Jingnan Liu

Precise orbit determination of Beidou Satellites with precise positioning

Chinese Beidou satellite navigation system constellation currently consists of eight Beidou satellites and can provide preliminary service of navigation and positioning in the Asia-Pacific Region. Based on the self-developed software Position And Navigation Data Analysis(PANDA) and Beidou Experimental Tracking Stations (BETS), which are built by Wuhan University, the study of Beidou precise orbit determination, static precise point positioning (PPP), and high precision relative positioning, and differential positioning are carried out comprehensively. Results show that the radial precision of the Beidou satellite orbit determination is better than 10 centimeters. The RMS of static PPP can reach several centimeters to even millimeters for baseline relative positioning. The precision of kinematic pseudo-range differential positioning and RTK mode positioning are 2–4 m and 5–10 cm respectively, which are close to the level of GPS precise positioning. Research in this paper verifies that, with support of ground reference station network, Beidou satellite navigation system can provide precise positioning from several decimeters to meters in the wide area and several centimeters in the regional area. These promising results would be helpful for the implementation and applications of Beidou satellite navigation system.

Total least squares adjustment in partial errors-in-variables models: algorithm and statistical analysis

The weighted total least squares (TLS) method has been developed to deal with observation equations, which are functions of both unknown parameters of interest and other measured data contaminated with random errors. Such an observation model is well known as an errors-in-variables (EIV) model and almost always solved as a nonlinear equality-constrained adjustment problem. We reformulate it as a nonlinear adjustment model without constraints and further extend it to a partial EIV model, in which not all the elements of the design matrix are random. As a result, the total number of unknowns in the normal equations has been significantly reduced. We derive a set of formulae for algorithmic implementation to numerically estimate the unknown model parameters. Since little statistical results about the TLS estimator in the case of finite samples are available, we investigate the statistical consequences of nonlinearity on the nonlinear TLS estimate, including the first order approximation of accuracy, nonlinear confidence region and bias of the nonlinear TLS estimate, and use the bias-corrected residuals to estimate the variance of unit weight.

Recent Development of PANDA Software in GNSS Data Processing

Under the financial support of several Chinese national scientific projects, PANDA (Positioning And Navigation Data Analyst) software developed originally by Wuhan University has achieved the advanced level in the world. PANDA is currently recognized as a main research tool in several famous institutes in the GNSS community. In this paper, the recent development of PANDA software is introduced, including the COSMIC orbit determination in low Earth orbits, the real-time GPS satellite orbit and clock determination and precise point positioning with ambiguity resolution. It is concluded that PANDA is of great improvement in the past five years, and more advancement will be made in its pragmatic aspect especially in engineering applications.

Crustal motion of Chinese mainland monitored by GPS

To measure and monitor the crustal motion in China, a GPS network has been established with an average side length of 1 000 km and with more than 20 points on the margins of each major tectonic block and fault zone in China. Three campaigns were carried out in 1992,1994 and 1996, respectively by this network. Here we present, for the first time, the horizontal displacement rates of 22 GPS monitoring stations distributed over the whole China and global IGS stations surrounding China, based on these GPS repeated measurements. From these results by GPS, we have obtained the sketch of crustal motion in China.

Variance components in errors-in-variables models: estimability, stability and bias analysis

Although total least squares has been substantially investigated theoretically and widely applied in practical applications, almost nothing has been done to simultaneously address the estimation of parameters and the errors-in-variables (EIV) stochastic model. We prove that the variance components of the EIV stochastic model are not estimable, if the elements of the random coefficient matrix can be classified into two or more groups of data of the same accuracy. This result of inestimability is surprising as it indicates that we have no way of gaining any knowledge on such an EIV stochastic model. We demonstrate that the linear equations for the estimation of variance components could be ill-conditioned, if the variance components are theoretically estimable. Finally, if the variance components are estimable, we derive the biases of their estimates, which could be significantly amplified due to a large condition number.

Real-time detection and repair of cycle slips in triple-frequency GNSS measurements

Cycle slip detection and repair are prerequisites to the use of the global navigation satellite system (GNSS) carrier phases for precise positioning. Modern GNSS techniques introduce triple- or multi-frequency signals that are beneficial for cycle slip detection and repair. We present a new real-time cycle slip detection and repair method based on the independent linear combinations of undifferenced triple-frequency GNSS observations. The proposed method forms three types of linear combinations based on the original observations. These combinations are called extra-wide lane (EWL), wide lane (WL), and narrow lane (NL). Cycle slips on the combinations are determined sequentially in three cascaded steps. The first step employs the geometry-free and ionosphere-free Hatch–Melbourne–Wübbena combination to determine and repair the EWL cycle slips. The second step subtracts the cycle-slip-repaired EWL combination from the WL combination to eliminate the geometry part of the WL combination. This subtraction results in a new function that contains the WL ambiguity and residual ionospheric delay. This function is differenced at two consecutive epochs to determine the WL cycle slips. The residual ionospheric delay difference is ignored because of its small magnitude relative to WL wavelength. The third step determines the NL cycle slips in the same manner as in the second step. The difference is that the cycle-slip-repaired EWL combination is replaced with the more accurate cycle-slip-repaired WL combination. Moreover, the residual ionospheric delay difference is compensated by the ionospheric delay rate derived from the original carrier phase observations. When the EWL, WL, and NL cycle slips are determined, cycle slips on the original carrier phase observations can be uniquely identified. The proposed approach has been tested on 30-s triple-frequency BeiDou navigation satellite system data under different levels of ionospheric variations, and on 30-s triple-frequency global positioning system and quasi-zenith satellite system data. Results indicate that the approach can effectively detect and correct cycle slips even for one cycle under low sampling rate or active ionospheric conditions on each frequency in real time.

Three-carrier ambiguity resolution using the modified TCAR method

Abstract Multi-frequency technique is expected to be widely adopted with the new generations of global navigation satellite system, which is anticipated to benefit ambiguity resolution (AR). Three-carrier AR (TCAR) is a classical AR method based on triple-frequency observations, which is efficient for AR of short baseline. However, this method ignores the residual ionospheric delay, which degrades the reliability in active ionosphere situations and reduces the success of AR for medium and long baselines. We investigate the classical TCAR method and identify the major deficiency that hampers its application, especially to medium and long baselines. To improve this algorithm, the second and third steps of the classical TCAR are modified accordingly. In step 2, the ambiguity-resolved extra-wide-lane (EWL) combination and three pseudorange observations are employed to eliminate or reduce the residual ionospheric delay, in addition to the geometry term. In step 3, besides the EWL combination and pseudoranges, the ambiguity-resolved wide-lane (WL) combination is used to completely eliminate the ionosphere and geometry terms. The combination coefficients of these pseudoranges and combinations are optimized to minimize the noise of the ambiguity estimates. In order to assess the performances, real triple-frequency observations of BeiDou navigation system of baselines with different lengths are processed by the two methods. Results show that, the classical TCAR method is very sensitive to ionospheric delay and limited to short baseline application, while the modified TCAR method is free from ionospheric delay and can be applied to AR of median and long baselines. For WL AR, the modified TCAR method shows a comparable performance with the classical TCAR method, and a better performance can be expected when the baseline becomes longer, e.g., from 100s to 1,000s kilometers. For narrow-lane AR, the modified TCAR method performs much better than the classical TCAR method for median and long baselines.

Analysis of BeiDou Satellite Measurements with Code Multipath and Geometry-Free Ionosphere-Free Combinations.

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.

Using Allan variance to analyze the error characteristics of GNSS positioning

Currently, we evaluate the positioning accuracy of GNSS mainly by providing statistical values that can represent the overall error level, such as CEP, RMS, 2DRMS, and maximum error. These are solid indicators of the general performance of the GNSS positioning. But some applications like GNSS/INS integrated system require a detailed analysis of the error characteristics and knowledge of the precise error model. This requirement necessitates the modeling of the error components of the GNSS positioning solutions. In our research, the Allan variance method is proposed to analyze the GNSS positioning errors, describe the error characteristics, and build the corresponding error models. Based on our research, four dominant noise terms are identified in the GNSS positioning solutions, that is, 1st order Gauss-Markov process, Gaussian white noise, random walk noise, and flicker noise, which indicates that white noise is not always enough and appropriate to model GNSS positioning errors for some applications. The results show that the Allan variance is a feasible and effective way to analyze the error characteristics of the GNSS positioning solutions.

Integer estimation methods for GPS ambiguity resolution: an applications oriented review and improvement

Abstract The integer least squares (ILS) problem, also known as the weighted closest point problem, is highly interdisciplinary, but no algorithm can find its global optimal integer solution in polynomial time. We first outline two suboptimal integer solutions, which can be important either in real time communication systems or to solve high dimensional GPS integer ambiguity unknowns. We then focus on the most efficient algorithm to search for the exact integer solution, which is shown to be faster than LAMBDA in the sense that the ratio of integer candidates to be checked by the efficient algorithm to those by LAMBDA can be theoretically expressed by rm, where r⩽1 and m is the number of integer unknowns. Finally, we further improve the searching efficiency of the most powerful combined algorithm by implementing two sorting strategies, which can either be used for finding the exact integer solution or for constructing a suboptimal integer solution. Test examples clearly demonstrate that the improved methods can perform significantly better than the most powerful combined algorithm to simultaneously find the optimal and second optimal integer solutions, if the ILS problem cannot be well reduced.