# Comparing GRACE-FO KBR and LRI Ranging Data with Focus on Carrier Frequency Variations
## 2 Phase Observable  
The phase of microwave or optical radiation can be described in a general-relativistic context as  
\[\Phi=\
u_{0}\cdot\tau_{\text{osc}}(t)+q_{0}=\
u_{0}\cdot\left[\tau(t)+\tau_{\text{ FV}}(\tau(t))\right]+q_{0}, \tag{1}\]  
where \(t\) is the coordinate time or Global Positioning System (GPS) time, \(\
u_{0}\) is the nominal frequency and \(q_{0}\) is a constant that cannot be resolved in general using interferometry, as phase observations exhibit an integer ambiguity. \(\tau_{\text{osc}}\) is a virtual oscillator time given by the sum of the proper time \(\tau\) of the satellite or emitting device (e.g., laser) and a contributor \(\tau_{\text{FV}}\) describing frequency variations or, in other words, deviations of the oscillator time from proper time due to imperfections, noise, etc. The oscillator time for the KBR is realized inside the instrument processing unit (IPU) of the microwave ranging instrument (MWI) assembly and can be called the ultra-stable oscillator (USO) time \(\tau^{\text{USO}}\) or the IPU receiver time. The difference between USO time and GPS time is the clock error, which is estimated during precise orbit determination (POD) together with the satellite's position and velocity state. The clock error is reported as time series in the clock offset level1b data product (CLK1B, [17]) as eps_Time (\(\epsilon_{\text{time}}\)), i.e.,  
\[\epsilon_{\text{time}}=t-\tau^{\text{USO}}(t). \tag{2}\]  
\(\epsilon_{\text{time}}\) contains the proper time, clock variations and a frequency offset, as can be seen from Figure 1. The proper time for a satellite can easily be computed by numerically integrating [42]  
\[1-\frac{\text{d}\tau}{\text{d}t}=\frac{GM}{rc_{0}^{2}}-\frac{GM}{rc_{0}^{2}} \cdot J_{2}\cdot\left(\frac{a_{e}}{r}\right)^{2}\cdot\frac{3z^{2}-r^{2}}{2r^{ 2}}+\frac{v^{2}}{2c_{0}^{2}}+\frac{\phi_{0}}{c_{0}^{2}}, \tag{3}\]  
where \(c_{0}\) is the speed of light, \(GM\) is Earth's gravitational parameter, \(a_{e}\) is Earth's mean radius, \(J_{2}\) is Earths oblateness coefficient, \(r\) is the distance between geocenter and satellite, \(z\) is the z-component of the position vector, \(v\) is the satellite velocity and \(\phi_{0}\) is a potential constant. Figure 1: (**Left**) Typical time-domain plot of the proper time \(\tau\) (red) determined from the satellite orbit product and the CLK1B \(\epsilon_{\text{time}}\)(light and dark blue), both in terms of the first derivative. The proper time is clearly visible in \(\epsilon_{\text{time}}\). (**Right**): Amplitude spectral density (ASD) of the time-series shown on the left side together with an estimate of the USO stability based on [10]. The traces labeled refined USO model and POD/Clock noise model are described in Section 6.4. See Figure 6 for a representation as Allan deviations. The Equivalent Noise Bandwidth (ENBW) given here and in other spectra can be used to convert the spectral density units to spectrum (rms) units [43]. The frequency offsets of the USOs w.r.t. their nominal frequency yield y-offsets in the left plot of Figure 1. As apparent from the legend, they are determined to be 7.4 ppb on satellite 1 (GF-1) and 6.6 ppb on satellite 2 (GF-2) on that particular day. The oscillator time of the laser is not accessible and is usually expressed in terms of the laser frequency. The instantaneous frequency reads  
\[\
u(\tau):=\frac{\mathrm{d}\Phi}{\mathrm{d}\tau}(\tau)=v_{0}\cdot\left(1+\frac{ \mathrm{d}\tau_{\mathrm{FV}}}{\mathrm{d}\tau}\right), \tag{4}\]  
where \(v_{0}\cdot\mathrm{d}\tau_{\mathrm{FV}}/\mathrm{d}\tau\) describes the frequency variations or noise with respect to a nominal frequency \(v_{0}\).