## 3 Results and Discussion
### ILAM for Detecting Cycle Slips  
The detection results of ILAM depend on the number of satellites and cycle slips. To facilitate a statistical analysis, we merged the periods with the same number of satellites together for analysis. Figure 4 shows the number of satellites in this period. There are 7-10 GPS satellites and 16-21 GPS + BDS satellites at the epochs.  
The epochs with 21 satellites were chosen to test ILAM. Two satellites are selected to add random cycle slips at every epoch. First, each satellite will be selected to obtain the repetitive rate. Figure 5 shows the repetitive rates of every satellite.  
The results shown in Figure 5 demonstrate that the repetitive rates of valid satellites are near the theoretical value, while the repetitive rates of abnormal satellites are close to zero. We then use Equation (20) to calculate the standard deviation by different numbers of cycle slips. Figure 6 shows changes in the standard deviation by different assumptions for the numbers of cycle slips at one epoch. When the number of cycle slips is two, which is the correct value, the standard deviation reaches its minimum value. We can determine the number of cycle slips by the standard deviation method. The final step is determining the corresponding satellites with lower repetition rates.  
To ensure the correct selection of the satellites with cycle slips, the maximum repetitive rate of abnormal satellites should be smaller than the minimum repetitive rate of valid satellites. The repetition rates of the valid satellites are much larger than those of the anomalous satellites, as shown in Figure 5, and cycle slips can be detected successfully when there are 21 satellites and two cycle slips. Therefore, two conditions must be met to detect cycle slips successfully with ILAM: one is that determining the correct number of cycle slips; the other one is, that the minimum repetitive rate of satellites without cycle slips is greater than the maximum repetitive rate of satellites with cycle slips.  
Figure 4: Satellite numbers for GPS and GPS + BDS.  
Figure 5: Repetitive rates of 21 satellites with two cycle slips (blue dots for valid satellites, red dots for satellites with cycle slips, and black line for theoretical value).  
According to Equation (19), the repetition rates of valid satellites will decrease when there are more cycle slips, which may lead to a failure to detect cycle slips. The detectable cycle slips number is an effective indicator to test detection capacity. By putting two conditions as criteria, we can judge if cycle slips can be detected successfully in one epoch. The proportion of successful epochs represents the success rates of cycle slip detection. The success rates under the conditions of different numbers of satellites are calculated by adding different numbers of cycle slips to the GPS + BDS, as shown in Figure 7.  
To compare the conditions of fewer satellites, the success rates with GPS are calculated by the same methods, as shown in Figure 8.  
Figure 8: Success rates of detecting cycle slips by different numbers of satellites and cycle slips with GPS.  
Figure 6: Standard deviations with different numbers of cycle slips at one epoch.  
Figure 7: Success rates of detecting cycle slips by different numbers of satellites and cycle slips with GPS + BDS.  
If 99% is set as the threshold value of the success rate to determine whether the cycle slips can be reliably detected, the detectable cycle slips numbers for different numbers of satellites are estimated from Figures 7 and 8, and these results are listed in Table 1. Obviously, with GPS + BDS, there are more satellites and the detectable cycle slips number is more than that with GPS. The mean numbers satellites are 18 and eight, respectively with GPS + BDS and GPS. Compared to GPS, the mean detectable cycle slips number increases by four-fold, from 2 to 10 with GPS + BDS. Therefore, ILAM method can take advantage of the multi-constellation.