Satellite Phase Biases Research Articles

Estimation of GNSS satellite bias corrections using non-common frequency model with application to PPP-AR

Abstract Corrections of satellite phase biases (SPBs) enable precise point positioning with ambiguity resolution (PPP-AR), which shortens the convergence time and improves positioning precision compared with precise point positioning (PPP). Extending PPP-AR from dual-frequency to multi-frequency can further enhance the positioning performance by correcting satellite code biases (SCBs) at the third frequency and above. In formulating a multi-frequency model to estimate satellite bias corrections (SBCs), including SPBs and SCBs, existing studies use only satellite observations collected at common frequencies (CF). This study proposes a non-common frequency (NCF) model that utilizes dual- and multi-frequency satellites to estimate SBCs simultaneously. To validate the NCF model, we collect data from an Australian network to estimate the SBCs and apply these corrections to the PPP-AR user positioning. The results show that the NCF model can provide SBCs for more satellites and frequencies than the CF model. After correction of SBCs, the proportion of fractional parts of float PPP ambiguities, which are less than 0.1 cycles, has a significant increase from 20% to
70%, indicating the successful restoration of the integer properties of PPP ambiguities by the
SBCs. Moreover, a comparable experiment between the NCF and CF PPP-AR models is conducted. The NCF PPP-AR model demonstrates superior performance over the CF model using triple-frequency satellites in convergence speed and positioning precision, achieving improvements of 41.16% and 39.52%, respectively. Meanwhile, compared to the CF model using dual-frequency satellites, the NCF PPP-AR model shows improvements of 17.16% and 7.46% in these aspects. The positioning performance of the NCF PPP-AR model is evaluated across ten user stations sampled over three days. The results indicate the root mean square
(RMS) positioning errors for the east, north and up components are 0.41, 0.51 and 2.46 cm, respectively. Additionally, the average convergence time for achieving 3D positioning errors within 1 dm is 9.75 min.

A method to assess the quality of GNSS satellite phase bias products

As part of the International GNSS Service (IGS), several analysis centers provide GPS and Galileo satellite phase bias products to support precise point positioning with ambiguity resolution (PPP-AR). Due to the high correlation with satellite orbits and clock offsets, it is difficult to assess directly the precision of satellite phase bias products. Once outliers exist in satellite phase biases, PPP-AR results are no longer reliable and the combination of satellite phase bias products from IGS analysis centers also gets difficult. In this contribution, we propose a method independent of ground measurements to detect outliers in satellite phase biases by computing the total Difference of satellite Orbits, Clock offsets and narrow-lane Biases at the midnight epoch between two consecutive days. Results over 180 days show that about 0.2, 1.1, 2.0 and 0.1% of the total DOCB values for GPS satellites exceed 0.15 narrow-lane cycles for CODE final, CODE rapid, CNES/CLS final and WUHN rapid satellite products, respectively, while the same outlier-ratios for Galileo satellites are 0.1, 0.9, 0.4 and 0.1%, respectively. As an important contribution to the orbit, clock and bias combination task, we check the consistency of satellite phase bias products between two analysis centers before and after removing these detected outliers from individual analysis centers. It is convincing that the number of large differences of satellite phase biases between two analysis centers is notably reduced.

Sequential Generation of Multi-GNSS Multi-Frequency PPP-RTK Products and Their Performance Using the EUREF Permanent GNSS Network

In the classic Precise Point Positioning (PPP) approach, observations of Global Navigation Satellite Systems (GNSS) are processed at the network level to generate satellite clocks and positions. This information can be used to enable accurate point positioning for a single GNSS receiver. In the PPP Real-Time Kinematic (PPP-RTK) approach, satellite phase biases are considered as well, enabling ambiguity resolution at the network and user levels. In this research, 30 s multi-frequency raw GPS, Galileo, and BDS-2/3 observations are processed at the network and user levels in a sequential Kalman filter. PPP-RTK enabling products are generated for up to five frequencies, and ambiguity resolution is performed at the network and user levels using a flexible ambiguity reparameterization approach, comparable to wide- and narrow-laning, which has shown to yield a significantly improved single epoch coordinate solution when multi-frequency observations are available. Different assumptions regarding the time stability of receiver and satellite phase biases have been made and compared. The availability of a precise user coordinate solution when multi-frequency and dual-frequency observations are processed is assessed and compared. A precise ambiguity-fixed solution is available in three epochs or fewer in 77% of all cases with an average of 24 visible satellites for static and kinematic receivers when multi-frequency observations are processed. When only dual-frequency observations are considered, a fixed solution is available in seven epochs or fewer in 71% of all cases. The fastest fixed solution was found in two epochs with multi-frequency observations and in six epochs with dual-frequency observations. Estimating a reference phase clock did not lead to an improvement in the coordinate solution. The findings indicate that a fixed solution can potentially be found faster than often suggested, with potential for further improvement when more satellites or regional atmospheric corrections are considered.

Absolute and relative POD of LEO satellites in formation flying: Undifferenced and uncombined approach

Absolute or relative precise orbit determination (POD) is an essential prerequisite for many low earth orbit (LEO) missions. The POD of LEO satellites typically relays on processing the onboard global navigation satellite system (GNSS) measurements. The absolute POD is usually based on an ionosphere-free (IF) combination, and currently, integer ambiguity resolution (IAR) can be achieved only when external GNSS satellite phase bias (SPB) products are used. The use of these products is not flexible in multi-frequency/multi-constellation scenarios and is difficult to achieve in real-time missions. For relative POD, the double-differenced (DD) with IAR is the most general method. However, the differencing process amplifies observation noise and loses the opportunity to impose dynamic constraints on some eliminated parameters. In this contribution, based on the use of undifferenced and uncombined (UDUC) observations, a new model for both absolute and relative POD is proposed. In this model, the ambiguities of common-view satellites are constructed into DD form, thus IAR can be achieved without any external SPB products. Working with the UDUC observations, multi-frequency scenarios can be easily applied, and residuals can be separated for each frequency. In addition, with precise GNSS satellite clock/orbit products, both the absolute and relative orbits can be derived, which supports absolute and relative LEO POD. Based on onboard GPS observations of T-A and T-B satellites in formation flying, the performance of the UDUC POD model with DD ambiguity was evaluated. With the UDUC algorithm and IAR, the proposed model presented a consistency of 2.8–3.8 cm in 3D with the reference orbits, and the orbit difference was reduced by 16.3% and 10.6% for T-A and T-B compared with the IF-based POD, respectively. In addition, the relative orbit of the two satellites derived from the proposed model showed a consistency of 1.1–1.5 mm, which proved the feasibility of the UDUC POD model with DD ambiguity for formation flying missions.

Performance of Galileo satellite products determined from multi-frequency measurements

Each Galileo satellite provides coherent navigation signals in four distinct frequency bands. International GNSS Service (IGS) analysis centers (ACs) typically determine Galileo satellite products based on the E1/E5a dual-frequency measurements due to the software limitation and the limited tracking capability of other signals in the early time. The goal of this contribution is to evaluate the quality of Galileo satellite products determined by using different dual-frequency (E1/E5a, E1/E5b, E1/E5, E1/E6) and multi-frequency (E1/E5a/E5b/E5/E6) measurements based on different sizes of ground networks. The performance of signal noise, the consistency of frequency-specific satellite phase center offsets and the stability of satellite phase biases are assessed in advance to confirm preconditions for multi-frequency processing. Orbit results from different dual-frequency measurements show that orbit precision determined from E1/E6 is clearly worse (about 35%) than that from other dual-frequency solutions. In view of a similar E1, E5a, E5b and E6 measurement quality, the degraded E1/E6 orbit performance is mainly attributed to the unfavorable noise amplification in the respective ionosphere-free linear combination. The advantage of using multi-frequency measurements over dual-frequency for precise orbit determination is clearly visible when using small networks. For instance, the ambiguity fixing rate is 80% for the multi-frequency solution while it is less than 40% for the dual-frequency solution if 150 s data sampling is employed in a 15-station network. Higher fixing rates result in better (more than 30%) satellite orbits and more robust satellite clock and phase bias products. In general, satellite phase bias products determined from a 20-station (or more) network are precise enough to conduct precise point positioning with ambiguity resolution (PPP-AR) applications. Multi-frequency kinematic PPP-AR solutions always show 5–10% precision improvement compared to those computed from dual-frequency observations.

Assessment of Galileo High Accuracy Service (HAS) test signals and preliminary positioning performance

The Galileo High Accuracy Service (HAS) is a GNSS augmentation that provides precise satellite corrections to users worldwide for free directly through Galileo’s E6 signal. The HAS service provides free PPP corrections from the Galileo constellation and the Internet, with targeted real-time 95% positioning performance of better than 20 cm horizontal and 40 cm vertical error after 5 min of convergence time globally and shorter in Europe. The HAS initial service, under validation at the time of writing, provides these capabilities with a reduced performance (based on the current Galileo stations network). Live HAS test signals broadcasted from the Galileo satellites during summer 2022 have been decoded and analyzed. Corrections include Galileo and GPS orbit, clock, and code bias corrections, with SISRE of 10.6 cm and 11.8 cm for Galileo and GPS, respectively. Code bias corrections showed good performance as well, with rms of 0.28 ns, 0.26 ns, and 0.22 ns for Galileo C1C–C5Q, C1C–C7Q, and C1C–C6C, respectively, and 0.20 ns for GPS C1C–C2L. Float PPP positioning performance results show that the combined Galileo and GPS solution can already achieve the HAS full service accuracy performance target and is close in terms of convergence time, with 95% rms of 13.1 cm and 18.6 cm horizontally and vertically, respectively, in kinematic mode, and with a 95% convergence time of 7.5 min. The latter is expected to be improved with the inclusion of satellite phase bias and local atmospheric corrections. With these early Galileo HAS test signals, this preliminary analysis indicates that the HAS full service targets are attainable. Finally, a correction latency analysis is performed, showing that even with latency of up to 60 s, positioning can remain within the targeted HAS accuracy performance.

Improved PPP-RTK positioning performance by using the elevation-dependent weighting model for the atmospheric delay corrections

With satellite phase bias products, integer ambiguity resolution (AR) enabled precise point positioning (PPP) can be achieved, that is, PPP-real-time kinematic (RTK). However, PPP-AR still requires minutes to converge to centimeter-level accuracy, as atmospheric delay parameters need to be estimated in the user un-combined (UC) model. To shorten the convergence time, reference-network-derived atmospheric delay corrections are often directly corrected for raw code/phase observables, without considering their uncertainties, thereby causing model errors. In this study, we propose a new ionosphere-weighted model, in which an elevation-dependent weight is adopted. To evaluate the proposed model, on-board kinematic experiments were conducted based on GPS/Galileo/BDS2/ BDS3 dual-frequency measurements. Compared with the positioning results of the traditional determined variance weight, the proportion of horizontal errors less than 5 cm is increased from 70% to 80%, and horizontal accuracy is improved by 22% and 9.8% at the 68th and 95th percentile, respectively, indicating that the proposed model can improve the PPP-RTK performance.

A method of real-time long-baseline time transfer based on the PPP-RTK

Long-baseline time transfer can nowadays reach rather high frequency stability based on post-processed batch least-squares adjustment using the Precise Point Positioning (PPP) or Integer-PPP (IPPP) methods. For real-time PPP users, time transfer results are degraded due to the filter-based processing mode, and the degraded accuracy of the real-time satellite orbits and clocks compared to the final ones. The Real-Time Kinematic (RTK) time transfer can significantly reduce the satellite-related errors, but has limits on the baseline length similar to the RTK positioning. Also, the delivery of raw observations instead of State-Space Representation (SSR) products could result in pressure on data transfer and difficulties related to latency and prediction. In this study, the PPP-RTK technique, which combines the advantages of the PPP and the RTK methods, is tested for real-time long-baseline time transfer. As an alternative approach to the above two methods, it allows for the time transfer of long baselines, while not relying on external high-sampling and high-precision satellite clocks. By delivering the satellite clocks and satellite phase biases produced within the PPP-RTK regional network, time differences can be estimated between users and the reference network station, with which stable time transfer between users separated by long baselines can be realized. Using dual-frequency GPS and Galileo data, the PPP-RTK time transfer is tested using approximately a thousand-kilometer-scale network in Europe. The time transfer results between two hydrogen masers, i.e., those on the 884 km baseline BRUX-ONSA and the 920 km baseline WTZR-ONSA, are computed. At an averaging time of 105 s, Modified Allan Deviation (MDEV) at the level of sub-10-15 to 10-15 can be reached when processing the user coordinates in the station fixed, static, or kinematic modes. The median clock residuals can converge to 1 ns and 0.3 ns within 2 min and 15 min, respectively, in the kinematic mode, while in the static and fixed modes the convergence times are shorter. With the augmentation of 150 Low Earth Orbit (LEO) satellites having simulated observations, the clock residuals can converge to 1 ns and 0.3 ns within 30 s and 3.5 min, respectively, for all the three estimation modes.

SVR and ARIMA models as machine learning solutions for solving the latency problem in real-time clock corrections

Real-time precise point positioning (PPP) has become a prevalent technique in global navigation satellite systems (GNSS). However, GNSS real-time users must receive space state representation (SSR) products to correct for satellite clock, orbit, and phase biases. The International GNSS Service (IGS) provides GNSS users with real-time services (RTSs) through different real-time correction SSR products. These products arrive at the GNSS users with some latency, which affects the quality of real-time PPP positioning. The autoregressive integrated moving average (ARIMA) and support vector regression (SVR) models are used in this research to predict those corrections to eliminate the latency effect. ARIMA model reduces the standard deviation by 28% and 13% for GPS and GLONASS constellations, respectively, compared to the real-time solution, which includes the latency effect, the research simulated the latency effect and named it a forced-latency solution, and the SVR model reduces the standard deviation by 28% and 23% for GPS and GLONASS constellations, respectively. The results for the permanent GNSS stations used in this study across different years 2013, 2014, 2015, 2019, and 2021 show a mean reduction in the 3D positioning standard deviation by 13% compared with the forced-latency solution for the ARIMA solution and 9% for the SVR solution. The potential of both models to overcome the latency effect is apparent based on the findings.

GNSS observable-specific phase biases for all-frequency PPP ambiguity resolution

An unwritten rule to resolve GNSS ambiguities in precise point positioning (PPP-AR) is that users should follow faithfully the frequency choices and observable combinations mandated by satellite clock and phase bias providers. Switching to other frequencies of measurements requires that the satellite clocks be converted, albeit in a roundabout way, to agree with the new frequencies of code biases. Satellite phase biases, on the other hand, are prescribed conventionally as wide-lane and narrow-lane combinations, which prevents users from resolving other phase combinations in the case of multi-frequency observables. We therefore develop an approach to compute observable-specific phase biases (phase OSBs) in concert with the legacy, but ambiguity-fixed, satellite clocks to enable PPP-AR over any frequency choices and observable combinations at the user end, i.e., all-frequency PPP-AR. In particular, the phase OSBs on the baseline frequencies (e.g., L1/L2 for GPS and E1/E5a for Galileo) are estimated by decoupling the code OSBs pre-aligned with the satellite clocks; then satellite clocks are re-estimated by holding pre-resolved undifferenced ambiguities and phase OSBs on the baseline frequencies; finally, all third-frequency phase OSBs are determined by introducing the ambiguity-fixed satellite clocks above. We used a global network of multi-frequency GPS/Galileo data over a month to verify this approach. In dual-frequency PPP-AR using GPS L1/L2, L1/L5, Galileo E1/E5a, E1/E5b, E1/E5 and E1/E6 signals, over 95% of wide-lane and narrow-lane ambiguity residuals were within ±0.25 and ±0.15 cycles, respectively, after the code and phase OSB corrections on raw GNSS measurements. As a result, the ambiguity fixing rates reached around 95% in all PPP-AR tests, though it was only the satellite clocks aligned with the GPS L1/L2 and Galileo E1/E5a pseudorange that were applied throughout. We stress that the key to computing such phase OSBs for all-frequency PPP-AR is that the code OSBs have the same bias datum as that of the satellite clocks.

Comparisons of CODE and CNES/CLS GPS satellite bias products and applications in Sentinel-3 satellite precise orbit determination

To resolve undifferenced GNSS phase ambiguities, dedicated satellite products are needed, such as satellite orbits, clock offsets and biases. The International GNSS Service CNES/CLS analysis center provides satellite (HMW) Hatch-Melbourne-Wübbena bias and dedicated satellite clock products (including satellite phase bias), while the CODE analysis center provides satellite OSB (observable-specific-bias) and integer clock products. The CNES/CLS GPS satellite HMW bias products are determined by the Hatch-Melbourne-Wübbena (HMW) linear combination and aggregate both code (C1W, C2W) and phase (L1W, L2W) biases. By forming the HMW linear combination of CODE OSB corrections on the same signals, we compare CODE satellite HMW biases to those from CNES/CLS. The fractional part of GPS satellite HMW biases from both analysis centers are very close to each other, with a mean Root-Mean-Square (RMS) of differences of 0.01 wide-lane cycles. A direct comparison of satellite narrow-lane biases is not easily possible since satellite narrow-lane biases are correlated with satellite orbit and clock products, as well as with integer wide-lane ambiguities. Moreover, CNES/CLS provides no satellite narrow-lane biases but incorporates them into satellite clock offsets. Therefore, we compute differences of GPS satellite orbits, clock offsets, integer wide-lane ambiguities and narrow-lane biases (only for CODE products) between CODE and CNES/CLS products. The total difference of these terms for each satellite represents the difference of the narrow-lane bias by subtracting certain integer narrow-lane cycles. We call this total difference “narrow-lane” bias difference. We find that 3% of the narrow-lane biases from these two analysis centers during the experimental time period have differences larger than 0.05 narrow-lane cycles. In fact, this is mainly caused by one Block IIA satellite since satellite clock offsets of the IIA satellite cannot be well determined during eclipsing seasons. To show the application of both types of GPS products, we apply them for Sentinel-3 satellite orbit determination. The wide-lane fixing rates using both products are more than 98%, while the narrow-lane fixing rates are more than 95%. Ambiguity-fixed Sentinel-3 satellite orbits show clear improvement over float solutions. RMS of 6-h orbit overlaps improves by about a factor of two. Also, we observe similar improvements by comparing our Sentinel-3 orbit solutions to the external combined products. Standard deviation value of Satellite Laser Ranging residuals is reduced by more than 10% for Sentinel-3A and more than 15% for Sentinel-3B satellite by fixing ambiguities to integer values.

Assessing the Performance of Multi-GNSS PPP-RTK in the Local Area

The benefits of an increased number of global navigation satellite systems (GNSS) in space have been confirmed for the robustness and convergence time of standard precise point positioning (PPP) solutions, as well as improved accuracy when (most of) the ambiguities are fixed. Yet, it is still worthwhile to investigate fast and high-precision GNSS parameter estimation to meet user needs. This contribution focuses on integer ambiguity resolution-enabled Precise Point Positioning (PPP-RTK) in the use of the observations from four global navigation systems, i.e., GPS (Global Positioning System), Galileo (European Global Navigation Satellite System), BDS (Chinese BeiDou Navigation Satellite System), and GLONASS (Global’naya Navigatsionnaya Sputnikova Sistema). An undifferenced and uncombined PPP-RTK model is implemented for which the satellite clock and phase bias corrections are computed from the data processing of a group of stations in a network and then provided to users to help them achieve integer ambiguity resolution on a single receiver by calibrating the satellite phase biases. The dataset is recorded in a local area of the GNSS network of the Netherlands, in which 12 stations are regarded as the reference to generate the corresponding corrections and 21 as the users to assess the performance of the multi-GNSS PPP-RTK in both kinematic and static positioning mode. The results show that the root-mean-square (RMS) errors of the ambiguity float solutions can achieve the same accuracy level of the ambiguity fixed solutions after convergence. The combined GNSS cases, on the contrary, reduce the horizontal RMS of GPS alone with 2 cm level to GPS + Galileo/GPS + Galileo + BDS/GPS + Galileo + BDS + GLONASS with 1 cm level. The convergence time benefits from both multi-GNSS and fixing ambiguities, and the performances of the ambiguity fixed solution are comparable to those of the multi-GNSS ambiguity float solutions. For instance, the convergence time of GPS alone ambiguity fixed solutions to achieve 10 cm three-dimensional (3D) positioning accuracy is 39.5 min, while it is 37 min for GPS + Galileo ambiguity float solutions; moreover, with the same criterion, the convergence time of GE ambiguity fixed solutions is 19 min, which is better than GPS + Galileo + BDS + GLONASS ambiguity float solutions with 28.5 min. The experiments indicate that GPS alone occasionally suffers from a wrong fixing problem; however, this problem does not exist in the combined systems. Finally, integer ambiguity resolution is still necessary for multi-GNSS in the case of fast achieving very-high-accuracy positioning, e.g., sub-centimeter level.

Triple-Frequency GPS Un-Differenced and Uncombined PPP Ambiguity Resolution Using Observable-Specific Satellite Signal Biases

The new generations of global navigation satellite system (GNSS) space vehicles can transmit three or more frequency signals. Multi-frequency observations bring a significant improvement to precise point positioning ambiguity resolution (PPP AR). However, the multi-frequency satellite code and phase biases need to be properly handled before conducting PPP AR. The traditional satellite bias correction methods, for example, the commonly used differential code biases (DCB), are limited to the dual-frequency ionosphere-free (IF) case and become more and more difficult to extend to multi-GNSS and multi-frequency cases. In this contribution, we propose the observable-specific signal bias (OSB) correction method for un-differenced and uncombined (UDUC) PPP AR. The OSB correction method, which includes observable-specific satellite code and phase bias correction, can directly apply kinds of OSBs to GNSS original observation data, thus, it is more appropriate for multi-GNSS and multi-frequency cases. In order to verify the performance of multi-frequency UDUC-PPP AR based on the OSB correction method, triple-frequency GPS observation data provided by 142 Multi-GNSS Experiment (MGEX) stations were used to estimate observable-specific satellite phase biases at the PPP service end and some of them were also used to conduct AR at the PPP user end. The experiment results showed: the averaged time-to-first-fix (TTFF) of triple-frequency GPS kinematic UDUC-PPP AR with observable-specific satellite code bias (SCB) corrections could reach about 22 min with about 29% improvement, compared with that without observable-specific SCB corrections; TTFF of triple-frequency static UDUC-PPP AR with observable-specific phase-specific time-variant inter-frequency clock bias (IFCB) corrections took about 15.6 min with about 64.3% improvement, compared with that without observable-specific IFCB corrections.

Real-Time PPP-RTK Performance Analysis Using Ionospheric Corrections from Multi-Scale Network Configurations

The long convergence time required to achieve high-precision position solutions with integer ambiguity resolution-enabled precise point positioning (PPP-RTK) is driven by the presence of ionospheric delays. When precise real-time ionospheric information is available and properly applied, it can strengthen the underlying model and substantially reduce the time required to achieve centimeter-level accuracy. In this study, we present and analyze the real-time PPP-RTK user performance using ionospheric corrections from multi-scale regional networks during a day with medium ionospheric disturbance. It is the goal of this contribution to measure the impact the network dimension has on the ambiguity-resolved user position through the predicted ionospheric corrections. The user-specific undifferenced ionospheric corrections are computed at the network side, along with the satellite phase biases needed for single-receiver ambiguity resolution, using the best linear unbiased predictor. Such corrections necessitate the parameterization of an estimable user receiver code bias, on which emphasis is given in this study. To this end, we process GPS dual-frequency data from four four-station evenly distributed CORS networks in the United States with varying station spacings in order to evaluate if and to what extent the ionospheric corrections from multi-scale networks can improve the user convergence times. Based on a large number of samples, our experimental results showed that sub-10 cm horizontal accuracy can be achieved almost instantaneously in the ionosphere-weighted partially-ambiguity-fixed kinematic PPP-RTK solutions based on corrections from a network with 68 km spacing. Most of the solutions (90%) were shown to require less than 6.0 min, compared to the ionosphere-float PPP solutions that needed 68.5 min. In case of sparser networks with 115, 174 and 237 km spacing, 50% of the horizontal positioning errors are shown to become less than one decimeter after 1.5, 4.0 and 7.0 min, respectively, while 90% of them require 10.5, 16.5 and 20.0 min. We also numerically demonstrated that the user’s convergence times bear a linear relationship with the network density and get shorter as the density increases, for both full and partial ambiguity resolution.

Determination of optimal satellite phase bias combinations for BeiDou triple-frequency PPP-RTK

The precise point positioning real-time kinematic (PPP-RTK) technique enables a single receiver to achieve high-precision and real-time positioning services, mainly relying on integer ambiguity resolution based on satellite phase bias (SPB) information. For the PPP-RTK technique based on uncombined observations, the SPBs of basic frequencies are available only at low precision because of high correlation and therefore need to be updated frequently. Thus, the most appropriate linear combinations of BeiDou triple-frequency SPBs with low update rates for transmission to users must be identified. To acquire combinations of BeiDou triple-frequency SPBs, we use the Z-transformation to process the covariance matrixes corresponding to SPB estimates to generate decorrelated SPB combinations. To meet different circumstances, geometry-free, geometry-fixed and geometry-plus-satellite-clock-fixed variants of PPP-RTK are applied to process data from the regional network. The computed results indicate that the primary stable, secondary and tertiary stable SPB combinations of B1, B2 and B3 frequencies are (0, −1, 1), (1, 2, −3), and (−33, −129, 163) for the geometry-free variant, (0, −1, 1), (1, 2, −3), and (−21, −95, 117) for the geometry-fixed variant, and (0, −1, 1), (1, 3, −4), and (−18, −84, 103) for the geometry-plus-satellite-clock-fixed variant. Furthermore, the statistical results of the SPB series provide stable periods of the primary stable combinations of longer than 7 d. For the secondary and tertiary stable combinations, the stable periods for the geometry-free variant are 12 h and 5 h, for the geometry-fixed variant 12 h and 8 h, and for geometry-plus-satellite-clock-offset variant 12 h and 10 h. These stable periods suggest how often the SPB information should be updated and provide meaningful insight for the allocation of the computing resources of the PPP-RTK server.

Estimating multi-frequency satellite phase biases of BeiDou using maximal decorrelated linear ambiguity combinations

Improving ambiguity resolution (AR) in multi-frequency undifferenced and uncombined precise point positioning (PPP) benefits from accurate uncalibrated phase delays (UPD), which are often estimated from linear combinations of float ambiguities. The traditional linear ambiguity combinations for estimating these UPDs in case of triple-frequency observations are typically extra-wide-lane, wide-lane, and L1 combinations. We proposed the method for estimating UPDs from triple-frequency ambiguities using maximal decorrelated linear ambiguity combinations obtained by the least-squares ambiguity decorrelation adjustment Z-transformation. To validate the quality and availability of estimating UPDs for the BeiDou navigation satellite system, based on maximal decorrelated linear ambiguity combinations, tests using observations from stations of the Crustal Movement Observation Network of China and the Asia-Pacific Reference Frame project are performed using undifferenced and uncombined PPP-AR. The results show the internal precision of combined satellite UPDs estimated from the maximal decorrelated linear ambiguity combinations is better than that estimated from traditional combinations in terms of temporal stability and RMS of posteriori residuals. Furthermore, the statistical results also demonstrated that triple-frequency PPP-AR using the improved UPDs reduces the average convergence time by 8.9 and 12.3% in horizontal and vertical directions, and also improves the positioning accuracy for 3?h of observations by 11.1, 9.1 and 8.3% in the east, north and up directions, respectively, compared with triple-frequency PPP-AR using the UPDs derived from the traditional combinations.

PPP-RTK based on undifferenced and uncombined observations: theoretical and practical aspects

A synthesis of two prevailing global navigation satellite system positioning technologies, namely the precise point positioning and the network-based real-time kinematic, results in the emergence of the PPP-RTK, enabling single-receiver users to achieve high positioning accuracy with reasonable timeliness through integer ambiguity resolution. The realization of PPP-RTK needs to accomplish two sequential tasks. The first task is to determine a class of corrections including, among others, the satellite phase biases (SPBs) at the network level. With these corrections, the second task, then, is to solve for the ambiguity-fixed, absolute position at the user level. In this contribution, we revisit three variants (geometry-free, geometry-fixed and geometry-plus-satellite-clock-fixed) of the undifferenced and uncombined PPP-RTK network model and then point out their implications for practical use. We also carry out a case study using multi-day, dual-frequency global positioning system data from the crustal movement observation network of China stations, aiming to figure out what are the most appropriate linear combinations of the SPBs to be transmitted to the users from the viewpoint of decorrelation, and to assess the static and kinematic positioning performance.

A sequential network approach for estimating GPS satellite phase biases at the PPP-AR producer-side

Ambiguity resolution (AR) in precise point positioning (PPP) requires precise satellite orbit, clocks, and phase biases corrections. Satellite phase biases are fractional hardware corrections which help to retrieve the un-differenced integer carrier phase ambiguities. Satellite corrections can be obtained from the international GNSS service (IGS) or estimated by correction providers called producer-side. We introduce a new PPP-AR observation model and a new sequential network algorithm (SNA) to estimate satellite phase biases. The new model is fully compatible with standard IGS satellite correction products, and it takes advantage of currently available IGS global ionosphere maps to improve the stability of corrections estimation. Furthermore, the proposed model is full-rank per-frequency and per-site and this method simplifies the integration of any additional frequency or site observables in the system of equations. The per-site satellite phase biases method allows users to customize their network solution. In many cases, users only have to estimate the phase biases of a few satellites estimated by few stations to resolve ambiguities of their observed satellites. The novel two-step algorithm provides a good balance between the computational burden, the computer memory load, the efficiency of handling parameters, and the precise estimation of correction parameters. The proposed PPP-AR model and the SNA performance is then validated by estimating satellite phase biases with 1 year of GPS data from a sub-network of IGS stations. A rigorous a posteriori statistical test is performed using data from an independent GPS network. As a result, the precision of WL and L1 ambiguities was improved significantly with the confidence level of P > 99.99% by applying the estimated phase bias corrections to phase observables.

Satellite-Clock Modeling in Single-Frequency PPP-RTK Processing

The real-time kinematic precise point positioning (PPP-RTK) technique enables integer ambiguity resolution by providing singlereceiver users with information on the satellite phase biases next to the standard PPP corrections. Using undifferenced and uncombined observations, rank deficiencies existing in the design matrix need to be eliminated to formestimable parameters. In this contribution, the estimability of the parameters was studied in single-frequency ionosphere-weighted scenario, given a dynamic satellite-clock model in the network Kalman filter. In case of latency of the network corrections, the estimable satellite clocks, satellite phase biases, and ionospheric delays need to be predicted over short time spans. With and without satellite-clock models incorporated in the network Kalman filter, different approaches were used to predict the network corrections. This contribution shows how the predicted network corrections responded to the presence and absence of satellite-clock models. These differences in the predicted network corrections were also reflected in the user positioning results. Using three different 1-Hz global positioning system (GPS) single-frequency data sets, two user stations in one small-scale network were used to compute the positioning results, applying predicted network corrections. The latency of the network products ranges from 3 to 10 s. It was observed that applying strong satellite-clock constraints in the network Kalman filter (i.e., with the process noise of 1 or 0.5mm per square root of second) reduced the root-mean squares (RMS) of the user positioning results to centimeters in the horizontal directions and decimeters in the vertical direction for latencies larger than 6 s, compared to the cases without a satellite-clock model.

GPS receiver phase biases estimable in PPP-RTK networks: dynamic characterization and impact analysis

The integer ambiguity resolution enabled precise point positioning (PPP-RTK) has been proven advantageous in a wide range of applications. The realization of PPP-RTK concerns the isolation of satellite phase biases (SPBs) and other corrections from a network of Global Positioning System (GPS) reference receivers. This is generally based on Kalman filter in order to achieve real-time capability, in which proper modeling of the dynamics of various types of unknowns remains crucial. This paper seeks to gain insight into how to reasonably deal with the dynamic behavior of the estimable receiver phase biases (RPBs). Using dual-frequency GPS data collected at six colocated receivers over days 50–120 of 2015, we analyze the 30-s epoch-by-epoch estimates of L1 and wide-lane (WL) RPBs for each receiver pair. The dynamics observed in these estimates are a combined effect of three factors, namely the random measurement noise, the multipath and the ambient temperature. The first factor can be overcome by turning to a real-time filter and the second by considering the use of a sidereal filtering. The third factor has an effect only on the WL, and this effect appears to be linear. After accounting for these three factors, the low-pass-filtered, sidereal-filtered, epoch-by-epoch estimates of L1 RPBs follow a random walk process, whereas those of WL RPBs are constant over time. Properly modeling the dynamics of RPBs is vital, as it ensures the best convergence of the Kalman-filtered, between-satellite single-differenced SPB estimates to their correct values and, in turn, shortens the time-to-first-fix at user side.