# A Parallel Approach for Multi-GNSS Ultra-Rapid Orbit Determination
## 2 Methodology
### Multi-Arc Parallel Normal Equations Construction Based on MPI
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Item** & **Description** & **Remark** \\ \hline Observations & Double-differenced phase and & External a \\ Satellite system & PS, GLONASS, BDS, Galileo & External \\ Sampling rate & 30 s & External \\ Elevation cut-off angle & 3\({}^{\circ}\) & External \\ Ionosphere model & Ionosphere-free combination with & \\ Ionosphere model & higher-order-ionosphere correction & External \\  & GMF [34] with 2-h resolution of & \\ Tropospheric model & station-specific troposphere parameters, & Estimated b \\  & gradients [35] with daily resolution & \\ PCO/PCV & IGS14.txt & External \\ A priori orbit & MGEX broadcast ephemeris & External \\ Stations coordinates & IGS SINEX products & External \\ Orbital model & 3 position and 3 velocity coordinates at & \\  & the start of the orbital arc, without & Estimated \\ Solar Radiation Parameter (SRP) & pseudo-stochastic pulses & \\ model & ECOM1 model with five parameters [36] & External \\  & xp, yp, xp-rate, yp-rate and LOD are & \\ ERP & estimated with tight constraint of & Estimated \\  & UT1-UTC [37] & \\ Gravity field model & EGM2008, 12 \(\times\) 12 degree [38] & External \\ N-body perturbation & JPL DE405 ephemeris & External \\ Solid earth tides, pole & IERS Conventions 2010 [39] & External \\ tides/Semidiurnal tide & & \\ Ocean Tidal model & FES2004 [40] & External \\ Precession and nutation models & IAU2006/IAU2000A & External \\ Relativistic effects & IERS Conventions 2010 & External \\ \hline \hline \end{tabular} a The remark of External means the data source from models or products provided by other agencies; b The remark of Estimated means the data source generated by our software.  
\end{table} 
Table 2: Multi-GNSS ultra-rapid orbit determination strategies. Equation (2) is the basic observation equation for ultra-rapid orbit determination. In order to determine the ultra-rapid orbit, the traditional sequential batch processing method traverses the observation data in all arcs by epoch to construct the normal equations. In this study, a parallel method based on MPI is used in this process. The main idea is to establish the multi-arc normal equation in the orbit determination by adopting the multi-processing method. Assuming that the single-day data is divided into three arcs, the length of each arc is 8 h. Each sub-process processes the measured data of the current arc and then calculates the observation equation coefficients. Based on the current time period, sub-normal equations including initial satellite state, force model parameters, earth rotation parameters, atmospheric parameters, station coordinates and ambiguity parameters are generated. Each process sends the sub-normal equation matrices and auxiliary parameters to the main process to obtain the total normal equations. The parameter sequence of each period is consistent with that of the entire original period. The mathematical model and the parameters have not changed, so there is no additional conversion and connection. Supposing that the entire arc is divided into k sub-arcs, the least squares problem can be solved by accumulating the normal equations. The sub-normal equations and the entire normal equation can be expressed as:  
\[\left\{ \begin{array}{l}N_{1}\delta x=W_{1}\\ N_{2}\delta x=W_{2}\\ \cdots\\ N_{k}\delta x=W_{k}\end{array} \right. \tag{3}\]  
\[(N_{1}+N_{2}+\cdots+N_{k})\delta x=(W_{1}+W_{2}+\cdots+W_{k}) \tag{4}\]  
where \(N_{k}\) and \(W_{k}\) are the related matrix of sub-normal equations and \(\delta x\) is unknown parameter vector. Based on this entire normal equation, the elimination of parameters and least squares (LS) estimation can be then carried out. It is worth noting that the number of time periods are designed according to the computer hardware and time requirements.