# Joint Timekeeping of Navigation Satellite Constellation with Inter-Satellite Links
## 4 Performance Evaluation of ISLT
### Clock Noise Identification  
MPOD clock products were collected by the International GNSS Service (IGS) Multi-GNSS Experiment (MGEX) analysis centers (ACs) from 1 March to 31 March 2019--31 days in total for satellite clock identification.  
The short-term prediction precision of the three-state Kalman filter relied on the modeling of WFM, RWFM and RRFM and the estimation of the corresponding diffusion coefficients based on the clock products. The clock deviation outliers were always submerged because of the large scale of clock deviations. Clock deviations were therefore transformed to frequency deviations through first-order difference for gross error detection:  
\[y_{k}=\frac{x_{k+1}-x_{k}}{\tau_{0}},\quad k=1,\ldots,N-1 \tag{47}\]  
where \(N\) is the amount of clock deviation data and \(\tau_{0}\) is the sample interval of 30 s. The median method (MAD) is used to detect gross errors, and when the frequency deviation:  
\[\left|y_{k}\right|>m+p\cdot\mathrm{MAD}, \tag{48}\]  
\(y_{k}\) is treated as a gross error and eliminated from the original data. \(m=\mathrm{Median}(y_{k})\), \(\mathrm{MAD}=\mathrm{Median}\Big{(}\big{|}y_{k}-m\big{|}/0.6745\Big{)}\). Median is the operation of estimating the median. \(p\) was taken as 5 in this research. After removing the gross errors, the missing data were assigned by interpolation.  
The MPOD clock products have absorbed the orbit mismodeling errors because of the correlation between the satellite orbits and clocks. In order to reliably identify the background noises of BeiDou spaceborne clocks, the periodic signal at 1, 2, 3, and 4 cycle(s) per revolution (cpr) is removed from clock deviations [16]. According to the detrended clock deviations, Allan deviations (ADEVs) were computed and are shown in Figure 2. The three noise components are related to Allan variances of the clocks by:  
\[\sigma_{y}^{2}(\tau)=\frac{\sigma_{1}^{2}}{\tau}+\frac{\sigma_{2}^{2}\tau}{3} +\frac{\sigma_{3}^{2}\tau^{3}}{20} \tag{49}\]  
which means that the Allan deviation is a function of the averaging time that is formulated as \(\sigma_{y}(\tau)=A\tau^{\mu}\) or the logarithm form \(\mathrm{lg}\sigma_{y}(\tau)=\mathrm{lg}A+\mu\mathrm{lg}\tau\), \(\mu\) is the curve slope of \(\mathrm{lg}\sigma_{y}(\tau)\)-\(\mathrm{lg}\tau\), equaling \(-0.5\), \(0.5\) and \(1.5\) for WFM, RWFM and RRFM, respectively, and the constant term \(\mathrm{lg}A\) relates to the  
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Serial Number** & **Satellite PRN** & **Nominal Clock** \\ \hline
1 & C19 & Rb \\
2 & C20 & Rb \\
3 & C21 & Rb \\
4 & C22 & Rb \\
5 & C26 & PHM \\
6 & C27 & PHM \\
7 & C29 & PHM \\
8 & C30 & PHM \\
9 & C37 & Rb \\ \hline \hline \end{tabular}
\end{table}
Table 1: Atomic clocks of nine BeiDou-3 satellites.  
diffusion coefficients. If the curve \(\text{lg}\sigma_{y}(\tau)\)-\(\text{lg}\tau\) is fit piecewise with a linear model, the clock noises are identified as in Table 2.  
In general, the onboard Rb clocks are mainly affected by white frequency noises and random walk noises, while the PHMs are only influenced by white frequency noises. Since no random run frequency noise was observed for all the nine clocks, with an averaging time of less than 100,000 s, the corresponding diffusion coefficients were taken as zeros and the same to the RWFM diffusion coefficients of PHMs.