# Estimation of Inter-System Biases between BDS-3/GPS/Galileo and Its Application in RTK Positioning
## 3 Datasets and Experiments
### Performance of RTK Positioning
#### 3.4.2 LCM RTK vs. TCM RTK  
The ambiguity dilution of precision (ADOP) is an easy-to-compute scale diagnostic that can reflect the precision of ambiguity resolution, which is defined as [36]:  
\[ADOP=|Q_{\hat{a}}|^{\frac{1}{2\hat{a}}} \tag{10}\]  
where \(Q_{\hat{a}}\) is the variance-covariance matrix of ambiguity \(\hat{a}\) with a dimension of \(n\). The ADOP unit (cycles) is the same as that of ambiguity. A lower ADOP corresponds to a  
Figure 7: Horizontal scatters (**top**) and vertical time series (**bottom**) of the positioning errors of BDS-3 (**a**), GPS (**b**), Galileo (**c**), and triple-system (TS) (**d**) dual-frequency (DF) RTK of baseline GODE-USN7 on DOY 120, 2021 with a cutoff elevation angle of 10\({}^{\circ}\). On the right of the horizontal scatterplot and on the bottom of the vertical time series, the details of the fixed solutions are displayed with the 95% confidence interval. The float solutions, correct solutions, and incorrect solutions are indicated in grey, green, and red, respectively. higher probability of correctly fixed solutions. [PERSON] [37] described more characteristics of ADOP and provided one conclusion: an ADOP smaller than 0.12 cycles may have a success rate above 99.9%. In this part, the ADOP is used as one of the criteria used to evaluate the ambiguity resolution to assess the performance of TCM RTK with respect to LCM RTK. In Section 3.4.1, we analyzed the performance of SS and TS LCM RTK positioning with a cutoff elevation angle of 10\({}^{\circ}\). Considering the larger visible satellites (about 22.89, as shown in Figure 1) with an elevation angle over 10\({}^{\circ}\), in this section, we limit the cutoff elevation angle to 20 and 40\({}^{\circ}\) to compare the positioning performance of LCM and TCM with higher cutoff elevation angles. Figure 8 shows the positioning errors of TS LCM and TCM RTK of the baseline GODE-USN7 on DOY 120, 2021 with cutoff elevation angles of 20 and 40\({}^{\circ}\). Due to the relatively sufficient number of satellites with an elevation over 20\({}^{\circ}\), the LCM and TCM RTK can achieve an SR near 100%. The positioning accuracy of the float solution of TCM is higher than that of LCM, whereas the positioning accuracies of the fixed solutions remains similar. With the cutoff elevation angle raised to 40\({}^{\circ}\), not only the positioning accuracy, but the SRs of both LCM and TCM decrease obviously. Under such a high cutoff elevation angle, both positioning accuracy and SR in TCM significantly improve compared with LCM. When TS observations were applied, three pivot satellites are necessary in LCM, whereas only one pivot satellite is enough in TCM. Therefore, if only one satellite exists in a system, this satellite will be abandoned in LCM RTK. In TCM RTK, this single satellite can also work in positioning through the inter-system DD. This means TCM RTK helps with using more satellites and a significant improvement in positioning performance occurs when the satellites are insufficient. From the results of the experiments, with a cutoff elevation angle of 40\({}^{\circ}\), 7.7 and 13.4% improvements in the SR and accuracy of the float solution with TCM RTK were achieved, respectively, compared with LCM RTK. Figure 9 shows the ratio, ADOP, and number of visible satellites of LCM and TCM RTK with cutoff elevation angles of 20 and 40\({}^{\circ}\) for baseline GODE-UNS7. From the plots and legends, it can be found that the ratio of TCM is apparently higher than that of LCM. The percentage of the ratio over 3.0 remains at a high level, about 95%, even with a high cutoff elevation angle of 40\({}^{\circ}\), whereas that of LCM is lower than 82%. With the higher cutoff elevation angle, the value of the ratio increases more in TCM than in LCM (10.7% increase in ratio over 2.0, and 16.3% increase in ratio over 3.0). For the ADOP, the TCM RTK corresponds to a lower value of ADOP over the entire observation period, which means the TCM has a higher precision of ambiguity than LCM.