# Comparing GRACE-FO KBR and LRI Ranging Data with Focus on Carrier Frequency Variations
## 6 Instrument Noises and Corrections
### Phase Readout Noise  
The KBR and LRI both track the phase of an interference beatnote that is present as an electrical signal in the corresponding processing unit. Phase changes are proportional to changes in the light or microwave propagation time up to first order and, thus, to distance changes between spacecraft. The readout noise of the phase is usually determined by the carrier-to-noise ratio (CNR) entering the phase-tracking loop, which can be expressed in terms of a ranging error as [10; 21]  
\[\mathrm{ASD}[\rho_{\mathrm{CNR}}](f)=\frac{\lambda\cdot 1\,\mathrm{rad}}{2 \cdot 2\pi\cdot\sqrt{\mathrm{CNR}}},\quad\mathrm{CNR}=\frac{l_{\mathrm{rms}}^{2}} {\mathrm{PSD}[\mathcal{N}](f_{\mathrm{beat}})},\quad[\mathrm{ASD}[\rho_{ \mathrm{CNR}}]]=\mathrm{m}/\sqrt{\mathrm{Hz}}, \tag{56}\]  
where the half-wavelength \(\lambda/2\) is used to rescale the white phase noise level to units of half-round-trip range. The CNR is defined as the ratio of root-mean-square (rms) carrier power \(l_{\mathrm{rms}}^{2}\), i.e., squared rms amplitude of the beatnote photocurrent, to the noise power spectral density (\(\mathrm{PSD}[\mathcal{N}]\)) of the photocurrent at the beatnote frequency. Typical values for the LRI are above \(80\,\mathrm{dBHz}=10^{8}\,\mathrm{Hz}\) for both the reference and transponder satellite, which is well above the minimum requirement of \(70\,\mathrm{dBHz}=10^{7}\,\mathrm{Hz}\) of the instrument. The transponder phase measurement is used as a sensor in a feedback control loop to control the transponder laser frequency with high gain and bandwidth, e.g., the phase variations in terms of range on the transponder side are below \(10\,\mathrm{pm}/\sqrt{\mathrm{Hz}}\) at \(1\,\mathrm{Hz}\) and even smaller for lower frequencies. Thus, it is an in-loop measurement close to zero. Any phase variations arising on the transponder, e.g., from laser noise or readout noise, are imprinted onto the laser beam and transmitted to the reference satellite. Hence, the reference satellite measures the phase readout noise of transponder and reference satellite  
\[\mathrm{ASD}[\rho_{\mathrm{LRI,CNR}}](f)=\frac{\lambda_{\mathrm{R}}\cdot 1 \,\mathrm{rad}}{2\cdot 2\pi}\cdot\sqrt{\frac{1}{\mathrm{CNR}_{\mathrm{T}}}+ \frac{1}{\mathrm{CNR}_{\mathrm{M}}}}. \tag{57}\]  
This formula is actually independent of the loop gain of the frequency controller, i.e., even with low gain and significant signal in the transponder phase, the ranging signal \(\rho_{\mathrm{LRI}}\) is usually formed by the combination of both satellites in order to remove the common phase ramp (cf. Equation (9)) and to recover the complete ranging signal. Thus, it will exhibit the phase readout noise of both satellites. The CNR-limited phase readout noise \(\mathrm{ASD}[\rho_{\mathrm{CNR}}]\) is, strictly speaking, non-stationary, as the CNR is time-dependent. Therefore, we plot short segments of data in order to prevent potential artifacts due to non-stationarity. The left plot of Figure 5 shows the calculated phase readout noise compared to the ranging signal of the LRI (magenta trace). A CNR value of \(77.5\,\mathrm{dBHz}=10^{7.75}\,\mathrm{Hz}\) from a single satellite yields approx. \(11\,\mathrm{pm}/\sqrt{\mathrm{Hz}}\) of white phase readout noise, or \(\sqrt{2}\cdot 11\,\mathrm{pm}/\sqrt{\mathrm{Hz}}\) from two satellites. Such low values are not observable in the ranging phase of the LRI (magenta trace in left plot of Figure 5) due to other noise sources, mainly laser frequency variations from the cavity. However, the LRI system measures the interference beatnote on a satellite with four channels in order to make use of the Differential Wavefront Sensing (DWS) technique [18]. The ranging information is given by the average of the four phase measurements, which is the linear combination of four channels with multipliers \(+1/4\). The CNR values reported by the LRP of the LRI refer to the coherent sum (average) of four channels. However, if the noise among different channels is uncorrelated and the phase difference among different channels is small (cf.