## 3 Two-Way Ranging Formulations
 \tag{14}\]  
where the first term on the right-hand side is the scaled value of the instantaneous range, the second term corresponds to the pathlength error due to time delays, the third term is caused by the laser frequency noise and is scaled by the range, the fourth term is due to the time tag error, the fifth term represents a constant bias, for which the measured range is called the biased range, and the last term is due to the phase readout noise. The instantaneous range in the first term is scaled for the reason that there is a small difference between the nominal laser frequency \(f_{0}\) and the actual constant part of the laser frequency \(\bar{f}_{1}\) in flight. Since the cavity is unable to provide information of \(\bar{f}_{1}\), if there is no other method to obtain it in orbit, a scaled factor \(s\) close to one will be estimated [21], i.e.,  
\[s=1+\alpha\approx\frac{f_{0}}{\bar{f}_{1}}, \tag{15}\]  
where \(\alpha\) is of the order of \(10^{-6}\) for the GRACE-FO mission and is provided in the RL04 LR1B data product. After applying the scale correction, the measured biased range reads  
\[ \begin{split}\rho_{\text{LRI}}(t_{1})&\approx \rho_{\text{inst}}+\delta\rho_{\text{path}}+\frac{\delta f_{1}}{\bar{f}_{1}} \rho_{\text{inst}}+c\frac{\bar{f}^{\text{OPLL}}}{2\bar{f}_{1}}(\delta t_{2}- \delta t_{1})\\ &+\frac{c}{2\bar{f}_{1}}(N_{1}+N_{2})+\frac{c}{2\bar{f}_{1}}( \varepsilon_{1}+\varepsilon_{2}).\end{split}  \tag{16}\]  
Since the influence of the scale factor is beyond the scope of this paper and is not important to the study here, we assume that we have obtained the value of \(\bar{f}_{1}\) in orbit and, thus, there is no need to estimate the scale factor in the following sections. The pathlength error \(\delta\rho_{\text{path}}\) consists of all the physical effects related to the light path, including the delays due to media, TTL coupling, and relativistic effects [25], and one should correct it by applying the so-called light time correction (LTC), taking all these effects into account. In practice, however, the light time correction is usually calculated without considering the TTL coupling, and an additional TTL correction term is estimated individually. In the GRACE-FO RL04 KBR1B data product, the light time correction and the antenna offset correction, which is referred to as the TTL correction for KBR, are provided separately in two columns. The TTL correction in the GRACE-FO RL04 LR1B data product is set to zero since it is much smaller than the one for KBR and can be omitted. Therefore, we have  
\[ \begin{split}\delta\rho_{\text{path}}&\approx \delta\rho_{\text{TOF}}+\delta\rho_{\text{TTL}}\\ &\approx-\delta\rho_{\text{LTC}}-\delta\rho_{\text{TTLC}}\end{split}  \tag{17}\]  
where \(\delta\rho_{\text{TOF}}\) denotes the time-of-flight effect including the pathlength error due to media and relativistic effects; \(\delta\rho_{\text{TTL}}\) denotes the TTL coupling error; \(\delta\rho_{\text{LTC}}\) and \(\delta\rho_{\text{TTLC}}\) denote the estimated light time correction and the TTL correction, respectively. Once the light time correction term and the TTL correction term are estimated, we can obtain the corrected biased range [13]  
\[ \begin{split}\rho_{\text{corr}}(t_{1})&=\rho_{\text {LRI}}(t_{1})+\delta\rho_{\text{LTC}}+\delta\rho_{\text{TTLC}}\\ &\approx\rho_{\text{inst}}+\frac{\delta f_{1}}{\bar{f}_{1}}\rho_ {\text{inst}}+c\frac{\bar{f}^{\text{OPLL}}}{2\bar{f}_{1}}(\delta t_{2}-\delta t _{1})+\frac{c}{2\bar{f}_{1}}(N_{1}+N_{2})+\frac{c}{2\bar{f}_{1}}(\varepsilon _{1}+\varepsilon_{2}),\end{split}  \tag{18}\]  
which is one of the inputs to the gravity field recovery.