Optimisation of an Automatic Online Post-Processing Service for Static Observations as Realised in the Polish ASG-EUPOS System

Abstract

The paper presents an assessment of the accuracy and precision of point positioning using a currently available online service for automatic post-processing (APPS) of GNSS observations made by the Polish Active Geodetic Network—European Position Determination System (ASG-EUPOS). The ASG-EUPOS network is part of the national EUPOS project and was selected for the study because of its substantial spatial reach in Europe. A modification to the current APPS algorithm is proposed to limit the number of baseline vectors used in positioning and change the method for adjusting post-processing results. Currently, the APPS of the ASG-EUPOS determines point coordinates based on the method for adjusting direct and equally accurate observations with the least-squares method (equally accurate method, EAM). The modification proposes to determine the coordinates based on post-processing results adjustment using the unequally accurate observations method (unequally accurate method, UAM). Our analysis diagnoses the uncertainty of the ASG-EUPOS’s APPS, and the modification optimises the adjustment. The proposed modification facilitates a much lower standard deviation of positioning in relation to reference positions. It is demonstrated that, if modified, the ASG-EUPOS service can provide automatic post-processing for 1 h GPS observations with a 1 s interval at 0.03 m accuracy for 99% of cases.

1. Introduction

The development of the GNSS (Global Navigation Satellite System) has always allowed for follow-on improvements in measuring techniques and methods, leading to the need for continuous registration of satellite signals sent to the surface of the Earth. Such physical sensors for continuous registration of satellite signals are permanent reference stations with precise GNSS antennas and receivers. These networks are the underpinnings of global, regional, and local reference frames [1,2,3]. This makes research on GNSS time series a relevant issue both in terms of the determination of periodic components [4] and station velocity [5,6]. Active geodetic networks that realise the spatial reference system in their respective areas are replacing classical geodetic control networks. The contribution of GNSS reference networks also includes ground-based augmentation systems (GBAS), which facilitate relative positioning with real-time kinematic (RTK), network real-time kinematic (NRTK), and the static approach.

The European EUREF Permanent GNSS Network (EPN) realises the European Terrestrial Reference System (ETRS89). The next level of active geodetic networks in Europe is national reference networks in individual countries. Most of them are consistent with the guidelines of the European Position Determination System (EUPOS) initiated in 2002 by the Berlin Senate Department for Urban Development and the European Academy of the Urban Development, Berlin. The EUPOS encompasses Central European and Eastern European institutions (Figure 1) which undertook to build and develop national networks of reference stations (national EUPOS GNSS networks) based on unified standards and guidelines [7,8,9,10]. The EUPOS consists of the following member states with their national EUPOS GNSS networks: Slovakia (SKPOS), the Czech Republic (CZEPOS), Hungary (GNSSNet.hu), Estonia (ESTPOS), Poland (ASG-EUPOS), Bulgaria (BULiPOS), Romania (ROMPOS), Latvia (LATPOS), Moldova (MOLDPOS), Macedonia, Germany (SAPOS), Lithuania (LITPOS), and Slovenia (SLOVENIA EUPOS). The total area covered by these national EUPOS GNSS networks is approx. 34% of the EU area and approx. 14% of Europe.

The primary principles of the EUPOS, which apply to national networks, are: the network consists of differential GNSS (DGNSS) permanent multifunctional reference stations; the average distance between the stations does not exceed 75 km; the stations’ coordinates are determined with the highest precision in a geodetic reference frame, which is an implementation of ETRS89, and in national coordinate systems; and, in order to improve the geometry of the network and ensure sufficient accuracy near borders, the national networks use observations from stations of other positioning systems participating in the EUPOS that are located in border areas to improve network geometry and ensure sufficient accuracy in borderlands.

National EUPOS GNSS networks are obliged to offer three services [9]: EUPOS NRTK, EUPOS DGNSS, and EUPOS Geodetic. Therefore, national systems have to offer DGNSS corrections, which makes it possible to support precise positioning and navigation down to metre (EUPOS DGNSS), centimetre accuracy in real time (EUPOS NRTK), and centimetre and sub-centimetre accuracy in the online automated post-processing (APPS) of user GNSS observations (EUPOS Geodetic).

The EUPOS Geodetic is a sub-service for automatic online post-processing of code and carrier phase static or kinematic measurements based on Receiver Independent Exchange Format 2.11 (RINEX) or 3.0 observation files sent to it. The service should guarantee decimetre or even sub-centimetre accuracy depending on the measuring instrument, the duration of the session, and measurement conditions. EUPOS members took individualised approaches to this matter. Some national EUPOS GNSS networks employ off-the-shelf APPS solutions by geodesy industry leaders, such as Trimble, Geo++, or Leica. Others developed their own EUPOS Geodetic algorithms. As the EUPOS Geodetic is intended to guarantee decimetre and sub-centimetre accuracy, it needs to be continuously validated, upgraded, and modified. Note further that, in addition to EUPOS Geodetic services in national EUPOS GNSS networks, there are many other international services with APPS for static GPS observations, such as AUSPOS Online GPS Processing Service (https://gnss.ga.gov.au/auspos (accessed on 30 July 2022)) by Geoscience Australia, the Scripps Coordinate Update Tool (SCOUT) by Scripps Orbit and Permanent Array Center (SOPAC), and the Online Positioning User Service (OPUS) by the National Geodetic Survey (NGS) (https://www.ngs.noaa.gov/OPUS (accessed on 30 July 2022)). Most of these services require at least 1 h static observations, but the session should take at least two hours for centimetre accuracy [11]. APPS services have been validated by such researchers such as El-Mowafy [12], Jamieson and Gillins [13], and Gökdaş and Özlüdemir [14].

The present paper validates the EUPOS Geodetic service of a selected national EUPOS GNSS network. We selected the Polish Active Geodetic Network EUPOS (ASG-EUPOS) as an example of a national EUPOS GNSS network due to its significant spatial coverage in Europe. The objective of the study is to apply a change of the method for adjusting post-processing results. Actually, the APPS of the ASG-EUPOS determines point coordinates based on the method for adjusting direct and equally accurate observations with the least-squares method. The proposed modification moves towards a weighting of the unequally accurate baseline vector components. Additionally, validation of the accuracy and precision of the results of the currently used, as well as of the modified version of APPS, is carried out in the event of limiting the number of baseline vectors used in positioning. Our analyses demonstrate whether a reduction in the number of baseline vectors and modified adjustment procedure yield more accurate positioning compared to the current methods (in post-processing of 1 h dual-frequency GPS observations). It is particularly important in the case of a failure of stations near the measurement area. The research presented here is an expansion of the studies by Kudas and Wnęk [15] and Kudas and Wnęk [16].

2. Materials and Methods

2.1. Study Area and Object

The ASG-EUPOS network consists of 106 reference stations in Poland capable of receiving GPS, GLONASS, and Galileo signals, with 100 stations compatible with BDS signals [17]. Most of these stations are able to track BDS III signals. The ASG-EUPOS can exchange border area data with systems owned by four neighbours of Poland, SKPOS, CZEPOS, SAPOS, and LITPOS, with access to 24 reference stations. Fifteen national reference stations and four foreign stations in the Polish ASG-EUPOS belong to the EPN. The ASG-EUPOS provides the user with real-time (NAWGEO, KODGIS/NAWGIS) and post-processing (POZGEO and POZGEO-D) services (

Table 1. Services offered by ASG-EUPOS [17].

Service	Method of Position Determination	Assumed Positioning Accuracy	Application
NAWGEO	RTK and NRTK with use of dual-frequency GNSS observations	0.03 m horizontal 0.05 m vertical	land surveying, precision agriculture
KODGIS NAWGIS	real-time positioning with single-frequency code receivers	down to 0.25–3.00 m in 3D	construction, spatial information systems, agriculture, tourism and recreation, automatic navigation, etc.
POZGEO	post-processing of dual-frequency GPS observations	depends on the measurement conditions and duration. It ranges from 0.01 to 0.10 m in 3D	land surveying
POZGEO–D			provide users with historical GPS or GLONASS observation data from ASG-EUPOS or virtual reference stations

).

The data recorded by ASG-EUPOS stations are used in research such as geodynamic research [18], atmosphere and ionosphere monitoring [19], and agricultural research [20]. The ASG-EUPOS services had about 4.2 thousand subscribers in late 2019.

The ASG-EUPOS was founded on the Trimble Pivot Platform and upgraded with the RTX Network processing module in 2018. The ASG-EUPOS has hardware and software for calculating correction data based on GPS, GLONASS, Galileo, and BDS, which is particularly valuable in network areas with dual-system receivers only. Some stations are equipped with Trimble NetR9 receivers which can track all signals from BDS III generation satellites.

The APPS module provides relative positioning based on a network of permanent reference stations of the ASG-EUPOS. The APPS requires at least 720 measurement epochs in the user’s observation file. However, it is recommended that the observation files contain at least 40 min with 1 s intervals (2400 measurement epochs). The POZGEO service uses an algorithm developed in Poland [21,22] which is linked to a sub-service of Trimble. To determine the position of a point, the APPS of the ASG-EUPOS selects the six shortest baseline vectors between a point and the neighbouring reference stations [23].

While there are quite a lot of publications on the real-time services of the ASG-EUPOS [24,25,26,27,28], the assessment of the POZGEO has not been a popular research topic. Analyses of measurement results with the POZGEO can be categorised as those tackling the session duration [29], the number of satellites available [30], and the impact of GNSS antenna calibration models on geometrical heights measurement [31,32].

2.2. Potential Modifications of the ASG-EUPOS’s APPS

Currently, the APPS of the ASG-EUPOS determines point coordinates based on the method for adjusting direct and equally accurate observations with the least-squares method (equally accurate method, EAM). In this method of adjusting, the components of the basis vectors should be treated as observations. Three-dimensional point coordinates (X, Y, Z) are determined with the EAM (including accuracy evaluation after adjustment) following the procedure below (Equations (1)–(7)).

Coordinates X, Y, Z of point P are determined from measured baseline vectors to reference stations (Figure 2):

$$\left\{\begin{array}{c}{X}_P^i=X_i+\Delta X_{Pi}\\ Y_P^i=Y_i+\Delta Y_{Pi}\\ Z_P^i=Z_i+\Delta Z_{Pi}\end{array}\right.$$

(1)

where:

• $X_P^i, Y_P^i,Z_P^i$—coordinates of determined point P based on baseline vector to the i reference station;
• $X_i, Y_i, Z_i$—coordinates of the i reference station;
• $\Delta X_{Pi}, \Delta Y_{Pi},\Delta Z_{Pi}$—components of baseline vector between point P and I reference station.

For a GPS vector (ΔX, ΔY, ΔZ) between two points P, i (i = 1, 2, 3, …, n; n—the number of tie points), the following observation equations may be written:

$$\left\{\begin{array}{c}\Delta X_{Pi}+v_{Pi}^{\Delta X}=X_P-X_i\\ \Delta Y_{Pi}+v_{Pi}^{\Delta Y}=Y_P-Y_i\\ \Delta Z_{Pi}+v_{Pi}^{\Delta Z}=Z_P-Z_i\end{array}\right.$$

(2)

where:

• $X_P, Y_P,Z_P$—coordinates of determined point P;
• $v_{Pi}^{\Delta X}, v_{Pi}^{\Delta Y},v_{Pi}^{\Delta Z}$—corrections for components of the baseline vector.

As Equation (2) is linear, the correction equations are derived directly (without expansion into a Taylor series):

$$\left\{\begin{array}{c}{v}_{Pi}^{\Delta X}=X_P-X_i-\Delta X_{Pi}\\ v_{Pi}^{\Delta Y}=Y_P-Y_i-\Delta Y_{Pi}\\ v_{Pi}^{\Delta Z}=Z_P-Z_i-\Delta Z_{Pi}\end{array}\right.$$

(3)

In keeping with the least-squares principle, the corrections expressed in Equation (3) should conform (for each coordinate X, Y, Z separately) to:

$${\displaystyle \sum }_{i=1}^n{\left(v_{Pi}^{\Delta X}\right)}^2=min.;{\displaystyle \sum }_{i=1}^n{\left(v_{Pi}^{\Delta Y}\right)}^2=min.;{\displaystyle \sum }_{i=1}^n{\left(v_{Pi}^{\Delta Z}\right)}^2=min.$$

(4)

The conditions in Equation (4) are sound assuming that the components of the baseline vector are considered as equally accurate observations. These conditions are satisfied if the coordinates of point P are calculated as simple arithmetic means of the values from Equation (1):

$$X_P=\frac{{{\displaystyle \sum }}_{i=1}^n{X}_P^i}{n};Y_P=\frac{{{\displaystyle \sum }}_{i=1}^n{Y}_P^i}{n};Z_P=\frac{{{\displaystyle \sum }}_{i=1}^n{Z}_P^i}{n}$$

(5)

The accuracy of the adjusted coordinates from Equation (5) is determined by the average error of the arithmetic mean according to the propagation of mean error [33]:

$$m_X=\pm \frac{m^{\Delta X}}{\sqrt{n}};m_Y=\pm \frac{m^{\Delta Y}}{\sqrt{n}};m_Z=\pm \frac{m^{\Delta Z}}{\sqrt{n}}$$

(6)

We calculate the a posteriori observation mean errors in Equation (6) i.e., $m^{\Delta X}$, $m^{\Delta Y}$, $m^{\Delta Z}$ from the corrections of Equation (3) obtained by substituting adjusted coordinates into Equation (5).

$$m^{\Delta X}=\pm \sqrt{\frac{{{\displaystyle \sum }}_{i=1}^n{\left(v_{Pi}^{\Delta X}\right)}^2}{n-1}};m^{\Delta Y}=\pm \sqrt{\frac{{{\displaystyle \sum }}_{i=1}^n{\left(v_{Pi}^{\Delta Y}\right)}^2}{n-1}};m^{\Delta Z}=\pm \sqrt{\frac{{{\displaystyle \sum }}_{i=1}^n{\left(v_{Pi}^{\Delta Z}\right)}^2}{n-1}}$$

(7)

The current model for adjusting baseline vectors in the APPS (Equations (1)–(7)) employs a simplification; the post-processing results are, in fact, not equally accurate, because not all requirements for considering them equally accurate are satisfied. Namely, the observations used to determine the vectors in post-processing are recorded using various GNSS antennas and receivers in different observation and environmental conditions. Therefore, we validated a method for determining coordinates based on the adjustment of post-processing results (components of baseline vectors) with the unequally accurate observations method (UAM) with baseline vectors to the six nearest reference stations. The spatial distribution of the points in a network can also affect measurements that employ a network of reference stations. Hence, we wondered whether the quality of solutions is significantly affected if the number of baseline vectors is reduced from six to five for the EAM and UAM. Three-dimensional point coordinates are determined with the UAM using a procedure similar to the EAM, with Equations (4)–(7) replaced with Equations (8)–(12) below. Observation weights $p_i^{\Delta X}, p_i^{\Delta Y}, p_i^{\Delta Z}$ in Equation (8) are determined using the mean errors of the vector’s components ($m_{Pi}^{\Delta X}$, $m_{Pi}^{\Delta Y}$, $m_{Pi}^{\Delta Z}$) from post-processing.

$$p_i^{\Delta X}=\frac{1}{{\left(m_{Pi}^{\Delta X}\right)}^2};p_i^{\Delta Y}=\frac{1}{{\left(m_{Pi}^{\Delta Y}\right)}^2};p_i^{\Delta Z}=\frac{1}{{\left(m_{Pi}^{\Delta Z}\right)}^2}$$

(8)

$${\displaystyle \sum }_{i=1}^n\left[p_i^{\Delta X}\cdot {\left(v_{Pi}^{\Delta X}\right)}^2\right]=\min .;\hspace{1em}{\displaystyle \sum }_{i=1}^n\left[p_i^{\Delta Y}\cdot {\left(v_{Pi}^{\Delta Y}\right)}^2\right]=\min .; {\displaystyle \sum }_{i=1}^n\left[p_i^{\Delta Z}\cdot {\left(v_{Pi}^{\Delta Z}\right)}^2\right]=\min .$$

(9)

In this case, the coordinates of point P are calculated as weighted means as follows (Equation (10)):

$$X_P=\frac{{{\displaystyle \sum }}_{i=1}^n\left(p_i^{\Delta X}\cdot X_P^i\right)}{{{\displaystyle \sum }}_{i=1}^n{p}_i^{\Delta X}};\hspace{1em}{Y}_P=\frac{{{\displaystyle \sum }}_{i=1}^n\left(p_i^{\Delta Y}\cdot Y_P^i\right)}{{{\displaystyle \sum }}_{i=1}^n{p}_i^{\Delta Y}};\hspace{1em}{Z}_P=\frac{{{\displaystyle \sum }}_{i=1}^n\left(p_i^{\Delta Z}\cdot Z_P^i\right)}{{{\displaystyle \sum }}_{i=1}^n{p}_i^{\Delta Z}}$$

(10)

$$m_X=\pm \frac{m_0^{\Delta X}}{\sqrt{{{\displaystyle \sum }}_{i=1}^n{p}_i^{\Delta X}}};\hspace{1em}{m}_Y=\pm \frac{m_0^{\Delta Y}}{\sqrt{{{\displaystyle \sum }}_{i=1}^n{p}_i^{\Delta Y}}};\hspace{1em}{m}_Z=\pm \frac{m_0^{\Delta Z}}{\sqrt{{{\displaystyle \sum }}_{i=1}^n{p}_i^{\Delta Z}}}$$

(11)

$$m_0^{\Delta X}=\pm \sqrt{\frac{{{\displaystyle \sum }}_{i=1}^n\left[p_i^{\Delta X}\cdot {\left(v_{Pi}^{\Delta X}\right)}^2\right]}{n-1}};\hspace{0.33em}\hspace{0.33em}{m}_0^{\Delta Y}=\pm \sqrt{\frac{{{\displaystyle \sum }}_{i=1}^n\left[p_i^{\Delta Y}\cdot {\left(v_{Pi}^{\Delta Y}\right)}^2\right]}{n-1}};\hspace{0.33em}\hspace{0.33em}{m}_0^{\Delta Z}=\pm \sqrt{\frac{{{\displaystyle \sum }}_{i=1}^n\left[p_i^{\Delta Z}\cdot {\left(v_{Pi}^{\Delta Z}\right)}^2\right]}{n-1}}$$

(12)

Note that, if the observation weights in Equation (8) are equal, the UAM adjustment is identical to the EAM adjustment. Furthermore, in the future, weighting based on the full covariance matrix may also be considered.

To assess the spatial distribution of network points, we subdivided the territory of Poland by the mean baseline length and its standard deviation.

2.3. Division of the Study Area by Tie Station Location

We employed a discrete model of the land territory of Poland comprising 1,250,008 points in a 0.5 × 0.5 km grid to characterise the study area in terms of the number and lengths of baseline vectors. The spatial analysis involved two dimensions (x, y) only because of insignificant altitude differences (approx. 75% of Poland is lowlands with a mean elevation of 173 m AMSL) and a smooth north–south altitude gradient. We employed the EPSG 2180 system, which is a single-zone Gauss–Krüger projection of the GRS80 ellipsoid. According to the literature [34], the lengths of vectors between ASG-EUPOS stations in Poland are similar in 2D and 3D coordinate systems, and any differences are negligible. Thus, in this investigation, we omitted the height component for calculating the baseline lengths. The territory of Poland was divided in QGIS with the Distance Matrix (summary). Each point in the model was assigned its distance to the n nearest ASG-EUPOS stations. We further determined the mean length of a baseline vector to the n nearest stations, standard deviation, and maximum and minimum baseline vector length. The division of the Polish territory was performed for six baseline vectors, as currently used in the APPS of the ASG-EUPOS, and for five and three baseline vectors.

2.4. Validation of the ASG-EUPOS’s APPS and the Proposed Modification

The POZGEO was validated by analysing coordinates from the post-processing of GPS observations from approx. 30 days in three areas of different ASG-EUPOS network densities (Figure 3).

The study involved 1 h static GPS observations with 1 s intervals from 318 to 349 DOY 2020 (13 November to 14 December 2020), totalling 767 h of observations. We selected research points with various baseline vector lengths to the six nearest ASG-EUPOS stations. The data were recorded at points in a commercial network of reference stations, TPI NETpro: BROD (B 53°15′28.44299″, L 19°24′25.82764″ E), OPOC (B 51°22′39.36412″ N, L 20°16′50.73161″ E), and JASL (B 49°44′49.52674″ N, L 21°28′16.92604″ E). More information about the TPI NETpro can be found in the paper of Kudas and Wnęk [35]. The data are collected with TPSCR-G5 antennas and Topcon NET-G5 receivers. The APPS employs a general model for calibrating the antenna phase centre developed by the equipment manufacturer, and, in the case of the ASG-EUPOS service, it is possible to enter the type and model of the antenna in the observation file sent to the service. The investigated points are points in the horizontal detailed national control network. Their positions are determined precisely as required by Polish regulations [36]. The OPOC station is situated in a low-density area, while JASL is in a zone of higher network density (Figure 3). In the case of the approach currently employed in the APPS of the ASG-EUPOS, the length of baseline vectors is, for BROD, from 38.2 to 70.5 km ($\overline{d}=$52.6 km, ${\sigma }_d=$10.9 km), for OPOC, from 41.1 to 89.9 km ($\overline{d}=$61.6 km, ${\sigma }_d=$18.3 km), and, for JASL, from 23.2 to 60.6 km ($\overline{d}=$48.0 km, ${\sigma }_d=$13.3 km). If the five nearest stations are used, the mean baseline vector length is reduced, as is the deviation (for JASL, ${\sigma }_d=$13.2 km, for OPOC, ${\sigma }_d=$13.3 km, and, for BROD, ${\sigma }_d=$7.2 km).

The sets of three-dimensional coordinates (X, Y, Z) of the research points determined with the APPS and the proposed modified method (UAM) were converted to a local N, E, U system and analysed. Basic accuracy measures, such as the 99% Spherical Accuracy Standard (SAS99), 90% Spherical Accuracy Standard (SAS90), Mean Radial Spherical Error (MRSE), and Twice the Distance Root Mean Squared (2DRMS), in relation to the reference positions of the points were determined to assess positioning accuracy (Equations (13)–(16)).

$$MRSE=\sqrt{{\sigma }_N^2+{\sigma }_E^2+{\sigma }_U^2}$$

(13)

$$SAS90=0.833 \left({\sigma }_N^{ }+{\sigma }_E^{ }+{\sigma }_U^{ }\right)$$

(14)

$$SAS99=1.122 \left({\sigma }_N^{ }+{\sigma }_E^{ }+{\sigma }_U^{ }\right)$$

(15)

$$2DRMS=2\sqrt{{\sigma }_N^2+{\sigma }_E^2}$$

(16)

where:

• ${\sigma }_N, {\sigma }_E, {\sigma }_U$—standard deviation of coordinate differences with respect to the reference position in the directions of the coordinate axis, calculated from Equation (17).

$$\sigma =\sqrt{\frac{\sum {\left(x-\overline{x}\right)}^2}{\left(n-1\right)}}$$

(17)

where:

• $n$—the sample size;
• $\overline{x}$—the sample mean.

In order to characterise the distance between determined points and reference positions, the distance coefficient of variation ($C{V}_d$) was determined in accordance with the Equation (18).

$$C{V}_d=\frac{{\sigma }_d}{\stackrel{=}{d}}\cdot 100\%$$

(18)

where:

• ${\sigma }_d$—standard deviation of sample distance between determined points and reference position;
• $\stackrel{=}{d}$—mean distance.

3. Results

3.1. Division of the Study Area

The division of Poland in relation to the number of baseline vectors and the mean baseline vector length allowed us to characterise the following spatial relations between the points being determined and the positions of the reference ASG-EUPOS stations (Figure 4). If six baseline vectors were used, rover points in most areas were connected via mean vector lengths from 50 to 60 km (approx. 53.6%) and from 40 to 50 km (approx. 34.5%). When five baseline vectors were chosen, the percentage of areas with a mean vector length from 40 to 50 km increased to 57.8%, and the share of areas with a 50 to 60 km average length was 31.07%. Having analysed the variant with three baseline vectors, we found out that Poland is dominated by areas with a mean baseline vector length from 30 to 40 km (approx. 56.0%) and from 40 to 50 km (approx. 31.1%). Mean baseline vector lengths did not exceed 60 km in 89.9% of Poland when six baseline vectors were used, 96.9% if five baseline vectors were, and 100% if three baseline vectors were used. Therefore, five baseline vectors offer a shorter baseline vector length in APPS than the current method. It achieves better modelling of observation conditions (such as troposphere and ionosphere impact) thanks to shorter distances between the sought point and reference stations.

We further divided the territory of Poland by baseline vector standard deviation to better characterise the geometric variability of the proposed variants (Figure 4). An analysis of changes in baseline vector length standard deviation demonstrated that the standard deviation in the length of six baseline vectors is below 20 km for 63.7% of Poland. For five baseline vectors, it is 75.3%, and, for three baseline vectors, it is 91.1% of Poland. A baseline vector deviation over 25 km affects 8.5% of Poland for the six nearest stations and 4.5% for five baseline vectors.

If baseline vectors to the five nearest stations were used, the area with the mean vector length below 50 km increased by approx. 55% compared to the variant with six baseline vectors. A switch from six to five baseline vectors caused a 14% increase in the area with the baseline vector length standard deviation from 5 to 15 km. Therefore, it seems only reasonable to restrict the number of baseline vectors to five when considering modifying the APPS algorithm in the ASG-EUPOS. This notion is consistent with the literature [15], where it has been suggested that it is possible to identify configurations of as few as three baseline vectors among the six baseline vectors currently used in the APPS of the ASG-EUPOS which yield results of positioning accuracy and solution consistency similar to those achieved with the current APPS method of six baseline vectors.

3.2. Adjustment Results Using the EAM and UAM for Five and Six Baseline Vectors

We processed 766 h of observations for the BROD station. For JASL and OPOC, it was 767 h and 763 h, respectively.

Table 2. Mean discrepancies with respect to reference coordinates, their standard deviation, and extreme values obtained with the EAM and UAM for six and five baseline vectors.

Station	Solution	Statistics	Number of Baseline Vectors
			6			5
			Coordinate			Coordinate
			N	E	U	N	E	U
BROD	EAM	Mean (m)	0.005	0.005	0.008	0.004	0.000	0.009
		Standard deviation (m)	0.009	0.011	0.017	0.010	0.006	0.016
		Max. (m)	0.190	0.043	0.118	0.229	0.022	0.143
	UAM	Mean (m)	0.004	0.001	0.008	0.004	0.002	0.008
		Standard deviation (m)	0.006	0.007	0.014	0.006	0.006	0.014
		Max. (m)	0.023	0.037	0.056	0.019	0.026	0.048
OPOC	EAM	Mean (m)	0.000	0.008	0.001	0.001	0.000	0.000
		Standard deviation (m)	0.008	0.012	0.017	0.006	0.008	0.014
		Max. (m)	0.172	0.112	0.256	0.103	0.116	0.162
	UAM	Mean (m)	0.001	0.003	0.001	0.001	0.000	0.001
		Standard deviation (m)	0.004	0.006	0.012	0.004	0.006	0.011
		Max. (m)	0.028	0.025	0.053	0.034	0.052	0.053
JASL	EAM	Mean (m)	0.004	0.009	0.001	0.005	0.004	0.000
		Standard deviation (m)	0.018	0.021	0.020	0.021	0.022	0.022
		Max. (m)	0.257	0.233	0.234	0.300	0.273	0.267
	UAM	Mean (m)	0.005	0.007	0.000	0.005	0.005	0.000
		Standard deviation (m)	0.011	0.010	0.010	0.011	0.008	0.009
		Max. (m)	0.180	0.053	0.072	0.190	0.061	0.072

shows the values of mean standard deviations and values of extreme absolute linear discrepancies in relation to reference coordinates for the EAM and UAM adjustment of post-processing results using six and five baseline vectors respectively for all the stations.

In most cases, the mean linear discrepancies were slightly larger for the UAM than for the EAM. In the case of the UAM and six baseline vectors, the mean discrepancies for coordinate N assumed values from −0.005 to −0.001 m, for coordinate E, from −0.007 to 0.003 m, and, for coordinate U, from −0.001 to 0.008 m, whereas, in the case of the UAM and five baseline vectors, the mean discrepancies for coordinate N assumed values from −0.005 to −0.001 m, for coordinate E, from −0.005 to 0.000 m, and, for coordinate U, from 0.000 to 0.008 m. Therefore, the results of a post-processing adjustment with the UAM and six or five baseline vectors were similar in terms of mean linear discrepancy ranges (Table 2). In some cases of the EAM, the max values of linear discrepancies were larger for five baseline vectors than for six in the N-component. This was probably caused by a wrong ambiguity resolution in at least one of the shorter baselines. Regarding the extreme values for the six and five baseline vectors using the EAM and UAM, the extreme value of the discrepancy was apparently lower for the UAM. It was particularly evident if six baseline vectors were used for the OPOC station for component U, as the extreme value dropped from 0.256 m to 0.053 m, and, for station JASL, the U extreme discrepancy fell to 0.072 from 0.234 m. Similarly, the extreme discrepancy for coordinate U was reduced in the approach with five baseline vectors from 0.162 to 0.053 m for OPOC and from 0.267 to 0.072 m for JASL. For the UAM adjustment, the values of the extreme discrepancies were comparable regardless of the number of baseline vectors.

Regarding the position of the points on the horizontal plane in relation to the reference position (Figure 5), the precision and accuracy of the coordinates improved thanks to the UAM. The 2DRMS for the UAM adjustment with five baseline vectors was 0.015 to 0.028 m and, for six vectors, 0.015 to 0.029 m, whereas the 2DRMS, in the case of the EAM adjustment with five baseline vectors, was 0.021 to 0.061 m and, for six vectors, 0.029 to 0.055 m.

When six and five baseline vectors were used with the proposed UAM adjustment, we noticed a decrease in outliers for all stations and elimination of linear discrepancy outliers for coordinate U (Figure 6), which were recorded for the EAM adjustment.

In most cases, the values of mean discrepancies of coordinates N, E, and U were within ±0.002 m regardless of the adjustment method. Nevertheless, when the UAM method was employed, the standard deviation of coordinates evidently improved. The largest values of the maximum discrepancies were recorded for EAM adjustments. The UAM adjustment significantly reduced the maximum linear discrepancies of each coordinate. Available case studies also confirmed the stability and accuracy of the solution with five baseline vectors [15,16].

4. Discussion

If six baseline vectors and the original APPS algorithm were used, the mean distance of the determined position of the BROD station to the reference position was 0.018 m, while, for the proposed modification with the UAM, it was 0.015 m. The distance coefficient of variation was reduced to 61% from 70%. For OPOC, the mean distance from the positions determined with the APPS to the reference position was 0.016 m. Using the proposed modification, we achieved 0.012 m. The distance coefficient of variation was then reduced to 62% from 94% in this case. The mean distance from the determined positions of JASL to its reference positions was 0.018 m for APPS; for the modified approach, it was 0.013 m. Additionally, in this case, the distance coefficient of variation was reduced to 85% from 155%.

The percentage of positions within 0.02 m of the reference positions varied from 66% to 76% for the EAM with six baseline vectors, while, for the UAM, it amounted to between 80% and 89% (Figure 7). The UAM and five baseline vectors yielded the most precise results (Figure 7). Therefore, the proposed modification improves service precision.

The mean distance of BROD to its reference position was 0.015 m for the EAM and five baseline vectors and 0.014 m for the UAM. The distance coefficient of variation was reduced to 58% from 82%. When we used five vectors and the EAM, the mean distance to the reference position of the OPOC station was 0.012 m, while, with the UAM, it was 0.011 m. The distance coefficient of variation for OPOC declined to 66% from 101%. For five baseline vectors and the EAM adjustment, the mean distance of JASL to the reference position was 0.016 m, which changed to 0.012 m when the UAM was applied. The positioning distance coefficient of variation was reduced to 97% from 214% for JASL. A distance coefficient above 100% indicates that the value of the standard deviation of vector length deviations exceeds the mean value, which proves the presence of outliers in the solution set.

Accuracy measures for 3D positioning for sets of positions indicated an improvement in accuracy when the UAM was applied (Figure 8). MRSE, SAS90, and SAS99 were the lowest for each analysed point when the proposed modification was employed with the UAM and five baseline vectors. The accuracy measures also improved when the UAM was used with six baseline vectors for BROD and OPOC. When the APPS of the ASG-EUPOS employed the current EAM with five vectors, the accuracy measures for the analysed points improved as well.

The current AUSPOS service yields the mean linear discrepancy of 0.05 m for horizontal coordinates and 1 h static observations. When the session is extended to four hours, the planar mean linear discrepancies reach 0.02 m for AUSPOS and OPUS [6]. Ocalan et al. [37] noted that the accuracy of the AUSPOS service and PPP automatic post-processing services such as APPS JPL or Trimble RTX increases significantly for 6 h satellite observations in good conditions. Mean differences in coordinates in relation to coordinates from Bernese 5.2 amount to between 0.001 and 0.013 m for N, 0.005 and 0.011 m for E, and 0.009 and 0.108 m for U in such a case [37]. Based on the analysis of the spatial distribution of the points positioned in relation to their reference positions in the horizontal plane, calculated mean linear discrepancies, and 2DRMS, we can conclude with 95% probability that the horizontal plane accuracy for 1 h GPS observations is not worse than 0.03 m for the UAM adjustment with six or five baseline vectors. Therefore, the modified APPS of the ASG-EUPOS is capable of reaching results similar to those of global APPS with shorter satellite observation sessions. The value of the MRSE for all the analysed points indicates that the proposed modification with the UAM and five baseline vectors yields an accuracy of the POZGEO ASG-EUPOS of 0.016 m. The discrepancy of the geometrical height coordinate U of the analysed points regarding its reference value ranged from 0.01 to 0.014 m for the UAM and five baseline vectors with the linear discrepancy extreme values from 0.048 to 0.072 m. Therefore, if the APPS of the ASG-EUPOS is modified, it offers better accuracy than that of the APPS JPL, AUSPOS, and Trimble RTX.

The following issue should be considered in light of the analyses. The reduction in the number of vectors will, beyond any doubt, lead to the rejection of vectors to the furthest stations, which will decrease the mean baseline vector length. Nevertheless, future research should investigate whether it is better to stick to the five baseline vector principle if one station fails and use the next nearest station or use only four baseline vectors instead. A recent case study for a point in the southern ASG-EUPOS network [15] indicated that the EAM with five or four baseline vectors yields an MRSE of up to 0.03 m. It has been further demonstrated that the APPS POZGEO algorithm is immune to failures of stations nearby because solutions based on five baseline vectors (with one station down) and the EAM give results similar to the standard solution (the EAM and six baseline vectors) [16]. Currently used global APPS, such as the SCOUT, OPUS, and AUSPOS, use GPS carrier phase observations [38]. Positioning accuracy is also affected by the number of GNSS constellations involved in the measurement and post-processing. According to Omer et al. [39], the positioning accuracy increases for all three components and the session duration can be shortened for multi-GNSS observations. Therefore, the potential benefits of expanding the APPS of the ASG-EUPOS with multi-GNSS observations should be investigated further.

5. Conclusions

The paper diagnosed the uncertainty inherent to the APPS service of the Polish reference station network ASG-EUPOS and attempted to validate a proposed modification to improve it.

The proposed modification facilitated a much lower standard deviation of determined positions in relation to reference positions than that of the currently employed APPS algorithm in the ASG-EUPOS.

Accuracy measures (MRSE, SAS90, and SAS99) indicated that the UAM with six or five baseline vectors yields more accurate and precise coordinates than the currently implemented APPS algorithm. The number of baseline vectors, their length, and the uncertainty of their components do not affect the result as much with the UAM as with the EAM (currently used in the APPS of the ASG-EUPOS). The accuracy measures reached similar values for all analysed points regardless of whether six or five baseline vectors were used with the UAM.

It has been demonstrated that, if modified, the EUPOS Geodetic service of the ASG-EUPOS can provide automated post-processing for user-provided 1 h GPS observations with 1 s intervals (3600 epochs) and positioning with 0.025 m or better accuracy for 90% of cases and 0.03 m accuracy for 99% of cases.

It is advisable to modify the software of the EUPOS Geodetic service of the ASG-EUPOS to improve its accuracy and reliability through the proposed and validated modification.