# Comparing GRACE-FO KBR and LRI Ranging Data with Focus on Carrier Frequency Variations
## 3 LRI Ranging Phase  
For the LRI, the ranging phase \(\varphi_{\mathrm{TWR}}\) is referred to as a two-way ranging (TWR) quantity in the GRACE-FO project and it is determined from the phase difference between transponder \(\varphi_{\mathrm{T}}\) and reference \(\varphi_{\mathrm{R}}\) satellite [17]. The phase measurements on both satellites are performed with phase meters. The transponder satellite uses a phase-locked loop to lock its laser to the incoming light using a high gain and high bandwidth control loop [19]. This means the transponder phase is a rather trivial measurement given by a phase ramp with constant slope ([21], Equation (2.207))  
\[\varphi_{\mathrm{T}}=10\,\mathrm{MHz}\cdot\tau_{\mathrm{T}}^{\mathrm{USO}}, \quad[\varphi_{\mathrm{T}}]=\mathrm{cycles}. \tag{8}\]  
On the reference satellite, the laser is stabilized using an optical cavity and the phase meter output \(\varphi_{\mathrm{R}}\) contains both the ranging information and the transponder phase ramp. Since the same phase ramp is present on both satellites, it cancels out to a large extent in the differential measurement \(\varphi_{\mathrm{TWR}}\). A small residual of the phase ramp might be present due to uncertainties in the propagation time \(\Delta t_{\mathrm{TR}}\) and in the timing products (CLK1B), which are covered by the timetag error, as discussed in Section 6.4. Hence, one can write the TWR phase as [45]  
\[\varphi_{\mathrm{TWR}}=\varphi_{\mathrm{T}}(t-\Delta t_{\mathrm{TR}})- \varphi_{\mathrm{R}}(t)=\Phi_{\mathrm{R}}(t)-\Phi_{\mathrm{R}}(t-\Delta t_{ \mathrm{RTR}})+q_{\mathrm{R}}, \tag{9}\]  
where \(\Phi_{\mathrm{R}}\) is the optical phase of the reference laser in units of cycles and \(\Delta t_{\mathrm{RTR}}\) is the round-trip light propagation time from reference to transponder and back to the reference satellite. The round-trip propagation time \(\Delta t_{\mathrm{RTR}}\) contains the ranging information, while the propagation time from transponder to reference satellite \(\Delta t_{\text{TR}}\) is less critical, since it is just used in the time argument of the smooth transponder phase in order to remove the phase ramp from \(\varphi_{\text{TWR}}\). \(\Delta t_{\text{TR}}\) can be derived from GPS orbit data and is needed to account for the propagation of the phase value from transponder to reference. The constant \(q_{\text{R}}\) arises from the inability of interferometers to measure the absolute phase or distance. Using the definition of the optical phase from Equation (7), one obtains  
\[\varphi_{\text{TWR}} =\int_{\tau-\Delta t_{\text{TRTR}}}^{\tau}v_{\text{R}}(\tau^{ \prime})\;\mathrm{d}\tau^{\prime}+q_{\text{R}} \tag{10}\] \[=v_{\text{R}}\cdot[\tau-\tau+\Delta t_{\text{TRTR}}(\tau)+\tau_{ \text{FV}}(\tau)-\tau_{\text{FV}}(\tau-\Delta t_{\text{TRTR}})]+q_{\text{R}}\] (11) \[\approx v_{\text{R}}\cdot\Delta t_{\text{TRTR}}+v_{\text{R}}\cdot \frac{\mathrm{d}\tau_{\text{FV}}}{\mathrm{d}\tau}\cdot\Delta t_{\text{TRTR}}+ q_{\text{R}}\] (12) \[=v_{\text{R}}(\tau)\cdot\Delta t_{\text{TRTR}}(\tau)+q_{\text{R}} =v_{\text{R}}(t)\cdot\Delta t_{\text{TRTR}}(t)+q_{\text{R}}, \tag{13}\]  
where the last line shows the phase observable in terms of proper or local quantities on the satellite (\(v(\tau)\cdot\Delta t_{\text{TRTR}}(\tau)\)) and in terms of GCRS quantities (\(v(t)\cdot\Delta t_{\text{RTR}}(t)\)). We use \(\varphi\) to indicate phase quantities that have a slope in the radio-frequency domain (kHz or MHz), i.e., which are tracked and recorded on GRACE-FO, while the phase quantities \(\Phi\) have phase slopes in the THz or GHz domain, which cannot be directly recorded. The LRI level1a (LRI1A) data of GRACE-FO contain the raw phases of reference \(\varphi_{\text{R}}\) and transponder \(\varphi_{\text{T}}\) satellite, while the LRI level1b (LRI1B) product contains the raw biased range \(\rho_{\text{TWR}}\) and the light-time correction (LTC) \(\rho_{\text{LTC}}\).