Performance of Atlas GNSS Global Correction Service for High-Accuracy Positioning

The first data set consists of 135 h of GNSS data acquired in June 2019 over the roof of the Engineering Department of the University of Bologna at the sampling frequency of 1 Hz, thus providing a representative statistical sample for a general assessment about the Atlas global correction service during a standalone positioning. The absence of significant obstacles that could obstruct the proper satellites view from the receiver guaranteed ideal conditions. Data were acquired using all the available constellations in 41 sessions with a time span ranging from about 1–5 h. The receiver was switched off and restarted after each session with the aim to simulate a number of surveys in the most similar conditions and allowing a statistical evaluation of the convergence time the system needs to provide the declared performances. The upper-bound limit of about 5 h was not a choice but a technical limit of the instrument controller coupled with the S900A, which was not designed for acquiring autonomously real-time positions for very long sessions. Moreover, in some cases, we shortened the observing sessions to obtain a higher number of new converging phases and better evaluate this parameter. Once the instrument had converged, the acquisition continued in order to evaluate repeatability of the solutions, also on long acquisition times.

In three cases, the raw GNSS observables (codes and phases) were recorded too, in order to calculate the postelaborated coordinates with a PPP approach by using GIPSY-OASIS II version 6.4 software. This was done by continuously acquiring data for about 12 h because the record of raw data does not undergo the limitations occurring to Atlas real-time solutions, and it allowed us to have a reference position for the chosen point.

The second data set was acquired during September 2019 to consider the heterogeneous conditions of a real situation. In this case, the surveyed test points were those of the Coastal Geodetic Network (RGC), realized in 2016 thanks to the collaboration between the Coastal Monitoring Unit of regional agency for prevention, environment and energy of emilia-romagna (ARPAE) and the Department of Civil, Chemical and Environmental Engineering (DICAM) of the University of Bologna (Vecchi et al. 2020). These benchmarks, located along the Adriatic coast, are used for local monitoring surveys and their coordinates are aligned to the official national reference system ETRS89-ETRF2000 (2008.0 epoch) (Gandolfi et al. 2017a; Vecchi et al. 2020). RGC monographs available on the ARPAE Cartographic Portal (ARPAE 2021), have been used as a reference for the following analysis. These measures involved 15 RGC benchmarks, and for each one, a 10-min-long observing session was performed acquiring data at a 1-Hz frequency. Also in this case, signals of all the constellations were used.

The campaign involved three distinct days, surveying five points each. The time spent between the instrument turn on and when it reached converged condition (CONV marked on the screen) was accounted for, but no data were acquired in the meanwhile. Moving from a point to another, the receiver was not powered off in order to avoid new initializations of the system, in the perspective of a time optimization required by operational conditions. In several cases, the convergence was lost when moving, and the time necessary to reach it again was accounted for after having reached the next RGC benchmark. In only one case (RICE0300), it was impossible to perform the Atlas positioning because of an obstruction of the sky visibility in the direction of the geostationary satellite.

Assessment of the Atlas system’s accuracy required a set of reference coordinates expressed in the same reference system stated by Hemisphere, which is the ITRF2008. By comparing these reference coordinates with the ones obtained by the survey, it is possible to describe the agreement of the observed values with their true values. In order to compute the most representative ITRF2008 coordinates of the lab test benchmark estimated using the Atlas service, we averaged the values of all the coordinates acquired under the condition of having a formal horizontal error (HRMS) less than $4\mathrm{cm}$. Because the system is declared converged when the HRMS goes under $8\mathrm{cm}$, this should allow us to consider only positions recorded in best operating conditions. The number of such positions actually used was 254,376, meaning more than 70 h of observations.

The reference position for the same point was estimated by computing three different PPP solutions using the GIPSY-OASIS II software [developed by jet propulsion laboratory (JPL)]. These solutions were calculated starting from raw data acquired during long observing sessions (about 12 h), and they should guarantee an accuracy within $1\mathrm{cm}$ (Gandolfi et al. 2017b). The formal reference frame to which the PPP solutions are aligned is IGb08, a coherent update of the ITRF2008, therefore fully comparable with the Atlas reference. The three independent PPP solutions were averaged, obtaining the reference position with which to compare the one estimated using Atlas. The comparison between the postprocessed PPP coordinates and the ones obtained from Atlas is given in Table 1, where positions are expressed in the universal transverse mercator (UTM) system. The agreement of the mean Atlas position with the reference one (IGb08) is at the centimeter level, thus proving the correct realization of the Atlas reference system. In light of this, the averaged Atlas coordinates will be used a reference for all the following evaluations.

As for the field test, the reference coordinates of each benchmark were compared with the averaged value of the coordinates surveyed in the related $10\text{-}\min$ session performed using Atlas in converged conditions. Because the RGC reference coordinates are expressed in the ETRF2000 (2008.0)-UTM32 frame, a transformation was performed to align the Atlas solutions and allow the comparison. In order to compare coherently the field test results with those of lab test, for the latter, we also evaluated the accuracy by considering the mean coordinates recorded in the first $10\min$ after convergence for each of the 41 sessions.

The evaluation of the convergence time of the solutions is a very significant aspect, especially for the practical point of view, because it is the time that a surveyor has to wait before starting to work, and it could depend on several elements, such as the satellites view and the multipath effect (Seepersad and Bisnath 2015; Abou Galala et al. 2018). When the receiver is turned on, the system needs some time to provide the declared precision. The used H10 service fixes a threshold of $8\mathrm{cm}$ for the HRMS value to consider a solution as converged. The controller displays a mark CONV to state the converged status of the system, and a surveyor should use this information to start operating expecting the performance declared by Hemisphere. The manufacturers declare a convergence time ranging from 10 to $40\min$ (Hemisphere 2020). Because all the solutions with a HRMS value equal or smaller to $8\mathrm{cm}$ are considered as converged, we computed the time necessary to reach this condition for each measuring session.

Coordinate repeatability is the parameter that can be used to evaluate the positioning precision. It refers to the agreement between results obtained by measuring the same point in the same conditions in relation to a certain range of time. In the analysis, we distinguish between two different concepts: precision and pass-to-pass repeatability.

Precision has been evaluated by computing the STD of the coordinates acquired during the same session in converged conditions. This evaluation followed the same procedure for both the data sets, with sessions of different length: a few hours for the lab test and $10\min$ for the field test. The precision aspect can be analyzed in deeper detail by observing the histogram showing the distribution of the residuals, or in other words the probability of occurrence of a certain value. The histogram of the horizontal residuals was obtained by considering the Euclidean distance of the coordinates with respect to the real value, choosing classes of $0.5\mathrm{cm}$. The histogram of the height residuals have been defined using classes of $1\mathrm{cm}$. Both the described graphs have been obtained by considering only the residuals related to converged solutions for all the acquired sessions.

The pass-to-pass repeatability has been defined for agricultural applications (ISO 2012; Hemisphere 2018), and it represents the capability of the system to provide coherent coordinates within a short time range. Therefore, we simulated a test comparing each couple of observations spaced out by $15\min$ to obtain a distribution of these biases. We calculated their standard deviation, which actually represents the pass-to-pass precision, and the maximum bias between these couples of coordinates. The declared pass-to-pass repeatability for the Atlas service is about $2.5\mathrm{cm}$ ($1\sigma$) (Hemisphere 2018).

Another analysis focused on the precision, which could be obtained from static surveys with different acquisition times (1, 5, 10, 15, and $30\min$). The result of a static survey is typically the average value of the coordinates acquired during the observing session; therefore, we used moving averages to simulate several of these surveys. The STD of the time series of averaged coordinates was computed to define the precision of the measuring sessions performed using different time spans. This computation concerned only the time series with at least $90\min$ of data in converged conditions, and the standard deviation has been calculated for each time duration.

During a real-time positioning, users can have feedback on the coordinate quality thanks to two statistical parameters: the formal horizontal and vertical errors (HRMS and VRMS). Therefore, these values should be reliable to avoid accepting solutions that are actually less precise or waiting too long to record the coordinates. In particular, the formal error should be reliable, especially after reaching the convergence; otherwise, the user might not realize they were working without the expected precision.

The lab test allowed the evaluation of the reliability of the formal errors, thanks to the availability of a reference position to calculate the real errors of the coordinates. For each of the 41 sessions, a time series of the residuals between Atlas coordinates and the reference position was calculated. The value of each residual has been compared with its related formal error, expressed in terms of RMS ($\sigma$), multiplied by a scalar $n$ in order to define a certain confidence level.

A reliability parameter ${\nu }_k^i$ was calculated for any epoch $i$ by usingwhere $k=N$, $E$, and $h$ components; $P_k^{\mathrm{ref}}$ = reference coordinates; and $P_k^i$ = measured ones. The formal error RMS is the HRMS in case of northing and easting, whereas it is the VRMS for the height. We used $n=2$ to define the 95% confidence level or $n=3$ for the 99.73% one. If the absolute value of the residual is higher than the threshold ($n\times {\mathrm{RMS}}^i$) this means that the formal error has been underestimated at the considered confidence level. The percentages of solutions affected by an underestimation of the formal error have been evaluated throughwhere $n_k$ = number of observations of the session used to weight the value.

The analysis of the repeatability focused on three different possible scenarios: measures repeated after long periods, measures acquired within a short time range, and static surveys. Fig. 3 provides an example of a 4.5-h time series, which could be useful to understand the behavior of the Atlas positioning over time. The time series shows the residuals with respect to a topocentric reference system centered on the reference position. The thick lines represent data filtered by a moving average.

In Fig. 3(a), the residuals of the plan components are represented, and different shades are used to represent their time variation. Looking at the whole data set, coordinates seem to be grouped around the reference value. Nevertheless, by filtering the time series using a simple moving average (thick line), a drift over time can be observed. This is probably due to tidal effects or tropospheric delays that are not completely modeled, maybe because of coarse density of the ground network used to compute Atlas corrections. Consequently, if we focus on short time spans, coordinates are scattered around the local mean, which is less than what they were around the mean over the whole period. This can be easily observed in the north–east graph reported in Fig. 3 for the plan components, but the same can be said as for the Up component by looking at its time series.

Figs. 5 and 6 represent two examples of time series of the residuals in northing, easting, and height. A light shade is used when the real error is smaller than $n$ times the formal error according to Eq. (1), and a dark shade is used to highlight cases of underestimations. In particular, Fig. 5 relates to a session acquired in favorable conditions, which provided optimal results, whereas Fig. 6 shows an example of time series related to a session affected by some anomalies. In the latter case, Fig. 6(a) was obtained by considering $2\sigma$ confidence interval, and Fig. 6(b) relates to $3\sigma$.

Basing on Eq. (2), the average percentages of solutions for which the associated HRMS or VRMS value are underestimated were computed for all data sessions and are reported in Table 6. Considering the $2\sigma$ confidence interval, the formal error is reliable for the northing, with only the 2% of the solutions affected by underestimation of the formal error. Differently, for the easting and the height, the underestimations of the HRMS and VRMS occur in 9% and 10% of the cases, respectively. These values are higher than the 5% expected for the defined confidence interval. In Table 6, the first line refers to all the data, including some sessions for which anomalous percentages of underestimated errors were found (11 sessions out of a total of 41). Excluding these sessions from the statistics, the results are presented in the second line of Table 6. In this case, the formal errors are compliant with the expected probabilities for all coordinate components. Furthermore, the same evaluations have been done for all data considering a $3\sigma$ confidence interval, and the results are given in Line 3 of Table 6. Also in this case, the percentages of underestimated formal errors are higher than the expected significance level.