MultiGNSS orbit determination using satellite laser ranging
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Abstract
Galileo, BeiDou, QZSS, and NavIC are emerging global navigation satellite systems (GNSSs) and regional navigation satellite systems all of which are equipped with laser retroreflector arrays for range measurements. This paper summarizes the GNSSintensive tracking campaigns conducted by the International Laser Ranging Service and provides results from multiGNSS orbit determination using solely SLR observations. We consider the whole constellation of GLONASS, all active Galileo, four BeiDou satellites: 1 MEO, 3 IGSO, and one QZSS. We analyze the influence of the number of SLR observations on the quality of the 3day multiGNSS orbit solution. About 60 SLR observations are needed for obtaining MEO orbits of sufficient quality with the root mean square (RMS) of 3 cm for the radial component when compared to microwavebased orbits. From the analysis of a minimum number of tracking stations, when considering the 3day arcs, 5 SLR stations do not provide a sufficient geometry of observations. The solution obtained using ten stations is characterized with RMS of 4, 9, and 18 cm in the radial, alongtrack, and crosstrack direction, respectively, for MEO satellites. We also investigate the impact of the length of orbital arc on the quality of SLRderived orbits. Hence, 5 and 7day arcs constitute the best solution, whereas 3day arcs are of inferior quality due to an insufficient number of SLR observations and 9day arcs deteriorate the alongtrack component. The median RMS from the comparison between 7day orbital arcs determined using SLR data with microwavebased orbits assumes values in the range of 3–4, 11–16, and 15–27 cm in radial, alongtrack, and crosstrack, respectively, for MEO satellites. BeiDou IGSO and QZSS are characterized by RMS values higher by a factor of 8 and 24, respectively, than MEO orbits.
Keywords
MultiGNSS SLR Precise Orbit Determination GLONASS Galileo BeiDou QZSS1 Introduction
1.1 Role of SLR in space geodesy
Satellite laser ranging (SLR) is a precise spacegeodetic technique that provides range measurements to artificial satellites. Due to both the characteristics of geodetic satellites and precise devices for range measurements installed at SLR tracking stations, SLR plays an important role in the realization of the International Terrestrial Reference Frame (ITRF, Altamimi et al. 2016). Orbits of geodetic satellites, such as LAGEOS or Etalon, are determined with subcentimeter accuracy using range measurements (Sośnica et al. 2014; Appleby et al. 2016). As a result, SLR contributes to the determination of geocenter coordinates, defining thus the origin of ITRF, the global scale, and station coordinates. Apart from the realization of ITRF, SLR provides the most accurate value of the standard gravitational parameter, GM, and lowdegree spherical harmonics of the Earth’s gravity field (Thaller et al. 2011; Bloßfeld et al. 2015; Sośnica et al. 2015; Cheng and Ries 2017). The International Laser Ranging Service (ILRS, Pearlman et al. 2002) unifies and coordinates all activities of SLR stations that represent the ground segment of SLR. The ILRS does not only collect, archive, analyze, and distribute SLR and Lunar Laser Ranging data, but also supports different space missions by providing special SLR tracking campaigns and the priority list^{1} including satellites to be tracked by the SLR stations. The priority list contains both passive geodetic satellites and various active spacecraft equipped with Laser Retroreflector Arrays (LRAs) for range measurements.
1.2 SLR tracking of GNSS
Due to the fact that all new active navigation satellites are equipped with LRAs, SLR serves as a validation tool for microwavebased GNSS orbits (Zhu et al. 1997; Appleby et al. 1999; Urschl et al. 2007; Fritsche et al. 2014; Montenbruck et al. 2015a; Steigenberger et al. 2015; Zajdel et al. 2017). The SLR validation performed by Sośnica et al. (2015) was characterized by the root mean square (RMS) at the level of 2.4 and 3.3 cm for GPS and GLONASS satellites, respectively, which coincides with the orbit accuracy declared by the International GNSS Service (IGS, Dow et al. 2009). SLR residuals play a crucial role in the quality evaluation of both operational products of the official multiGNSS experiment (MGEX, Montenbruck et al. 2017; Prange et al. 2017) and the realtime MGEX products (Kazmierski et al. 2018). Finally, SLR residuals may serve as a validation tool for the empirical models designed for the absorption of solar radiation pressure (SRP) such as the Empirical CODE Orbit Model (ECOM, Arnold et al. 2015) or boxwing models (RodriguezSolano et al. 2014).
Range measurements can also be used for an independent determination of satellite orbits. Pavlis (1995) determined GPS orbits solely from SLR observations obtaining the RMS at the level of 7.7, 75.1, and 56.5 cm for GPS35 and 9.8, 72.9, and 90.9 cm for GPS36 in the radial, alongtrack, and crosstrack direction, respectively. A joint adjustment of GNSS and SLR observations performed by Urschl et al. (2007) provided an improvement of the determination of the semimajor axis of GNSS orbits. Urschl et al. (2008) calculated the preliminary 9day orbits of the very first Galileo, i.e., GIOVEA, using solely SLR measurements and achieved the accuracy at the level of 0.1, 0.5, and 1.0 m in the radial, alongtrack, and crosstrack direction, respectively. Montenbruck et al. (2015b) calculated 14day orbits using solely SLR data for two NavIC satellites, IRNSS1A and IRNSS1B. Due to the poor geometry of SLR observations provided by 8 stations, out of which only 2 were located out of Europe, and an insufficient number of SLR observations, NavIC orbits were determined with the accuracy at the level of 2, 15, and 10 m in the radial, alongtrack, and crosstrack direction, respectively. The combination of GNSS and SLR observations performed by Hackel et al. (2015) resulted in a mitigation of systematic errors in GalileoIOV solutions. The degradation of the internal consistency between GNSS and SLR combination was solved by adding an offset of 5 cm to Galileo’s LRA which should rather be assigned to mismodeling of the microwave antenna thrust (Steigenberger et al. 2017), albedo (RodriguezSolano et al. 2012), and the existence of the satellite signature effect (Otsubo et al. 2001; Sośnica et al. 2015).
1.3 The new GNSS constellations
Characteristics of the MGEX constellation considered in the analysis
System  GLONASS  Galileo  BeiDou  QZSS  

Type  GLONASSM  GLONASSK  GalileoIOV  GalileoFOC  GalileoFOC (extended)  MEO  IGSO  GEO  QZS1 
R01–R08  E11, E12  E26, E22, E24, E30  C11, C12, C14  C06, C07, C08  C01, C02, C03  
PRN number  R10–R19  R09, R20 (spare)  E19, E20  E08, E09, E01, E02  E14, E18  C33, C34, C35  C09, C10, C15 (C13)  C04, C05, C17  J01 
R21–R24  E07, E03, E04, E05  C31, C32  
CODE products  ALL  ALL  E11, E12, E19  ALL  ALL  C11–C14  C06–C10, C15 (C13)  –  J01 
ILRS tracking  ALL  ALL  ALL  ALL  ALL  C11, C33, C34  C08, C10, C15, C31, C32  C01  J01 
Orbit type  MEO  MEO  MEO  MEO  MEO  MEO  Geosynch.  Geostat.  Geosynch. 
No. of retroreflectors  112  123  84  60  60  42  42  90  56 
Back coating  N (launched after 2010)  Y  N  N  N  N  N  N  N 
Corner cube height/dim. (mm)  19.7/29.0  19.1/28.3  23.3/33.0  19.1/28.2  19.1/28.2  24.0/33.0  24.0/33.0  24.0/33.0  29.7/40.6 
Size of LRA (mm)  311.0/510.8  Outer dimension: 633.7  430.0/470.0  331.0/248.7  331.0/248.7  316.0/280.0  316.0/280.0  490.0/430.0  400.0/400.0 
Inner dimension: 342.5  
Mass (dry) (kg)  1415  935  695–697  706–709  661/662  1900  1900  1550  1800 
Semimajor axis (km)  25,507  25,507  29,600  29,600  27,978  27,904  42,165  42,165  42,169 
Altitude (km)  19,132  19,132  23,225  23,226  17,178–26,019  21,529  35,790  35,790  32,000–40,000 
Revolution period (h)  11.26  11.26  14.08  14.08  12.94  12.89  23.93  23.93  23.93 
Inclination (\(^{\circ }\))  64.8  64.8  54.9–55.6  54.9–57.3  50.1  56.2  53.3–57.7  0.9–1.6  40.7 
Eccentricity  0.0009–0.0015  0.0015  0.0001–0.0002  0.0001–0.0004  0.1585/0.1584  0.0023  0.0025–0.0043  0.0002  0.0740 
Max beta  Plane 1: 87.75, Plane 2: 53.20  Plane A: 75.50, Plane B: 63.55  48.0/48.5  55.52  Plane C: 39.98, Plane D: 56.51  24.50  63.75  
Plane 3: 70.60  Plane C 39.0  
Draconitic year (day)  353  353  356  356  351  354  362  360  361 
Revolution of node (year)  25.6  25.6  36.9–37.5  37.4–39.1  25.8  31  125.0  73.3  95.5 
Revolution of perigee (year)  274.7  274.7  65.3–70.9  65.3–70.0  31.4  63.1  205  36.6  77.4 
The GLONASS constellation consists of 24 operational MEO satellites with the majority of Mtype satellites, and two experimental Ktype satellites, out of which only one is operational. GLONASS satellites are equipped with the highest number of corner cubes, i.e., 112 and 123 for GLONASSM and GLONASSK, respectively. Retroreflectors mounted on GLONASSK satellites surround the GNSS transmit antenna and are the only ringshaped LRAs.
Galileo consists of 18 out of 30 planned satellites decomposed into three MEO orbits. The Galileo segment contains two types of spacecraft, i.e., four inorbit validation (IOV) spacecraft that were launched after two, inactive now, GIOVE satellites, and 14 fully operational capability (FOC) spacecraft, out of which, two were launched into highly elliptic orbits (Montenbruck et al. 2017). Despite the fact that the satellites in elliptic orbits cannot be used in navigation, they can serve for the investigation of a gravitational redshift (Delva et al. 2015). Galileo satellites are significantly lighter than other satellites (see Table 1); thus, they are more affected by the transmit antenna thrust (Steigenberger et al. 2017) and albedo (RodriguezSolano et al. 2012). GalileoIOV are equipped with arrays of the largest area (\(430.0\times 470.0\) mm) that contain one of the biggest corner cubes (23.3/33.0 mm). As a result, it is easier for stations to get good returns from the early Galileo rather than from GalileoFOC satellites which are equipped with LRAs of a reduced number (60) of smaller (19.1/28.2 mm) corner cubes. As a result, GalileoFOC LRAs have an area smaller (\(331.0 \times 248.7\) mm) by a factor more than 2 compared to GalileoIOV which makes GalileoFOC one of the hardest satellites to track by SLR stations. The current operational BeiDou2 constellation consists of six GEO, six IGSO, and three MEO satellites. BeiDou MEO and IGSO carry fewer corner cubes (42) than GalileoFOC satellites; however, the corner cubes are larger. As a result, the whole array has a bigger surface (316.0/280.0 mm) than GalileoFOC. BeiDou GEO is equipped with significantly larger LRAs compared to other satellites (490.0/430.0 mm). Larger LRAs are required for geosynchronous satellites due to their high altitude. Japanese QZSS is an augmentation system for GPS and consists of three geosynchronous and one geostationary spacecraft. Similar to BeiDou GEO, all QZSS are equipped with larger LRAs than MEO satellites (400.0/400.0 mm). The other regional navigation system considered by MGEX is the Indian NavIC that consists of 3 GEO satellites and 4 IGSO satellites.
1.4 The goal of this study
This study discusses the results of precise multiGNSS orbit determination using solely SLR observations as the first step of the combined SLRGNSS solution. We analyze both the internal and external accuracies of the solution in order to evaluate the influence of SLR observations to GNSS satellites on the multiGNSS solution and evaluate the role of laser ranging in combination with microwave observations. We concentrate mostly on active MEO GNSS satellites. We also evaluate the utility of SLRderived multiGNSS orbits according to the current requirements of Global Geodetic Observing System (GGOS, Plag and Pearlman 2009).
In existing literature on the determination of GNSS orbits using SLR, there is little information about unambiguous strategies in terms of the number and geometry of observations that are sufficient for the precise multiGNSS orbit determination. Information about the optimal calculation strategy and the bestsuited arc length is missing as well. Moreover, SLR stations are capable of tracking more than 50 GNSS satellites (Kirchner and Koidl 2015); thus, this analysis can contribute to the development of tracking strategy, in which multiGNSS constellations would be tracked homogeneously.

How many SLR observations are necessary to determine precise multiGNSS orbits using SLR data only?

What is an optimal geometry of observations, and thus, how many stations should track navigation satellites to provide a homogeneous coverage with observations for the whole GNSS constellation?

What is an optimal length for an orbital arc, i.e., how much can we extend an orbital arc in order to both, gather the largest number of SLR observations to GNSS satellites without degradation of the orbit?
1.5 Structure of the paper
The paper is structured as follows. Section 2 describes the methodology of the solution. Section 3 summarizes GNSS special tracking campaigns held by ILRS in the period of 2014–2017. Section 4 discusses the results. After the investigation of the efficiency of solutions, we assess internal accuracy by the analysis of the mean error of the orbital semimajor axis. Then, we investigate the external accuracy of the orbit solution as a function of the number of SLR observations and the number of tracking stations. Section 5 investigates different arc lengths in order to develop an optimal processing strategy. Section 6 contains comments and summarizes the paper in terms of the improvement of the consistency of solutions based on the microwave (GNSS) and optical (SLR) observations.
2 Methodology
The calculations of multiGNSS orbits using solely range measurements are performed in the modified version of Bernese GNSS Software 5.2 (Dach et al. 2015). In order to determine a precise orbit, a set of a priori orbit positions and Earth rotation parameters (ERPs) is needed. In our calculations, we use official ERPs and orbits from the Center for Orbit Determination in Europe (CODE, Prange et al. 2017).^{2} In order to increase the consistency of orbit solutions, we use the same SRP model as CODE—the new ECOM2 (Arnold et al. 2015). The solution provided by CODE is based on the same software package and is one of the most accurate for GPS, GLONASS and Galileo constellations within MGEX (Montenbruck et al. 2017) as well as for Galileo satellites launched into highly eccentric orbits (Sośnica et al. 2018). The CODE products, however, do not contain BeiDou GEO satellites. Both SRP models developed by CODE, i.e., ECOM and ECOM2, were designed to absorb solar radiation pressure for satellites that work strictly in the yawsteering attitude mode, whereas BeiDou GEO is maintained almost continuously in the orbitnormal mode. In our analysis, we process data from the period between 2014.0 and 2016.9, during which two changes in using orbit models occurred. Till the end of 2014, the classical ECOM was used (Beutler et al. 1994; Springer et al. 1999). At the beginning of 2015, the new ECOM2 was proposed. In August 2015, the number of estimated ECOM2 parameters in CODE products was reduced from 9 to 7 by excluding 4timesperrevolution parameters.
The strategy of the orbit solution is similar to the official 5system CODE solution for MGEX (Prange et al. 2017), i.e., 3day orbital arcs are generated, and the solution for a particular day refers to the middle day of the arc. Such an approach greatly stabilizes the solution, especially for new and incomplete GNSS systems, and thus is used not only for MGEX solutions, but also for operational and reprocessed products at CODE (Lutz et al. 2016).
However, during the 3day period, the number of SLR observations may be insufficient to provide a solution of the best possible quality. An extension of the arc length results in the increase in the number of SLR observations. On the other hand, the solution can suffer from the degradation of both the Keplerian and empirical orbit parameters as the external forces acting upon satellites may change over time. Also, with the extension of the orbital arc the risk of multiple satellite maneuvers occurs which may degrade the solution. Due to that fact, we make an attempt to arrange a strategy for the orbit determination, testing 3, 5, 7, and 9daylong arcs in order to both maximize the number of SLR observations and not allow orbit parameters to degrade.
Characteristics of performed solutions: solution A with the estimation of all parameters, solution B with the estimation of orbit parameters only
Parameter  Solution A  Solution B 

Orbit parameters  6 Keplerian (nday)  6 Keplerian (nday) 
7 ECOM2 parameters (nday)  7 ECOM2 parameters (nday)  
(D0, Y0, B0, B1C, B1S, D2C, D2S)  (D0, Y0, B0, B1C, B1S, D2C, D2S)  
Station coordinates  
Core stations  Verification of core stations with rejection of outliers exceeding 45 mm from Helmert transformation. Network constrains  Fixed to the SLRF2008 
No net rotation  
No net translation  
Other stations  Estimated as free parameters (nday)  Fixed to the SLRF2008 
Range biases  Annual mean biases calculated for each stationsatellite pair and resubstituted  
Earth rotation parameters  X pole (1day)  Fixed to a priori values 
Y pole (1day)  Fixed to a priori values  
LoD (1day)  Fixed to a priori values  
Geocenter coordinates  Estimated as free parameters (nday)  Fixed to a priori values 
After the data screening, during which observations exceeding the maximum sigma of residuals at the level of 25 cm are marked as outliers, the set of normal equations is saved. In order to minimize the influence of SLR range biases on estimated parameters, we calculate mean annual range biases for each stationsatellite pair. A similar approach was used for LAGEOS satellites by Appleby et al. (2016). The computed range biases are resubstituted to a priori data for further calculations; thus, the systematic errors caused by range biases are diminished and the solution becomes more stable due to a reduced number of estimated parameters. In further calculations, resubstituted range biases are strongly constrained to a priori values. The penultimate step is the validation of SLR core stations by calculating the Helmert transformation parameters between a priori coordinates from SLRF2008 and computed coordinates. Stations with residuals from the Helmert transformation greater than 45 mm in the north, east, or up component are marked and not taken as core stations in further calculations. Coordinates of noncore stations are estimated as free parameters without any constraints imposed thereon. Having provided lists of verified core stations, we proceed to the final parameter estimation in which we test different arc length strategies by stacking 1day normal equations into 3, 5, 7, and 9day arcs, after which we compare the middle days of orbital arcs with microwavebased orbits from CODE.
3 The ILRSintensive tracking campaigns
At the 18th International Workshop on Laser Ranging in Japan in November 2013, ILRS agreed to increase efforts on tracking GNSS constellations and initiated a special study group called Laser Ranging to GNSS s/c Experiment (LARGE). The goal of the LARGE group was to define an operational strategy for tracking GNSS satellites and to improve the consistency between solutions provided by ILRS and IGS. In the frame of the LARGE project, three special tracking campaigns were announced between 2014 and 2017. Figure 1 illustrates the number of SLR observations to GNSS satellites registered for each 3day orbit solution. The three special tracking campaigns are marked with red boxes.
The first special campaign lasted from August 1 to September 30, 2014. During the campaign, stations were asked to track all GNSS satellites or at least spacecraft from the current priority list of ILRS. The purpose of the pilot campaign was to check the capability of SLR stations to track the whole constellation of GNSS satellites, which was far more than an ordinary procedure at SLR stations. Through the first campaign the most intensively tracked GLONASS satellites were R02, R07, R18, and R21. The number of SLR observations to the Galileo and BeiDou satellites increased only slightly. For QZS1 no changes in the number of observations were registered. One of the most important outcomes of the campaign was that the intensification of GNSS satellite tracking did not have a negative influence on the tracking of geodetic satellites, including low orbiting spacecraft (Pearlman et al. 2015).
The second campaign was held from November 22, 2014, to the end of February 2015. Instead of tracking all GNSS satellites, the scenario for the second campaign included in the first place six GLONASS satellites, i.e., R02, R07, R12 R17, R18, and R20. The tracking of GalileoIOV and BeiDou satellites was given the second priority. The number of SLR normal points increased by 107, 154, and 107% for GLONASS R02, R12, and R18, respectively, as compared to the similar period of 100 days of the noncampaign period, i.e., between the end of April 2014 and August 1, 2014. Apart from Galileo E19, for which the number of observations increased only by 7%, the increase of SLR observations was neither registered for the other Galileo satellites, nor for the BeiDou MEO satellites. Instead, stations were focused on tracking GLONASS satellites included in the third priority list: R03, R04, R08, and R09.
4 Results
4.1 The effectiveness of multiGNSS orbit solutions using SLR
The efficiency of the orbit solutions is evaluated as a ratio between the number of successful SLR solutions to the number of determined orbit solutions using microwave observations at CODE for the same period.
The median efficiency of the orbit solution equals 87, 86, 79, 63, and 39% for GLONASS, Galileo, BeiDou MEO, BeiDou IGSO, and QZSS, respectively. The effectiveness of GLONASS, Galileo and BeiDou MEO owes to the fact that those constellations consist only of global coverage satellites which provides an even geometry of SLR observations, whereas BeiDou IGSO and QZSS suffer from a poor geometry of observations due to their regional attitude. We distinguish satellites that were included in the priority list of ILRS during the period of analysis (see Fig. 2). Only one GLONASS satellite (R07) was included in the ILRS priority list during the whole period between 2014 and 2016. For more than a half of the GLONASS constellation, it was possible to provide precise orbit solutions for more than 80% of cases, even if the satellites were not included in the priority list. In total, 10 spacecraft of the Russian constellation were included in the priority list only for a short period or were not included at all. As a result, for some of those satellites it was impossible to determine a reliable orbit solution for about 42% cases (see Fig. 2). These satellites were tracked only by the European stations; thus, their observational geometry was inferior. Galileo satellites are placed in the priority list right after their launch. As a consequence, orbit solutions could be provided for more than 80% cases for almost all Galileo spacecraft. BeiDou and QZSS satellites were included in the priority list for the whole period of the analysis; however, all of these satellites (apart from C11) are regional, geosynchronous spacecraft. As a result, due to the insufficient number of SLR observations, the efficiency of solution is rather poor (see Sect. 4.3).
4.2 Impact of the observation number on the internal orbit quality
The formal error of orbit semimajor axis converges quickly in the range between 13 and 50 observations (Fig. 3). In order to determine GNSS orbits with the accuracy of the semimajor axis at the level of 6 cm, we need approximately 50 SLR observations. This can be read from Fig. 3 in the case of all MEO satellites. The formal error of the semimajor axis when using 60 SLR observations equals 5.0, 4.6, 4.2, and 5.9 cm for R18, E19, E30, and C11 satellites, respectively.
4.3 Comparison to the microwavebased orbits
4.3.1 Dependency on the number of observations
Figure 4 shows three examples of the comparison between SLRbased and microwavebased orbits of GLONASS R18 for three selected days in 2015. When the number of SLR observations is 54 (Fig. 4, middle), the differences are at the centimeter level for the radial and alongtrack components and at the decimeter level for crosstrack. When doubling the number of SLR observations, the RMS in the radial and alongtrack directions does not improve significantly (Fig. 4, top). However, the crosstrack component improves by the factor of 4. When the number of SLR observations is just 25, the solutions becomes unstable with differences up to several meters (Fig. 4, bottom). The alongtrack component exhibits a secular drift, whereas all orbit components show large periodic variations of the period corresponding to the satellite revolution and the second harmonic of the satellite revolution period.
The accuracy of an orbit based on 25 SLR observations is insufficient for geodetic purposes. However, such an orbit can be sufficient for the determination of space debris (Cordelli et al. 2016), including inactive GLONASS satellites, all of which are equipped with retroreflectors for SLR measurements. A good observational geometry, i.e., including observations collected by stations from different continents, can improve the orbit quality of inactive satellites even in a solution based on just 25 SLR observations.
Summary of RMS between microwave and SLR orbits obtained using 60 SLR observations for 3day arcs (all values are given in cm)
Component  GLONASS  Galileo  BeiDou  QZSS  

IOV  FOC  MEO  IGSO  
Radial  2.9  4.0  4.3  2.3  4.6  – 
Alongtrack  8.8  10.4  15.7  10.2  24.5  – 
Crosstrack  16.7  18.1  20.3  12.3  49.8  – 
3D  21.4  23.3  26.2  17.9  55.6  – 
More than 100 observations to one spacecraft do not significantly improve the solution. In terms of the orbit determination using SLR, stations should focus on providing a constant number of observations to the whole multiGNSS constellation rather than focus on particular satellites.
The distribution of RMS of differences for GalileoFOC E30 is slightly better (2.9, 7.5, and 20.7 cm in radial, alongtrack, and crosstrack, respectively) than for GLONASS R10 (6.2, 11.1, and 20.5 cm in the radial, alongtrack, and crosstrack, respectively), despite a similar number of observations. However, R10 is simultaneously tracked only by 5 SLR stations, whereas E30 is tracked by more than 10 stations. The high number of stations itself does not provide a recipe for an accurate orbit solution. Stations that provide range measurements have to be homogeneously and globally distributed. High quality of Galileo E30 orbit results from the fact that this satellite is tracked by worldwide distributed stations, which provide a proper geometry of observations.
Orbit solutions vary for different types of BeiDou satellites. BeiDou MEO switches from the yawsteering mode to normal mode when the \(\left \beta \right \) angle, i.e., the elevation angle of the Sun above the satellite’s orbital plane, is below \(4^{\circ }\) (Montenbruck et al. 2015c). Despite the fact that the normal mode is not modeled correctly, the RMS of differences for BeiDou MEO is similar to GLONASS and Galileo constellations. The regional attitude of IGSO satellites significantly limits the number of observations and weakens the geometry of range measurements. The reference microwave orbits for geosynchronous satellites are not of the highest quality; as well, thus further investigations in terms of the geosynchronous orbit parameter estimation have to be performed.
4.3.2 Dependency on the number of SLR tracking stations
The quality of the orbit solution depends also on the number of SLR tracking stations. Our analysis confirms that the accuracy of orbit solution increases with the growth of the number of SLR stations as reported by Hugentobler (2016) who used simulated SLR data. According to Fig. 6, the solution based on fewer than 5 SLR stations provides orbits of a poor quality with the median RMS of differences at the level of 7.8, 23.0, and 48.6 cm in the radial, alongtrack, and crosstrack, respectively, for all satellites considered in our calculations. GLONASS R10 was tracked by up to 5 SLR stations—all of which are located in Europe, which does not allow for obtaining highquality orbits (see Fig. 6). BeiDou IGSO C08 is tracked at maximum by 6 stations (Fig. 6), whereas the QZS1 is simultaneously tracked by only up to 4 stations and on average by 1–2 stations. Moreover, the geometry of observations is barely changeable in time. With the increase in the number of stations from 5 to 10, the value of the RMS starts to decline and reaches 3.6, 9.4, and 18.4 cm in the radial, alongtrack, and crosstrack direction, respectively, which is by the factor of 2 better compared to the case of employing 5 SLR stations. For more than 10 SLR stations, the orbit solution stabilizes. In the case of 12 tracking stations, the accuracy of the orbit solution reaches the level of 2.9, 8.4, and 14.9 cm in the radial, alongtrack, and crosstrack component, respectively.
The high number of SLR stations itself does not provide a reliable orbit without a sufficient observation geometry, which is a reason for the good quality of the orbit solutions for R07, R18, and E19. The median value of the RMS for GLONASS R07 tracked by 14 stations equals 2.6, 5.2, and 9.9 cm in the radial, alongtrack, and crosstrack direction, respectively (see Fig. 6). GLONASS R18 is characterized by even smaller values of RMS, i.e., 1.7, 3.8, and 6.8 cm in the radial, alongtrack, and crosstrack direction, respectively, while being tracked by 15 SLR stations. For GalileoIOV E19 the best solution is provided by 14 stations and equals 2.8, 8.2, and 13.1 cm in the radial, alongtrack, and crosstrack direction, respectively. Unexpectedly, for GalileoFOC E30 the lowest values of RMS are obtained from the solution based on 10 tracking stations and equals 2.3, 5.7, and 6.3 cm for the radial, alongtrack, and crosstrack direction, respectively. However, the median values are obtained from only one solution; thus, it should not be considered as representative without further investigations.
In conclusions, at least 60 SLR observations collected by 10 different SLR stations are needed to obtain 3day SLR orbits with a quality of about 3, 9, and 18 cm for the radial, alongtrack, and crosstrack, respectively. With the increase in the number of SLR observations and the number of stations, the solution improves and stabilizes when more than 10 homogeneously distributed stations provide at least 100 observations.
5 The impact of the arc length
5.1 Orbit overlaps
We check the internal consistency of different arc length strategies by the analysis of the orbit overlaps which are calculated as an RMS between the middle day of the arc and the corresponding day of the consecutive arc. For the longer arcs, the internal accuracy of the determined orbits increases, because for the 3day arcs the middle (second) day is compared to the first day from the consecutive arc, whereas for the 5day arcs the middle (third) day is compared to the second day from the consecutive arc.
5.2 The distribution of solutions for selected satellites
Percentage of solutions with the accuracy higher than 10 cm in the radial, alongtrack, and crosstrack component for R02, E19, and C11 for different processing strategies. Values for solution A and B are shown
PRN  Component  3Day (%)  5Day (%)  7Day (%)  9Day (%)  

A  B  A  B  A  B  A  B  
R02  Radial  71  78  90  88  92  90  89  90 
Alongtrack  38  39  57  42  57  45  52  47  
Crosstrack  9  14  26  21  32  29  35  33  
E19  Radial  62  66  80  82  81  83  80  83 
Alongtrack  18  23  27  27  25  25  22  22  
Crosstrack  4  7  11  10  10  12  10  14  
C11  Radial  37  42  57  59  62  63  64  66 
Alongtrack  11  16  26  21  37  25  39  32  
Crosstrack  3  6  9  8  12  12  14  15 
Galileo E19 is tracked by almost the same set of SLR stations as R02; however, the distribution of solutions is slightly worse as compared to the GLONASS satellites. The number of solutions below the level of 10 cm in the radial direction is higher by 16, 17 and 17% for 5, 7, and 9day arcs, respectively, as compared to the 3day solutions. In contrary to the radial component, for the crosstrack component, 5day arcs constitute the best solution (see Table 4).
BeiDou C11 is the least intensively tracked satellite in the collection (Table 4). The geometry of observations is the weakest as well, as it is tracked intensively by the European and Australian stations and Changchun. As a result, the solution improves with the extension of the orbital arc. The number of solutions below the level of 5 cm in the radial direction is greater by 20% when compared 7day solutions to the 3day solutions.
In conclusion, for the most intensively tracked satellites, e.g., R02 and E19, 5 or 7day arcs constitute the optimal solution. For the 9day solution the degradation in the alongtrack component can be noticed (see Table 4). For R02, the increase in solutions with accuracy below the level of 10 cm in the alongtrack component is more spectacular (from 38 to 57%) than for E19 (from 18 to 27%) when extending the arc from 3 to 5 days. For satellites that are tracked less intensively (mostly by the European stations), we conclude that 7 and 9daylong arcs constitute the best solution. However, for the most intensively tracked satellites the 9day solution causes a degradation in the alongtrack component. The crosstrack component improves, not much enough though, to compensate for the degradation of other components. Moreover, one has to remember that the longer the arc is, the bigger the risk occurs that satellites maneuvers take place, due to which, in our case, almost 10% of solutions could not be obtained for the 9day arcs.
5.3 Strategy development for all satellites
Median RMS for 5 and 7day SLR solutions compared to the internal consistency of all MGEX Analysis Centers reported by Montenbruck et al. (2017)
5Day (SLR)  7Day (SLR)  Montenbruck et al. (2017) (GNSS)  

Radial  Along  Cross  3D  Radial  Along  Cross  3D  Radial  Along  Cross  3D  
GLONASS  ALL  7.4  21.5  39.0  48.1  ALL  6.2  18.2  29.7  38.2  4–11  4–12  3–9  6–17 
ILRS  4.6  13.3  19.4  25.6  ILRS  4.1  11.8  15.7  21.6  
R07  0.8  2.4  1.9  3.2  R18  1.1  1.2  2.6  3.1  
Galileo  ALL  4.8  15.1  30.9  36.0  ALL  4.0  15.6  27.3  32.8  4–10  10–19  6–20  14–29 
E12  1.6  3.7  1.8  4.4  E18  1.6  2.0  2.3  3.4  
BeiDou  MEO  5.0  15.4  28.2  34.9  MEO  4.4  13.4  22.9  28.3  3–11  10–21  6–10  12–26 
C11  1.2  1.8  5.1  5.5  C11  0.9  2.6  7.0  7.5  
IGSO  24.4  64.7  268.8  302.0  IGSO  15.9  43.0  185.2  195.9  11–23  24–39  17–23  32–51  
QZSS  J01  72.7  201.1  560.6  764.2  J01  64.5  156.6  610.5  714.8  10–71  28–133  16–156  40–240 
Due to the fact that all Galileo satellites are tracked evenly, the European constellation seems to be the most reliable one for the development of the optimal arc length strategy. According to Fig. 8, the most reliable length of the arc is 7day. However, as in the case of 9day solutions, the satellites maneuvers still interfere with the process of orbit determination. As a result, given the compromise between the quality of the solution and the efficiency of calculations we consider the 5day solution B as the most reliable. The accuracy for this solution is at the level of 4.8, 15.1, and 30.9 cm in the radial, alongtrack, and crosstrack direction, respectively.
BeiDou MEO segment is represented only by one spacecraft, i.e., C11. This satellite is not tracked as intensively as the other MEO satellites; thus, the longer the arc is the more observations supply the solution (see Fig. 8). A similar situation occurs for the BeiDou IGSO constellation. The extension of the orbital arc does not, however, improve the solution for QZSS.
Table 5 gives the median results from the 5 and 7day orbit determination using SLR with the consistency of all Analysis Centers (AC) of MGEX provided by Montenbruck et al. (2017) which are based on microwave GNSS observations. GLONASS satellites are divided into two groups, i.e., all satellites from the Russian constellation and satellites included in the priority list of ILRS. The RMS of SLR solutions for intensively tracked GLONASS satellites (i.e., 4 cm) is only slightly higher than the bottom threshold of consistency of MGEX ACs for the radial component. A similar situation occurs for all MEO satellites, especially for Galileo and BeiDou MEO whose accuracy of the determination of the radial and alongtrack components of about 4 and 15 cm respectively, fits well the orbit accuracy provided by MGEX ACs. Due to the poor SLR observation geometry, the solution for BeiDou IGSO and QZSS is significantly worse than the MGEX performance. Table 5 contains also the best solutions obtained for GLONASS, Galileo, and BeiDou MEO satellites for the 5 and 7day solutions. In all cases, the RMS is slightly lower, albeit in the threshold of the consistency of MGEX solutions.
The most consistent solution was provided for the GLONASS R07 on July 15, 2015. The 5day solution was calculated using 129 SLR observations provided by 12 homogeneously distributed SLR stations. During the 5day period R07 was tracked by 2 stations from North America, 3 from Asia, 2 from Australia, and 5 from Europe all of which provided an even and sufficient geometry of observations. The RMS of the best solution for GLONASS R07 equals: 0.8, 2.4, and 1.9 cm in the radial, alongtrack, and crosstrack direction, respectively (see Table 5).
6 Discussion and conclusions
Thanks to the great effort of ILRS and SLR tracking stations, the number of SLR observations to multiGNSS has grown significantly since late 2013. This enables us to determine GNSS orbits based on SLR observations in 87, 86, 46, and 39% cases for GLONASS, Galileo, BeiDou, and QZSS, respectively. Based on the orbit solutions, we performed analyses which answered the questions: what is the sufficient number of SLR observations and what is the optimal geometry of SLR observations in order to provide an orbit solution at the cm level. We also investigated an optimal arc length in order to both increase the considered number of observations and to avoid the orbit parameters to degrade.
The internal consistency of the orbit solutions based solely on SLR observations to multiGNSS satellites is at a satisfying level, i.e., the median RMS of orbit overlaps reached the level of 4, 15, and 33 cm in the radial, alongtrack, and crosstrack direction, respectively. In order to determine multiGNSS orbits with the mean error of the semimajor axis at the level of 6 cm, one needs about 50 SLR observations. As in the case of the external accuracy, with 20–30 observations one may obtain an orbit of a meter accuracy which can be sufficient for inactive navigation satellites equipped with LRAs; thus, range measurements to GNSS satellites may contribute to tracking of space debris.
On average, 60 observations are sufficient to provide a precise multiGNSS orbits of a scientifically useful quality at the level of: 3, 11, and 17 cm in the radial, alongtrack, and crosstrack direction, respectively, for all MEO satellites, whereas the quality of the solution for BeiDou IGSO was worse by the factor of 4 and for QZSS the number of 60 observations has never been reached. If the number of observations increases to 100, the RMS of both the alongtrack and crosstrack components decreases to the cm level. More than 100 observations improve the solution only marginally such that it is not worth putting effort on tracking a small subset of particular satellites in an attempt to increase the data beyond this level. It would be more efficient to track the whole MGEX constellation homogeneously.
The number of SLR stations that provide SLR observations is not without significance. More than 5 stations are necessary to provide a decent geometry of observations. In total, 10 SLR stations provide a sufficient geometry to determine multiGNSS orbits of a useful quality. Stations that provide observations have to be, however, homogeneously distributed in order to provide a sufficient geometry of observations. BeiDou IGSO and QZSS satellites are observed by fewer than 6 stations, which is one of the reasons why we cannot calculate highquality orbits for those satellites due to a poor geometry of observations. Moreover, the normal attitude mode, which is used by various satellites during parts of the eclipse periods, causes problems with orbit modeling for both the microwave and SLR solutions.
Figure 9 summarizes the diversity of the achieved orbit quality. The accuracy of SLRderived orbits depends on: (1) the number of SLR observations, (2) the number and distribution of SLR tracking stations, (3) the length of the orbital arc, (4) the generation of the satellite, (5) type of orbital plane, (6) \(\beta \) angle, (7) empirical models used in the calculation (ECOM1/ ECOM2).
The analysis performed in this paper provides information about the required number of SLR observations and their optimal geometry; thus, this work can serve as a recommendation for SLR stations for collecting SLR observations in terms of precise multiGNSS orbit determination using SLR data.
Firstly, one should investigate the accuracy and capabilities of the orbit determination using solely range measurements in order to use SLRderived multiGNSS orbits in studies, such as the determination of global geodetic parameters. The quality of the orbit solution is at the satisfying level, with the best case of the 5day solution for the GLONASS R07 which was calculated using 129 observations delivered by 12 homogeneously distributed SLR stations. RMS of the solution reached the level of 0.8, 2.4, and 1.9 cm in the radial, alongtrack, and crosstrack direction, respectively, which fulfills the criteria of the precise orbit.
MultiGNSS orbits derived using solely SLR data serve as an independent orbit solution; thus, they can contribute as an evaluation tool for the orbit modeling problems indicated by microwave solutions. Figure 9 validates the correctness of the new ECOM2 model for Galileo satellites. The influence of low \(\beta \) angle does not affect Galileo satellites to such an extent during the use of ECOM2. However, the methodology presented in this paper performs well for MEO satellites but still requires further investigation in the case of geosynchronous satellites. After that, the methodology can be used for BeiDou3 satellites, i.e., C31, C32 (IGSO) and C34, C33 (MEO) that are not currently considered by ACs, and for the determination of GalileoIOV E20 that broadcasts microwave signal on just one frequency since 2014, and thus, the precise MGEX orbits are not available. Finally, multiGNSS orbit determination using solely SLR data is the first step for a combined microwaveSLR orbit solution which is crucial for the SLRGNSS colocation in space onboard GNSS satellites independent from groundbased local ties.
Footnotes
Notes
Acknowledgements
This work has been realized in the frame of the OPUS project founded by the National Science Center Poland (NCN) Grant: UMO2015/17/B/ST10/03108. The authors would like to thank the CODE for the microwave orbits and the ILRS for SLR data. We also thank ILRS for organizing special tracking campaigns and SLR tracking stations for putting an extent effort to collect observations to GNSS satellites. We also thank the Wroclaw Center of Networking and Supercomputing (http://www.wcss.wroc.pl): MATLAB Software License No: 101979. The authors would like to thank three anonymous reviewers and the editor for valuable remarks.
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