# Comparing GRACE-FO KBR and LRI Ranging Data with Focus on Carrier Frequency Variations
## 3 LRI Ranging Phase
### Model Errors: Scale Factor and Light Propagation Time
When using both ways at the same time, and comparing the range with model errors (\(\rho_{\text{...}}^{\text{mod}}\)) to the original formula, we obtain after expansion to leading order and neglecting terms that contain \(\Delta t_{\text{RTR}}^{\text{GPS,e}}\) in Equation (38):  
\[\rho_{\text{TWR,1 corr}}^{\text{mod}}-\rho_{\text{TWR,1 corr}}\] \[\approx+\frac{c_{0}}{2}\delta_{\text{SCF}}(t)\cdot\Delta t_{ \text{RTR}}(t)-\frac{c_{0}}{2}\epsilon_{\text{SCF}}(t)\cdot\left(\Delta t_{ \text{RTR}}(t)+\Delta t_{\text{RTR}}(0)\cdot\left(\frac{\
u_{\text{R}}(0)}{ \
u_{\text{R}}(t)}-1\right)\right)\] \[\qquad+\frac{c_{0}}{2}\cdot\Delta t_{\text{RTR}}^{\text{GPS,e}}( 0)\cdot\left(\frac{\
u_{\text{R}}(0)}{\
u_{\text{R}}(t)}-1\right)+\frac{c_{0} }{2}\cdot\epsilon_{\text{SCF}}(0)\cdot\Delta t_{\text{RTR}}(0)\cdot\frac{\
u_ {\text{R}}(0)}{\
u_{\text{R}}(t)}+\text{const.} \tag{36}\] \[\approx+\frac{c_{0}}{2}\delta_{\text{SCF}}(t)\cdot\Delta t_{ \text{RTR}}(t)-\frac{c_{0}}{2}\epsilon_{\text{SCF}}(t)\cdot\Delta t_{\text{ RTR}}(t)+\text{const.}\] (37) \[\rho_{\text{TWR,22 approx}}^{\text{mod}}-\rho_{\text{TWR,2 approx}}\] \[\Delta t^{\mathrm{GPS}}_{\mathrm{RTR}}(t)\longrightarrow\Delta t ^{\mathrm{GPS}}_{\mathrm{RTR}}(t)+\Delta t^{\mathrm{GPS,e}}_{\mathrm{RTR}}(t) \tag{33}\]0 (38) \[\Delta t^{\mathrm{GPS}}_{\mathrm{RTR}}(t)\longrightarrow\Delta t ^{\mathrm{GPS}}_{\mathrm{RTR}}(t)+\Delta t^{\mathrm{GPS,e}}_{\mathrm{RTR}}(t) \tag{33}\]1  
where the scale factor (SCF) error was introduced in two versions as  
\[\Delta t^{\mathrm{GPS}}_{\mathrm{RTR}}(t)\longrightarrow\Delta t ^{\mathrm{GPS}}_{\mathrm{RTR}}(t)+\Delta t^{\mathrm{GPS,e}}_{\mathrm{RTR}}(t) \tag{33}\]2  
Both formulas (\(\rho_{\text{TWR,1 corr}}\) and \(\rho_{\text{TWR,2 approx}}\)) and both ways to apply the error yield the same result in the leading terms: the scale factor error (\(\epsilon_{\text{SCF}}\) and \(\tilde{\epsilon}_{\text{SCF}}\)) arising from the limited knowledge of the frequency \(\
u_{\text{R}}\) produces ranging errors proportional to the satellite separation \(c_{0}\Delta t_{\text{RTR}}(t)/2\approx L(t)\). This is the expected result and the usual coupling of frequency noise into the range measurement [18] (Equation (11)) at least to leading order. The exact error coupling can become rather complicated. In order to assess the magnitude of scale factor errors, we once more assume an oscillatory and a drifting behavior as in the previous subsection. For the drift, we reuse the value of \(\
u_{d}/\
u_{0}=3.6\cdot 10^{-15}\,1\,\text{/sec}\) (cf. Table 1), which results in a scale factor change \(\Delta\epsilon_{\text{SCF}}=3.1\cdot 10^{-10}\) per day. Such a scale factor error is present if the cavity resonance frequency in the LRI drifts by 87 kHz per day without knowledge about the drift, which we regard to be potentially possible. The resulting ranging error per day is then  
\[\Delta t^{\mathrm{GPS}}_{\mathrm{RTR}}(t)\longrightarrow\Delta t ^{\mathrm{GPS}}_{\mathrm{RTR}}(t)+\Delta t^{\mathrm{GPS,e}}_{\mathrm{RTR}}(t) \tag{33}\]3  
A realistic error for the oscillatory behavior arises when the proper time induced modulation is neglected, as currently in the regular LRI data processing. This means the sinusoidal modulation with peak amplitude \(\
u_{\text{R}}(t)/\langle\
u_{\text{R}}\rangle\approx 4\cdot 10^{-12}\) at \(1/\text{rev}\) frequency is neglected, which implies an oscillating scale factor error \(\epsilon_{\text{SCF}}\approx 4\cdot 10^{-12}\) at the orbital frequency. The resulting ranging error is  
\[\Delta t^{\mathrm{GPS}}_{\mathrm{RTR}}(t)\longrightarrow\Delta t ^{\mathrm{GPS}}_{\mathrm{RTR}}(t)+\Delta t^{\mathrm{GPS,e}}_{\mathrm{RTR}}(t) \tag{33}\]4  
The calculated changes of \(68\,\upmu\text{m}\) per day and \(0.9\,\upmu\text{m}_{\text{peak}}\) at \(1/\text{rev}\) frequency are actually apparent from the red traces in Figure 2 for the error from Equation (17). This equation exhibits a time-dependent frequency \(\
u_{\text{R}}(t)\) only in the divisor, which has very little effect on the range, as will be shown later.