## 3 DDPR-Dependent Model Based on Optimal Satellite Subset
In Scheme 3, the influences of both the integrated C/N0 and satellite elevation were taken into account. To scale both datasets to the same range and remove the impact of dimensions, the corresponding integrated C/N0 and satellite elevation were initially normalized. Subsequently, a comprehensive quality assessment indicator CQ, which combines the C/N0 and satellite elevation, was formulated, with its expression being  
\[\text{CQ}=a\times\widehat{c/N0}+b\times\widehat{ele} \tag{9}\]  
where \(\widehat{c/N0}\) and \(\widehat{ele}\) represent the normalized integrated C/N0 and satellite elevation, respectively, while a and b are the corresponding weight coefficients. Given that the integrated C/N0 is more reflective of the quality of satellite observations than the satellite elevation, this paper set the values of a and b to 0.6 and 0.4, respectively. In each satellite selection scheme, we will select the N satellites with the optimal quality evaluation indicators to participate in the positioning calculation. The value of N can be determined based on the surrounding environment of the smartphone and the total number of satellites that can be observed. For example, in an open environment, the value of N can usually be set between 12 and 15, while in an obstructed environment, it can be set between 8 and 10. For comparing the performance of the three mentioned schemes, observational data were statically collected for nearly 3 h using Huawei Mate40 and Huawei P40 smartphones in a relatively open environment, using a geodetic receiver as the reference station. Utilizing the gathered experimental data, single-frequency pseudorange relative positioning was conducted using the three aforementioned schemes, and the positioning results for each epoch were compared with the reference values. Figure 4 illustrates the distributions of positioning errors on the horizontal plane for each scheme. As can be seen, for the Huawei  
Figure 3: Mean values of absolute DDPRs changing with (**a**) elevation angles and (**b**) the integrated C/N0 for a smartphone. Mate40 smartphone, when using Scheme 1 for positioning, the distribution of error points is more concentrated in the east (E) direction than in the north (N) direction. Overall, the horizontal positioning errors are relatively centralized, predominantly falling within the range of [\(-\)5 m, 5 m]. In comparison to Scheme 1, Scheme 2 exhibits a noticeably larger range of positioning errors in both the E and N directions, with the distribution primarily concentrated within the [\(-\)10 m, 10 m] range. The interval of concentrated point error distribution in Scheme 3 falls between that of Schemes 1 and 2. Table 1 presents the root mean square (RMS) of the positioning results for each scheme. The RMS is mainly used to measure the deviation between a set of observed values and the true values, and it can well reflect the overall accuracy of measurements. From Table 1, it is evident that Scheme 1 exhibits the smallest RMS values in both the N and E directions across the three schemes, with Scheme 3 coming next. On the whole, in the horizontal direction, the positioning RMS values for the three schemes are 3.55 m, 6.11 m, and 4.96 m, respectively. This demonstrates that the satellite selection scheme based on the integrated C/N0 is capable of selecting a relatively optimal subset of satellites, resulting in better positioning outcomes. Conversely, the satellite selection scheme based on elevation yields the lowest positioning accuracy. Satellite selection Scheme 3 based on the C/N0 and satellite elevation, which takes into account both factors, has a positioning effect that lies between the two. Figure 4: Distributions of positioning errors on the horizontal plane for each satellite selection scheme. (**a**) Huawei Mate 40 smartphone. (**b**) Huawei P40 smartphone. For the Huawei P40 smartphone, Scheme 1 also demonstrates a better concentration of position errors for each epoch compared to the other two schemes.