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

Abstract

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.

The Extraction and Characterization of Pseudorange Multipath Based on BDS-3 Multi-Frequency Observations

“…For example, the Ω of B1I/B1C can be 25-35 times that of other dual-frequency combinations and about 45 times that of quad-and five-frequency combinations. Related studies have pointed out that, when the Ω is large, it will amplify the observation error implied by the carrier phase observations [20,34]. Therefore, combinations of Ω similar to B1I/B1C should be careful not to be used for extracting MPs.…”

“…Reasonably eliminating or reducing the effects of these errors can effectively improve the positioning accuracy [11][12][13][14]. Previous studies have shown that a reasonable combination of multiple frequency observations can reduce the impact of these errors, such as ionosphere-free combinations, ionosphere-reduced combinations, and geometric-free combinations [15][16][17][18][19][20]. With the increasing number of transmission signals, there are more favorable combinations available.…”

“…However, it is worth noting that, when extracting the MP of each BDS-3 satellite frequency, the traditional dual-frequency pseudorange and carrier phase observations combination method is still used and the multi-frequency signals provided by BDS-3 are not fully utilized. In theory, introducing additional carrier phase observations can better extract the true signal of an MP [20,34]. Therefore, how to reasonably use multi-frequency observations to extract the MPs for each frequency should be studied.…”

A mixed multi-frequency precise point positioning strategy based on the combination of BDS-3 and GNSS multi-frequency observations

“…However, to reduce the impacts of the second order of ionospheric delays, the combination coefficients can be easily obtained based on updating the A matrix with the aid of adding a column of ionospheric factors. Moreover, an example of BDS-3 five-frequency combination is presented in table 1, where the coefficients based on eliminating the second-order ionospheric delays are also listed [39]. According to the above method, an optimal IF combination of multi-frequency observations for a single GNSS system can be obtained, where all received frequencies are used to construct the observation equations.…”

Single-Epoch Decimeter-Level Precise Point Positioning with a Galileo Five-Frequency Ionosphere-Reduced Combination

“…Usually in the ionosphere-free (IF) combination case, in order to eliminate the influence of ionosphere, the combined ambiguity lose its integer nature; in contrast, the IR combination based on integer coefficients can still guarantee the integer solvability of ambiguity under the condition of almost eliminating ionosphere, which provides new opportunities for GNSS data processing. Both Feng [22] and Guo et al [23] systematically studied the selection criteria of IR combinations. Based on BDS-3 multi-frequency signals, Li et al [24,25] studied an IR combination suitable for medium-/long-baseline RTK ambiguity resolution, whose ambiguity maintains integer characteristics, and the noise level is comparable to the NL combination.…”