## 2 Methodology
### GPS/BDS-Combined Single-Frequency KOD Model  
For the GPS/BDS-combined KOD model, the equation of single-frequency pseudororange and carrier phase observation from a satellite to the LEO onboard GPS/BDS receiver \(r\) is given as follows:  
\[\left\{ \begin{array}{l}P_{r}^{s}=\rho_{r}^{s}+c\left(t_{r}-t^{s}\right)+d_{r} -d^{s}+I_{r}^{s}+\epsilon_{p}^{s}\\ L_{r}^{s}=\rho_{r}^{s}+c\left(t_{r}-t^{s}\right)-I_{r}^{s}+\lambda^{s}(B_{r}^{ s}+b_{r}-b^{s})+\epsilon_{l}^{s}\end{array} \right. \tag{1}\]  
where \(s\) represents the satellite of the GPS or the BDS system; respectively, \(P_{r}^{s}\) and \(L_{r}^{s}\) are the pseudorange and carrier phase observation in meters; \(\rho_{r}^{s}\) is the geometry distance between the satellite and receiver; \(c\) is the speed of light in a vacuum; \(t_{r}\) is the receiver offset and \(t^{s}\) is the satellite clock offset; \(d_{r}\) is the receiver uncalibrated code delay (UCD) and \(d^{s}\) is the satellite UCD; \(I_{r}^{s}\) is the ionospheric delay; \(B_{r}^{s}\) is the integer phase ambiguity in the unit of cycles and \(\lambda^{s}\) is the wavelength; \(b_{r}\) is the receiver uncalibrated phase delay (UPD) and \(b^{s}\) is the satellite UPD; \(\varepsilon_{p}^{s}\) and \(\epsilon_{l}^{s}\) are the multipath and measurement noises of the pseudorange and phase observation, respectively. In this model, the tropospheric delay error does not exist, since most LEO satellite orbits operate on the top of the troposphere and the tropospheric impact on the GNSS signal is negligible. The inter-system bias between GPS and BDS can be treated as white noise; thus, the independent receiver clock parameter is set for the GPS or BDS system. The corresponding receiver UCD can be combined into the receiver clock since it is usually stable. For the ionosphere delay, there are many correction techniques, such as the NeQuick [30], the TIEGCM [31], the NTCM-GL [32], and the Klobuchar model [33], for single-frequency GNSS observation from a LEO satellite. A downscaled version of the Klobuchar model is used for convenience to mitigate the ionosphere effect in the pseudorange observation, and the scale factor is empirically given as 0.3 [34]. The pseudorange observation is then combined with phase GRAPHIC observation to estimate the receiver's position, receiver clock offset, and ambiguity parameters. Therefore, the GPS/BDS-combined KOD observation model is as follows:  
\[\left\{ \begin{array}{l}\overline{p}_{r}^{s}=\mu_{r}^{s}\Delta x_{r}+c \overline{t}_{r}+\varepsilon_{p}^{s}\\ \overline{s}_{r}^{s}=\frac{p_{r}^{s}+l_{r}^{s}}{2}=\mu_{r}^{s}\Delta x_{r}+c \overline{t}_{r}+\overline{B}_{r}^{s}+\frac{\epsilon_{p}^{s}+c_{l}^{s}}{2} \end{array} \right. \tag{2}\]  
where \(p_{r}^{s}\) and \(l_{r}^{s}\) are the observed minus computed (OMC) values of the pseudorange and phase observation in meters, respectively; \(\overline{p}_{r}^{s}\) is the OMC value of the pseudorange observation after mitigating the ionosphere effect by the downscaled Klobuchar model; \(\overline{g}_{r}^{s}\) is the OMC value of the GRAPHIC observation; \(\mu_{r}^{s}\) is the unit row vector from receiver to satellite after linearization; \(\Delta x_{r}\) is the column vector of the receiver's position increments relative to the a priori position; \(\overline{t}_{r}\) is the corresponding receiver clock offset after merging the receiver UCD; \(\overline{B}_{r}^{s}\) is the corresponding ambiguity parameter in meters after merging the UPD. A bias term will be introduced into the single-frequency KOD model if the precise satellite clock product provided by the international GNSS services (IGS) is used to correct the satellite clock error [35]. The precise satellite clock offsets are usually obtained with GNSS ionospheric-free observations of two frequencies [36]. For the single-frequency GPS/BDS-combined KOD observation model, a correction term computed by the differential code biases (DCB) should be subtracted to eliminate the introduced bias.