Ambiguity Resolution Success Research Articles

Single-station PPP-RTK: correction latency and ambiguity resolution performance

Single-station PPP-RTK is a special case of PPP-RTK in that corrections are computed, instead of a network, by only one single GNSS receiver. The present contribution aims to develop a framework to generate multi-epoch, single-station corrections, thereby providing PPP-RTK users the capability to time-predict corrections that are subject to time delay or latency. By presenting analytical expressions of the user ambiguity variance matrix, we address how the ambiguity resolution performance is driven by the correction latency and therefore by the uncertainty involved in the time-prediction of single-station PPP-RTK corrections. Supported by numerical results, our analytical study shows that the number of satellites and number of frequencies work in tandem to enable one to increase the correction latency, yet ensuring successful single-receiver ambiguity resolution.

Impact of robot antenna calibration on dual-frequency smartphone-based high-accuracy positioning: a case study using the Huawei Mate20X

The access to Android-based Global Navigation Satellite Systems (GNSS) raw measurements has become a strong motivation to investigate the feasibility of smartphone-based positioning. Since the beginning of this research, the smartphone GNSS antenna has been recognized as one of the main limitations. Besides multipath (MP), the radiation pattern of the antenna is the main site-dependent error source of GNSS observations. An absolute antenna calibration has been performed for the dual-frequency Huawei Mate20X. Antenna phase center offset (PCO) and variations (PCV) have been estimated to correct for antenna impact on the L1 and L5 phase observations. Accordingly, we show the relevance of considering the individual PCO and PCV for the two frequencies. The PCV patterns indicate absolute values up to 2 cm and 4 cm for L1 and L5, respectively. The impact of antenna corrections has been assessed in different multipath environments using a high-accuracy positioning algorithm employing an undifferenced observation model and applying ambiguity resolution. Successful ambiguity resolution is shown for a smartphone placed in a low multipath environment on the ground of a soccer field. For a rooftop open-sky test case with large multipath, ambiguity resolution was successful in 19 out of 35 data sets. Overall, the antenna calibration is demonstrated being an asset for smartphone-based positioning with ambiguity resolution, showing cm-level 2D root mean square error (RMSE).

Models, methods and assessment of four-frequency carrier ambiguity resolution for BeiDou-3 observations

One of the most significant advantages of multiple frequencies is that it can improve the success rate of ambiguity resolution, such as the three-frequency carrier ambiguity resolution (TCAR) method. So far, the global BeiDou-3 navigation satellite system already provides four frequencies. Hence, we systematically study the models and methods of four-frequency carrier ambiguity resolution (FCAR) by using real BeiDou-3 data. First, the models of four-frequency linear combinations are given, and the optimal linear combinations are found out based on certain optimal criteria. Second, two typical methods, including geometry-free and geometry-based methods, are studied. In the end, the real BeiDou-3 data are used to evaluate the performance of the FCAR method, where two strategies, i.e., single-epoch and multi-epoch solutions, are both applied. The results indicate that the FCAR method can offer more high-quality virtual signals in quantity and quality than the TCAR method. By using the real BeiDou-3 data with different lengths ranging from 4.9 m to 61.6 km, three high-quality and independent signals can fix the ambiguities instantaneously with an approximately 100% success rate. Then the fourth independent signal can be fixed with high efficiency and success rate. Therefore, the FCAR method is promising in large-scale real-time precise positioning, where the successful ambiguity resolution may take tens of minutes to hours.

The ADOP and PDOP: Two Complementary Diagnostics for GNSS Positioning

Ambiguity dilution of precision (ADOP) and position dilution of precision (PDOP) are two popular scalar-diagnostics used in Global Navigation Satellite System (GNSS) positioning. Where the ADOP is a predictor for carrier-phase ambiguity resolution performance, the PDOP is meant to predict the receiver-satellite geometry’s capability for precise positioning. We will show, however, that although the PDOP works well for code-based positioning, one has to exercise great care in using the PDOP for real-time kinematic (RTK) positioning. We show that the ADOP and PDOP have distinct behaviors, an important consequence of which is that one can have time periods with small PDOPs, and thus seemingly good geometry for precise positioning, but at the same time large ADOPs, thus showing that successful ambiguity resolution and therefore precise positioning will not be possible. Also, the reverse situation may occur, i.e., having large PDOPs with small ADOPs. In such a situation, the large PDOPs should not automatically lead to the conclusion of poor position performance, because the large gain that ambiguity resolution brings will often still make precise positioning possible. We will analyze and explain this complementary behavior of the PDOP and ADOP and demonstrate this both analytically and empirically. For this analysis we use real Global Positioning System (GPS) single- and multifrequency signals and GPS/Quasi-Zenith Satellite System (QZSS), GPS/Navigation with Indian Constellation (NAVIC) L5 signals of two baselines located in Perth, Australia.

A 4D tomographic ionospheric model to support PPP-RTK

Successful implementation of integer ambiguity resolution enabled precise point positioning (aka PPP-RTK) algorithms is inextricably linked to the ability of a user to perform near real-time positioning by quickly and reliably resolving the integer carrier-phase ambiguities. In the PPP-RTK technique, a major barrier to successful ambiguity resolution is the unmodelled impact of the ionosphere. We present a 4D ionospheric tomographic model that computes in real time the ionospheric electron density as a linear combination of basis functions, namely B-splines. The results show that when the ionospheric estimates are provided as atmospheric corrections for a PPP-RTK end-user, the time to fix its horizontal position below 10 cm is around 20 epochs (the sample rate is 30 s) at the 90 % of the cumulative distribution function (CDF), as opposed to the time it takes when no external corrections are provided, which is around 80 epochs at 90 % of the CDF.

GLONASS ambiguity resolution

A new integer-estimable GLONASS FDMA model is studied and analysed. The model is generally applicable, and it shows a close resemblance with the well-known CDMA models. The analyses provide insights into the performance characteristics of the model and concern a variety of different ambiguity-resolution critical applications. This will be done for geometry-free, geometry-fixed and several geometry-based formulations. Next to the analyses of the model’s instantaneous ambiguity-resolved positioning and attitude determination capabilities, we show the ease with which the model can be combined with existing CDMA models. We thereby present the instantaneous ambiguity-resolution performances of integrated L1 GPS + GLONASS, both for high-grade geodetic and mass-market receivers. We also consider the potential of the single-frequency combined model for mixed-receiver processing, particularly for the case the between-receiver GLONASS pseudorange data are biased. In all cases, the speed of successful ambiguity resolution is studied as well as the precision with which positioning is determined. Software routines for constructing the model are also provided.

Improving the Stochastic Model for VRS Network-Based GNSS Surveying

Abstract The VRS network-based technique has become the main precise GNSS surveying method especially for medium-range baselines (approximately 20-70 km). The key concept of this approach is to use the observables of multiple reference stations to generate the network correction in the form of a virtual reference station for mitigating distance-dependent errors including atmospheric effects and orbital uncertainty at the user’s location. Numerous GNSS data processing strategies have been adopted in the functional model in order to improve both the positioning accuracy and the success of ambiguity resolution. However, it is impossible to completely model the aforementioned errors. As a result, the unmodelled residuals still remain in the virtual reference station observables when the least squares estimation is employed. An alternative approach to deal with these residuals is to construct a more realistic stochastic model whereby the variance-covariance matrix is assumed to be homoscedastic. This research aims to investigate a suitable stochastic model used for the VRS technique. The rigorous statistical method, MINQUE has been applied to estimate the variance-covariance matrix of the double-difference observables for a virtual reference station to rover baseline determination. The findings of the comparison to the equal-weight model and the satellite elevation-based model indicated that the MINQUE procedure could enhance the positioning accuracy. In addition, the reliability of ambiguity resolution is also improved.

Australia-first high-precision positioning results with new Japanese QZSS regional satellite system

The Japanese Quasi-Zenith Satellite System (QZSS) has recently (October 2017) reached its first 4-satellite constellation. In this contribution, the standalone performance of this 4-satellite QZSS constellation is assessed by means of its triple-frequency (L1 + L2 + L5) real-time kinematic (RTK) integer ambiguity resolution and precise positioning capabilities. Our analyses are carried out for data collected in Perth, Australia, and include a study of the noise characteristics of the QZSS code and phase data, particularly concerning their precision, time correlation and multipath. Our results show that while the phase observations on different frequencies are of similar precision, the code observations on different frequencies show considerably different precisions and can be ordered, from high to low, as L5, L2 and L1. As to positioning and ambiguity resolution, we demonstrate that the Position Dilution Of Precision (PDOP) and the Ambiguity Dilution Of Precision (ADOP) exhibit complementary characteristics, both of which are important for predicting precise positioning capabilities. We show that despite the large PDOPs, the ADOPs are sufficiently small to indicate (almost) instantaneous successful ambiguity resolution. This is confirmed by our empirical data analyses, demonstrating that instantaneous ambiguity resolution is feasible, despite the relatively poor 4-satellite receiver-to-satellite positioning geometry over Australia, thus showing that already now centimeter-level stand-alone QZSS positioning is possible with the current 4-satellite constellation (February–March 2018).

Low-cost, 4-system, precise GNSS positioning: a GPS, Galileo, BDS and QZSS ionosphere-weighted RTK analysis

With the combination of emerging GNSSs, single-frequency (SF) precise RTK positioning becomes possible. In this contribution we evaluate such low-cost ublox receiver and antenna performance when combining real data of four CDMA systems, namely L1 GPS, E1 Galileo, L1 QZSS, and B1 BDS. Comparisons are made to more expensive dual-frequency (DF) GPS receivers and antennas. The formal and empirical ambiguity success rates and positioning precisions will first be evaluated while making use of L1+E1, so as to investigate whether instantaneous SF RTK is possible without the need of B1 BDS or L1 QZSS. This is followed by an analysis of the SF 4-system model performance when the residual ionosphere can be ignored and modeled as a function of the baseline length, respectively. The analyses are conducted for a location in Dunedin, New Zealand, and compared to Perth, Australia with the better visibility of BDS and QZSS. The results indicate that successful instantaneous and precise RTK positioning is feasible while using L1 GPS and E1 Galileo data, and that the SF 4-system model is competitive to DF GPS even when residual ionospheric delays are present. We finally demonstrate that when the impact from the ionosphere increases and more than one epoch is needed for successful ambiguity resolution, the SF 4-system model performance can still remain competitive with the DF GPS receivers. This is particularly true in Perth with more satellites and when higher than customary elevation cut-off angles need to be used to avoid low-elevation multipath.

Mitigating BeiDou Satellite-Induced Code Bias: Taking into Account the Stochastic Model of Corrections

The BeiDou satellite-induced code biases have been confirmed to be orbit type-, frequency-, and elevation-dependent. Such code-phase divergences (code bias variations) severely affect absolute precise applications which use code measurements. To reduce their adverse effects, an improved correction model is proposed in this paper. Different from the model proposed by Wanninger and Beer (2015), more datasets (a time span of almost two years) were used to produce the correction values. More importantly, the stochastic information, i.e., the precision indexes, were given together with correction values in the improved model. However, only correction values were given while the precision indexes were completely missing in the traditional model. With the improved correction model, users may have a better understanding of their corrections, especially the uncertainty of corrections. Thus, it is helpful for refining the stochastic model of code observations. Validation tests in precise point positioning (PPP) reveal that a proper stochastic model is critical. The actual precision of the corrected code observations can be reflected in a more objective manner if the stochastic model of the corrections is taken into account. As a consequence, PPP solutions with the improved model outperforms the traditional one in terms of positioning accuracy, as well as convergence speed. In addition, the Melbourne-Wübbena (MW) combination which serves for ambiguity fixing were verified as well. The uncorrected MW values show strong systematic variations with an amplitude of half a wide-lane cycle, which prevents precise ambiguity determination and successful ambiguity resolution. After application of the code bias correction models, the systematic variations can be greatly removed, and the resulting wide lane ambiguities are more likely to be fixed. Moreover, the code residuals show more reasonable distributions after code bias corrections with either the traditional or the improved model.

GLONASS ionosphere-free ambiguity resolution for precise point positioning

Current GLONASS satellites transmit signals based on the frequency division multiple access (FDMA) technology. Due to equipment delays occurring within GNSS receivers, GLONASS carrier phase and code observations are contaminated by inter-frequency biases. As a consequence, GLONASS ambiguity parameters in long-baseline processing are typically estimated as float values. In this paper, a strategy is investigated which benefits from the frequency spacing of GLONASS frequencies on the L\(_{1}\) and L\(_{2}\) bands, allowing for an ionosphere-free ambiguity with a wavelength of approximately 5 cm to be defined; therefore, avoiding the problematic wide-lane ambiguity resolution. Based on 12 independent baselines with a mean inter-station distance of about 850 km over a 1-week period, it is demonstrated that close to 95 % of the estimated double-differenced ionosphere-free ambiguities are within 0.15 cycles of an integer, thereby suggesting that long-baseline ambiguity resolution can be achieved for GLONASS. Applying between-station ambiguity constraints in precise point positioning (PPP) solutions was found to improve longitudinal repeatability in static mode by more than 20 % for sessions between 2 and 6 h in duration. In kinematic mode, only limited improvements were made to the initial convergence period since the short wavelength of GLONASS ionosphere-free ambiguities requires the solution to be nearly converged before successful ambiguity resolution can be achieved.

A new approach to modernized GPS phase-only ambiguity resolution over long baselines

With the advent of modernized GPS, triple-frequency phase measurements (L1, L2, and L5) are available for civil use. The successful ambiguity resolution of the integer ambiguities of the phase measurements will be the key to centimeter-level positioning. In order to achieve ambiguity resolution over long baselines, code measurements (pseudorange) are regularly incorporated with the phase measurements in the observation model. However, code multipath affects ambiguity resolution and thus completely eliminating the influence is an important issue. Therefore, the present study proposes an approach that uses only the phase measurements in the observation model. The proposed approach has three steps and focuses on resolving the integer ambiguities of the triple-frequency phase measurements. Simulation baseline data were processed by the proposed approach and the results show that the integer ambiguities of the phase measurements can be successfully resolved and that satellite geometry is an important factor for the phase-only ambiguity resolution performance. Real triple-frequency GPS data from currently available Block IIF satellites were also processed to demonstrate the feasibility of the proposed approach.

Enhancement of the accuracy of single-epoch GPS positioning for long baselines by local ionospheric modelling

Single-epoch relative GPS positioning has many advantages, especially for monitoring dynamic targets. In this technique, errors occurring in previous epochs cannot affect the position accuracy at the current epoch, but careful processing is required, and resolving carrier phase ambiguities is essential. Statistical ambiguity resolution functions have been used to determine the best values of these ambiguities. The function inputs include as a minimum the known base station position, the approximate roving antenna position, and the dual-frequency carrier phase measurements from both receivers. We investigate different solutions to find the ambiguity function inputs that achieve the highest ambiguity resolution success rate. First, we address the rover seed position. A regionally filtered undifferenced pseudorange coordinate solution proves better than a double-differenced one. Multipath errors approximately repeat themselves every sidereal day in the case of static or quasi-static antennas; applying a sidereal filter to the pseudorange-derived positions mitigates their effects. Second, we address the relative carrier phase measurements, which for medium to long baselines are significantly affected by ionospheric propagation errors imperfectly removed during differencing. In addition to the International GNSS Service ionospheric model, we generate a local pseudorange-based ionospheric correction. Applying this correction improves the quality of the phase measurements, leading to more successful ambiguity resolution. Temporally smoothing the correction by means of a Kalman filter further improves the phase measurements. For baselines in the range 60–120 km, the mean absolute deviation of single-epoch coordinates improves to 10–20 cm, from 30–50 cm in the default case.

Correlation Analysis of Multipath Effects in GPS-Code and Carrier Phase Observations

Multipath error is a limiting factor for successful ambiguity resolution in precise real-time kinematic GPS positioning, both in the carrier phase and pseudorange measurements. Understanding the temporal characteristics of the multipath errors enables the stochastic modelling of these errors. To do this, dual frequency data as well as data from short baselines of several metres apart are observed and analyzed under different multipath conditions. Further, a covariance model (first order Gauss-Markov in this study), accounting for the multipath errors can be created if the temporal characteristics of the multipath errors are derived. The correlation time parameter is an important factor in the analysis of a random process hence this study. Different baselines are observed to estimate the correlation time of the multipath errors. Longer correlation times are obtained in the carrier phase observations due to the double difference process. Not accounting for the long correlation time usually leads to an overestimation of the accuracy and may affect the determination of the ambiguity parameters. This has been shown through a simple numerical investigation.

Local ionospheric modelling of GPS code and carrier phase observations

A limiting factor for successful ambiguity resolution in precise GPS positioning is the existence of un-modelled ionospheric errors in both the carrier phase and the pseudorange measurements. In this study, the ionospheric delay is modelled using a four-parameter polynomial utilizing dual frequency observations in the region of study in southern Iran. Thereafter, the corrected and non-corrected observations for ionospheric delay are compared with the ionospheric-free solution of the same observations. Two baselines (15 km and 72 km) are used for this comparison. An improvement of 0.22 ppm and 0.07 ppm is achieved, respectively, for the two baselines. Further, the ionospheric delay is modelled using two other methods, i.e. the Klobuchar and divergence models. The divergence model uses single frequency observations. The improvement of the results is restricted as the noise level in code observations is high. An improvement of 0.15 ppm for the 72 km baseline and no improvement for the 15 km baseline are observed using the divergence model. The Klobuchar model corrects 50%-60% of the ionospheric errors in this study.

On InSAR Ambiguity Resolution For Deformation Monitoring

On InSAR Ambiguity Resolution For Deformation Monitoring

Impact of systematic errors on precise long-baseline kinematic GPS positioning

Many kinematic GPS applications rely on high accuracy, which usually requires the ambiguities to be fixed. Normally, a reference station in the rover’s vicinity is needed for successful ambiguity resolution. Alternatively, a network surrounding the rover and allowing one to derive area correction parameters is needed. Unfortunately, both approaches are not feasible in certain situations. This paper is a contribution to precise kinematic positioning over long baselines. Atmospheric refraction becomes critical in the error budget, but progress has been made to use numerical weather models to derive tropospheric corrections, for instance. The spatial correlation of both ionospheric and tropospheric propagation delays is investigated in this paper and special attention is paid on the systematic error behavior of tropospheric refraction. The principles developed are applied to an extended reliability test of the ambiguities. Finally, it is demonstrated in positioning experiments that kinematic positioning retrieval with fixed ambiguities is actually possible for baselines between 150 and 300 km with an accuracy of approximately 2 cm in post-mission processing.

On the approximation of the integer least-sqaures success rate: which lower or upper bound to use?

The probability of correct integer estimation, the success rate, is an important measure when the goal is fast and high precision positioning with a Global Navigation Satellite System. Integer ambiguity estimation is the process of mapping the least-squares ambiguity estimates, referred to as the float ambiguities, to an integer value. It is namely known that the carrier phase ambiguities are integer-valued, and it is only after resolution of these parameters that the carrier phase observations start to behave as very precise pseudorange measurements. The success rate equals the integral of the probability density function of the float ambiguities over the pull-in region centered at the true integer, which is the region in which all real values are mapped to this integer. The success rate can thus be computed without actual data and is very valuable as an a priori decision parameter whether successful ambiguity resolution is feasible or not. The pull-in region is determined by the integer estimator that is used and therefore the success rate also depends on the choice of the integer estimator. It is known that the integer least-squares estimator results in the maximum success rate. Unfortunately, it is very complex to evaluate the integral when integer least-squares is applied. Therefore, approximations have to be used. In practice, for example, the success rate of integer bootstrapping is often used as a lower bound. But more approximations have been proposed which are known to be either a lower or upper bound of the actual integer least-squares success rate. In this contribution an overview of the most important lower and upper bounds will be given. These bounds are compared theoretically as well as based on their performance. The performance is evaluated using simulations, since it is then possible to compute the ’actual’ success rate. Simulations are carried out for the two-dimensional case, since its simplicity makes evaluation easy, but also for the higher-dimensional geometry-based case, since this gives an insight to the performance that can be expected in practice.

A comparison of P code and high performance C/A code GPS receivers for on the fly ambiguity resolution

Carrier phase ambiguity resolution on the fly is investigated using two receiver technologies, namely dual-frequency P code and high performance, single frequency, C/A code receivers. Both receiver types were used simultaneously in a series of land kinematic trials. A least-squares search technique is used to find the correct double difference carrier phase ambiguities. Both C/A and single frequency P code technologies are found to be equivalent and capable of resolving the integer ambiguities on the fly using some 30 to 200 seconds of data under benign multipath conditions. Successful ambiguity resolution on the fly results in cm-level accuracy kinematic positioning. The ambiguity resolution time required and success rate are however found to be strongly dependent on the level of carrier phase multipath and, as a consequence, on the error variance assigned to the carrier phase measurements. The use of widelaning with the dual frequency P code results in ambiguity resolution in seconds. The performance of widelaning is also superior in a comparatively high carrier phase multipath environment.