## 2 Materials and Methods  
### The Multi-GNSS Observation Model of the New Prototype Software Platform  
In this paper, during the processing of the ultra-rapid POD solution, dual-frequency ionosphere-free (IF) combined observations were used to eliminate the first order of ionospheric delays. [PERSON] et al. showed that the impact of higher order ionosphere delays can reach several millimeters for the radial, along-track, and cross-track orbit components [29]. Considering that this paper only plans to get a preliminary result for five-system ultra-rapid orbits, the impact of higher order ionosphere delays was not considered. The observation equation of the IF combined observations can be linearized and expressed as follows:  
\[ \begin{array}{l}l_{r,IF}^{s}=\mu_{r}^{s}\cdot\left(\Phi(t,t_{0})^{s}\cdot \cdot\cdot\cdot_{0}^{s}-R_{r,rep}\cdot r_{r}\right)-\Delta t_{IF}^{s}+\Delta t _{r,IF}+\partial_{r}^{s}\cdot\delta_{err}\\ +\lambda_{IF}\left(b_{r,IF}-b_{IF}^{s}+N_{r,IF}^{s}\right)+m_{r,trop}^{s}\cdot \delta_{r,trop}+\varepsilon_{r,IF}^{s}\\ \\ p_{r,IF}^{s}=\mu_{r}^{s}\cdot\left(\Phi(t,t_{0})^{s}\cdot\cdot_{0}^{s}-R_{r,rep }\cdot r_{r}\right)-\Delta t_{IF}^{s}+\Delta t_{r,IF}+\partial_{r}^{s}\cdot \delta_{err}\\ \\ +c\left(d_{r,IF}-d_{IF}^{s}\right)+m_{r,trop}^{s}\cdot\delta_{r,trop}+\omega_{r, IF}^{s}\end{array}  \tag{1}\]  
where  
\[\partial_{o}^{s}=\left(a,e,i,\Omega,\omega,u_{0},\mathbb{P}_{1\sim 9}^{s} \right)^{T} \tag{3}\]  
\[\delta_{err}=\left(x_{0},y_{0},\theta_{0}\right)^{T} \tag{4}\]  
in addition, \(l_{r,IF}^{s}\) and \(p_{r,IF}^{s}\) denote the \"observed minus computed\" IF phase and pseudo-range observables from satellite \(s\) to receiver \(r\), respectively; \(\mu_{r}^{s}\) represents the unit vector in the direction from receiver \(r\) to satellite \(s\); and \(\Phi(t,t_{0})^{s}\) is the state transition matrix from the initial epoch \(t_{0}\) to current epoch \(t\). \(\sigma_{o}^{s}\) denotes the initial orbit state vector of satellite \(s\), including the semimajor axis of the orbit \(a\), the eccentricity of the orbit \(e\), the inclination of the orbital plane with respect to the equatorial plane \(i\), the right ascension of the ascending node \(\Omega\), the argument of perigee \(\omega\), the argument of latitude \(u_{0}\), and the solar radiation pressure (SRP) parameters of the empirical CODE orbit model (ECOM) developed by the Astronomical Institute, University of Bern \(\left(\text{P}_{1\sim 9}^{\text{s}}\right)\)[30; 31]. \(R_{r,rep}\) denotes the satellite coordinates rotation matrix from the Earth-fixed frame to the inertial frame, according to the Earth rotation parameters (ERP); \(r_{r}\) is the receiver position increments relative to the a priori position; \(\Delta t_{IF}^{\text{s}}\) is the IF combined clock bias of satellite \(s\); \(\Delta t_{r,IF}\) is the IF combined clock bias of receiver \(r\); \(\partial_{r}^{s}\) is the partial derivative of the IF combined observation to ERP; \(\delta_{rep}\) is the ERP, including polar motion \((x_{0},y_{0})\) and length of day change \(\theta_{0}\); \(c\) is the speed of light in vacuo; \(\lambda_{IF}\) is the wave-length of the IF combination; \(b_{r,IF}\) and \(b_{IF}^{\text{s}}\) are the receiver \(r\) and satellite \(s\) phase delays of IF combination, respectively; \(N_{r,IF}^{\text{s}}\) is the ambiguity of the IF combination; \(d_{r,IF}\) and \(d_{IF}^{\text{s}}\) are the receiver \(r\) and satellite \(s\) code biases of the IF combination; \(m_{r,trop}^{s}\) is a related mapping function with respect to the zenith tropospheric wet delays \(\delta_{r,trop}\); and \(\epsilon_{r,IF}^{s}\) and \(\omega_{r,IF}^{s}\) are the sum of measurement noise and multipath effects for the IF combined phase and pseudo-range observations, respectively. In the same navigation satellite system, the observation equation of the linearized double-difference IF combination observation is as follows:  
\[