# Comparing GRACE-FO KBR and LRI Ranging Data with Focus on Carrier Frequency Variations
## 3 LRI Ranging Phase
### Formula 1: Phase-Frequency Ratio  
The direct naive approach using Equation (17) with a time-dependent frequency and approximation in Equation (13) yields  
\[\rho_{\text{TWR,1}}(t):=c_{0}\cdot\frac{\varphi_{\text{TWR}}(t)}{2\
u_{\text{R }}(t)}\approx c_{0}\frac{\Delta t_{\text{RTR}}(t)}{2}+c_{0}\frac{q_{\text{R}}} {2\
u_{\text{R}}(t)}. \tag{19}\]  
The divisor is the apparent frequency in the GCRS \(\
u(t)\), since \(\
u(\tau)\) would result in a contracted range \(c_{0}\Delta t_{\text{RTR}}\). The naive formula \(\rho_{\text{TWR,1}}\) yields the correct result of Equation (18), if \(q_{\text{R}}=0\), i.e., if the phase measurement \(\varphi_{\text{RTR}}\) is an absolute measurement without integer ambiguity due to indistinguishable cycles. In actual interferometry, the phase time-series \(\varphi_{\text{RTR}}(t)\) exhibits an unknown arbitrary bias and it is debiased in order to avoid large numerical values, loss of numerical precision due to floating-point arithmetic or overflow in finite-sized registers of a computer. We assume without loss of generality that phase time-series start at 0 cycles at an initial epoch, i.e., \(\varphi_{\text{RTR}}(t=0)=0\) in a stretch of data that is being converted from phase to range. This implies that  
\[q_{\text{R}} =-\
u_{\text{R}}(t=0)\cdot\Delta t_{\text{RTR}}(t=0) \tag{20}\] \[\rho_{\text{TWR,1}}(t) \approx c_{0}\frac{\Delta t_{\text{RTR}}(t)}{2}-c_{0}\Delta t_{ \text{RTR}}(0)\cdot\frac{\
u_{\text{R}}(t=0)}{2\
u_{\text{R}}(t)} \tag{21}\]  
where the first summand on the right-hand side in line (21) is the desired ranging signal and the second term describes laser frequency variations that couple via the baseline length \(c_{0}\Delta t_{\text{RTR}}(0)/2\approx 220\,\text{km}\). Since Equation (21) shows the deviation of \(\rho_{\text{TWR,1}}\) from the correct result, we can directly define an improved formula as  
\[\rho_{\text{TWR,1 corr}}(t) :=c_{0}\cdot\frac{\varphi_{\text{TWR}}(t)}{2\
u_{\text{R}}(t)}+c_ {0}\Delta t_{\text{RTR}}^{\text{GPS}}(0)\cdot\left(\frac{\
u_{\text{R}}(0)}{ 2\
u_{\text{R}}(t)}-\frac{1}{2}\right), \tag{22}\] \[\approx c_{0}\cdot\frac{\varphi_{\text{TWR}}(t)}{2\
u_{\text{R}}(t)}- \frac{c_{0}\Delta t_{\text{RTR}}^{\text{GPS}}(0)}{2}\cdot\left(\frac{\
u_{ \text{R}}(t)-\
u_{\text{R}}(0)}{\
u_{\text{R}}(0)}\right), \tag{23}\]  
where the second correction term on the right-hand side in line (22) is usually much smaller than the first ranging term, as the expression in the bracket is close to zero due to the introduced constant \(-1/2\). The round-trip propagation time at the initial epoch received the superscript GPS (\(c_{0}\Delta t_{\text{RTR}}^{\text{GPS}}(0)\)) in order to indicate that the value can be derived from GPS orbit data and algorithms to compute the light propagation time [45]. The millimeter to centimeter precision of GPS is sufficient to derive the absolute distance \(c_{0}\Delta t_{\text{RTR}}^{\text{GPS}}(0)\approx 440\,\text{km}\) without spoiling the high precision from LRI in \(\rho_{\text{TWR,1 corr}}\), since the correction term is small due to \(\
u_{\text{R}}(0)/\
u_{\text{R}}(t)\approx 1\). The approximation in line (23) was based on \(1/(1+x)\approx 1-x\), with \(x=\
u_{\text{R}}(t)/\
u_{\text{R}}(0)-1\), and it illustrates more clearly that the correction term is the product of inter-satellite distance and fractional frequency deviations.