Article citation information:
Krasuski,
K., Kirschenstein, M. Examination of
different models of troposphere delays in SBAS positioning in aerial navigation.
Scientific Journal of Silesian University
of Technology. Series Transport. 2023, 118,
123-137. ISSN: 0209-3324. DOI: https://doi.org/10.20858/sjsutst.2023.118.9.
Kamil KRASUSKI[1],
Małgorzata KIRSCHENSTEIN[2]
EXAMINATION OF DIFFERENT MODELS OF TROPOSPHERE DELAYS IN SBAS POSITIONING
IN AERIAL NAVIGATION
Summary. This paper
presents the results of a study on the use of different tropospheric correction
models in SBAS positioning for air navigation. The paper, in particular,
determines the influence of the Saastamoinen troposphere and RTCA-MOPS models
on the determination of aircraft coordinates and mean coordinate errors in the
SBAS positioning method. The study uses real kinematic data from a GPS
navigation system recorded by an onboard GNSS satellite receiver as well as
SBAS corrections. In the experiment, the authors include SBAS corrections from
EGNOS and SDCM augmentation systems. The navigation calculations were performed
using RTKLIB v.2.4.3 and Scilab 6.1.1 software. Based on the conducted
research, it was found that the difference in aircraft coordinates using
different troposphere models can reach up to ±2.14 m. Furthermore, the
use of the RTCA-MOPS troposphere model improved the values of mean coordinate
errors from 5 to 9% for the GPS+EGNOS solution and from 7 to 12% for the
GPS+SDCM solution, respectively. The obtained computational findings confirm
the validity of using the RTCA-MOPS troposphere model for SBAS positioning in
aerial navigation.
Keywords: troposphere
delay, SBAS, aircraft coordinates, mean errors
1. INTRODUCTION
SBAS
satellite positioning plays a key role in air navigation in determining the
position of an aircraft. The main purpose of using SBAS in air navigation is to
improve the positioning performance of an aircraft. In particular, the improvement
of GNSS satellite positioning performance should be understood as a
determination of positioning quality parameters in the form of accuracy,
continuity, availability and reliability parameters [1]. Of the above four
quality parameters for SBAS satellite positioning in air navigation, accuracy
and reliability appear to be the most important ones [1]. However, to be able to improve GNSS positioning
performance, SBAS corrections must be applied to the Single Point Positioning
(SPP) method [2]. Among the SBAS corrections, it is
possible to distinguish the following: GNSS satellite position corrections,
GNSS satellite clock error corrections, an ionospheric correction and a
tropospheric correction [3]. While GNSS satellite position corrections,
GNSS satellite clock error corrections and an ionospheric correction are
included in the SBAS message, the model of the troposphere has to be calculated
empirically [4]. Therefore, a selection of a suitable
troposphere model for SBAS positioning is crucial, primarily for determining
the ellipsoidal height of an aircraft.
2. SCIENTIFIC KNOWLEDGE ANALYSIS
This
second section describes examples of research papers regarding the subject of
determining the tropospheric correction in SBAS positioning or the influence of
the tropospheric correction in determining the coordinates. Paper [5]
shows the significance of a systematic error, that is, the tropospheric
correction within the GPS+Galileo, SBAS and GBAS systems. Moreover,
publication [6] shows the results of determining the tropospheric correction
using the GPT model and the SBAS model. The conducted
research proved that the SBAS model is better than the GPT model for the
calculation of the tropospheric correction. Also, the paper
[7] discusses the impact of using different tropospheric correction models,
including the SBAS model, in precise GPS, GLONASS, and GPS+GLONASS positioning
for reference station networks. The lowest positioning accuracy was
obtained in the GLONASS solution. Publication [8] shows
the impact of the tropospheric correction model on the algorithm of determining
HPL/VPL integrity parameters in SBAS satellite positioning for aerial
navigation.
Furthermore, the authors of the publication [9] have developed an algorithm for
evaluating the influence of troposphere parameters on SBAS navigation signals
using fuzzy functions. The troposphere parameter
results obtained were related to the state of the ionosphere to determine the
levels of reliability in GPS+SBAS positioning. An
interesting study was conducted in [10], in which a new model for determining
the tropospheric correction for the SBAS system for the East Asian area was
shown. Next, the paper [11] published the results
of a study on the application of a tropospheric correction model for the MSAS
augmentation system for the area of Japan. Furthermore,
the paper [12] shows the application of the tropospheric correction, calculated
for the BDSBAS augmentation system, in the PPP measurement technique for
single- and dual-frequency observations. A similar
study was also conducted in the paper [13], in which different tropospheric
correction models were investigated in the PPP measurement technique for a
network of reference stations over Asia. Paper [14]
presents the impact of different tropospheric correction models in the SPP code
method in global, seasonal and geographical terms. A troposphere model
dedicated to SBAS systems was also used in the study.
Based on the available data, it
appears that:
- the problem of using an appropriate tropospheric correction
model for SBAS positioning is relevant [5],
-
until now, the
problem of determining an appropriate tropospheric correction model has mainly
concerned GNSS satellite navigation [6, 7, 12, 13, 14],
As can be observed in the analysis of the current expertise, there is a
lack of research work on the actual implementation of the given tropospheric
correction algorithm in SBAS positioning for air navigation. In particular, there is no information about the
impact of the proposed tropospheric correction model on the determination of
aircraft coordinates and their mean errors. Therefore, this paper presents the findings of a study on
the application of two tropospheric delay models, that is, the Saastamoinen
model [15] and the RTCA-MOPS model [16], in the process of determining the
aircraft position. Owing to the developed study
results, it is possible to determine which tropospheric delay model is optimal
for the SBAS positioning method in air navigation, especially since the
research will be conducted for two operationally independent SBAS augmentation
systems, that is, EGNOS and SDCM [17]. The flight
tests were conducted in north-eastern Poland. The work is universal in nature
and may be extended to include other SBAS augmentation systems available in
Poland, such as the Indian GAGAN system [17].
3. RESEARCH METHOD
The basic
algorithm of the SBAS positioning method in air navigation can be expressed as
follows [18, 19]:
|
|
(1) |
where:
Based on equation (1), aircraft coordinates are determined from the GPS+SBAS solution using the least squares method. The algorithm of the least squares method is presented below [20]:
where:
In equation (1),
there is a tropospheric delay factor in the form of the parameter
4. RESEARCH EXPERIMENT
The test
experiment was conducted on real GNSS kinematic data recorded by an onboard
receiver mounted on a Diamond DA 20-C1 aircraft. The test flight took place
during the autumn period of 2020 in north-eastern Poland on the
Olsztyn-Suwałki-Olsztyn route. A Septentrio AsterRx2i geodetic
receiver was fixed onboard the aircraft [21]. The satellite receiver recorded
GNSS observations, including GPS code observations with a time interval of 1
second. In addition, owing to the real-time service:
ftp://serenad-public.cnes.fr/SERENAD0 [22], it was possible to collect
corrections from EGNOS and SDCM augmentation systems, which were used in the
navigation calculations in equation (1). As a first
step, navigation calculations for equation (1) were performed in the RTKLIB
v.2.4.3 software [23]. In RTKLIB software, the position of the aircraft was
determined from the GPS+EGNOS and GPS+SDCM solutions. Two
tropospheric delay models were considered in the calculations, that is, the Saastamoinen model and the RTCA-MOPS
model. The configuration of the navigation calculation in RTKLIB was
set as follows [24]:
- positioning
mode: single,
- elevation
mask: 5°,
- source of
ionosphere delay: SBAS corrections using ionosphere GRID maps,
- source of
troposphere delay: Saastamoinen model and RTCA-MOPS model,
- source of
satellite coordinates and clocks: broadcast ephemeris and SBAS message,
- GNSS
system: GPS+EGNOS and GPS+SDCM,
- source of
GPS observations: RINEX format,
- source of
EGNOS and SDCM corrections: EMS file,
- reference
frame of coordinates: WGS-84 frame,
- interval
of computations: 1 s,
- final
coordinates: geocentric XYZ coordinates.
Thus, the
RTKLIB programme ultimately generated four independent determinations of the
aircraft position in the form of:
- GPS+EGNOS
solution from the Saastamoinen model for the
Septentrio AsterRx2ireceiver (EGNOS-SAAS
designation),
- GPS+EGNOS
solution from the RTCA-MOPS model for the
Septentrio AsterRx2ireceiver (EGNOS-RTCA
designation),
- GPS+SDCM
solution from the Saastamoinen model for the
Septentrio AsterRx2ireceiver (SDCM-SAAS
designation),
- GPS+EGNOS
solution from the Saastamoinen model for the Septentrio AsterRx2ireceiver
(EGNOS-RTCA designation).
The aircraft coordinates were finally expressed in the XYZ geocentric
coordinates [20]. The computations findings in graphic, tabular and descriptive
forms are presented in Section 5. Scilab v.6.1.1 software [25] was used to
present the obtained results.
5. RESULTS AND DISCUSSION
To determine the influence of the proposed
tropospheric correction model in the SBAS positioning, the difference of the
determined coordinates from the GPS+EGNOS and GPS+SDCM solutions for the
Saastamoinen troposphere model and RTCA-MOPS is shown first. For this purpose, the parameters
where:
Fig. 1. Difference of aircraft coordinates based on
the GPS+EGNOS solution with
the applied Saastamoinen and RTCA-MOPS model
Figure 1
shows the results of the
Figure 2 shows the results of the
Fig. 2. Difference of aircraft coordinates based on
the GPS+SDCM solution
with the applied Saastamoinen and RTCA-MOPS model
Next, the
average errors of the aircraft coordinates in the form of
Fig. 3. Mean errors of aircraft position along the X-axis based
on the GPS+EGNOS solution with the applied Saastamoinen and RTCA-MOPS model
Figures 5 and 6 show
the mean error values along the Y-axis from the GPS+EGNOS and GPS+SDCM
solutions. Mean error values
Fig. 5. Mean errors of aircraft position along the Y-axis
based on the GPS+EGNOS solution with the applied Saastamoinen and RTCA-MOPS
model
Fig. 6. Mean errors of aircraft position along the Y-axis based
on the GPS+SDCM solution with the applied Saastamoinen and RTCA-MOPS
model
Figures 7 and 8 show
the mean error values along the Z-axis for the GPS+EGNOS and GPS+SDCM
solutions. Mean error values
Fig. 7. Mean errors of aircraft position along the Z-axis based
on the GPS+EGNOS solution with the applied Saastamoinen and RTCA-MOPS model
When developing the results of the obtained coordinates and their mean errors, the ellipsoid of the point position error is additionally determined [20]. The values of the ellipsoid parameter of the point position error are determined from the relationship [20]:
where:
(
Fig. 8. Mean errors of aircraft position along the Z-axis based
on the GPS+SDCM solution with the applied Saastamoinen and RTCA-MOPS model
Figures 9 and 10 show the results of parameters
The
obtained results show the high efficiency of the RTCA-MOPS model over the
Saastamoinen model in SBAS positioning for air navigation. Using a selected
tropospheric correction model in the positioning method significantly affects
the aircraft coordinate findings, as shown in Figures 1 and 2. Most
importantly, the RTCA-MOPS troposphere model reduced the values of the mean
errors of the determined aircraft coordinates in both the GPS+EGNOS and
GPS+SDCM solutions. The repeatability of the test method is true for the two
augmentation systems - EGNOS and SDCM. Compared to the state of expertise, similar
conclusions were drawn in related works [6, 10, 11, 13], in which the RTCA-MOPS
troposphere model in GNSS positioning was also examined. It can, therefore, be
stated that the RTCA-MOPS troposphere model is optimal for the SBAS positioning
method in air navigation.
Fig. 9. Ellipsoid error of
point position based on the
GPS+EGNOS solution
with the applied Saastamoinen and RTCA-MOPS model
Fig. 10. Ellipsoid error of point position based on the
GPS+SDCM solution
with the applied Saastamoinen and RTCA-MOPS model
6. CONCLUSIONS
This paper presents the results of a study on
the use of different tropospheric correction models in the SBAS precise
positioning for aerial navigation. Specifically, this
paper determines the influence of the Saastamoinen troposphere and RTCA-MOPS
models on the determination of aircraft coordinates and mean coordinate errors
in the SBAS positioning method. The flight test was executed in
north-eastern Poland in 2020. The test used real kinematic data from a GPS
navigation system recorded by an onboard GNSS satellite receiver and SBAS
corrections downloaded from a real-time server. In the experiment, the authors
included SBAS corrections from EGNOS and SDCM augmentation systems. The
navigation calculations were performed using RTKLIB v.2.4.3 and Scilab 6.1.1
software. Based on the conducted research, it was found that the difference in
aircraft coordinates using different troposphere models can reach up to
±2.14 m. Furthermore, the use of the RTCA-MOPS
troposphere model improved the values of the mean coordinate errors from 5 to
9% for the GPS+EGNOS solution and from 7 to 12% for the GPS+SDCM solution,
respectively. Additionally, the ellipsoid values of
the point position error were improved from 5 to 9% in the GPS+EGNOS solution
and from 9 to 11% in the GPS+SDCM solution, assuming the use of the RTCA-MOPS
troposphere model in the SBAS positioning method. The
obtained computational findings confirm the validity of using the RTCA-MOPS
troposphere model for SBAS positioning in air navigation. The computational strategy presented in this paper is
universal and can be extended by its implementation into the GAGAN positioning
in air navigation in Poland. The authors intend to
conduct further research into the impact of the tropospheric correction in the
GNSS positioning for air navigation.
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Received 19.09.2022; accepted in
revised form 02.11.2022
Scientific Journal of Silesian University of Technology. Series
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[1] Institute of Navigation, Polish Air Force University,
Dywizjonu 303 nr 35 Street, 08-521 Dęblin, Poland. Email: k.krasuski@law.mil.pl. ORCID: https://orcid.org/0000-0001-9821-4450
[2]
Institute of Navigation, Polish Air Force University, Dywizjonu 303 nr 35
Street, 08-521 Dęblin, Poland. Email: m.kirschenstein@law.mil.pl.
ORCID: https://orcid.org/0000-0002-4817-083X