Cycle slip detection and repair based on the unmodeled-error-constrained geometry-free combining geometry-based models for a single-frequency receiver

For the current problem of cycle slips in the observation data of the BDS-2 and BDS-3 (Bei Dou Navigation Satellite System), in this paper, BDS B1I and B3I signals are used as research objects to study the detection of cycle slips, and their repair algorithm. The Geometry-free (GF) and Melbourne–Wübeena (MW) combination algorithm are used for the detection of cycle slips. A new method of arc partition is proposed in this work to detect cycle slips as the boundary to delimit two different observation arcs. In this way, the different values of cycle slips can be divided and marked. Moreover, the gross errors can be removed. Finally, the detection of cycle slips and the analysis of all epochs can be completed and repaired. This work also analyzes the dual-frequency data effect of cycle slips on code multipath observation. The results showed that this method greatly improved the speed of detection of cycle slips.

The current cycle slip detection methods of Global Navigation Satellite System (GNSS) were mostly proposed on the basis of assuming the ionospheric delay varying smoothly over time. However, these methods can be invalid during active ionospheric periods, e.g., high Kp index value and scintillations, due to the significant increase of the ionospheric delay. In order to detect cycle slips during high ionospheric activities successfully, this paper proposes a method based on two modified Hatch–Melbourne–Wübbena combinations. The measurement noise in the Hatch–Melbourne–Wübbena combination is minimized by employing the optimally selected combined signals, while the ionospheric delay is detrended using a smoothing technique. The difference between the time-differenced ambiguity of the combined signal and this estimated ionospheric trend is adopted as the detection value, which can be free from ionospheric effect and hold the high precision of the combined signal. Five threshold determination methods are proposed and compared to decide the cycle slip from the magnitude aspect. This proposed method is tested with triple-frequency Global Navigation Satellite System observations collected under high ionospheric activities. Results show that the proposed method can correctly detect and fix cycle slips under disturbed ionosphere.

Successful resolution of cycle ambiguities plays a crucial to high accuracy and high integrity navigation by carrier phase differenced positioning architecture. However, a cycle slip caused by loss-of-lock of signal changes the fixed ambiguity, which will affect the cycle resolution integrity. It is, therefore, necessary for detecting cycle slip and further satisfying its integrity. In general, dual frequency measurements are deployed to construct the carrier phase differenced positioning architecture for satisfying the stringent accuracy and integrity performance for safety-critical applications, such as automatic shipboard landing. Thus, dual-frequency-based cycle slip detection methodologies can be introduced in this positioning architecture. The geometry-free (GF) modeling uses only dual-frequency phase measurements to construct the statistical residuals, which are sensitive to small cycle slips that should be detected in high-integrity applications. At present, the threshold of the GF modeling is generally designed as an empirical constant, such as 5 cm, lacking theoretical derivation to quantify its integrity risk. Therefore, this paper proposes an integrity monitoring method for GF modeling to satisfy the integrity performance. Determining the threshold value of the cycle slip statistical test is designed as a function of a required false alarm probability and the quality of raw phase measurements. A required missed detection probability can be used to construct the minimal detectable error (MDB), which can evaluate the availability of the integrity monitoring method to control the missed detection probability. As noted above, the threshold and the MDB are jointly deployed to provide the high-integrity requirements. This paper collects simulated and real-world dual-frequency GNSS datasets to verify the performance of the proposed method. The results show that this method can dynamically adjust the detection threshold to control the false alarm probability and adjust the MDB to evaluate the availability of the integrity monitoring method to control the missed detection probability. Consequently, the proposed integrity monitoring method of cycle slip detection can effectively satisfy the high integrity requirements of cycle slip detection.

Cycle slips are discontinuity events in a receiver’s phase lock on a Global Navigation Satellite System (GNSS) signal. If cycle-slip events are not detected and repaired, the quality of positioning using carrier phase processing suffers. In this paper, we present an algorithm for flagging cycle slips during data preprocessing. This new algorithm is part of a new GNSS software that replaces a legacy software at the National Geodetic Survey (NGS). It uses the Hatch–Melbourne–Wübbena (HMW), ionosphere-free and geometry-free linear combinations to detect cycle slips. The use of the three observables in parallel helps improve sensitivity to cycle slips and overcome the challenge of detecting cycle slips of equal size and same sign on both/multiple frequencies. Different detection threshold values were evaluated in a well-controlled environment using GNSS data acquired from continuously operating reference stations (CORS) networks and artificially simulated cycle slips that were introduced into the datasets. The goal of evaluation was to determine the optimal values for cycle-slip detection parameters. In the absence of formal confidence limits, the results of the study show that, in a worst-case scenario involving small slips of magnitudes ranging from 1 to 3 cycles, there is a range of parameter values that resulted in at least 97% success rate without any false detections. Based on these and other findings, we provide a set of optimal values from which the results are an improvement of at least 34% in cycle-slip detection, compared to results from the legacy algorithm’s settings. The results of the study will support the National Oceanic and Atmospheric Administration’s (NOAA) GNSS services and tools for the public, such as the Online Positioning User Service (OPUS).

A novel method of ambiguity resolution and cycle slip processing for single-frequency GNSS/INS tightly coupled integration system

This paper presents a new cycle slip detection and repair method using Total Electron Content Rate (TECR) information derived from individual satellite dual-frequency data of a single Global Positioning System (GPS) receiver while pseudorange measurements are subject to arbitrarily large range errors. Sudden Increase of Pseudorange Error (SIPE), similar to cycle slips in nature, is quite common in various data acquisition scenarios. The basic principle of this method is to take advantage of the fact that the ionospheric TECR does not exceed certain threshold, which is set as 0.35 TECU/s in this study. Analytic expressions to evaluate the effect of SIPE on cycle slip detections have been developed. The search spaces for cycle slip candidate pairs are defined, given a predefined (sufficiently large) SIPE value. Two cycle slip validation rules are proposed to validate the cycle slip candidates. Over 99.9% of candidates can be rejected with the application of two validation rules. The theoretically maximal number of remaining cycle slip candidate pairs (NRCP) can be exactly calculated based on the magnitude of SIPE, TECR threshold, and the data sampling interval. After applying validation rules, the correct cycle slip pairs can be identified using a modified low-order polynomial fitting method. This method is tested on 13 high rate (1-Hz) dual-frequency datasets recorded by both ground-based static and satellite-borne high dynamic GPS receivers under various levels of ionospheric activities. Simulated cycle slips in 12 different possible cases and varying SIPE magnitudes are introduced into the data sets. In each test scheme, averagely 600–750 pairs of cycle slips are simulated. The SIPE magnitudes are set to vary from 50.0 m to 1000.0 m. Test results show that all the cycle slips in all the test schemes and all the datasets have been successfully detected and fixed even with a maximum SIPE of 1000.0 m in pseudoranges. A distinct advantage of this method is that it works in real-time with individual satellite’s data from a single dual-frequency receiver, even if the carrier phases have virtually any size of cycle slips and the pseudoranges have virtually arbitrarily large errors.

Multi-frequency (quad/pent-frequency) observations improve the success rate for BeiDou Navigation Satellite System (BDS-3) cycle slip detection and repair. Three factors are comprehensively considered: the combination of observation wavelength, the amplification factor of ionospheric delay, and the insensitivity, to select the optimal combination coefficients for BDS-3 multi-frequency cycle slip detection. Due to the ill-conditioned equation in multi-frequency cycle slip repair, the generalized-cross-validation regularization auxiliary LAMBDA method is proposed. To validate the effectiveness of the proposed detection and repair method under different observation environments, two stations are selected for the test. Using real data, the effectiveness of the selected optimal combination coefficients and the proposed method are tested. The conclusions are as follows: (1) for quad-frequency, one geometry-free and ionosphere-free (GFIF) combination [−1, 2, −4, 3] and three linearly independent geometry free carrier phase (GF) combinations [−1, 1, 0, 0], [−1, 1, 1, −1], and [−1, 2, −1, 0] are employed to detect cycle slips. For pent-frequency, one GFIF combination [−2, 2, 1, −2, 1] and four linearly independent GF combinations [−1, 1, 0, 0, 0], [0, 0, −1, 1, 0], [0, 0, −1, 0, 1], and [0, −1, 1, 1, −1] are adopted for cycle slip detection. (2) The algorithm proposed in this paper has been demonstrated to achieve a quad/pent-frequency cycle slip detection success rate of 99.99% and 100% in the coastal environment, and both 100% for quad/pent-frequency detection success in the urban environment. (3) The success rate of repair were 99.27% and 99.94% for the coastal stations, and 99.83% and 100% for the urban stations, respectively. The proposed algorithm offers a dependable approach for the identification and rectification of multi-frequency cycle slips, while also providing a point of reference for the future advancement of multi-frequency cycle slip repair techniques.

Global navigation satellite systems (GNSS) products are integral to a wide array of scientific and commercial applications such as pecise orbit determination of low Earth orbit satellites, earthquake monitoring, GNSS reflectomrety, tropospheric and ionospheric research, surveying and much more. These products consisting of GNSS orbit, clock, phase biases and more are generated by the analysis centres of the International GNSS Service (IGS) by processing observations from a global network of ground stations to one or more GNSS constellations. The processing consists of a combined station position and GNSS satellite orbit determination through a least squares approach donated as global multi-GNSS processing.Within global multi-GNSS processing it is assumed that the observation noise is elevation-dependent and any spatial and temporal correlations are disregarded. Within numerous studies it has been shown that this assumption is incorrect while several studies additional pointed out that a sophisticated stochastic modelling has a positiv impact on GNSS processing and resulting products. In past reseach we have shown to exploit the post-fit residuals to derive temporal correlations for a sophisticated stochastic modeling. However, there have not been any large-scale investigations regarding the impact of stochastic modelling of observation noise on global GNSS processing products. Furthermore, to guarantee the quality of the GNSS products global multi-GNSS processing requires a sophisticated cycle slip detection and repairing algorithm. Cycle slips are discontinuities in the phase observations and if not corrected can lead to degrading quality of GNSS products.   We present our advancements in global multi-GNSS processing by exploiting post-fit residuals for stochastic modeling and cycle slip detection. We used several years of observations and a selected IGS network of ground stations to generate GNSS products. Based on this data we analysed the impact our newly integrated approaches have on GNSS products such as orbits, clocks, phase biases and station coordinate time series.  

Global Satellite Navigation System (GNSS) has become one of the most important infrastructural facilities for major countries in the world. GPS carrier phase observations are often used for high precision positioning, but are susceptible to the influence of cycle slips. The EWL (extra-wide-lane) combination, for longer combination wavelengths, is frequently used for detecting and repairing cycle slips. The EWL combination adopts carrier phase and pseudorange observations; but in some cases, pseudorange observations have proven to be imprecise or inapplicable. Based on the traditional EWL model, a Doppler-aided EWL cycle slip detection and repair model is proposed herein. The model is applicable to situations where the ionosphere is stable and the sampling rate is high. Analyzing the static experimental data with 1s sampling interval, it could be observed that combining the proposed new method with traditional EWL method before adding any cycle slip will lower the first-order difference of integer ambiguity down to less than 0.5 cycles. This implies that the new method is also applicable to cycle slips detectable and reparable by traditional EWL methods. However, the ionospheric delay term reintroduced by the new method still requires further research.

Cycle slip detection in the carrier phase is important for achieving precise relative positioning with global navigation satellite systems (GNSSs) in urban environments. The false alarm rate (FAR) and missing detection rate (MDR) of the detection approach will significantly affect the positioning error. This paper proposes a novel cycle slip detection approach based on inter-frequency cross-validation (IFCV), and designs corresponding test statistics and thresholds. In contrast to conventional cycle slip detection methods, the proposed IFCV approach has the advantages of higher accuracy and efficiency, which is validated by the airborne dynamic test. For cycle slips of specific proportions (such as five-cycle and four-cycle slips), the IFCV method significantly reduces the MDR from 3.89% to 0% and decreases the relative positioning error from 4.69 cm to 3.17 cm. In tests involving randomly injected cycle slips, the IFCV method reduces the MDR from 0.02% to 0%, reduces the relative positioning error from 3.83 cm to 3.02 cm, and decreases the average computational time from 1.02 ms to 0.53 ms.

Cycle slips are a common error source in Global Navigation Satellite System (GNSS) carrier phase measurements. In this paper, the cycle slip problem is approached using Receiver Autonomous Integrity Monitoring (RAIM) methodology. Carrier phase measurements are used here in a single-receiver time-differential positioning method where integer ambiguities are canceled, but any cycle slips remain. The performance of the method was assessed by comparing the detection results to a Real-Time Kinematic (RTK) solution and by manual data examination. Postprocessing results obtained using authentic Global Positioning System (GPS) measurements logged by low-cost single-frequency receivers show that the method is able to reliably detect and identify single errors but fails in an exemplary multiple outlier scenario. As no reference receiver is needed, the method is a potential means to produce cycle-slip-corrected data usable in any postprocessing application.

The multi-frequency signals of the Beidou system are very helpful for positioning, but in data processing, the detection and repair of multi-frequency cycle slips also face certain challenges. A recursive filtering and integer constraint cycle slip detection and repair method for the Beidou six-frequency is proposed in this paper. This method utilises five sets of combination coefficients of six-frequency and three sets of dual-frequency combinations, and detects and repairs cycle slips through integer constraints of cycle slip estimation. The proposed method has been validated by data with a sampling rate of 1 Hz, and can detect the position and size of cycle slips at various frequencies, whether it is random cycle slips or small cycle slips of equal size less than 2 cycles, with a detection success rate of 100%.

In the preprocessing of high-precision navigation and positioning data, the most widely used MW combination cycle slip detection method is greatly affected by pseudorange noise. It has issues such as missing small cycle slips and failing to promptly reset the recursive averaging process after cycle slip detection failure, which leads to subsequent threshold divergence. This paper proposes an improved MW combination cycle slip detection method based on Complete Ensemble Empirical Mode Decomposition (CEEMDAN), permutation entropy, and wavelet denoising, which uses CEEMDAN to decompose the cycle slip signal into a series of intrinsic modal functions (IMFs) and then selects the IMFs that require denoising through the permutation entropy algorithm, and the wavelet denoising technique is combined to eliminate the residual noise further, so that the noise can be removed more accurately. Experimental results show that compared with the original MW algorithm, the proposed improved method can effectively reduce the influence of pseudo-range noise, and reduce the false detection rate of cycle slip from 1.6% and 6–0%. All small period slips can be successfully detected in complex noise environments, avoiding the missed detection of the original MW algorithm and the related threshold divergence problems.

Ionospheric delay prediction based on online polynomial modeling for real-time cycle slip repair of undifferenced triple-frequency GNSS signals

Global Navigation Satellite System (GNSS) plays an increasingly important role in all walks of life. In order to meet the demands of different users, it is crucial to establish corresponding correct mathematical models, especially in the field of precise positioning and navigation. However, due to the spatiotemporal complexity of and limited knowledge on GNSS errors, some residual errors would inevitably remain even after being corrected with differencing and linear combination, empirical model correction, and traditional parameterization. These residual observation errors are referred to as unmodeled errors. In fact, the unmodeled errors have adverse impacts on high-precision GNSS positioning. However, most existing studies mainly focus on handling the part of systematic errors that can be adequately modeled and then simply ignore the part of unmodeled errors that may actually exist. To make a breakthrough in the precision and reliability of GNSS applications currently, this study focuses on the theory and method for processing the GNSS unmodeled errors based on the mathematical model compensation, including resilient functional model adjustment and resilient stochastic model optimization. Specifically, according to the significance and properties of unmodeled errors, the unmodeled-error-ignored model, unmodeled-error-corrected model, unmodeled-error-fixed model, unmodeled-error-float model, and unmodeled-error-weighted model are proposed, where the approaches of significance testing, geometry-free/based/fixed models, hemispherical/hierarchy maps, composite stochastic model, multi-epoch partial parameterization, and inequality and equality constraints are adopted. Ultimately, an unmodeled error processing flow that can adaptively adjust as the external conditions change is proposed, then one can obtain high-precision GNSS positioning solutions.