# Comparing GRACE-FO KBR and LRI Ranging Data with Focus on Carrier Frequency Variations
## 7 Impact of Frequency Variations on KBR-LRI Residuals  
We have corrected the relativistic (proper-time) frequency variations in LRI and the KBR frequency variations, as discussed in Section 6.5, on a daily basis for most of the days between December 2018 and March 2022 in the GRACE-FO v04 data. Days for which LRI1B, KBR1B or CLK1B data were incomplete, e.g., due to gaps from instrument reboots or diagnostic events, were skipped. For each day, we evaluated the spectrum of KBR-LRI residuals as shown exemplarily in Figure 8. On the day of 10 January 2019, the spectrum of KBR-LRI residuals was reduced when applying the correction, while they day of 28 October 2021 shows an increase in the residual level (compare red and blue traces). Figure 7: Difference between the official KBR1B biased range (v04) and a corrected KBR1B data product, from which USO frequency variations were removed. The red trace serves as a sanity check to show that our self-derived KBR1B product can reproduce the official KBR1 Bv04 results up to some negligible deviations, which probably stem from different interpolation methods. The blue curve shows the difference after frequency variations have been fully removed, while the black trace shows the difference calculated if only the division of the phase with a time-dependent frequency is considered without \(\Delta t_{\text{FV,DOWR}}\)-term. Since the black trace is close to the red one and much lower than the blue one, one can conclude that the \(\Delta t_{\text{FV,DOWR}}\)-term has much more relevance than dividing the phase by the time-variable frequency. The spectrum was computed with GRACE-FO data from 1 January 2019. As a metric, we use the root-mean-square (rms) value of KBR-LRI residuals at low frequencies, which is obtained by integrating the power spectral density (squared ASD) for frequencies below 1 mHz. The plots entitled _With Rescale_ (right column) indicate that we re-estimated the differential scale and time-shift between LRI and KBR together with a linear trend. This is in particular necessary for days before 12 August 2019, as the LRI1B data is sometimes not properly scaled and time-shifted in this period. The change in daily rms value of LRI-LBR residuals at the low frequency due to LRI and KBR frequency correction is shown in Figure 9. For January 2019, the rms value could be lowered noticeably, but, the values oscillate with the beta angle of the orbit, i.e., the angle between orbital plane and Sun direction. The absolute rms values are depicted in Figure 10 and exhibit the oscillatory pattern as well. Both plots indicate only a marginal reduction in rms due to the frequency correction, e.g., a change in mean rms from 3.97 um to 3.90 um. We repeated the analysis with more empirical parameters being estimated in the daily rescale step, namely, the 1/rev, 2/rev and 2/day oscillation amplitudes in terms of sine and cosine components and a quadratic trend, in addition to the differential scale, time-shift and linear trend already in use. The 1/rev and 2/rev frequencies in the ranging data are likely to have significant errors, as environmental parameters, such as temperature, inertial orientation and magnetic field, predominately vary at these frequencies. The 2/day parameter was added due to the CLK1B \(\epsilon_{\text{time}}\) exhibiting a pronounced 2/day feature on several days (cf. Figure 1), which could be related to Earth's rotation period or to the orbit period of satellites in the GPS constellation. Figure 8: KBR-LRI residuals at low frequencies without applying the frequency corrections (blue) and with applied LRI and KBR frequency correction (red). The corrections are shown as yellow and purple traces. We then estimated the KBR-LRI residuals together with the many empirical parameters again, once without the frequency correction and once with the correction for individual days. The results are shown in Figure 11. The additional empirical parameters absorbed the oscillatory beta angle-related behavior in the rms value to a large extent and lowered the rms value from approx.