A static precise single-point positioning method based on carrier phase zero-baseline self-differencing

Satellite navigation positioning has become an indispensable component of everyday life, where precise pinpointing and rapid convergence are crucial in delivering timely and accurate location information. However, due to the damping of integer ambiguities and system residual errors, the rapid convergence of Precise Point Positioning (PPP) implementation is a significant challenge. To address this, this paper proposes a novel Carrier Phase Zero-Baseline Self-Differencing Precise Point Positioning (CZS-PPP) technique and its ionosphere-free fusion model. By employing the proposed CZS-PPP approach in separate scenarios involving BDS-3, GPS, and dual-system settings, we systematically validate the efficacy of the method. The experimental results indicate that the convergence time of the method is less than 4 min in a single-system scenario. Furthermore, in a dual-system scenario, the method can achieve rapid convergence in less than 3 min. The CZS-PPP technique presented demonstrates the elimination of integer ambiguities and the effective suppression of system residuals, in comparison to the conventional method. The proposed approach has demonstrated remarkable performance across different systems, offering a promising new pathway for achieving PPP fast convergence in BDS/GNSS.


CZS-PPP principle
The core principle entails employing itself as a reference, essentially considering the epochs before and after as the reference station and the mobile station.This involves a static baseline length of zero and a dynamic baseline length approaching zero, forming a zero-baseline self-differencing mode.Changes in position relative to the previous epoch are computed by analyzing the carrier phase variations between the current and preceding epochs.This involves subtracting the carrier phase from the previous epoch from the carrier phase of the current epoch.Subsequently, the self-differencing values for carrier phase at each frequency are calculated.The coordinates are then determined using a dual-frequency ionosphere-free combination model.
The basic procedure of our method is as follows: In the first step, Single-Point Positioning (SPP) is used to determine the receiver coordinates at the initial epoch.In the second step, taking the receiver coordinates at the initial epoch as a virtual station, epoch differencing is performed between the second epoch and the initial epoch.The solution provides the coordinate changes relative to the virtual station, so the receiver coordinates at the second epoch are obtained by adding the virtual station coordinates and the coordinate changes.In the third step, the virtual station is updated, taking the receiver coordinates at the second epoch as the new virtual station.Similarly, epoch differencing is performed between the third epoch and the second epoch to obtain new coordinate changes and the receiver coordinates at the third epoch, updating the virtual station.This process is iteratively applied to subsequent epochs until convergence to the true receiver coordinates is achieved.The carrier phase observation equation (converted to distance), as shown in the equation, is used.
In the above equation, L i represents the carrier phase observations (converted to distance), where the subscript i denotes the carrier frequency.ρ is the satellite-to-ground geometric distance, δt k represents the receiver clock bias, δt j represents the satellite clock bias, and c is the speed of light, all in units of seconds.G h and G w are the tropospheric zenith direction dry and wet mapping functions, respectively.T h represents the zenith direction dry delay component, obtained from the Saastamoinen model 31 .T w represents the zenith direction wet delay component and is treated as an estimated parameter alongside position parameters.I i represents the ionospheric delay on L i .ε L i represents the observation noise for carrier phase, all in units of meters.N i represents the integer ambiguity on L i .
The Eq. ( 1) is differenced between consecutive epochs to obtain the zero-baseline self-differencing equation (converted to distance) as follows: In the equation, represents the differencing operator between consecutive epochs.The carrier phase zerobaseline self-differencing eliminates the integer ambiguity parameters.�ρ represents the change in satellite-to- receiver geometric distance between consecutive epochs, and �ρ is defined as follows: (1) In the equation, t and t − 1 represent two consecutive epochs; t | represents the unit direction vector from the receiver to the satellite; R s t represents the satellite coordinate vector, and R r t represents the receiver coordinate vector.Additionally, R r t = R r t−1 + R , where R represents the change in receiver coordinates.There- fore, �ρ can be expressed as follows: Substituting Eq. ( 4) into Eq.( 2) yields: and calculate its error equation as: where l is: X J , Y J , Z J represents the satellite coordinates, while X 0 K , Y 0 K , Z 0 K represents the approximate coordinates of the receiver.
In Eqs.(6), v represents the residual vector, l denotes the known parameters, and the parameters to be esti- mated include the changes in receiver coordinates R for both before and after epochs, the variation in tropo- spheric zenith wet delay component T w , and the receiver clock bias change �δt k .When a sufficient number of visible satellites is available, these parameters can be estimated through the least-squares method or Kalman filtering.

Reconstructed Doppler method
Presently, the unprocessed Doppler observations obtained from receivers display substantial noise, making them unsuitable for accurate GNSS positioning or velocity estimation due to elevated error levels and diminished accuracy.As a result, a method has been suggested to reconstruct Doppler frequency shift values, aiming to produce noise-free theoretical values.The reconstructed alterations in carrier phase demonstrate remarkable precision and are devoid of cycle slips.
The reconstructed Doppler frequency shift values depend on various parameters, such as satellite positions, satellite clock biases, tropospheric delays, ionospheric delays, receiver velocity, receiver position, and receiver clock biases, among others.These parameters constitute the variables to be estimated, establishing a theoretical basis for the pseudorange single-point positioning model utilizing reconstructed Doppler values.
For satellite positioning, satellites orbit around the Earth, making them mobile signal sources.When extending the formula to single-point positioning, one can derive the reconstructed Doppler frequency shift value, denoted as f d .In Fig. 1, the observation vector − → ρ of a satellite points from the receiver to the satellite, while the − → v vector represents the receiver's velocity.Let − → v s = (x vs , y vs , z vs ) be the satellite velocity, P s = (x s , y s , z s ) be the satellite position, and P u = (x u , y u , z u ) be the receiver's position, obtained through pseudorange positioning.Therefore, � ρ = (x ρ , y ρ , z ρ ) = x s − x u , y s − y u , z s − z u .The reconstructed Doppler value can be calculated as follows: In the equation, l is the unit observation vector, � l = � ρ/|� ρ| , where |� ρ| represents the satellite-to-earth geometric distance, is the wavelength.The dot symbol ". "denotes vector dot product.
Due to the presence of cycle slips, clock jumps, and gross errors in traditional carrier phase-smoothed pseudorange methods, this paper introduces a method to calculate the carrier phase change quantity • ij (converted to distance) through the integration of the reconstructed Doppler frequency shift values.This method can be expressed as follows: In the equation, represents the wavelength of the carrier signal transmitted by the satellite, t i and t j are two epochs in time, f d (t i ) and f d t j represent the reconstructed Doppler frequency shift values at the two epochs, i and j denote the carrier phase observations at the two epochs, and • ij represents the carrier phase change between the two epochs.The time difference between t j and t i represents the time interval between the two epochs.This approach effectively mitigates the effects of carrier phase observations including cycle slips, clock jumps, and gross errors.
Moreover, considering the carrier phase change between epochs as an indicator of the pseudorange rate, and in an ideal scenario where factors like ionospheric effects, tropospheric effects, and receiver clock bias are disregarded, the theoretical equivalence of pseudorange change and carrier phase change between the same epochs holds true.Nevertheless, in real-world scenarios, owing to diverse errors and observational noise, the carrier phase change exhibits greater precision compared to the pseudorange change.Consequently, incorporating the reconstructed Doppler frequency shift values to substitute the pseudorange change can result in a more smoothed pseudorange and enhanced pseudorange accuracy, as outlined below: In the equation, δρ j t i , t j represents the change in pseudorange between two epochs, t i and t j .The initial epoch positioning relies on single-point pseudorange positioning, where pseudorange measurements are susceptible to significant noise.This noise is a random variable conforming to a zero-mean normal distribution, typically ranging between 5 and 10 m.Consequently, mitigating pseudorange noise becomes imperative.While traditional carrier phase-smoothed pseudorange techniques 32 with Doppler observations show no signs of cycle slips, Doppler observations from GNSS receivers often harbor considerable noise, leading to heightened errors and diminished accuracy.In light of this, a novel approach based on reconstructed Doppler for carrier phase-smoothed pseudorange is proposed to effectively suppress pseudorange noise and enhance accuracy.The underlying principle of this method is elucidated below: The pseudorange observations at the first epoch can be deduced by integrating the carrier phase change quantities obtained from different epochs' reconstructed Doppler values.A smoothing window of length k is defined, which means there are k observations within the window: ρ j (t 1 ) , ρ j (t 2 ) , …, ρ j (t k ) .By utilizing the inte- grated reconstructed Doppler frequency shift values, carrier phase change quantities δρ j (t 1 , t 2 ) , δρ j (t 1 , t 3 ) , …, δρ j (t 1 , t k ) can be calculated from t 1 to t k and retroactively applied to the first epoch.Consequently, k pseudorange observations are available for the first epoch, namely: As the sliding window encompasses k epochs, the pseudorange values computed for these k epochs are averaged to derive the smoothed pseudorange value for the initial epoch:

CZS-PPP ionosphere-free combination model
The conventional PPP method employs the dual-frequency ionosphere-free model (PPP-IF), where the combined model of pseudorange and carrier phase observations is as follows:  In eqs.( 16) and ( 17), P IF represents the ionosphere-free combined observations of dual-frequency pseu- dorange observations P 1 and P 2 ; L IF represents the ionosphere-free combined observations of dual-frequency carrier phase observations L 1 and L 2 ; f 1 and f 2 are the frequencies of the observations; δt k denotes the receiver clock bias; δt j represents the satellite clock bias; c is the speed of light in vacuum; G h and G w are the mapping functions for dry and wet tropospheric delays, respectively; T h and T w are the dry and wet tropospheric delays; N IF is the integer ambiguity of the ionosphere-free combined carrier phase observations; IF is the wavelength of the ionosphere-free combined observations; ε P IF and ε L IF denote the noise errors of the pseudorange and carrier phase observations, respectively.
The dual-frequency ionosphere-free combination model 33 for CZS-PPP is presented as follows: In Eqs. ( 18) and ( 19), P IF and L IF represent the ionosphere-free linear combinations of pseudorange obser- vations and carrier phase self-differencing values on BDS-3 satellites B1C and B2A, respectively.ρ is the geo- metric distance from the receiver to the satellite, and �ρ represents the change in satellite-to-receiver geometric distance between the previous and current epochs.G h and G w are the tropospheric zenith direction dry and wet mapping functions, respectively.T h is the change in tropospheric zenith direction dry delay component between the previous and current epochs, and T w is the change in tropospheric zenith direction wet delay component between the previous and current epochs.T w is treated as an estimated parameter along with position parameters.ε P IF represents the pseudorange observation noise for the ionosphere-free combination, while �ε L IF represents the phase observation noise for the ionosphere-free combination between the previous and current epochs.All the above units are in meters.c denotes the speed of light.�δt k is the change in receiver clock bias between the previous and current epochs, and �δt j is the change in satellite clock bias between the previous and current epochs, both in seconds.Other error terms in the observation equation (such as antenna phase center corrections 34 , phase wind-up effects 35 , tidal loading deformations 36 , relativistic effects 37 , and Earth rotation 38 ) are corrected using respective models.The dual-frequency ionosphere-free combination model offers the advantages of eliminating first-order ionospheric effects with fewer estimated parameters, resulting in stable positioning performance, a simple model, and ease of operation.
In our research, we utilized L 1 and L 2 dual-frequency observations from the BDS-3 and GPS dual-systems 39 to compute the intermediate-frequency combinations.The ionosphere-free combination equation, established based on pseudorange observations and carrier phase self-differencing values, can be expressed as: ( 16) www.nature.com/scientificreports/ In the equations, represents the differencing operator between the previous and current epochs.The superscripts g and b denote GPS and BDS-3 satellites, respectively.δt b,g sys represents the system time difference parameter between GPS and BDS-3, measured in seconds.In the dual-constellation PPP model combining GPS and BDS-3, in this paper, GPS time is used as the reference and BDS-3 time is converted to GPS time.In addition, we include the parameter �δt b,g sys in the set of parameters to be determined for estimating the system clock difference between GPS and BDS-3.The dual-frequency ionosphere-free combination model consists of five parameters: position changes ( x, y, z ), receiver clock bias �δt k , tropospheric zenith wet component T w , and system time difference parameter �δt b,g sys .
In this paper, a conventional random model is used, and the noise ( σ ) in the observation equation is weighted according to the elevation angle.
In the Eq. ( 25), ele is the elevation angle.σ 0 is commonly set to 1 m for pseudorange observations and 1 cm for phase observations.The random model is as follows: In the Eq. ( 26), w signifies the weight matrix of the observations, while R denotes the covariance matrix associated with the observations.Due to the different observation accuracies of GPS and BDS, the corresponding observation weights are also different.In this paper the observation weights of GPS and BDS are set to 1:2.

Algorithm validation and performance analysis
The method in this paper is applicable to static mode, in order to verify the effectiveness of the above CZS-PPP algorithm for static data processing, experiments will be carried out utilizing a Beidou Xingtong receiver at the Intelligent Navigation and Remote Sensing Research Center of Xiangtan University.All receivers employed in these experiments will utilize both the GPS and BDS-3 satellite navigation systems for precise positioning.
As depicted in Fig. 5, the algorithmic experiments employed a dual-frequency GNSS receiver (BeiDou Xingtong) positioned on the rooftop of the Information Building at Xiangtan University to capture satellite signals.
( www.nature.com/scientificreports/This setup significantly reduced multipath effects, which could be considered negligible.The Earth Centered, Earth Fixed (ECEF) coordinates of its antenna were accurately measured using standard precision instruments, achieving an accuracy better than 3 mm.Original observation data spanning days 017-023 of the year 2023, totaling 7 days, was acquired.The data was sampled at 1-s intervals, encompassing simultaneous reception of BDS-3 and GPS satellite data.Raw observation data was collected daily from 9:00 to 10:00, over a one-hour period.The convergence criteria in this paper are as follows: achieving convergence thresholds simultaneously in the East (E), North (N), and Up (U) directions for PPP, and ensuring stability within the thresholds for 120 consecutive epochs (with a sampling interval of 60 min).The convergence threshold is set at 10 cm, and the convergence time is measured from the initial epoch to reaching the convergence threshold.The experiment utilized dualfrequency raw observation data from BDS-3 and GPS satellite systems to validate the superiority of CZS-PPP.

Experiment 1
In conditions where only a single system was considered, distinct experiments were conducted to validate and compare the individual BDS-3 system 40,41 and the standalone GPS system.Precise satellite coordinates and accurate satellite clock biases were sourced from products provided by the International GNSS Service (IGS) center.CZS-PPP utilized 5-min precise satellite coordinates and 30-s precise satellite clock bias products from the German Research Centre for Geosciences (GFZ).Raw observations in dual frequency, encompassing BDS-3 B1C, B2A 42,43 , and GPS L1C, L2W, were collected over seven consecutive days, from the 17th to the 23rd day of the year 2023, as acquired by the receiver.Figure 6 showcases the positioning results for a single BDS-3 system.In the diagram, (E, N, U) represent a specific station coordinate system, known as the East-North-Up (ENU) coordinate system.The ENU coordinate system captures the positional changes of the receiver in the East (E), North (N), and Up (U) directions.The error curves in the figure depict the absolute differences between the coordinates obtained through CZS-PPP calculations and the receiver's standard coordinates in the ENU coordinate systems.Over time, convergence is achieved.At 217 s, the convergence reaches 0.07 m in the E direction, 0.06 m in the N direction, and 0.08 m in the U direction.Figure 8 depicts the positioning results for a single GPS system.Over time, convergence is attained, reaching 0.03 m in the E direction, 0.08 m in the N direction, and 0.09 m in the U direction at 201 s.Following convergence, the positioning accuracy remains consistently stable.In contrast, Fig. 9 illustrates that the number of GPS satellites is approximately equivalent to the number of BDS-3 satellites.
Table 1 presents information on the convergence time and three-dimensional RMS positioning accuracy in the ENU coordinates for both the individual BDS-3 satellite system and the standalone GPS satellite system.Based on continuous observations spanning 7 days, it is evident that the average convergence time for the single BDS-3 system is 220 s, accompanied by an average ENU three-dimensional RMS positioning accuracy of 0.08 m.In comparison, the single GPS system demonstrates an average convergence time of 214 s, with an average ENU three-dimensional RMS positioning accuracy of 0.076 m.The two satellite systems display comparable average convergence times and average positioning accuracy.In the context of the BDS-3/GPS dual-satellite system 44,45 , the positioning results of CZS-PPP are compared with those of the widely employed PPP method.The conventional PPP method employs the PPP-IF model 46,47 .Both methodologies utilize identical observational data and precise ephemeris.A comparative analysis is conducted using data continuously collected for 7 days, drawn from the dual-frequency raw data received by the receiver over the consecutive 7 days, spanning from day 17 to day 23 in 2023.In Fig. 10, in the context of a dual-satellite navigation system, a notable increase in the count of visible satellites is depicted, leading to an enhanced satellite spatial geometry.
In Fig. 11, it is evident that the convergence time for PPP-IF is notably extended, requiring approximately 700 s to converge, and achieving a three-dimensional average positioning accuracy of 0.15 m.
The outcomes illustrated in Fig. 12 clearly demonstrate the superior performance of CZS-PPP over PPP-IF.CZS-PPP achieves convergence in 152 s and sustains a three-dimensional average positioning accuracy of 0.06 m.This highlights the superior performance of CZS-PPP in terms of both convergence time and positioning accuracy when compared to PPP-IF.
In Fig. 13, this paper presents a statistical comparison of the system residual errors between CZS-PPP and the traditional PPP-IF method under the dual-constellation conditions of GPS and BDS-3.The horizontal axis represents satellite numbers, and the vertical axis represents system residual errors.It can be observed that the system residual errors of the proposed method are significantly smaller than those of the traditional PPP-IF method.
Table 2 presents data spanning seven consecutive days.The average convergence time for PPP-IF is 709 s, accompanied by an average three-dimensional RMS positioning accuracy of 0.132 m.In contrast, CZS-PPP attains an average convergence time of 156 s, with an average three-dimensional RMS positioning accuracy of 0.068 m.This underscores the superior performance of CZS-PPP.

Conclusions
This study delves into the critical factors impeding the rapid convergence of PPP, such as integer ambiguity and residual errors within the system.Notably, it addresses these challenges without relying on external sources like ground augmentation networks or enhancements from low Earth orbit satellite navigation systems, resulting in a significant reduction in PPP convergence time.The proposed method, termed Carrier Phase Zero-Baseline Self-Differencing PPP (CZS-PPP), initially employs principles of satellite precise orbits and Doppler generation  to reconstruct error-free Doppler measurements.It then utilizes Doppler integration equations to smooth pseudorange, minimizing cumulative residual errors and enhancing pseudorange accuracy for precise initial epoch coordinates.Subsequently, through self-differencing between consecutive epochs, system residual errors and integer ambiguities are further mitigated, leading to a substantial reduction in PPP convergence time.The singlesystem case achieves convergence in less than 4 min, while the dual-system case converges in under 3 min.The accuracy of these findings is effectively validated using actual measurements from BDS-3, GPS, and BDS-3/ GPS.This research contributes crucial theoretical and technical support for real-world applications of BDS-3/ GNSS real-time PPP.

Figure 1 .
Figure 1.Doppler model of satellites and receivers.

Figure 3
compares the original pseudorange observations with the pseudorange observations smoothed by reconstructed Doppler.The positioning performance of these two types of observations is further validated in Fig. 4. It can be observed from Fig. 4 that the SPP-Reconstruct Doppler (the SPP with Reconstructed Doppler smoothed pseudorange) demonstrates significantly better positioning accuracy and stability compared to SPP with original pseudorange observations.

Figure 2 .
Figure 2. Comparison of the integrated changes between observed Doppler values and reconstructed Doppler values.

Figure 3 .
Figure 3.Comparison between original observations and newly generated observations.

Figure 4 .
Figure 4. Comparison of three-dimensional positioning errors between SPP and SPP-reconstruct doppler.
Figure 7 illustrates the number of BDS-3 satellites observed within an hour.During the specified time interval, approximately 8 satellites are observed, meeting the minimum requirement of at least 4 satellites for positioning.

Figure 6 .
Figure 6.Positioning results for a single BDS-3 system.

Figure 8 .
Figure 8. Presents the positioning results for a single GPS system.

Figure 9 .
Figure 9. Displays the number of observed GPS satellites.

Figure 13 .
Figure 13.Comparison of system residual errors between CZS-PPP and PPP-IF.

Table 1 .
Comparison between BDS-3 and GPS systems.

Table 2 .
Comparison between CZS-PPP and PPP-IF.