors used a model of the
tropospheric correction as the Saastamoinen model. The calculations were made in the RTKLIB
v.2.4.2 programme [17]. The calculations are based on the GPS data derived from the on-board
receiver mounted in a Cessna 172. The data comes from a test flight over Dęblin. Moreover, in
the DGPS differential method, the authors used the data derived from the GPS receiver mounted
on the reference station in Dęblin. The SBAS method used corrections from the EGNOS S120
satellite. The calculations performed with an interval and time synchronisation were equal to
1 s. In addition, the remaining comparative analyses were performed in the Scilab v.6.0.0
programme [18].
In the first stage of the research, the authors determined the position of the Cessna 172 for
the SPP method, twice: initially with the tropospheric correction, and later without its use. In
the second stage of the research, it was possible to designate the position of the Cessna 172,
first, for the SBAS method, and second, without it. In the next step, the authors designated the
reference position of the Cessna 172 for the DGPS differential method, using the tropospheric
correction. The accuracy analysis is presented in section 4.
4. RESULTS
The analysis of accuracy was made to compare the designated aircraft coordinates in the SPP,
SBAS and DGPS solutions. The comparative analysis was performed for geocentric XYZ
aircraft coordinates. In the second stage, the authors specified the accuracy of coordinates of
the aircraft in the SPP code solution, as below [6]:
SPP DGPS
SPP DGPS
SPP DGPS
dx x x
dy y y
dz z z



(4)

Methods of precise aircraft positioning in the GPS system with… 77.
where:
 
, ,
SPP SPP SPP
x y z
- obtained aircraft coordinates from Equation 1,
 
, ,
DGPS DGPS DGPS
x y z
- obtained aircraft coordinates from Equation 3.
Furthermore, for the results of Equation 4, the authors determined
a statistic quantity, which determines the positioning accuracy in the form of RMS parameter,
as below [13]:
2
2
2
dx
dy
dz
dx
RMS N
dy
RMS N
dz
RMS N









(5)
where:
N
- number measurement epochs.
In the second stage, the authors specified the accuracy of aircraft coordinates in the SBAS
code solution, as below [6]:
SBAS DGPS
SBAS DGPS
SBAS DGPS
rx x x
ry y y
rz z z



(6)
where:
 
, ,
SBAS SBAS SBAS
x y z
- obtained aircraft coordinates from Equation 2,
 
, ,
DGPS DGPS DGPS
x y z
- obtained aircraft coordinates from Equation 3.
Furthermore, for the results of Equation 6, the authors determined
a statistic quantity, which determines the positioning accuracy in the form of RMS parameter,
as below [13]:
2
2
2
rx
ry
rz
rx
RMS n
ry
RMS n
rz
RMS n









(7)
where:
n
- number measurement epochs.

78 M. Kirschenstein, K. Krasuski, A. Goś
Fig. 1 shows the results of the aircraft positioning accuracy using the SPP code method. The
results in Fig. 1 considers the impact of the tropospheric correction in the process of computing
the position of the aircraft in the SPP method. The aircraft positioning accuracy along the X-
axis ranged from -4.2 to +3.4 m. Next, the aircraft positioning accuracy along the Y-axis ranged
from -1.5 to +1.2 m. In addition, the accuracy of aircraft positioning along the Z-axis ranged
from -1.7 to +2.2 m. It is worth emphasising that the average positioning accuracy is equal to
+0.4 m along the X-axis, +0.2 m along the Y-axis, and +0.1 m along the Z-axis.
Fig. 2 shows the results of the aircraft positioning accuracy using the SPP code method. The
results in Fig. 2 do not consider the effect of the tropospheric correction in the computational
process of the aircraft position in the SPP method. The aircraft positioning accuracy along the
X-axis ranged from -1.1 to +15.6 m. Next, the aircraft positioning accuracy along the Y-axis
ranged from -0.1 to +6.9 m. In addition, the accuracy of aircraft positioning along the Z-axis
ranged from +4.5 to +18.5 m. It is worth to note that the average positioning accuracy is equal
to +7.7 m along the X-axis, +2.3 m along the Y-axis, and +8.4 m along the Z-axis.
Fig. 3 shows the results of the aircraft positioning accuracy using the SBAS code method.
The results in Fig. 1 do not include the effects of the tropospheric correction in the
computational process of the aircraft in the SBAS positioning method. The aircraft positioning
accuracy along the X-axis ranged from +2.0 to +6.9 m. Next, the aircraft positioning accuracy
along the Y-axis ranged from -1.0 to +0.5 m. In addition, the accuracy of aircraft positioning
along the Z-axis ranged from +1.1 to +5.7 m. It is imperative to note that the average positioning
accuracy is equal to +3.9 m along the X-axis, -0.3 m along the Y-axis, and +3.2 m along the Z-
axis.
Fig. 1. The accuracy of aircraft position based on the SPP solution with troposphere correction
[Source: Based on Scilab software]

Methods of precise aircraft positioning in the GPS system with… 79.
Fig. 2. The accuracy of aircraft position based on the SPP solution without troposphere
correction [Source: Based on Scilab software]
Fig. 3. The accuracy of aircraft position based on the SBAS solution with troposphere
correction [Source: Based on Scilab software]

80 M. Kirschenstein, K. Krasuski, A. Goś
Fig. 4 shows the results of the aircraft positioning accuracy using the SBAS positioning
method. The results in Fig. 1 do not include the impact of the tropospheric correction in the
process of computing the position of the aircraft in the SBAS positioning method. The aircraft
positioning accuracy along the X-axis ranged from +4.8 to +18.5 m. Next, the aircraft
positioning accuracy along the Y-axis ranged from -0.6 to +5.2 m. In addition, the accuracy of
aircraft positioning along the Z-axis ranged from +7.2 to +23.2 m. It is of considerable note that
the average positioning accuracy is equal to +9.9 m along the X-axis, +1.4 m along the Y-axis,
and +11.8 m along the Z-axis.
Fig. 4. The accuracy of aircraft position based on the SBAS solution without troposphere
correction [Source: Based on Scilab software]
Figs. 5 and 6 illustrate the results of 3D-error aircraft position in a 3D plane. The shift value
of the designated aircraft position in the SPP and the SBAS solutions against the reference
position determined by the DGPS technique is defined as follows [14]:
2 2 2
2 2 2
3dx dy dz
D error rx ry rz






(8)
The values of the 3D-error parameter for the SPP method range from 0.1 to 4.4 m, using the
troposphere correction in navigation computations for the aircraft position. The 3D-error
parameter values for the SPP method range from 4.7 to 22.4 m, when the tropospheric
correction is not included in the navigation computations of the aircraft position. The 3D-error
parameter value for the SBAS method ranges from 2.6 to 8.0 m, using the tropospheric
correction in the navigation calculations of the aircraft position. Moreover, the values of the 3D-

Methods of precise aircraft positioning in the GPS system with… 81.
error parameter for the SBAS method range from 9.2 to 28.1 m, when the tropospheric
correction is not included in navigation computations of the aircraft position. Based on the 3D-
error parameter findings, it can be observed that disregarding the tropospheric correction in
calculations results in massive degradation of an aircraft position against the reference
trajectory, for example, even up to 22.4 m in the SPP method and 28.1 m in the SBAS method,
respectively. Therefore, it can be concluded that the tropospheric correction is of huge
importance in determining an aircraft position during flight operations.
Fig. 5. The values of 3D-error of aircraft position accuracy based on the SPP solution
[Source: Based on Scilab software]
Tab. 1
Comparison of obtained RMS parameter [Authors’ study]
Positioning method
RMS along to X-
axis [m]
RMS along to Y-
axis [m]
RMS along to Z-
axis [m]
SPP (with troposphere
correction)
1.7
0.5
0.6
SPP (without
troposphere correction)
8.4
2.6
8.6
SBAS (with
troposphere correction)
4.0
0.3
3.2
SBAS (without
troposphere correction)
10.4
1.6
12.2

82 M. Kirschenstein, K. Krasuski, A. Goś
Fig. 6. The values of 3D-error of aircraft position accuracy based on the SBAS solution
[Source: Based on Scilab software]
Tab. 1 shows the results of accuracy in the form of the statistical parameter RMS, in
accordance with Equations 5 and 7. It can be observed that disregarding the tropospheric
correction in the SPP method and the SBAS method causes significant degradation of the
aircraft position. In the SPP method, the lowest RMS accuracy equals 1.7 m, using the
tropospheric correction in the observation model. Next, the lack of troposphere correction in
the SPP method causes degradation in aircraft position accuracy, even to the level of 8.6 m. In
the SBAS method, the lowest RMS accuracy equals 4.0 m when using the tropospheric
correction in the observation model. Besides, the lack of tropospheric correction in the SBAS
method leads to the degradation of aircraft position accuracy, even to the level of 12.2 m.
5. CONCLUSION
This paper demonstrates the results of research into the aircraft positioning accuracy by
means of the SPP, SBAS and DGPS methods. In particular, the study focuses on examining the
impact of the tropospheric correction on aircraft positioning accuracy in air navigation. In
practice, this work presents research tests showing how the use of the tropospheric correction
in the SPP and the SBAS methods affect the accuracy of determining aircraft coordinates. The
study uses real observation and GPS navigation data derived from an experimental air test made
by the Cessna 172. Based on the conducted calculations, it was found that:
- the lack of use of the tropospheric correction in the SPP method results in a drop in accuracy
of determining the XYZ aircraft coordinates even to the level of 18.5 m;

Methods of precise aircraft positioning in the GPS system with… 83.
- the lack of use of the tropospheric correction in the SBAS method results in a drop in
accuracy of determining the XYZ aircraft coordinates even to the level of 23.2 m;
- the lack of use of the tropospheric correction in the SPP method results in a rise in the RMS
error to the level of 8.6 m;
- the lack of use of the tropospheric correction in the SBAS method results in a rise in the
RMS error to the level of 12.2 m.
The obtained research findings, in the work, indicate that the systematic error in the form of
the tropospheric correction exerts a tremendous influence on the designation of an aircraft
position. Furthermore, ignoring the systematic error in an observational equation of determining
the aircraft position may cause large degradation of accuracy in the designated coordinates of
a moving object in air navigation. Hence, it is important to make a correct interpretation of
observational equations in a given method for determining the position of a moving object in
the aspect of modelling systematic errors.
Acknowledgements
This paper was supported by the Military University of Aviation in 2020.
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Received 12.07.2020; accepted in revised form 29.10.2020
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