Bofeng Li

Ps-LAMBDA: Ambiguity success rate evaluation software for interferometric applications

Integer ambiguity resolution is the process of estimating the unknown ambiguities of carrier-phase observables as integers. It applies to a wide range of interferometric applications of which Global Navigation Satellite System (GNSS) precise positioning is a prominent example. GNSS precise positioning can be accomplished anytime and anywhere on Earth, provided that the integer ambiguities of the very precise carrier-phase observables are successfully resolved. As wrongly resolved ambiguities may result in unacceptably large position errors, it is crucial that one is able to evaluate the probability of correct integer ambiguity estimation. This ambiguity success rate depends on the underlying mathematical model as well as on the integer estimation method used. In this contribution, we present the Matlab toolbox Ps-LAMBDA for the evaluation of the ambiguity success rates. It allows users to evaluate all available success rate bounds and approximations for different integer estimators. An assessment of the sharpness of the bounds and approximations is given as well. Furthermore, it is shown how the toolbox can be used to assess the integer ambiguity resolution performance for design and research purposes, so as to study for instance the impact of using different GNSS systems and/or different measurement scenarios.

Robustness of GNSS integer ambiguity resolution in the presence of atmospheric biases

Both the underlying model strength and biases are two crucial factors for successful integer GNSS ambiguity resolution (AR) in real applications. In some cases, the biases can be adequately parameterized and an unbiased model can be formulated. However, such parameterization will, as trade-off, reduce the model strength as compared to the model in which the biases are ignored. The AR performance with the biased model may therefore be better than with the unbiased model, if the biases are sufficiently small. This would allow for faster AR using the biased model, after which the unbiased model can be used to estimate the remaining unknown parameters. We assess the bias-affected AR performance in the presence of tropospheric and ionospheric biases and compare it with the unbiased case. As a result, the maximum allowable biases are identified for different situations where CORS, static and kinematic baseline models are considered with different model settings. Depending on the size of the maximum allowable bias, a user may decide to use the biased model for AR or to use the unbiased model both for AR and estimating the other unknown parameters.

Improved method for estimating the inter-frequency satellite clock bias of triple-frequency GPS

Considering the contribution of the hardware biases to the estimated clock errors, an improved method for estimating the satellite inter-frequency clock bias (IFCB) is presented, i.e., the difference in the satellite clock error as computed from ionospheric-free pseudorange and carrier phase observations using L1/L2 and P1/P2 versus L1/L5 and P1/P5. The IFCB is composed of a constant and a variable part. The constant part is the inter-frequency hardware bias (IFHB). It contains the satellite and receiver hardware delays and can be expressed as a function of the DCBs [DCB (P1xa0−xa0P2) and DCB (P1xa0−xa0P5)]. When a reference satellite is selected, the satellite IFHB can be computed but is biased by a reference satellite IFHB. This bias will not affect the utilization of IFCB in positioning since it can be absorbed by the receiver clock error. Triple-frequency observations of 30 IGS stations between June 1, 2013, and May 31, 2014, were processed to show the variations of the IFHB. The IFHB values show a long-term variation with time. When a linear and a fourth-order harmonic function are used to model the estimated IFCB, which contains contributions of the hardware delays and clock errors, the results show that 89xa0% of the IFCB can be corrected given the current five triple-frequency GPS satellites with the averaged fitting RMS of 1.35xa0cm. Five days of data are processed to test the estimated satellite clock errors using the strategy presented. The residuals of P1/P5 and L1/L5 have a STD of <0.27xa0m and 0.97xa0cm, respectively. In addition, most predicted satellite IFCBs reach an accuracy of centimeter level and its mean accuracy of 5xa0days is better than 7xa0cm.

GNSS antenna array-aided CORS ambiguity resolution

Array-aided precise point positioning is a measurement concept that uses GNSS data, from multiple antennas in an array of known geometry, to realize improved GNSS parameter estimation proposed by Teunissen (IEEE Trans Signal Process 60:2870–2881, 2012). In this contribution, the benefits of array-aided CORS ambiguity resolution are explored. The mathematical model is formulated to show how the platform-array data can be reduced and how the variance matrix of the between-platform ambiguities can profit from the increased precision of the reduced platform data. The ambiguity resolution performance will be demonstrated for varying scenarios using simulation. We consider single-, dual- and triple-frequency scenarios of geometry-based and geometry-free models for different number of antennas and different standard deviations of the ionosphere-weighted constraints. The performances of both full and partial ambiguity resolution (PAR) are presented for these different scenarios. As the study shows, when full advantage is taken of the array antennas, both full and partial ambiguity resolution can be significantly improved, in some important cases even enabling instantaneous ambiguity resolution. PAR widelaning and its suboptimal character are hereby also illustrated.

High Dimensional Integer Ambiguity Resolution: A First Comparison between LAMBDA and Bernese

The LAMBDA method for integer least-squares ambiguity resolution has been widely used in a great variety of Global Navigation Satellite System (GNSS) applications. The popularity of this method stems from its numerical efficiency and its guaranteed optimality in the sense of maximising the success probability of integer ambiguity estimation. In the past two decades, the LAMBDA method has been typically used in cases where the number of ambiguities is less than several tens. With the advent of denser network processing and the availability of multi-frequency, multi-GNSS systems, it is important to understand LAMBDA’s performance in high dimensional spaces. In this contribution, we will address this issue using real GPS data based on the Bernese software. We have embedded the LAMBDA method into the Bernese software and compared their ambiguity resolution performances. Twelve day dualfrequency GPS data with a sampling interval of 30 s was used in the experiment, which was collected from a network of 19 stations in the Perth area of Western Australia with an average baseline length of 380 km. Different experimental scenarios were examined and tested with different observation spans, which represent the different ambiguity dimensions. The results showed that LAMBDA is still efficient even when the number of ambiguities is more than 100, and that the baseline repeatability obtained with the ambiguities resolved from the LAMBDA method agreed well with that of Bernese. Therefore, for future dense network processing, the easy-to-use LAMBDA method should be considered as an alternative to baseline-per-baseline methods as those used in e.g. the Bernese software.

GNSS windowing navigation with adaptively constructed dynamic model

The conventional dynamic model in the Kalman filtering-based GNSS navigation usually contains the state information of only one previous epoch, which can hardly reflect the real complex dynamic characteristics of motion behaviors. To improve the adaptability and accuracy of GNSS applications, a window-based polynomial fitting method is used to construct the dynamic model. In the given window with multiple state epochs, all candidate dynamic models with different model orders are self-constructed by using these multiple state epochs in real time. Then, based on the model selection theory, a model evaluation criterion is derived in the Bayesian framework to choose the optimal dynamic model. With this optimal constructed dynamic model, the improved navigation solution then can be obtained by a window-recursive approach. We test the proposed strategies by using the simulation and real GNSS vehicular experiments. All results demonstrate the validity and efficiency of the presented method.

Impacts of BeiDou stochastic model on reliability: overall test, w-test and minimal detectable bias

Extensive studies have concluded that the GNSS observations are heteroscedastic and physically correlated. Typically, the observation precisions are elevation dependent and between-frequency cross-correlations and time correlations exist. The influence of these stochastic characteristics on the GNSS positioning has been numerically well understood. However, their influence on the statistic tests of reliability has been rarely studied. We will systematically study the influence of GNSS stochastic characteristics on the statistic tests involved in reliability. With BeiDou as an example, the realistic elevation-dependent model, cross-correlations and time correlations are estimated. Then their impacts on the reliability are numerically analyzed by comparing with the empirical stochastic model where the stochastic characteristics, i.e., elevation-dependent precisions, cross-correlations and time correlations, are not adequately specified. Besides the overall test and w-test, the minimal detectable bias (MDB) and the separability of two w-test statistics are examined. The results show that the realistic elevation-dependent model will reduce probabilities of both false alarm and wrong detection for both overall test and w-test. Introducing the cross-correlations and time correlations properly can obtain the realistic MDBs together with reasonable separability measures, which all are helpful for users to make objective decisions in quality control of real GNSS applications.

A new differential code bias (C1---P1) estimation method and its performance evaluation

The current satellite clock products are computed using the ionosphere-free phase (L1/L2) and code (P1/P2) observations. Thus, if users conduct undifferenced positioning using these clock products together with C1 and P2 observations, the differential code bias (DCB) (C1–P1) should be properly compensated. The influence of DCB (C1–P1) on the undifferenced ambiguity solutions is investigated. Based on the investigation, we propose a new DCB (C1–P1) estimation method. Using it, the satellite DCB (C1–P1) can be computed. A 30-day (DOY 205–234, 2012) dual-frequency GPS data set is processed to estimate the DCB (C1–P1). Comparing the estimated results with that of IGS DCB products, the accuracy is better than 0.13xa0m. The performances of DCB (C1–P1) in the code-based single-point positioning, precise point positioning (PPP) convergence and wide-lane uncalibrated phase delay (UPD) estimation are investigated using the estimated DCB (C1–P1). The results of the code-based single-point positioning show that the influence of DCB (C1–P1) on the up direction is more evident than on the horizontal directions. The accuracy is improved by 50xa0% and reaches to decimeter level with DCB (C1–P1) application. The performance of DCB (C1–P1) in PPP shows that it can accelerate PPP convergence through improving the accuracy of the code observation. The computed UPD values show that influence of DCB (C1–P1) on UPD of each satellite is different, and some values are larger than 0.3 cycles.

Challenges in ambiguity resolution: Biases, weak models, and dimensional curse

Next generation Global Navigation Satellite Systems will open the door to a whole new field of applications, for example in Earth observation, construction, and safety-of-life navigation. This implies very high requirements not only on precision and availability, but also on reliability. Integer carrier phase ambiguity resolution is the key to (near) real-time and high-precision GNSS positioning and navigation. The reliability of integer ambiguity estimation depends on the strength of the underlying GNSS model and on the applied integer estimation method. This brings certain challenges and limitations that need to be addressed and have not all been solved so far. The aim of this contribution is to address these remaining challenges and limitations: it will be explained why it is important to do so, and how solutions can be obtained. Experimental results will be used to underpin the importance and potential improvement in terms of precision and/or reliability.

Optimal Doppler-aided smoothing strategy for GNSS navigation

AbstractnCarrier-phase-smoothed code (CPSC), i.e., smoothing of the code using carrier phases, has widely been used to reduce the code noise in GNSS applications. However, the efficiency of CPSC suffers significantly from cycle slips, interruptions and jitters. The GNSS Doppler, as an instantaneous measurement, is robust and immune to cycle slips and proven useful in GNSS-challenged environments. We develop optimal Doppler-smoothed code based on the principle of minimum variance using the Hatch filter for two typical applications, which are called pure Doppler-smoothed code (PDSC) and continued Doppler-smoothed code. PDSC results from smoothing the code using only Doppler, whereas in case of continued Doppler-smoothed code, the smoothing continues using Doppler once the carrier phase becomes unavailable. Furthermore, in order to refine the Doppler-smoothed code model, a balance factor is introduced for adjusting the contributions of raw code and Doppler measurements to the smoothed code in case the Doppler noise is relatively large. Finally, experiments are carried out to demonstrate the performance of the theory, which verifies its validity and efficiency.