## 2 Methodology
### Joint Detection of Clock Jump and Cycle Slip for the Single-Frequency Data  
The clock steering is a routine operation to constrain the receiver clock bias since the large bias affects the signal tracking and orbit computation. During the clock steering event, the pseudorange and carrier phase measurements from all satellites are shifted by the same amount, which is equivalent to a receiver clock jump. It has no impact on computing orbit if the receiver clock is modeled as the white noise process. However, frequent clock steering events will be a challenge in cycle slip detection. The clock steering events can be reflected in the GPS and BDS receiver clock series, which are shown in Figure 3. To improve the readability of the figure, only five-minute receiver clock series are represented. It can be seen that about 90 clock steering events happen in five minutes. Consequently, a constant bias is introduced in the phase observation and will be incorrectly identified as one cycle slip if the detection algorithm does not consider the clock steering events.  
A high-order time difference (HTD) approach based on the carrier phase is capable of handling the high-dynamic single-frequency cycle slip detection, while it cannot cope with the receiver clock steering. Unfortunately, the Luojia-1A onboard receiver employs frequent clock steering for maintaining reasonable time synchronization. The clock steering means that the pseudorange and carrier phase measurements of all satellites are adjusted to control the receiver clock bias, which may affect cycle slip detection. Therefore, the receiver clock jump must be considered in a single-frequency cycle slip detection algorithm. To solve this problem, the optimized algorithm flowchart is shown in Figure 4.  
The real-time single-frequency receiver clock jump and cycle slip detection algorithm with only several adjacent data contains three steps: (1) the data preprocessing, (2) the cycle slip and clock jump detection, and (3) the clock jump estimation. In the data preprocessing, some abnormal observations are first screened by judging whether the carrier phase observation L1 or the pseudorange observation P1 is zero. The epoch-differenced L1-P1 (dLP) observable is used to detect outliers since it is insensitive to clock jumps if the single-frequency pseudorange and phase observations are consistent. The noise of the dLP observable is typically 3-5 m [43], and its threshold value (threLP) can be set as four times this noise. The second-order differenced carrier phase time series is generated in the second step, and the linear prediction model with the previous several epochs is used for the cycle slip and clock jump detection. The predicted residual with respect to the real second-order differenced phase (Lres) reflects the impact of the receiver-satellite kinematic and the remaining ionospheric delay, which is much smaller than the phase jump such as the clock cycle, outlier, or clock jump. Cycle slips or outliers will be identified if the predicted residual falls between the threshold values of cycle slip (threCS) and clock jump (threCI), which are set as 3.0 cycles and 10.0 cycles based on the noise of observable and the steering mechanism, respectively. Since the phase outlier causes two adjacent abnormal detection results and the cycle slip only influences the current result, they can be further discriminated. In the third step, the clock jump can be estimated robustly with the possible  
Figure 3: Frequent clock steering events captured in the GPS and BDS receiver clock series.  
clock jump (CJ) observables whose total number is defined as the ncJ. The cycle slips and outliers are also detected according to the residuals of CJ observables.