# Comparing GRACE-FO KBR and LRI Ranging Data with Focus on Carrier Frequency Variations
## 7 Impact of Frequency Variations on KBR-LRI Residuals
4 \(\upmu\)m to approx. 1 \(\upmu\)m, as had been expected. A change in rms value due to the frequency correction thereby became visible more clearly, i.e., from mean rms of 1.14 \(\upmu\)m to 0.99 \(\upmu\)m (cf. legend in figure). This 0.15 \(\upmu\)m change in rms in the 1 mHz bandwidth, which would imply a spectral density improvement of 0.15 \(\upmu\)m/\(\sqrt{1\,\mathrm{mHz}}=4.7\,\upmu\)m/\(\sqrt{\mathrm{Hz}}\) assuming white noise behavior, excludes effects at 1/rev, 2/rev and 2/day because KBR-LRI residuals at these frequencies are highly reduced due  
Figure 10: Daily absolute value of low frequency (<1 mHz) rms of the KBR-LRI residuals. The rms value of residuals is low when the beta angle is large, i.e., when satellites are close to or in full-sun phase and not entering shadow. The left plot uses original LRIIbv04 data with their scale and time-shift, while the right plot assumes re-estimation (rescale) of scale, time-shift and linear trend on a daily basis. Figure 9: Daily change in the low-frequency rms value (<1 mHz) of the KBR-LRI residuals when applying the KBR and LRI frequency variation correction. Both plots show the difference in rms value before and after the correction. The left plot uses original LRIIbv04 data without re-estimating the scale and time-shift between LRI and KBR, while the right plot assumes re-estimation (rescale) of scale, time-shift and linear trend on a daily basis. Values above the cyan dashed line indicate a reduction in rms value due to the correction. On days with a high absolute value of beta angle, i.e., close to or in full-sun phase of the satellites, the frequency correction yields the largest rms reductions, e.g., in January 2019.  
the empirical parameters. Therefore, this change should be understood as a lower bound as it neglects the frequencies where the correction has most of its signal. The reduction in rms demonstrates that applying the frequency corrections, i.e., considering intraday variability of the carrier frequencies, slightly improves the agreement of LRI and KBR ranging data. We also emphasize that the relativistic effect due to the proper time, i.e., 0.9 \(\upmu\)m-peak at 1/rev (cf. Equation (42)), is omitted in both instruments, if no frequency correction is applied. If the corresponding frequency correction is applied to both instruments, the 0.9 \(\upmu\)m-peak is present in both instruments. Thus, it never appears in the KBR-LRI residuals. However, neglecting the effect would introduce an error compared to a true error-free range for both measurements. The effect of the frequency corrections on LRI and KBR is small, i.e., at the micron scale at very low frequencies, which means it should have negligible impact on gravity fields, as typical pre-fit and post-fit residuals of gravity fields are much larger than the KBR-LRI residuals. However, these corrections reduce the errors in the instruments and are a necessary step towards a better understanding of the KBR-LRI residuals at low frequencies, e.g., when studying the tone errors of instruments at 1/rev and 2/rev frequencies.