Mitigation of Short-Term Temporal Variations of Receiver Code Bias to Achieve Increased Success Rate of Ambiguity Resolution in PPP

: Ambiguity resolution (AR) is critical for achieving a fast, high-precision solution in precise point positioning (PPP). In the standard uncombined PPP (S-UPPP) method, ionosphere-free code biases are superimposed by ambiguity and receiver clock o ﬀ sets to be estimated. However, besides the time-constant part of the receiver code bias, the complex and time-varying term in receivers destroy the stability of ambiguities and degrade the performance of the UPPP AR. The variation of receiver code bias can be conﬁrmed by the analysis in terms of ionospheric observables, code multipath (MP) of the Melbourne–Wübbena (MW) combination and the ionosphere-free combination. Therefore, the e ﬀ ect of receiver code biases should be rigorously mitigated. We introduce a modiﬁed UPPP (M-UPPP) method to reduce the e ﬀ ects of receiver code biases in ambiguities and to decouple the correlation between receiver clock parameters, code biases, and ambiguities parameters. An extra receiver code bias is set to isolate the code biases from ambiguities. The more stable ambiguities without code biases are expected to achieve a higher success rate of ambiguity resolution and a shortened convergence time. The variations of the receiver code biases, which are the unmodeled errors in measurement residuals of the S-UPPP method, can be estimated in the M-UPPP method. The maximum variation of the code biases is up to 16 ns within two-hour data. In the M-UPPP method, the averaged epoch residuals for code and phase measurements recover their zero-mean features. For the ambiguity-ﬁxed solutions in the M-UPPP method, the convergence times are 14 and 43 min with 17.7% and 69.2% improvements compared to that in the S-UPPP method which are 17 and 90 min under the 68% and 95% conﬁdence levels.


Introduction
A major problem in facilitating precise point positioning (PPP) ambiguity resolution is that the undifferenced ambiguities are not integer values, due to the existence of the code and phase delay biases [1][2][3][4][5][6]. Fractional-cycle biases (FCB) in global navigation satellite system (GNSS) phase measurements must therefore be corrected or removed in order to recover the integer property of ambiguities [2,[7][8][9][10][11][12]. In fact, these biases are hardware-dependent and exist in all receivers and which is known as the decoupled clock model (DC) method, to estimate satellite code and phase clock offsets with integer ambiguity. Compared to the FCB method, the IRC and DC models regard the receiver code, the satellite code, and the phase biases as non-constant clock-like terms [16,34]. However, the estimation of the integer phase clock offset or decoupled satellite clock offset increases the computational burden, and these clock offset products are also not compatible with the IGS official clock offset products. These two methods are not flexible for common users. Though the satellite code bias has shown a high stability over a period of one day, the stability of receiver code bias is heavily dependent on receiver temperature variation and firmware quality [31,35,36]. Hence, only specific receivers whose code biases are stable over one day by preliminary analysis are used to estimate the satellite ephemeris and clock offsets or FCBs in a network solution [10]. However, a wide variety of receiver types are adopted for users, and their code biases properties are not explicit known, especially in real-time applications.
To confirm the variation of the receiver code bias, the ionospheric observables obtained from carrier-to-code leveling (CCL) and UPPP were analyzed to confirm the stability of the between-receiver differential code bias (BR-DCB). To exclude the effects of the code multipath (MP) and ionospheric delays, the Melbourne-Wübbena (MW) and ionosphere-free combinations can be used to present the code multipath [37]. To eliminate the adverse impact of the variability of receiver code biases on ambiguity estimations in PPP, a modified uncombined PPP (M-UPPP) model is introduced to separate the receiver code biases from the ambiguity parameters. By using a similar treatment as in the IRC and DC models for receiver clock offsets, the independent clock offset terms are introduced in the model for the code and phase measurements, respectively.
The following sections start with detailed formulations of the CCL method with the MW combination, the ionosphere-free code multipath, and the UPPP model. Then, the introduced M-UPPP method is presented with the key point of ambiguity datum. The data and experimental setup are introduced in the next section. In the subsequent section, the analysis of receiver code bias variations and positioning performance of the M-UPPP method are presented. The statistic results of the convergence time and the ambiguity fixing success rate when using the M-UPPP method applied to over 220 stations are also shown in this section. Finally, a discussion of the results, conclusions, and perspectives are provided.

Methods
Firstly, the linearized code and phase basic observation equations are derived. Then, the carrier-to-code leveling method and the uncombined PPP method are introduced to extract ionospheric observations for detecting receiver code bias variations. The geometry-free and ionosphere-free combinations are also adopted to evaluate the multipath measurements for detecting the receiver code bias variations. Additionally, through an analysis of related formulas, the M-UPPP method is be introduced to improve the performance of positioning and ambiguity resolution. In the following, the ambiguity datum of the M-UPPP method is explained.

Basic Code and Phase Observation Equations
The dual-frequency basic code and phase observation equations are described as: where P s r, f is the code measurement at frequency f ( f = 1, 2) for satellite s (m); Φ s r, f is the carrier phase measurement at frequency f ( f = 1, 2) (m), ρ is the geometric range between receiver r and satellite s; c is the speed of light in vacuum; dt r is the receiver clock offset; dt s is the satellite clock offset; T is the slant troposphere delay; I 1 is the ionospheric delay along the line-of-sight of receiver and satellite at the first frequency and γ f = λ 2 f /λ 2 1 ; λ f is the wavelength at frequency f (m); N s r, f is the phase ambiguity at frequency f (cycle); d r, f and d s f are the constant code biases at frequency f for the receiver and the satellite in meters, respectively; δd r, f and δd s f are corresponding time-varying terms; b r, f and b s f are the phase biases at frequency f for receiver and satellite in meters, respectively; δb r, f and δb s f are corresponding time-varying terms; m s P, f and m s Φ, f are the code and the phase multipath, respectively; and ε P, f and ε Φ, f are the noises of the code and carrier phase measurements.

Carrier-to-Code Leveling (CCL) Method
To analyze the receiver code biases, differences between two receivers' ionospheric observables estimated from the CCL method are useful for detecting bias variations. The CCL method is commonly adopted to directly extract the ionospheric observables with code and phase geometry-free (GF) combinations. This geometry-free combination is illustrated in the following [38]: P s r,4 = P s r,2 − P s r,1 = (γ 2 − 1)I s r,1 + (DCB s + δDCB s − (DCB r + δDCB r ) + m s r,P 4 + ε P 4 Φ s r,4 = Φ s r,1 − Φ s r,2 = (γ 2 − 1)I s r,1 + (λ 2 N s r,1 − λ 1 N s r,2 ) + (DPB r + δDPB r ) − (DPB s + δDPB s ) + m s r,Φ 4 where P s r, 4 and Φ s r,4 are the GF combinations for code and phase measurements, respectively; DCB r = d r,1 − d r,2 and DCB s = d s 1 − d s 2 are the constant differential code biases (DCB) for the receiver and the satellite, respectively, δDCB r = δd r,1 − δd r,2 and δDCB s = δd s 1 − δd s 2 are the time-varying terms; DPB r = b r,1 − b r,2 and DPB s = b s 1 − b s 2 are the differential phase biases (DPB) for the receiver and the satellite, respectively, with the time-varying terms δDPB r = δb r,1 − δb r,2 and δDPB s = δb s 1 − δb s 2 . Hence, we formulate the Melbourne-Wübbena (MW) combination in detail to analyze the GF ambiguity as: In the CCL method, the phase measurement can be corrected by the averaged GF ambiguities. The carrier-phase smoothed code measurement can be formulated as: where d leveling denotes the leveling errors, which means the averaged ambiguity bias that is caused by the code biases and the multipath. For one satellite, it is a constant value over a continuous observation arc. We also define the receiver and satellite biases, the multipath and measurement noise terms in detail. The receiver bias is denoted as: Similarly, we denote the satellite biases as: Additionally, we denote the multipath and the measurements noises as: Remote Sens. 2020, 12, 796

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Hence, when considering all bias terms in the formulation, the ionospheric observables obtained in the CCL method include the slant total electron content (STEC), the receiver and satellite biases, leveling errors, and the multipath and measurements noises. The phase-multipath and measurement noise can be neglected because of the high precision of the phase measurements. Here, we also excepted the carrier phase multipath and measurement noise in our following analysis. Code multipath was eliminated due to averaging processing of GF combined ambiguities. As such, Equation (4) can be represented as: L s r,ccl = (γ 2 − 1)I s r,1 + D s ccl − D r,ccl + d leveling (8) Consider that two receivers (marked A and B) create a short or a zero baseline. A single difference that cancels the STEC and satellite bias is built to evaluate the variation of the receiver bias, which mainly covers the receiver code bias, over a continuous observation arc. Hence, this single difference, called the between receiver differential code bias (BR-DCB), is presented as: From Equation (5), we know that if the time-varying part of differential phase bias is insignificant, the BR-DCB in Equation (9) for one satellite over a continuous arc is constant. All observations of satellites are useful to detect the behavior of the receiver code biases [26,29].

Ionosphere-Free Code and Phase Combinations
In ionosphere-free code and phase combinations, the first-order ionospheric delays are removed. We define the following terms as: where α 12 and β 12 are ionosphere-free combination coefficients on each frequency that are used to formulate the ionosphere-free code and phase bias, respectively, for satellite, receiver constant, and time-varying terms. We used the ionosphere-free phase combination minus the code combination to analyze the variation of ionosphere-free code biases and the ionosphere-free code multipath. Here, we also neglected the carrier phase multipath and measurements noise.
We used this difference to analyze the ionosphere-free code bias variations and be free of ionospheric delays effects.

Uncombined PPP Method
The unknown parameters, receiver and satellite clock offsets, code and phase delay biases, and ambiguities in Equation (1) are correlated. Hence, these parameters cannot be simultaneously estimated in a least-squares adjustment. In practice, only the coordinates, the tropospheric zenith wet delay, receiver clock error, ionospheric delays and ambiguity parameters are estimated in the UPPP method. In practice, the satellite clock offsets are corrected by IGS precise clock products. The satellite clock parameters are still exhibited in equations for the convenient analysis of other parameters. The code biases are assimilated into the clock offsets. Due to just one clock offset parameter for code and phase measurements, the ambiguities, code and phase biases are combined as one parameter. Considering the time-varying phase bias, the estimated clock parameters are defined as [16,39] where d t r and d t s are reparametrized clock offsets for the receiver and the satellite, respectively. While ionospheric delays are eliminated by the ionospheric-free combination, these delays have to be estimated in case of raw observations in the UPPP method without the extra ionospheric information constraints [40]. In the UPPP method, the ionospheric delays and ambiguities are reparametrized as where I s r,1 is the reparametrized ionospheric delay parameter and N s r, f is the reparametrized ambiguity parameter. Hence, the S-UPPP model with code and phase measurements can be expressed as [16,39]: where e P, f denotes the code measurement residuals. In Equations (13) and (14), the ionospheric delays and ambiguity parameters are estimated together with the receiver constant code bias. This assumption is based on the fact that the time-varying code bias should be assimilated into the code residuals due to the far-weak weights posed on code measurements. Hence, the time-varying code bias on each frequency is completely assimilated into the code residuals e P, f . However, the code biases d r, f , δd r, f , and ambiguity N s r, f are strongly correlated. The effects on ambiguity estimation of code bias variation should not be neglected. With extreme variations of code bias in a short time, the model in Equation (14) would not achieve an optimal solution. In fact, large variation of code bias in short time degrade the estimation accuracy of ionospheric delays and ambiguities. The unmodeled variations of the receiver code bias in this S-UPPP must be rigorously taken into consideration. Hence, we introduce the M-UPPP model according to the PPP-RTK method [11,12] and the DC model [6].

Modified Uncombined PPP Model
In the S-UPPP model, we assume that the constant code bias is assimilated into the clock offsets and ambiguities, while the time-varying code bias is assimilated into code residuals [16]. To consider the effects of code bias on estimation of clock offsets and ambiguities, the M-UPPP method is introduced to decrease the correlation between code bias and ambiguity. In fact, the hardware biases for satellites are stable enough to meet the previous assumption, while the variability of the receiver code biases is relatively conspicuous. Consequentially, the time-constant character of ambiguities are affected by the variational receiver code biases. Therefore, it is important to separate most code bias components from ambiguities in carrier phase observation. Similar as in the DC method for decoupling the code and phase clock parameters, we isolated receiver code bias from the phase ambiguities to improve the performance of PPP or ambiguity resolution. Here, we define the d r,P as the difference of the code Remote Sens. 2020, 12, 796 7 of 25 and phase receiver clocks, which denotes the relative receiver code bias. Hence, Equation (14) can be expressed as: where the ambiguity term N s r, f is denoted as: where e P, f is the new code residuals in the M-UPPP method. Compared to ambiguity parameter in Equation (13), the ambiguity in Equation (16) is free of the effects of ionosphere-free receiver combined code bias. As already proposed [11,18], if ionospheric delay corrections are available, the receiver DCB can be estimated as an unknown parameter in the UPPP method. Hence, the receiver code bias, DCB r is separated from ambiguities parameters. With ionospheric constrained information, ambiguities are rigorously treated as constant and are free of all receiver code bias content [11,12].

Ambiguity Datum Fixing and Receiver Code Bias Content
Though the code biases in the M-UPPP method have been separated from the ambiguities, the lack of datum for receiver phase clock offset and ambiguities should be taken into consideration. If the ambiguities have been fixed to a determined value, the phase clock offset parameter can be fixed to a specific value without the help of code measurement. With this specific datum of ambiguity, a full-rank solution can be established from the first epoch of the PPP processing. Furthermore, the new receiver phase clock offset is not the real value of the receiver clock offset for phase measurements because of the referenced ambiguity datum shift of satellite s from its real value to an initial value. Hence, the phase clock offset dt r,Φ is denoted as: where the ambiguity shift difference is ∆N between real value N s real and datum value referent N s re f satellite s, and dt r,Φ is the real phase clock offset. Similarly, the code clock offset dt r,P is denoted as: dt r,P = dt r,P + (δb r,i f + d r,i f + δd r,i f )/c (18) where dt r,P is the real code clock offset and d r,i f and δd r,i f denote the constant ionosphere-free combined code bias and its time-varying term, respectively. The code clock offset is set as the combination of phase clock offset and their difference. Hence, the code clock offset is denoted as: where the estimable d r,P is the difference between the code clock offset and phase clock offset that was introduced in Equation (15). Hence, the new combined code bias d r,P mainly contains the ionosphere-free code bias and the ambiguity datum difference. Note that the time-varying code bias term, δd r, f , which is assimilated in code residuals in Equation (14), is estimated in d r,P . The remaining code bias in code residuals is the time-varying receiver DCB, in which fluctuation is less insignificant. The variable quantity of the time-varying code bias d r,P is not determinable because it is dependent on the quality of receiver hardware. Hence, it is estimated as random parameter as the receiver clock offset. For the introduction of estimable receiver code bias and existence of ambiguity datum, the estimable ambiguities are denoted as: where N s est and N i est are the estimable ambiguities for reference satellite s and rover satellite i, respectively. Note that N s est is fixed to the datum value of reference satellite s. With the float ambiguities solution, the FCB products are used to recover the integer nature of ambiguities [2,10,41]. Firstly, the float ambiguities solution is estimated in the Kalman filter by using the UPPP method. Then, the FCB products are used to correct these float ambiguities and recover their integer nature. Based on these corrected ambiguities, the search strategy based on the least-squares ambiguity decorrelation adjustment (LAMBDA) method is applied to search for the optimal integer solution by using the integer least-squares (ILS) estimator [41,42]. Moreover, the ratio-test upon the estimated ambiguities is adopted to decide whether to accept or not the integer solution [43]. Finally, the integer solution of the ambiguities is used to constrain the observation equation as virtual observation. In this study, the empirical threshold of the ratio-test was 2.0. To improve the ambiguity success fixing rate, a partial ambiguity resolution was applied with the LAMBDA method [44].

Data and Experiments
In this study, the datasets were collected from IGS Global Data Center at CDDIS (Crustal Dynamics Data Information System) for FCB estimation, ionospheric observables extraction, and PPP process on 10 April 2017 (DOY 100, 2017). At the server end, 162 globally distributed stations were selected to estimate the FCB corrections that are displayed in Figure 1. With these reference stations data, the uncombined PPP was processed by utilizing GPS L1/L2 and P1/P2 observations with the process strategies listed in Table 1. Based on the data processing results, the float undifferenced and uncombined ambiguities on each station were put together into the FCB estimator. A Kalman filter was used to estimate the FCB products with a 15 min interval by using the linear combinations of uncombined ambiguities by the coefficient (4, −3) and (1, −1) for the dual frequency data [2,9,10,42]. The details of the FCB estimation are presented in Figure 2. Furthermore, without the estimation of FCB product, the published products, e.g., the phase clock/bias products estimated in Wuhan University (ftp://igs.gnsswhu.cn/pub/whu/phasebias), were also useful to directly achieve the ambiguity resolution [45]. With the float ambiguities solution, the FCB products are used to recover the integer nature of ambiguities [2,10,41]. Firstly, the float ambiguities solution is estimated in the Kalman filter by using the UPPP method. Then, the FCB products are used to correct these float ambiguities and recover their integer nature. Based on these corrected ambiguities, the search strategy based on the leastsquares ambiguity decorrelation adjustment (LAMBDA) method is applied to search for the optimal integer solution by using the integer least-squares (ILS) estimator [41,42]. Moreover, the ratio-test upon the estimated ambiguities is adopted to decide whether to accept or not the integer solution [43]. Finally, the integer solution of the ambiguities is used to constrain the observation equation as virtual observation. In this study, the empirical threshold of the ratio-test was 2.0. To improve the ambiguity success fixing rate, a partial ambiguity resolution was applied with the LAMBDA method [44].

Data and Experiments
In this study, the datasets were collected from IGS Global Data Center at CDDIS (Crustal Dynamics Data Information System) for FCB estimation, ionospheric observables extraction, and PPP process on 10 April 2017 (DOY 100, 2017). At the server end, 162 globally distributed stations were selected to estimate the FCB corrections that are displayed in Figure 1. With these reference stations data, the uncombined PPP was processed by utilizing GPS L1/L2 and P1/P2 observations with the process strategies listed in Table 1. Based on the data processing results, the float undifferenced and uncombined ambiguities on each station were put together into the FCB estimator. A Kalman filter was used to estimate the FCB products with a 15 min interval by using the linear combinations of uncombined ambiguities by the coefficient (4, −3) and (1, −1) for the dual frequency data [2,9,10,42]. The details of the FCB estimation are presented in Figure 2. Furthermore, without the estimation of FCB product, the published products, e.g., the phase clock/bias products estimated in Wuhan University (ftp://igs.gnsswhu.cn/pub/whu/phasebias), were also useful to directly achieve the ambiguity resolution [45].      To access the variation of receiver code biases, we calculated the ionospheric observable differences between two stations by using CCL and the UPPP method of a short baseline, as described in Table 2. Without the ionosphere corrections to constrain the ionospheric delay parameters in the uncombined PPP method, the receiver DCB was lumped into the ionospheric delay terms in Equation (13). In a short baseline, we assumed that the ionospheric delay for the same satellite within a similar environment was identical. Hence, the different ionospheric observables between receivers indicated the relative variation of receiver code biases, as presented in Equations (9) and (11). In order to avoid the effect of severe ionospheric disturbances, the disturbance storm index (Dst) and Kp index during this period are presented in Figure 3 and are used to show the geomagnetic activities. In Figure 3, the absolute Dst index maximum is below 16 nT, while the Kp index maximum is below 2, which indicates a relatively quiet geomagnetic activity condition.   Considering the effects of receiver code bias variation on ambiguities and ionospheric delays estimation, the M-UPPP method was introduced to evaluate the improved positioning performance with a large number of globally distributed stations. The data from 220 globally distributed stations (which were not used in FCB estimation), shown in Figure 4, were processed by utilizing the S-UPPP and M-UPPP methods with ambiguity resolution. The 24-h data were divided into three-hour sessions. Each session was treated as an independent arc. Afterwards, the processing was reinitialized. The positioning accuracy, convergence time and ambiguity success fixing rate were evaluated with these data.
For the data processing, the final orbit products with 15 min intervals and clock offset products with 30 s intervals, which were downloaded from the CODE (Center for Orbit Determination in Europe) Analysis Center, were used to keep compatibility. The fractional cycle biases, estimated as shown in Figure 2, were used to achieve the ambiguity resolution [8,10]. The satellite DCBs were corrected by IGS products. It has to be noted that no external ionosphere information was taken into account.  Considering the effects of receiver code bias variation on ambiguities and ionospheric delays estimation, the M-UPPP method was introduced to evaluate the improved positioning performance with a large number of globally distributed stations. The data from 220 globally distributed stations (which were not used in FCB estimation), shown in Figure 4, were processed by utilizing the S-UPPP and M-UPPP methods with ambiguity resolution. The 24-h data were divided into three-hour sessions. Each session was treated as an independent arc. Afterwards, the processing was re-initialized. The positioning accuracy, convergence time and ambiguity success fixing rate were evaluated with these data.  Considering the effects of receiver code bias variation on ambiguities and ionospheric delays estimation, the M-UPPP method was introduced to evaluate the improved positioning performance with a large number of globally distributed stations. The data from 220 globally distributed stations (which were not used in FCB estimation), shown in Figure 4, were processed by utilizing the S-UPPP and M-UPPP methods with ambiguity resolution. The 24-h data were divided into three-hour sessions. Each session was treated as an independent arc. Afterwards, the processing was reinitialized. The positioning accuracy, convergence time and ambiguity success fixing rate were evaluated with these data.
For the data processing, the final orbit products with 15 min intervals and clock offset products with 30 s intervals, which were downloaded from the CODE (Center for Orbit Determination in Europe) Analysis Center, were used to keep compatibility. The fractional cycle biases, estimated as shown in Figure 2, were used to achieve the ambiguity resolution [8,10]. The satellite DCBs were corrected by IGS products. It has to be noted that no external ionosphere information was taken into account.  For the data processing, the final orbit products with 15 min intervals and clock offset products with 30 s intervals, which were downloaded from the CODE (Center for Orbit Determination in Europe) Analysis Center, were used to keep compatibility. The fractional cycle biases, estimated as shown in Figure 2, were used to achieve the ambiguity resolution [8,10]. The satellite DCBs were corrected by IGS products. It has to be noted that no external ionosphere information was taken into account.

Results
Firstly, the code bias variations were analyzed with CCL and the UPPP method by using the single different ionospheric observables. Furthermore, to exclude the multipath effects as reasons for code bias variations, the MW and ionosphere-free combination code multipaths were used. Then, we analyzed the variations of ambiguities and residuals from code and phase measurements that were caused by the code bias in detail with one station data. With the M-UPPP method, the ionospheric observables were reprocessed and compared with that obtained from the S-UPPP method. To further analyze the effects of code biases on ambiguity resolutions from the UPPP method, we accessed 220 permanent station data on 10 April 2017, from the globally distributed IGS network. We processed these data with the S-UPPP and M-UPPP methods. Statistic results were generated to analyze the positioning accuracy with different methods. Meanwhile, comparisons of the convergence time and ambiguity fixing success rate are also presented.

Leveling Errors for Analysis of Receiver Biases
Here, the CCL and S-UPPP methods were used to directly extract the ionospheric observables to analyze the BR-DCB by using the stations in Table 2. For Figure 5, the BR-DCBs were estimated with the CCL method, S-UPPP float-ambiguity solutions, and fixed-ambiguity solutions, which are marked as "CCL," "PPP-float," and "PPP-AR," respectively, over three consecutive days from DOY 100 to 102, 2017. In the CCL solutions, the significant fluctuations in one satellite continuous observation arc indicated that the time-varying amplitude of differential phase biases were noticeable. The main trend also showed the shift of the receiver code bias. In Equation (5), the receiver code bias of each satellite was averaged to remove the multipath effects and fluctuations of the time-varying term. In the CCL method, we could not confirm the details of receiver time-varying code biases. In the S-UPPP method, the ionospheric observables were estimated epoch by an epoch with receiver code bias. For the S-UPPP float solutions, the fluctuation of the BR-DCB is more significant than that in CCL method. In S-UPPP fixed-ambiguity resolution, the ambiguity resolution failed during periods of large variations of code biases. However, as seen in Figure 6, without a large variation of receiver bias, the BR-DCBs in the CCL and S-UPPP solutions was highly consistent and the fixed-ambiguity solutions had the best performance. Compared with the solutions in Figure 6 from DOY 123 to 125, 2017, it can be seen that the fluctuations of the code bias in Figure 5 had clear negative effects for extracting ionospheric delays and ambiguity resolutions in the S-UPPP method. Additionally, the fluctuation of receiver code bias in three consecutive days did not show periodic variations. Some researchers have shown that this variation is caused by local receiver equipment and temperature changes [30,[46][47][48].

Multipath Analysis
In the CCL method, the BR-DCB is free of the code multipath. Hence, we did not confirm that the DCB variations were mainly caused by the code multipath. Free of ionospheric delays effects, the MW combination in Equation (3) and ionosphere-free combination in Equation (11) were adopted to calculate the code multipath. For Figure 7, the satellite PRN 12 was selected to present the MW code multipath over three consecutive sidereal days in stations TSK2 and TSKB. Compared with TSK2, the MW code multipath results in TSKB with large fluctuation indicated that there must have been some bias in the TSKB code multipath. In Equation (3), this variation may have been caused by the receiver phase and code time-varying terms or the code multipath. The effect of the code multipath was mainly created from certain low-elevation directions. The measurements in this experiment were collected from two static IGS receivers. Therefore, the geometry of the satellite-receiver rays repeated every sidereal day. However, in Figure 7, the station TSKB did not show the MP-repeatability of sidereal days like in the TSK2 results. In Figure 8, six satellite results indicate that this variation must have been a receiver-dependent bias.      In Equation (15), the M-UPPP method was mainly used to remove the ionosphere-free code bias from ambiguities. Hence, the ionosphere-free combination was the proper method to analyze the code bias and the multipath. For Figure 9, the satellite PRN 12 was selected to show the ionospherefree code multipath over three consecutive days. Similarly, as in the MW combination, the inconsistent fluctuation in TSKB on each sidereal day indicated that the receiver code bias variation must have existed in this multipath. In Figure 10, six satellite results show consistent trends in one day, and these indicated that the receiver-dependent code bias had to be taken into consideration.  In Equation (15), the M-UPPP method was mainly used to remove the ionosphere-free code bias from ambiguities. Hence, the ionosphere-free combination was the proper method to analyze the code bias and the multipath. For Figure 9, the satellite PRN 12 was selected to show the ionospherefree code multipath over three consecutive days. Similarly, as in the MW combination, the inconsistent fluctuation in TSKB on each sidereal day indicated that the receiver code bias variation must have existed in this multipath. In Figure 10, six satellite results show consistent trends in one day, and these indicated that the receiver-dependent code bias had to be taken into consideration. In Equation (15), the M-UPPP method was mainly used to remove the ionosphere-free code bias from ambiguities. Hence, the ionosphere-free combination was the proper method to analyze the code bias and the multipath. For Figure 9, the satellite PRN 12 was selected to show the ionosphere-free code multipath over three consecutive days. Similarly, as in the MW combination, the inconsistent fluctuation in TSKB on each sidereal day indicated that the receiver code bias variation must have existed in this multipath. In Figure 10, six satellite results show consistent trends in one day, and these indicated that the receiver-dependent code bias had to be taken into consideration. Remote Sens. 2019, 11, x FOR PEER REVIEW 14 of 25  Furthermore, to analyze the effect of the multipath errors, signal-to-noise ratio (SNR) measurements were used to analyze the difference between the different receivers. For Figure 11a, the satellite PRN 6 was selected to present the SNR measurements. The major trends could be removed by polynomial fit so that the multipath errors were mainly reflected in the periodic fluctuations of SNR residuals. In Figure 11b, the SNR residuals, after removing the major trends, had the same distributions and variations. At the same time, the two stations had the same satellite distributions with a distance of less than 36.2 m. With the statistical analysis of all satellite in one day, the SNR residuals of all satellites for the two receivers did not have significantly differences. Hence, the variations of the single different ionospheric observables were mainly caused by receiverdependent code biases.  Furthermore, to analyze the effect of the multipath errors, signal-to-noise ratio (SNR) measurements were used to analyze the difference between the different receivers. For Figure 11a, the satellite PRN 6 was selected to present the SNR measurements. The major trends could be removed by polynomial fit so that the multipath errors were mainly reflected in the periodic fluctuations of SNR residuals. In Figure 11b, the SNR residuals, after removing the major trends, had the same distributions and variations. At the same time, the two stations had the same satellite distributions with a distance of less than 36.2 m. With the statistical analysis of all satellite in one day, the SNR residuals of all satellites for the two receivers did not have significantly differences. Hence, the variations of the single different ionospheric observables were mainly caused by receiverdependent code biases. Furthermore, to analyze the effect of the multipath errors, signal-to-noise ratio (SNR) measurements were used to analyze the difference between the different receivers. For Figure 11a, the satellite PRN 13 was selected to present the SNR measurements. The major trends could be removed by polynomial fit so that the multipath errors were mainly reflected in the periodic fluctuations of SNR residuals. In Figure 11b, the SNR residuals, after removing the major trends, had the same distributions and variations. At the same time, the two stations had the same satellite distributions with a distance of less than 36.2 m. With the statistical analysis of all satellite in one day, the SNR residuals of all satellites for the two receivers did not have significantly differences. Hence, the variations of the single different ionospheric observables were mainly caused by receiver-dependent code biases. Similar variations of the single different receiver code biases in Figure 5 have also been found in previous research [24,29,30]. Variations of about 9 ns in [24] and 6.5 ns in [29] for receiver code bias fluctuations within two consecutive hours were already published. Hence, the fluctuation amplitude of receiver code bias depends on receiver-specific conditions, such as hardware version or temperature variations [23]. The effect on positioning of this code bias variations is analyzed in detail in the following.

The Receiver Biases Effect on PPP with Single Station
The time-varying parts of the code biases are unmodeled errors, a fact that is reflected in the residuals or assimilated into other parameters after adjustment. The TSKB station can be used to analyze the difference between the standard and modified UPPP methods.

Measurements Residuals
With the assumption that the code biases are constant over time, the S-UPPP model cannot capture the receiver clock offset to a level of accuracy that corresponds to phase observations [49]. The receiver clock offset datum is determined by the code observations, but its accuracy is mainly improved by phase measurements. However, constant combined parameters of ambiguities and code biases cannot represent the variations of code biases over time. Consequentially, the unmodeled errors are assimilated into the residuals, and it is improper to represent combined ambiguity parameters as constants. This is shown in Figure 12.
In the S-UPPP method, the obvious variations in residuals destroyed the Gaussian white noise distribution of the code measurement residuals, although the residuals were expected to be zeromean. Meanwhile, the abnormal residuals also showed in phase measurements from 16:00 to 20:00 and at the beginning and ending stages. In the M-UPPP method, the unmodeled errors were estimated, and the code and phase residuals had an ideal distribution with zero-mean, as seen in Figure 13.
In Equation (15), the ionosphere-free combined code biases can be directly estimated by M-UPPP. As can be seen in Figure 14, the estimated code biases from the M-UPPP method showed large variations over 4.8 m within two hours, a trend which is the same as in the code residuals of the S-UPPP method. This variation trend is also presented in the BR-DCB of Figure 5. This indicates that the time-varying code bias actually has a negative effect on ambiguity resolution. Figure 11. The signal-to-noise ratio (SNR) measurements (a) and their residuals after removing the major trends (b) for satellite PRN 13 of TSKB and TSK2. Similar variations of the single different receiver code biases in Figure 5 have also been found in previous research [24,29,30]. Variations of about 9 ns in [24] and 6.5 ns in [29] for receiver code bias fluctuations within two consecutive hours were already published. Hence, the fluctuation amplitude of receiver code bias depends on receiver-specific conditions, such as hardware version or temperature variations [23]. The effect on positioning of this code bias variations is analyzed in detail in the following.

The Receiver Biases Effect on PPP with Single Station
The time-varying parts of the code biases are unmodeled errors, a fact that is reflected in the residuals or assimilated into other parameters after adjustment. The TSKB station can be used to analyze the difference between the standard and modified UPPP methods.

Measurements Residuals
With the assumption that the code biases are constant over time, the S-UPPP model cannot capture the receiver clock offset to a level of accuracy that corresponds to phase observations [49]. The receiver clock offset datum is determined by the code observations, but its accuracy is mainly improved by phase measurements. However, constant combined parameters of ambiguities and code biases cannot represent the variations of code biases over time. Consequentially, the unmodeled errors are assimilated into the residuals, and it is improper to represent combined ambiguity parameters as constants. This is shown in Figure 12.
In the S-UPPP method, the obvious variations in residuals destroyed the Gaussian white noise distribution of the code measurement residuals, although the residuals were expected to be zero-mean. Meanwhile, the abnormal residuals also showed in phase measurements from 16:00 to 20:00 and at the beginning and ending stages. In the M-UPPP method, the unmodeled errors were estimated, and the code and phase residuals had an ideal distribution with zero-mean, as seen in Figure 13.
In Equation (15), the ionosphere-free combined code biases can be directly estimated by M-UPPP. As can be seen in Figure 14, the estimated code biases from the M-UPPP method showed large variations over 4.8 m within two hours, a trend which is the same as in the code residuals of the S-UPPP method. This variation trend is also presented in the BR-DCB of Figure 5. This indicates that the time-varying code bias actually has a negative effect on ambiguity resolution. Remote Sens. 2019, 11, x FOR PEER REVIEW 16 of 25

Ambiguity Solutions
At the same time, the isolation of the code biases from ambiguity recovers the strongly stable feature of ambiguities that can be estimated as constant parameters. In Figure 15

Ambiguity Solutions
At the same time, the isolation of the code biases from ambiguity recovers the strongly stable feature of ambiguities that can be estimated as constant parameters. In Figure 15

Ambiguity Solutions
At the same time, the isolation of the code biases from ambiguity recovers the strongly stable feature of ambiguities that can be estimated as constant parameters. In Figure 15     From the static solution, the effects of varying receiver code biases in the UPPP method could be confirmed. These effects were also found in the positioning solutions of the dynamic mode in Figure 17. Significant improvements were found in the M-UPPP method. The large positioning errors were caused by the variation of receiver code biases. The variation of positioning errors was consistent with the variation of the code bias that is shown in Figure 14. The large fluctuations of positioning errors and code bias started at about 16:00. This indicates that the estimation of the ambiguities and coordinate parameters was strongly affected by the large code bias variations in the dynamic positioning mode. Unlike in the static model, the positioning errors were recovered to a regular level along with diminishing receiver code bias variations. The maximum of the positioning errors was about 3.9 m from 16:00 to 20:00.

Extraction of Ionospheric Observable from M-UPPP
With the above analysis, the accuracy of ionospheric observables that were extracted from the S-UPPP method was degraded by the time-varying receiver code bias. By using the M-UPPP method, we isolated the ionosphere-free combined receiver code bias from ambiguities and ionospheric delays in order to improve the positioning results in TSKB. Hence, the ionospheric observables estimated in the M-UPPP method should be improved. In Figure 18, the new BR-DCB results are presented to illustrate the high consistency between the CCL and M-UPPP methods in extracting the ionospheric observables. Though the time-varying differential phase bias remained in the ionospheric From the static solution, the effects of varying receiver code biases in the UPPP method could be confirmed. These effects were also found in the positioning solutions of the dynamic mode in Figure 17. Significant improvements were found in the M-UPPP method. The large positioning errors were caused by the variation of receiver code biases. The variation of positioning errors was consistent with the variation of the code bias that is shown in Figure 14. The large fluctuations of positioning errors and code bias started at about 16:00. This indicates that the estimation of the ambiguities and coordinate parameters was strongly affected by the large code bias variations in the dynamic positioning mode. Unlike in the static model, the positioning errors were recovered to a regular level along with diminishing receiver code bias variations. The maximum of the positioning errors was about 3.9 m from 16:00 to 20:00. From the static solution, the effects of varying receiver code biases in the UPPP method could be confirmed. These effects were also found in the positioning solutions of the dynamic mode in Figure 17. Significant improvements were found in the M-UPPP method. The large positioning errors were caused by the variation of receiver code biases. The variation of positioning errors was consistent with the variation of the code bias that is shown in Figure 14. The large fluctuations of positioning errors and code bias started at about 16:00. This indicates that the estimation of the ambiguities and coordinate parameters was strongly affected by the large code bias variations in the dynamic positioning mode. Unlike in the static model, the positioning errors were recovered to a regular level along with diminishing receiver code bias variations. The maximum of the positioning errors was about 3.9 m from 16:00 to 20:00.

Extraction of Ionospheric Observable from M-UPPP
With the above analysis, the accuracy of ionospheric observables that were extracted from the S-UPPP method was degraded by the time-varying receiver code bias. By using the M-UPPP method, we isolated the ionosphere-free combined receiver code bias from ambiguities and ionospheric delays in order to improve the positioning results in TSKB. Hence, the ionospheric observables estimated in the M-UPPP method should be improved. In Figure 18, the new BR-DCB results are presented to illustrate the high consistency between the CCL and M-UPPP methods in extracting the ionospheric observables. Though the time-varying differential phase bias remained in the ionospheric

Extraction of Ionospheric Observable from M-UPPP
With the above analysis, the accuracy of ionospheric observables that were extracted from the S-UPPP method was degraded by the time-varying receiver code bias. By using the M-UPPP method, we isolated the ionosphere-free combined receiver code bias from ambiguities and ionospheric delays in order to improve the positioning results in TSKB. Hence, the ionospheric observables estimated in the M-UPPP method should be improved. In Figure 18, the new BR-DCB results are presented to illustrate the high consistency between the CCL and M-UPPP methods in extracting the ionospheric observables. Though the time-varying differential phase bias remained in the ionospheric observables, it was quite smaller than the code bias. Additionally, we also obtained the best performance with the ambiguity resolution. observables, it was quite smaller than the code bias. Additionally, we also obtained the best performance with the ambiguity resolution.

Receivers Biases Effects on PPP in Statistics Solutions
It has been proven that the M-UPPP method can improve positioning performance in one station, TSKB, from positioning experiments in the static and dynamic modes. To further analyze the effects of this method, we processed all 220 stations for statistic performance.

The Performance of Static Positioning
The histogram distribution, which was constructed from the mean value of different UPPP positioning results with a total of 1760 three-hour sessions, is displayed in Figure 19. The significant improvements are presented in the PPP AR, as compared with the float solutions. Compared with the S-UPPP method, the M-UPPP method had significant improvements for the float and fixed ambiguity solutions. The differences between the S-UPPP and M-UPPP methods are presented in Table 3. Compared with the standard UPPP AR, the accuracy was improved from 1.77 to 1.45 cm for the M-UPPP AR.

Receivers Biases Effects on PPP in Statistics Solutions
It has been proven that the M-UPPP method can improve positioning performance in one station, TSKB, from positioning experiments in the static and dynamic modes. To further analyze the effects of this method, we processed all 220 stations for statistic performance.

The Performance of Static Positioning
The histogram distribution, which was constructed from the mean value of different UPPP positioning results with a total of 1760 three-hour sessions, is displayed in Figure 19. The significant improvements are presented in the PPP AR, as compared with the float solutions. Compared with the S-UPPP method, the M-UPPP method had significant improvements for the float and fixed ambiguity solutions. The differences between the S-UPPP and M-UPPP methods are presented in Table 3. Compared with the standard UPPP AR, the accuracy was improved from 1.77 to 1.45 cm for the M-UPPP AR.  The positioning performance was evaluated at the 68% and 95% confidence levels in the static PPP model for the S-UPPP and M-UPPP methods in Figure 20. Inevitably, the fixed-ambiguity solutions had higher accuracies than the float solutions in most of periods. Compared with the S-UPPP method, significant improvements could be found over the processing period, especially in the initial phase. Furthermore, the improvements in the horizontal component were more significant than those in the vertical component. With the modified methods, a fast convergence could be achieved for the fixed-ambiguity solutions. The ambiguity fixing success rate is defined as the ratio of the number of the fixed solutions to that of all solutions in one epoch. The fixed-ambiguity resolution in one epoch can be confirmed by a ambiguity resolution ratio that is larger than 2 and a difference of the absolute fixed positioning errors minus the float positioning errors of less than 3 cm. In Figure 21, similar slight improvements are presented for UPPP methods. In Table 4, the significant improvements in terms of convergence time are presented. Here, we define the convergence time as to when the averaged three-dimensional RMS was smaller than 10 cm in the static mode, smaller than 20 cm in the dynamic mode, and lasting for 10 continuous epochs [18,50,51]. Compared with the S-UPPP methods with the fixed-ambiguity solutions in which the convergence times were 90 and 17 min under the 95% and 68% confidence levels, it took 43 and 14 min for the M-UPPP method to converge to the defined accuracy. The convergence times of the float solutions were 111 and 36 min for the S-UPPP method and 81.5 and 31 The positioning performance was evaluated at the 68% and 95% confidence levels in the static PPP model for the S-UPPP and M-UPPP methods in Figure 20. Inevitably, the fixed-ambiguity solutions had higher accuracies than the float solutions in most of periods. Compared with the S-UPPP method, significant improvements could be found over the processing period, especially in the initial phase. Furthermore, the improvements in the horizontal component were more significant than those in the vertical component. With the modified methods, a fast convergence could be achieved for the fixed-ambiguity solutions. The positioning performance was evaluated at the 68% and 95% confidence levels in the static PPP model for the S-UPPP and M-UPPP methods in Figure 20. Inevitably, the fixed-ambiguity solutions had higher accuracies than the float solutions in most of periods. Compared with the S-UPPP method, significant improvements could be found over the processing period, especially in the initial phase. Furthermore, the improvements in the horizontal component were more significant than those in the vertical component. With the modified methods, a fast convergence could be achieved for the fixed-ambiguity solutions. The ambiguity fixing success rate is defined as the ratio of the number of the fixed solutions to that of all solutions in one epoch. The fixed-ambiguity resolution in one epoch can be confirmed by a ambiguity resolution ratio that is larger than 2 and a difference of the absolute fixed positioning errors minus the float positioning errors of less than 3 cm. In Figure 21, similar slight improvements are presented for UPPP methods. In Table 4, the significant improvements in terms of convergence time are presented. Here, we define the convergence time as to when the averaged three-dimensional RMS was smaller than 10 cm in the static mode, smaller than 20 cm in the dynamic mode, and lasting for 10 continuous epochs [18,50,51]. Compared with the S-UPPP methods with the fixed-ambiguity solutions in which the convergence times were 90 and 17 min under the 95% and 68% confidence levels, it took 43 and 14 min for the M-UPPP method to converge to the defined accuracy. The convergence times of the float solutions were 111 and 36 min for the S-UPPP method and 81.5 and 31 The ambiguity fixing success rate is defined as the ratio of the number of the fixed solutions to that of all solutions in one epoch. The fixed-ambiguity resolution in one epoch can be confirmed by a ambiguity resolution ratio that is larger than 2 and a difference of the absolute fixed positioning errors minus the float positioning errors of less than 3 cm. In Figure 21, similar slight improvements are presented for UPPP methods. In Table 4, the significant improvements in terms of convergence time are presented. Here, we define the convergence time as to when the averaged three-dimensional RMS was smaller than 10 cm in the static mode, smaller than 20 cm in the dynamic mode, and lasting for 10 continuous epochs [18,50,51]. Compared with the S-UPPP methods with the fixed-ambiguity solutions in which the convergence times were 90 and 17 min under the 95% and 68% confidence levels, it took 43    The dynamic experiments were processed with same data as in static experiments. Similar improvements to those in the static mode are presented in Figure 22. The improvement in the horizontal component was more significant than that in the vertical component. Slight improvements are also presented in Figure 23 for the M-UPPP method. The improvements in terms of convergence time are also shown in Table 5. For the dynamic mode, the threshold value of the convergence time was defined as 20 cm. Compared with the S-UPPP methods with the fixed-ambiguity solutions in which the convergence times were 70.5 and 20 min under the 95% and 68% confidence levels, it took 55.5 and 16 min for the M-UPPP method to converge to the defined accuracy, respectively. The convergence times of the float solutions were 89.5 and 34.5 min for the S-UPPP method and 77 and 28 min for the M-UPPP method under the 95% and 68% confidence levels, respectively. The M-PPP method also significantly improved the dynamic positioning accuracy and shortened the convergence time.

The Positioning Performance in Dynamic Mode
The dynamic experiments were processed with same data as in static experiments. Similar improvements to those in the static mode are presented in Figure 22. The improvement in the horizontal component was more significant than that in the vertical component. Slight improvements are also presented in Figure 23 for the M-UPPP method. The improvements in terms of convergence time are also shown in Table 5. For the dynamic mode, the threshold value of the convergence time was defined as 20 cm. Compared with the S-UPPP methods with the fixed-ambiguity solutions in which the convergence times were 70.5 and 20 min under the 95% and 68% confidence levels, it took 55.5 and 16 min for the M-UPPP method to converge to the defined accuracy, respectively. The convergence times of the float solutions were 89.5 and 34.5 min for the S-UPPP method and 77 and 28 min for the M-UPPP method under the 95% and 68% confidence levels, respectively. The M-UPPP method also significantly improved the dynamic positioning accuracy and shortened the convergence time.   The dynamic experiments were processed with same data as in static experiments. Similar improvements to those in the static mode are presented in Figure 22. The improvement in the horizontal component was more significant than that in the vertical component. Slight improvements are also presented in Figure 23 for the M-UPPP method. The improvements in terms of convergence time are also shown in Table 5. For the dynamic mode, the threshold value of the convergence time was defined as 20 cm. Compared with the S-UPPP methods with the fixed-ambiguity solutions in which the convergence times were 70.5 and 20 min under the 95% and 68% confidence levels, it took 55.5 and 16 min for the M-UPPP method to converge to the defined accuracy, respectively. The convergence times of the float solutions were 89.5 and 34.5 min for the S-UPPP method and 77 and 28 min for the M-UPPP method under the 95% and 68% confidence levels, respectively. The M-PPP method also significantly improved the dynamic positioning accuracy and shortened the convergence time.

Discussion
A high precision ambiguity solution is the main goal of ambiguity resolution in the PPP AR. In this study, we focused on the effect of receiver code biases in ambiguity resolution. With more reliable and stable ambiguity solutions, a faster PPP AR was achieved with a higher ambiguity fixing success rate. Long-term and short-term variations of receiver code biases have been demonstrated [23,24,[29][30][31]35,36]. Due to the difficulty in the isolation of receiver code biases from observables and their receiver-dependent features, many researchers have not studied their absolute values and their effects on positioning performance. With the introduced method, the variation of receiver code bias can be directly estimated. Its effects on ambiguity resolution can be removed, which benefits the PPP AR. In Figure 10, it can be seen that variation of code bias was up to 4.8 m within the two-hour data for the PPP AR. After removing the code bias, the ambiguity parameters quickly achieved stable solutions. The improved performance of positioning could be found in the static and dynamic modes for the M-UPPP method. The variations of receiver code bias are dependent on the quality of equipment and on location environment. The effect of code bias on a single receiver is detailed here to present the effective performance of eliminating the effect of receiver code bias on positioning and ambiguity resolution. This method has been confirmed that it is effective in static and dynamic mode for PPP. Hence, the performance of the PPP AR in real-time applications should be further studied in next work.

Conclusions
This study introduced a modified method to mitigate the short-term temporal variations of receiver code bias for an increased ambiguity success fixing rate in PPP and improving positioning performance. We decoupled the receiver clock offset parameters with ambiguities and isolated the code biases from ambiguities. The effects of code biases can be removed from ambiguities, and the integer nature of ambiguities are easier to be recovered. We performed experiments by using the standard and modified methods in the UPPP model to present the code biases and evaluate the improvement in terms of convergence time and success rate of ambiguity resolution in static and dynamic PPP.

Discussion
A high precision ambiguity solution is the main goal of ambiguity resolution in the PPP AR. In this study, we focused on the effect of receiver code biases in ambiguity resolution. With more reliable and stable ambiguity solutions, a faster PPP AR was achieved with a higher ambiguity fixing success rate. Long-term and short-term variations of receiver code biases have been demonstrated [23,24,[29][30][31]35,36]. Due to the difficulty in the isolation of receiver code biases from observables and their receiver-dependent features, many researchers have not studied their absolute values and their effects on positioning performance. With the introduced method, the variation of receiver code bias can be directly estimated. Its effects on ambiguity resolution can be removed, which benefits the PPP AR. In Figure 10, it can be seen that variation of code bias was up to 4.8 m within the two-hour data for the PPP AR. After removing the code bias, the ambiguity parameters quickly achieved stable solutions. The improved performance of positioning could be found in the static and dynamic modes for the M-UPPP method. The variations of receiver code bias are dependent on the quality of equipment and on location environment. The effect of code bias on a single receiver is detailed here to present the effective performance of eliminating the effect of receiver code bias on positioning and ambiguity resolution. This method has been confirmed that it is effective in static and dynamic mode for PPP. Hence, the performance of the PPP AR in real-time applications should be further studied in next work.

Conclusions
This study introduced a modified method to mitigate the short-term temporal variations of receiver code bias for an increased ambiguity success fixing rate in PPP and improving positioning performance. We decoupled the receiver clock offset parameters with ambiguities and isolated the code biases from ambiguities. The effects of code biases can be removed from ambiguities, and the integer nature of ambiguities are easier to be recovered. We performed experiments by using the standard and modified methods in the UPPP model to present the code biases and evaluate the improvement in terms of convergence time and success rate of ambiguity resolution in static and dynamic PPP.
Based on the analysis of the measurement residuals, remarkable short-term temporal variations of the code biases up to 16 ns within two hours have been detected. These unmodeled errors also affect the phase measurement residuals. With the improved method, the code biases are removed from ambiguities and the ambiguities become more stable. For some special stations, the improvement of the positioning accuracy for the ambiguity float and fixed solutions is significant. It is valuable to rigorously take variations of code biases into consideration in UPPP processing.
With the observations from the IGS network, the improvement in terms of convergence time and success rate of ambiguity resolution are presented. Significant improvements of final ambiguity-float and ambiguity-fixed solutions for the M-UPPP method were achieved with three-hour data. In the S-UPPP method, the convergence times were 36 and 17 min for ambiguity float and fixed solutions, respectively, and these were 31 and 14 min in the M-UPPP method under the 68% confidence level. Higher ambiguity fixing success rates were also achieved at different times for the M-UPPP method. Under the 95% confidence level, the same trends are also clearly presented. By the dynamic PPP processing, similar improvements were found for the M-UPPP method. Hence, the improvement was significant for isolating the code biases from ambiguities and crucial to achieving a rapid ambiguity-fixed solution. When it comes to using the satellite clock offsets and phase biases products, the effects of code biases in network solutions should also be taken into heavier consideration, as will be done in future work.
Author Contributions: J.W., G.H. and P.Z. conceived and designed the experiments; J.W. performed the experiments, analyzed the data and wrote the paper; Y.Y., Q.Z. and Y.G. reviewed the paper. All authors have read and agreed to the published version of the manuscript.