## 2 Improved Cycle-Slip Detection and Repair Algorithm
In Table 2, the average and STD of the fractional parts of the estimated \(L_{2}\) are less than 0.2 cycles, whereas some of the maximum values are beyond 0.5 cycles, especially for WHU1. Moreover, based on the improved algorithm, a few wrong cycle slips were found in this study. However, the properness is beyond 99% for all experiments, in which the incorrect estimation for iGMAS is slightly higher than for MGEX observations. It can be explained that the larger noise of iGMAS observations than for MGEX causes the inaccurate estimation of ionospheric delay from the analysis of Table 1.  
In Table 3, the properness of four days for different BeiDou satellites on LHA1 reached upwards of 100%. Meanwhile, the maximum values are smaller than for G01 in Table 2, which can ensure the properness of rounding the fractional parts. However, it should be noted that the results of C32 seem to be better than for BDS-2. According to the experiments, it proved that the improved algorithm  
In Figure 2, the fractional parts of the G01 (DOY 183) at three stations are drawn; it could be found that the fractional parts of iGMAS observations are convergent as the epoch increases. Since the improved algorithm is epoch-wise processing, the results are influenced by the noise of observations at the beginning. In addition, the MGEX observations are almost less than 0.3 cycles. Similarly, Figure 3 shows the results of three types BeiDou satellites and BDS-3 experimental satellites, which are based on LHA1 observations on DOY 183. Simultaneously, the corresponding fractional parts are less than 0.2 cycles. However, the result was slightly worse for MEO than for IGSO and GEO satellites and its trend could be caused by the BeiDou satellite-induced biases [31]. The accuracy of the improved algorithm proposed in this study can meet the requirements of orbit determination by testing iGMAS and MGEX observations (without cycle slips). The properness of rounding reached 100%, except for a few errors. Then, a group of cycle slips was inserted into the observations to test the reliability of the improved algorithm. Due to the fact that the Turboedit could not solve the special combinations on different frequencies, such as 1:1 or 9:7 for cycle slips on \(L_{1}\) and \(L_{2}\) of GPS, the experiment selected six types of combinations ((0,1), (1,0), (1,1), (9,7), (100,1), and (790,563)) to analyze the improved algorithm [32]. The observations from the last experiment were inserted cycle slips, and the repaired results are based on rounding the float estimates. Table 4 only summarizes the results of G01, C01, C06, C14, and C32 based on LHA1 on DOY 183. In Table 4, the cycle slips in GPS or BeiDou observations, correspondingly, were completely corrected based on the improved algorithm. However, the estimated cycle slips at epoch 2600 were (0.525,0.698), which is difficult to fix as the real values in the ambiguity resolution. In this study, from the results of all experiments, this phenomenon need not be taken into consideration as the very few numbers did not impact the corresponding conclusions. Moreover, to further analyze the improved algorithm, the results of Turboedit were also listed in Table 4. The experimental results show that the ability of the improved algorithm to detect and repair cycle slips outperforms the Turboedit approach, especially for repairing values. Furthermore, the differences between the estimated and the predicted ionospheric delays are also calculated to test the accuracy of the predicted polynomial model. Figure 4 displays the ionospheric delay residuals for G01, C06, C14, and C32 (LHA1) calculated from the same data as Figure 2. In Figure 4, the errors of the predicted ionospheric  
Figure 3: Time series of the fractional parts of B2 (C01, C06, C14) and B3 (C32) observations on DOY 183 based on LHA1 ((**a**) C01 (BDS_GEO); (**b**) C06 (BDS_IGSO); (**c**) C14 (BDS_MEO); and (**d**) C32 (BDS-3)). delays are below 0.02 m as time increases, which can meet the requirements of the cycle-slip detection and repair.