# High-Accuracy Clock Offsets Estimation Strategy of BDS-3 Using Using Multi-Source Observations
## 2 Materials and Methods
### The New Strategy
#### 2.2.1 Estimation of Inter-Satellite Clock Offsets  
As shown in Figure 5a, BDS-3 follows the TDMA structure. One satellite connects with different satellites at different intervals to perform dual, one-way ranging measurements. In one connectivity cycle whose duration is 60 s, multiple Ka-band two-way inter-satellite clock offsets between one specific satellite and those on other satellites can be obtained by processing measurements using Equation (1). In fact, BDS-3 onboard atomic clocks perform high stability in the short timescale. The above redundant Ka-band two-way inter-satellite clock offsets can be further reduced to the same time (see Figure 5b) [29]. After that, the  
Figure 4: The procedure of the BDS-3 new clock offsets estimation strategy.  
redundant Ka-band inter-satellite two-way clock can be used to estimate the clock offsets with higher accuracy (see Figure 5c) [19].  
After selecting a satellite as a reference satellite, \"z.\" In the process of indirect adjustments, the observations are redundant Ka-band two-way inter-satellite clock offsets, and the parameters to be estimated are the Ka-band two-way inter-satellite clock offsets with respect to the reference satellite. The derivatives of observations of parameters to be estimated are given in Equation (2).  
\[\left\{\begin{array}{l}\frac{\partial\mathcal{C}lk^{Ka}AB}{\partial\overset{ \rightarrow}{x}}=[1,-1,0\ldots 0]\\ \ldots\\ \frac{\partial\mathcal{C}lk^{Ka}A_{\underline{k}}}{\partial\overset{ \rightarrow}{x}}=[1,0,0\ldots 0]\\ \ldots\\ \frac{\partial\mathcal{C}lk^{Ka}B}{\partial\overset{\rightarrow}{x}}=[0,-1,0 \ldots 0]\\ \ldots\\ \end{array}\right. \tag{2}\]  
where, \(\overset{\rightarrow}{x}\) are the clock offsets of all satellites with respect to a reference satellite, \(Clk^{Ka}{}_{AB}\) are the Ka-band two-way inter-satellite clock offsets between the \(A\) and \(B\). The results from the trial suggest that if the Ka-band two-way inter-satellite clock offsets within 1 min of the whole constellation are used to reduce, there will be more than 300 two-way links, and the reduced errors will be less than 7 mm. The comparison of the clock offsets before and after indirect adjustment is shown in Figure 6. The dark blue curves present the clock offsets after indirect adjustments, and the azury blue present the original Ka-band two-way inter-satellite clock offsets. Both of them deduct the same trends. From Figure 6. the advantages of the further indirect adjustments processing are that it not only can estimate inter-satellite clock offsets when they are not linked. Moreover, compared with Ka-band two-way inter-satellite clock offsets, the clock offsets, by further estimated, have higher accuracy and less noise.  
Figure 5: The procedure of the inter-satellite clock offset estimation. Wherein, t0, t1, t2 are different epoch within 60 s. Ka two-way is the Ka-band two-way clock offsets. Despite their epoch are different (as (**a**)), they can be reduced to the same epoch by prior message (as (**b**)) for further adjustment processing in the new strategy (as (**c**)).