## 2 STAR Algorithms to Convert Carrier Phase to Excess Phase and Bending Angle Profiles
### COSMIC-2 Excess Phase Calculation
#### 2.4.2 COSMIC-2 Receiver and GNSS Transmitter Clock Bias Removal  
We used a coarse resolution of the receiver clock bias obtained from the COSMIC-2 POD solution to correct the carrier phase time tag. The POD receiver clock bias solution determines the true time approximately (with errors of equipment group delay) at antenna position for geometric range calculation and the interpolation for various analyses. During the excess phase calculation, the receiver clock bias from the POD solution is used to compute the coordinate time on orbit at which the receiver carrier phase was measured. Then, all the positioning, velocity, and carrier phase error removal will be based on this time. Given that the LEO satellite's orbital movements are 7 km per second, the inaccurate time representation of 0.1 ms can yield the antenna position error of 70 cm (not POD error), much larger than the orbital requirement of 10 cm. This position error can be projected onto the time, phase, and Doppler frequency domain. The accuracy of the bending angle derived from the excess phase is greatly affected by how the clock bias is effectively removed [25; 28; 43]. However, the receiver clock bias removal from the carrier phase needs to be re-determined in the excess phase calculation using single differencing for more accurate results (compared with the direct interpolation from the POD clock solution) given the significant clock bias of COSMIC-2. The single difference clock bias solution was derived using the paired observation from POD antennas' reference link and different satellites' occultation observations. The difference between the clock bias correction using the single differencing and Bernese LEO clock solution is \(10^{-6}\) s level over the occultation periods (about 2-3 min). COSMIC-1 has a high rate of reference link observations with 50 Hz on POD antennas. These observations from a GNSS satellite, usually in antenna zenith direction, can form an ionospheric free, distance free phase combined with the occultation phase measurement from another satellite, generally in a limb direction, to be used for the receiver clock removal for single differencing. Since there are no high-rate observations from COSMIC-2 POD antennas, one alternative is to use the low-rate RINEX observations with a sampling rate of 1 Hz. The linear combination of carrier phase L1 and L2 can be used as:  
\[L_{L^{\prime}G^{\prime}}^{3}\ =\ L_{L^{\prime}G^{\prime}}^{1}+\frac{f_{2}^{2}}{f_{1}^ {2}-f_{2}^{2}}\Big{(}L_{L^{\prime}G^{\prime}}^{1}-L_{L^{\prime}G^{\prime}}^{2} \Big{)} \tag{2}\]  
where \(L_{L^{\prime}G^{\prime}}^{2}\) is the L2 signal from GNSS G' received by receiver \(L^{\prime}\),\(f1\) and \(f2\) are frequencies of the GNSS signal L1 and L2. This ionosphere-free L3 must also be geometric-free, where the direct distance between the POD antenna and the reference GNSS has been removed. Note that the frequency of each GLONASS constellation may be different. Some RINEX carrier phases display a jump every other epoch with less than around \(\frac{1}{4}\) cycle. A low pass filter and reconnection of the residual phase from a polynomial fitting is used to remove the oscillation from the constructed L3, which otherwise will be introduced into the excess Doppler as noise. The GPS time tag of the receiver observation (after correction) is also compared with that from UCAR. Generally, the difference is in the order of \(10^{-7}\) s. This reflects the difference in receiver clock time calculation from UCAR. The magnitude of time tag difference on this level will not affect the later conversion from excess phase to bending angle since the clock bias is effectively removed using the single differencing method during the excess phase calculation. The zero-differencing can also be used effectively to remove the clock bias if the clock bias can be accurately evaluated. All the time errors and corrections in the equation must be accounted for. COSMIC-2 satellites were designed with very stable atomic clocks.