BDS triple-frequency carrier-phase linear combination models and their characteristics

Using GNSS observable from some stations in the Asia-Pacific area, the carrier-to-noise ratio (CNR) and multipath combinations of BeiDou Navigation Satellite System (BDS), as well as their variations with time and/or elevation were investigated and compared with those of GPS and Galileo. Provided the same elevation, the CNR of B1 observables is the lowest among the three BDS frequencies, while B3 is the highest. The code multipath combinations of BDS inclined geosynchronous orbit (IGSO) and medium Earth orbit (MEO) satellites are remarkably correlated with elevation, and the systematic “V” shape trends could be eliminated through between-station-differencing or modeling correction. Daily periodicity was found in the geometry-free ionosphere-free (GFIF) combinations of both BDS geostationary Earth orbit (GEO) and IGSO satellites. The variation range of carrier phase GFIF combinations of GEO satellites is −2.0 to 2.0 cm. The periodicity of carrier phase GFIF combination could be significantly mitigated through between-station differencing. Carrier phase GFIF combinations of BDS GEO and IGSO satellites might also contain delays related to satellites. Cross-correlation suggests that the GFIF combinations’ time series of some GEO satellites might vary according to their relative geometries with the sun.

Extension of the BDS IGSO and MEO satellite antenna patterns with FengYun-3C and LuTan-1 onboard data

The triple-frequency linear combination with a low noise, a long wavelength, and a weak ionosphere is beneficial to effectively eliminate or weaken the common errors, advance the reliability of cycle slip detection and repair, and speed up the convergence time of fixed ambiguity. By establishing the Galileo triple-frequency carrier linear combination model, three types of linear combinations are derived: Geometry-free (GF) combinations, minimum noise (MN) combinations, and ionosphere-free (IF) combinations. The geometric relationships of these linear combinations are displayed in the form of image. The results indicate that the angle formed by the IF combinations and the MN combinations is between 75.02° and 86.01°, which also illustrates that it is more difficult to meet the carrier phase combinations with a low noise and a weak ionosphere. Moreover, to guarantee the integer cycle characteristics of ambiguity, the combination coefficient must be an integer. Galileo triple-frequency linear combination is solved utilizing the extremum method. To sum up, the sum of the coefficients of the extra wide lane (EWL) combinations and wide lane (WL) combinations is zero, and the sum of the coefficients of the narrow lane (NL) combinations is one. (0, 1, −1) is the optimal triple-frequency linear combination in Galileo. Three independent linear combinations are selected separately from the EWL, WL, and NL to jointly solve the integer ambiguity. Further, it creates a prerequisite for high-precision and real-time kinematic positioning.

Beidou satellite navigation system (BDS) signal is blocked during dynamic positioning which can lead to poor geometry structure of visible satellites and serious multipath effect. In this contribution, the advantages of inertial navigation system (INS), autonomy and short-term accuracy, are used to speed up the BDS ambiguity resolution (AR). However, the accuracy of general INS cannot meet the fixed BDS original carrier ambiguity requirement. BDS can broadcast three frequency observations, so the virtual observations of long wavelength and short wavelength are formed by linear combination, which make the INS precision meet the fixed ambiguity requirement and achieve high precision positioning, respectively. The short wavelength observations ambiguity due to the relatively large noise is difficult to reliably fixed. To solve this problem, Empirical mode decomposition (EMD) can be used to decompose the error of the short wavelength observations into two parts: systematic error and random error. Systematic error is used to correct the short wavelength ambiguity float solution and reduce the ambiguity search space, to achieve a reliable fixed purpose, finally solving the baseline solution. The experimental results show that the algorithm is more fast and reliable than the traditional algorithms, and more suitable for dynamic positioning in the complex environment.

Aiming at the characteristics of the Beidou satellite navigation system (BDS), we introduce a new relative positioning method based on un-differenced observation. Unlike the relative positioning which always uses conventional double-difference approach to eliminate the satellite and receiver side errors and the common errors on the propagation path, this method preprocesses data similar to Precise Point Positioning (PPP) and uses ionosphere-free combination to relative positioning. During this process, one hand satellite clock errors are estimated as unknown parameters and use to establish the relationship between base station and rover, on the other hand ambiguity don’t need to be fixed. The experiment result which based on resolving one week BDS observation data shows, the coordinate accuracy can reach millimeter and centimeter-level precision in horizontal and vertical direction when using this method for relative positioning over thousands of kilometers.

GNSS is now being speedily expanded to our daily life, but the positioning precision still can hardly meet the demands of many high-precision applications, such as approaching landing system on airports. Due to the development of GNSS, quadruple-frequency signals are now available in China's BeiDou Navigation Satellite System (BDS) and the European Galileo system, which can contribute to positioning precision. Positioning precision can not be improved obviously by quadruple-frequency carrier phases until cycle slips are detected and repaired. A method using four linear combinations to detect and repair quadruple-frequency cycle slips is proposed in the paper. The choices of the four linear combinations are conducted in cascaded steps in accordance to the cycle slip fixing probability. When the four detection combinations are determined, cycle slips on original carrier phase observations can be uniquely determined. The proposed algorithm has been tested on real 30-second quadruple-frequency static observations of BDS and Galileo and on real 0.05-second quadruple-frequency kinematic observations of BDS and Galileo. Simulated and real cycle slips are tested. The results show that the proposed algorithm can detect and repair cycle slips even for one cycle effectively.

To reduce the uncertainty in location information supplied by GNSS receiver, the range errors (clock bias, troposphere, ionosphere, multipath etc.,) have to be eliminated. The linear combinations of multi-frequency GNSS observables, will aid in eliminating most of the errors. The ionospheric error is treated as predominant error and can be mitigated by using ionosphere-free linear combination. In this paper, the attainable accuracy using ionosphere-free linear combination of combined GPS L11/L5 and Galileo E1/E5a is evaluated for single point positioning. Taking the advantages of availability of civilian codes on signal frequencies, code-phase measurements are used instead of carrier-phase. The 95th percentile horizontal, vertical and 3D position accuracies are 1.08m, 0.80m and 1.81m respectively

High precision positioning using carrier phase observations is the main method and approach of Beidou navigation satellite system (BDS) positioning. Especially, the ambiguity fixing has always been the focus, and it is always the key problem of the related research. In the medium/long baseline, the ionospheric delay error will affect the efficiency of ambiguity resolution seriously, and then affect the precision and efficiency of precise positioning. To solve this problem, this paper put forward a method of medium/long baseline ambiguity resolution based on the classical three-carrier ambiguity resolution (TCAR) model. To improve the reliability of wide-lane (WL) ambiguity fixing, a factor is introduced into the optimal linear combination for reducing noise, and the ionospheric priori standard deviation is used to minimize the effect of the length of the baseline. In order to verify the feasibility and effectiveness of the algorithm, the proposed algorithm is compared with the classical TCAR algorithm in several aspects such as the ambiguity fixed success rate. The results show that the proposed algorithm has the highest success rate in the case of medium/long baseline, and can fix the ambiguity instantly.

In this section we introduce what we call “Positive Code-Phase” linear combination or the LP linear combination (phase and code added) to eliminate the first-order ionosphere effect and estimate LEO orbits using single-frequency GPS measurements, (see Svehla and Rothacher 2003a; 2005b). We do not smooth code measurements with the linear model as proposed by the GRAPHIC (Group and Phase Ionospheric Calibration) linear combination in (Yunck 1993; Gold et al. 1994; Muellerschoen et al. 2004). We show that in the case of the GRACE-B satellite it is possible to estimate LEO orbits to an accuracy of 2–3 cm RMS (1.3 cm radial) using single-frequency GPS measurements only, (see also Svehla et al. 2010a). This is similar to the orbit accuracy of 1–2 cm one can typically achieve with dual-frequency carrier-phase measurements. This is possible due to the very low noise level of the code measurements from the GRACE-B satellite and recent gravity field models from the GRACE and GOCE missions that provide very accurate gravity field coefficients up to degree and order 120 allowing an orbit parameterization with a very modest number of empirical parameters. In addition, thanks to the excellent precision of the real-time GPS satellite clock parameters provided by the IGS, we show that this cm-orbit accuracy can be achieved even in real-time. Subsequently, we introduce an estimation of the group delay pattern of GNSS satellite antennae based on the LP linear combination. We show that the LP linear combination can be used to estimate single-code group delay variations (GDV) for GNSS satellite antennae at the single-frequency level and present the first GDV pattern based on GPS measurements from the GRACE-B satellite. The GDV pattern based on LP linear combination is related to a single code observable and not to an ionosphere-free linear combination, a strong advantage in the presence of multi-GNSS data. After that, we present the concept of using single-frequency GPS radio-occultations (RO) as a very promising alternative to standard GPS-RO based on dual-frequency measurements. The advantage of this approach is that carrier and code measurements on the same GPS frequency follow the same path in the ionosphere. This is not the case for the bended carrier-phase GPS-RO measurements on different GPS frequencies that can reach a vertical separation of up to 500 m in some cases. Since the antenna used for GPS-RO is typically a high-gain antenna, the noise level of the code measurements is very low and, with an additional smoothing, this approach could be used for GPS-RO with SBAS satellites in GEO. The same approach could also be applied to GNSS reflectometry (GNSS-R).

This paper proposes an enhanced algorithm to estimate the differential code biases (DCB) on three frequencies of the BeiDou Navigation Satellite System (BDS) satellites. By forming ionospheric observables derived from uncombined precise point positioning and geometry-free linear combination of phase-smoothed range, satellite DCBs are determined together with ionospheric delay that is modeled at each individual station. Specifically, the DCB and ionospheric delay are estimated in a weighted least-squares estimator by considering the precision of ionospheric observables, and a misclosure constraint for different types of satellite DCBs is introduced. This algorithm was tested by GNSS data collected in November and December 2013 from 29 stations of Multi-GNSS Experiment (MGEX) and BeiDou Experimental Tracking Stations. Results show that the proposed algorithm is able to precisely estimate BDS satellite DCBs, where the mean value of day-to-day scattering is about 0.19 ns and the RMS of the difference with respect to MGEX DCB products is about 0.24 ns. In order to make comparison, an existing algorithm based on IGG: Institute of Geodesy and Geophysics, China (IGGDCB), is also used to process the same dataset. Results show that, the DCB difference between results from the enhanced algorithm and the DCB products from Center for Orbit Determination in Europe (CODE) and MGEX is reduced in average by 46 % for GPS satellites and 14 % for BDS satellites, when compared with DCB difference between the results of IGGDCB algorithm and the DCB products from CODE and MGEX. In addition, we find the day-to-day scattering of BDS IGSO satellites is obviously lower than that of GEO and MEO satellites, and a significant bias exists in daily DCB values of GEO satellites comparing with MGEX DCB product. This proposed algorithm also provides a new approach to estimate the satellite DCBs of multiple GNSS systems.

The study of BDS RTK algorithm based on zero-combined observations and ionosphere constraints

A Precise Point Positioning (PPP) method is presented that uses raw triple-frequency GPS measurements augmented with a regional ionospheric model (RIM) for faster solution convergence. The proposed method has two characteristics. Firstly, there is no noise amplification since raw uncombined carrier phase and code measurements are used. This reduces the solution convergence time, contrary to the ionosphere-free linear combination with a noise amplification factor of 3 that is used in traditional dual-frequency model. Secondly, the ionosphere delay remaining in the uncombined equations is modelled using externally provided precise ionosphere corrections, hence reducing the number of parameters to be estimated. Triple frequency simulated data was used to validate the model using a fully operational GPS constellation at four sites in Australia for one week duration. The performance of 2-hourly PPP solutions were compared for three cases including (1) the proposed triple frequency raw uncombined model with treatment of the ionosphere delay using an externally provided model (2) triple frequency raw uncombined model with estimation of ionosphere error (i.e. without RIM) and (3) traditional dual frequency PPP model. The three cases were compared in terms of solution convergence time taken to achieve and maintain 3- dimensional (3D), horizontal and vertical positional accuracies of 1 decimeter and 5cm. The tests for the first case considered the use of varying precisions of the externally provided ionosphere model with standard deviations (std) of 0.1, 0.25, 0.5 and 1.0 TECU. Results from the comparison show that the best improvement was achieved with the proposed raw model when using a high precision ionosphere model with std = 0.1 TECU, where the convergence time to achieve sub-decimeter accuracy was reduced by 37%, 84% and 33% in the 3D, horizontal and vertical components. The most significant improvement was in the horizontal component since the ionospheric error mostly affects the height. The second model also outperformed the standard dual frequency model, but had less improvement in convergence time by 13%, 3% and 21% in the three components.

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.

Centimetre-level RTK solutions are mainly influenced by satellite orbit errors, ionospheric and tropospheric delays, and measurement noise (including multipath effects). Estimation and mitigation of the main errors for the CM-level Compass RTK solutions over medium-long baselines are investigated. Tests conducted for this research lead to the following conclusions: 1.For 100 km baselines, a 4 cm error in height component will be induced by a 10 m orbit error. For longer baselines, rapid precise ephemeris will be needed for CM-level accuracy RTK solutions.2.The residual ionospheric delay error can be eliminated using the optimal triple-frequency ionosphere-free linear combination with the coefficients of 2·6087, −0·5175 and −1·0912 respectively for observations on f1, f2 and f3 frequencies. This combination is optimal in terms of its noise level, e.g., the noise is only amplified three times. It can be used for high accuracy RTK positioning.3.The residual tropospheric delay can be resolved for the introduced relative zenith tropospheric delay (RZTD) parameters.It is shown that the RTK solutions estimated from the least squares (LS) with the RZTD parameters are worse than that without these parameters. For instance, the errors in the height components are amplified approximately three times, which may be caused by the strong correlation between the introduced RZTD parameters and the height components. However, considering the fact that the residual zenith tropospheric delays vary slowly with time and the variation can be assumed to follow a random walk process, the RTK solutions can be improved using the Kalman filter and a priori information for the RZTD parameters.

In this paper, we propose a scheme referred to as integer-forcing message recovering (IFMR) to enable receivers to recover their desirable messages in interference channels. Compared to the state-of-the-art integer-forcing linear receiver (IFLR), our proposed IFMR approach needs to decode considerably less number of messages. In our method, each receiver recovers independent linear integer combinations of the desirable messages each from two independent equations. We propose an efficient polynomial-time algorithm to sequentially find the equations and integer combinations with maximum rates and analyze its complexity. We evaluate the performance of our scheme and compare the results with the minimum mean-square error linear receiver (MMSELR) and lattice-reduction-aided successive interference cancellation with signal-to-interference-plus-noise ratio maximizing preprocessing (LaR-aided SIC with SINR-Max), as well as the IFLR schemes. The results indicate that our IFMR scheme outperforms the MMSELR and LaR-aided SIC with SINR-Max schemes, in terms of achievable rate, considerably. Also, compared to the IFLR, the IFMR scheme achieves slightly less rates in moderate signal-to-noise ratios, with significantly less overall implementation complexity.