# Comparing GRACE-FO KBR and LRI Ranging Data with Focus on Carrier Frequency Variations
## 8 Conclusions  
GRACE Follow-On offers the unique opportunity to compare two almost independent ranging instruments, the KBR and the LRI. Both instruments rely on readout of phase changes present in the beat note of interfering microwave or near-infrared laser radiation. The measured phase values can be converted to a range with units of meter, using the respective carrier frequencies \(\
u_{\rm A/B}^{\rm K/Ka}\) or \(\
u_{\rm R}\). In this paper, we have derived and analyzed different formulas for converting the phase to a range when the time-dependency of the carrier frequency is taken into account. It turned out that the previously suggested approach of differentiating the phase, rescaling the derivative using the frequency and performing an integration afterwards is not exact, since a summand was neglected. In this work, we present the exact formula, which is also likely to be relevant for future laser ranging systems. In addition to the analytically exact formula, several approximate formulas can be used that yield results with picometer precision, which is usually sufficient in a GRACE-like context. We stressed that the time-variability of the carrier frequency \(\
u\), if it is neglected in the phase-to-range conversion, yields an error that is proportional to the satellite separation \(L\), commonly referred to as laser frequency noise in the LRI. The time-variability is, on the one hand, caused by relativistic effects from the proper time, which is common to both LRI and KBR, and on the other hand, by the instability of the optical cavity for LRI and by the instability of the USO for KBR. In the case of the KBR, the microwave carrier frequencies can be determined in post-processing from GPS precise orbit and clock error determination. Figure 11: Results as in Figures 9 and 10, but now using more empirical parameters for the daily estimates: 1/rev, 2/rev, 2/day, scale, time-shift and with linear and quadratic trends. In the second part of this paper, the difference between measured range of LRI and KBR in the official v04 data products was compared to some potential noise and error contributors. It was pointed out that the KBR-LRI residuals are bound at frequencies above a few mHz to a level of approx. \(0.6\,\mathrm{\SIUnitSymbolMicro m}/\sqrt{\mathrm{Hz}}\) due to the CNR-limited phase readout noise from KBR, although the analytical model predicts a noise level of approx. \(0.4\,\mathrm{\SIUnitSymbolMicro m}/\sqrt{\mathrm{Hz}}\). This small discrepancy has not been understood so far and is caused by the K-band observations, while Ka-band measurements seem to be consistent with the model. According to our analysis, two effects are limiting the KBR-LRI residuals at low frequencies. These effects are residual timetag errors in the KBR system after applying the GPS-derived clock correction, which cannot be further improved with post-processing, and the intraday carrier frequency variations in the KBR and LRI, which can be corrected using the formulas introduced in the first half of this paper. The LRI frequency variations can only be corrected for the proper time effect using GNI1B orbit data, which contain dominant 1/rev and 2/rev components. In GRACE-FO, there is no direct measurement of the absolute laser frequency, and thus, the laser frequency is assumed to be static on a daily basis and its daily values are derived from correlating LRI and KBR observations. For KBR, the microwave frequencies are derived from the USO and potential frequency variations can be determined with respect to GPS data. The corresponding KBR frequency correction accounting for intraday variability can be computed directly from the CLK1B data product and exhibits a continuous spectrum as well as a 1/rev peak due to the proper time effect. Finally, we applied the KBR and LRI frequency corrections and were able to demonstrate a slight improvement in the agreement between KBR and LRI range, i.e., the rms value of KBR-LRI residuals below 1 mHz was reduced by \(0.15\,\mathrm{\SIUnitSymbolMicro m}\).