## 2 Method
### The Clock Model  
Generally, the clock offset parameter is estimated using white noise. In this contribution, we estimate the clock offset using the between-epoch constraint model (random-walk model). If the receiver clock offset exists in the IGS clock file, a prior variance is calculated from the known clock offset. If not, the receiver clock offset is calculated using the PPP method and a prior variance is computed according to the Allan variance. Considering the white frequency modulation (WFM) noise \(q_{1}\), the random walk frequency modulation (RWFM) noise \(q_{2}\), and the random run frequency modulation (RRFM) noise \(q_{3}\), we obtain the following equation [23]:  
\[\delta^{2}_{Allan}(\tau)=q_{1}\tau^{-1}+\frac{q_{2}\tau}{3}+\frac{1}{20}q_{3} \tau^{3} \tag{5}\]  
where \(\delta^{2}_{Allan}(\tau)\) is the Allan variance at a time interval of \(\tau\); \(q_{1}\), \(q_{2}\) and \(q_{3}\) are estimated first by the least square method; and a prior noise variance \(Q_{w}\) is then determined [24]. In the RT-PPP static mode, the value of the coordinate is known. Additionally, the tropospheric delay can be constrained by a prior value because of its small short-term change. Therefore, the data interruption does not induce any influence on these two parameters. However, as we mentioned previously, the clock offset parameter will re-converge because of the ambiguity in initialization when the data is interrupted. Hence, based on the between-epoch constraint model, an adaptive prediction clock model is applied to mitigate the re-convergence error of the clock offset when experiencing a data discontinuity. A standard model for a clock offset is described as follows:  
\[d\overline{t}_{r}=a_{0}+a_{1}(t_{i}-t_{0})+\frac{a_{2}}{2}(t_{i}-t_{0})^{2}+ \sum_{l=0}^{k}A_{l}\sin(\omega t_{i}+\varphi_{l}) \tag{6}\]  
where \(a_{0}\), \(a_{1}\) and \(a_{2}\) are the coefficients of the low-degree polynomial; \(t_{i}\) is the time at epoch \(i\); \(A_{l}\) denotes the coefficients of the periodic functions; \(\omega\) is the frequency of the periodic items; and \(\varphi_{l}\) is the phase deviation of the periodic term. The first-order, second-order polynomial model is employed in this contribution, and \(a_{0}\), \(a_{1}\) and \(a_{2}\) are estimated as constant parameters using the least square method. Note that the weight of the clock is determined by the corresponding variance. Furthermore, an adaptive prediction model is introduced for predicting the clock offset to ensure the continuity of the clock offset time series. 'Adaptive' signifies that the selection of the polynomial model is dynamic, depending on the fitting residuals. A detailed diagram of the adaptive prediction clock model is shown in Figure 1. The frequency spectrum of the clock offset is analysed using a fast Fourier transform (FFT). The process of the RT-PPP time and frequency transfer using the clock model is shown in Figure 2. It mainly includes two parts, as follows:  
1. We obtain the observations from the receiver (connected to their time and frequency generator, a 1 PPS signal and a 5/10 MHz frequency signal), broadcast ephemeris and state space representation (SSR) corrections from IGS/ACs [25; 26; 27]. 2. If the data is not interrupted, it will go directly to RT-PPP processing to calculate the difference between the local time UTC (i) and the reference time (ref). Then, the clock offset, and the corresponding variance are stored. We use a sliding window with a window length of 240 data points to store the clock offset data. However, when data interruption occurs, we use the stored clock offset data to calculate the coefficient of the adaptive prediction model, and then predict the clock offset of the current epoch, with the current clock offset as a prior value, while setting the corresponding variance. Importantly, an a priori clock offset of 20 consecutive epochs is set to the predicted clock offset starting from the data, and the products are re-received to help ambiguity parameters to recover quickly and accurately.