# Comparing GRACE-FO KBR and LRI Ranging Data with Focus on Carrier Frequency Variations
## 6 Instrument Noises and Corrections
### Timetag Errors
The noise level of DSAC is consistent with the GRACE-FO USO Allan deviations in Figure 6 at lowest integration times (i.e., highest frequencies), though it bears mentioning that the GRACE-FO traces correspond to a single day and exhibit some day-to-day variability (not shown). The noise models for the timetag error in Equations (59) and (60) can produce an error in the phase measurement of KBR and LRI. In order to assess its magnitude, we replace the argument of the phase measurement from correct time \(t\) with an erroneous time \(t+\delta t\):  
\[\varphi(t)\longrightarrow\varphi(t+\delta t)\approx\varphi(t)+\dot{\varphi} \cdot\delta t. \tag{61}\]  
The effect of the timetag error on the LRI ranging phase can be derived from Equation (9), whereas its effect on the LRI range can be determined from Equation (17) as  
\[\mathrm{ASD}[\delta\varphi_{\mathrm{TWR}}] =\mathrm{ASD}[\dot{\varphi}_{\mathrm{T}}\cdot\delta t_{\mathrm{T} }-\dot{\varphi}_{\mathrm{R}}\cdot\delta t_{\mathrm{R}}]\approx\phi\cdot\sqrt{2 }\cdot\mathrm{ASD}[\delta t]\approx 10\,\mathrm{MHz}\cdot\sqrt{2}\cdot\mathrm{ASD}[ \delta t] \tag{62}\] \[\mathrm{ASD}[\delta\varphi_{\mathrm{TWR}}] \approx\mathrm{ASD}[\varphi_{\mathrm{TWR}}\cdot\lambda_{\mathrm{ R}}/2]\approx 5\,\mathrm{m/s}\cdot\sqrt{2}\cdot\mathrm{ASD}[\delta t], \tag{63}\]  
using the fact that the LRI phases measured on transponder and reference satellites have a common and dominant slope of \(10\,\mathrm{MHz}\) (cf. Equation (8)), and we assumed that the timetag noise on both spacecraft is uncorrelated and has the same stochastic characteristics, such that the differential combination yields \(\sqrt{2}\delta t\) in the spectral domain. The same approach for the KBR at K- and Ka-band yields a timetag-induced noise of  
\[\mathrm{ASD}[\delta\varphi_{\mathrm{DDWR}}^{\mathrm{K/Ka}}] =\mathrm{ASD}[\dot{\varphi}_{\mathrm{B}}^{\mathrm{K/Ka}}\cdot \delta t_{\mathrm{B}}-\dot{\varphi}_{\mathrm{A}}^{\mathrm{K/Ka}}\cdot\delta t _{\mathrm{A}}] \tag{64}\] \[\approx\dot{\varphi}\cdot\sqrt{2}\cdot\mathrm{ASD}[\delta t] \approx\sqrt{2}\cdot\mathrm{ASD}[\delta t]\cdot \begin{cases}500\,\mathrm{kHz },&\text{for K-band}\\ 670\,\mathrm{kHz},&\text{for Ka-band}\end{cases} \] (65) \[\mathrm{ASD}[\delta\varphi_{\mathrm{DDWR}}^{\mathrm{K/Ka}}] \approx\mathrm{ASD}[\varphi_{\mathrm{DDWR}}^{\mathrm{K/Ka}}\cdot c _{0}/(\
u_{\mathrm{A}}^{\mathrm{K/Ka}}+\
u_{\mathrm{B}}^{\mathrm{K/Ka}})] \approx 3071\,\mathrm{m/s}\cdot\sqrt{2}\cdot\mathrm{ASD}[\delta t]. \tag{66}\]  
for the phase \(\varphi\) and range \(\rho\). Although the timetag error is different for the K- and Ka-phase due to the difference in beatnote frequencies (\(500\,\mathrm{kHz}\) vs. \(670\,\mathrm{kHz}\)), it is the same in terms of range. When forming the final ionosphere-free DOWR combination, one has to consider that the timetag error is correlated between the K- and Ka-band, such that both contributors add linearly as  
Figure 6: Allan deviations of the most of the traces shown in the right plot of Figure 1. The additional green trace shows the in-orbit stability of the trapped-ion atomic clock [42], which is dominated by GPS measurement noise for the shown values of \(\tau\). \[\text{ASD}[\delta\rho_{\text{DOWR}}]=a^{\text{K}}\cdot\text{ASD}[\delta\rho_{ \text{DOWR}}^{\text{K}}]+a^{\text{Ka}}\cdot\text{ASD}[\delta\rho_{\text{DOWR}}^{ \text{Ka}}]\approx 3071\,\text{m}/\text{s}\cdot\sqrt{2}\cdot\text{ASD}[ \delta t], \tag{67}\]  
with the coefficients \(a^{\text{K}/\text{Ka}}\) given in Table 1.  
Equation (63) and (67) depend on \(\delta t\) and have been used to produce the black timetag curves of LRI and KBR, respectively, in Figure 4. Traces labeled with _timetag raw_ are based on \(\delta t=\delta t_{\text{USO,mod}}\) given in Equation (59), while _timetag corrected_ use the \(\delta t=\delta t_{\text{c}}\) noise from Equation (60). As apparent from Figure 4, the timetag noise in KBR likely is a significant contributor to the increase of KBR-LRI residuals below \(10\,\text{mHz}\).