For four systems combined solutions with the M-MHM model, can reach an accuracy of 0.75, 0.55, and 2.08 cm in the east, north, and up components.
The positioning accuracy is on average 12% improvement compared to MHM and 29% improvement compared to SF. Compared with sidereal filtering and original MHM model, the M-MHM brings the highest improvement in both residual variance reduction and positioning accuracy. Although GPS L1 frequency is identical to Galileo E1, the model still has some systematic bias between GPS and Galileo. In addition, the multipath manifests great consistency among satellites for GPS, GLONASS, and Galileo systems when elevation angles are higher than 15°, while is more satellite dependent for BDS. Based on the multipath distribution, we propose a modified Multipath Hemispherical Map model (M-MHM), which constructs grids from residuals and is divided into three equal-elevation angle parts with an optimal resolution 0.2° × 0.2° × 1° from numerous experiments. These regularities double-reduce the complexity of data processing. For GLONASS, the residuals of R1 frequency recovered from R2 frequency with the mathematical relationship are better than 0.1 mm accuracy, which means the effect of inter-frequency bias can be neglected. Besides BDS Geostationary Earth Orbit satellites, the residuals for other satellites can establish accurate mathematic relationship between the frequencies. The results indicate that the residuals between frequencies have a significant linear negative correlation and synchronous time lag for each system. Although the residuals for one frequency assimilate the errors from other frequencies, which is caused by error adjustment by the least squares estimator, the primary component of residuals is multipath.
Hence, we extract the multipath directly from raw carrier-phase residuals of GPS, GLONASS, Galileo, and BDS, by using PPP technique. Multipath effect on carrier-phase observation is one of the bottlenecks for mm-level applications when using precise point positioning (PPP).