## 4 Results and Discussion
### Equation Validation  
Figure 2 depicts the linear correlation between the phase residuals on two frequencies for GPS, GLONASS, and BDS at station SCPW on DOY 219. As can be seen, the residuals manifest strong negative correlation. All the correlation coefficients are -1.0, and the slopes of lines are -1.647, -1.652, and -1.673 for GPS, GLONASS, and BDS IGSO and MEO satellites, respectively, which are very close to the corresponding negative squares of ratios of the two frequencies (-1.648, -1.653, -1.672). Nonetheless, for BDS GEO satellites, the correlation coefficient is only about -0.4, and the slope of -0.458 shows a pronounced discrepancy with respect to the theoretical value of -1.672. This indicates that the phase residuals of the BDS GEO satellites on two frequencies have weak correlation and could not be properly described by Equation (12). The cause for this phenomenon is the pseudo-range bias, which degrades the precision of ionospheric parameters, consequently contaminating the residuals.  
Figure 1: Distribution of the high-rate GNSS stations around the epicenter of the 2017 Mw 6.5 Jiuzhaigou earthquake event. The red star is the epicenter location.  
To investigate the pseudo-range weighting effects on carrier-phase residuals, the a priori precision of pseudo-range was set lower by a factor of two and three to 1.8 and 2.7 m, respectively. The results are shown in Figure 3. A higher a priori precision results in a higher observation weight. It is clear that the correlations rapidly increase from -0.4 to -0.9 with the decrease of pseudo-range weight. As expected, the corresponding slopes of the regression line become closer and closer to the square of the ratio of frequencies B1 and B2. Since the pseudo-range primarily provides the initial value for the least-squares estimator, to avoid contaminating the phase residuals, it is advisable to lower the weight of the pseudo-ranges. In this paper, the a priori precision of the BDS GEO pseudo-range is set to 2.7 m.  
Figure 3: Correlation between the residuals on frequencies B1 and B2 with different a priori precision of BDS GEO pseudo-range. The different color dots represent the residuals for different satellites. The regression lines, represented by the blue dashed lines, and the correlation coefficients (R) are also shown.  
Figure 2: Correlation between the phase residuals from GPS (L1/L2), GLONASS (R1/R2), and BDS (B1/B2). The different color dots represent the residuals for different satellites. The regression lines, represented by the blue dashed lines, and the correlation coefficients (R) are also shown.  
To validate the correctness of Equation (12), the residuals on frequency two are recovered from frequency one, and then differenced with the observed residuals. The percentage error is based on the equation of \((1.0-\text{a2/a1})\times 100\) %, where a1 and a2 are, respectively, the observed and projected residuals. As shown in Figure 4, the mean percentage errors for GPS, GLONASS, and BDS IGSO/MEO are all under 0.3 %, whereas for GEO they increase by about 6%-10 % to 2.5-4 mm. Although the recovery for GEO residuals performs not as well as that of other satellites, it is still acceptable, which gives a powerful proof of the correctness of Equation (12).