Nothing Special   »   [go: up one dir, main page]
Next Article in Journal
Estimation of Maize Biomass at Multi-Growing Stage Using Stem and Leaf Separation Strategies with 3D Radiative Transfer Model and CNN Transfer Learning
Previous Article in Journal
Modelling Water Depth, Turbidity and Chlorophyll Using Airborne Hyperspectral Remote Sensing in a Restored Pond Complex of Doñana National Park (Spain)
Previous Article in Special Issue
Performance of Ground-Based Global Navigation Satellite System Precipitable Water Vapor Retrieval in Beijing with the BeiDou B2b Service
You seem to have javascript disabled. Please note that many of the page functionalities won't work as expected without javascript enabled.
 
 
Search for Articles:
Title / Keyword
Author / Affiliation / Email
Journal
Article Type
 
 
Advanced Search
 
Section
Special Issue
Volume
Issue
Number
Page
 
Journals
Remote Sensing
Volume 16
Issue 16
10.3390/rs16162999
remotesensing-logo
Submit to this Journal Review for this Journal Propose a Special Issue
► Article Menu

Article Menu

share Share announcement Help format_quote Cite question_answer Discuss in SciProfiles thumb_up
...
Endorse textsms
...
Comment
first_page
settings
Order Article Reprints
Font Type:
Arial Georgia Verdana
Font Size:
Aa Aa Aa
Line Spacing:
Column Width:
Background:
Article

Multi-GNSS Precise Point Positioning with Ambiguity Resolution Based on the Decoupled Clock Model

by
Shuai Liu
1,*,
Yunbin Yuan
2,
Xiaosong Guo
1,
Kezhi Wang
1 and
Gongwei Xiao
3
1
The 54th Research Institute of CETC, Shijiazhuang 050081, China
2
State Key Laboratory of Geodesy and Earth’s Dynamics, Innovation Academy for Precision Measurement Science and Technology, Chinese Academy of Sciences, Wuhan 430071, China
3
School of Communications and Information Engineering (School of Artificial Intelligence), Xi’an University of Posts and Telecommunications, Xi’an 710121, China
*
Author to whom correspondence should be addressed.
Remote Sens. 2024, 16(16), 2999; https://doi.org/10.3390/rs16162999
Submission received: 23 July 2024 / Revised: 11 August 2024 / Accepted: 14 August 2024 / Published: 15 August 2024
(This article belongs to the Special Issue Beidou/GNSS Positioning, Navigation and Timing: Methods and Technology (Second Edition))
Browse Figures
Figure 1
<p>A schematic diagram for multi-GNSS decoupled clock estimation.</p> "> Figure 2
<p>Global distribution MGEX stations involved in multi-GNSS decoupled clock estimation. The blue dots indicate that the station can observe GPS signals, the green dots indicate that Galileo can be observed, and the red dots indicate that BDS-3 can be observed.</p> "> Figure 3
<p>Statistics on the number of stations participating in the multi-GNSS decoupled clock estimation.</p> "> Figure 4
<p>Comparison of computational efficiency for multi-GNSS decoupled clock estimation.</p> "> Figure 5
<p>Mean STD statistics for multi-GNSS decoupled clock products.</p> "> Figure 6
<p>Distribution map of MGEX stations for multi-GNSS PPP-AR. All of the selected stations support GPS, Galileo, and BDS-3.</p> "> Figure 7
<p>Comparison of PPP-AR kinematic positioning errors for station GANP based on different schemes.</p> "> Figure 8
<p>Kinematic positioning error distribution of GE scheme.</p> "> Figure 9
<p>Kinematic positioning error RMS of each station in the GE scheme.</p> "> Figure 10
<p>Convergence time, time to first fix, and fixing rate of each station in the GE scheme.</p> "> Figure 11
<p>Kinematic positioning error distribution of GEC scheme.</p> "> Figure 12
<p>Kinematic positioning error RMS of each station in the GEC scheme.</p> "> Figure 13
<p>Convergence time, time to first fix, and fixing rate of each station in the GEC scheme.</p> ">
Versions Notes

Abstract

:
Ambiguity resolution (AR) can markedly enhance the precision of precise point positioning (PPP) and accelerate the convergence process. The decoupled clock model represents a pivotal approach for ambiguity resolution, yet current research on this topic is largely confined to GPS. Consequently, in this study, we extend the investigation of the decoupled clock model to multi-GNSS. Firstly, based on the conventional model, we derive the multi-GNSS decoupled clock estimation model and the precise point positioning with ambiguity resolution (PPP-AR) model. Secondly, we provide a detailed explanation of the estimation process for the multi-GNSS decoupled clock estimation. To validate the efficacy of the proposed model, we conduct multi-GNSS decoupled clock estimation and PPP-AR experiments using six days of observation data. The results demonstrate that the decoupled clocks of GPS, Galileo, and BDS-3 can all achieve high accuracy, thus fully meeting the requirement of ambiguity resolution. In terms of positioning performance, the joint three systems have higher positioning accuracy, reaching 3.10 cm and 6.13 cm in horizontal and vertical directions, respectively. Furthermore, the convergence time (CT) and time to first fix (TTFF) are shortened, to 23.13 min and 13.65 min, respectively. The experimental findings indicate that the proposed multi-GNSS decoupled clock model exhibits high precision and rapid positioning service capabilities.

1. Introduction

Precise point positioning (PPP) [1,2,3], as a significant breakthrough in GNSS (Global Navigation Satellite System), has been extensively researched in recent years. However, conventional single-system PPP still faces numerous challenges in achieving high-precision positioning, such as long convergence time and low initial positioning accuracy. These issues severely limit the application of PPP in real-time, kinematic, and high-precision positioning scenarios. To overcome these limitations, researchers have begun exploring the use of multi-GNSS fusion in PPP [4,5,6]. With the continuous development and improvement of multi-GNSS, as well as the ongoing optimization of data processing algorithms, significant progress has been made in the research of multi-GNSS PPP [7,8,9]. Studies have shown that compared to the GPS single-system, joint multi-GNSS PPP significantly improves filtering convergence time, positioning reliability, and availability [10].
The integration of multi-GNSS offers a new solution for achieving high-precision positioning in a timely manner. In comparison to ambiguity-float solutions, PPP ambiguity resolution (PPP-AR) can further enhance positioning accuracy [11,12,13,14]. Researchers have conducted research on multi-GNSS PPP-AR. For instance, HU et al. [15] investigated the estimation methods of fractional cycle bias products for various systems, including GPS, Galileo, BDS-2, and QZSS. They also conducted PPP-AR experiments using these products and found that the four-system PPP-AR significantly reduces the convergence time (CT) and time to first fix (TTFF) of positioning, particularly in kinematic positioning scenarios. Geng et al. [16] developed a method for generating multi-GNSS observation-specific bias (OSB) products, evaluated their accuracy, and conducted PPP-AR experiments. The results demonstrated that the product’s fluctuation range remained within 2 ns, and it could achieve a positioning accuracy of better than 1 cm. Furthermore, researchers have delved into multi-GNSS PPP-AR from various perspectives, such as overlapping frequency signals [17], harsh environments [18], and additional atmospheric correction [19]. They have also made some ambiguity resolution software open-source [20,21] and publicly released ambiguity resolution products [22,23,24], which have greatly advanced the development of multi-GNSS PPP-AR.
The research on PPP-AR discussed above is based on the assumption that the phase bias remains constant throughout the day. However, studies have shown that the receiver phase bias is not a constant [25] and can vary by approximately 6 cm within a span of 3 h due to factors such as temperature [26]. This highlights the need to consider it a time-varying parameter. Furthermore, the coupling processing of clock parameters in pseudorange and carrier-phase observations can result in a “day-boundary” in the solution results [27,28], which greatly limits the application of PPP in precision measurement fields like precision time transfer [29]. To avoid these issues, it is important to not align the pseudorange and carrier-phase observations during positioning calculation and instead construct separate clock parameters for each type of observation. This is the main principle of the decoupled clock model.
Currently, there is a lack of research on decoupled clock models, with existing studies primarily focusing on three aspects. At the theoretical level, Shi et al. [30] compared the decoupled clock model, integer-recovery clock model, and fractional cycle bias model and found that the only difference between the three models is the processing strategy for phase bias. These models are consistent in terms of the integer property recovery, the system redundancy, and the necessary correction products required for ambiguity resolution. In a detailed study on decoupled satellite clock estimation and PPP-AR, Liu et al. [31] examined the selection strategy for ambiguity datum, ambiguity resolution strategy, decoupled satellite clock accuracy, and positioning performance evaluation. The results indicate that the decoupled phase clock can achieve high accuracy without interference from pseudorange bias, and the PPP-AR based on the decoupled clock model can produce positioning results consistent with the integer-recovery clock model. Naciri et al. [32] extended the dual-frequency decoupled clock model to three frequencies and conducted experiments using Galileo as an example. While the positioning results based on the three-frequency-GNSS decoupled clock model showed little improvement in accuracy compared to dual frequency, they were able to reduce the convergence time by 40% to 50%, significantly enhancing the positioning performance of PPP-AR based on the decoupled clock model.
In order to enhance the positioning performance of PPP-AR, this study expands the decoupled clock model from a single system to multiple systems. Firstly, a function model suitable for estimating multi-GNSS decoupled satellite clocks and PPP-AR is derived based on the single-system decoupled clock model. Secondly, in order to address the issue of low efficiency in the decoupled clock estimation for multi-GNSS, we put forward a series of improvement strategies and provide a comprehensive account of the data processing for the modified model. Finally, experiments are conducted to verify the accuracy of the estimated multi-GNSS decoupled satellite clocks and the positioning performance of PPP-AR. The article is organized as follows: Section 2 describes the derivation process of the multi-GNSS decoupled clock model, Section 3 presents the experiments for decoupled satellite clock estimation and PPP-AR, and Section 4 summarizes the study.

2. Methodology

In this section, we first derive the conventional decoupled clock model. Then, the conventional model is extended to the multi-GNSS decoupled clock estimation and PPP-AR models. Finally, the implementation scheme of multi-GNSS decoupled clock estimation is introduced in detail.

2.1. The Conventional Decoupled Clock Model

The raw GNSS dual-frequency observation equations in units of length can be written as follows:
P r , j Q , s = ρ r Q , s + c d t r Q − c d t Q , s + T r Q , s + μ j Q , s I r , 1 Q , s + d r , j Q − d j Q , s + ε r Q L r , j Q , s = ρ r Q , s + c d t r Q − c d t Q , s + T r Q , s − μ j Q , s I r , 1 Q , s + λ j Q , s N r , j Q , s + b r , j Q − b j Q , s + e r Q
where the superscripts Q and s refer to the satellite system (GPS, Galileo, or BDS-3) and satellite number, respectively; the subscripts r and j denote the receiver number and the frequency, respectively; P is pseudorange, and L is the carrier-phase observables; ρ is the receiver-satellite geometric distance, and c is the speed of light in vacuum; d t is the clock offsets of receiver or satellite; T is the slant tropospheric delay; μ represents the conversion factor of the ionosphere, and μ i = f 1 2 / f i 2 where f 1 and f j are frequencies; I is the first-order slant ionospheric delay; λ is the wavelength of frequency, and N is the integer ambiguity; d denotes the pseudorange hardware bias of the receiver or satellite, while b denotes the phase hardware bias.
For simplicity, unnecessary superscripts and subscripts are omitted. Then, the ionosphere-free (IF) combination observables can be expressed as follows:
P 3 = α 1 P 1 + β 1 P 2 = ρ + c d t r − c d t s + T + d r , 3 − d 3 s L 3 = α 1 L 1 + β 1 L 2 = ρ + c d t r − c d t s + T + α 1 λ 1 N 1 + β 1 λ 2 N 2 + b r , 3 − b 3 s
where P 3 and L 3 refers to the ionosphere-free pseudorange and carrier-phase observables, respectively; α 1 and β 1 represent the ionospheric combination coefficients and can be formulated as follows:
α 1 = f 1 2 f 1 2 − f 2 2 , β 1 = − f 2 2 f 1 2 − f 2 2
d r , 3 and d 3 s are the ionosphere-free pseudorange hardware biases at the receiver and satellite, respectively, while b r , 3 and b 3 s are the ionosphere-free phase hardware biases. The above hardware biases can be expressed as follows:
d r , 3 = α 1 d r , 1 + β 1 d r , 2 d 3 s = α 1 d 1 s + β 1 d 2 s b r , 3 = α 1 b r , 1 + β 1 b r , 2 b 3 s = α 1 b 1 s + β 1 b 2 s
The most significant difference between the decoupled clock model and other models is that it constructs clock parameters for pseudorange and carrier-phase observation equations, avoiding coupling between the two types of observables. Defining N w = N 1 − N 2 as the wide-lane (WL) ambiguity and λ n = c / ( f 1 + f 2 ) as the narrow-lane (NL) wavelength, the decoupled clock model can be obtained as follows:
P 3 = ρ + c d t r , P − c d t P s + T L 3 = ρ + c d t r , L − c d t L s + T + ( λ n N 1 − β 1 λ 2 N w )
with
d t P , r = d t r + d r , 3 c d t P s = d t s + d 3 s c d t r , L = d t r + b r , 3 c d t L s = d t s + b 3 s c
We know that the Melbourne–Wübbena (MW) combination observable is defined as follows:
A m = ( α 2 L 1 + β 2 L 2 ) − ( α 3 P 1 + β 3 P 2 ) = λ w ( N w + b r , w − b w s )
with
α 2 = f 1 f 1 − f 2 , β 2 = − f 2 f 1 − f 2 α 3 = f 1 f 1 + f 2 , β 3 = − f 2 f 1 + f 2
Through simultaneous Equations (5) and (7), we can get the decoupled clock model as follows:
P 3 = ρ + c d t r , P − c d t P s + T L 3 = ρ + c d t r , L − c d t L s + T + ( λ n N 1 − β 1 λ 2 N w ) A m = λ w ( N w + b r , w − b w s )

2.2. Multi-GNSS Decoupled Clock Model for Clock Estimation

Expanding the decoupled clock model expressed in Equation (8) to multi-GNSS , the following can be obtained:
P 3 G = ρ + c d t r , P G − c d t P G , s + T P 3 E = ρ + c d t r , P E − c d t P E , s + T P 3 C = ρ + c d t r , P C − c d t P C , s + T L 3 G = ρ + c d t r , L G − c d t L G , s + T + ( λ n G N 1 G − β 1 G λ 2 G N w G ) L 3 E = ρ + c d t r , L E − c d t L E , s + T + ( λ n E N 1 E − β 1 E λ 2 E N w E ) L 3 C = ρ + c d t r , L C − c d t L C , s + T + ( λ n C N 1 C − β 1 C λ 2 C N w C ) A m G = λ w G ( b r , w G − b w G , s + N w G ) A m E = λ w E ( b r , w E − b w E , s + N w E ) A m C = λ w C ( b r , w C − b w C , s + N w C )
where the superscripts G, E, and C denote GPS, Galileo, and BDS-3, respectively, and the meanings of the other parameters are the same as in Equation (8).
Here, we define Equation (9) as the conventional multi-mode GNSS decoupled clock model. Due to the high number of satellites, a large number of ambiguity parameters will be introduced, resulting in a sharp increase in the dimension of the design matrix. The time complexity of the least squares method is O ( n 3 ) , so when using least squares filtering for decoupled clock estimation, the inverse process will become very complicated, which seriously reduces the computational efficiency. In addition, integer least squares (ILS) has been proven to be an NP-hard problem [33], and thus, when using the LAMBDA strategy based on the ILS algorithm for ambiguity resolution, it will be an extremely time-consuming process.
In order to improve the computational efficiency of multi-GNSS satellite clock estimation, the matrix dimension is usually reduced by eliminating the ambiguity parameter using the inter-epoch difference [34]. However, the purpose of the decoupled clock estimation is to obtain the clock offset that absorbs the ambiguity of fractional parts, so a satellite clock estimation scheme based on differential observations is not applicable. In the computation of high-dimensional matrices, the purpose of improving computational efficiency can usually be achieved by chunking. Based on this idea, this paper proposes a scheme to improve the efficiency of decoupled clock estimation.
For ionosphere-free models, generally, only one ambiguity parameter is estimated for one satellite. The decoupled clock model combines the pseudorange observations with the carrier-phase observations and the MW combination observations for joint solving, leading to the introduction of an additional ambiguity parameter for each satellite. In addition, the wide-lane ambiguity bias is also introduced at the receiver and satellite ends, increasing the dimension of the normal equations. Therefore, in this paper, we propose to solve the MW virtual observation equation in the decoupled clock model separately and estimate only one ambiguity parameter for the carrier-phase observation equation, which achieves the purpose of improving the computational efficiency. The modified multi-GNSS decoupled clock model is as follows:
P 3 G = ρ + c d t r , P G − c d t P G , s + T P 3 E = ρ + c d t r , P E − c d t P E , s + T P 3 C = ρ + c d t r , P C − c d t P C , s + T L 3 G = ρ + c d t r , L G − c d t L G , s + T + A 3 G L 3 E = ρ + c d t r , L E − c d t L E , s + T + A 3 E L 3 C = ρ + c d t r , L C − c d t L C , s + T + A 3 C
where the form of the ambiguity parameter to be estimated can be expressed as follows:
A 3 Q = λ n Q N 1 Q − β 1 Q λ 2 Q N w Q
and the MW combination virtual observation equation can be expressed as follows:
A m G = λ w G ( b r , w G − b w G , s + N w G ) A m E = λ w E ( b r , w E − b w E , s + N w E ) A m C = λ w C ( b r , w C − b w C , s + N w C )
Outside of that, observations from different systems are independent of each other, and the joint treatment of the net solution does not lead to significant gains. Thus, in order to further reduce the dimensionality of the design matrix in a single filtering, decoupled clock products for different systems are estimated separately in sequence. Thus, we get the following equations:
P 3 Q = ρ + c d t r , P Q − c d t P Q , s + T L 3 Q = ρ + c d t r , L Q − c d t L Q , s + T + A 3 Q
and
A m Q = λ w Q ( b r , w Q − b w Q , s + N w Q )
The strategy of separate processing of the virtual observations for MW combinations of multi-GNSS decoupled clock model, as well as the separate processing of different systems, is equivalent to decomposing a single high-dimensional problem into multiple low-dimensional problems, which avoids the extraordinarily high computational complexity associated with the one-time estimation of all the unknown parameters and can significantly improve the computational efficiency.

2.3. Multi-GNSS Decoupled Clock Model for PPP-AR

At the user end, the multi-GNSS decoupled clock model expressed in Equation (9) estimates the receiver clock offsets for different systems separately. Because of the high randomness of the receiver clock offset, it is generally estimated as white noise. However, the biases between different systems are time-stable, so the inter-system bias (ISB) parameter is introduced for the above model. Afterwards, according to Equation (9), the PPP-AR model based on the decoupled clock model and considering ISB parameters for GPS/Galileo/BDS-3 joint processing can be obtained as follows:
P 3 G = ρ + c d t r , P G − c d t P G , s + T P 3 E = ρ + c d t r , P G − c d t P E , s + c ISB r , P E , G + T P 3 C = ρ + c d t r , P G − c d t P C , s + c ISB r , P C , G + T L 3 G = ρ + c d t r , L G − c d t L G , s + T + ( λ n G N 1 G − β 1 G λ 2 G N w G ) L 3 E = ρ + c d t r , L G − c d t L E , s + c ISB r , L E , G + T + ( λ n E N 1 E − β 1 E λ 2 E N w E ) L 3 C = ρ + c d t r , L G − c d t L C , s + c ISB r , L C , G + T + ( λ n C N 1 C − β 1 C λ 2 C N w C ) A m G = λ w G ( b r , w G − b w G , s + N w G ) A m E = λ w E ( b r , w G + ISB r , w E , G − b w E , s + N w E ) A m C = λ w C ( b r , w G + ISB r , w C , G − b w C , s + N w C )
where ISB Q , G denotes the inter-system bias between system Q and GPS, and the meanings of the other parameters are consistent with the previous section. The ISB Q , G for each type of observation can be expressed as follows:
ISB r , P Q , G = d t r , P Q − d t r , P G = ( d t r Q − d t r G ) + d r , 3 Q − d r , 3 G c ISB r , L Q , G = d t r , L Q − d t r , L G = ( d t r Q − d t r G ) + b r , 3 Q − b r , 3 G c , ( Q ≠ G ) ISB r , w Q , G = b r , w Q − b r , w G
The first term of both the pseudorange and the carrier-phase inter-system biases of the decoupled clock model is the time difference between system Q and GPS. When using external satellite clock products, differences in time benchmarks for different products will also be introduced. The differences are stable and independent of the receiver and have the same impact on all receivers. The second term of the bias between the two types of systems is the difference in hardware delay between types of systems, which is station-dependent, less random, and relatively stable within a certain period of time. As for the inter-system bias of wide-lane ambiguity, the modified multi-GNSS decoupled clock estimation model proposed in this paper takes the wide-lane ambiguity bias as an intra-day constant estimate, so the bias should also have high stability.

2.4. Implementation of Decoupled Clock Estimation

In light of the construction of the GPS/Galileo/BDS-3 decoupled clock model put forth in the preceding section, the data processing scheme for the estimation of the multi-GNSS decoupled clock product can be summarized as illustrated in Figure 1. And the following is a summary of the content depicted in Figure 1.
  • Prepare the observation data from evenly distributed observation stations around the world for a network solution. In addition, multi-GNSS precise satellite orbit products, broadcast ephemeris, and other dependent files must be obtained.
  • Read all the correction files. To save memory and increase the speed of the software, only the observation data of a single epoch are read in one calculation cycle.
  • Quality control methods such as gross error detection, clock jump detection and repair, and cycle slip detection are performed on GNSS observations.
  • According to the strategy of system-by-system estimation, the observation data and related information of a certain system are selected for processing, and the process is as follows.
    (a)
    Construct observation equations based on Equations (13) and (14), respectively.
    (b)
    Choose appropriate clock and ambiguity datum for the two sets of observation equations. To determine the narrow-lane ambiguity, consistent datum must be used for both Equations (13) and (14).
    (c)
    Perform filtering for each set of observation equations to acquire the ionosphere-free ambiguity and float solutions for the wide-lane ambiguity.
    (d)
    Round the wide-lane ambiguity to an integer to produce the wide-lane ambiguity bias product.
    (e)
    By combining the ionosphere-free and integer wide-lane ambiguities, the float narrow-lane ambiguity can be determined. Further, by fixing the narrow-lane ambiguity to an integer, one can obtain the decoupled pseudorange and phase clocks, thereby completing the single-epoch, single-system decoupled clock calculation.
  • Determine whether all the systems of the current epoch have been solved. If not completed, then, continue to solve the next system. If completed, then, output the current epoch of the multi-GNSS decoupled clock and other incidental products.
  • Check if all epochs have been solved. If not, continue to solve the next epoch. If completed, end the run.
In the conventional decoupled clock model, the MW virtual observations are computed using only the observations of the current epoch, which results in a more pronounced variation in the virtual observations between epochs. Consequently, the wide-lane ambiguity bias is generally required to be estimated as white noise. In the modified multi-GNSS decoupled clock estimation model, the MW virtual observation is no longer jointly calculated with the pseudorange and carrier-phase observation. Therefore, in order to reduce its inter-epoch variation, a smooth observation is obtained by averaging each epoch, and the wide-lane ambiguity bias is estimated as a daily constant to enhance the strength of the model.

3. Results

3.1. Processing Strategies

The details of data processing strategies are listed in Table 1. It is important to mention that GBM products are commonly used as the benchmark for assessing the precision of pseudorange and carrier-phase clocks. When evaluating the accuracy of positioning, the IGS weekly solutions should be used as the benchmark. Additionally, when applying rounding to resolve wide-lane ambiguity, the decimal portion and standard deviation of the ambiguity parameter should be less than 0.15 cycles. The MLAMBDA algorithm is adopted to fix narrow-lane ambiguities. A success rate with a threshold of 0.9999 and ratio-test with a threshold of 2.0 are simultaneously used to check the validity of the narrow-lane ambiguity resolution.

3.2. Evaluation of Decoupled Clock Products

To validate the multi-GNSS decoupled clock estimation based on the modified model, 90 globally distributed multi-GNSS experiment (MGEX) stations are selected for the experiment. The time interval is from 15 to 20 February 2021, spanning a total of six days. The distribution of the selected stations is shown in Figure 2, and the color of the dot represents the satellite system signals that the station can observe. All stations support GPS, 87 stations support Galileo, and 74 stations support BDS-3.
After the multi-GNSS decoupled clock estimation, the number of stations from each system actually participating in the solution is counted as shown in Figure 3. As illustrated in the figure, the number of stations participating in the GPS decoupled clock estimation remains relatively stable throughout the process of solving, with only a few stations being rejected for some epochs. In contrast, the number of stations participating in the BDS-3 decoupled clock estimation is lower than that of GPS but still relatively stable across all time periods. However, there are more frequent rejections of stations when solving the Galileo decoupled clock. It is important to note that the number of stations participating in the BDS-3 decoupled clock estimation is significantly lower than the 74 shown in Figure 2; this is due to some stations not supporting the signals selected in this section.
In order to verify the improvement of the modified multi-GNSS decoupled clock model in terms of computational efficiency, decoupled clock estimation is carried out on the same computer (Windows 10, 3.6 GHz CPU, 16 GB RAM) based on the proposed model and the conventional model shown in Equation (9). Record the time required for two models to solve each epoch and compare them to obtain the statistical results shown in Figure 4.
As illustrated in the figure, the modified model exhibits a markedly superior computational efficiency compared to the conventional model. The modified model demonstrates a consistent and stable computational efficiency throughout the entire solving process. However, in the time period between 12 h and 19 h, the conventional model exhibits a markedly reduced computational efficiency. At the point in time where the conventional model is most significantly time-consuming, the efficiency improvement of the modified model can reach approximately 20 times that of the conventional model. The aforementioned results demonstrate that a more pronounced enhancement in computational efficiency can be achieved when the multi-GNSS decoupled clock estimation is conducted in accordance with the modified decoupled clock model.
Due to the differing meanings of the decoupled clock and conventional clock, the standard deviation (STD) metrics are employed to assess the accuracy of the decoupled clock products. In this case, the reference satellites are G01 for GPS, E01 for Galileo, and C19 for BDS-3. The accuracy of the decoupled clock product is statistically calculated with the GBM product as the reference, and the statistical results are presented in Figure 5. It should be noted that the statistics presented in the figure are the values of the various types of clock/bias products for each satellite as averaged over the days.
The figure illustrates that the phase clock products of GPS exhibit the highest stability, while the stability of the wide-lane ambiguity bias products of some satellites is somewhat inferior, and the stability of the pseudorange clocks is intermediate between the two systems. The phase clocks of Galileo exhibit a lower level of stability than those of the other two systems, while the pseudorange clocks and the wide-lane ambiguity bias demonstrate a notable degree of stability. The precision of the pseudorange clocks of BDS-3 is relatively low, with the phase clocks and the wide-lane ambiguity bias falling somewhere in between. The phase clock and wide-lane ambiguity bias are of an intermediate level.
For Galileo, some stations frequently reinitialize during the clock estimation process, so it is necessary to re-fix the ambiguity. Due to the decoupling of the clock, the pseudorange clock is basically unaffected, and the wide-lane ambiguity is also relatively easy to fix. As a result, Galileo’s pseudorange clock and wide-lane bias accuracy are relatively high, while the accuracy of the carrier-phase clock is slightly lower. In comparison to GPS and Galileo, Beidou-3 involves a smaller number of stations in clock estimation, resulting in slightly poorer stability of the decoupled clock products. The product accuracy statistics for each system satellite are presented in Table 2, which is derived from Figure 5.
The statistical results presented in the table demonstrate that the phase clock of GPS exhibits the highest accuracy, with a precision of approximately 0.027 ns. The phase clock of BDS-3 also demonstrates good stability, with a precision of approximately 0.073 ns. In contrast, the stability of Galileo is the lowest, with a precision of approximately 0.107 ns. Additionally, the pseudorange clock of Galileo demonstrates the highest accuracy, with a precision of approximately 0.861 ns, while the accuracy of GPS and BDS-3 is slightly lower, with values of 1.137 ns and 2.185 ns, respectively. The wide-lane ambiguity bias products of the three systems have high stability due to the smoothing of the MW virtual observations. The bias of Galileo has the highest stability, with a value of 0.003 cycles, followed by BDS-3 and GPS at 0.007 cycles and 0.010 cycles, respectively.
The results presented above demonstrate that the modified multi-GNSS decoupled clock estimation model proposed in this paper is capable of accurately estimating decoupled clock/bias products with greater precision and efficiency.

3.3. Positioning Experiments

To verify multi-GNSS PPP-AR based on the decoupled clock model, 16 globally distributed MGEX stations are selected for the positioning experiments. The time interval is from 15 to 20 February 2021, spanning a total of six days. The distribution of the selected stations is shown in Figure 6. All of the selected stations support GPS, Galileo, and BDS-3. The signal and error correction model of the satellite navigation system used in the experiment, as well as the processing strategy for related parameters, remain consistent with the previous section. It is worth noting that in the positioning experiments, as in the decoupled clock estimation, we also smooth the MW virtual observations, so that the receiver wide-lane ambiguity bias should be estimated as an intra-day constant.
The multi-GNSS PPP-AR experiments are divided into three groups according to the systems used, namely the combined GPS + BDS-3 positioning scheme (GC), the combined GPS + Galileo positioning scheme (GE), and the combined GPS + Galileo + BDS-3 positioning scheme (GEC). The tests include the assessment of positioning accuracy, convergence time, time to first fix, and fixing rate (FR) for different schemes. Figure 7 shows the kinematic positioning errors of station GANP under the three schemes on 15 February 2021. The figure illustrates that the positioning accuracy under the three schemes after convergence is comparable, with the positioning results being relatively stable. All three schemes achieve a high level of accuracy, indicating that the multi-GNSS PPP-AR based on the decoupled clock model has a robust positioning performance.
In order to visually reflect the positioning performance of each station, the positioning results under the GE scheme on 15 February 2021 are statistically analyzed. The results are presented in Figure 8, Figure 9 and Figure 10. The statistical results of positioning accuracy include errors in the convergence process. The convergence time is defined as the time required for 10 consecutive epochs of positioning errors within 10 cm. The time to first fix is defined as the time when an ambiguity-fixed solution is obtained for 10 consecutive epochs. And the fixing rate is defined as the proportion of the ambiguity-fixed epochs to the total number of epochs. From Figure 8, it can be seen that the positioning errors in each direction satisfy the normal distribution, and the positioning errors in the east and north directions are basically concentrated within 1 cm, and most of them are concentrated around 0 cm, while the positioning errors in the vertical direction are basically concentrated within 2 cm.
The root mean square (RMS) of kinematic positioning errors in the east, north, and vertical directions under the GE positioning scheme are approximately 2.57 cm, 1.60 cm, and 6.40 cm, respectively. According to the statistical results in Figure 9 and Figure 10, it can be seen that the horizontal positioning error of each station is about 2 cm to 3 cm, and the vertical positioning error of some stations is larger, but most of them are about 5 cm to 6 cm. The convergence time and time to first fix of most of the stations are less than 20 min, which is a relatively fast convergence and ambiguity resolution speed. In addition, the fixing rate of each station is more than 90%. The above results show that the joint GPS/Galileo positioning based on the decoupled clock model can get good positioning results.
For purposes of comparison, Figure 11, Figure 12 and Figure 13 illustrate the positioning results obtained under the GEC scheme on 15 February 2021. After joining BDS-3, the positioning accuracy has slightly improved, but the effect is not particularly significant. The RMS of positioning errors in the east, north, and vertical directions are approximately 2.37 cm, 1.59 cm, and 6.02 cm, respectively. The high level of ambiguity fixing rate under the GE scheme makes it challenging to improve it again after joining BDS-3. It is evident that the convergence time and time to first fix of certain stations have been markedly diminished. For instance, the FAA1 station and the KIRI station have exhibited notable improvements, with convergence times of over 20 min reduced to less than 20 min, respectively. The aforementioned outcomes demonstrate that the proposed multi-GNSS PPP-AR, based on the decoupled clock model, exhibits a satisfactory positioning efficacy.
In order to enhance the dependability of the experiments, the positioning indicators of the 6-day solutions of 16 stations under distinct schemes in this experiment are subjected to statistical analysis, as illustrated in Table 3. The values presented in the table represent the average of all stations.
The statistical results presented in Table 2 demonstrate that the kinematic positioning accuracies achieved under the various schemes can attain a horizontal positioning accuracy of approximately 4 cm and vertical positioning accuracy of 7 cm. Additionally, the positioning accuracy of the GPS/Galileo scheme is superior to that of the GPS/BDS-3 combination. And the joint solution of the three systems, namely GPS/Galileo/BDS-3, exhibits the highest positioning accuracy, with comparable findings in terms of convergence time and time to first fix. Nevertheless, the GPS/BDS-3 solution exhibits the highest ambiguity fixing rate. In general, ambiguity resolution can achieve higher positioning accuracy. However, the positioning accuracy of this solution is not satisfactory. Therefore, it cannot be ruled out that there may be ambiguity-fixed errors in this combination.
The aforementioned results demonstrate that multi-GNSS PPP-AR based on the decoupled clock model proposed in this paper is capable of achieving a superior positioning effect under diverse system combinations.

4. Discussion

The decoupled clock model is known for its ability to solve the pseudorange, carrier-phase, and MW combination equations simultaneously, taking into account the time-varying characteristics of wide-lane ambiguity bias and avoiding the negative impact of pseudorange time-varying bias on the carrier-phase observation equation. However, when applying this approach to multi-GNSS, a large number of parameters need to be estimated, which significantly reduces computational efficiency. To address this issue, we propose a method of separately solving each satellite navigation system and smoothing the MW combination observations between epochs. This raises the question of whether it is necessary to estimate the wide-lane ambiguity bias products epoch by epoch or if a set of intra-day wide-lane ambiguity bias products can be used for the user-end algorithm of the decoupled clock model.
From Figure 5, it can be seen that Galileo has the most stable wide-lane ambiguity bias, within 0.01 cycles. Although the stability of GPS wide-lane bias is slightly lower, it is still within 0.04 cycles. We typically use the direct rounding method to fix the wide-lane ambiguity, but a stricter strategy can be implemented by considering the decimal part of the wide-lane ambiguity and its standard deviation. This decimal threshold is usually much greater than 0.04 cycles, so using the wide-lane ambiguity bias as an intra-day constant in this paper is sufficient for ambiguity resolution. This strategy significantly reduces the data volume of products in the decoupled clock model and improves compatibility and interoperability with other ambiguity resolution products in IGS.
There are two issues worth discussing when using the decoupled clock model for positioning experiments. The first issue is the positioning error. From Figure 9 and Figure 12, it can be seen that the horizontal error of some stations is close to or greater than the vertical error, which may not reflect the actual situation. We found that during the convergence stage of each station, the positioning error is relatively large, leading to deviations in the positioning statistical results. For example, at the GANP station shown in Figure 9, the positioning errors in the east, north, and up directions are 3.76 cm, 1.65 cm, and 3.57 cm, respectively, when considering all epochs. However, when ignoring the first 15 min, the positioning errors in these directions are only 0.38 cm, 0.44 cm, and 2.34 cm, respectively. We consider that decoupling the clock parameters of the pseudorange and carrier-phase observation equations is the main reason for the large initial positioning errors. How to handle the relationship between decoupled equations (such as by improving stochastic models) to reduce this adverse effect is a topic worthy of attention.
The second issue is the convergence time. From Figure 10 and Figure 13, it is evident that the convergence time of PPP-AR experiments at some stations exceeds the time to first fix. Typically, the ambiguity can only be resolved after the positioning results have converged; thus, the convergence time should be shorter than the time to first fix. Therefore, it can be concluded that there are ambiguity-fixed errors at the aforementioned stations. As shown in Table 1, despite implementing a strict strategy for ambiguity resolution, we are still unable to avoid the problem of ambiguity-fixed errors. It is our contention that this is due to positioning errors. Due to significant initial positioning errors, the positioning results converge to incorrect values, which ultimately results in an incorrect fixation of ambiguity.

5. Conclusions

In this study, we developed multi-GNSS decoupled clock estimation and PPP-AR methods based on the decoupled clock model. To verify the effectiveness of the two methods, the decoupled clock estimation and PPP-AR experiments were conducted over a six-day observation period using 90 and 16 MGEX stations, respectively. The following conclusions were obtained. Firstly, the computational efficiency of satellite clock estimation based on the modified multi-GNSS decoupled clock model is significantly enhanced, exhibiting a factor of 20 improvement over the conventional model. Secondly, the accuracy of the multi-GNSS decoupled clock based on the modified model is high. For the decoupled phase clock, GPS has the highest accuracy, reaching about 0.027 ns, and Galileo’s accuracy is slightly lower, about 0.107 ns. But Galileo’s pseudorange clock and wide-lane ambiguity bias are the most stable, reaching 0.861 ns and 0.003 cycles, respectively. The various clocks of BDS-3 also achieve high accuracy. Finally, the RMS of the kinematic positioning error in the east, north, and vertical directions when combining GPS/Galileo/BDS-3 is 2.62 cm, 1.66 cm, and 6.13 cm, respectively. The convergence time is approximately 23 min, while the time to first fix is approximately 14 min. The ambiguity fixation rate is as high as 95.8%.

Author Contributions

S.L. authored the manuscript. Y.Y. and G.X. assisted in the writing of the manuscript. X.G. and K.W. provided constructive feedback and assisted in revising the manuscript. All authors have read and agreed to the published version of the manuscript .

Funding

This work was supported by the Natural Science Foundation of Hebei Province, China (No. D2024523002), the National Key Research and Development Program of China (No. 2023YFE0208400, No. 2022YFC2204601, No. 2023YFC2206100), and the Natural Science Basic Research Project of Shaanxi Province (No. 2023-JC-QN-0278).

Data Availability Statement

The datasets analyzed in this study can be made available by the corresponding author on request.

Acknowledgments

The authors would like to acknowledge the IGS for providing the observation data and precise products.

Conflicts of Interest

The authors declare no conflicts of interest.

References

  1. Kouba, J.; Héroux, P. Precise point positioning using IGS orbit and clock products. GPS Solut. 2001, 5, 12–28. [Google Scholar] [CrossRef]
  2. Malys, S.; Jensen, P.A. Geodetic point positioning with GPS carrier beat phase data from the CASA UNO Experiment. Geophys. Res. Lett. 1990, 17, 651–654. [Google Scholar] [CrossRef]
  3. Zumberge, J.F.; Heflin, M.B.; Jefferson, D.C.; Watkins, M.M.; Webb, F.H. Precise point positioning for the efficient and robust analysis of GPS data from large networks. J. Geophys. Res. Solid Earth 1997, 102, 5005–5017. [Google Scholar] [CrossRef]
  4. Xia, F.; Ye, S.; Xia, P.; Zhao, L.; Jiang, N.; Chen, D. Assessing the latest performance of Galileo-only PPP and the contribution of Galileo to Multi-GNSS PPP. Adv. Space Res. 2018, 63, 2784–2795. [Google Scholar] [CrossRef]
  5. Zhou, F.; Dong, D.; Li, W.; Jiang, X.; Wickert, J.; Schuh, H. GAMP: An open-source software of multi-GNSS precise point positioning using undifferenced and uncombined observations. GPS Solut. 2018, 22, 33. [Google Scholar] [CrossRef]
  6. Xiao, G.; Liu, G.; Ou, J.; Liu, G.; Wang, S.; Guo, A. MG-APP: An open-source software for multi-GNSS precise point positioning and application analysis. GPS Solut. 2020, 24, 66. [Google Scholar] [CrossRef]
  7. Liu, T.; Yuan, Y.; Zhang, B.; Wang, N.; Tan, B.; Chen, Y. Multi-GNSS precise point positioning (MGPPP) using raw observations. J. Geod. 2017, 91, 253–268. [Google Scholar] [CrossRef]
  8. Fu, W.; Huang, G.; Zhang, Q.; Gu, S.; Ge, M.; Schuh, H. Multi-GNSS real-time clock estimation using sequential least square adjustment with online quality control. J. Geod. 2019, 93, 963–976. [Google Scholar] [CrossRef]
  9. Mirmohammadian, F.; Asgari, J.; Verhagen, S.; Amiri-Simkooei, A. Improvement of multi-GNSS precision and success rate using realistic stochastic model of observations. Remote Sens. 2021, 14, 60. [Google Scholar] [CrossRef]
  10. Cai, C.; Gao, Y.; Pan, L.; Zhu, J. Precise point positioning with quad-constellations: GPS, BeiDou, GLONASS and Galileo. Adv. Space Res. 2015, 56, 133–143. [Google Scholar] [CrossRef]
  11. Li, P.; Zhang, X.; Ren, X.; Zuo, X.; Pan, Y. Generating GPS satellite fractional cycle bias for ambiguity-fixed precise point positioning. GPS Solut. 2016, 20, 771–782. [Google Scholar] [CrossRef]
  12. Ge, M.; Gendt, G.; Rothacher, M.; Shi, C.; Liu, J. Resolution of GPS carrier-phase ambiguities in Precise Point Positioning (PPP) with daily observations. J. Geod. 2008, 82, 389–399. [Google Scholar] [CrossRef]
  13. Laurichesse, D.; Mercier, F.; Berthias, J.; Cerri, L. Integer ambiguity resolution on undifferenced GPS phase measurements and its application to PPP and satellite precise orbit determination. Navig. J. Inst. Navig. 2009, 56, 135–149. [Google Scholar] [CrossRef]
  14. Liu, T.; Chen, H.; Song, C.; Wang, Y.; Yuan, P.; Geng, T.; Jiang, W. Beidou-3 precise point positioning ambiguity resolution with B1I/B3I/B1C/B2a/B2b phase observable-specific signal bias and satellite B1I/B3I legacy clock. Adv. Space Res. 2023, 72, 10. [Google Scholar] [CrossRef]
  15. Hu, J.; Zhang, X.; Li, P.; Ma, F.; Pan, L. Multi-GNSS fractional cycle bias products generation for GNSS ambiguity-fixed PPP at Wuhan University. GPS Solut. 2020, 24, 15. [Google Scholar] [CrossRef]
  16. Geng, J.; Zhang, Q.; Li, G.; Liu, J.; Liu, D. Observable-specific phase biases of Wuhan multi-GNSS experiment analysis center’s rapid satellite products. Satell. Navig. 2022, 3, 23. [Google Scholar] [CrossRef]
  17. Chen, G.; Li, B.; Zhang, Z.; Liu, T. Integer ambiguity resolution and precise positioning for tight integration of BDS-3, GPS, GALILEO, and QZSS overlapping frequencies signals. GPS Solut. 2022, 26, 26. [Google Scholar] [CrossRef]
  18. Wang, X.; Li, X.; Shen, Z.; Li, X.; Zhou, Y.; Chang, H. Factor graph optimization-based multi-GNSS real-time kinematic system for robust and precise positioning in urban canyons. GPS Solut. 2023, 27, 200. [Google Scholar] [CrossRef]
  19. Cui, B.; Wang, J.; Li, P.; Ge, M.; Schuh, H. Modeling wide-area tropospheric delay corrections for fast PPP ambiguity resolution. GPS Solut. 2022, 26, 56. [Google Scholar] [CrossRef]
  20. Geng, J.; Chen, X.; Pan, Y.; Mao, S.; Li, C.; Zhou, J.; Zhang, K. PRIDE PPP-AR: An open-source software for GPS PPP ambiguity resolution. GPS Solut. 2019, 23, 91. [Google Scholar] [CrossRef]
  21. Li, X.; Han, X.; Li, X.; Liu, G.; Feng, G.; Wang, B.; Zheng, H. GREAT-UPD: An open-source software for uncalibrated phase delay estimation based on multi-GNSS and multi-frequency observations. GPS Solut. 2021, 25, 66. [Google Scholar] [CrossRef]
  22. Loyer, S.; Perosanz, F.; Mercier, F.; Capdeville, H.; Marty, J.C. Zero-difference GPS ambiguity resolution at CNES–CLS IGS Analysis Center. J. Geod. 2012, 86, 991–1003. [Google Scholar] [CrossRef]
  23. Schaer, S.; Villiger, A.; Arnold, D.; Dach, R.; Prange, L.; Jäggi, A. The CODE ambiguity-fixed clock and phase bias analysis products: Generation, properties, and performance. J. Geod. 2021, 95, 81. [Google Scholar] [CrossRef]
  24. Guo, J.; Geng, J.; Zeng, J.; Song, X.; Defraigne, P. GPS/Galileo/BDS phase bias stream from Wuhan IGS analysis center for real-time PPP ambiguity resolution. GPS Solut. 2024, 28, 67. [Google Scholar] [CrossRef]
  25. Zhang, B.; Teunissen, P.J.G.; Yuan, Y.; Zhang, X.; Li, M. A modified carrier-to-code leveling method for retrieving ionospheric observables and detecting short-term temporal variability of receiver differential code biases. J. Geod. 2019, 93, 19–28. [Google Scholar] [CrossRef]
  26. Zhang, B.; Teunissen, P.J.G.; Yuan, Y. On the short-term temporal variations of GNSS receiver differential phase biases. J. Geod. 2017, 91, 563–572. [Google Scholar] [CrossRef]
  27. Defraigne, P.; Bruyninx, C. On the link between GPS pseudorange noise and day-boundary discontinuities in geodetic time transfer solutions. GPS Solut. 2007, 11, 239–249. [Google Scholar] [CrossRef]
  28. Collins, P.; Bisnath, S.; Lahaye, F.; Héroux, P. Undifferenced GPS ambiguity resolution using the decoupled clock model and ambiguity datum fixing. Navig. J. Inst. Navig. 2010, 57, 123–135. [Google Scholar] [CrossRef]
  29. Zhang, X.; Guo, J.; Hu, Y.; Zhao, D.; He, Z. Research of eliminating the day-boundary discontinuities in GNSS carrier phase time transfer through network processing. Sensors 2020, 20, 2622. [Google Scholar] [CrossRef]
  30. Shi, J.; Gao, Y. A comparison of three PPP integer ambiguity resolution methods. GPS Solut. 2014, 18, 519–528. [Google Scholar] [CrossRef]
  31. Liu, S.; Yuan, Y. Generating GPS decoupled clock products for precise point positioning with ambiguity resolution. J. Geod. 2022, 96, 6. [Google Scholar] [CrossRef]
  32. Naciri, N.; Bisnath, S. An uncombined triple-frequency user implementation of the decoupled clock model for PPP-AR. J. Geod. 2021, 95, 60. [Google Scholar] [CrossRef]
  33. Chang, X.W.; Yang, X.; Zhou, T. MLAMBDA: A modified LAMBDA method for integer least-squares estimation. J. Geod. 2005, 79, 552–565. [Google Scholar] [CrossRef]
  34. Chen, Y.; Yuan, Y.; Zhang, B.; Liu, T.; Ding, W.; Ai, Q. A modified mix-differenced approach for estimating multi-GNSS real-time satellite clock offsets. GPS Solut. 2018, 22, 72. [Google Scholar] [CrossRef]
Remotesensing 16 02999 g001
Figure 1. A schematic diagram for multi-GNSS decoupled clock estimation.
Figure 1. A schematic diagram for multi-GNSS decoupled clock estimation.
Remotesensing 16 02999 g001
Remotesensing 16 02999 g002
Figure 2. Global distribution MGEX stations involved in multi-GNSS decoupled clock estimation. The blue dots indicate that the station can observe GPS signals, the green dots indicate that Galileo can be observed, and the red dots indicate that BDS-3 can be observed.
Figure 2. Global distribution MGEX stations involved in multi-GNSS decoupled clock estimation. The blue dots indicate that the station can observe GPS signals, the green dots indicate that Galileo can be observed, and the red dots indicate that BDS-3 can be observed.
Remotesensing 16 02999 g002
Remotesensing 16 02999 g003
Figure 3. Statistics on the number of stations participating in the multi-GNSS decoupled clock estimation.
Figure 3. Statistics on the number of stations participating in the multi-GNSS decoupled clock estimation.
Remotesensing 16 02999 g003
Remotesensing 16 02999 g004
Figure 4. Comparison of computational efficiency for multi-GNSS decoupled clock estimation.
Figure 4. Comparison of computational efficiency for multi-GNSS decoupled clock estimation.
Remotesensing 16 02999 g004
Remotesensing 16 02999 g005
Figure 5. Mean STD statistics for multi-GNSS decoupled clock products.
Figure 5. Mean STD statistics for multi-GNSS decoupled clock products.
Remotesensing 16 02999 g005
Remotesensing 16 02999 g006
Figure 6. Distribution map of MGEX stations for multi-GNSS PPP-AR. All of the selected stations support GPS, Galileo, and BDS-3.
Figure 6. Distribution map of MGEX stations for multi-GNSS PPP-AR. All of the selected stations support GPS, Galileo, and BDS-3.
Remotesensing 16 02999 g006
Remotesensing 16 02999 g007
Figure 7. Comparison of PPP-AR kinematic positioning errors for station GANP based on different schemes.
Figure 7. Comparison of PPP-AR kinematic positioning errors for station GANP based on different schemes.
Remotesensing 16 02999 g007
Remotesensing 16 02999 g008
Figure 8. Kinematic positioning error distribution of GE scheme.
Figure 8. Kinematic positioning error distribution of GE scheme.
Remotesensing 16 02999 g008
Remotesensing 16 02999 g009
Figure 9. Kinematic positioning error RMS of each station in the GE scheme.
Figure 9. Kinematic positioning error RMS of each station in the GE scheme.
Remotesensing 16 02999 g009
Remotesensing 16 02999 g010
Figure 10. Convergence time, time to first fix, and fixing rate of each station in the GE scheme.
Figure 10. Convergence time, time to first fix, and fixing rate of each station in the GE scheme.
Remotesensing 16 02999 g010
Remotesensing 16 02999 g011
Figure 11. Kinematic positioning error distribution of GEC scheme.
Figure 11. Kinematic positioning error distribution of GEC scheme.
Remotesensing 16 02999 g011
Remotesensing 16 02999 g012
Figure 12. Kinematic positioning error RMS of each station in the GEC scheme.
Figure 12. Kinematic positioning error RMS of each station in the GEC scheme.
Remotesensing 16 02999 g012
Remotesensing 16 02999 g013
Figure 13. Convergence time, time to first fix, and fixing rate of each station in the GEC scheme.
Figure 13. Convergence time, time to first fix, and fixing rate of each station in the GEC scheme.
Remotesensing 16 02999 g013
Table 1. Detailed strategies for multi-GNSS decoupled clock estimation and PPP-AR.
Table 1. Detailed strategies for multi-GNSS decoupled clock estimation and PPP-AR.
ItemsStrategies
FrequenciesGPS: L1 & L2; Galileo: E1 & E5b; BDS-3: B1 & B3
ObservationsPseudorange and carrier-phase
A priori noisePseudorange: 0.3 m, carrier phase: 0.003 m
Station datumLCK3
Cut-off elevation10°
Phase wind-upCorrected
Relativistic effectCorrected
Differential code biasCODE P1-C1 products
Tidal displacementsSolid earth tide, ocean tide loading, and pole tide
Phase center offset and variationsigs14.atx
Station coordinatesFixed to IGS weekly solutions at the server end and estimated at the user end. In kinematic mode, coordinates are estimated as white noise parameters.
Earth rotation parametersIGS products
Satellite orbitsGBM products
Satellite clocksEstimated as white noises at the server end and fixed to the estimated products in this study at the user end
Receiver clocksEstimated as white noises
Zenith troposphere delaysEstimated as random-walk noises with respect to the Saastamoinen model, and the Niell Mapping Function is used
AmbiguitiesEstimated as constants over each continuous session
Integer ambiguity fixingRounding directly at the server end, rounding and MLAMBDA are applied to fix ambiguity at the user end
EstimatorLeast square filter
Table 2. Multi-GNSS satellite decoupled clock accuracy statistics.
Table 2. Multi-GNSS satellite decoupled clock accuracy statistics.
SystemPhase Clock (ns)Pseudorange Clock (ns)Wide-Lane Ambiguity Bias (c)
GPS0.0271.1370.010
Galileo0.1070.8610.003
BDS-30.0732.1850.007
Table 3. Statistics on positioning performance indicators under different schemes.
Table 3. Statistics on positioning performance indicators under different schemes.
SchemeAmbiguity-Fixed Solution (cm)CT (min)TTFF (min)FR (%)
EastNorthHorizontalVertical
GC3.332.003.886.2728.2022.0697.30
GE2.811.773.326.5323.7617.5995.28
GEC2.621.663.106.1323.1313.6595.80
Disclaimer/Publisher’s Note: The statements, opinions and data contained in all publications are solely those of the individual author(s) and contributor(s) and not of MDPI and/or the editor(s). MDPI and/or the editor(s) disclaim responsibility for any injury to people or property resulting from any ideas, methods, instructions or products referred to in the content.

Share and Cite

MDPI and ACS Style

Liu, S.; Yuan, Y.; Guo, X.; Wang, K.; Xiao, G. Multi-GNSS Precise Point Positioning with Ambiguity Resolution Based on the Decoupled Clock Model. Remote Sens. 2024, 16, 2999. https://doi.org/10.3390/rs16162999

AMA Style

Liu S, Yuan Y, Guo X, Wang K, Xiao G. Multi-GNSS Precise Point Positioning with Ambiguity Resolution Based on the Decoupled Clock Model. Remote Sensing. 2024; 16(16):2999. https://doi.org/10.3390/rs16162999

Chicago/Turabian Style

Liu, Shuai, Yunbin Yuan, Xiaosong Guo, Kezhi Wang, and Gongwei Xiao. 2024. "Multi-GNSS Precise Point Positioning with Ambiguity Resolution Based on the Decoupled Clock Model" Remote Sensing 16, no. 16: 2999. https://doi.org/10.3390/rs16162999

Note that from the first issue of 2016, this journal uses article numbers instead of page numbers. See further details here.

Article Metrics

Back to TopTop