# Integrated Precise Orbit Determination of Multi-Gnss and Large LEO Constellations
## 3 Integrated POD Method and Strategy  
### Integrated POD Method  
Based on the simulated GNSS observations, the LEO-augmented four-system POD, also known as the integrated POD, can be performed. The ionosphere-free (IF) combination is adopted to eliminate the ionospheric delay in the dual-frequency code and phase measurements. The linearized observation equation for IF combinations can be written as follows:  
\[\left\{ \begin{array}{c}p_{g,IF}^{s}=u_{g}^{s}\cdot\varphi(t,t_{0})^{s}O_{0}^ {s}+c(\overline{\delta t}_{g}-\overline{\delta t}^{s})+M_{g}^{s}\cdot Z_{g}+ \varepsilon_{g,IF}^{s}\\ p_{lo,IF}^{s}=u_{lo}^{s}\cdot\varphi(t,t_{0})^{s}O_{0}^{s}-u_{lo}^{s}\cdot \varphi(t,t_{0})_{lo}\mathcal{O}_{lo,0}+c(\overline{\delta t}_{lo}-\overline{ \delta t}^{s})+\varepsilon_{lo,1F}^{s}\\ p_{g,IF}^{s}=u_{g}^{s}\cdot\varphi(t,t_{0})^{s}O_{0}^{s}+c(\overline{\delta t}_{ g}-\overline{\delta t}^{s})+M_{g}^{s}\cdot Z_{g}+\lambda_{IF}\overline{N}_{g,IF}^{s}+ \omega_{g,IF}^{s}\\ l_{lo,IF}^{s}=u_{loc}^{s}\cdot\varphi(t,t_{0})^{s}O_{0}^{s}-u_{lo}^{s}\cdot \varphi(t,t_{0})_{lo}\mathcal{O}_{lo,0}+c(\overline{\delta t}_{lo}-\overline{ \delta t}^{s})+\lambda_{IF}\overline{N}_{loo,1F}^{s}+\omega_{loo,IF}^{s}\end{array} \right. \tag{4}\]with  
\[\left\{ \begin{array}{c}O_{0}^{6}=\left(x_{0}^{s},y_{0}^{s},z_{0}^{s},v_{x}^{s},v_{ x}^{s},v_{z}^{s},v_{1}^{s},p_{1}^{s},\cdots,p_{n}^{s}\right)^{T}\\ O_{l\omega,0}=\left(x_{l\omega,0},y_{l\omega,0},z_{l\omega,0},v_{l\omega,x},v_{ l\omega,y},v_{l\omega,z},p_{l\omega,1},P_{l\omega,2},\cdots,p_{l\omega,n} \right)^{T}\\ \overline{N}_{g,\mathrm{IF}}^{s}=N_{g,\mathrm{IF}}^{s}+B_{g,\mathrm{IF}}-B_{ \mathrm{IF}}^{s}-\frac{b_{g,\mathrm{IF}}+b_{\mathrm{IF}}^{s}}{A_{\mathrm{IF}}^{ s}}\\ \overline{N}_{l\omega,\mathrm{IF}}^{s}=N_{l\omega,\mathrm{IF}}^{s}+B_{l\omega, \mathrm{IF}}-B_{\mathrm{IF}}^{s}-\frac{b_{l\omega,\mathrm{IF}}+b_{\mathrm{IF} }^{s}}{A_{\mathrm{IF}}^{s}}\end{array} \right. \tag{5}\]  
where \(u_{g}^{s}\) and \(u_{l\omega}^{s}\) denote the unit vector of the direction from ground receiver to satellite and from LEO to satellite respectively. \(\varphi(t,t_{0})^{s}\) and \(\varphi(t,t_{0})_{l\omega}\) refer to the state-transition matrix for satellite and LEO, which transfer the satellites state from reference epoch \(t_{0}\) to epoch \(t\). \(O_{0}^{s}\) and \(O_{l\omega,0}\) are the initial state vector for GNSS and LEO satellites respectively, which consist of the position and velocity of the satellite at the initial epoch and dynamics parameters. For GNSS satellites, the estimated dynamics parameters mainly refer to the solar radiation pressure (SRP) parameters, while for LEO satellites, these dynamics parameters are usually made up of SRP parameters, atmosphere drag parameters, and empirical accelerations. \(M_{g}^{s}\) is the mapping function for tropospheric delay, and \(Z_{g}\) denotes the zenith delay of the tropospheric wet component. \(\overline{N}_{g,\mathrm{IF}}^{s}\) and \(\overline{N}_{l\omega,\mathrm{IF}}^{s}\) represent the float ambiguity for ground station and LEO satellites respectively. The other symbols have similar meanings as those described in Equation (2), except for the IF linear combination. In the case of the multi-GNSS integrated POD, the estimated parameters can be divided into satellite-dependent parameters, ground station-dependent parameters, and LEO-dependent parameters, which can be expressed as follows:  
\[X=\left[X^{s},X_{g},X_{l\omega}\right]^{T} \tag{6}\]  
with  
\[\left\{