Geometry-free Combination Research Articles

Undifferenced Cycle Slip Estimation of Triple-Frequency BeiDou Signals with Ionosphere Prediction

ABSTRACTWith the rapid development of BeiDou satellite navigation system (BDS), high-quality service has been provided in the Asia-Pacific region currently, which will be extended to the whole world very soon. BDS is the first Global Navigation Satellite System that all satellites broadcast the triple-frequency signals. The triple-frequency signals in theory can improve the cycle slip detection that is one of the preconditions in precise positioning by making use of carrier phase. This paper discusses the development of a cycle slip detection method for undifferenced BDS triple-frequency observations in kinematic scenario. In this method, two geometry-free extra-wide-lane combinations and one geometry-free narrow-lane (NL) combinations are employed. The key is to mitigate the between-epoch ionospheric biases in the geometry-free NL combinations. We propose to predict the ionospheric biases of current epoch by using those from its consecutive foregoing epochs. The method is tested with extensive experiments in varying observation scenarios. The results show that in case of sampling interval as small as 5 s, the between-epoch ionospheric biases can be ignored and the correct cycle slips can be determined. Meanwhile in case of lower sampling frequency, one needs to compensate the ionospheric biases of current epoch by using the predicted ionospheric biases. The presented method can correctly detect all cycle slips even if they are as small as 1 cycle.

A modified un-combined model to improve the performance of precise point positioning: model and test results

Precise point positioning (PPP) can achieve high-accuracy point solution using un-differenced code and carrier phase observations. Many PPP models have a higher measurement noise and multipath effects, resulting in a long time to obtain sub-decimeter to centimeter positioning accuracy during a short duration. This paper will address a modified un-combined (MUC) model based on phase–phase geometry-free combination in addition to code–phase ionosphere-free combination and original carrier phase observations. The observation system composed by the new combinations has a lower measurement noise and orbit error level compared with the un-combined (UC) model, University of Calgary (UofC) model and standard un-differenced ionosphere-free combined (UD) model. Moreover, it can capture the changing characteristics of atmospheric delays between epochs as constraints to accelerate the filter convergence. We use the data sets from about 114 international GNSS service reference stations to analyze the performance of the MUC model. Numerical results show that the MUC model generally yields the best performance when the observation duration is short. For the 0.5 h duration observation data sets, about 70.2 % of all 6912 convergence tests are convergent, which is an increase by 19.2, 2.8 and 49.3 % points compared with the UC, UofC and UD models, respectively. For the common convergence parts of all the test data sets, the mean convergence time of the MUC model is significantly reduced by 22.9, 9.6 and 35.3 %, and the percentage of the 2D position biases within 0–5 cm is increased by 16.0, 3.4 and 55.0 %, respectively, compared with the UC, UofC and UD models. Therefore, the proposed PPP model is more beneficial for the PPP user to quickly obtain sub-decimeter to centimeter positioning accuracy during a short duration.

Real-time cycle-slip detection and repair for BeiDou triple-frequency undifferenced observations

A new method of real-time cycle-slip detection and repair for BeiDou navigation satellite system (BDS) triple-frequency undifferenced observations is proposed. The method, which is based on code–phase and geometry-free phase combinations, composes linearly independent combination observations to uniquely determine the cycle-slip of original frequency phase observations in two independent steps. First, two extra-wide lane (EWL) code–phase combinations are utilised to determine the EWL combination cycle-slips. The sum of the carrier phase combination coefficients is zero. Second, the geometry-free phase combination with a non-zero sum of combination coefficients composes linearly independent combination observations with two code–phase combinations. Ionospheric delay variance systematic deviation is corrected with several previous carrier-phase observations without cycle-slip, which seriously affect the accuracy of the geometry-free phase combination observations. The BDS triple-frequency carrier-phase and code observations of different satellite types at 30 s sampling interval, which involve increased active ionospheric conditions in 1 day, are utilised to test the method. Results show that the proposed method can efficiently detect and accurately repair any cycle-slip combinations in real-time for different satellite types under active ionospheric conditions. No erroneous and leakage judgments are observed in the test.

A double-differenced cycle slip detection and repair method for GNSS CORS network

The difficulty to detect and repair cycle slip of carrier phase measurements is a key limit for continuously high accuracy of GNSS positioning and navigation services. We propose an automated cycle slip detection and repair method for data preprocessing of a CORS network. The method jointly uses double-differenced (DD) geometry-free (GF) combination and ionospheric-free observation corrected for the computed geometrical distance (IF-OMC) to estimate the cycle slips in dual-frequency observations. The DD GF combination, which is only affected by the ionospheric residual, can be used to detect cycle slips with high reliability except for special pairs such as (77, 60) on GPS L1/L2 frequencies. The detection principle of the IF-OMC observable is such that there is a large discontinuity related to the previous epoch when cycle slips occur at the present epoch. The disadvantages of these two combinations can be overcome employing the proposed detection method. The cycle slip pair (77, 60) has no effect on the GF combination, while a change of 14.65 m is derived from GPS L1/L2 observations using the IF-OMC algorithm. Using pre-determined station coordinates as precise values, we found that the accuracy of the DD IF-OMC combination was 18 mm for a 200-km CORS baseline. Therefore, cycle slips in dual-frequency observations can be correctly and uniquely determined using DD GF and IF-OMC equations. The proposed method was verified by adding simulated cycle slips in observations collected from the CORS network under a quiet ionosphere and shown to be effective. Moreover, the method was assessed with observations made during intense ionospheric activity, which generated extensive cycle slips. The results show that the algorithm can detect and repair all cycle slips apart from two exceptions relating to long data gaps.

GNSS triple-frequency geometry-free and ionosphere-free track-to-track ambiguities

During the last few years, more and more GNSS satellites have become available sending signals on three or even more frequencies. Examples are the GPS Block IIF and the Galileo In-Orbit-Validation (IOV) satellites. Various investigations have been performed to make use of the increasing number of frequencies to find a compromise between eliminating different error sources and minimizing the noise level, including the investigations in the triple-frequency geometry-free (GF) and ionosphere-free (IF) linear combinations, which eliminate all the geometry-related errors and the first-order term of the ionospheric delays.In contrast to the double-difference GF and IF ambiguity resolution, the resolution of the so-called track-to-track GF and IF ambiguities between two tracks of a satellite observed by the same station only requires one receiver and one satellite. Most of the remaining errors like receiver and satellite delays (electronics, cables, etc.) are eliminated, if they are not changing rapidly in time, and the noise level is reduced theoretically by a factor of square root of two compared to double-differences. This paper presents first results concerning track-to-track ambiguity resolution using triple-frequency GF and IF linear combinations based on data from the Multi-GNSS Experiment (MGEX) from April 29 to May 9, 2012 and from December 23 to December 29, 2012. This includes triple-frequency phase and code observations with different combinations of receiver tracking modes. The results show that it is possible to resolve the combined track-to-track ambiguities of the best two triple-frequency GF and IF linear combinations for the Galileo frequency triplet E1, E5b and E5a with more than 99.6% of the fractional ambiguities for the best linear combination being located within ±0.03 cycles and more than 98.8% of the fractional ambiguities for the second best linear combination within ±0.2 cycles, while the fractional parts of the ambiguities for the GPS frequency triplet L1, L2 and L5 are more disturbed by errors as e.g. the uncalibrated Phase Center Offsets (PCOs) and Phase Center Variations (PCVs), that have not been considered. The best two GF and IF linear combinations between tracks are helpful to detect problems in data and receivers. Furthermore, resolving the track-to-track ambiguities is helpful to connect the single-receiver ambiguities on the normal equation level and to improve ambiguity resolution.

Influence of Ionospheric Irregularities on GNSS Remote Sensing

We have used numerical simulation to study the effects of ionospheric irregularities on accuracy of global navigation satellite system (GNSS) measurements, using ionosphere-free (in atmospheric research) and geometry-free (in ionospheric research) dual-frequency phase combinations. It is known that elimination of these effects from multifrequency GNSS measurements is handi-capped by diffraction effects during signal propagation through turbulent ionospheric plasma with the inner scale being smaller than the Fresnel radius. We demonstrated the possibility of reducing the residual ionospheric error in dual-frequency GNSS remote sensing in ionosphere-free combination by Fresnel inversion. The inversion parameter, the distance to the virtual screen, may be selected from the minimum of amplitude fluctuations. This suggests the possibility of improving the accuracy of GNSS remote sensing in meteorology. In the study of ionospheric disturbances with the aid of geometry-free combination, the Fresnel inversion eliminates only the third-order error. To eliminate the random TEC component which, like the measured average TEC, is the first-order correction, we should use temporal filtering (averaging).

Ambiguity resolution for triple-frequency geometry-free and ionosphere-free combination tested with real data

The recent GPS Block IIF satellites SVN62 and SVN63 and the Galileo satellites GIOVE-A, GIOVE-B, PFM and FM2 already send signals on more than two frequencies, and more GNSS satellites will provide tracking data on at least three frequencies in the near future. In this paper, a simplified general method for ambiguity resolution minimizing the noise level for the triple-frequency geometry-free (GF) and ionosphere-free (IF) linear combinations is presented, where differently scaled code noise on the three frequencies was introduced. For the third of three required linear combinations, the most demanding one in triple-frequency ambiguity resolution, we developed a general method using the ambiguity-corrected phase observations without any constraints to search for the optimal GF and IF linear combination. We analytically demonstrate that the noise level of this third linear combination only depends on the three frequencies. The investigation concerning this frequency-dependent noise factor was performed for GPS, Galileo and Compass frequency triplets. We verified the theoretical derivations with real triple-frequency GPS and Galileo data from the Multi-GNSS Experiment (M–GEX) of the International GNSS Service (IGS). The data of about 30 M–GEX stations around the world over 11 days from 29 April 2012 to 9 May 2012 were used for the test. For the third linear combinaton using Galileo E1, E5b and E5a, which is expected to have the worst performance among all the GNSS frequency triplets in our investigation, the formal errors of the estimated ambiguities are in most cases below 0.2 cycles after 400 observation epochs. If more GPS satellites sending signals on three frequencies or more stations tracking Galileo E6 signal are available in the future, an improvement by a factor of two to three can be expected.

Improved precise point positioning in the presence of ionospheric scintillation

Ionospheric disturbances can be detrimental to accuracy and reliability of GNSS positioning. We focus on how ionospheric scintillation induces significant degradation to Precise Point Positioning (PPP) and how to improve the performance of PPP during ionospheric scintillation periods. We briefly describe these problems and give the physical explanation of highly correlated phenomenon of degraded PPP estimates and occurrence of ionospheric scintillation. Three possible reasons can contribute to significant accuracy degradation in the presence of ionospheric scintillation: (a) unexpected loss of lock of tracked satellites which greatly reduces the available observations and considerably weakens the geometry, (b) abnormal blunders which are not properly mitigated by positioning programs, and (c) failure of cycle slip detection algorithms due to the high rate of total electronic content. The latter two reasons are confirmed as the major causes of sudden accuracy degradation by means of a comparative analysis. To reduce their adverse effect on positioning, an improved approach based on a robust iterative Kalman filter is adopted to enhance the PPP performance. Before the data enter the filter, the differential code biases are used for GNSS data quality checking. Any satellite whose C1---P1 and P1---P2 biases exceed 10 and 30 m, respectively, will be rejected. Both the Melbourne---Wubbena and geometry-free combination are used for cycle slip detection. But the thresholds are set more flexibly when ionospheric conditions become unusual. With these steps, most of the outliers and cycle slips can be effectively detected, and a first PPP estimation can be carried out. Furthermore, an iterative PPP estimator is utilized to mitigate the remaining gross errors and cycle slips which will be reflected in the posterior residuals. Further validation tests based on extensive experiments confirm our physical explanation and the new approach. The results show that the improved approach effectively avoids a large number of ambiguity resets which would otherwise be necessary. It reduces the number of re-parameterized phase ambiguities by approximately half, without scarifying the accuracy and reliability of the PPP solution.

Geometry-free linear combinations for Galileo

Global navigation satellites of the European Galileo system transmit code signals on four carriers in the L1, E5a, E5b and E6 band. New geometry-free linear combinations are presented that eliminate the geometry terms (user to satellite ranges and orbital errors), the clock errors of the user and satellites and the tropospheric delay. The remaining parameters of these carrier phase combinations include integer ambiguities, ionospheric delays, carrier phase multipath and phase noise. The weighting coefficients are designed such that the integer nature of ambiguities is maintained. The use of four frequency combinations is highly recommended due to a noise reduction of up to 14.4 dB and an ionospheric reduction of up to 25.6 dB compared to two frequency geometry-free combinations. Moreover, a modified Least-squares Ambiguity Decorrelation Adjustment (LAMBDA) algorithm is suggested, which differs in two points from the traditional approach: the baseline is replaced by the ionospheric delay and the correlation is caused by linear combinations instead of double differences. For correct ambiguity resolution, the ionospheric delay can be determined with millimeter accuracy. This is quite beneficial as the ionosphere represents the largest source of error for absolute positioning.

Wide-lane Assisted Long Baseline High Precision Kinematic Positioning by GNSS

In the previous report (Isshiki, 2004b), theory and algorithm of a new dual frequency long baseline kinematic positioning method were discussed. In the theory, the wide-lane coordinates are used as a constraint for obtaining the correct L1 ambiguities by solving the ionosphere free equations. The effectiveness was verified by some numerical examples. A precise positioning for baseline of several hundred kilometeres are possible. In the present report, the effect of epoch interval and observation length is investigated by numerical calculations. The relationship between the positioning error and the baseline length is also discussed. Furthermore, an algorithm for the real time application is shown. In the previous report (Isshiki, 2004b), new theory and algorithm for dual frequency long baseline kinematic positioning were discussed. The theory is based on the facts that the initial phase ambiguities of the wide-lane combinations can be obtained correctly irrespective of the baseline length by using HMW (Hatch-Melbourne- Wubbena) combinations (Hatch, 1982; Melbourne, 1985; Wubbena, 1985), the slowly varying part of the ionospheric delays can be estimated with an appropriate accuracy from an external information source such as IONEX (Hugeltobler et al., 2001), and the rapidly varying part can be obtained correctly by using the geometry-free combinations. Specifically, the coordinates of the observation receiver can be obtained rather precisely by solving the wide-lane combinations by using the above-mentioned ambiguities and the ionospheric delays. And the coordinates are used to impose an constraint to the least squares solution of the ionosphere free combinations. This corresponds to the constraint in the case of static positioning where the coordinates are constant. In the previous report, the validity of the new solution was verified by using data with the epoch interval of thirty seconds downloaded from the homepage of GSI (Geographical Survey Institute, http://terras.gsi.go.jp /inet_NEW/). This new solution makes kinematic positioning for the baseline length of several hundred kilometers possible. In the present report, the effects of the number of the epoch and observation length are discussed by using data with the epoch interval of one second which can't be downloaded directly from the above-mentioned homepage. The relationship between the baseline length and the positioning error is also investigated. And the algorithm for the real time processing is discussed too.

Reliability of the TEC Computed Using GPS Measurements — The Problem of Hardware Biases

In this paper, we outline the procedure used at the Royal Observatory of Belgium in order to compute the TEC using GPS measurements with a precision of 2–3 TECU. This procedure requires the determination of the so-called receiver and satellite differential group delays. The combined biases (receiver + satellite) are determined on a daily basis (i.e. one solution a day) using the geometry-free combination of code observations. The method is applied to a network of 7 permanently operating Turbo Rogue receivers; the reliability of these computed biases is discussed.