Modeling and Performance Evaluation of Precise Positioning and Time-Frequency Transfer with Galileo Five-Frequency Observations

The present Global Navigation Satellite System (GNSS) can provide at least double-frequency observations, and especially the Galileo Navigation Satellite System (Galileo) can provide five-frequency observations for all constellation satellites. In this contribution, precision point positioning (PPP) models with Galileo E1, E5a, E5b, E5 and E6 frequency observations are established, including a dual-frequency (DF) ionospheric-free (IF) combination model, triple-frequency (TF) IF combination model, quad-frequency (QF) IF combination model, four five-frequency (FF) IF com-bination models and an FF uncombined (UC) model. The observation data of five stations for seven days are selected from the multi-GNSS experiment (MGEX) network, forming four time-frequency links ranging from 454.6 km to 5991.2 km. The positioning and time-frequency transfer performances of Galileo multi-frequency PPP are compared and evaluated using GBM (which denotes precise satellite orbit and clock bias products provided by Geo Forschung Zentrum (GFZ)), WUM (which denotes precise satellite orbit and clock bias products provided by Wuhan University (WHU)) and GRG (which denotes precise satellite orbit and clock bias products provided by the Centre National d’Etudes Spatiales (CNES)) precise products. The results show that the performances of the DF, TF, QF and FF PPP models are basically the same, the frequency stabilities of most links can reach sub10−16 level at 120,000 s, and the average three-dimensional (3D) root mean square (RMS) of position and average frequency stability (120,000 s) can reach 1.82 cm and 1.18 × 10−15, respectively. The differences of 3D RMS among all models are within 0.17 cm, and the differences in frequency stabilities (in 120,000 s) among all models are within 0.08 × 10−15. Using the GRG precise product, the solution performance is slightly better than that of the GBM or WUM precise product, the average 3D RMS values obtained using the WUM and GRG precise products are 1.85 cm and 1.77 cm, respectively, and the average frequency stabilities at 120,000 s can reach 1.13 × 10−15 and 1.06 × 10−15, respectively.


Introduction
In recent decades, Global Navigation Satellite System (GNSS) have made great progress, including Global Positioning System (GPS) modernization by the USA, the BeiDou Navigation Satellite System (BDS) developed by China, the Galileo Navigation Satellite System (Galileo) constructed by the European Union (UN), the Global Navigation Satellite System in Russia (GLONASS) restored by Russia, and the Quasi-Zenith Satellite System (QZSS) implemented by Japan, which have realized interoperability and multifrequency signal support [1][2][3][4][5]. The Galileo system operated by the European Space Remote Sens. 2021, 13, 2972 3 of 22 in the multi-frequency PPP model. In addition, based on the evaluation the multipath combination noise of Galileo multi-frequency observations, the convergence time and positioning accuracy of DF, TF, QF and FF models were evaluated by using precision products of different ACs, and the performance of time-frequency transfer of Galileo E1, E5a, E5b, E5 and E6 multi frequency signals is compared and analyzed systematically for the first time. In Section 2, five mathematical PPP models of the Galileo E1, E5a, E5b, E5 and E6 frequency observations are established; observation types, combination coefficients, ionospheric coefficients and noise amplification coefficients of the Galileo DF, TF, QF, and FF PPP models are also summarized and compared. Section 3 introduces the experimental data and their processing strategy. In Section 4, we use the GBM, WUM and GRG (which denote precise satellite orbit and clock bias products provided by the Centre National d'Etudes Spatiales (CNES)) precise products to evaluate the positioning and the time-frequency transfer performance results of the Galileo DF, TF, QF, and FF PPP models. Finally, we summarize this work and draw some conclusions.

Methodology
In this section, starting from GNSS raw observations, the Galileo FF UC PPP model, the single ionosphere-free (IF) PPP model among five frequency observations, two QF IF PPP model among five frequency observations, three TF IF PPP model among five frequency observations and four DF IF PPP model among five frequency observations are established, respectively. Then, the observation types, combination coefficients, ionospheric coefficients and noise amplification coefficients of the Galileo DF, TF, QF, and FF PPP models are summarized and compared.

GNSS Observation Model
GNSS pseudo range and carrier phase observations can be expressed as [28]: P i r,j = ρ i r + cdt r − cdt i + MF i r ZWD r + γ j I i r,1 + d r,j − d i j + ζ i j L i r,j = ρ i r + cdt r − cdt i + MF i r ZWD r − γ j I i r,1 + λ j N i j + b r,j − b i j + ξ i j (1) where the P and L denote pseudo range and carrier phase observation values in meters, respectively; superscript i denotes the i-th satellite of the GNSS system; subscripts r and j denote the receiver and frequency identifiers, respectively; for convenience, the frequency identifiers 1, 2, 3, 4 and 5 in the Galileo system represent E1, E5a, E5b, E5 and E6 frequencies, respectively; λ is wavelength corresponding to frequency j; ρ represents the geometric distance between the satellite and receiver; c represents the speed of light in a vacuum; dt r and dt s denote receiver and satellite clock offsets in seconds, respectively; MF denotes the wet mapping function; ZWD is zenith troposphere wet delay (ZWD); I denotes the slant ionospheric delay; γ j denotes frequency dependent ionospheric delay amplification factors, γ j = λ 2 j /λ 2 1 ; N is carrier phase integer ambiguity; d denotes the receiver uncalibrated code delay (UCD) in meters; b denotes uncalibrated phase delay (UPD) in cycles; and ζ and ξ represent the pseudo orange and carrier phase observation noises, respectively.

Five-Frequency UC PPP Model
In the five-frequency UC PPP model, the ionospheric delay is related to E1/E5a IF combined with DCB and cannot be separated, and the IFB parameters are required to Remote Sens. 2021, 13,2972 4 of 22 compensate for the inconsistency of receiver UCDs at the E5b, E5 and E6 frequencies.
Then, the Galileo five-frequency UC pseudo range and carrier phase observations can be expressed as: r,j = µ i r · x + c dt r + MF i r ZWD r + γ j I i r,j + ζ i j (j = 1, 2) p i r,j = µ i r · x + c dt r + MF i r ZWD r + IFB UC j + γ j I i r,1 + ζ i j (j = 3, 4, 5) l i r,j = µ i r · x + c dt r + MF i r ZWD r − γ j I i r,1 + λ j N i j + ξ i j (j = 1, 2, 3, 4, 5) with c dt r = cdt r + d r,IF 12 I i r,1 = I i r,1 + DCB r,12 /(1 − γ 2 ) IFB UC j = d r,j − d r,IF 12 − γ j DCB r,12 / 1 − γ j (j = 3, 4, 5) (4) where p and l are pseudo range and carrier phase observed minus computed (OMC) values in meters, respectively; subscript IF denotes ionospheric-free combination; DCB r,12 denotes E1 and E5a DCB of the receiver side; IFB uc denotes UC IFBs; µ denotes the unit vector of the component from the receiver to the satellites; x denotes the vector of the receiver position increments; and hat "~" denotes the reparametrized estimate.

Single Five-Frequency IF Combination (FF1) PPP Model
Galileo five-frequency observations can be combined according to the geometry-free, ionospheric-free and minimum noise principles with a single ionospheric combination observation and can be expressed as:    e 1 + e + 2e 3 + e 4 + e 5 = 1 e 1 γ 1 + eγ2 2 + e 3 γ 3 + e 4 γ 4 + e 5 γ 5 = 0 e 2 1 + e 2 2 + e 2 3 + e 2 4 + e 2 5 = min The coefficients following the above criteria can be uniquely determined as [25]: with where e 1 , e 2 , e 3 , e 4 and e 5 denote the five-frequency IF combination coefficients, e is the coefficient in the denominator. When the five pseudo range measurements are combined within a single equation, the FF1 PPP observation equation can be expressed as: with c dt r = cdt r + d r,IF 12345 where the receiver clock offset will absorb the UCD of E1/E5a/E5b/E5/E6 IF combination.

Two Quad-Frequency IF Combinations (FF2) PPP Model
In the FF2 PPP model, E1/E5a/E5b/E5 and E1/E5a/E5b/E6 observations were selected to form two quad-frequency IF combinations. Similarly, additional IFB parameters were required to maintain compatibility between the two quad-frequency IF combinations. Hence, the two quad-frequency IF combination PPP models can be formulated as follows: with where IFB IF1235 denotes the IFB parameters between the E1/E5a/E5b/E6 and E1/E5a/E5b/E6 code IF combinations.

Three Triple-Frequency IF Combinations (FF3) PPP Model
The FF3 PPP model for the Galileo five-frequency was implemented by combining the three triple-frequency IF observations, and we chose E1/E5a/E5b, E1/E5a/E5 and E1/E5a/E6 observations for combination. In this case, we needed to add two estimated IFB parameters, which can be expressed as: with where IFB IF124 and IFB IF125 denote the IFB parameters between the E1/E5a/E5b, E1/E5a/E6 and E1/E5a/E5b code IF combinations.

Four Dual-Frequency IF Combinations (FF4) PPP Model
The FF4 PPP model consists of the observation equation implemented by combining the four DF IF observations. Theoretically, any two Galileo E1, E5a, E5b, E5, and E6 signals can form a dual-frequency ionospheric-free combination. However, considering the noise amplification coefficients after combination, we chose E1/E5a, E1/E5b, E1/E5 and E1/E6 signals for combination. In addition, it was necessary to note that the clock difference of each combination is not consistent. In this case, we needed to add estimated IFB parameters, which can be expressed as: Remote Sens. 2021, 13, 2972 where IFB IF13 , IFB IF14 and IFB IF15 denote the IFB parameters between the E1/E5b, E1/E5, E1/E6 and E1/E5a code IF combinations. Table 1 summarizes the types of observation values, combination coefficients, ionospheric delay coefficients and noise amplification coefficients of the dual-frequency, triplefrequency, quad-frequency, and five-frequency PPP models. Among them, the FF2, FF3 and FF4 PPP models are composed of DF, TF and QF IF combinations, and additional IFB parameters need to be estimated to maintain consistency among different equations. The noise amplification factor of FF1 is smaller than that of other IF combinations. After the filter converges and the ionosphere and ambiguity parameters are separated, the FF1, FF2, FF3, FF4 and UC PPP models are equivalent.

Date Selection and Processing Strategies
Five stations that can receive the Galileo E1/E5a/E5b/E5/E6 signal observations over a period of 190-196 days in 2020 were selected from the international GNSS service (IGS) multi-GNSS experiment (MGEX) network. In order to ensure the reliability of timefrequency transfer, the stations of UTC (Coordinated Universal Time) laboratories were preferred in our research. The sampling interval of the observations was 30 s. These stations were all equipped with high-precision atomic clocks. With the BRUX station as the reference station, four time-frequency links ranging from 454.6 km to 5591.2 km were established. Figure 1 shows the distribution of the selected MGEX stations, the information of which is listed in Table 2.
ferred in our research. The sampling interval of the observations was 30 s. These s were all equipped with high-precision atomic clocks. With the BRUX station as th ence station, four time-frequency links ranging from 454.6 km to 5591.2 km were lished. Figure 1 shows the distribution of the selected MGEX stations, the informa which is listed in Table 2.  The code and phase observation noises were set to 0.3 m and 3.0 mm, respe and elevation-dependent weighting for the observations was applied. Since the r phase center offset (PCO) and phase center variation (PCV) corrections for Galile  The code and phase observation noises were set to 0.3 m and 3.0 mm, respectively, and elevation-dependent weighting for the observations was applied. Since the receiver phase center offset (PCO) and phase center variation (PCV) corrections for Galileo were not available, we used the GPS signal corrections for Galileo signals. Such a strategy has been used effectively by various authors [29,30]. In addition, for the E5b, E5, and E6 observables, receiver PCO and PCV corrections for the GPS L2 frequency were used. Since different observations were used in the same satellite clock estimation, there were IFCBs in the E5b, E5 and E6 observations [31]. Relevant research shows that the consistency of Galileo multifrequency time-dependent UCDs can be ensured; thus, IFCBs in the Galileo system can be neglected [30]. The detailed processing strategies are summarized in Table 3.

Results
First, we evaluated the number of visible satellites, time dilution of precision (TDOP) and multipath noises of BRUX, CEBR, PTBB, ROAG and USN7 stations with Galileo satellites. Then, the positioning performances and time-frequency transfer stabilities of four models (DF, TF, QF and FF1) composed of dual-frequency, triple-frequency, quadfrequency, and five-frequency observations were compared by PPP solution; the positioning accuracies, IFB parameters and time-frequency transfer stabilities of the five-frequency observation models (FF1, FF2, FF2, FF3, FF4 and UC) were also evaluated. Finally, we used different AC precise products to compare and evaluate the performances of different PPP models. It should be noted that the positioning performance was assessed with respect to the coordinates from the IGS SNX file, and the principle of convergence was that the three-dimensional (3D) bias of twenty successive epochs is better than 10.0 cm; in addition, the 3D root mean square (RMS) values after convergence were counted. The modified Allan deviation (MADEV) was used as the stability index of time-frequency transfer.

Number of Visible Satellites, TDOP and Multipath Combination Noise Analysis
Until July 2020, the Galileo system had twenty-four valid satellites in orbit, including four in-orbit validation (IOV) satellites and twenty fully operational capability (FOC) satellites. The distribution of the average number of visible satellites and average position dilution of precision (PDOP) values for the GPS and Galileo constellations with a cutoff elevation of 5.0 degrees with a 1 • × 1 • grid on days of year (DOYs) 190 to 196 in 2020 are shown in Figure 2.
As shown in Figure 2, the number of visible satellites and the PDOP value of the GPS and Galileo constellations are symmetrically distributed with the equator, which shows a strong correlation with the dimension, and the positioning performance is better near the equator; the average number of visible satellites of Galileo ranges from 7.0 to 10.0, and the average PDOP value of Galileo ranges from 1.6 to 2.1. By comparison, the average number of visible satellites of Galileo is slightly less than that of GPS, and the PDOP value of Galileo is slightly larger than that of GPS. Figure 3 presents the number of visible satellites and TDOP values of the five stations. When the cutoff satellite elevation is 7 degrees, the average numbers of visible satellites at stations BRUX, CEBR, PTBB, ROAG and USN7 are 7.08, 6.6, 7.05, 6.68 and 8.85, respectively. Among them, the number of visible satellites with gross error removed at some moments of CEBR and USN7 stations are less than 4.0, which cannot be a PPP solution. It can be observed from the right figure that the TDOP values of the five stations all have some noise phenomenon, which will affect the accuracy of the time-frequency transfer of this epoch. The average TDOPs of the BRUX, CEBR, PTBB, ROAG and USN7 stations are 1.12, 1.29, 1.14, 1.26 and 1.26, respectively. dilution of precision (PDOP) values for the GPS and Galileo constellations with a cutoff elevation of 5.0 degrees with a 1° × 1° grid on days of year (DOYs) 190 to 196 in 2020 are shown in Figure 2. As shown in Figure 2, the number of visible satellites and the PDOP value of the GPS and Galileo constellations are symmetrically distributed with the equator, which shows a strong correlation with the dimension, and the positioning performance is better near the equator; the average number of visible satellites of Galileo ranges from 7.0 to 10.0, and the average PDOP value of Galileo ranges from 1.6 to 2.1. By comparison, the average number of visible satellites of Galileo is slightly less than that of GPS, and the PDOP value of Galileo is slightly larger than that of GPS. Figure 3 presents the number of visible satellites and TDOP values of the five stations. When the cutoff satellite elevation is 7 degrees, the average numbers of visible satellites at stations BRUX, CEBR, PTBB, ROAG and USN7 are 7.08, 6.6, 7.05, 6.68 and 8.85, respectively. Among them, the number of visible satellites with gross error removed at some moments of CEBR and USN7 stations are less than 4.0, which cannot be a PPP solution. It can be observed from the right figure that the TDOP values of the five stations all have some noise phenomenon, which will affect the accuracy of the time-frequency transfer of this epoch. The average TDOPs of the BRUX, CEBR, PTBB, ROAG and USN7 stations are 1.12, 1.29, 1.14, 1.26 and 1.26, respectively.  As shown in Figure 4, the multipath combination noise of each frequency is within ± 2.0 m, among which the multipath combination noise of the E1 frequency is the largest and that of the E5 frequency is the smallest. When the cutoff elevation is 7 degrees, the BRUX station E1, E5a, E6, E5 and E5b multipath combination noise RMS values are 0.  As shown in Figure 4, the multipath combination noise of each frequency is within ± 2.0 m, among which the multipath combination noise of the E1 frequency is the largest and that of the E5 frequency is the smallest. When the cutoff elevation is 7 degrees, the BRUX station E1, E5a, E6, E5 and E5b multipath combination noise RMS values are 0.   As shown in Table 4, the average values among the four time-frequency links in each model are completely consistent, and the RMS values have only slight deviations, which indicates that the multifrequency stability is not significantly improved compared with the dual-frequency stability. The RMS value of the inter-epoch difference of the BRUX-CEBR link is approximately 10.5 ps, and those of the other three time-frequency links are less than 8.2 ps, which indicates that the frequency stability of the BRUX-CEBR link is worse. In addition, it is noted that the hydrogen atomic clock attached to the BRUX-CEBR link has obvious clock drift, which is related to the performance of the CEBR station atomic clock.
in the TF PPP model. The convergence times of the BRUX, CEBR and ROAG stations are less than 15.0 min, and those of the PTBB and USN7 stations are approximately 25.0 min. According to the statistics, the average 3D RMS values of the DF, TF, QF and FF1 PPP models are 1.88 cm, 1.95 cm, 1.85 cm, and 1.78 cm, respectively; the average convergence times are 16.6 min, 16.6 min, 16.7 min and 17.3 min, respectively.  As shown in Table 4, the average values among the four time-frequency links in each model are completely consistent, and the RMS values have only slight deviations, which indicates that the multifrequency stability is not significantly improved compared with the dual-frequency stability. The RMS value of the inter-epoch difference of the BRUX            Figure 9 presents the clock offsets of five-frequency PPP models, and each clock offset sequence has the same trend, which proves the correctness of the algorithm. In addition, there are some deviations among the clock offsets, among which the FF1 and FF2 PPP models and the FF4 and UC PPP models have minimum deviations.   As shown in Table 5   As shown in Figure 11, the IFB sequences are basically stable, and the IFB values of the FF4 model are the largest. The average RMS values of IFB IF13 , IFB IF14 , and IFB IF15 in the FF4 PPP model are 7.8 ns, 4.1 ns, and 9.9 ns, respectively; and the average STD values are 0.04 ns, 0.03 ns, and 0.15 ns, respectively. The average RMS values of IFB UC3 , IFB UC4 , and IFB UC5 in the UC PPP model are 5.49 ns, 3.09 ns, and 5.15 ns, respectively, and the corresponding average STD values are 0.03 ns, 0.02 ns, and 0.04 ns, respectively. As mentioned, the IFB IF15 STD value of the FF4 PPP model is the largest, leading to the receiver clock bias, and IFB IF15 cannot be accurately separated, which will affect the time-frequency transfer performance.  As shown in Figure 11, the IFB sequences are basically stable, and the IFB values of the FF4 model are the largest. The average RMS values of IFBIF 13 , IFBIF 14 , and IFBIF 15 in the FF4 PPP model are 7.8 ns, 4.1 ns, and 9.9 ns, respectively; and the average STD values are 0.04 ns, 0.03 ns, and 0.15 ns, respectively. The average RMS values of IFBUC 3 , IFBUC 4 , and IFBUC 5 in the UC PPP model are 5.49 ns, 3.09 ns, and 5.15 ns, respectively, and the corresponding average STD values are 0.03 ns, 0.02 ns, and 0.04 ns, respectively. As mentioned, the IFBIF 15 STD value of the FF4 PPP model is the largest, leading to the receiver clock bias, and IFBIF 15 cannot be accurately separated, which will affect the time-frequency transfer performance.

PPP Performance by Using Different AC Products
To further evaluate the performance of Galileo's five-frequency PPP model, the WUM and GRG precise ephemeris and clock products are used for experiments. It should be noted that the GRG clock product has absorbed the UPD of the satellite side, and the

PPP Performance by Using Different AC Products
To further evaluate the performance of Galileo's five-frequency PPP model, the WUM and GRG precise ephemeris and clock products are used for experiments. It should be noted that the GRG clock product has absorbed the UPD of the satellite side, and the GBM and WUM clock products have absorbed the UCD of the satellite side, with some differences. The average convergence times and average 3D RMS values at five stations of the DF, TF, QF, FF1, FF2, FF3, FF4 and UC PPP models are shown in Figure 12.

PPP Performance by Using Different AC Products
To further evaluate the performance of Galileo's five-frequency PPP model, the WUM and GRG precise ephemeris and clock products are used for experiments. It should be noted that the GRG clock product has absorbed the UPD of the satellite side, and the GBM and WUM clock products have absorbed the UCD of the satellite side, with some differences. The average convergence times and average 3D RMS values at five stations of the DF, TF, QF, FF1, FF2, FF3, FF4 and UC PPP models are shown in Figure 12. The convergence times of the WUM product are significantly longer than those of the GBM and GRG products, and the average convergence time of the GRG product is the shortest. Compared with the dual-frequency PPP model, the multifrequency PPP model does not significantly improve the convergence speed. The average convergence times using the GBM, WUM and GRG products are 16.8 min, 22.2 min, and 15.7 min, respectively. Compared with the results using the GRM and WUM products, the average convergence speed obtained using the GRG product is increased by 6.8% and 29.1%, respectively. The average 3D RMS values obtained using the GBM, WUM and GRG products The convergence times of the WUM product are significantly longer than those of the GBM and GRG products, and the average convergence time of the GRG product is the shortest. Compared with the dual-frequency PPP model, the multifrequency PPP model does not significantly improve the convergence speed. The average convergence times using the GBM, WUM and GRG products are 16.8 min, 22.2 min, and 15.7 min, respectively. Compared with the results using the GRM and WUM products, the average convergence speed obtained using the GRG product is increased by 6.8% and 29.1%, respectively. The average 3D RMS values obtained using the GBM, WUM and GRG products are 1.82 cm, 1.85 cm, and 1.77 cm, respectively. Compared with the results using the GRM and WUM products, the positioning accuracy achieved using the GRG product is improved by 2.7% and 4.3%, respectively.
Limited by space, Figure 13 only shows the clock offsets of the FF1 PPP model by using the GBM, WUM and GRG precise products. The clock offsets of the BRUX-CEBR, BRUX-PTBB and BRUX-ROAG time-frequency links are completely consistent, and there is only a small constant deviation term. Although the BRUX-USN7 clock offset trend is consistent, the deviation is slightly larger, which may be related to the quality of the observations of USN7. According to statistics, the deviations of the BRUX-CEBR link using GBM products compared with WUM and GRG products are 0.02 ns and 0.03 ns, respectively; the deviations of the BRUX-PTBB link using GBM products compared with WUM and GRG products are all 0.01 ns; the deviations of the BRUX-ROAG link using GBM products compared with WUM and GRG products are all 0.03 ns; and the deviations of the BRUX-USN7 link using GBM product compared with WUM and GRG products are 0.08 ns and 0.09 ns, respectively.
tively; the deviations of the BRUX-PTBB link using GBM products compared with WUM and GRG products are all 0.01 ns; the deviations of the BRUX-ROAG link using GBM products compared with WUM and GRG products are all 0.03 ns; and the deviations of the BRUX-USN7 link using GBM product compared with WUM and GRG products are 0.08 ns and 0.09 ns, respectively.  Table 6 shows the RMS values of the epoch difference by using WUM and GRG precision products for the PPP solution. The average RMS values of the BRUX-CEBR, BRUX-PTBB, BRUX-ROAG and BRUX-USN7 time-frequency links with the WUM product are 10.54 ps, 7.01 ps, 7.14 ps, and 8.24 ps, respectively, while those of the BRUX-CEBR, BRUX-PTBB, BRUX-ROAG and BRUX-USN7 time-frequency links with the GRG product are 10.57 ps, 7.02 ps, 7.22 ps, and 8.18 ps, respectively, which is consistent with the solution results obtained using the precise GBM product.    Table 6 shows the RMS values of the epoch difference by using WUM and GRG precision products for the PPP solution. The average RMS values of the BRUX-CEBR, BRUX-PTBB, BRUX-ROAG and BRUX-USN7 time-frequency links with the WUM product are 10.54 ps, 7.01 ps, 7.14 ps, and 8.24 ps, respectively, while those of the BRUX-CEBR, BRUX-PTBB, BRUX-ROAG and BRUX-USN7 time-frequency links with the GRG product are 10.57 ps, 7.02 ps, 7.22 ps, and 8.18 ps, respectively, which is consistent with the solution results obtained using the precise GBM product. As shown in Figure 14, the frequency stabilities of the BRUX-CEBR, BRUX-PTBB and BRUX-USN7 links can reach the 10 −16 level, and the frequency stability of the BRUX-USN7 link using the GRG product is the most significantly improved compared to those using WUM and GRG products, which may be related to the observation quality of USN7 and the absorption of satellite UPD by GRG clock products. The average frequency stabilities over four links using GBM, WUM and GRG products in 120,000 s are 1. 18  and 1.10 × 10 −15 , respectively. All these results demonstrate that the frequency stability using the GRG product is better than those using WUM and GBM products at 120,000 s. It can also be found that compared with the DF model, the multifrequency model's frequency stabilities are not significantly improved, which may be limited by the additional estimation of IFB parameters and the accuracy of multifrequency DCB products [40].
WUM and GRG products, which may be related to the observation quality of USN7 and the absorption of satellite UPD by GRG clock products. The average frequency stabilities over four links using GBM, WUM and GRG products in 120,000 s are 1.18 × 10 −15 , 1.13 × 10 −15 and 1.06 × 10 −15 , respectively. Using WUM products, the average frequency stabilities of the DF, TF, QF, FF1, FF2, FF3, FF4 and UC PPP models are 1.12 × 10 All these results demonstrate that the frequency stability using the GRG product is better than those using WUM and GBM products at 120,000 s. It can also be found that compared with the DF model, the multifrequency model's frequency stabilities are not significantly improved, which may be limited by the additional estimation of IFB parameters and the accuracy of multifrequency DCB products [40]. To deeply analyze the frequency transfer stability of the BRUX-USN7 link, Figure 15 shows the frequency stability of the BRUX-USN7 link by using different ACs precision products. The frequency stability obtained using the GBM, WUM and GRG products before 30,000 s is completely consistent, but the frequency stability achieved using GRG products between 30,000 s and 120,000 s is significantly improved compared with using the GBM and WUM products. Compared with GBM and WUM, the stability obtained using GRG products is increased by 33.83% and 40.62% on average, respectively, and with the increase in time, the improvement range of stability gradually increases, which indicates that using GRG products has certain advantages over using GBM and WUM products in long-term frequency stability. To deeply analyze the frequency transfer stability of the BRUX-USN7 link, Figure 15 shows the frequency stability of the BRUX-USN7 link by using different ACs precision products. The frequency stability obtained using the GBM, WUM and GRG products before 30,000 s is completely consistent, but the frequency stability achieved using GRG products between 30,000 s and 120,000 s is significantly improved compared with using the GBM and WUM products. Compared with GBM and WUM, the stability obtained using GRG products is increased by 33.83% and 40.62% on average, respectively, and with the increase in time, the improvement range of stability gradually increases, which indicates that using GRG products has certain advantages over using GBM and WUM products in long-term frequency stability.

Discussion
With GNSS interoperability and support for multiple signal frequencies, the era of multi frequency and multi system PNT services has come. Compared with the traditional DF model, the multi frequency combination will be able to look for smaller noise coefficients, improve the reliability of the system, and help to achieve fast fixed ambiguity. At the same time, it will also face the challenge of IFB and IFCB processing in multi frequency signals. This research is a new exploration of Galileo five frequency signal PPP models, which systematically realizes the comparison and evaluation of Galileo TF, DT, FF, and DF models in terms of precision positioning and time-frequency transfer performance. Although the noise amplification factor of some multi frequency combination models decreases, the estimated IFB parameters are increased with the increase in the component observation equation. The precise separation of IFBs from clock bias and ambiguity parameters, the setting of multi frequency observation noise and the accuracy of multi frequency DCB products also affect the performance of five frequency PPP models. With the improvement of multi frequency DCB products, combined with the advantages of multi-frequency signals in redundancy, low noise, and long wavelength, it is reasonable to consider that Galileo five frequency signal will be more widely used in the future. In subsequent research, we will determine the noise of multi frequency observations and explore the performance of multi frequency signal in terms of ambiguity resolution.

Discussion
With GNSS interoperability and support for multiple signal frequencies, the era multi frequency and multi system PNT services has come. Compared with the tradition DF model, the multi frequency combination will be able to look for smaller noise coe cients, improve the reliability of the system, and help to achieve fast fixed ambiguity. the same time, it will also face the challenge of IFB and IFCB processing in multi frequen signals. This research is a new exploration of Galileo five frequency signal PPP mode which systematically realizes the comparison and evaluation of Galileo TF, DT, FF, a DF models in terms of precision positioning and time-frequency transfer performan Although the noise amplification factor of some multi frequency combination models d creases, the estimated IFB parameters are increased with the increase in the compone observation equation. The precise separation of IFBs from clock bias and ambiguity p rameters, the setting of multi frequency observation noise and the accuracy of multi f quency DCB products also affect the performance of five frequency PPP models. With t improvement of multi frequency DCB products, combined with the advantages of mu frequency signals in redundancy, low noise, and long wavelength, it is reasonable to co sider that Galileo five frequency signal will be more widely used in the future. In sub quent research, we will determine the noise of multi frequency observations and explo the performance of multi frequency signal in terms of ambiguity resolution. It should be noted that in order to meet the performance of time-frequency transfer, we need to select the stations that support the Galileo five-frequency observation signals and are connected the external high-precision hydrogen atomic clocks. After eliminating stations with poor data quality and choosing the preferred stations of UTC laboratories, there are only a few proper suitable stations in the MGEX network. At the same time, scientists are more concerned about the index of the daily stability, which can basically reflect the best performance of the atomic clocks. The time-frequency links selected in this study are very representative, and the length of the observations can meet the requirements of short-term stability and long-term stability. In addition, our study evaluates the performance of positing and time-frequency transfer of double-, triple-, quad-and five-frequency PPP models, focusing on the comparison of results among different PPP models. Increasing the length of observations has no obvious practical significance and has little effect; the observations selected in our study can fully support our conclusions. The application of Galileo multi-frequency PPP solutions during different ionospheric active periods requires further investigation.

Conclusions
In this contribution, the mathematical PPP model of Galileo five-frequency observations is established; observation types, combination coefficients, ionospheric coefficients, and noise amplification coefficients from the DF to FF PPP models are also compared.