Experimental Validation of a GNSS Receiver Antenna Absolute Field Calibration System

Abstract

Carrier-phase measurements are essential in precise Global Navigation Satellite System (GNSS) positioning applications. The quality of those observations, as well as the final positioning result, is influenced by an extensive list of GNSS error sources, one of which is the receiver antenna phase center (PC) model. It has been well established that the antenna PC exhibits variability depending on the frequency, direction, and intensity of the incoming GNSS signal. To mitigate the corresponding range errors, phase center corrections (PCCs) are determined through a specialized procedure known as receiver antenna calibration and subsequently applied in data processing. In 2023, the Laboratory for Measurements and Measuring Technique (LMMT) of the Faculty of Geodesy, University of Zagreb, Croatia, initiated the development of a new robotic GNSS receiver antenna calibration system. The system implements absolute field calibration and PCC modeling through triple-difference (TD) carrier-phase observations and spherical harmonics (SH) expansion. This study presents and documents dual-frequency (L1 and L2) Global Positioning System (GPS) calibration results for several distinct receiver antennas. Furthermore, the main goals of this contribution are to evaluate the accuracy of dual-frequency GPS calibration results on the pattern level with respect to independent calibrations obtained from Geo++ GmbH and to extensively experimentally validate LMMT calibration results in the spatial (coordinate) domain, i.e., to investigate how the application of LMMT PPC models reflects on geodetic-grade GNSS positioning. Our experimental research results showed a submillimeter calibration accuracy, i.e., 0.36 mm for GPS L1 and 0.54 mm for the GPS L2 frequency. Furthermore, our field results confirmed that the application of LMMT PCC models significantly increases baseline accuracy and GNSS network solution accuracy when compared to type-mean PCC models of the International GNSS Service (IGS).

1. Introduction

Achieving high-accuracy Global Navigation Satellite System (GNSS) positioning, essential for various fundamental applications such as the realization of geodetic reference frames [1], monitoring and geodynamics [2], large-scale engineering projects [3], and national Continuously Operating Reference Station (CORS) networks, as well as for the determination of consistent GNSS products including satellite orbit solutions, clock corrections, atmospheric parameters, and Earth orientation parameters [4,5], requires precise GNSS antenna absolute phase center correction (PCC) determination as a crucial precondition. In simplified terms, PCCs are a direct consequence of the antenna’s design properties and electromagnetic characteristics [6]. The geometric location of GNSS signal reception within the antenna, or the so-called receiver antenna phase center (PC), varies based on the direction and frequency of the incoming signal, therefore causing delays and advances in carrier-phase observations and corresponding azimuth-, elevation-, and frequency-dependent range errors [7,8,9,10,11]. This measurement error source is solved through a special procedure known as GNSS receiver antenna calibration, within which PCC grids (models) for the entire antenna upper hemisphere and per frequency are determined. Calibration results, i.e., PCC values, are stored in a separate file, in line with the IGS ANTenna EXchange (ANTEX) file format [12], to be applied in GNSS data processing algorithms.

It is also essential to clarify that two distinct types of antenna calibration models exist: individual and type-mean [13]. An individual receiver antenna PCC model is derived from one or more calibrations of a specific receiver antenna (defined by its unique serial number). This model applies only to that specific antenna. In contrast, a type-mean PCC model is created by averaging calibration results from multiple antennas of equal type but different manufacturer serial numbers. Such a type-mean model is thus representative of that entire antenna type. The IGS publishes its official ANTEX file [14], which includes type-mean multi-GNSS multi-frequency PCC models for almost all geodetic-grade antenna types. The file is regularly updated, and, at the time of this research, the latest official version is from GPS Week number 2335 (igs20_2335.atx).

The topic of antenna calibration has been researched for nearly 30 years since the early beginnings of Global Positioning System (GPS) positioning [15]. Today, the absolute calibration approach represents the state-of-the-art standard in antenna calibration methodologies. Two mutually independent absolute calibration methods have been developed and are in use: chamber and robot. The chamber calibration method [16,17] is a research laboratory technique in which the receiver antenna undergoing calibration is set up within an anechoic chamber alongside an artificial signal source. In contrast, the robot calibration method [7,8,11,17,18,19,20], also referred to as the absolute field method, is an in situ technique in which the receiver antenna undergoing calibration is set up at the open-sky calibration field alongside a reference antenna in a short-baseline setup. During the calibration, raw carrier-phase observations are registered. This method is the de facto standard for GNSS receiver antenna calibrations. This claim is supported by the fact that out of 430 receiver antenna type-mean PCC models contained in the current official IGS ANTEX file (igs20_2335.atx) [14], 331 PCC models are the result of absolute field, i.e., robot, calibrations by different calibration institutions.

Given the importance of this field of research for geodesy as well as many other related scientific and professional disciplines, several institutions and working groups worldwide have developed, or are developing, their own GNSS receiver antenna robotic calibration systems. Given the fact that, in partnership with the University of Hannover, Institute of Geodesy (Institut für Erdmessung—IfE), it developed the first robotic GNSS receiver antenna calibration system in 2000 and has contributed to the IGS ANTEX since then, the German company Geo++ GmbH is globally regarded as a leading authority in the field [21]. Furthermore, other robotic systems have been developed or are under further development at IfE, Germany [22,23,24], Geoscience Australia in Canberra, Australia [25], National Geodetic Survey (NGS) in Virginia, USA [26], GNSS Research Center of the Wuhan University in China [10,27,28], the Institute of Geodesy and Photogrammetry at ETH Zürich in Switzerland [18,19,29], the Institute of Geodesy and Civil Engineering of the University of Warmia and Mazury in Olsztyn, Poland [20], and Topcon Positioning System in Concordia, Italy [30].

Recent publications and results of the IGS “RingCalVal” global campaign, which includes almost all the mentioned institutions, have shown that current contributors to the official IGS ANTEX file have more variability in terms of calibration results than previously appreciated [24]. Moreover, considering constant developments and modernizations of GNSS [31], the main current issues [17], the importance of this open research area to the international scientific antenna community, and the high level of interest in it, the development of a new robotic calibration system is highly desirable. Additionally, a significant scientific and practical value of a new calibration system to the Republic of Croatia lies in the continued development of the Croatian Positioning System (CROPOS) [32]. Therefore, due to such a strong impetus, in Croatia, at the Laboratory for Measurements and Measuring Technique (LMMT), University of Zagreb—Faculty of Geodesy, in 2023, the development of a new GNSS receiver antenna absolute field calibration system began [17,32,33,34,35].

In Tupek et al. [17], the authors initially presented the newly developed absolute field calibration system at LMMT in Croatia, along with the underlying calibration methodology and PCC estimation model. They also published preliminary antenna calibration results supported only by a simple repeatability analysis and validation through comparison with independent calibration results obtained by Geo++ GmbH in Germany. Analyses have been conducted on the PCC pattern level and only for the GPS L1 frequency (frequency code G01 consistent with ANTEX). In summary, the obtained results have shown themselves to be extremely promising and demonstrated the potential and feasibility of the developed absolute field calibration system. However, a comprehensive validation and investigation of the influence of LMMT calibration results on GNSS positioning has not been conducted.

Therefore, this study builds upon previous research efforts and advances the developed calibration system to dual-frequency GPS calibrations. The primary goals of this research are to conduct a more comprehensive and in-depth evaluation of the accuracy of dual-frequency GPS calibration results on the pattern level with respect to absolute calibrations obtained by Geo++ GmbH and to extensively validate LMMT dual-frequency GPS calibration results in the spatial (coordinate) domain, i.e., to investigate how the application of LMMT PPC models reflects on geodetic-grade GNSS positioning.

The structure of this paper is as follows. In Section 2 a concise theoretical overview of the standard PCC model and description of the LMMT antenna calibration system are provided. Furthermore, applied methodological strategies are elaborated and conducted experimental validation tests are explained. Section 3 details the obtained experimental research results, providing a thorough analysis and discussion. Lastly, Section 4 assesses the research objectives, synthesizes the key insights, and provides the concluding remarks.

2. Materials and Methods

2.1. Receiver Antenna Phase Center Corrections (PCCs)

Consistent with the convention of IGS regarding the ANTEX definition of GNSS receiver antenna corrections [12], as illustrated in Figure 1, the total PCC value is decomposed into the phase center offset (PCO) vector and the frequency- and direction-dependent phase center variation (PCV) [10,11,12,17,18,19,20,27,28,29,30,32]. The well-known receiver antenna PCC function that connects PCC, PCO, and PCV is:

$$PCC\left({\alpha }^i,z^i\right)=-{\mathrm{e}}^{\mathrm{T}}\left({\alpha }^i,z^i\right)\cdot PCO+PCV\left({\alpha }^i,z^i\right)+r$$

(1)

where $\mathrm{e}\left({\alpha }^i,z^i\right)$ is the line-of-sight unit vector, ${\alpha }^i$ and $z^i$ are the azimuth and zenith angle of the satellite in the antenna frame (AF), and r is a constant component that is uniform in all directions, arising from the relative character of GNSS observations [11,17,32]. The PCO vector components and tabulated PCV values are stored in an ANTEX file.

The PCCs, i.e., the PCO and PCVs, are defined within a three-dimensional (3D), left-handed coordinate system fixed to the receiver antenna (antenna frame—AF). The origin of the system is the antenna reference point (ARP), with the z-axis aligned along the antenna vertical symmetry axis. The x-axis is perpendicular to the z-axis and aligned with the antenna’s north reference point (NRP), while the system’s y-axis is defined as perpendicular to both the x- and z-axes, completing the left-handed system. In the field, during a survey, the x-axis is north oriented, and the xy-coordinate plane is aligned with the local horizon [17,32,33,34,35].

Because PCCs can be converted to any PCO vector and PCV set, as can be seen from Equation (1), the standard approach to address this rank deficiency is to fix the PCV at zenith to zero, commonly referred to as the zero-zenith constraint, given by [5,17,32,36]:

$$PCV\left({\alpha }^i,z^i=0°\right)=0.$$

(2)

Therefore, when comparing different calibration results, the analysis must be conducted exclusively at the PCC level. Independent analysis at the PCO/PCV level is not meaningful.

2.2. LMMT Robotic Calibration System

The LMMT robotic calibration system comprises a six-axis industrial robot Mitsubishi MELFA RV-4FLM-Q, its corresponding controller Mitsubishi MELFA CR750, GNSS receivers with antennas, a personal computer (laptop), and the necessary wiring. An overview of the system components is shown in Figure 2, with the on-site calibration setup illustrated in Figure 3.

As the most prominent hardware component of the system, the industrial robot is employed to precisely and efficiently sample the entire upper hemisphere of the AUT, ensuring homogeneous coverage in accordance with the concepts of absolute receiver antenna calibration. The robot is a multiple-joint-type robot arm of vertical structure and six degrees of freedom, with a rated load of 4 kg [37]. Communication between the robot controller and the personal laptop is facilitated via the TCP/IP protocol, enabling two-way data exchange. Prior to antenna calibration, the industrial robot is installed at the calibration field, i.e., the position and orientation of the robot base frame (RBF) in the global geocentric frame (e.g., ITRF2020) is determined according to the defined setup methodology and is based on a precise total station survey. The calibration field of the LMMT calibration facility consists of two concrete pillars (part of the Calibration Baseline of the Faculty of Geodesy [38]), thus forming a 5 m long short-baseline. From a software perspective, the LMMT system operates through three independent modules, each custom-built in-house using Python (version number 3.10): the Antenna Calibration Module (ACM), the PCC Estimation Module (PEM), and the Time Synchronization module (TiSy) [17,32].

Time synchronization is an important issue according to the receiver antenna robotic calibration method. This is mainly because carrier-phase observations registered during AUT rotations, i.e., robot travel time, must be temporally filtered and removed from further data processing and PCC estimation. Within LMMT’s calibration system, the one pulse-per-second signal (1PPS) and the corresponding TimeTag message, which are outputted by the receiver, are utilized. This time synchronization method ensures that the computer’s internal clock is disciplined and aligned with GPS Time (GPST). During system development, a comprehensive time synchronization test has been conducted, within which time differences between the computer’s internal clock and Coordinated Universal Time (UTC), generated by the GNSS receiver, have been calculated. The duration of the synchronization test was 8000 epochs (seconds), i.e., approximately 2 h and 13 min, and was divided into three phases based on the working status of TiSy: TiSy OFF, TiSy ON, and TiSy OFF. Corresponding computer clock errors have been calculated and are depicted in Figure 4. Blue lines at epochs $t=1900$ and $t=7300$ indicate the change in TiSy’s working status. Based on the obtained synchronization test results, and a comprehensive analysis of other influential factors, the time synchronization uncertainty of the developed calibration system has been experimentally estimated to be 1.02 ms [32]. The resulting data largely fulfill the requirements of the robotic calibration system.

To estimate the PCC model of the AUT, the antenna is positioned by the industrial robot in a total of 2088 distinct spatial orientations with 2.5 s stationary at every single attitude. The calibration rotation sequence is defined by an azimuthal increment of 5° within the 0° to 360° range and a zenithal increment of 5° within the −70° to 70° range. Consequently, the full calibration procedure lasts approximately 2 h. On both stations simultaneously (AUT and REF), under identical receiver settings and an observation interval of 10 Hz, raw carrier-phase observations are registered.

Generally, the LMMT calibration system and PCC modeling are based on the triple-difference (TD) approach. Simultaneously registered raw carrier-phase observations are differentiated both spatially and temporally to generate the well-known time-differences of carrier-phase double-differences. By forming TDs over a short-baseline between epochs $t_k$ and $t_{k+1}$, corresponding to two time-adjacent AUT orientations (attitudes) k and (k+1) and satellites $i$ and $j$, the high temporal and spatial correlation of GNSS signals is leveraged. This approach effectively cancels out most of the GNSS error budget, leaving the final TDs to contain primarily the AUT PCCs and the differential carrier-phase noise $\partial {\epsilon }_{\mathrm{T},\mathrm{R}}^{ij}$ [17,32,33,34,35]:

$$T{D}_{\mathrm{T},\mathrm{R}}^{ij}\left(t_k,t_{k+1}\right)=-PC{C}_{\mathrm{T}}^j\left(t_{k+1}\right)+PC{C}_{\mathrm{T}}^i\left(t_{k+1}\right)+PC{C}_{\mathrm{T}}^j\left(t_k\right)-PC{C}_{\mathrm{T}}^i\left(t_k\right)+\partial {\epsilon }_{\mathrm{T},\mathrm{R}}^{ij}.$$

(3)

In line with the standard approach, the parametrization of the AUT’s PCC model using the precalculated TD values is most effectively achieved through spherical harmonics (SH) expansion. The basic equation for SH expansion is given by [17,32,33,34,35]:

$$PCC\left({\alpha }^i,z^i\right)={\displaystyle \sum _{m=0}^{m_{\max }}{\displaystyle \sum _{n=0}^m{\tilde{P}}_{mn}\left[\cos \left(z^i\right)\right]}}\cdot \left[a_{mn}\cos \left(n{\alpha }^i\right)+b_{mn}\sin \left(n{\alpha }^i\right)\right],$$

(4)

where ${\tilde{P}}_{mn}$ is the fully normalized Legendre function [39], ${\alpha }^i$ and $z^i$ are the azimuth and zenith angles in the AF, $a_{mn}$ and $b_{mn}$ are the SH coefficients. At LMMT, the SH resolution is defined as $m=n=8$, resulting in a total of 91 unknown coefficients, which are determined through a conditional least squares (CLSQ) adjustment. Following the determination of SH coefficients, the PCCs for the entire AUT hemisphere are calculated, converted into PCO/PCV, and exported in accordance with the ANTEX file format.

A comprehensive description of the GNSS receiver antenna calibration facility at LMMT in Croatia, along with the associated calibration methodology, is presented in Tupek et al. [17] and Tupek [32].

2.3. Validation Methodology

In general, two approaches are best suited for validating GNSS receiver antenna calibration results, i.e., PCC patterns. The first approach is to compare calibration results with independent results obtained by a different calibration institution, the so-called “ring calibration”. With such a validation test, the compatibility of two, or more, calibration institutions can be evaluated. Currently, the most comprehensive global “ring calibration” test is the IGS “RingCalVal” campaign, which includes eight calibration facilities globally and six different receiver antennas [24]. The second approach is to validate PCC patterns in a field test using short-baselines [11,29,40]. The advantage of such an approach is the fact that ground-truth, i.e., reference values of baseline components, is known and atmospheric effects (ionosphere, troposphere), as well as other GNSS error sources (satellite orbit and clock errors, satellite PCV), are effectively eliminated on a short-baseline.

Thus, in accordance with the goals of this research, i.e., to experimentally test the developed LMMT robotic calibration system, to assess the accuracy of dual-frequency GPS calibration results, and to extensively validate LMMT calibration results in the spatial domain, the following experimental tests and analyses have been performed:

• Accuracy analysis at the PCC pattern level by comparison with independent calibration results provided by Geo++ GmbH;
• Short-baseline validation and analysis in the spatial domain;
• Validation and analysis at the GNSS/GPS network solution level.

In the subsequent subsections of this section, each experimental test is elaborated from a methodological perspective. Obtained experimental research results are documented in Section 3.

To acquire data for validation and analysis purposes, from late May to mid-June of 2023, four calibration campaigns were carried out within which six distinct GNSS receiver antennas were individually calibrated using the developed robotic calibration system. A recapitulation of the basic information of the calibration campaigns is given in

Table 1. Summary of key details from the calibration campaigns conducted.

Campaign Number	Date of Calibration	Day of Year (DoY)
1	26 May 2023	146
2	16 June 2023	167
3	19 June 2023	170
4	20 June 2023	171

. The six GNSS antennas involved are four reference-station-grade Trimble Zephyr 2 Geodetic antennas (Figure 5a) and two rover-grade antennas—a Leica Geosystems AX1202 GG antenna (Figure 5b) and a Topcon PG-A1 antenna (Figure 5c). No radome was used during calibration with any antenna. A summarized overview of the individually calibrated receiver antennas is given in

Table 2. Overview of individually calibrated GNSS receiver antennas at the LMMT calibration facility; S/N—antenna manufacturer’s serial number; I/L—LMMT internal antenna label added for simpler referencing.

Antenna	S/N	IGS Antenna Code	I/L	Calibration Date/ Campaign No.
Trimble Zephyr 2 Geodetic	30739001	TRM57971.00 NONE	Z1	26 May 2023/1
				16 June 2023/2
				19 June 2023/3
				20 June 2023/4
Trimble Zephyr 2 Geodetic	30734472	TRM57971.00 NONE	Z2	19 June 2023/3
Trimble Zephyr 2 Geodetic	30738967	TRM57971.00 NONE	Z3	19 June 2023/3
Trimble Zephyr 2 Geodetic	30278601	TRM55971.00 NONE	Z4	16 June 2023/2
Leica Geosystems AX1202 GG	08190116	LEIAX1202GG NONE	LG	16 June 2023/2
				20 June 2023/4
Topcon PG-A1	308-1874	TPSPG_A1 NONE	TP	19 June 2023/3
				20 June 2023/4

.

2.3.1. Validation by Comparison with Geo++ GmbH

Experimental validation and accuracy analysis of the LMMT receiver antenna individual absolute calibration results were performed by comparing them with independent individual calibration results provided by Geo++ GmbH in Germany. Considering the importance of the Geo++ GmbH calibration system, its contribution to IGS, the fact that it is globally regarded as a leading authority in the field, and that this is basically the best in terms of calibration that one can do, it is completely justified to take Geo++ GmbH calibration results as reference values in the analysis.

Validation and accuracy analysis was conducted at the PCC pattern level for the antenna Trimble Zephyr 2 Geodetic (S/N: 30739001), which was individually calibrated by Geo++ GmbH on 5 August 2022. The GPS L1 (frequency code G01 consistent with ANTEX) and GPS L2 (frequency code G02 consistent with ANTEX) calibration results (PCO and PCV) for the TRM57971.00 NONE (S/N:30739001) antenna obtained by Geo++ GmbH are given in Figure 6.

The comparison of the two individual antenna calibration results (Geo++ and LMMT), along with the quantification of the corresponding scalar measures, was performed according to the PCC comparison methodology elaborated in detail by Tupek et al. [17] and Tupek [32]. The primary objective of the test is to assess the compatibility of LMMT calibration results with those obtained by Geo++ GmbH and to evaluate the practical significance of the developed LMMT system.

2.3.2. Short-Baseline Validation

The second experimental validation was conducted in a short-baseline setup. Four GNSS receiver antennas (I/L: Z1, TP, LG, Z2) were utilized for the validation test, forming three short-baselines of approximate lengths: 5 meres (baseline T–R), 10 m (baseline T–A), and 20 m (baseline T–ST). The short-baseline validation setup is depicted in Figure 7.

Such a short-baseline validation test is based on the evaluation of the developed antenna calibration system in the spatial domain and on the product level, i.e., antenna PCC model. The accuracy of GNSS-derived short-baseline solutions is determined by comparison with reference values of baseline components that have been predetermined with independent methods of significantly higher accuracy. The short-baseline approach is applied because GNSS carrier-phase data can be processed in a single-frequency mode, and thus PCC models can be evaluated per calibrated frequency.

Within this experimental validation, short-baseline processing was performed multiple times with two distinct antenna calibrations: type-mean from IGS and individual from LMMT. Furthermore, processing was performed in a single-frequency mode independently for GPS L1 frequency (G01), GPS L2 frequency (G02), and, lastly, in a dual-frequency mode by using the well-known ionosphere-free linear combination GPS L0 [41] of GPS L1 and GPS L2. Therefore, the main goals of this experimental validation test are to determine how the application of different PCC models transfers to the spatial domain and subsequently impacts geodetic-grade positioning in relative positioning mode and to evaluate if individual antenna calibration at the LMMT calibration facility results in higher baseline accuracy when compared to IGS type-mean antenna PCC models. A two-tailed F-test was conducted to objectively evaluate the equivalence of the obtained IGS and LMMT short-baseline accuracy results.

The realization in the test field, i.e., the determination of reference values of short-baseline components, was performed through an independent measurement campaign, which included a GNSS survey and a high-precision terrestrial survey of a micro-network using a total station and precise leveling. Precise coordinates of stations T, R, A, and ST were determined with submillimeter accuracy at a level of 0.1–0.2 mm [32]. Using these coordinates, reference values of short-baseline components were derived and are presented in

Table 3. Reference values of short-baseline components in ITRF2020 and topocentric frame (TF), i.e., local-level frame, predetermined by an independent measurement campaign.

Short-Baseline	$\begin{array}{l}\Delta \mathit{X}\\ \left[\mathbf{m}\right]\end{array}$	$\begin{array}{l}\Delta \mathit{Y}\\ \left[\mathbf{m}\right]\end{array}$	$\begin{array}{l}\Delta \mathit{Z}\\ \left[\mathbf{m}\right]\end{array}$	$\begin{array}{l}\Delta \mathit{e}\\ \left[\mathbf{m}\right]\end{array}$	$\begin{array}{l}\Delta \mathit{n}\\ \left[\mathbf{m}\right]\end{array}$	$\begin{array}{l}\Delta \mathit{u}\\ \left[\mathbf{m}\right]\end{array}$
T–R	0.1785	−4.8749	1.1078	−4.7350	1.6132	−0.0262
T–A	0.3986	−9.7471	2.2109	−9.4789	3.1940	−0.0274
T–ST	0.8633	−19.4700	4.4171	−18.9529	6.3345	−0.0091

.

2.3.3. GNSS/GPS Network Validation

The third experimental validation was conducted at a GNSS/GPS network solution level, i.e., based on the adjustment results of a GNSS/GPS network. Four GNSS receiver antennas (I/L: Z1, Z2, Z3, Z4) were utilized for the validation test, forming a network of four stations: CB1 (antenna Z1), CB2 (antenna Z2), BRSK (antenna Z3), and MDVG (antenna Z4). The GNSS/GPS network is depicted in Figure 8.

Validation at a network solution level is based on the evaluation of the developed LMMT antenna calibration system in the spatial domain and at the final product level. Experimental validation fundamentally relies on the analysis of the internal, or relative, network accuracy. Because only dual-frequency GPS antenna calibration results (L1 and L2) are validated, the network was processed in a dual-frequency GPS-only mode. The GPS data registered on all four network stations were processed two times using identical input data and processing parameters, different only in the applied antenna PCC models: firstly, with type-mean antenna PCC models by IGS and, secondly, with individual antenna PCC models by LMMT. All other processing parameters and input data were not changed. Therefore, a difference in the network solution and estimated station position accuracy is a direct consequence of the antenna patterns applied. As a scalar measure of local network accuracy, the standard deviation of station position is used. Furthermore, the global accuracy of the GNSS/GPS network is characterized by the standard deviation of point position $s_{\mathrm{M}}$, as defined by Mittermayer [42]:

$$s_{\mathrm{M}}={\overline{s}}_0\sqrt{{\displaystyle \frac{\mathrm{tr}\left(Q_{\mathrm{XX}}\right)}{u}}},$$

(5)

where $Q_{\mathrm{XX}}$ is the network’s cofactor matrix, ${\overline{s}}_0$ is the a posteriori standard deviation of unit weight, and $u$ is the number of network unknowns.

The main goals of this validation test are to determine how the application of different antenna PCC models transfers to the spatial domain and reflects on a GNSS network solution, and to evaluate if individual antenna calibration at the LMMT calibration facility results in higher local and global GNSS network accuracy when compared to type-mean antenna models provided by IGS.

3. Results and Discussion

According on the experimental validation methodology detailed in Section 2.3, and the receiver antenna calibration campaigns conducted in May and June of 2023, dual-frequency calibration results, i.e., PCC models (PCO and PCVs), for G01 and G02 were estimated for all six individually calibrated GNSS receiver antennas (Table 2). The final PCC model of antennas that have been calibrated multiple times (see Table 2) was determined according to the methodology presented in Tupek et al. [17] and Tupek [32], and is based on a conversion to a common datum and mean PCO vector. Since four antennas of the Trimble Zephyr 2 Geodetic type (LMMT internal labels Z1, Z2, Z3, and Z4) have very similar phase patterns, to maintain simplicity, here only results for the Z1 (TRM57971.00 NONE, S/N: 30739001) antenna are presented (Figure 9). Furthermore, Figure 10 and Figure 11 present calibration results of antennas LEIAX1202GG NONE (I/L: LG) and TPSPG_A1 NONE (I/L: TP).

Calibration results, i.e., PCC models, of the three main antenna types included in the validation show dominant elevation-dependent variations (PCV values). However, for antennas LEIAX1202GG NONE (Figure 10) and TPSPG_A1 NONE (Figure 11), substantial azimuth-dependent variations, for both G01 and G02 frequencies, are noticeable. This PCC’s azimuth dependency is particularly pronounced for the TPSPG_A1 NONE antenna, primarily attributed to the antenna’s square-shaped design and lack of a ground plane (Figure 5c). The TRM57971.00 NONE antenna (S/N: 30739001, I/L: Z1) exhibits a significantly more stable phase pattern, as to be expected given its classification as a reference-station-grade antenna and its superior quality among the three analyzed. Moreover, a difference in the PCO Up component between G01 and G02 PCC models of 7.8 mm is notable and interesting (Figure 9).

3.1. Results of Validation with Geo++ GmbH

To experimentally validate the LMMT GNSS receiver antenna calibration facility and its calibration results, a comparative analysis was conducted with independent calibrations obtained by Geo++ GmbH. Dual-frequency calibration results for the TRM57971.00 NONE antenna (S/N: 30739001, I/L: Z1) were compared at the pattern level. PCC difference patterns $\Delta PCC$, per frequency (G01 and G02), were calculated and are illustrated in Figure 12. PCC difference values were calculated for the entire upper antenna hemisphere, i.e., $\alpha =\left[0°,360°\right]$ and $z=\left[0°,90°\right]$, using an angular grid defined by azimuthal and zenithal increments of 5°. To quantitatively assess the accuracy of the LMMT calibrations, scalar metrics were derived from the calculated PCC difference patterns. These metrics are presented, per frequency, in

Table 4. Scalar metrics characterizing the difference between individual absolute calibrations from Geo++ GmbH and LMMT.

Calibration Frequency	Full Antenna Hemisphere (0° Elevation Cut-Off) [mm]					Reduced Antenna Hemisphere (10° Elevation Cut-Off) [mm]
	Min.	Max.	RMS	Range	IQR	Min.	Max.	RMS	Range	IQR
GPS L1 (G01)	−1.85	0.61	0.36	2.46	0.41	−1.13	0.61	0.31	1.74	0.38
GPS L2 (G02)	−2.15	1.01	0.54	3.16	0.67	−1.26	0.85	0.47	2.11	0.54

.

Analyzing the full receiver antenna hemisphere, Geo++ and LMMT PCC differences are within a range of 2.46 mm for G01 and 3.16 mm for G02. The middle half of PCC differences are within 0.41 mm for G01 and 0.67 mm for G02. The overall agreement between the individual PCC patterns of Geo++ and LMMT, quantified by the root-mean-square (RMS) deviation, per frequency, is:

$$\begin{array}{c}{\mathrm{RMS}}_{\mathrm{G}01}=0.36\mathrm{mm},\\ {\mathrm{RMS}}_{\mathrm{G}02}=0.54\mathrm{mm}.\end{array}$$

(6)

The GPS L1 calibration results show slightly better agreement with Geo++ GmbH reference values than GPS L2 results. However, as the G01 and G02 metrics differences are in the order of tenths of a millimeter, their potential impact on practical applications remains to be thoroughly analyzed. Furthermore, by analyzing scalar metrics for the elevation-reduced antenna hemisphere, a higher calibration accuracy is evident. However, this is to be expected because such an analysis does not include the antenna’s PCC model near the horizon (below 10° of elevation in the AF) which will, in general, always be of lower accuracy due to SH expansion properties and the fact that few or no observations are available below the antenna’s horizon.

In line with the obtained experimental validation results, it is evident that the developed GNSS receiver antenna absolute field calibration system is in agreement with Geo++ GmbH within 0.54 mm for dual-frequency GPS (L1 and L2) calibrations.

3.2. Results of Short-Baseline Validation

In line with the goal of this research to experimentally validate the LMMT calibration facility and corresponding calibration results by means of a short-baseline test, and according to the described validation methodology (Section 2.3.2), a 6 h long static GNSS survey on four stations (Figure 7), three independent short-baselines, was conducted on 7 June 2023. Each of the four antennas was connected to a Trimble NetR5 infrastructure GNSS receiver whereby only GPS carrier-phase observations were registered with a standard elevation mask of 10° and an observation interval of 1 s (1 Hz).

Table 5. Recapitulation of basic information on the GNSS (GPS) survey of short-baselines.

Date/DoY/Epoch.	7 June 2023/158/2023.43
Time of survey (UTC)/Survey duration	6:45–12:45/6:00
Observation interval	1 s
Elevation mask	10°
Average number of GPS satellites	9

gives a summarized overview of the GNSS (GPS) survey.

To estimate the impact of the receiver antenna PCC model (type-mean by IGS and individual by LMMT) on baseline accuracy, short-baseline solutions, i.e., geocentric (ITRF2020) and topocentric coordinate differences, were calculated two times using identical input data and processing parameters, different only in the applied antenna PCC models. Short-baselines were calculated in a relative positioning approach in single-frequency (only GPS L1 and only GPS L2) and dual-frequency (ionosphere-free linear combination GPS L0) modes utilizing IGS precise orbits and ionospheric corrections based on the IGS global ionospheric maps (GIMs). Processing of GPS-only observations was performed using the WaSoft software package [43]. Results of short-baseline postprocessing are presented in

Table 6. Short-baseline solutions, i.e., geocentric (ITRF2020) and topocentric coordinate differences, per frequency (GPS L1, GPS L2, and ionosphere-free linear combination L0 of L1 and L2) and per receiver antenna model (LMMT individual and IGS type-mean).

Short Baseline	Freq.	PCC Model	$\begin{array}{l}\Delta \mathit{X}\\ \left[\mathbf{m}\right]\end{array}$	$\begin{array}{l}\Delta \mathit{Y}\\ \left[\mathbf{m}\right]\end{array}$	$\begin{array}{l}\Delta \mathit{Z}\\ \left[\mathbf{m}\right]\end{array}$	$\begin{array}{l}\Delta \mathit{e}\\ \left[\mathbf{m}\right]\end{array}$	$\begin{array}{l}\Delta \mathit{n}\\ \left[\mathbf{m}\right]\end{array}$	$\begin{array}{l}\Delta \mathit{u}\\ \left[\mathbf{m}\right]\end{array}$
T–R	L1	IGS	0.1787	−4.8769	1.1069	−4.7370	1.6129	−0.0271
		LMMT	0.1786	−4.8759	1.1086	−4.7360	1.6139	−0.0258
	L2	IGS	0.1695	−4.8753	1.1006	−4.7329	1.6145	−0.0375
		LMMT	0.1808	−4.8747	1.1106	−4.7355	1.6136	−0.0226
	L0	IGS	0.1935	−4.8799	1.1180	−4.7440	1.6110	−0.0098
		LMMT	0.1760	−4.8780	1.1073	−4.7373	1.6152	−0.0289
T–A	L1	IGS	0.3982	−9.7479	2.2085	−9.4796	3.1928	−0.0295
		LMMT	0.3998	−9.7481	2.2119	−9.4802	3.1941	−0.0260
	L2	IGS	0.3967	−9.7477	2.2128	−9.4790	3.1968	−0.0274
		LMMT	0.3979	−9.7485	2.2120	−9.4801	3.1956	−0.0273
	L0	IGS	0.4023	−9.7483	2.2041	−9.4811	3.1870	−0.0300
		LMMT	0.4042	−9.7475	2.2138	−9.4809	3.1923	−0.0216
T–ST	L1	IGS	0.8639	−19.4707	4.4185	−18.9537	6.3352	−0.0078
		LMMT	0.8634	−19.4707	4.4186	−18.9536	6.3356	−0.0081
	L2	IGS	0.8659	−19.4720	4.4180	−18.9555	6.3337	−0.0071
		LMMT	0.8641	−19.4715	4.4197	−18.9545	6.3361	−0.0070
	L0	IGS	0.8618	−19.4689	4.4212	−18.9514	6.3382	−0.0069
		LMMT	0.8632	−19.4698	4.4187	−18.9526	6.3357	−0.0080

.

Furthermore, based on the obtained short-baseline solutions (Table 6) and the predetermined reference values of significantly higher accuracy (Table 3), short-baseline errors, per frequency and PCC model, were calculated and are presented in

Table 7. Short-baseline errors expressed along each coordinate axis individually, as well as in terms of the horizontal (2D) and total spatial (3D) errors, per frequency (GPS L1, GPS L2, and ionosphere-free linear combination L0 of L1 and L2) and per receiver antenna model (LMMT individual and IGS type-mean).

Short Baseline	Freq.	PCC Model	$\begin{array}{c}{\mathit{\epsilon }}_{\mathit{X}}\\ \left[\mathbf{mm}\right]\end{array}$	$\begin{array}{c}{\mathit{\epsilon }}_{\mathit{Y}}\\ \left[\mathbf{mm}\right]\end{array}$	$\begin{array}{c}{\mathit{\epsilon }}_{\mathit{Z}}\\ \left[\mathbf{mm}\right]\end{array}$	$\begin{array}{c}{\mathit{\epsilon }}_{\mathit{e}}\\ \left[\mathbf{mm}\right]\end{array}$	$\begin{array}{c}{\mathit{\epsilon }}_{\mathit{n}}\\ \left[\mathbf{mm}\right]\end{array}$	$\begin{array}{c}{\mathit{\epsilon }}_{\mathit{u}}\\ \left[\mathbf{mm}\right]\end{array}$	$\begin{array}{c}{\mathit{\epsilon }}_{\mathbf{2}\mathbf{D}}\\ \left[\mathbf{mm}\right]\end{array}$	$\begin{array}{c}{\mathit{\epsilon }}_{\mathbf{3}\mathbf{D}}\\ \left[\mathbf{mm}\right]\end{array}$
T–R	L1	IGS	−0.2	2.0	0.9	2.0	0.4	0.9	2.0	2.2
		LMMT	−0.1	1.0	−0.8	1.0	−0.7	−0.4	1.2	1.3
	L2	IGS	9.0	0.4	7.2	−2.1	−1.2	11.3	2.4	11.5
		LMMT	−2.3	−0.2	−2.8	0.4	−0.3	−3.6	0.6	3.6
	L0	IGS	−15.0	5.0	−10.2	8.9	2.2	−16.4	9.2	18.8
		LMMT	2.5	3.1	0.5	2.3	−2.0	2.6	3.0	4.0
T–A	L1	IGS	0.4	0.8	2.4	0.7	1.2	2.1	1.4	2.6
		LMMT	−1.2	1.0	−1.0	1.3	−0.1	−1.3	1.3	1.9
	L2	IGS	1.9	0.6	−1.9	0.1	−2.8	0.0	2.8	2.8
		LMMT	0.7	1.4	−1.1	1.2	−1.5	0.0	1.9	1.9
	L0	IGS	−3.7	1.2	6.8	2.2	7.1	2.6	7.4	7.8
		LMMT	−5.6	0.4	−2.9	1.9	1.7	−5.8	2.6	6.3
T–ST	L1	IGS	−0.6	0.7	−1.4	0.8	−0.7	−1.3	1.1	1.7
		LMMT	−0.1	0.7	−1.5	0.7	−1.1	−1.0	1.3	1.7
	L2	IGS	−2.6	2.0	−0.9	2.6	0.8	−2.0	2.7	3.4
		LMMT	−0.8	1.5	−2.6	1.7	−1.6	−2.1	2.3	3.1
	L0	IGS	1.5	−1.1	−4.1	−1.5	−3.7	−2.1	4.0	4.5
		LMMT	0.1	−0.2	−1.6	−0.2	−1.1	−1.1	1.2	1.6

. Short-baseline errors are expressed along each coordinate axis individually, as well as in terms of the horizontal (2D) and total spatial (3D) errors.

From a simple overview of the resulting short-baseline errors, consistently lower values obtained with the application of LMMT individual PCC models are evident. Furthermore, short-baseline errors derived by forming the ionosphere-free linear combination GPS L0 of GPS L1 and GPS L2 show a change correlated with the ionosphere-free linear combination’s coefficients, i.e., approximately 2.54 for G01 (GPS L1) and −1.54 for G02 (GPS L2). This results in a significant amplification of short-baseline errors based on the application of IGS type-mean PCC models.

To objectively evaluate the equivalence of the obtained short-baseline accuracies with respect to the antenna phase center model used (LMMT individual or IGS type-mean) all short-baseline validation results were considered, regardless of processing frequency, and a two-tailed F-test was conducted. The general formula to calculate baseline accuracy $\sigma$ is given by the following expression:

$$\sigma =\sqrt{{\displaystyle \frac{1}{n}}{\displaystyle \sum _{i=1}^n{\epsilon }_i^2}}$$

(7)

Short-baseline accuracy was quantified in 1D (height or equivalent to the PCO Up component), 2D (horizontal component), and 3D (spatial component) per antenna model (LMMT individual or IGS type-mean). The goal of such an analysis is to evaluate if the application of different antenna PCC models has a different impact on baseline determination in the height direction, horizontal direction, and spatial direction. Therefore, in line with this analysis, based on the determined short-baseline errors (Table 7) and Equation (7), the accuracies of short-baselines were calculated and are presented in

Table 8. Short-baseline accuracy results in the height direction, 2D, and 3D per antenna PCC model applied to baseline processing (LMMT individual or IGS type-mean). All values are in millimeters.

PCC Model	$\mathbf{Baseline}\mathbf{Accuracy}\left(\mathit{\sigma }\right)$ Per Component
	Height	2D	3D
IGS	6.82	4.51	8.18
LMMT	2.62	1.87	3.21

.

The obtained short-baseline accuracy results imply a higher achieved accuracy based on the application of LMMT individual antenna phase center models. Therefore, to test the hypothesis of equality of the gained accuracies, a two-tailed F-test was performed. The corresponding null hypothesis $H_0$ and alternative hypothesis $H_{\mathrm{A}}$ were set:

$$\begin{array}{c}{H}_0:\hspace{1em}{\sigma }_{\mathrm{IGS}}={\sigma }_{\mathrm{LMMT}},\\ H_{\mathrm{A}}:\hspace{1em}{\sigma }_{\mathrm{IGS}}\ne {\sigma }_{\mathrm{LMMT}}.\end{array}$$

(8)

The tests were performed at a significance level of $\alpha =0.05$, with the numerator and denominator degrees of freedom $f_{\mathrm{IGS}}=f_{\mathrm{LMMT}}=8$. Key statistical parameters for the analysis are summarized in

Table 9. Key statistical parameters of the performed two-tailed F-test for equality of variances per short-baseline component.

Short-Baseline Component	Test Statistics	p-Value	Lower Critical Value	Upper Critical Value	Hypothesis Accepted
Height	6.793	0.014	0.226	4.433	$H_{\mathrm{A}}$
2D	5.850	0.022	0.226	4.433	$H_{\mathrm{A}}$
3D	6.475	0.016	0.226	4.433	$H_{\mathrm{A}}$

.

Results of the performed two-tailed F-tests provide strong evidence to reject the null hypothesis. With a probability of 0.95, it can be concluded that empirical accuracies (standard deviations) of the two independent experimental data sets are significantly different. A significant difference in short-baseline accuracies with respect to the antenna phase center models used (individual by LMMT and type-mean by IGS) was achieved. Moreover, statistical testing results showed that, by applying LMMT individual PCC models in baseline processing, higher short-baseline accuracy was achieved in the height baseline component, horizontal (2D) baseline component, and spatial (3D) baseline component. To summarize, in the short-baseline validation test conducted, the application of LMMT individual receiver antenna PCC models positively transferred into the spatial domain and yielded approximately 2.5 times higher baseline accuracy compared to the use of IGS type-mean antenna PCC models. The developed LMMT robotic calibration system results in higher baseline accuracy compared to IGS type-mean antenna calibration models.

3.3. Results of GNSS/GPS Network Validation

In the last, third, validation test within this research, an analysis at the GNSS/GPS network solution level was conducted. According to the experiment methodology (Section 2.3.3), a static GNSS survey on four stations (Figure 8) was conducted on 12 June 2023. Each of the four Trimble Zephyr 2 Geodetic GNSS antennas (I/L: Z1, Z2, Z3, Z4) was connected to a Trimble NetR5 infrastructure GNSS receiver, whereby only GPS carrier-phase observations was registered with an observation interval of 15 s and a standard elevation mask of 10°. A summarized overview of the GNSS (GPS) survey is given in

Table 10. Recapitulation of basic information on the GPS survey of the GNSS/GPS network.

Date/DoY/Epoch	12 June 2023/163/2023.45
Time of survey (UTC)/Survey duration	6:20–13:00/6:40
Registration interval	15 s
Elevation mask	10°
Average number of GPS satellites	7

.

To estimate the influence of the receiver antenna PCC model (type-mean by IGS or individual by LMMT) on the GNSS/GPS network’s local and global accuracy, the network solution was computed two times using identical input data and processing parameters, different only in the applied antenna PCC models.

After the field survey, processing of GPS-only observations was performed using the WaSoft software package [43]. Baseline processing in the GNSS/GPS network was conducted in the standard dual-frequency mode by utilizing IGS precise (final) satellite orbits, IGS global ionospheric maps (GIMs), and tropospheric corrections (tropospheric zenith delay) based on the Saastamoinen model. The network was adjusted as a minimum constraint network and in a “each-to-each” approach by processing all six baselines between all four network stations. Furthermore, the non-standard approach without involvement of fixed stations was applied. The reasoning behind such an approach is that inclusion of reference points of higher order implies inclusion of antennas that have not been individually calibrated by the LMMT calibration system. This is contrary to the goals of this analysis. Furthermore, to support this research design approach, the primary focus of this analysis is not on station positions (coordinates), but rather on the achieved local and global network’s accuracy.

Results of GNSS/GPS network postprocessing and adjustment, per antenna model applied (type-mean by IGS or individual by LMMT), are presented in

Table 11. Earth-Centered–Earth-Fixed (ECEF) GNSS/GPS network station positions in ITRF2020 (epoch 2023.45) with corresponding positional accuracies expressed as standard deviations, based on the application of IGS type-mean antenna PCC models.

Station	$\begin{array}{c}\mathit{X}\\ \left[\mathbf{m}\right]\end{array}$	$\begin{array}{c}\mathit{Y}\\ \left[\mathbf{m}\right]\end{array}$	$\begin{array}{c}\mathit{Z}\\ \left[\mathbf{m}\right]\end{array}$	$\begin{array}{c}{\mathit{s}}_{\mathit{X}}\\ \left[\mathbf{mm}\right]\end{array}$	$\begin{array}{c}{\mathit{s}}_{\mathit{Y}}\\ \left[\mathbf{mm}\right]\end{array}$	$\begin{array}{c}{\mathit{s}}_{\mathit{Z}}\\ \left[\mathbf{mm}\right]\end{array}$	$\begin{array}{c}{\mathit{s}}_{\mathbf{3}\mathbf{D}}\\ \left[\mathbf{mm}\right]\end{array}$
CB1	4,288,747.3357	1,230,643.5523	4,542,659.7027	0.99	1.15	0.75	1.69
CB2	4,288,755.2340	1,230,468.2036	4,542,699.6583	0.99	1.15	0.75	1.69
MDVG	4,278,144.0819	1,221,927.0432	4,555,592.3033	1.58	1.40	1.69	2.70
BRSK	4,307,965.4474	1,200,393.7500	4,532,779.1697	2.44	2.98	1.38	4.09

(IGS PCC model) and

Table 12. Earth-Centered–Earth-Fixed (ECEF) GNSS/GPS network station positions in ITRF2020 (epoch 2023.45) with corresponding positional accuracies expressed as standard deviations, based on the application of LMMT individual antenna PCC models.

Station	$\begin{array}{c}\mathit{X}\\ \left[\mathbf{m}\right]\end{array}$	$\begin{array}{c}\mathit{Y}\\ \left[\mathbf{m}\right]\end{array}$	$\begin{array}{c}\mathit{Z}\\ \left[\mathbf{m}\right]\end{array}$	$\begin{array}{c}{\mathit{s}}_{\mathit{X}}\\ \left[\mathbf{mm}\right]\end{array}$	$\begin{array}{c}{\mathit{s}}_{\mathit{Y}}\\ \left[\mathbf{mm}\right]\end{array}$	$\begin{array}{c}{\mathit{s}}_{\mathit{Z}}\\ \left[\mathbf{mm}\right]\end{array}$	$\begin{array}{c}{\mathit{s}}_{\mathbf{3}\mathbf{D}}\\ \left[\mathbf{mm}\right]\end{array}$
CB1	4,288,747.3376	1,230,643.5514	4,542,659.7015	0.75	0.88	0.57	1.30
CB2	4,288,755.2354	1,230,468.2027	4,542,699.6572	0.75	0.88	0.57	1.30
MDVG	4,278,144.0785	1,221,927.0438	4,555,592.3017	1.21	1.07	1.29	2.07
BRSK	4,307,965.4476	1,200,393.7511	4,532,779.1737	1.87	2.29	1.06	3.14

(LMMT PPC model).

Furthermore, according to the elaborated validation methodology (Section 2.3.3), for both GNSS/GPS network realizations, the global accuracy of the network, defined by the standard deviation of point position according to Equation (5), is calculated:

$$\begin{array}{c}{s}_{\mathrm{M}}^{\mathrm{IGS}}=1.57\mathrm{mm},\\ s_{\mathrm{M}}^{\mathrm{LMMT}}=1.21\mathrm{mm}.\end{array}$$

(9)

By analyzing the local network accuracy, i.e., the standard deviations of station positions (Table 11 and Table 12), a higher achieved network accuracy based on the usage of LMMT individual antenna PCC models is evident. Furthermore, the higher global accuracy of the network, as quantified by the standard deviation of the point position following Mittermayer’s methodology, further confirms that individual GNSS receiver antenna calibration at LMMT leads to improved accuracy in network realization.

To summarize, in the GNSS/GPS network validation test conducted, the application of LMMT individual receiver antenna PCC models positively transferred into the spatial domain and yielded approximately 23% higher network realization accuracy compared to the use of type-mean antenna PCC models provided by IGS.

4. Conclusions

This research builds upon previous studies and further substantiates that the newly developed GNSS receiver antenna robotic calibration system, designed and implemented de novo at the Laboratory for Measurements and Measuring Technique, University of Zagreb—Faculty of Geodesy, Croatia, delivers meaningful and reliable dual-frequency GPS calibration results.

With the goal of extensively evaluating the antenna calibration system at LMMT, three independent validation tests were conducted. To acquire data for validation and analysis purposes, four calibration campaigns were carried out, within which six distinct GNSS receiver antennas were individually calibrated using the LMMT robotic calibration system. An independent accuracy analysis at the PCC pattern level was performed by comparing LMMT calibration results with Geo++ GmbH results. In line with that, Geo++ GmbH and LMMT individual dual-frequency PCC models for the TRM57971.00 NONE antenna (S/N:30739001) were compared and PCC differences calculated. An estimated overall agreement between the dual-frequency GPS (L1 and L2) antenna PCC patterns determined by Geo++ GmbH and LMMT was achieved, with an RMS deviation of up to 0.45 mm. These experimental results are in line with previous research and further prove the compatibility of LMMT and Geo++ GmbH calibration facilities at the dual-frequency GPS level.

Furthermore, a short-baseline validation test was conducted to evaluate how the application of different PCC models transfers to the spatial domain and thus reflects on geodetic-grade positioning in relative positioning mode, and to determine if individual antenna calibration at the LMMT calibration facility results in higher baseline accuracy when compared to type-mean antenna PCC models provided by IGS. Research results showed that a significant difference in short-baseline accuracies, with respect to the antenna phase center models applied (LMMT individual and IGS type-mean), was achieved. The developed GNSS receiver antenna robotic calibration system at LMMT results in approximately 2.5 times higher baseline accuracy compared to IGS type-mean models.

Moreover, in the last validation test, the impact of the receiver antenna PCC model (LMMT individual and IGS type-mean) on GNSS/GPS network solution was evaluated. Because of the different PCC models applied, differences in station positions reach 4 mm in terms of total spatial (3D) offset. The developed GNSS receiver antenna robotic calibration system at LMMT resulted in approximately 23% higher network realization accuracy with respect to the use of IGS type-mean models. The obtained results are in line with the established scientific consensus and confirm the practical importance and significance of GNSS receiver antenna individual absolute calibration in the context of high-accuracy GNSS positioning. Individual calibration is particularly important to users dealing with the realization of national, regional, and global reference networks, geodynamics, monitoring, engineering projects, etc.

Multi-GNSS is undoubtedly the standard in contemporary practical and scientific applications requiring high-accuracy satellite positioning. Moreover, the need for high-precision multi-GNSS receiver antenna phase center models will increase in future. The advancement of the precise point positioning (PPP) technique is expected to achieve a level of accuracy where the accuracy of type-mean antenna models will become a limiting factor. Therefore, future research at LMMT will primarily focus on further calibration system developments in terms of providing multi-GNSS and multi-frequency calibration results to the antenna community. Also, the system will be developed following the activities of the IGS Antenna Working Group (IGS AWG) and the development of the new ANTEX 2.0 file format.