# Comparing GRACE-FO KBR and LRI Ranging Data with Focus on Carrier Frequency Variations
## 6 Instrument Noises and Corrections
### Carrier Frequency Variations  
Carrier frequencies are the laser frequency \(\
u_{\text{R}}\) in LRI and the K- and Ka-band frequencies \(\
u_{\text{A}/\text{B}}^{\text{K}/\text{Ka}}\) in KBR. Variations in these frequencies correspond to changes in the conversion factor from phase to range, as discussed in Section 3. If the frequency variations are measured and known, they can be accounted for in the conversion and do not falsify the range measurement when employing Equation (28). For the application in GRACE-FO, the approximation formulas Equations (22), (23) and (31) usually are sufficiently accurate. For the LRI, we assume the following model for the laser frequency at the reference satellite and for a time-series of one day  
\[\
u_{\text{R}}(t)\approx\langle\
u_{\text{R}}(t)\rangle+\langle\
u_{\text{R}} (t)\rangle\cdot\left(\frac{\text{d}\tau_{\text{R}}(t)}{\text{d}t}-1\right)+ \delta\
u_{\text{cav}}(t), \tag{68}\]  
where \(\langle\
u_{\text{R}}(t)\rangle\) is the daily mean frequency determined from correlating LRI phase to KBR range (cf. Section 5), the second term contains the modulation of the laser frequency due to the proper time \(\tau_{\text{R}}\) (cf. Equation (6)) and the last term accounts for the instability of the cavity, which was modelled in Section 2 as \(\delta\
u_{\text{cav}}(t)=\text{d}\tau_{\text{FV}}(t)/\text{d}t\cdot\langle\
u _{\text{R}}(t)\rangle\). The daily mean frequency \(\langle\
u_{\text{R}}(t)\rangle\) is used as the conversion factor from phase to range (cf. Equation (17)) in the LRI1 bv04 processing. The error arising from neglecting the intraday time-variability of the frequency represents a scale factor error as discussed in Section 3.4. Such errors couple into the range to first order as the product of fractional frequency variation and inter-satellite distance \(L(t)\). The ranging error due to \(\delta\
u_{\text{cav}}\) is usually referred to as laser frequency noise in the LRI and is caused by the limited stability of the cavity resonance frequency, e.g., due to thermal fluctuations. It can be written as  
\[\text{ASD}[\delta\rho_{\text{FV,corr,LRI}}](f)=\frac{\text{ASD}[\delta\
u_{ \text{cav}}](f)}{\langle\
u_{\text{R}}(t)\rangle}\cdot L(t)\quad\text{with} \quad\text{ASD}[\delta\
u_{\text{cav}}](f)=\frac{10^{-15}}{\sqrt{f}}\cdot \langle\
u_{\text{R}}(t)\rangle, \tag{69}\]  
which is based on an extrapolation of the LRI ranging noise from high to low frequencies [52] and wherein \(L\approx 220\,\text{km}\pm 50\,\text{km}\) is the satellite separation. This noise level is in good agreement with pre-flight measurements of the cavity stability [19]. ASD\([\delta\rho_{\text{FV,corr,LRI}}]\) is shown as a solid blue trace in the left plots of Figure 4. It is labeled as frequency variationscorrected because this trace assumes that the second proper time term in Equation (68) has been removed. This removal can be accomplished using  
\[\rho^{\rm corr}_{\rm LRI1 Bv04}(t)=\rho_{\rm LRI1 Bv04}(t)+\left(\frac{\mathrm{d} \tau_{\rm R}(t)}{\mathrm{d}t}-1\right)\cdot L(t). \tag{70}\]  
The proper time of the reference satellite \(\tau_{\rm R}\) and absolute distance \(L\) can be computed from GRACE-FO orbit data (GN1B data product). The correction is added to the biased range of LRI1B in order to obtain a corrected biased range, which accounts for the variations of proper time in the reference laser frequency. The spectrum of the raw laser frequency noise can be approximated as the sum of the proper-time term and the cavity stability as  
\[\mathrm{PSD}[\delta\rho_{\rm FV,raw,LRI}](f)\approx\mathrm{PSD}\!\left[ \frac{\mathrm{d}\tau_{\rm R}}{\mathrm{d}t}\right]\cdot L^{2}+\mathrm{PSD}[ \delta\rho_{\rm FV,corr,LRI}], \tag{71}\]  
which is shown as a dashed blue trace in the left plots of Figure 4.  
For the KBR, the carrier frequency variations at low frequencies can be computed directly from \(\epsilon_{\rm time}\) in the CLK1B data product. Currently, the KBR1B data processing uses daily mean values of the carrier frequencies, which neglects intraday variations.