## 4 Positioning Performance of GPS/QZSS and BDS/QZSS in the Asia-Pacific Region
### ADOP, Success Rate and Baseline Accuracy  
The concept of ambiguity dilution of precision (ADOP) was first proposed by [PERSON] in 1997 to measure ambiguities by computing the geometric mean of the ambiguity conditional  
Figure 3: Availability and PDOP trends with increases in the mask angle. Figure 2: Availability and position dilution of precision (PDOP) of the single BDS/GPS and integrated BDS/QZSS or integrated GPS/QZSS with different mask angles for the HKLT station from day of year (DOY) 54 to 61.  
standard deviations [25]. ADOP was introduced based on the determinant of the ambiguity variance-covariance matrix:  
\[ADOP=\sqrt{|Q_{a}|^{\frac{1}{a}}} \tag{12}\]  
where \(Q_{a}\) is the variance-covariance matrix of the float ambiguities, n is the dimension of the matrix, and \(|\cdot|\) denotes the determinant of the matrix-. The success rate is the index that indicates whether integer ambiguity resolution can be expected to be successful. Based on the ambiguity estimator theory, the success rate of ambiguity resolution is determined by computing the numerical integration of the probability density function of float ambiguities in the 'pull-in region', which is very rigorous in theory [26]. However, it is difficult to describe the 'pull-in region' so the integration process is very complex and the success rate cannot be calculated directly. We can only approximately approach the theoretical success rate through the calculation of the upper and lower success rate bound [27]. The bootstrapped success rate (Bstr) is a lower bound of the integer least-squares success rate and is defined based on the variance-covariance matrix of the decorrelated float ambiguities [28]. \[Bstr=\prod_{i=1}^{n}\left[2\Phi(\frac{1}{2\sigma_{z_{i|}}})-1\right] \tag{13}\]  
where \(\sigma_{z_{i|}}\) is the conditional standard deviations of the estimated ambiguities. We simulated the ADOP and Bstr of a 10 km baseline at HKLT station in Hong Kong as the base station for a week. Figure 4 shows the mean ADOP and Bstr of different systems with different mask angles which are counted in Table 3. The single GPS ADOP cannot be computed above a 20\({}^{\circ}\) mask angle, and GPS/QZSS ADOP cannot be computed above a 30\({}^{\circ}\) mask angle because the number of visible satellites is not sufficient. We can clearly draw the following conclusions: (1) The single GPS baseline resolution is limited by the number of satellites, especially when the mask angle is above 20\({}^{\circ}\), and the combination of GPS/QZSS will improve the limitation; (2) GPS/QZSS achieves a great decrease in ADOP and a great increase in Bstr compared to the single GPS; (3) The ADOP and Bstr of BDS/QZSS are both better than that of GPS and GPS/QZSS; however, compared with the single BDS, the improvement of BDS/QZSS is relatively slight. In addition, it can be easily found that the Bstr of a 0\({}^{\circ}\) mask angle is slightly lower than that of 5\({}^{\circ}\). The signal from a low-elevation satellite is easily affected by shelter, such as buildings, surface fog, and haze. The noise error of the satellite signal at 0\({}^{\circ}\) is relatively larger than that at 5\({}^{\circ}\), which will mitigate the corresponding bootstrapped success rate. Figure 4: Ambiguity dilution of precision (ADOP) and bootstrapped success rate (Bstr) of different systems with different mask angles. In addition, there are some differences between the bootstrapped success rate and the actual success rate. In some cases, despite the very high bootstrapped success rate of the ambiguity estimation, the actual ambiguity can also be wrong. The main reason may be that ambiguity estimation depends on the precision of the float ambiguities; the estimation could be wrong due to the residual ionosphere, troposphere, or multipath errors in float ambiguities [29]. This indicates that the Bsr could become inaccurate when DD residuals become larger with the increase of the baseline length.