Integer Ambiguity Research Articles

A study on three commonly used GNSS integer estimators with probability of correct estimation

Integer ambiguity resolution plays a key role in applications related to GNSS precise positioning. This contribution focuses on three commonly used integer estimators (IEs), i.e., integer rounding (IR), integer bootstrapping (IB) and integer least-squares (ILS). Contributions are mainly of four aspects. First, the objective function of IR and IB is given, respectively. Second, an upper bound for ILS is proposed. Third, a sorting technique is introduced to tighten the upper bound of IR/ILS after decorrelation. Fourth, the success-rate approximation for IR and ILS with bounded error is developed, respectively. Finally, real-collected data in PPP validate the following arguments. (i) Applying sorting technique after decorrelation can improve the tightness of IR upper bound a lot, but can only slightly tighten the ILS upper bound. (ii) The proposed ILS upper bound is almost as the same tight as the one in PS-LAMBDA software, much tighter than other known upper bounds. Unlike the PS-LAMBDA upper bound, the proposed ILS upper bound has the advantage of time efficiency for real-time applications. (iii) The proposed approximations produce more accuracy approximated success-rate than the frequently-used ambiguity dilution of precision (ADOP) based approximation.

A novel cascading partial ambiguity resolution method of BDS triple-frequency with inertial aiding for kinematic-to-kinematic relative positioning

Abstract The integer ambiguity resolution (AR) of the carrier phase is crucial for high-precision positioning. To solve the problem that the three carrier ambiguity resolution method will not be reliable when the observation quality cannot meet the requirement under the kinematic-to-kinematic condition, a novel cascading partial AR method of BeiDou navigation satellite system triple-frequency with inertial aiding for kinematic-to-kinematic relative positioning is proposed. At first, inertial data is utilized to improve the reliability and robustness of the extra wide lane AR in dynamic environments. Meanwhile, to improve the fixed success rate, the wide lane ambiguity is obtained by a linear combination of dual extra wide lane with inertial aiding indirectly. For the most challenging narrow-lane ambiguity, a Geometry-based enhanced mode is individually developed for the model constraint, in which a partial AR method for narrow lane with float solution filtered is designed to improve the fixed success rate under the GB model. Finally, the simulation and practical experiment were conducted to verify the effectiveness of the proposed method. The results show that the proposed method can effectively improve the fixed success rate of ambiguity under the kinematic-to-kinematic conditions, which can reach about 99.54% and achieve cm-level relative positioning in dynamic environments.

Highly accurate absolute optical transfer delay measurement over a long distance assisted by the pulse time signal

In this paper, an absolute optical transfer delay measurement method based on pulse time signal, pseudo-random code phase, and microwave phase is proposed. By employing two-stage integer ambiguity resolution, not only can a measurement range of several hundred kilometers be achieved, but sub-picosecond level measurement accuracy can also be attained. A test system was built in the laboratory and experimentally verified on a fiber optic link. The experimental results verify an accuracy of ±0.1 ps in measuring an ultrahigh-accuracy optical delay line. In addition, long fiber is also tested, which proves that a measurement range of at least 100 km can be achieved. The refresh rate of the measurement results can reach 100 ms each time.

Estimating observable-specific phase biases of GEO satellites from regional CORS to analyze its contribution in PPP ambiguity resolution

Achieving high-precision positioning and fast convergence has been the objective of Precision Point Positioning (PPP), which is restricted by the integer ambiguity resolution (AR). Recently, the emergence of observable specific code/phase bias (OSB) has gratefully alleviated this phenomenon. However, for the BeiDou Satellite Navigation System (BDS), only MEO/IGSO satellites are applied in BDS PPP-AR, while GEO satellites are generally ignored. It will largely limit the utilization in China or Asia-Pacific region, influencing both the convergence time and the positioning accuracy. To investigate the feasibility of applying GEO satellites for AR, the phase OSB of GEO satellites is estimated through the raw observations in this research. Based on the observations data from regional network in China region, the characteristics of BDS mixed-constellation OSBs are analyzed, and PPP-AR with different processing strategies in both static and kinematic mode are performed. The results indicate that the phase OSBs were relatively stable and the percentage of the estimated residuals for B1I/B3I signals within ±0.15/0.25 cycles were over 95/99% and 92/97% for IGSO and MEO satellites, while GEO satellites were slightly poor, with 89 and 96%, respectively. The average positioning accuracy of static PPP-AR containing GEO satellites was 0.55, 0.56 and 1.43 cm in the east, north and up directions, with improvements of 14.06, 3.44 and 9.48% respect to fixed solutions without GEO satellites. For kinematic mode, when the ambiguity of GEO satellites was solved, the convergence time and the average time to first fix were considerably reduced to 25.51 and 27.12 min, exhibiting the improvement of 26.16 and 14.80% compared to fixed solutions without GEO satellites, respectively. This further demonstrated the considerable applications potential of GEO satellites to regional users.

Compact RTK for expanded area (COREA): a new method for carrier-phase-based satellite augmentation system

This study proposes a new concept of carrier-phase-based satellite augmentation system named “Compact Real-time Kinematic for Expanded Area (COREA),” which provides centimeter-level positioning services across nationwide coverage. The proposed system’s architecture is very similar to that of the satellite-based augmentation system (SBAS), a meter-level aviation safety service. While network real-time kinematic and precise-point-positioning-RTK (PPP-RTK) rely on several densely positioned reference stations, COREA provides carrier-phase-based corrections using a few reference stations with a distance of 400–1000 km. Furthermore, the COREA corrections can be transmitted by satellite signals with extremely low-speed data links of 250 bps, similar to SBAS. This study focused on the generation method for satellite code/phase clock (CPC) corrections, which is the most significant part of the system. We analyzed the user performance of the COREA system constructed in the Midwest and South of the United States with six reference stations. Consequently, satellite CPC corrections are resilient to communication failures and highly accurate in identifying user integer ambiguity. The 95% position accuracy is approximately 1.8 cm horizontally and 7.1 cm vertically, with an average convergence time of 1–3 min using only GPS triple-frequency signals. In summary, the COREA system preserves the hardware architecture of the legacy SBAS while providing centimeter-level services with fast convergence times by utilizing extremely low-speed satellite data links across the country.

An Effective Scheme for Modeling and Compensating Differential Age Errors in Real-Time Kinematic Positioning

In many real-time kinematic (RTK) positioning applications, reference observations are transmitted over wireless links that can experience frequent interruptions or substantial delays. This results in large differential ages between base and rover observations, which, in turn, leads to a deterioration in positioning performance. To bridge the significant age difference, in this work, we propose a simple and effective scheme for modeling and compensating for such errors. Firstly, the overall differential age error was modeled using truncated Taylor expansion. Then, a time-differenced carrier phase (TDCP)-based observation model was established to estimate the errors with the Kalman framework. Since estimating the receiver’s clock error is unnecessary, equivalent transformation and sequential filtering technology were adopted to significantly reduce the computational complexity. Furthermore, a predictor performance monitor was introduced to mitigate the integrity risks that may occur due to model mismatches. The effectiveness of this scheme was validated by static and dynamic field experiments. The static experiment results showed that when the differential age was 60 s, the GPS and BDS satellites’ overall root mean square error (RMSE) with the asynchronous RTK (ARTK) prediction method was 2.8 and 5.5 times that of the proposed method, respectively. Moreover, when the differential age was 120 s, these values were 3.3 and 5.4 times that of the proposed method, respectively. The field experiment results showed that when the differential age was 60 s, the integer ambiguity fixed rate and false fixed rate of the ARTK method were 0.90 and 1.63 times that of the proposed method, respectively. Finally, at a 120 s differential age, these values were 0.78 and 4.78 times that of the proposed, respectively.

Investigation of the accuracy of BeiDou, QZSS and QZSS/BeiDou satellites configuration for short, medium and long baselines in the Asia-Pacific regions

Abstract The field of satellite navigation has seen significant advancements due to the fast development of multi-constellation Global Navigation Satellite Systems (GNSS). Around 150 satellites will be in service when all six systems – GPS, GLONASS, Galileo, BeiDou, QZSS, and NAVIC – are launched by 2030, offering both enormous potential and advantages for research and engineering applications. This study used an experiment on the accuracy, particularly for short, medium, long baselines (Wide Lane ambiguity solution) of the BeiDou, QZSS and QZSS/BeiDou combinations. It showed that with the integration of BeiDou/QZSS static measurements in the study region millimetre-centimetre accuracy for short, medium, and long baselines can be attained. Based on the results of this study, it can be concluded that the 1st (QZSS/BeiDou), 2nd (BeiDou), and 3rd (QZSS) strategies feature different horizontal accuracies for all categories. The obtained results with different satellite configurations for the Fixed-Wide-Lane integer ambiguity solution are compared with each other. Accuracy at the short baseline (BeiDou, QZSS, and BeiDou/QZSS satellites) was obtained in the range of 0.5–0.7 cm. For the medium baseline, it was computed around 1.8–82 cm. For the long baseline, the accuracy was 5.6–13.3 cm.

The Modified Ambiguity Function Approach with regularization for instantaneous precise GNSS positioning

Abstract The Modified Ambiguity Function Approach (MAFA) implicitly conducts the search procedure of carrier phase GNSS integer ambiguity resolution (IAR) in the coordinate domain using the integer least squares (ILS) principle, i.e. MAFA-ILS. One of the still open scientific problems is an accurate definition of the search region, especially in the context of instantaneous IAR. In doing so, the float solution results, which encompass float position (FP) and its variance-covariance (VC) matrix, must be improved as these are necessary for defining the search region. For this reason, the ambiguity parameters are separately regularized, and then the baseline parameters are conditioned on regularized float ambiguities. The conditional-regularized estimation is thus designed, obtaining the regularized FP (RFP) and its VC-matrix. This solution is promising because its accuracy is enhanced in the sense of mean squared error (MSE) thanks to the improved precision at the cost of regularized bias. The optimal regularization parameter (RP) values obtained for ambiguity parameters balance the contributions of improved precision and bias in the regularized float baseline solution’s MSE. Therefore, the regularized search region is defined accurately in the coordinate domain to contain such approximate coordinates that more frequently give the correct ILS solution. It also contains fewer MAFA-ILS candidates, improving the search procedure’s numerical efficiency. The regularized ILS estimator performs well with the presence of bias, increasing the probability of correct IAR in the coordinate domain.

Study of high-precision time transfer method enhanced by PPP-AR/PPP-RTK

Abstract With the continuous development and improvement of Global Navigation Satellite Systems (GNSS), the use of GNSS for high-precision time transfer has become increasingly mature. To delve into the contribution of information-enhanced GNSS PPP to time transfer performance, this paper focuses on the comprehensive evaluation and analysis of the time transfer performance of PPP-AR and PPP-RTK. Using GPS as a case study, experimental results demonstrate that the average success fixing rate of PPP integer ambiguity resolution across five stations is 93.57%, with initial fixing times consistently below 20 minutes. The stability of the station clock offset sequence of PPP-AR is better than that of PPP floating solution. Compared with the PPP floating solution, the average improvement of PPP-AR stability is 16.54%. Furthermore, using PPP-AR for time transfer not only enhances the stability of the time transfer link clock offset sequence, but also improves its noise level. The stability is increased by 14.41% on average, and the noise level is improved by 9.11% on average. PPP-RTK also improves the stability and noise level of time transfer even better than PPP-AR. However, as the reference network scale increases, the accuracy of the interpolated atmospheric delay correction decreases, impacting the performance of PPP-RTK.

Satellite orientation modelling with quaternions and its impact on BDS-3 PPP-AR

Satellite orientation is a fundamental component of processing high-precision data from the Global Navigation Satellite System (GNSS). However, insufficient handling of satellite orientation may substantially affect the quality of the precise point positioning (PPP) solutions. This research investigates the influence of satellite orientation using satellite attitude quaternions on the third-generation BeiDou Navigation Satellite System (BDS-3) PPP. To illustrate the impact of satellite attitude, both float and PPP with integer ambiguity resolution (PPP-AR) were undertaken using observations from a multi-GNSS Experiment (MGEX) network. To fully evaluate the effect of the quaternions on PPP, a comprehensive comparison between BDS-3 and Galileo constellations was conducted using nominal attitude and quaternions. The findings indicate a significant performance difference between BDS-3 and Galileo satellites in phase observations after resolving integer ambiguities, with Galileo consistently demonstrating superior phase residuals across various receivers. When it comes to positioning accuracy, the utilisation of quaternions improves coordinate precision across both constellations. Furthermore, the quaternions enhance the stability of time series for the positioning biases and observational residuals.

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

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

GNSS carrier phase common-view precision time transfer using the ionospheric-weighted single-differenced model

ABSTRACT High-precision global navigation satellite system (GNSS) time transfer depends on the carrier phase observations, and ionospheric delay affects rapid integer ambiguity resolution. Therefore, this work applies the ionospheric-weighted single-differenced model. Experimental results indicate that for a time-link of 67.9 km, the time transfer accuracy of different single-system is better than 0.03 ns. Compared with single-system, multi-system demonstrates 8% to 14% improvement in accuracy and improvement of no more than 31% in frequency stability. Additionally, compared with the case without ionospheric constraint, imposing ionospheric constraints can improve performance of time transfer, but the gain decreases as the time-link length increases.

An improved wide-lane ambiguity resolution method for kinematic smartphone positioning

Abstract The release of GNSS raw data on Android smartphones provides the potential for high-precision smartphone positioning using multi-constellation and multi-frequency signals. However, severe multipath and low observation quality in kinematic environments make double-differenced uncombined ambiguities difficult to resolve reliably. To address this, the paper proposes an improved wide-lane (WL) integer ambiguity resolution (IAR) method that combines integer rounding and the Least-Square AMBiguity Decorrelation Adjustment (LAMBDA) methods. The proposed method achieved fix rates of 57% to 70% in challenging environments, with an average improvement of 7 · 7% in horizontal positioning accuracy compared to the float solution. The traditional partial integer rounding method only improved accuracy by 1 · 1%.

Crowdsourcing RTK: a new GNSS positioning framework for building spatial high-resolution atmospheric maps based on massive vehicle GNSS data

High-quality spatial atmospheric delay correction information is essential for achieving fast integer ambiguity resolution (AR) in precise positioning. However, traditional real-time precise positioning frameworks (i.e., NRTK and PPP-RTK) depend on spatial low-resolution atmospheric delay correction through the expensive and sparsely distributed CORS network. This results in limited public appeal. With the mass production of autonomous driving vehicles, more cost-effective and widespread data sources can be explored to create spatial high-resolution atmospheric maps. In this study, we propose a new GNSS positioning framework that relies on dual base stations, massive vehicle GNSS data, and crowdsourced atmospheric delay correction maps (CAM). The map is easily produced and updated by vehicles equipped with GNSS receivers in a crowd-sourced way. Specifically, the map consists of between-station single-differenced ionospheric and tropospheric delays. We introduce the whole framework of CAM initialization for individual vehicles, on-cloud CAM maintenance, and CAM-augmented user-end positioning. The map data are collected and preprocessed in vehicles. Then, the crowdsourced data are uploaded to a cloud server. The massive data from multiple vehicles are merged in the cloud to update the CAM in time. Finally, the CAM will augment the user positioning performance. This framework forms a beneficial cycle where the CAM’s spatial resolution and the user positioning performance mutually improve each other. We validate the performance of the proposed framework in real-world experiments and the applied potency at different spatial scales. We highlight that this framework is a reliable and practical positioning solution that meets the requirements of ubiquitous high-precision positioning.

An adaptive fusion algorithm for coastal sea altimetry based on dual-frequency Beidou-R carrier phase.

This article proposes an integer ambiguity determination method based on Beidou system-reflectometry (Beidou-R) observations of the carrier phase at the B1I and B3I frequencies. To enhance the accuracy of sea surface height (SSH) estimation, this study introduces a parallel filtering algorithm and an adaptive iterative fusion algorithm, enabling data fusion based on the variance at B1I and B3I frequencies. To validate and evaluate the proposed method, a coastal experiment was conducted at the Shenxian River. In this experiment, reflected signals from GEO and IGSO satellites were collected. Data analysis reveals that the method is effective, demonstrating that the root mean square error (RMSE) of SSH achieves 2.85 cm and 2.89 cm for PRN 04 and PRN 33, respectively. Furthermore, the impact of the elevation angle on measurement accuracy is analyzed. This study aims to propose a method to enhance coastal sea surface height estimation, offering potential advancements in sea surface altimetry.

On the optimality of DIA-estimators: theory and applications

In this contribution, we introduce, in analogy to penalized ambiguity resolution, the concept of penalized misclosure space partitioning, with the goal of directing the performance of the DIA-estimator towards its application-dependent tolerable risk objectives. We assign penalty functions to each of the decision regions in misclosure space and use the distribution of the misclosure vector to determine the optimal partitioning by minimizing the mean penalty. As each minimum mean penalty partitioning depends on the given penalty functions, different choices can be made, in dependence of the application. For the DIA-estimator, we introduce a special set of penalty functions that penalize its unwanted outcomes. It is shown how this set allows one to construct the optimal DIA-estimator, being the estimator that within its class has the largest probability of lying inside a user specified tolerance region. Further elaboration shows how these penalty functions are driven by the influential biases of the different hypotheses and how they can be used operationally. Hereby the option is included of extending the misclosure partitioning with an additional undecided region to accommodate situations when it will be hard to discriminate between some of the hypotheses or when identification is unconvincing. By extending the analogy with integer ambiguity resolution to that of integer-equivariant ambiguity resolution, we also introduce the maximum probability estimator within the similar larger class.

Improved GRACE-FO kinematic orbit determination with epoch-difference KBR during the GPS flex power period

The flex power technology in satellite navigation systems enhances anti-jamming capabilities but can impact the quality of GPS observations and the accuracy of low Earth orbit determination, such as GRACE Follow-On (GRACE-FO) mission. This study investigates the influence of GPS flex power on Hatch-Melbourne-Wübbena (HMW) linear combinations and GRACE-FO kinematic orbits from January 1 to September 30, 2020. Epoch-differenced K-Band Ranging (KBR) data is introduced in orbit determination during the flex power period to improve both absolute and inter-satellite relative accuracy. The analysis indicates that the influence of early flex power (before February 13, 2020) on HMW combinations and orbits is minimal, whereas the effect of later flex power (after February 14, 2020) is significant: (1) HMW combinations exhibit notable systematic discontinuities even with elevation angles greater than 50 degrees, causing the fixing rate of wide-lane ambiguities to drop from 96 % to 80 %. (2) Kinematic absolute orbits show significant deteriorations of approximately 9 mm and 4 mm in the three-dimensional direction for float and integer ambiguity resolution (FAR and IAR), while relative accuracy of FAR and IAR orbits decreases by 50 % and 46 %, respectively. However, using epoch-differenced KBR (DKBR) data, the accuracy of absolute orbits could be increased by up to 15 % and the accuracy of relative orbits could be improved by at least 69 %, which showcases a positive effect. Thus, this can be considered as an alternative method to improve the accuracy of GRACE-FO orbit during the flex power period.

A novel stop-and-go kinematic positioning method for PPP–RTK

ABSTRACT The technology of precise point positioning – real time kinematic (PPP – RTK) can realize regional accurate positioning. However, poor quality atmospheric corrections can greatly impact the integer ambiguity resolution. The low reliability of interpolated corrections may arise from blocked transmission of the updated corrections, atmospheric disturbance, and when the user is located outside the service area. Therefore, to maintain a continuous ambiguity-fixed state in an inaccurate correction observation environment, we propose a novel stop-and-go (SAG) kinematic positioning method. Integer ambiguities inherited from the last epoch are used to estimate the ionospheric errors of the current epoch using an inversion calculation method. Unmodeled errors contained in the ionospheric corrections are also considered. We then experimentally compare the proposed method and traditional PPP – RTK and SAG methods with the results of the PPP-ambiguity resolution calculation using geomagnetic storm-influenced data. All indicators, including the number of fixed ambiguities, errors of corrections, successful fix rate, and correct fix rate, confirm the superiority of the proposed method. The proposed method achieves a horizontal correct fix rate of 98% and maintains the vertical correct fix rate above 96% throughout the day. It is proved that for most of the time the novel method can provide continuous ambiguity-fixed solutions for users in an inaccurate correction observation environment.

Low-cost GNSS antennas in precise positioning: a focus on multipath and antenna phase center models

The rapid growth of the GNSS equipment market has put affordable receivers and antennas capable of receiving satellite signals into the hands of users. High positioning accuracy, previously achievable only with high-grade devices, is becoming possible with low-cost ones. However, simplifications in the design of these devices, intended to reduce the manufacturing cost, affect their capabilities. This study analyzes the positioning accuracy that may be achieved with recent low-cost antennas. We put particular stress on investigating the susceptibility of such antennas to the multipath effect and implications from the quality of the antenna phase center models. The positioning performance is assessed by employing the Precise Point Positioning method with the integer ambiguity resolution of phase observations. The results obtained with three low-cost antennas are validated against three high-grade antennas. We reveal a two-to threefold decrease in positioning performance with low-cost antennas compared to high-quality equipment. However, positioning accuracy increased when a low-cost antenna with a phase correction model was used, particularly for the eastern component of coordinate bias. In addition, a significant susceptibility of low-cost antennas to the multipath effect was confirmed, especially for GPS L2 and Galileo E5a signals.

Orbit and clock products for quad-system satellites with undifferenced ambiguity fixing approach

Integer Ambiguity Resolution (IAR) can significantly improve the accuracy of GNSS Precise Orbit Determination (POD). Traditionally, the IAR in POD is achieved at the Double Differenced (DD) level. In this contribution, we develop an Un-Differenced (UD) IAR method for Global Positioning System (GPS)+ BeiDou Navigation Satellite System (BDS) + Galileo navigation satellite system (Galileo)+ Global'naya Navigatsionnaya Sputnikovaya Sistema (GLONASS) quad-system POD by calibrating UD ambiguities in the raw carrier phase and generating the so-called carrier range. Based on this method, we generate the UD ambiguity-fixed orbit and clock products for the Wuhan Innovation Application Center (IAC) of the International GNSS Monitoring and Assessment System (iGMAS). One-year observations in 2020 from 150 stations are employed to investigate performance of orbit and clock products. Notably, the UD Ambiguity Resolution (AR) yields more resolved integer ambiguities than the traditional DD AR, scaling up to 9%, attributable to its avoidance of station baseline formation. Benefiting from the removal of ambiguity parameters, the computational efficiency of parameter estimation undergoes a substantial 70% improvement. Compared with the float solution, the orbit consistencies of UD AR solution achieve the accuracy of 1.9, 5.2, 2.8, 2.1, and 2.7 cm for GPS, BeiDou-2 Navigation Satellite System (BDS-2), BeiDou-3 Navigation Satellite System (BDS-3), Galileo, and GLONASS satellites respectively, reflecting enhancements of 40%, 24%, 54%, 34%, and 42%. Moreover, the standard deviations of Satellite Laser Ranging (SLR) residuals are spanning 2.5–3.5 cm, underscoring a comparable accuracy to the DD AR solution, with discrepancies below 5%. A notable advantage of UD AR lies in its capability to produce the Integer Recovered Clock (IRC), facilitating Precise Point Positioning (PPP) AR without requiring additional Uncalibrated Phase Delay (UPD) products. To assess the performance of quad-system kinematic PPP based on IRC, a network comprising 120 stations is utilized. In comparison to the float solution, the IRC-based PPP AR accelerates convergence time by 31% and enhance positioning accuracy in the east component by 54%.