# Comparing GRACE-FO KBR and LRI Ranging Data with Focus on Carrier Frequency Variations
## 6 Instrument Noises and Corrections
### Timetag Errors  
The telemetry of KBR and LRI contains recorded instrument timetags, which are equally sampled at their respective nominal rate. Timetag errors arise when the timetags in the instrument timeframe are converted to another time system, usually to GPS time. The KBR and LRI are driven by the same ultra-stable oscillator (USO). Thus, the KBR time, which is often referred to as IPU receiver time, and the LRI time are both realizations of the USO time, but KBR/IPU and LRI time have a quasi-static offset relative to each other. It is quasi-static because it changes whenever one of the instruments reboots. Additionally, the offset is usually less than a second, since the instruments synchronize their respective time to the integer GPS second after the reboot has occurred. Since KBR and GPS measurements are performed within the IPU, the KBR timetags can be directly converted to GPS time using the \(\epsilon_{\mathrm{time}}\) value from the CLK1B product (cf. Equation (2)) that is determined during precise orbit determination for each satellite. For the LRI, the conversion requires additional steps. First, data reports, as given in the LHK1A/B data product, provide the timetag difference between LRI time and OBC time. Second, the TIM1B data product contains the offset between OBC time and KBR/IPU time, which is virtually zero during nominal operation, as the OBC time is steered towards the IPU time. In the end, the \(\epsilon_{\mathrm{time}}\) from the CLK1B data product yields the final relation needed to convert the timetag to GPS time. In Figure 4, the curves labeled _timetag raw_ shown in dashed black indicate a hypothetical noise calculated under the assumption of using a daily mean value of \(\epsilon_{\mathrm{time}}\), e.g., no \(\epsilon_{\mathrm{time}}\) information is employed for frequencies above \(\approx\)1/86,400 s. Thereby, the error in the timetag conversion from instrument to GPS time would be given by the variability of \(\epsilon_{\mathrm{time}}\). The in-flight variability was plotted as spectral density in the right plot of Figure 1 for GF-1 (blue) and GF-2 (cyan blue). The curves increase below 3 mHz, mainly due to the USO instability and proper time modulations at 1/rev and 2/rev frequencies, while the increase above 3 mHz is likely caused by limitations of precise orbit determination and clock solution precision, i.e., the measurement precision of \(\epsilon_{\mathrm{time}}\). The analytical formula for the USO stability from [10] (B-1), shown in solid magenta in Figure 1, is in poor agreement with the GRACE-FO in-flight data, justifying a refined USO model that we derive ad-hoc as  
\[\mathrm{ASD}[\delta t_{\mathrm{USO,mod}}](f)=1\,\mathrm{s}\cdot\sqrt{2.556 \cdot 10^{-33}\,\mathrm{Hz}^{4}/f^{5}+3.325\cdot 10^{-26}\,\mathrm{Hz}/f^{2}}. \tag{59}\]  
The first time derivative of this USO model is shown by the magenta dashed trace in the right plot of Figure 1. It is only a rough model which only approximately describes the variability in \(\epsilon_{\mathrm{time}}\) in the spectral domain, as the spectra differ slightly between satellites and the \(\epsilon_{\mathrm{time}}\) spectra change slightly from day to day (not shown). Furthermore, we take an educated guess for the measurement precision for eps-time due to POD/Clock errors of the GPS measurement system. The model reads  
\[\mathrm{ASD}[\delta t_{\epsilon}](f)=\frac{1.5\,\mathrm{mm}}{c_{0}\cdot\sqrt {f}}+\frac{10\,\mathrm{\SIUnitSymbolMicro m}\cdot\mathrm{Hz}}{c_{0}\cdot\sqrt{ f^{25}}}, \tag{60}\]  
and is shown as a black trace in the spectral domain in Figure 1 and as an Allan deviation in Figure 6. The model has a rms of \(11\,\mathrm{mm}/c_{0}\) in the band \(0.1\,\mathrm{mHz}-0.05\,\mathrm{Hz}\) and (roughly) agrees with the GPS POD/clock measurement noise in the Deep Space Atomic Clock(DSAC) experiment conducted in a low-Earth orbit [42] with respect to Allan deviations.