## 3 Results and Discussion
### Success Probability and Decimal Test for Cycle Slip Repair  
To test the cycle slip repair method, two GPS satellites are added with one and 1.5 cycle slips at every epoch. From Table 1, ILAM can detect two cycle slips, if the number of satellites is greater than seven. Because the cycle slips should be detected before analyzing the cycle slip repair method, the condition of seven satellites with GPS is ignored in the analysis. After detecting the cycle slips with the ILAM method, the float solutions of the cycle slips can be calculated by Equation (21). The float solutions of the same two cycle slips with GPS +BDS and GPS are shown in Figures 9 and 10. As shown in Figures 9 and 10, the float solutions are scattered around the true value, and the mean value of the float solutions is very close to the true value, which means that the TDCP model eliminates most of the common mode errors. The Root Mean Square (RMS) values with GPS + BDS are smaller than those with GPS. The mean RMSs are 0.018 and 0.024 cycles respectively with GPS + BDS and GPS. The result demonstrates that the accuracy of float solutions with GPS + BDS improves by 25% compared with GPS. The next step is to obtain the integer cycle slips. Obviously, the one cycle slip is an integer cycle slip and should be repaired, and the 1.5 cycle slips is a non-integer outlier. The validation of the success probability and decimal test are utilized for rounding to obtain the integer cycle slips. According to Equations (22) and (23), the success probability and decimal test threshold value with GPS + BDS and GPS can be acquired at every epoch and shown in Figures 11 and 12. As shown in Figures 11 and 12, the success probabilities of one cycle slip and 1.5 cycle slips are greater than 99.99%, which shows that the precision of float solutions can pass the success probability test but the non-integer outliers are not excluded by success probability. However, the decimals of the float solutions of 1.5 cycles are greater than the threshold value at all of the epochs, and the decimals of float solutions of one cycle are less than the threshold value at almost all of the epochs. The results of the GPS + BDS are similar to those of the GPS. Therefore, the decimal test can detect the non-integer outlier. Figure 11: Validation by success probability and decimal test for GPS + BDS: 1.5 Cycles (a); 1 Cycle (b). Figure 12: Validation by success probability and decimal test for GPS: 1.5 Cycles (a); 1 Cycle (b). If the decimal of a non-integer outlier is greater than the threshold value, the decimal test can detect it successfully. In order to evaluate the validity of the decimal test, we make the value of 1.5 cycle slips gradually reduced from 1.5 to 1, with a decrease of 0.05 each time, and detect it at every epoch. We define the proportion of successful epochs represents the detection rate of outlier detection. Then, we can calculate the detection rates under different decimals shown in Figure 13.  
As shown in Figure 13, when the decimal is 0 which means the cycle slip is an integer, the detection rates are both 0 with GPS + BDS and GPS, and the decimal test do not exclude the integer cycle slips incorrectly. When the decimal is greater than 0 and less than 0.3, the detection rates with GPS + BDS are far greater than that with GPS, and reach to 100% when the decimal is greater than 0.2. We can define the decimal when the detection rate is greater than 99.9% for the first time as the maximum decimal bias in repair. Compared to GPS, the maximum decimal bias reduces by 1/3, from 0.3 to 0.2 with GPS + BDS. Therefore, the decimal test with GPS + BDS is more effective than it is with GPS. The relative positions between consecutive epochs are important for velocity estimation, which is one of the major TDCP applications. It is sensitive to cycle slips and gross errors. Cycle slip detection and repair can improve the accuracy of relative positions.