## 2 Methodology
### Cycle Slip Detection with ILAM  
There are many methods for eliminating or reducing the impact of outliers. In these methods, robust estimation methods adjust the stochastic model to reduce the impact of outliers on the results; other methods are based on statistical principles [25], and have been extended to multiple outliers [26] and applied [27; 28]. However, these methods are based on residuals of adjustment, while the residuals of observations with outliers may be not larger [29]. To detect multiple outliers is still an open problem [26]. The local analysis method (LAM) is proposed to address the shortcomings of the existing outliers detecting methods based on residuals, and it does not depend on residuals of adjustment. The core idea is that an observed quantity can be expressed as combinations of other observed quantities, and the observed quantity and the combinations can be compared to detect outliers [29]. This method  
Figure 1: Statistics of time-differenced inter-system bias (TDISB) for five sites on DOY 353 in 2015. Equation (11) reveals that the true value of \(w\) is small. The true error of \(w\) is computed by Equation (17). According to the law of error propagation, the error of mean squares \(\sigma_{w}\) of \(w\) can be calculated based on the precision of the TDCP observations. In the experiment, we assume the precision of raw TDCP is 5 mm including the errors relative to multipath, receiver noise, and the residuals of the partially corrected error sources. This value is related with receivers and troposphere and ionosphere environment. Additionally, the elevation-dependent measurement weighting method is considered. When \(|w|\leq 2\sigma_{w}\) (assuming random errors are normally distributed), the combination of these four satellites can be treated as a repeat observation of satellite \(i\). Except for satellite \(i\), any combination of four satellites can constitute a combination of satellite \(i\). Any satellite has the same number of combinations as an observed quantity. If the numbers of combinations and repeat observations of one observed quantity are k and m, respectively, the repetitive rate \(P_{A}\) is defined as:  
\[P_{A}=\frac{m}{k} \tag{18}\]  
Theoretically, there is no repeat observation in the combinations if cycle slips exist in the observed quantity of one satellite. The number of repeat observations in the combinations is determined by the number of satellites and cycle slips if there is no cycle slip in the observed quantity of one satellite. Therefore, we can get the theoretical value of the repetitive rate by:  
\[P_{T}= \begin{cases}\frac{C_{n-t-1}^{4}}{C_{n-1}^{4}}&(\text{there is no cycle slip})\\ 0&(\text{there is cycle slip})\end{cases}  \tag{19}\]  
where \(n\) and \(t\) are the number of satellites and cycle slips, respectively. We can determine the number of cycle slips based on the repetition rates of the satellites that have no cycle slips and detect the satellites that have cycle slips based on the repetitive rates. Because of the influence of random factors, \(P_{A}\) should be close to \(P_{T}\), but not equal. \(P_{T}\) increases as the number of satellites increases and decreases as the number of cycle slips increases. Figure 2 shows the \(P_{T}\) changes under incremental numbers of satellites and cycle slips. The detection process can be divided into two steps. The first step is to search the number of cycle slips \(t\), and the second step is to determine which satellites have cycle slips. Each satellite will be calculated in all combinations to obtain the repetitive rate. The repetitive rates of all of the satellites will be sorted by descending order. Hypothetically, the numbers of satellites and cycle slips are \(n\) and \(t\), respectively, and the number of valid satellites is \(n-t\). Because the repetition rates of valid satellites would certainly be larger than those of abnormal satellites, which are close to zero, the larger repetition  
Figure 2: Repetitive Rate under Different Numbers of Satellites and Cycle Slips.