# High-Accuracy Clock Offsets Estimation Strategy of BDS-3 Using Using Multi-Source Observations
## 2 Materials and Methods
### The New Strategy
#### 2.2.2 Estimation of Clock Offsets with Respect to BDT  
The new strategy can estimate high-accuracy inter-satellite clock offsets by least square estimation processing for redundant Ka-band two-way inter-satellite clock offsets. However, the reference of satellite clock offsets parameter is BDT. The inter-satellite two-way estimated clock offsets need to be further traced to BDT. Despite no good ideas to directly solve the problems shown in Figure 1, we find that these problems can be effectively circumvented with some changes in the estimated clock offsets strategy. Firstly, the BDS-3 GEO is always over China, and MCS's antennas can track BDS-3 GEO anytime. BDS-3 can estimate GEOs clock offsets without passing through Ka-band two-way inter-satellite clock offsets and C-band two-way inter-station clock offsets. Then, the velocity of GEO related to antenna ground is slow, and the altitude angle and azimuth angle of tracked antenna nearly remain unchanged. Lastly, GEOs need to broadcast the messages of other services, including Satellite Based Augmentation System Service (SBAS) [30,31] and Precise Point Position Service in B2b signal (PPP-B2b) [32]. The above services require MCS to use antennas with a larger diameter and greater signal transmission power to track GEOs full time, which will help suppress the multipath effect in two-way observation. Therefore, in the satellite-ground clock offsets estimation, only the L-band two-way clock offsets of the large-diameter antennas following GEOs are considered the satellite-ground clock offsets estimation. Similar to the inter-satellite clock offsets estimation, in order to filter the noise and errors, the GEO's clock offsets estimation also adopts the least square estimation method. However, in the estimation, the zero value of different L-band antennas should be calibration and aligned. Set a GEO satellite as a reference \"Q\" the parameter to be estimated includes Q's clock offsets with respect to BDT, and others are the differences of zero value of L-band larger-diameter antennas. After that, the observation equations in the least square estimation are given as Equation (3). \[\left\{ \begin{array}{l}\rho_{Q}(t)=clk_{Q}{}^{L}(t)=clk_{Q}(t)\\ \rho_{A}(t)=clk_{A}{}^{L}(t)-clk_{AQ}{}^{Two-way~{}estimated}(t)=clk_{Q}(t)+ Bias_{A}{}^{L}\\ \rho_{B}(t)=clk_{B}{}^{L}(t)-clk_{BQ}{}^{Two-way~{}estimated}(t)=clk_{Q}(t)+ Bias_{B}{}^{L}\end{array} \right. \tag{3}\]  
Figure 6: The improvements after using redundant Ka-band inter-satellite clock offsets to estimate Ka-band inter-satellite clock offsets (since 19 May 2022). where the derivatives of observation of the parameters to be estimated are given as Equation (4). \[\left\{ \begin{array}{l}\frac{\partial{{}_{Q}}}{\partial{{}_{\overrightarrow{x}} }}=[(0,0)_{{}_{\overrightarrow{x}}_{0}}(1)_{{}_{\overrightarrow{x}}_{1}}]\\ \frac{\partial{{}_{PA}}}{\partial{{}_{\overrightarrow{x}}}}=[(1,0)_{{}_{ \overrightarrow{x}}_{0}}(1)_{{}_{\overrightarrow{x}}_{1}}]\\ \frac{\partial{{}_{BB}}}{\partial{{}_{\overrightarrow{x}}}}=[(0,1)_{{}_{ \overrightarrow{x}}_{0}}(1)_{{}_{\overrightarrow{x}}_{1}}]\end{array} \right. \tag{4}\]  
wherein, \({clk_{A}}^{L}(t)\), \({clk_{B}}^{L}(t)\), \({clk_{Q}}^{L}(t)\) are the L-band two-way clock offsets of three GEO satellites. \({clk}^{\rm Two-way\ {estimated}}\) are the Ka-band two-way inter-satellite estimated clock offsets. are the parameters to be estimated, which include the global parameter \(\overrightarrow{x}_{0}\) and local parameter \(\overrightarrow{x}_{1}\). Global parameters are the zero value differences, including the \({Bias_{A}}^{L}\) and \({Bias_{B}}^{L}\). Local parameters include the clock offsets of the reference satellite at different epochs. In the least square estimation processing, the global and local parameters can be effectively separated by operating the normal equation to eliminate parameters. Thus, the update of the antenna's bias and the estimation of higher-accuracy clock offsets of BDS-3 GEOs can be both real-time realized.