## 2 STAR Algorithms to Convert Carrier Phase to Excess Phase and Bending Angle Profiles
### COSMIC-2 Excess Phase Calculation
#### 2.4.3 Geometric Range Corrections  
Precise position estimates of the LEO and GNSS satellites are essential for accurate excess phase retrieval. The GNSS orbits were resampled using Bernese GNSS software from 15 min to 30 s through refitting its orbital dynamic parameters from CODE orbit solutions. These 30 s datasets are then used to calculate high-resolution GNSS satellite mass center position through polynomial interpolation (11 th order) at the GNSS signal transmitting time. The clock bias results are simply polynomial interpolation of the CODE 30 s datasets.  
The POD and clock bias for LEO and GNSS satellites are stored in the IGS Extended Standard Product 3 Orbit format (SP3-C). The receiver clock bias is used to convert the observation time on the local receiver time to the coordinate time. The corrected coordinate time is then used to calculate (interpolate) the LEO orbit in the ECI coordinate system.  
The antenna offset and the phase center variation has been converted from instrument coordinate to ECI using COSMIC-2 attitude files provided by UCAR. They were added to the mass center position (also converted to ECI) to retrieve the position and velocity. UCAR provides the COSMIC-2 attitude quaternions in a True of Epoch (TOE) ECI; we tested different ECI schemes. The difference between the TOE and other ECI for the antenna offset (usually less than a millimeter) can be ignored, given the antenna length of less than 1 m. For the GNSS satellite, the antenna parameters are obtained from the GNSS satellite parameter files (satellite.I14, PCV_COD.I14), which are updated online [44].  
To calculate the direct link between the LEO satellite (RO antenna) and the GNSS satellite (transmitter antenna), the time delay \(\tau\) of the ray path from the transmitter to the receiver in a vacuum needs to be calculated. The first guess is the interpolated GNSS antenna position at time t. A Newton's relaxation method is applied to iterate the propagation time and the GNSS antenna position (with both updated in each iteration), given the initial positions of GNSS satellites at the observation time on the LEO receiver until the time delay is converged and the time difference between two iterate steps is less than ten ns. A preliminary evaluation of the excess phase (on the order of 6 km near-surface, corresponding to a 20 us time delay) has been used to count the time delay caused by the excess path. Given the converged time delay \(\tau\) and the GNSS antenna position (and velocity), the direct link distance \(\|\overset{\rightarrow}{x}^{R}(t)-\overset{\rightarrow}{x}^{T}(t-\tau)\|\) can be calculated at time t. The range correction term due to the relativistic effects \(\delta D(\overset{\rightarrow}{x}^{R}(t),\overset{\rightarrow}{x}^{-T}(t-\tau))\) ([25,42]) has also been considered, and the general magnitude is on the order of 0.2 m, which is nearly constant and has negligible effects on the excess phase.  
The validation is carried out as the direct radial distance from the earth center or between the transmitter and receiver phase center is independent of the coordinate system. The direct antenna radial distance (from the earth center) is compared with that from UCAR, and the difference is generally within 20 cm, and the time variation can be neglected. Note that a direct comparison between antenna positions can yield a difference in the order of meters in ECI horizontal direction due to the different coordinate systems used. Sometimes, this can be more than meters due to POD differences, which have been discussed later in the discussion section.