Characterization of Inter-System Biases in GPS + BDS Precise Point Positioning

: With the combination of multi-GNSS data, the precise-point positioning (PPP) technique can improve its accuracy, availability and reliability: Inter-system bias (ISB) is the non-negligible parameter in multi-GNSS PPP. To further enhance the performance of multi-GNSS PPP, it is crucial to analyze the characterization of inter system biases (ISBs) and model them properly. In this contribution, we comprehensively investigate the characterization of ISBs between global positioning system (GPS) and BeiDou navigation satellite system (BDS) in di ﬀ erent situations. (1) We estimate ISB by using di ﬀ erent precise products from the Center for Orbit Determination (CODE), Deutsches GeoForschungsZentrum (GFZ) and Wuhan University (WHU). The results indicate that the one-day estimates of ISB are stable when using CODE and WHU products, whereas the estimates based on GFZ products vary remarkably. As for the three-day time series of ISB, a sudden jump exists between two adjacent days, which is due to the change of satellite clock datum; (2) We investigate the ISB characterization a ﬀ ected by the ambient environments of the receivers. The result shows that the ISBs estimated from receivers (and antennas) with same type are still inconsistent, which indicates that the ambient environment, probably the temperature, inﬂuences the ISB characterization as well, since the receivers are in di ﬀ erent areas; (3) We analyze the ISB characterization a ﬀ ected by receiver and antenna type with the same ambient environment. To ensure the same ambient environment, the ultra-short baselines were applied to investigate the ISB characterization a ﬀ ected by the receiver and antenna type. With the insights into ISB characterizations, we carry out combined GPS and BDS PPP with modeling the ISB as time constant, random walk process and white noise. The results suggest that the random walk process outperforms in most cases, since it strengthens the model to some extend and, at the same time, considers the variation of ISBs.


Introduction
With the rapid developments and modernizations of the global navigation satellite systems (GNSSs) including the American GPS, Chinese BDS, European Galileo and Russian GLONASS, the integration of multi-GNSS brings a better and wider range of positioning, navigation, timing (PNT) and atmospheric retrieval applications [1][2][3][4][5][6]. As compared to GPS-only case, the multiple constellation increases the satellites in common view, thus enabling a higher accuracy and more reliability in precise-point positioning (PPP). The higher redundancy of the multi-GNSS model also shortens the initialization time of PPP. The combination of multi-GNSS opens up a new stage of its applications, whereas processing multi-GNSS data are more challenging [7,8]. P T,s r, f (i) = ρ T,s r (i) + c ∆t T r (i) − ∆t T,s (i) + µ T f I T,s r,1 (i) + T s r + b T r, f + b T,s f + ε T P (i) L T,s r, f (i) = ρ T,s r (i) + c ∆t T r (i) − ∆t T,s (i) − µ T f I T,s r,1 (i) + T s r + λ T f N T,s r, f + B T r, f + B T,s f + ε T L (i) (1) where superscript T represents the system (G for GPS and C for BDS). ρ T,s r (i) is the geometric distance; c is the speed of light; ∆t T r (i) and ∆t T,s (i) are the receiver and satellite clock offsets; I T,s r,1 (i) the (first-order) slant-ionosphere delay on the first frequency with its coefficient (µ T f = λ 2 f /λ 2 1 , in which λ f is the wavelength); T s r is the tropospheric slant delay; b T r, f and b T,s f are the receiver and satellite code biases; N T,s r, f is the integer ambiguity; B T r, f and B T,s f are receiver and satellite phase biases; ε P (i) and ε L (i) are noise in code and phase observation. Except N T,s r, f (unit: cycle), λ f (unit: m/cycle) and µ T f (without unit), all quantities are expressed in units of range.
Note that, the effects of phase windup, solid tide, ocean loading, relativistic effects, earth rotation have been corrected by corresponding models [23].
Conventionally, the IGS precise clock products absorb the ionosphere-free (IF) combination of satellite code biases as following [24]: Equation (1) represents a rank-deficient system. In order to deal with the linear correlation between parameters (the receiver clock offsets with the receiver-code bias, the slant-ionosphere delay with the receiver-code bias, the ambiguities with the receiver and satellite phase biases, hardware biases between frequencies), the idea is to lump some of them together [25]. The reparameterization process is as below: whereˆpresents that the parameter after recombination. The full-rank GPS/BDS PPP observation equations can be rewritten as We select the estimable GPS receiver clock∆t G r as reference, then the GPS/BDS ISB can be denote as where ∆t G r and ∆t C r respectively represent the GPS and BDS receiver clock offset which absorb the satellite clock datum. ∆D G and ∆D C are GPS and BDS satellite clock datum, respectively. For the reason that the receiver and satellite clock parameters are linear dependent, a reference of clock offset must be set for each system [26]. In the meantime, the variation of the reference will be absorbed in the receiver clock offset. Therefore, the ISB obtained by PPP contains the offsets of inter-system satellite clock datums and the receiver-code biases.
The state vector X of GPS/BDS undifferenced and uncombined PPP can be written as

Processing Strategy
The final precise orbit (5 min interval for CODE and GFZ, 15 min interval for WHU) and clock products (30 s interval) from Wuhan University (WUM), center for orbit determination (CODE) and Deutsches GeoForschungsZentrum (GFZ) were employed. The absolute antenna phase-center correction models (igs14.atx) were applied to correct the phase-center offset and variation, which for BDS satellites obtained from European Space Agency (ESA). In order to analyze the characteristics of ISB, the ISB is modeled as white noise. The UNB3 model was used to correct the tropospheric dry delay and the tropospheric zenith wet delay is estimated as parameter, with Neil as the mapping function. The float phase ambiguities were estimated as constant for each continuous satellite arc. The traditional methods-turbo-edit detection-namely the geometry-free and MW observations were applied in the detection of cycle slip [27]. The elevation cutoff angle was set to 7 • and an elevation-dependent weighting scheme (sin 2 (elevation)) for the observation was applied [28]. The weight ratio of GPS and BDS is set at 1:1.5 [29]. Due to the BDS GEO satellite orbit and clock accuracy, the weight of BDS GEO satellite observations is reduced by 10 times [30]. In order to strengthen the model, we estimated the ISBs with static PPP processing. The detail processing strategies are shown in Table 1. The distribution of the selected stations is shown in Figure 1.

Characterization of ISB
According to Equation (6), the ISB estimated by PPP contains the satellite clock datum and t ceiver-code bias. In the following, undifferenced and uncombined PPP was adopted to analyze t fluence factors of ISB. The technical features of the receiver and antenna types mentioned in t periments are available in the official websites, TRIMBLE (https://geospatial.trimble.com), LEIC ttps://leica-geosystems.com), SETPENTRIO (https://www.septentrio.com).

ISB Characterization Affected by the Satellite Clock Datum
In this experiment, randomly, we collect data from three IGS stations including PNGM, HKS d GAMG from day of year (DOY) 034-036, day of year (DOY) 128-130, day of year (DOY) 22 9, 2018, respectively. The observation data are available over FTP (ftp://cddis.gsfc.nasa.gov) f ee.

Characterization of ISB
According to Equation (6), the ISB estimated by PPP contains the satellite clock datum and the receiver-code bias. In the following, undifferenced and uncombined PPP was adopted to analyze the influence factors of ISB. The technical features of the receiver and antenna types mentioned in the experiments are available in the official websites, TRIMBLE (https://geospatial.trimble.com), LEICA (https://leica-geosystems.com), SETPENTRIO (https://www.septentrio.com).

ISB Characterization Affected by the Satellite Clock Datum
In this experiment, randomly, we collect data from three IGS stations including PNGM, HKSL and GAMG from day of year (DOY) 034-036, day of year (DOY) 128-130, day of year (DOY) 227-229, 2018, respectively. The observation data are available over FTP (ftp://cddis.gsfc.nasa.gov) for free.
The absolute ISBs estimated with different precise products differ a lot in magnitude. To show the results clearly, all estimates are subtracted by the ISB estimated in first epoch. Figure 2 shows the ISB time series of three stations using WUH, CODE and GFZ precise products, respectively. The ISBs suddenly jump between two adjacent days, and the amplitude of the fluctuation is different-with-different precise products. This fluctuation is caused by the change of satellite clock datum [31]. Moreover, the daily ISBs based on CODE and WHU products are stable, while the variation of daily ISBs corresponding to GFZ products is remarkable. As we can see, the time series of ISBs from three stations using same products are consistent. It further proves that the change of satellite clock datum affects the ISBs. the fluctuation is different-with-different precise products. This fluctuation is caused by the change of satellite clock datum [31]. Moreover, the daily ISBs based on CODE and WHU products are stable, while the variation of daily ISBs corresponding to GFZ products is remarkable. As we can see, the time series of ISBs from three stations using same products are consistent. It further proves that the change of satellite clock datum affects the ISBs.

ISB Characterization Affected by the Ambient Environment
Four IGS stations (CEDU HOB2 KARR and PNGM) are selected on DOY 002, 2018. Stations CEDU and HOB2 are equipped with SEPT POLARX5 receivers while stations KARR and PNGM are equipped with TRIMBLE NETR9 receivers. For more detailed information, see Table 2.

Station
Receiver type Antenna type CEDU SEPT POLARX5 AOAD/M_T HOB2 SEPT POLARX5 AOAD/M_T KARR TRIMBLE NETR9 TRM59800.00 PNGM TRIMBLE NETR9 TRM59800.00 Figure 3 shows that the ISBs estimated with the WUM and CODE precise products showed little fluctuation. It is worth noting that even though stations KARR and PNGM were equipped with the same type of receivers and antennas, the ISB estimate were still different when using WUM and CODE precise products. We surmised that the ambient environment of the receiver, probably the temperature, may have affected the characterization of ISBs [10]. However, for the reason that the

ISB Characterization Affected by the Ambient Environment
Four IGS stations (CEDU HOB2 KARR and PNGM) are selected on DOY 002, 2018. Stations CEDU and HOB2 are equipped with SEPT POLARX5 receivers while stations KARR and PNGM are equipped with TRIMBLE NETR9 receivers. For more detailed information, see Table 2.  Figure 3 shows that the ISBs estimated with the WUM and CODE precise products showed little fluctuation. It is worth noting that even though stations KARR and PNGM were equipped with the same type of receivers and antennas, the ISB estimate were still different when using WUM and CODE precise products. We surmised that the ambient environment of the receiver, probably the temperature, may have affected the characterization of ISBs [10]. However, for the reason that the satellite clock datum of GFZ final products vary over time significantly, the difference between the different stations could not perform notably. Figure 3 shows that the ISBs obtained with even the same receiver and antenna type, they had little difference. Therefore, in order to investigate the receiver and antenna type affected the ISB characterization, we collected the data from ultra-short baseline (about several meters) stations to ensure the same ambient environments. We discuss the implication of receiver and antenna type in more detail below.

Discussion
Appl. Sci. 2020, 10, x FOR PEER REVIEW 7 of 13 satellite clock datum of GFZ final products vary over time significantly, the difference between the different stations could not perform notably.  Figure 3 shows that the ISBs obtained with even the same receiver and antenna type, they had little difference. Therefore, in order to investigate the receiver and antenna type affected the ISB characterization, we collected the data from ultra-short baseline (about several meters) stations to ensure the same ambient environments. We discuss the implication of receiver and antenna type in more detail below.

ISB Characterization Affected by the Receiver Type
As shown in Table 3, this experiment selects two stations (MAT1, MATE) equipped LEICA GR30 receivers and one station equipped SEPT POLARX5TR receiver. All of them were connected to LEIAR20 antennas. The distance between any two of them was less than 30 m. Therefore, we assumed that the external temperature of three stations was identical. For the reason that the observed BDS satellites were not enough, we only estimated the ISBs using WUM and GFZ products in this experiment due to the CODE precise products exclude the BDS GEO satellites. For the same reason, we skipped the data on DOY 254 and 256, 2018 and displayed the results on DOY 253, 255 and 257, 2018. Figure 4 shows the ISB estimated from station MAT1, MAE1 and MAZ1. We see that the ISB time series from stations MAT1 and MATE equipped receivers with same type were consistent. However, see the results from station MATZ, which was equipped with the other type of receiver, the ISB estimates differed from the results from stations MAT1 and MAE1 in some periods.

ISB Characterization Affected by the Receiver Type
As shown in Table 3, this experiment selects two stations (MAT1, MATE) equipped LEICA GR30 receivers and one station equipped SEPT POLARX5TR receiver. All of them were connected to LEIAR20 antennas. The distance between any two of them was less than 30 m. Therefore, we assumed that the external temperature of three stations was identical. For the reason that the observed BDS satellites were not enough, we only estimated the ISBs using WUM and GFZ products in this experiment due to the CODE precise products exclude the BDS GEO satellites. For the same reason, we skipped the data on DOY 254 and 256, 2018 and displayed the results on DOY 253, 255 and 257, 2018. Table 3. Stations selected for the investigation of ISB characterization affected by the receiver type.

Station
Receiver Type Antenna Type SEPT POLARX5TR LEIAR20 Figure 4 shows the ISB estimated from station MAT1, MAE1 and MAZ1. We see that the ISB time series from stations MAT1 and MATE equipped receivers with same type were consistent. However, see the results from station MATZ, which was equipped with the other type of receiver, the ISB estimates differed from the results from stations MAT1 and MAE1 in some periods. Typically, in the period of hour after 8, there was an offset about 0.1 m between ISBs from two types of receivers when using WUM products. As we can see, the gaps of ISB estimated by different receiver types were similar, it due to the variation of the receiver-code bias was closely related to temperature [10], while the Appl. Sci. 2020, 10, 4968 8 of 13 temperatures were comparable in the adjacent days. In addition, the ISBs based on WUM products were more stable than those based on GFZ products, which was in accordance with what we find in Figure 2.
ISBs based on WUM products were more stable than those based on GFZ products, which was in accordance with what we find in Figure 2.

ISB Characterization Affected by the Antenna Type
This experiment aimed to find out whether the ISB characterization was related to antenna type or not. See Table 4, we select stations YARR and YAR3 equipped with the SEPT PLOARX5 receivers. Station YARR was connected to LEIAT504 antenna, while was YAR3 connected to LEIAR25 antenna. Similar as before, the distance between YARR and YAR3 was less than 30 m, the external temperature of these two stations could be considered very similar.

ISB Characterization Affected by the Antenna Type
This experiment aimed to find out whether the ISB characterization was related to antenna type or not. See Table 4, we select stations YARR and YAR3 equipped with the SEPT PLOARX5 receivers. Station YARR was connected to LEIAT504 antenna, while was YAR3 connected to LEIAR25 antenna. Similar as before, the distance between YARR and YAR3 was less than 30 m, the external temperature of these two stations could be considered very similar. SEPT POLARX5 LEIAR25 Figure 5 shows, the ISBs from different antennas were inconsistent when using WUM and CODE products, due to the smaller scales, the differences of ISB estimated with CODE precise products were more significant. This indicate that the ISB was related to antenna type and it may be caused by antenna and cable induced delay [32], thus making the ISB antenna-specific. However, the ISBs based on GFZ products were consistent, this was because the ISB variation caused by the change of satellite clock datum were much larger than the variation caused by the antenna code bias.
ISBs based on GFZ products were consistent, this was because the ISB variation caused by the change of satellite clock datum were much larger than the variation caused by the antenna code bias. Based on the analysis of ISB characterizations, we carried out the GPS/BDS PPP with modeling the ISB as, constant (CT), random walk (RW) process and white noise (WN) [33], respectively. Data from twelve IGS stations on DOY 252-281, 2018 were used to analyze the performances of GPS/BDS float dynamic PPP using different ISB modeling strategies. Taking station PNGM as an example randomly, the GPS/BDS float dynamic PPP solutions were carried out with modeling ISB as constant, random walk and white noise process. As for the equipped receivers and antennas, refer to Table 2. Figure 6 shows that, for the PNGM station, the positioning errors in east, north and up were almost the same with different modeling strategies of ISB using WUM and CODE precise products, while the solution using GFZ precise products, the performance with constant ISB modeling strategies were much worse than random walk and white noise ISB modeling strategies. Taken altogether, as Tables 5 and 6, similar to the positioning performance, the convergence performances were almost uniform with different modeling strategies of ISB, the performances of GPS/BDS PPP based on CODE and WUM products were comparable. In these three cases, the CT process performs best, the RW process follows, and the WN process presents the worst performance. This was confirmed with the stability of ISBs based on CODE and WUM precise products. As for the results based on GFZ precise products, estimate the ISB as constant is not reasonable because it always Based on the analysis of ISB characterizations, we carried out the GPS/BDS PPP with modeling the ISB as, constant (CT), random walk (RW) process and white noise (WN) [33], respectively. Data from twelve IGS stations on DOY 252-281, 2018 were used to analyze the performances of GPS/BDS float dynamic PPP using different ISB modeling strategies. Taking station PNGM as an example randomly, the GPS/BDS float dynamic PPP solutions were carried out with modeling ISB as constant, random walk and white noise process. As for the equipped receivers and antennas, refer to Table 2. Figure 6 shows that, for the PNGM station, the positioning errors in east, north and up were almost the same with different modeling strategies of ISB using WUM and CODE precise products, while the solution using GFZ precise products, the performance with constant ISB modeling strategies were much worse than random walk and white noise ISB modeling strategies. Taken altogether, as Tables 5 and 6, similar to the positioning performance, the convergence performances were almost uniform with different modeling strategies of ISB, the performances of GPS/BDS PPP based on CODE and WUM products were comparable. In these three cases, the CT process performs best, the RW process follows, and the WN process presents the worst performance. This was confirmed with the stability of ISBs based on CODE and WUM precise products. As for the results based on GFZ precise products, estimate the ISB as constant is not reasonable because it always performs great variation. Therefore, the results using RW and WN processing strategies were comparable, which were better than the results using CT processing strategy.   Consistently, the receiver-code biases from all the multi-GNSS experiment (MGEX) tracking stations demonstrate consistent stability as those from the stations we selected. Therefore, according to Table 5 and Figure 6, we came up with the results consistent with previous studies. However, the receiver-code bias was not always stable.
Peculiarly, we found that some stations varied significantly in receiver-code bias. Take the data from BALA on DAY 002, 2018 for example (besides GASC, JLCK, LAMB, MTDN, UCAL, etc. having the similar performance). This is freely available at Geoscience Australia FTP (ftp://ftp.ga.gov.au/). The station BALA is equipped with TRIMBLE NETR9 receiver and LEIAR25.R3 antenna. Consistently, the receiver-code biases from all the multi-GNSS experiment (MGEX) tracking stations demonstrate consistent stability as those from the stations we selected. Therefore, according to Table 5 and Figure 6, we came up with the results consistent with previous studies. However, the receiver-code bias was not always stable.
Peculiarly, we found that some stations varied significantly in receiver-code bias. Take the data from BALA on DAY 002, 2018 for example (besides GASC, JLCK, LAMB, MTDN, UCAL, etc. having the similar performance). This is freely available at Geoscience Australia FTP (ftp://ftp.ga.gov.au/). The station BALA is equipped with TRIMBLE NETR9 receiver and LEIAR25.R3 antenna.
As Figure 7 shown, the ISBs estimated by using CODE and WUM precise products were not always that stable due to the variation of receiver-code bias. Wrongly estimated the ISB as constant would carry out a terrible positioning performance as Figure 8. Therefore, we suggest that estimating ISB as random walk or white noise was more reliable, instead of constant.
Appl. Sci. 2020, 10, x FOR PEER REVIEW 11 of 13 As Figure 7 shown, the ISBs estimated by using CODE and WUM precise products were not always that stable due to the variation of receiver-code bias. Wrongly estimated the ISB as constant would carry out a terrible positioning performance as Figure 8. Therefore, we suggest that estimating ISB as random walk or white noise was more reliable, instead of constant.

Summary and Conclusions
In this contribution, we comprehensively analyzed the characterization of inter-system biases (ISBs) with different precise products, receiver types and antenna types. We also investigated whether or not the ambient environment of the receiver affects the ISB characterization. Based on the analysis of ISB characterization, we implemented the GPS/BDS PPP with modeling the ISB as time constant, random walk process and white noise.
A sudden jump exists in the ISB estimates between two adjacent days. It is caused by the change of satellite clock datum. The ISBs estimated with the precise products from different analysis centers have significant divergence. It is found that, in most cases, the estimated ISBs using WUM and CODE products are stable, while the variation of ISBs estimated by using GFZ products is remarkable. This is determined by the data processing strategies adopted in by different analysis centers. Moreover, even the stations equipped with the same type of receivers and antennas, the ISBs are inconsistent. This is because the stations are located in different places where the ambient environments (e.g., temperature) are different and the receiver-code biases are closely related to the temperature. The data from ultra-short baselines are used to confirm the same ambient

Summary and Conclusions
In this contribution, we comprehensively analyzed the characterization of inter-system biases (ISBs) with different precise products, receiver types and antenna types. We also investigated whether or not the ambient environment of the receiver affects the ISB characterization. Based on the analysis of ISB characterization, we implemented the GPS/BDS PPP with modeling the ISB as time constant, random walk process and white noise.
A sudden jump exists in the ISB estimates between two adjacent days. It is caused by the change of satellite clock datum. The ISBs estimated with the precise products from different analysis centers have significant divergence. It is found that, in most cases, the estimated ISBs using WUM and CODE products are stable, while the variation of ISBs estimated by using GFZ products is remarkable. This is determined by the data processing strategies adopted in by different analysis centers. Moreover, even the stations equipped with the same type of receivers and antennas, the ISBs are inconsistent. This is because the stations are located in different places where the ambient environments (e.g., temperature) are different and the receiver-code biases are closely related to the temperature. The data from ultra-short baselines are used to confirm the same ambient environment, the result indicate that the receiver and antenna type both affects the characterization of ISBs. Therefore, besides the impact of different final precise products, the receiver type, antenna type and even the ambient environment affect the ISB characterizations.
For GPS/BDS PPP solutions with WUM and CODE precise products, the positioning performances agree well among the three ISB processing methods. However, when the receiver-code biases vary significantly and GFZ precise products are used, the results of GPS/BDS PPP obtained by random walk or white noise are more accurate than those using the constant method.