# Comparing GRACE-FO KBR and LRI Ranging Data with Focus on Carrier Frequency Variations
## 6 Instrument Noises and Corrections
### Carrier Frequency Variations
In order to assess the ranging error, we use the USO model introduced in Equation (59) and shown as a dashed magenta trace in the right plot of Figure 1, and then compute the contribution as  
\[\mathrm{PSD}[\delta\rho_{\rm FV,raw,KBR}]=\frac{\mathrm{PSD}[\delta t_{ \rm USO,mod}](f)\cdot(2\pi f)^{2}}{2}\cdot L^{2}+\mathrm{PSD}\!\left[\frac{ \mathrm{d}(\tau_{\rm A}+\tau_{\rm B})}{2\mathrm{d}t}\right]\cdot L^{2}, \tag{72}\]  
which is depicted as a dashed blue line in the right plot of Figure 4. The first term on the right side is again the product of fractional frequency variations and inter-satellite separation \(L\). The factor of 2 in the denominator (cf. [10], B-7) arises from the fact that two independent USOs are used in the KBR and we assume that both USOs exhibit the same amount of uncorrelated variations. The second summand accounts for the effect of the proper time. These frequency variations \(\mathrm{ASD}[\delta\rho_{\rm FV,raw,KBR}]\) can be corrected by using time-dependent carrier frequencies as defined in Table 1, which are related to the CLK1B \(\epsilon_{\rm time}\) by Equation (2), i.e.,  
\[\
u^{\rm K}_{\rm A/B}(t) =\hat{f}_{\rm A/B,USO}\cdot 5076\cdot\left(1-\frac{\mathrm{d} \epsilon_{\rm time,A/B}}{\mathrm{d}t}\right) \tag{73}\] \[\
u^{\rm Ka}_{\rm A/B}(t) =\hat{f}_{\rm A/B,USO}\cdot 6768\cdot\left(1-\frac{\mathrm{d} \epsilon_{\rm time,A/B}}{\mathrm{d}t}\right) \tag{74}\]  
The time-dependent carrier frequencies should be low-pass-filtered in order to remove high-frequency fluctuations above \(3\,\mathrm{mHz}\), which limit the measurement precision of \(\epsilon_{\rm time}\) and do not stem from real variations in the carrier frequency (cf. right plot in Figure 1). The corrected KBR range should ideally be computed in a modified level1A to level1B processing scheme using Equation (54), which assumes that the re-scaling from phase to range is performed with time-dependent carrier frequencies and also takes the coupling of the phase-bias (\(\Delta f_{\rm FV,DOWR}\)) given in Equation (52) into account. However, we point out that a direct correction of the KBR1 Bv04 range data  
\[\text{ASD}[\delta\rho_{\text{FV,corr,LRI}}](f)=\frac{\text{ASD}[\delta\
u_{ \text{cav}}](f)}{\langle\
u_{\text{R}}(t)\rangle}\cdot L(t)\quad\text{with} \quad\text{ASD}[\delta\
u_{\text{cav}}](f)=\frac{10^{-15}}{\sqrt{f}}\cdot \langle\
u_{\text{R}}(t)\rangle, \tag{69}\]0  
might be performed as well, as the magnitude of \(c_{0}\Delta t_{\rm FV,DOWR}\) has a much larger effect than dividing the phase by a time-dependent frequency compared to division by a daily mean frequency. This is illustrated in Figure 7, which also confirms that the difference between both correction approaches, i.e., the effect of dividing the phase by a time-dependent frequency (black trace), is well below the KBR noise level (\(\approx\)\(0.6\,\mathrm{\mu m}/\sqrt{\mathrm{Hz}}\)). In order to derive the residual effect of USO frequency fluctuations in the KBR ranging data, i.e., after applying the correction using the modified level1a to level1b conversion, we use the model for the precision of CLK1B \(\epsilon_{\text{time}}\) data derived in Section 6.4 as ASD\([\delta t_{e}]\). The solid blue trace shows the residual KBR frequency variations in the right plot of Figure 4 based on  
\[\text{ASD}[\delta\rho_{\text{FV,corr,LRI}}](f)=\frac{\text{ASD}[\delta\
u_{ \text{cav}}](f)}{\langle\
u_{\text{R}}(t)\rangle}\cdot L(t)\quad\text{with} \quad\text{ASD}[\delta\
u_{\text{cav}}](f)=\frac{10^{-15}}{\sqrt{f}}\cdot \langle\
u_{\text{R}}(t)\rangle, \tag{69}\]1  
We can conclude that correcting for KBR frequency variations might slightly reduce the KBR-LRI residuals at frequencies between 0.1 mHz and 1 mHz (dashed blue vs. solid blue trace). However, the change is expected to be rather small as the timetag error (solid black trace) becomes limiting.