## 2 Methods
### BDS Satellite Clock Offsets  
The precise satellite clocks products are generated by the ionospheric-free (IF) observation model, the precise clock offsets will absorb the satellite IF uncalibrated code delays (UCDs) [20; 21]. It is noteworthy that the BDS broadcast clock offsets are determined by Two-way Satellite Time Frequency Transfer (TWSTFT) [22; 23], and the equipment group delay of BDS B3I (1268.520 MHz) is treated as the reference equipment group delay. Whether BDS-2 or BDS-3, the equipment group delay of B3I will be absorbed by the satellite broadcast clock offsets. The broadcast clock offsets and precise clock offsets can be expressed as [22; 24]:  
\[\left\{ \begin{array}{l}\delta\mathrm{I}_{\mathrm{Broadcast}}^{\mathrm{S}} =\delta\mathrm{I}^{\mathrm{S}}+\mathrm{B}_{\mathrm{B3I}}+\mathrm{d}\mathrm{D }_{\mathrm{Broadcast}}\\ \delta\mathrm{I}_{\mathrm{Precise}}^{\mathrm{S}}=\delta\mathrm{I}^{\mathrm{S}} +\mathrm{IF}_{\mathrm{UCDs}}+\mathrm{d}\mathrm{D}_{\mathrm{Precise}}\end{array} \right. \tag{1}\]  
where \(\delta\mathrm{I}_{\mathrm{Broadcast}}^{\mathrm{S}}\) and \(\delta\mathrm{I}_{\mathrm{Precise}}^{\mathrm{S}}\) represent broadcast and precise clock offsets calculated by the recommendation model of BDS ICD and International GNSS Service (IGS) product descriptions [17; 18; 19; 25]; \(\mathrm{B}_{\mathrm{B3I}}\) is the equipment group delay of B3I. \(\mathrm{IF}_{\mathrm{UCDs}}\) is the satellite IF UCDs. \(\mathrm{d}\mathrm{D}_{\mathrm{Broadcast}}\) and \(\mathrm{d}\mathrm{D}_{\mathrm{Precise}}\) denote the receiver-independent bias introduced through broadcast and precise satellite clock estimation. As mentioned above, the BDS original broadcast clock offsets are determined by TWSTFT, not IF observation model [22; 23]. Therefore, \(\mathrm{d}\mathrm{D}_{\mathrm{Broadcast}}\) will absorb the datum of TGD when TGD correction is performed to convert the original clock offsets to the clock offsets with a certain equipment group delay. Previous studies have shown that the TGD of BDS-2 and BDS-3 are also calculated by two information processing centers [12] and the inconsistency of the TGD datum for BDS-2 and BDS-3 resulted from different solution conditions will cause the inconsistency of the datum of satellite clock offsets between BDS-2 and BDS-3. After TGD correction, the broadcast clock offsets absorbed the equipment group delay of B3I can be restored to the clock offsets absorbed the equipment group delay of IF combination between B1I (1561.098 MHz) and B3I (1268.520 MHz):  
\[\delta\mathrm{I}_{\mathrm{IF},\ \mathrm{Broadcast}}^{\mathrm{S}}=\delta \mathrm{I}^{\mathrm{S}}+\mathrm{IF}_{\mathrm{B}_{\mathrm{P}}}+\mathrm{d} \mathrm{D}_{\mathrm{Broadcast}}=\delta\mathrm{I}_{\mathrm{Broadcast}}^{ \mathrm{S}}+\frac{\delta\mathrm{I}_{\mathrm{B1I}}^{2}}{\delta\mathrm{I}_{ \mathrm{B1I}}^{2}-\delta\mathrm{I}_{\mathrm{B3I}}^{2}}\cdot\mathrm{TGD}_{1} \tag{2}\]where \(\delta\mathrm{I}_{\mathrm{IF},\ \mathrm{Broadcast}}^{\mathrm{S}}\) is the broadcast clock offsets absorbing the IF equipment group delay \(\mathrm{IF}_{\mathrm{B}_{\mathrm{P}}}\); \(f_{\mathrm{B1I}}\) and \(f_{\mathrm{B3I}}\) are the frequencies on the B1I and B3I signals, respectively; \(\mathrm{TGD}_{1}\) is the differential equipment group delay between the B1I and B3I. In order to get the accuracy of broadcast clock offsets, we make a difference between broadcast clock offsets and precise clock offsets, which can be expressed as:  
\[\Delta\mathrm{t}=\delta\mathrm{I}_{\mathrm{IF},\ \mathrm{Broadcast}}^{ \mathrm{S}}-\delta\mathrm{T}_{\mathrm{Precise}}^{\mathrm{S}}=\left(\left( \delta\mathrm{t}^{\mathrm{S}}+\mathrm{IF}_{\mathrm{B}_{\mathrm{P}}}\right)- \left(\delta\mathrm{T}^{\mathrm{S}}+\mathrm{IF}_{\mathrm{UCDs}}\right)\right)+ \mathrm{dD}_{\mathrm{Broadcast}}-\mathrm{dD}_{\mathrm{Precise}} \tag{3}\]  
where \(\Delta\mathrm{t}\) is the BMP clock value. As shown in Equation (3), the datum of the satellite clock offsets cannot be eliminated by the first-order difference, so we need the double difference to eliminate the datum.