Rapid and accurate ambiguity resolution is the core of high-precision precise point positioning (PPP) data processing. However, the ambiguity parameters in PPP observation models are easily affected by atmospheric residual and gross errors, which lead to the probability of successfully fixing decreases and computational burden increases in full ambiguity resolution. Therefore, an increasing number of partial ambiguity resolution (PAR) strategies have been proposed. The selection of the optimal subset of PAR is crucial in this method. The traditional optimal subset selection method of PAR commonly leads to a single judgment criterion and weakened geometric configuration strength because the satellites with low elevation angles are often easily eliminated during the optimal subset selection. In this paper, a multi-factor constrained optimal subset selection method for PAR was proposed, which incorporates the ambiguity variance, the ambiguity dilution of precision (ADOP), satellite position dilution of precision (PDOP) and ratio test values. In order to verify the feasibility of the proposed optimal subset selection method, PAR tests under two schemes were performed for GPS/Galileo based on the static observation data of 15 Multi-GNSS Experiment (MGEX) tracking stations. The results show that, compared with the ambiguity variance sorting method, the proposed subset selection method can further improve the accuracy of the coordinate solution and the strength of geometric figure positioning. The average root mean square of the coordinate residuals is found to decrease by about 12.90%, 6.83% and 9.39% in the eastern, northern and vertical directions, respectively. The increase in the fixed epoch rate ranged from 0.87% to 33.33%, with an average of about 8.71%.

The calculation of surface area is meaningful for a variety of space-filling phenomena, e.g., the packing of plants or animals within an area of land. With Digital Elevation Model (DEM) data we can calculate the surface area by using a continuous surface model, such as by the Triangulated Irregular Network (TIN). However, just as the triangle-based surface area discussed in this paper, the surface area is generally biased because it is a nonlinear mapping about the DEM data which contain measurement errors. To reduce the bias in the surface area, we propose a second-order bias correction by applying nonlinear error propagation to the triangle-based surface area. This process reveals that the random errors in the DEM data result in a bias in the triangle-based surface area while the systematic errors in the DEM data can be reduced by using the height differences. The bias is theoretically given by a probability integral which can be approximated by numerical approaches including the numerical integral and the Monte Carlo method; but these approaches need a theoretical distribution assumption about the DEM measurement errors, and have a very high computational cost. In most cases, we only have variance information on the measurement errors; thus, a bias estimation based on nonlinear error propagation is proposed. Based on the second-order bias estimation proposed, the variance of the surface area can be improved immediately by removing the bias from the original variance estimation. The main results are verified by the Monte Carlo method and by the numerical integral. They show that an unbiased surface area can be obtained by removing the proposed bias estimation from the triangle-based surface area originally calculated from the DEM data.

Different types of gravity observations are available over coastal areas. The main challenge for coastal geoid determination is the proper fusion of heterogeneous gravity data including land, shipborne, airborne, and altimetry-derived gravity data. This paper describes the gravity data fusion and the computation of the gravimetric quasigeoid in the coastal area of mainland China. An iterative procedure of the weighted least-squares prediction based on rectangular harmonic functions is used for merging the land, altimetric, shipborne, and airborne gravity data. Applying the analytical continuation method in Molodensky's theoretic frame, the merged gravity data are then used to determine the gravimetric quasigeoid model by using the generalized Stokes' integral in a remove-compute-restore fashion. The gravimetric quasigeoid model is compared with the height anomalies determined at 662 GPS leveling points over the coastal region of mainland China, where both the geodetic height and the normal height are known. The standard deviations of the differences in the coastal provinces range from 1.8 to 4.4 cm. For the entire computation area, the mean and standard deviation of the differences are 27.9 and 3.9 cm, respectively.

Many Beidou navigation satellite system (BDS) receivers or boards provide dual-frequency measurements to conduct precise positioning and navigation for low-power consumption. Cycle-slip processing is a primary work to guarantee consistent, precise positioning with the phase data. However, the cycle-slip processing of BDS dual-frequency phases still follows with those of existing GPS methods. For single-satellite data, cycle-slip detection (CSD) with the geometry-free phase (GF) is disturbed by severe ionospheric delay variations, while CSD or cycle-slip repair (CSR) with the Melbourne–Wubbena combination (MW) must face the risk of the tremendous disturbance from large pseudorange errors. To overcome the above limitations, a new cycle-slip repair method for BDS dual-frequency phases (BDCSR) is proposed: (1) An optimal model to minimize the variance of the cycle-slip calculation was established to the dual-frequency BDS, after correcting the ionospheric variation with a reasonable and feasible way. (2) Under the BDS dual-frequency condition, a discrimination function was built to exclude the adverse disturbance from the pseudorange errors on the CSR, according to the rankings of the absolute epoch-difference GFs calculated by the searched cycle-slip candidates after correcting the ionospheric variation. Subsequently, many compared CSR tests were implemented in conditions of low and medium elevations during strong geomagnetic storms. Comparisons from the results of different methods show that: (1) The variations of ionospheric delays are intolerable in the cycle-slip calculation during the geomagnetic storm, and the tremendous influence from the ionospheric variation should be corrected before calculating the cycle-slip combination with the BDS dual-frequency data. (2) Under the condition of real dual-frequency BDS data during the geomagnetic storm, the actual success rate of the conventional dual-frequency CSR (CDCSR) by employing the optimized combinations, but absenting from the discrimination function, is lower than that of BDCSR by about 2%; The actual success rate of the CSD with MW (MWCSD), is lower than that of BDCSR by about 2%. (3) After adding gross errors of 0.7 m to all real epoch-difference pseudoranges epoch-by-epoch, results of CDCSR and MWCSD showed many errors. However, BDCSR achieved a higher actual success rate than those of CDCSR and MWCSD, about 43% and 16%, respectively, and better performance of refraining the disturbance of large pseudorange error on the cycle-slip determination was achieved in the BDCSR methodology.

In advance of precise positioning with phase data, cycle slip detection (CSD) is a basic work that should be implemented in phase data possessing. When the cycle slip occurred, cycle slip repair (CSR) can be taken to rebuild the continuity of phase data. Unfortunately, the large pseudorange errors can contaminate the combinations with the pseudoranges and phases such as the Hatch–Melbourne–Wubbena combination (HMW) and cause false CSD or wrong CSR results. On the other hand, the severe ionospheric time variation can deteriorate the epoch-difference geometry-free phase (GF), and tremendously interfere with the performances of CSD and CSR. To handle the aforementioned limitations, a global position system (GPS) triple-frequency CSR method (GTCSR) is proposed with two efficient treatments: (1) the significant ionospheric variations are corrected, and the influences from the residual ionospheric effects are minimized along with the observational noises; and (2) the impacts of large pseudorange errors are refrained by designing a discrimination function with a geometry-free and ionosphere-free phase to identify the correct cycle slip values. Consequently, CSR tests were conducted with three monitoring stations at different regions. First, during a strong geomagnetic storm, without correcting the ionospheric variation of CSR (WICSR) displayed obvious failures, and many epochs of cycle slip values from WICSR deviated from the known values. However, the results of the GTCSR were correct, and GTCSR presented a higher success rate than that of WICSR. Furthermore, for the real triple-frequency data, by adding gross errors of 2.5 m on all epoch-difference pseudoranges epoch by epoch, the conventional triple-frequency CSR with the optimized combinations (CTCSR) and the CSD with HMW (HMWCSD) showed many mistakes, where the results of CTCSR and HMWCSD on numerous epochs were inconsistent with the actual situations, but the success rate of GTCSR was significantly higher than those of CTCSR and HMWCSD. In summary, in the condition of the cutoff elevation being larger than 10 degrees, improved performances and higher success rates were achieved from GTCSR under environments of large pseudorange errors and severe ionospheric variations.

The ionospheric delay is one of the important error sources in the Global Navigation Satellite System (GNSS) data processing. With the rapid construction and development of GNSS, the abundant satellite resources have brought new opportunities for ionospheric monitoring. To further investigate the performances and abilities of Galileo and BDS in ionosphere modeling, we study the ionosphere modeling based on the 15th order spherical harmonic function, and 364 stations around the world are selected for global ionospheric modeling of GPS, GLONASS, Galileo and BDS systems under ionospheric quiet and active conditions, respectively. The results show that the average biases of the ionospheric models built by GPS, GLONASS and Galileo are relatively small, which are within 2 Total Electron Content Unit (TECU) as compared to the Center for Orbit Determination in Europe (CODE) global ionospheric map (GIM), while the average biases of the models built by BDS are between 6 and 8 TECU during the ionospheric quiet and active days, respectively. In addition, in order to analyze the modeling performances before and after using BDS geostationary earth orbit (GEO) satellites, BDS is divided into two groups, in which one group contains medium earth orbit (MEO), inclined geosynchronous orbit (IGSO) and GEO satellites; and the other group contains only MEO and IGSO satellites. The results show that the influence of GEO satellites on ionospheric modeling is less than 1 TECU. Due to the distribution of the stations, the 0-value region in the ionospheric model is mainly distributed in the mid and high-latitude regions of the southern hemisphere. Since the ionospheric parameters are lumped with the Differential Code Bias (DCB), we also estimate the DCB parameters and analyze their performances. The DCB estimated in ionosphere modeling shows strong stability, with the average biases of GPS, GLONASS, Galileo and BDS under 0.25 ns, 0.25 ns, 0.2 ns and 0.42 ns, respectively. We also estimate other DCB types of the four GNSS systems. The results show that the DCB is stable and shows consistency with Chinese Academy of Sciences (CAS) DCB products.