Research on Linear Combination Models of BDS Multi-Frequency Observations and Their Characteristics

The linear combination of multi-frequency carrier-phase and pseudorange observations can form the combined observations with special properties. The type and number of combined frequencies will directly affect the characteristics of the combined observations. BDS-2 and BDS-3 broadcast three and five signals, respectively, and the study of their linear combination is of great significance for precision positioning. In this contribution, the linear combination form of multi-frequency carrier-phase observations in cycles and meters is sorted out. Seven frequency combination modes are formed, and some special combinations for positioning are searched. Then, based on the principle of minimum combined noise, a simpler search method for the optimal real coefficients of ionosphere-free (IF) combination based on the least squares (LS) principle is proposed. The general analytical expressions of optimal real coefficients for multi-frequency geometry-based and ionosphere-free (GBIF), geometry-free and ionosphere-free (GFIF), and pseudorange multipath (PMP) combinations with the first-order ionosphere delay taken into account are derived. And the expression derivation process is given when both the first-order and second-order ionospheric delays are eliminated. Based on this, the characteristics of the optimal real coefficient combination in various modes are compared and discussed. The various combinations reflect that the accuracy of the combined observations from dual-frequency (DF) to five-frequency (FF) is gradually improving. The combination coefficient becomes significantly larger after taking the second-order ionospheric delay into account. In addition, the combined accuracy of BDS-3 is better than that of BDS-2. When only the first-order ionosphere is considered, the combination attribute of (B1C, B1I, B2a) is the best among the triple-frequency (TF) combinations of BDS-3. When both the first-order and second-order ionospheric delays are considered, the (B1C, B3I, B2a) combination is recommended.