# Comparing GRACE-FO KBR and LRI Ranging Data with Focus on Carrier Frequency Variations
## 6 Instrument Noises and Corrections
### Phase Readout Noise
[21], Section 2.6.9 for details), the same CNR values would apply to other linear combinations, as well. These other three linear combinations with multipliers \(\pm 1/4\) can be referred to as DWSyaw, DWSpitch and DWSx. DWSyaw and DWSpitch represent the angular misalignment between the interfering laser beams, and are actually zeroed by a feedback control loop using a fine-steering mirror as actuator. The remaining DWSx combination is not zeroed and exhibits white noise behavior for frequencies above 0.1 Hz (cf. left plot in Figure 5) with a magnitude consistent with the expected CNR-limited phase readout noise. It is noteworthy that DWSx is a local quantity that needs to be compared against the single spacecraft CNR-limit (Equation (56)). If noise is transferred between satellites via the laser light, it will affect all channels on the receiver, thus appearing in the average phase of the four channels, but not in the DWS combinations of the receiver. The consistency between DWSx noise level and analytical CNR-limited phase readout noise level is at least an indicator that the CNR-limited phase readout noise is at the expected level in the ranging signal (magenta trace), though it is not directly measurable. The phase readout noise in the context of the KBR is usually called system noise [10; 11]. Based on Equation (56) and the DOWR combination discussed in Section 4, it is straightforward to express the CNR-limited phase readout noise in the ionosphere-free DOWR range as  
\[\text{ASD}[\rho_{\text{KBR,CNR}}](f)=\frac{1\,\text{rad}}{2\cdot 2\pi}\cdot \sqrt{\frac{(a^{\text{K}}\lambda_{A}^{\text{K}})^{2}}{\text{CNR}_{A}^{\text{K }}}+\frac{(a^{\text{K}}\lambda_{B}^{\text{K}})^{2}}{\text{CNR}_{B}^{\text{K }}}+\frac{(a^{\text{Ka}}\lambda_{A}^{\text{Ka}})^{2}}{\text{CNR}_{A}^{\text{ Ka}}}+\frac{(a^{\text{Ka}}\lambda_{B}^{\text{Ka}})^{2}}{\text{CNR}_{B}^{\text{ Ka}}}}, \tag{58}\]  
where coefficients \(a^{\text{K}}\) and \(a^{\text{Ka}}\) as well as wavelength \(\lambda=c_{0}/\
u\) can be obtained from Table 1. This expression is consistent with the derivation given in [10] (Section B-1) and shown as magenta-dashed trace in the right plot of Figure 5. Unfortunately, the actual DOWR ranging  
Figure 5: CNR-limited phase readout noise in LRI (**left**) and KBR (**right**). In the LRI, the phase readout noise is well below the actual ranging signal, which is limited by laser frequency noise (magenta, left plot). In KBR, the phase readout noise is likely limiting the KBR range at frequencies above 10 mHz. The measured noise level in the K-band is slightly higher than the expected one derived from the CNR-values. This also yields a small discrepancy in the final ionosphere-free DOWR combination (magenta, dashed 0.4 m/\(\sqrt{\text{Hz}}\) vs. solid \(\approx 0.6\) m/\(\sqrt{\text{Hz}}\)). noise is slightly higher than predicted based on the CNR values reported in the KBR1B data product. When plotting the individual range measurements at K- and Ka-band, we can identify that the discrepancy is caused by the K-band range, which exhibits slightly higher noise than predicted by the CNR. Although the reason for this discrepancy is unclear to the authors of this paper, the deviation is rather small. Based on Figure 4, we conclude that the phase readout noise is negligible for the LRI, while it likely limits the KBR, and therefore the KBR-LRI residuals at high frequencies, to a white noise level of approx. \(0.6\,\mathrm{\SIUnitSymbolMicro m}/\sqrt{\mathrm{Hz}}\).