# Comparing GRACE-FO KBR and LRI Ranging Data with Focus on Carrier Frequency Variations
## 6 Instrument Noises and Corrections
### Tilt-to-Length Coupling  
Satellite rotations cause path length changes when the reference points (RPs) of the LRI or the antenna phase centers (APCs) of KBR are not co-located with the respective center-of-mass of satellites, which are the pivot points for rotations in space [21] (section 2.6.3). The RPs and APCs are effectively the fiducial points used to determine the biased inter-satellite range from interferometric phase readout and are fixed in the satellite body frame, i.e., moving in inertial space when the satellites rotate. The main effect of the coupling can be expressed in terms of a vector describing the offset between center of mass and the RP or APC that is projected onto the line of sight as the measurement axis. To first order, lateral components perpendicular to the line of sight (\(y\) & \(z\)) couple linearly with yaw and pitch, i.e., 100 um offset produce a yaw and pitch coupling of 100 um/rad per satellite into the biased range. The \(x\)-offset couples quadratically with yaw and pitch as 100 um/rad\({}^{2}\). In the KBR context, the tilt-to-length coupling is referred to as antenna offset correction [17; 46] and it is provided in the KRB1B data set in terms of the DOWR-range correction, including effects from both spacecraft (cf. \(c_{0}\Delta t_{\text{AOC}}\) in Equation (54)). Since the KBR antenna is at the front panel of the satellite, the APCs are offset by approx. 1.47 m in \(x\) direction  
Figure 4: Noise contributors and corrections in LRI and KBR in terms of amplitude spectral densities (ASDs). The grey KBR-LRI residuals are described in Section 5 and replotted from Figure 3. The KBR tilt-to-length correction (Antenna-Offset-Correction) in orange is provided in the KBR1B dataset and already removed in the KBR range, while the corresponding effect in the LRI has a lower magnitude and was derived based on [38]—see Section 6.2 for more details. The ionospheric correction (red) is relevant for KBR, but negligible in LRI due to the scaling with wavelength. The smaller wavelength of the laser scales the phase readout noise (magenta) to a negligible level as well, while in KBR, it is dominant at high frequencies (cf. Section 6.3). Raw frequency variations (dashed blue) assume a static frequency (within one day) and are related to the stability of the optical cavity and proper time effect in the LRI, whereas they are related to the USO stability in KBR. These raw frequency variations can be corrected by using a time-dependent frequency (cf. Section 6.5). The effect of timetag errors is being corrected in LRI and KBR by employing CLK1B \(e_{\text{time}}\) information (dashed vs. solid black traces). For some legend entries, multiple days were plotted in order to illustrate the stationarity and variability of the traces. The mean satellite separation is \(L=194\) km for January 2019. The ENBW for both plots is 52 μmHz. from the center-of-mass. This means that the correction contains a significant quadratic contribution from the pointing angles. The coupling factors can be measured in-orbit using so-called KBR calibration maneuver during which the satellites are deliberately rotated. In the LRI, the RPs on both satellites were co-located to the center of mass to the level of the mechanical integration accuracy in the order of \(100\,\mathrm{\SIUnitSymbolMicro m}\) in \(y\) and \(z\), which yields the dominant linear contribution. The LRI tilt-to-length coupling factors can be measured using the center-of-mass calibration maneuvers [38] that are usually repeated every six months. As of version 04 of LRI1B data, the tilt-to-length correction is not derived or applied to LRI data, but this may change in future releases. More information in regard to LRI tilt-to-length coupling can be found in [51]. The magnitude of the tilt-to-length effect on the KBR and LRI range measurement is shown in orange in Figure 4. In the LRI case, the effect is approximately one order of magnitude smaller compared to the KBR.