# Comparing GRACE-FO KBR and LRI Ranging Data with Focus on Carrier Frequency Variations
## 3 LRI Ranging Phase
### Formula 2: Integral of Differentiated Phase  
An alternative way to compute the half round-trip range from the round-trip ranging phase \(\varphi_{\text{TWR}}\) was suggested in [45], Equation (60)  
\[\rho_{\text{TWR,2}}\approx c_{0}\int_{0}^{t}\frac{\mathrm{d}\varphi_{\text{TWR} }(t^{\prime})/\mathrm{d}t^{\prime}}{2\
u_{\text{R}}(t^{\prime})}\;\mathrm{d}t^ {\prime} \tag{24}\]  
although it is easy to see that the equation is not exact, since  
\[\frac{\mathrm{d}\varphi_{\text{TWR}}}{\mathrm{d}t} =\frac{\mathrm{d}}{\mathrm{d}t}\int_{t-\Delta t}^{t}\
u_{\text{R} }(t^{\prime})\;\mathrm{d}t^{\prime}=\
u_{\text{R}}(t)-\
u_{\text{R}}(t-\Delta t )\cdot(1-\mathrm{d}\Delta t/\mathrm{d}t) \tag{25}\] \[=\
u_{\text{R}}(t-\Delta t)\cdot\mathrm{d}\Delta t/\mathrm{d}t+ \
u_{\text{R}}(t)-\
u_{\text{R}}(t-\Delta t)\] (26) \[\approx\
u_{\text{R}}(t)\cdot\mathrm{d}\Delta t/\mathrm{d}t+(1- \mathrm{d}\Delta t/\mathrm{d}t)\cdot\
u_{\text{R}}(t)\cdot\Delta t \tag{27}\]  
The first term in the line (26) contains the ranging information that is multiplied with the optical frequency at the emission event and not at the reception event as the divisor in Equation (24) suggests. Moreover, the second and third terms indicate needed correction terms that are missing in Equation (24). Thus, the exact formula to convert the round-trip phase \(\varphi_{\text{TWR}}\) to a half round-trip range reads  
\[\rho_{\text{TWR,2 exact}}(t) :=\frac{c_{0}}{2}\int_{0}^{t}\frac{\mathrm{d}\varphi_{\text{TWR} }(t^{\prime})/\mathrm{d}t^{\prime}}{\
u_{\text{R}}(t^{\prime}-\Delta t_{\text{ RTR}}(t^{\prime}))}-\left(\frac{\
u_{\text{R}}(t^{\prime})}{\
u_{\text{R}}(t^{ \prime}-\Delta t_{\text{RTR}}(t^{\prime}))}-1\right)\;\mathrm{d}t^{\prime} \tag{28}\] \[=\frac{c_{0}\Delta t_{\text{RTR}}(t)}{2}-\frac{c_{0}\Delta t_{ \text{RTR}}(0)}{2}\;. \tag{29}\]  
The term  
\[\frac{c_{0}}{2}\int_{0}^{t}\frac{\
u_{\text{R}}(t^{\prime})}{\
u _{\text{R}}(t^{\prime}-\Delta t_{\text{RTR}}(t^{\prime}))}-1\,\mathrm{d}t^{\prime}\] \[\approx\frac{c_{0}}{2}\int_{0}^{t}\frac{\
u_{\text{R}}(t^{ \prime})}{\
u_{\text{R}}(t^{\prime})}\cdot\Delta t_{\text{RTR}}(t^{\prime}) \,\mathrm{d}t^{\prime}\approx\frac{c_{0}}{2}\langle\Delta t_{\text{RTR}} \rangle\cdot\frac{\
u_{\text{R}}(t)-\
u_{\text{R}}(0)}{\
u_{\text{R}}(0)} \tag{30}\]  
approximately describes the range correction due to laser phase changes on the reference satellite while the laser light propagates forth and back between satellites with average propagation time \(\langle\Delta t_{\text{RTR}}\rangle\). The last approximation of this expression resembles the correction term shown in Equation (23).  
A handy approximation of the exact form in Equation (28) with all quantities being evaluated at the same time is given as  
\[\rho_{\text{TWR,2 approx}}=c_{0}\int_{0}^{t}\frac{\mathrm{d}\varphi_{\text{ TWR}}(t^{\prime})/\mathrm{d}t^{\prime}}{2\
u_{\text{R}}(t^{\prime})}-\left(1- \frac{\mathrm{d}\Delta t_{\text{RTR}}^{\text{GPS}}}{\mathrm{d}t}\right)\cdot \frac{\dot{\
u}_{\text{R}}(t^{\prime})}{2\
u_{\text{R}}(t^{\prime})}\Delta t_{ \text{RTR}}^{\text{GPS}}(t^{\prime})\;\mathrm{d}t^{\prime}, \tag{31}\]  
which is based on the approximation shown in Equation (27). The light propagation time in the correction term is marked with a superscript GPS to indicate that this quantity can be obtained from the orbit product, as in Equation (22).