A method of undifferenced ambiguity resolution for GPS+GLONASS precise point positioning

Integer ambiguity resolution is critical for achieving positions of high precision and for shortening the convergence time of precise point positioning (PPP). However, GLONASS adopts the signal processing technology of frequency division multiple access and results in inter-frequency code biases (IFCBs), which are currently difficult to correct. This bias makes the methods proposed for GPS ambiguity fixing unsuitable for GLONASS. To realize undifferenced GLONASS ambiguity fixing, we propose an undifferenced ambiguity resolution method for GPS+GLONASS PPP, which considers the IFCBs estimation. The experimental result demonstrates that the success rate of GLONASS ambiguity fixing can reach 75% through the proposed method. Compared with the ambiguity float solutions, the positioning accuracies of ambiguity-fixed solutions of GLONASS-only PPP are increased by 12.2%, 20.9%, and 10.3%, and that of the GPS+GLONASS PPP by 13.0%, 35.2%, and 14.1% in the North, East and Up directions, respectively.

GLONASS ambiguity fixing [17][18][19][20][21][22] . Their results show that the IFPB is mainly related to the receiver type and has a linear relation with frequency. Residual error can be reduced to a few millimeters after calibration and has hardly influence on wide-lane ambiguity fixing. However, no obvious linear relationship exists between the IFCBs and the frequency; hence, no unified and effective correction model is known 18,[23][24][25][26] . Reussner and Wanninger indicated that IFCBs are difficult to correct. They proposed a method based on wide-lane carrier phase observations and an ionosphere model to fix wide-lane ambiguity. However, their method depends on a high-accuracy ionosphere model. To ensure reliable wide-lane ambiguity fixing, the ionosphere model must have a prediction accuracy higher than 1.4 total electron content units 27 . Hence, Geng and Bock recommend that for Europe if only IGS GIMs are available, GLONASS PPP-AR is best carried out within an area spanning less than 800 km in both longitudinal and latitudinal directions 28 . Banville et al. selected two GLONASS reference satellites with adjacent frequency numbers for integer ambiguities definition and thus absorb the linear dependency of the narrow-lane IFCBs with respect to frequency 22 . However, this method disregards the nonlinear parts of the narrow-lane IFCBs, even though these nonlinear parts are non-negligible. Therefore, the IFCBs are critical for processing undifferenced GLONASS ambiguity fixing.
In this study, we assume that IFPB can be precisely corrected in advance based on the exiting studies. To address the influence of the GLONASS IFCBs on wide-lane ambiguity fixing, we propose an undifferenced ambiguity resolution method for GPS+ GLONASS PPP. The data of reference network are used to estimate the IFCBs calibrations and the FCB products, which are then delivered to the client to implement GLONASS ambiguity fixing. The rest of this article is organized as follows: first introduces the ambiguity fixing method for PPP, then elaborates the influence of the IFCBs on GLONASS wide-lane ambiguity fixing and introduces how the IFCBs are processed in GLONASS ambiguity fixing, then the data of EUREF Permanent network (EPN) are used to verify examples, the corresponding conclusion is presented at the last section.

PPP data processing for ambiguity resolution
In dual-frequency PPP, ionosphere-free combination of observations is adopted to eliminate the effect of the first-order ionospheric delay 1,2 . The pseudorange and carrier phase observation equation of ionosphere-free combination are written as follows: indicate the pseudorange and carrier phase observation of the ionosphere-free combination (meter); ρ k i indicates the satellite-receiver geometric distance affected by errors, such as relativistic effects and earth tide; c indicates the speed of light; ∆t r indicates the receiver clock error; λ indicates the wavelength of carrier phase ionosphere-free combination of observations; B IF k i indicates the ambiguity of the ionosphere-free combination of observations (cycle), and it can be written as follows: i represents the wide-lane ambiguity and narrow-lane ambiguity, respectively. When attempting PPP ambiguity fixing, wide-lane ambiguity is initially fixed by MW combination observations, and then narrow-lane ambiguity is established by the fixed wide-lane integer ambiguity and the ionosphere-free combination ambiguity. Through narrow-lane ambiguity fixing, the fixed solution of ambiguity can be obtained for PPP 5 (3) are actually not integers because these are affected by the FCBs of the satellite and receiver 16 . Thus, the real-valued undifferenced wide-lane and narrow-lane ambiguities can be expressed as follows: The premise of the FCB-based ambiguity fixing method for PPP is using the observation data from the server's reference network to separate FCB products that cause undifferenced ambiguity to become a non-integer. FCB products are then delivered to the client to restore the integer nature of the client's undifferenced ambiguity as well as realize the search and fixation of undifferenced integer ambiguities at the client 6 .
Scientific RepoRts | 6:26334 | DOI: 10.1038/srep26334 Effects of GLONASS narrow-lane IFCBs on wide-lane FCB estimation. Given that MW combination observations are irrelevant to the satellite-receiver geometric distance, with the first-order ionospheric delay eliminated [14][15][16] , MW combination observations are generally adopted to estimate wide-lane FCBs. The real-valued wide-lane ambiguity can be expressed as follows: 2 indicate the pseudorange and carrier phase observations at L1 and L2 respectively; and δ δ , r k i indicate the receiver and satellite pseudorange hardware delays respectively. The pseudorange and carrier phase multipath delays and the observation noise can be reduced through the averaging among multiple epochs and are negligible.
As shown in equation (5), wide-lane FCBs that are estimated by MW combination observations including the effects of the pseudorange-related hardware delays. For GPS, δ i and δ r k are relatively stable. Moreover, δ i is the same for server and client stations, whereas δ r k is the same for all satellites tracked at the receiver. Therefore, the characteristics of φ w i and φ w k will not change if they are combined with δ i and δ r respectively. Accordingly, the following equation can be derived: For the client, when wide-lane ambiguity is fixed through MW combination observations, the integer nature of wide-lane ambiguity can still be restored by using the FCB products 6 . However, for GLONASS, the pseudorange observation contains IFCBs that are currently difficult to correct 27 . Accordingly, the real-valued wide-lane ambiguity can be derived as follows: indicates the narrow-lane IFCBs 29 and it should be noted that the IFPB is assumed to have been corrected in equation (7). In GLONASS, the estimation of wide-lane FCB by using MW combination observations will be affected by the narrow-lane IFCBs, as shown below.
however, the IFCBs are related to several factors, such as receiver type and receiver firmware version. Hence, when the server and the client use different types of receivers, variations are observed between the narrow-lane IFCBs of the server and that of the client. If the narrow-lane IFCBs are ignored, then wide-lane FCB products estimated by the server will be unable to restore the integer nature of the undifferenced wide-lane ambiguity at the client-end. Wide-lane ambiguity fixing will then be affected. Therefore, narrow-lane IFCBs processing is the key to GLONASS ambiguity fixing. In the next section, the processing method of narrow-lane IFCBs in the wide-lane FCB estimation will be discussed.

Narrow-lane IFCBs processing in wide-lane FCB estimation for GLONASS. In the FCB estimation,
wide-lane FCB is generally estimated first. After wide-lane ambiguity is fixed, the narrow-lane ambiguity, which is composed of wide-lane ambiguity and ionosphere-free combination ambiguity, is used to estimate narrow-lane FCB. Ge et al. 6 proposed that the substitution of wide-lane integer ambiguity can avoid the effect of wide-lane FCB estimation error on narrow-lane FCB estimation when the narrow-lane ambiguity is composed. According to equation (3), the following equation can be obtained: however, for GLONASS, the magnitude of the IFCBs can reach several meters at some frequencies 26,27 . This bias will not only affect wide-lane FCB estimation, but will also lead to integer deviation in wide-lane ambiguity fixing. However, the frequency ratio of GLONASS L1 and L2 observations is f 1 :f 2 = 9:7. This ratio is substituted into equation (9), which derives the following equation: Processing method of FRA-IFCB on MW combination observations. Considering the effects of the GLONASS IFCBs, the real-valued wide-lane ambiguity of the GLONASS satellite i at station k can be expressed as follows: Fixing the real-valued wide-lane ambiguity to the nearest integer ambiguity derives the following equation: taking the nearest integer. Assuming that s stations are present in the reference network, and j GLONASS satellites can be observed in each station, the following can be obtained: where R k is an × j 1 -order matrix with all elements being 1, I k is an × j j -order identity matrix, and l k i indicates the decimal part of the wide-lane ambiguity of the i satellite at k station. In equation (12), the receiver FCB, satellite FCB, and IFCBs are linearly dependent. In equation (13), the normal equation is rank-deficient and needs the addition of + j m datum constraint equations. The present study selects one station as the base station because only single-differenced satellite FCB and the IFCBs between satellites make sense. This base station conforms to the following assumptions: (1) The receiver FCB of base station is 0.
(2) The IFCBs of each satellite that can be observed at the base station is 0. by adjusting the combined equations (13) and (14), the wide-lane FCBs of receiver and satellite and the IFCBs can be obtained. Notably, given that equation (12) eliminates the integer ambiguity parameter by rounding, only the FRA-IFCB can be obtained and the ODD-IFCB needs to be further corrected.
Processing method of ODD-IFCB on MW combination observations. Supposing that after correcting the IFCBs for the station k, the wide-lane ambiguity no longer contains ODD-IFCB. From equation (10), the real-valued narrow-lane ambiguity of satellite i can be derived as After correcting the FRA-IFCBs for the station p, if the wide-lane ambiguity still contains ODD-IFCB, then the real-valued narrow-lane ambiguity is as follows: n p i n p i n p n i The station k p , is made up of a baseline to evaluate the differences between satellites. The following can then be derived: . When the trigonometric function operation is used to eliminate the integer ambiguity parameter, the following can be obtained: cos(2 ( )) cos(2 (0 5 )) cos (2  ) sin(2 ( )) sin(2 (0 5 )) sin(2 ) (18) n k i n p i n k p n k p n k i n p i n k p n k p , , , , φ ∆ n k p , represents the inter-station single-difference receiver FCB, which has the same effect on different satellites at the same baseline. This means that if the wide-lane ambiguity is fixed correctly, the sine and cosine values of real-valued narrow-lane ambiguities of different satellites, in theory, are equivalent. But as shown in equation (18), when the wide-lane ambiguity fixed for a satellite has even-cycle deviation, compared with the sine and cosine of other satellites, the values are equal but the signs are opposite. On this basis, the ODD-IFCB caused by narrow-lane IFCBs on the wide-lane ambiguity can be calibrated by the following steps: (1) Selecting one station as the reference station and assume that the wide-lane ambiguity no longer contains ODD-IFCB at the reference station. To get more precise IFCB calibrations, the coordinates of the stations can be fixed and the post-process method can be used. As the IFCB is mainly related to receiver type and receiver firmware version 26 , the IFCB for different receiver type and firmware version can be estimated as well as the FCB estimation and then provided to clients. Anyway, the receiver type or firmware version of the client is probably not included in the reference network. In this situation, we can use the follow steps to obtain the IFCB calibrations.
(1) Assume that the receiver wide-lane FCB is zero, after correcting the satellite wide-lane FCBs by the wide-lane FCB products provided by the server, as shown in equation (12), the FRA-IFCB can be easily obtained. (2) In equation (16), if the satellite narrow-lane FCB has been corrected, the remains are only integer ambiguity, receiver FCB, and the half-cycle deviation. Then the trigonometric function operation is used to eliminate the integer ambiguity. Similarly to equation (18), the ODD-IFCB should be set to 1, if the values of the sine and cosine are equal but signs are opposite.
Notably, the ODD-IFCB at the reference station will be absorbed into the narrow-lane FCB. But this will not affect the ambiguities fixing at client-end, as the datum of the wide-and narrow-lane FCB and IFCB products are unified at the server and client-end.
Experimental validation. The GPS+ GLONASS data continuously observed by 45 stations (32 servers, 13 clients) of EPN for day of year (DOY) 335-341 in 2013 were adopted. The station distribution is shown in Fig. 1 and the receiver and antenna types and the firmware version are listed in Table 1. In Fig. 1, the red pentagrams represent reference stations and the blue cycles represent the clients. In this study, the precise orbits and clock error products with 30-second sampling interval provided by the European Space Agency were adopted; the errors such as earth tide and phase windup were corrected with the precise model provided by the International GNSS Service (IGS); and the antenna phase center was corrected with the absolute antenna phase center model provided by IGS. The GLONASS IFPB are corrected using the method proposed by Wanninger 20 while the GLONASS IFCB are processed by the method proposed in this study. After the estimation of wide-lane and narrow-lane FCBs is completed by the server, these FCB products and IFCBs corrections are delivered to the client. To improve the FCB estimation precision, the server fixes station coordinates and the satellite's elevation mask angle is set to 15°. The client adopts the dynamic PPP model and the same error correction model as the server. The integer nature of undifferenced ambiguity is restored by the FCB provided by the server and the corrected IFCBs. Through least-square ambiguity decorrelation adjustment (LAMBDA) 30 , the wide-lane and narrow-lane ambiguities are searched and fixed, followed by the fixing of ionosphere-free combination ambiguity. Thus, the ambiguity-fixed solution can be obtained for PPP. This study adopted partial ambiguity fixing strategy to fix the ambiguity and adopts the ratio and probability P 0 to check if the ambiguity can be fixed 31,32 . The thresholds of ratio and P 0 are 4 and 0.99 respectively. The calculation equation of probability P 0 is as follows 32 : To analyze the stability of the IFCB estimation, 30 days data are used to estimate the IFCB (one parameter for each station and each satellite) and the standard deviations (STDs) of the IFCB for each station and each satellite are computed. For simplify data analysis, the average STDs of all 24 satellites for each station are obtained. Figure 2 shows the average STDs of each station. As shown in Fig. 2 superior to 0.08 cycle and that of most station are superior to 0.05 cycle. This means IFCB is stable and can be consider as the same over 30 days. So the average IFCB over 30 days is consider as the final results for each station. Then, the relationship between different stations is analyzed. Considering that IFCB is related to receiver type and firmware, so in this study, for each receiver type and firmware version, one station is selected as the reference, then the correlation coefficients of the other stations to the reference station are computed. Figure 3 show the correlation coefficient for each receiver type and firmware version. As shown in Fig. 3 the correlation coefficients of the Trimble NetR5 is about 0.85, while the others are all high up to 0.9, which means the stations with the same receiver type and firmware are strongly correlated even if the antenna type is different. Thus, when data are processed, IFCBs of the same receiver type and receiver firmware version can be considered as the same.   of GPS satellites in seven days are all superior to 0.03 cycle, and the average STD is only 0.012 cycle. GLONASS stability is relatively lower. However, the STD of wide-lane FCB estimations of each satellite in seven days is superior to 0.09 cycle, and the average STD is superior to 0.05 cycle. It is illustrated that GPS and GLONASS wide-lane FCBs are relatively stable in a week and can be forecasted for several days.
The residual analysis after the ambiguity fixed by FCB products is an effective method to evaluate the FCB estimation accuracy 6,10 . This study used FCB estimation products to fix ambiguities of the reference network. After correcting the satellite and receiver FCBs, the ambiguity was fixed through rounding and the residual was analyzed. Figure 5 Fig. 7. For GPS, 98.5% of the residuals of narrow-lane FCBs are less than 0.10 cycle, and 99.7% are less than 0.15 cycle; for GLONASS, around 93.5% of the residuals of narrow-lane FCBs are less than 0.10 cycle, and 98.7% are less than 0.15 cycle. When GPS and GLONASS narrow-lane ambiguities are fixed, the residuals are smaller. This confirm that the narrow-lane ambiguity fixing is reliable and the proposed method can effectively eliminate the effects of the GLONASS IFCBs to realize GLONASS ambiguity fixing. However, GLONASS residuals are still relatively large compared with GPS residuals. The first reason may be that the precision of GLONASS orbits and clock error products is slightly worse than the precision of GPS. And the second reason maybe that, for GLONASS, the IFPBs cannot be totally eliminated in this study and the IFPB residuals will affect the accuracy of the narrow-lane FCBs.
To further evaluate the effectiveness of the ambiguity fixing method in this study, 13 stations were selected to analyze the performance of PPP ambiguity-fixed solutions of GPS-only, GLONASS-only, and combined GPS+ GLONASS at the client. The true-value coordinates of the stations were provided by weekly SINEX     the GPS and GLONASS can be fixed using FCB products estimated in this study. After fixing ambiguities, the dynamic PPP accuracies of GPS-only, GLONASS-only, and the combined GPS+ GLONASS can all be improved. Figure 12 shows the average fixing rates of GPS and GLONASS ambiguities at 13 stations in seven days. The fixing rate is around 90% for GPS, but only around 75% for GLONASS. Figures 13-16     of each station is improved. The average RMS GPS-only PPP at three directions (N, E, U) is reduced respectively from 1.19, 1.26, and 2.65 cm to 0.79, 0.79, and 2.12 cm; the accuracy in the three directions increased by 33.9%, 37.6%, and 20.2%. The average RMS of GLONASS-only at three directions (N, E, U) is reduced respectively from 1.09, 1.30, and 2.65 cm to 0.96, 1.03 and 2.37 cm; the accuracy in the three directions increased by 12.2%, 20.9%, and 10.3%. Compared with that for GPS, the ambiguity fixed for GLONASS contributes less to the accuracy improvement. This result is probably because GLONASS has a lower fixing rate than GPS, and the accuracy of GLONASS narrow-lane FCB estimation is relatively lower than that of GPS. For the combined GPS+ GLONASS PPP, if only GPS ambiguities are fixed, the average RMS at three directions (N, E, U) is reduced respectively from 0.88, 1.14, and 2.41 cm to 0.78, 0.76, and 2.09 cm; the accuracy in the three directions increased by 11.7%, 33.3%, and 13.3%. And if both the GPS and GLONASS ambiguities are fixed, the average RMS is reduced to 0.77, 0.74, and 2.07 cm, increasing by 13.0%, 35.2%, and 14.1%. It can be seen that, for combined GPS+ GLONASS PPP, fixing both the GPS and GLONASS ambiguities contributes less to the accuracy improvement than that fixing only GPS ambiguities. This is because if GPS ambiguities can be well fixed, generally, there will be more than seven  ambiguities being fixed. The position accuracy can well be improved. In this situation, fixing GLONASS ambiguities will contributes less to the accuracy improvement.

Conclusions
This study proposes undifferenced ambiguity resolution methods for GPS+ GLONASS PPP, which consider the IFCBs estimation. This study also establishes the parameter estimation models for the IFCBs at the server. The client can utilize the satellite wide-lane and narrow-lane FCBs delivered by the server as well as IFCBs products to eliminate the effect of the GLONASS IFCBs. The example result demonstrates that IFCB are very stable over 30 days and the same receiver type and firmware are strongly correlated. Using the IFCB corrections, GLONASS undifferenced ambiguity can be fixed and the success rate can reach 75%. Compared with the ambiguity float solutions, the positioning accuracies of ambiguity-fixed solutions of GLONASS-only PPP at N, E, and U directions are increased respectively by 12.2%, 20.9%, and 10.3%; the accuracies of ambiguity-fixed solutions of the  combined GPS+ GLONASS PPP are increased by 13.0%, 35.2%, and 14.1% in the three directions. However, the method proposed in this article requires the IFCBs corrections of different types of receivers. The client need to select the IFCB corrections according their receiver type and firmware version and calculate their own IFCB corrections if their receiver type or firmware version is not included in table list of the IFCB corrections. Furthermore, apart from the selected receivers of Leica series, Nov type, and Trimble series in this article, more types of receivers need to be used for example verification 33 .