Ionosphere-free Combination Research Articles

GNSS signal phase, TEC, and phase unwrapping errors

<h3>Abstract</h3> Precise measurements of signal phase are essential for Global Navigation Satellite System (GNSS) position estimates. However, propagation through the earth’s ionosphere imposes frequency-dependent phase errors. The frequency dependence is exploited to correct phase errors proportional to total electron content (TEC) divided by frequency. Scintillation causes additional stochastic errors, which can become the largest phase error contribution. Both geometry-free (GFCs) and ionosphere-free (IFCs) frequency combinations are subject to such uncorrectable but generally small <i>phase scintillation</i> errors. A recent published study compared GPS TEC estimates derived from nominally identical L1-L2 and L1-L5 GFCs. The TEC errors establish an upper bound on uncorrelated single-frequency scintillation error contributions. The measured TEC errors were verified with phase-screen simulations. However, a known but largely overlooked phase unwrapping error affects the extraction of signal phase from phase-screen complex signal realizations. This paper demonstrates a procedure for detecting and correcting phase-unwrapping errors and discusses their ramifications.

Quantifying errors in GNSS antenna calibrations

We evaluated the performance of GNSS absolute antenna calibrations and its impact on accurate positioning with a new assessment method that combines inter-antenna differentials and laser tracker measurements. We thus separated the calibration method contributions from those attainable by various geometric constraints and produced corrections for the calibrations. We investigated antennas calibrated by two IGS-approved institutions and in the worst case found the calibration’s contribution to the vertical component being in excess of 1 cm on the ionosphere-free frequency combination L3. In relation to nearby objects, we gauge the 1sigma accuracies of our method to determine the antenna phase centers within pm ,0.38 mm on L1 and within pm ,0.62 mm on L3, the latter applicable to global frame determinations where atmospheric influence cannot be neglected. In addition to antenna calibration corrections, the results can be used with an equivalent tracker combination to determine the phase centers of as-installed individual receiver antennas at system critical sites to the same level without compromising the permanent installations.

A New Method of Real-Time Kinematic Positioning Suitable for Baselines of Different Lengths

This study introduces a new real-time kinematic (RTK) positioning method which is suitable for baselines of different lengths. The method merges carrier-phase wide-lane, and ionosphere-free observation combinations (LWLC) instead of using pseudo-range, and carrier-phase ionosphere-free combination (PCLC), or single-frequency pseudo-range and phase combination (P1L1). In a first step, the double-differenced wide-lane ambiguities were calculated and fixed using the pseudo-range and carrier-phase wide-lane combination observations. Once the double-differenced wide-lane integer ambiguities were known, the wide-lane combined observations were regarded as accurate pseudo-range observations. Subsequently, the carrier-phase wide-lane, and ionosphere-free combined observations were used to fix the double-differenced carrier-phase integer ambiguities, achieving the final RTK positioning. The RTK positioning analysis was performed for short, medium, and long baselines, using the P1L1, PCLC, and LWLC methods, respectively. For a short baseline, the LWLC method demonstrated positioning accuracy similar to the P1L1 method, and performed better than the PCLC method. For medium and long baselines, the positioning accuracy of the LWLC method was slightly higher than those of the PCLC and P1L1 methods. In conclusion, the LWLC method provided high-precision RTK positioning results for baselines with different lengths, as it used high-precision carrier-phase observations with fixed ambiguities instead of low-precision pseudo-range observations.

Benefits of BDS-3 B1C/B1I/B2a triple-frequency signals on precise positioning and ambiguity resolution

BDS-3 currently has 28 operational satellites in orbit, of which 27 IGSO/MEO satellites provide open services on five frequencies simultaneously. In particular, the linear combinations of the BDS-3 B1C/B1I/B2a signals have significant benefits in reducing the influence of ionospheric delay error as well as improving ambiguity estimation and positioning accuracy. The presented optimal ionosphere-free combination (242, 218, − 345) and ionosphere-reduced combination (2, 2, − 3) can improve the measurement accuracy by about 20% compared to the BDS-3 B1C/B2a or GPS L1/L5 dual-frequency combination. The ionosphere-reduced combination (2, 2, − 3) with a wavelength of 10.9 cm is almost immune to the ionospheric delay error and has a smaller noise amplification factor compared to the existing dual-frequency combinations. Therefore, its combined ambiguities can be fixed directly even in the case of a long baseline, which can simplify the traditional precise positioning process based on the ionosphere-free combination. The numerical results of BDS-3 real data show that the triple-frequency ionosphere-free or ionosphere-reduced combinations can improve the single-point positioning accuracy by 16–20% and the phase differential positioning accuracy by 7–9%, respectively. The ambiguity resolution of the ionosphere-reduced combination (2, 2, − 3) is achieved with a fixing rate of 88.4% over long baseline up to 1600 km. The presented ionosphere-free and ionosphere-reduced combinations are also very promising to be applied in current PPP applications to simplify the ambiguity fixing process as well as improve positioning accuracy and shorten convergence time.

An SBAS Integrity Model to Overbound Residuals of Higher-Order Ionospheric Effects in the Ionosphere-Free Linear Combination

The next generation of satellite-based augmentation systems (SBAS) will support aviation receivers that take advantage of the ionosphere-free dual-frequency combination. By combining signals of the L1 and L5 bands, about 99% of the ionospheric refraction effects on the GNSS (Global Navigation Satellite Systems) signals can be removed in the user receivers without additional SBAS corrections. Nevertheless, even if most of the negative impacts on GNSS signals are removed by the ionospheric-free combination, some residuals remain and have to be taken into account by overbounding models in the integrity computation conducted by safety-of-live (SoL) receivers in airplanes. Such models have to overbound residuals as well, which result from the most rare extreme ionospheric events, e.g., such as the famous “Halloween Storm”, and should thus include the tails of the error distribution. Their application shall lead to safe error bounds on the user position and allow the computation of protection levels for the horizontal and vertical position errors. Here, we propose and justify such an overbounding model for residual ionospheric delays that remain after the application of the ionospheric-free linear combination. The model takes into account second- and third-order ionospheric refraction effects, excess path due to ray bending, and increased ionospheric total electron content (TEC) along the signal path due to ray bending.

GPS triple-frequency undifferenced and uncombined precise orbit determination with the consideration of receiver time-variant bias

As the development of the global navigation satellite system (GNSS), the navigation satellite transmitting multi-frequency signals becomes a prevailing trend. Currently, the usual GNSS precise orbit determination (POD) strategy uses dual-frequency (DF) ionosphere-free (IF) linear combination to build up the observation equation. In the area of precise positioning, it has been validated that the third frequency can decrease the convergence time of ambiguity resolution and has a subtle improvement on the positioning accuracy. However, there is no report about the benefits of the third frequency on GNSS POD. Although the precision of dual-frequency POD can meet requirements for generating different types of products, an obvious advantage of multi-frequency observations is that more robust results can be derived. Besides, with the third frequency observations, the improvements of orbits and clocks need to be validated. Thus, a triple-frequency (TF) uncombined (UC) POD method of GNSS satellites is developed. The hardware delay of carrier phase is divided by time-invariant and -variant components. Then the UC observation model is given by re-parameterizing the unknown parameters. The datum of satellite clocks is aligned to IGS products. The step-by-step ambiguity fixing method, i.e. the extra-wide-lane, wide-lane and narrow-lane ambiguities being fixed in sequence, is deduced by using double-differenced ambiguities in a network. We select 74 ground stations and 12 GPS BLOCK IIF satellites to process POD with a length of 20 days. Results of TF-UC POD show that about 10% improvements on orbits and clocks are received compared to L1/L2 DF-IF POD. The time-variant characteristic of phase biases from satellites and stations at the third frequency is analyzed, which shows that the bias at the station has a slight influence on the derived products. These results imply that the precision of orbits and clocks can be improved with observations from the third frequency added.

Evaluation of the Effect of Higher-Order Ionospheric Delay on GPS Precise Point Positioning Time Transfer

In high-precision GPS precise point positioning (PPP) time transfer, errors caused by the effect of ionosphere delay have to be corrected. Usually the ionosphere-free combinations of the pseudo code and the carrier phase is used in GPS PPP data processing, and it effectively eliminates the effect of the first-order ionospheric delay. This study quantitatively analyzes the errors caused by higher-order ionospheric (Ion2+) delays in precise PPP time transfer. Data of two 7-day test periods, including low and moderate ionospheric conditions, from 20 stations located in middle- and low-latitude, were analyzed. The difference in clock solution with and without the Ion2+ correction, including the receiver clock solution and time-link clock solution, was deeply analyzed and discussed. The difference sequence shows a constant bias plus some variations with a diurnal variation. For the difference of the receiver clock solutions, the mean standard deviation of the variations is 3.92 ps in low-latitude, which is much larger than that of 2.59 ps in mid-latitude due to the influence of the larger ionospheric electron density on the low-latitude. The maximum constant bias reached more than 15 ps and was negative at most stations in the northern hemisphere, while it was positive at most stations located in the southern hemisphere. The difference in the time-link solutions correlates not only with time and region, but also with the length of the time-links. The largest difference in the long time-link SYDN-PTBB, BJNM-SYDN, AMC2-SYDN, etc., reaches more than 25 ps, while that of the short time-link IENG-PTBB, BRUX-PTBB, etc., is less than 3.5 ps. Therefore, the Ion2+ correction is necessary for high-precision PPP time transfer over long time-links, especially time-links made by one station located in the northern hemisphere and another located in the south hemisphere; however, it could be ignored for short time-links.

Estimation of Ionospheric Delay Influence on the Efficiency of Precise Positioning of Multi-GNSS Observations

Currently, Global Navigation Satellite Systems (GNSS) are developing at a fairly rapid pace. Over the last years US GPS and Russian GLONASS were modernizing, whilst new systems like European Galileo and Chinese BDS are launched. The modernizations of the existing and the deployment of new GNSS made a whole range of new signals available to the users, and create a new concept  multi-GNSS. Ionospheric delay is one of the major error sources in multi-GNSS observations. At present, GNSS users usually eliminate the influence of ionospheric delay of the first order items by dual-frequency ionosphere-free combinations. But there is still residual ionospheric delay error of higher orders. In this paper we present four different processing scenarios to exclude the higher orders ionospheric delay effects on multi-GNSS Precise Point Positioning (PPP) performance, including: “only GPS” and “GPS+GLONASS+Galileo+BDS” – without/with eliminating ionospheric delay error of higher orders. Dataset collected from one GNSS station BOR1 (Borowiec, Poland) over almost two years provided by multi-GNSS experiment (MGEX) were used for dual-frequency PPP tests with one- and quadconstellation signals. For the second pair of scenarios were used a IONosphere map EXchange format (IONEX) that supports the exchange of 2- and 3-dimensional TEC maps given in a geographic grid. Numeric experiments show that, the results of different pairs of scenarios differ at the submillimeter level. The results also show that the multi-GNSS processing are better than those based on “only GPS”.

Uncombined precise orbit and clock determination of GPS and BDS-3

There is increasing concern about the uncombined (UC) observation model in the field of global navigation satellite system (GNSS). Based on the global positioning system (GPS) and the third-generation BeiDou navigation satellite system (BDS-3), this study processed the UC precision orbit determination (POD) for single and dual systems. First, a UC observation model suitable for multi-GNSS POD was derived, and the ionospheric-free (IF) combination observation model was presented. Although the ambiguity parameters of UC and IF strategies were different after reparameterization, the difference could be removed when processing ambiguity resolution, and the equivalence was proved theoretically. To demonstrate the accuracy of BDS-3 orbits fully, the observation data of approximately 1 month were selected for determining the precise orbit for global positioning system (GPS) only, BDS-3 only, and GPS/BDS-3 systems based on the UC and IF models. The orbit precision of BDS-3 satellites was validated by using metrics, including comparison with precision products released by Wuhan University, orbit boundary discontinuity, and satellite laser ranging (SLR) residuals. The results show that the orbit accuracies of the IF and UC models are almost the same, the difference in orbits is approximately several millimeters, and the clock difference is within 0.01 ns. The GPS/BDS-3 combined solution shows better accuracy compared to other solutions. The average accuracies in the R and 3D directions are approximately 4 and 15 cm, and the clock standard deviation is approximately 0.2 ns compared to external orbit product. The root mean square of SLR residuals is approximately 4 cm.

Precision point positioning with additional baseline vector constraint

Abstract Traditional precise point positioning (PPP) based on undifferenced ionosphere-free linear combination of observations has many advantages such as high accuracy and easy operation. PPP usually uses the Kalman Filter (KF) to estimate state vector. However, the positioning performance depends on the accuracy of the kinematic model and initial value. The inaccurate kinematic model or initial value will lead to filter performance degradation or even divergence. To overcome this problem, this paper proposes a PPP method with an additional baseline vector constraint, which uses the direction information and length information of the baseline to correct the estimated position of the receiver. By reducing the error covariance matrix of the float solution, the algorithm improves the accuracy of the float solution. By using the collected real GPS static and kinematic data, the performance of the traditional model and the proposed model in this paper are compared. It is shown that the additional baseline vector constraint improves the PPP solution effectively in comparison with that of traditional PPP model. Additionally, the contribution of the additional constraint is up to the accuracy of the prior information.

A method for undifferenced and uncombined PPP ambiguity resolution based on IF FCB

The integer characteristics of the undifferenced ambiguity can be recovered after the correction of fractional cycle biases (FCB), which can significantly shorten the initialization time of precise point positioning (PPP). In the former work, the ambiguity resolution model of the client end needs to be consistent with the FCB products server. However, the client may face the problem that inconsistent FCB products are provided and cannot be used to fix the ambiguity. We propose a new method for client of undifferenced and uncombined PPP ambiguity resolution based on FCB product of ionosphere-free combination PPP model. The hardware delay in server of ionosphere-free combination FCB (IF FCB) and undifferenced and uncombined FCB (L1L2 FCB) are deduced in detail. Constructing the narrow-lane ambiguity by using the raw ambiguity and the wide-lane ambiguity, the FCB product of the ionosphere-free combination PPP model is used to fix the ambiguity step by step. A globally distributed network consisting of about 118 MGEX stations is selected as the server end. Two types of FCB products, i.e., IF FCB and L1L2 FCB, are generated. 50 additional MGEX stations in global are selected as client end for performance assessment. The experimental results show that the new method for client of UU PPP model based on the IF FCB products has a comparable performance with the traditional method based on L1L2 FCB products, which improves the utilization of FCB products.

Performance of ionospheric-free PPP time transfer models with BDS-3 quad-frequency observations

The BeiDou Navigation Satellite System (BDS) has as achieved the status of a global navigation satellite system. Compared to BDS-2 satellites, BDS-3 satellites transmit B1C, B2a, and B2b signals but not B1I and B3I signals. In this study, for the four open-services of BDS-3 satellites, four ionospheric-free combinations, namely (B1I/B3I), (B1I/B2a), (B1C/B2a) and (B1C/B3I) were first proposed. Then, the performance of the BDS-3 carrier-phase time transfer was assessed and investigated. For comparison and evaluation purposes, precise point positioning (PPP) time transfer with BDS-2, GPS, and Galileo was also assessed. Analogous to the BDS-3 signal structure, the performance of BDS-2 based on the B1I/B2I- and B1I/B3I- combined PPP solutions was first validated. Then, with respect to the GPS and Galileo PPP time transfer, the BDS-3 B1I/B3I, B1I/B2a, B1C/B2a, and B1C/B3I; BDS-2 B1I/B2I; and BDS-2 + BDS-3 B1I/B3I PPP time transfers were assessed and analysed. The experimental results demonstrated that the BDS-3 B1I/B3I performance was slightly better than the BDS-3 B1I/B2a, B1C/B3I and B1C/B2a performances due to the differential code bias (DCB) accuracy. Furthermore, the BDS-2 B1I/B3I PPP was worse than the other PPP solutions, while BDS-2 + BDS-3 B1I/B3I PPP outperformed the other PPP solutions. Compared with the BDS-2 B1I/B3I PPP, the mean reductions in the standard deviation (STD) values were approximately (77.2, 21.3, 1.8, 1.3, 13.1) % from (0.241, 0.687, 0.547, 0.218, 0.585) ns to (0.136, 0.566, 0.537, 0.215, 0.517) ns calculated from BDS-3 and (97.5, 23.5, 4.9, 4.8, 16.7)% from (0.241, 0.687, 0.547, 0.218, 0.585) ns to (0.122, 0.556, 0.521, 0.208, 0.501) ns calculated and BDS-2 + BDS-3, respectively. For the frequency stability, BDS-3 demonstrated better results, while BDS-2 + BDS-3 presented significant improvements with respect to BDS-2. The maximum improvement in BDS-2 + BDS-3 reached up to 56.1% from 3.3973E-14 to 1.49E-14. Additionally, the modified Allan deviation (MDEV) of BDS-2 + BDS-3 was analogous to that of GPS and Galileo.

Automatic Calibration of Process Noise Matrix and Measurement Noise Covariance for Multi-GNSS Precise Point Positioning

The Expectation-Maximization algorithm is adapted to the extended Kalman filter to multiple GNSS Precise Point Positioning (PPP), named EM-PPP. EM-PPP considers better the compatibility of multiple GNSS data processing and characteristics of receiver motion, targeting to calibrate the process noise matrix Qt and observation matrix Rt, having influence on PPP convergence time and precision, with other parameters. It is possibly a feasible way to estimate a large number of parameters to a certain extent for its simplicity and easy implementation. We also compare EM-algorithm with other methods like least-squares (co)variance component estimation (LS-VCE), maximum likelihood estimation (MLE), showing that EM-algorithm from restricted maximum likelihood (REML) will be identical to LS-VCE if certain weight matrix is chosen for LS-VCE. To assess the performance of the approach, daily observations from a network of 14 globally distributed International GNSS Service (IGS) multi-GNSS stations were processed using ionosphere-free combinations. The stations were assumed to be in kinematic motion with initial random walk noise of 1 mm every 30 s. The initial standard deviations for ionosphere-free code and carrier phase measurements are set to 3 m and 0.03 m, respectively, independent of the satellite elevation angle. It is shown that the calibrated Rt agrees well with observation residuals, reflecting effects of the accuracy of different satellite precise product and receiver-satellite geometry variations, and effectively resisting outliers. The calibrated Qt converges to its true value after about 50 iterations in our case. A kinematic test was also performed to derive 1 Hz GPS displacements, showing the RMSs and STDs w.r.t. real-time kinematic (RTK) are improved and the proper Qt is found out at the same time. According to our analysis despite the criticism that EM-PPP is very time-consuming because a large number of parameters are calculated and the first-order convergence of EM-algorithm, it is a numerically stable and simple approach to consider the temporal nature of state-space model of PPP, in particular when Qt and Rt are not known well, its performance without fixing ambiguities can even parallel to traditional PPP-RTK.

Precise point positioning with decimetre accuracy using wide-lane ambiguities and triple-frequency GNSS data

Abstract This paper proposes precise point positioning (PPP) methods that offer an accuracy of a few decimetres (dm) with triple frequency GNSS data. Firstly, an enhanced triple frequency linear combination is presented for rapid fixing of the extra wide-lane (EWL) and wide-lane (WL) ambiguities for GPS, Beidou-2 and Galileo. This has improved performance compared to the Melbourne-Wübbena (MW) linear combination, and has 6.7 % lower measurement noise for the GPS L1/L2 signals, 12.7 % for L1/L5 and 0.7 % for L2/L5. Analysis with tested data showed a 5–6 % reduction in time required to fix the N 21 {N_{21}} and N 51 {N_{51}} ambiguities. Once the EWL/WL ambiguities are fixed with the proposed linear combinations, three methods are presented that aim to provide positioning accuracy of a few dm. In the first approach, the three EWL/WL ambiguities in their respective phase equations are used to derive a low-noise ionosphere-free (IF) linear combination. The second method uses a low noise IF combination with two carrier-phase EWL/WL equations and a single pseudorange measurement. The third method uses a low noise IF combination with a single carrier phase EWL equation and two pseudorange measurements. These proposed methods can provide dm level positioning accuracy if carrier phase measurements with mm precision is tracked by the receiver. When comparing these combinations with a combination proposed in [22], it is found that superior performance is achieved with the third method when carrier phase noise is &gt;5–6 mm for GPS and Beidou-2 and &gt;2–3 mm for Galileo. This model only requires the EWL ambiguity to be fixed which typically takes just one epoch of data. Thus, the user achieves instant decimetre level PPP accuracy.

Mitigation of Short-Term Temporal Variations of Receiver Code Bias to Achieve Increased Success Rate of Ambiguity Resolution in PPP

Ambiguity resolution (AR) is critical for achieving a fast, high-precision solution in precise point positioning (PPP). In the standard uncombined PPP (S-UPPP) method, ionosphere-free code biases are superimposed by ambiguity and receiver clock offsets to be estimated. However, besides the time-constant part of the receiver code bias, the complex and time-varying term in receivers destroy the stability of ambiguities and degrade the performance of the UPPP AR. The variation of receiver code bias can be confirmed by the analysis in terms of ionospheric observables, code multipath (MP) of the Melbourne–Wübbena (MW) combination and the ionosphere-free combination. Therefore, the effect of receiver code biases should be rigorously mitigated. We introduce a modified UPPP (M-UPPP) method to reduce the effects of receiver code biases in ambiguities and to decouple the correlation between receiver clock parameters, code biases, and ambiguities parameters. An extra receiver code bias is set to isolate the code biases from ambiguities. The more stable ambiguities without code biases are expected to achieve a higher success rate of ambiguity resolution and a shortened convergence time. The variations of the receiver code biases, which are the unmodeled errors in measurement residuals of the S-UPPP method, can be estimated in the M-UPPP method. The maximum variation of the code biases is up to 16 ns within two-hour data. In the M-UPPP method, the averaged epoch residuals for code and phase measurements recover their zero-mean features. For the ambiguity-fixed solutions in the M-UPPP method, the convergence times are 14 and 43 min with 17.7% and 69.2% improvements compared to that in the S-UPPP method which are 17 and 90 min under the 68% and 95% confidence levels.

Characteristics Analysis of Raw Multi-GNSS Measurement from Xiaomi Mi 8 and Positioning Performance Improvement with L5/E5 Frequency in an Urban Environment

Achieving continuous and high-precision positioning services via smartphone under a Global Navigation Satellite System (GNSS)-degraded environment is urgently demanded by the mass market. In 2018, Xiaomi launched the world’s first dual-frequency GNSS smartphone, Xiaomi Mi 8. The newly added L5/E5 signals are more precise and less prone to distortions from multipath reflections. This paper discusses the characteristics of raw dual-frequency GNSS observations from Xiaomi Mi 8 in urban environments; they are characterized by high pseudorange noise and frequent signal interruption. The traditional dual-frequency ionosphere-free combination is not suitable for Xiaomi Mi 8 raw GNSS data processing, since the noise of the combined measurements is much larger than the influence of the ionospheric delay. Therefore, in order to reasonably utilize the high precision carrier phase observations, a time differenced positioning filter is presented in this paper to deliver continuous and smooth navigation results in urban environments. The filter first estimates the inter-epoch position variation (IEPV) with time differenced uncombined L1/E1 and L5/E5 carrier phase observations and constructs the state equation with IEPV to accurately describe the user’s movement. Secondly, the observation equations are formed with uncombined L1/E1 and L5/E5 pseudorange observations. Then, kinematic experiments in open-sky and GNSS-degraded environments are carried out, and the proposed filter is assessed in terms of the positioning accuracy and solution availability. The result in an open-sky environment shows that, assisted with L5/E5 observations, the root mean square (RMS) of the stand-alone horizontal and vertical positioning errors are about 1.22 m and 1.94 m, respectively, with a 97.8% navigation availability. Encouragingly, even in a GNSS-degraded environment, smooth navigation services with accuracies of 1.61 m and 2.16 m in the horizontal and vertical directions are obtained by using multi-GNSS and L5/E5 observations.

Antenna phase center correction differences from robot and chamber calibrations: the case study LEIAR25

In recent years, the Global Navigation Satellite Systems (GNSS) have been intensively modernized, resulting in the introduction of new carrier frequencies for GPS and GLONASS and the development of new satellite systems such as Galileo and BeiDou (BDS). For this reason, the absolute field antenna calibrations performed so far for the two legacy carrier frequencies, the GPS and GLONASS, seem to be insufficient. Hence, all antennas will require a re-calibration of their phase center variations for the new signals to ensure the highest measurement accuracy. Currently, two absolute calibration methods are used to calibrate GNSS antennas: field calibration using a robot and calibration in an anechoic chamber. Unfortunately, differences in these methodologies also result in a disparity in the obtained antenna phase center corrections (PCC). Therefore, we analyze the differences between individual PCC obtained with these two methods, specifically for the Leica AR-25 antenna model (LEIAR25). In addition, the influence of PCC differences on the GNSS-derived position time series for 19 EUREF Permanent GNSS Network (EPN) stations was also assessed. The results show that the calibration method has a noticeable impact on PCC models. PCC differences determined for the ionosphere-free combination may reach up over 20 mm and can be transferred to the position domain. Further tests concerning the positioning accuracy showed that for horizontal coordinates differences between solutions were mostly below 1 mm, exceeding 2 mm only at two stations for the GLONASS solution. However, the height component differences exceeded 5 mm for four, six and six stations out of 19 for the GPS, GLONASS and Galileo solutions, respectively. These differences are strongly dependent on large L2 calibration differences.

A New Cycle Slip Detection and Repair Method for Galileo Four-Frequency Observations

Galileo four-frequency cycle slip detection and repair (CDR) is susceptible to the noise of pseudo-range observations. In this study, the Galileo four-frequency carrier phase smoothed pseudo-range (CPSP) assisted CDR method was proposed. Such method was conducted in three steps in sequence. First, the four linear independent combinations of the Galileo four-frequency observation were taken for CDR. Second, the non-divergent Hatch filter was employed to carry out the pseudo-range observation smoothly. Third, the true cycle slip was determined by rounding the float value using the least square method. To take the optimal combination of the CDR and verify the feasibility of the proposed method, the Galileo observations with satellites in different types were performed. According to the experimental results, (1) the four linear independent combinations of the geometry-free carrier phase combination (0, 1, 0, −1), (1, 1, −1, −1), (1, 2, −2, −1) and geometry-free and ionosphere-free combination (0, 0, 1, −1) were adopted for the cycle slip detection; (2) The success rate of cycle slip repair reached over 99.99% after four-frequency CPSP processing. With the CPSP assisted CDR method, the differences of the root mean square (RMS) between float and true values were down-regulated by 79.61%, 70.03%, 66.25% and 72.75%, respectively. (3) The differences of the root mean square (RMS) between float and true values were down-regulated by 13.62%, 10.67%, 10.67% and 10.67% after smoothing, respectively. In summary, the Galileo four-frequency CDR was effectively performed by the proposed method in active ionospheric area.

The effect of the second-order ionospheric term on GPS positioning in Antarctica

ABSTRACT Global Positioning System (GPS) signals are delayed when passing through the ionosphere due to an ionospheric refraction effect. The ionospheric-free linear combination of GPS signals can eliminate most of the error caused by the first-order ionospheric term. However, the influence of higher-order ionospheric terms remains, and it should be accounted for when conducting high-precision geodetic applications. In this study, we use data from twenty-one GPS continuous operating stations from different observing networks distributed in Antarctica and analyze the effect of the second-order ionospheric term on GPS positioning during the whole year for 2012. Results show that the effect on these stations is at the level of submillimeters to millimeters, and the effect in summer is several times higher than it is in winter. In addition, this effect is found to be positively correlated to the change of the total electron content over the Antarctic continent. On the other hand, all of the stations show the southward and upward movements derived from the effect. The common and seasonal displacement trends displayed in the Antarctic region should not be ignored in future high-precision research or applications but should be brought to attention, especially when total electron content in the ionosphere is high.

Reducing convergence time of precise point positioning with ionospheric constraints and receiver differential code bias modeling

Long convergence time has limited the wide application of traditional precise point positioning (PPP) based on an ionosphere-free combination of dual-frequency observations. Different from the traditional PPP, the uncombined PPP method based on raw observations estimates ionospheric delays. When external ionospheric information is available, it can be applied as a constraint to help shorten the convergence time, as a result of the reduced correlation between the position and the ionospheric parameters. The receiver differential code biases (DCBs) will be a concern, however, when applying the external ionospheric information. For receiver DCBs, it is usually assumed that the biases can be absorbed by the receiver clock parameters. We have demonstrated that the receiver DCBs cannot be fully assimilated by one receiver code clock parameter because the receiver DCBs have different effects on the code and carrier phase measurements at any frequency. Additional parameters are necessary to model the receiver DCBs so that their effects on the positioning solution can be minimized. We developed an ionosphere-constrained PPP model to incorporate ionospheric total electron content (TEC) in the slant (STEC) and vertical (VTEC) when leveraging a regional network and global ionospheric maps (GIMs). Both static and kinematic experimental results show that the convergence time and the positioning accuracy can be improved significantly. Accuracies at the first epoch of 0.4 m for GIM constraints, and 0.2 m for the regional constraints, are achievable. The convergence time to 1 dm horizontal accuracy is reduced to 7.5 min at a 68% confidence level.