# Comparing GRACE-FO KBR and LRI Ranging Data with Focus on Carrier Frequency Variations
## 4 KBR Dual-One Way Ranging (DOWR)
This implies that the \(q\)-constants in Equation (45) are  
\[q^{\text{K/Ka}}_{\text{A/B}}=-\
u^{\text{K/Ka}}_{\text{A/B}}(t=0)\cdot\Delta t^{ \text{K/Ka}}_{\text{AB/BA}}(t=0). \tag{50}\]  
Finally, by employing \(q\)-constants and time-dependent frequencies in Equation (49), inserting Equation (46) into Equation (49), and considering the relation of carrier frequencies to USO frequencies from Table 1, one obtains  
\[\frac{a^{\text{K}}\cdot\varphi^{\text{K}}_{\text{DOWR}}(t)}{v^{\text{K}}_{ \text{A}}(t)+v^{\text{K}}_{\text{B}}(t)}+\frac{a^{\text{Ka}}\cdot\varphi^{\text {Ka}}_{\text{DOWR}}(t)}{v^{\text{Ka}}_{\text{A}}(t)+v^{\text{Ka}}_{\text{B}}(t) }=\Delta t_{\text{inst}}(t)-\Delta t_{\text{DOWR,FV}}(t)-\Delta t_{\text{LTC, DOWR}}(t)+\text{const}, \tag{51}\]  
where the first term \(\Delta t_{\text{inst}}\) on the right-hand side is the desired ranging signal expressed as propagation time, the second term describes the frequency variations that couple with the satellite separation (\(\Delta t_{\text{AB}}\approx\Delta t_{\text{BA}}\approx L/c_{0}\)) as  
\[\Delta t_{\text{FV,DOWR}}(t):=\frac{\Delta t_{\text{AB}}(0)\cdot f_{\text{A, USO}}(0)+\Delta t_{\text{BA}}(0)\cdot f_{\text{B,USO}}(0)}{f_{\text{A,USO}}(t)+f_{ \text{B,USO}}(t)}-\frac{\Delta t_{\text{AB}}(0)+\Delta t_{\text{BA}}(0)}{2} \tag{52}\]  
and the third term is the light-time correction [45]  
\[=\varphi^{\text{K/Ka}}_{\text{B}}(t)-\varphi^{\text{K/Ka}}_{ \text{B}}(t-\Delta t^{\text{K/Ka}}_{\text{BA}})+\Phi^{\text{K/Ka}}_{\text{A }}(t)-\Phi^{\text{[PERSON]/Ka}}_{\text{A}}(t-\Delta t^{\text{[PERSON]/Ka}}_{\text{AB}})\]0  
where the coefficients \(b_{\text{AB}}\) and \(b_{\text{BA}}\) are given in \[=\varphi^{\text{K/Ka}}_{\text{B}}(t)-\varphi^{\text{K/Ka}}_{ \text{B}}(t-\Delta t^{\text{K/Ka}}_{\text{BA}})+\Phi^{\text{K/Ka}}_{\text{A }}(t)-\Phi^{\text{[PERSON]/Ka}}_{\text{A}}(t-\Delta t^{\text{[PERSON]/Ka}}_{\text{AB}})\]1 The second and third term on the right-hand side of Equation (51) show the needed correction terms in order to obtain the range independent of USO frequency variations, i.e.,  
\[=\varphi^{\text{K/Ka}}_{\text{B}}(t)-\varphi^{\text{K/Ka}}_{ \text{B}}(t-\Delta t^{\text{K/Ka}}_{\text{BA}})+\Phi^{\text{K/Ka}}_{\text{A }}(t)-\Phi^{\text{[PERSON]/Ka}}_{\text{A}}(t-\Delta t^{\text{[PERSON]/Ka}}_{\text{AB}})\]2  
where only \(c_{0}\) and \(a_{\text{K/Ka}}\) are time-independent. The instantaneous biased range \(\rho_{\text{KBR}}\) can be directly employed in gravity field recovery. The so-called antenna offset correction \(\rho_{\text{AOC}}\) was added to be consistent with the official SDS data processing and is discussed in Section 6.2. The light-propagation times \(\Delta t_{\text{AB}}\) and \(\Delta t_{\text{BA}}\) needed to compute the frequency correction term in Equation (52) can be obtained from the GPS orbit products and are also used for the light-time correction computation. The quantities differ by \(c_{0}|\Delta t_{\text{AB}}-\Delta t_{\text{BA}}|\approx 10\,\text{m}\) in GRACE-FO due to the finite speed of light and the relative velocity between satellites [45]. Note that the USO frequencies \(f_{\text{A/B,USO}}\) in Equation (52) can be replaced with the carrier frequency of the K- or the Ka-band, since  
\[=\varphi^{\text{K/Ka}}_{\text{B}}(t)-\varphi^{\text{K/Ka}}_{ \text{B}}(t-\Delta t^{\text{K/Ka}}_{\text{BA}})+\Phi^{\text{K/Ka}}_{\text{A }}(t)-\Phi^{\text{[PERSON]/Ka}}_{\text{A}}(t-\Delta t^{\text{[PERSON]/Ka}}_{\text{AB}})\]3  
Compared to previous derivations and the official SDS processing described in [46], the novel aspect of Equation (54) is that it considers a time-variable frequency when converting the phase to range, and it contains a correction term \(\Delta t_{\text{FV,DOWR}}\) for the frequency variations. In the following sections, we show that this term might be of relevance when LRI and KBR data are compared at low frequencies.