# A Parallel Approach for Multi-GNSS Ultra-Rapid Orbit Determination
## 1 Introduction
In order to obtain accurate solutions and achieve higher computational efficiency, many scholars have optimized the theories and methods of orbit determination by dividing subnets, grouping combined systems, dividing arc length of observation, and reducing observations. [PERSON] (1999), for the first time, proposed the method of merging the past three consecutive single-day arcs to orbit determination in 1999 [26]. [PERSON] (2008) further used a dynamic smoothing algorithm to smooth the orbit which can improve the accuracy and reliability of the comprehensive algorithm solution. In order to improve the efficiency of ultra-rapid orbit determination [27], [PERSON] (2008) deeply studied the problem of ambiguity processing in real-time orbit determination and proposed a sliding window method [28]. Through the superposition of short-arc normal equations, the orbit can be calculated with higher efficiency, and the accuracy is improved when the ambiguities are fixed. In addition, TUM used a two-step approach for generating Galileo and QZSS products [29]. Furthermore, a three-step approach was applied by WHU to orbit product calculation: GPS and GLONASS orbit and ERPs are solved first, then station coordinates, zenith tropospheric delay (ZTD) and receiver clock are estimated, and the orbit of other satellites are solved in the last step. Then, a block recursive least squares method was proposed to improve \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Type** & **Accuracy** & **Latency** & **Updates** & **Sample Interval** \\ \hline Broadcast & orbits & \(\sim\)100 cm & real-time & - & daily \\ Ultra-rapid & orbits & \(\sim\)5 cm & real-time & 4 times & 15 min \\ Ultra-rapid & orbits & \(\sim\)3 cm & 3–9 h & 4 times & 15 min \\ Rapid & orbits & \(\sim\)2.5 cm & 17–41 h & daily & 15 min \\ Final & orbits & \(\sim\)2.5 cm & 12–18 days & weekly & 15 min \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Accuracy and latency of each IGS orbit products. the accuracy of GNSS ultra-rapid orbits. The update frequency of ultra-rapid orbits was increased from 6-h to 1-h, and the accuracy was significantly improved [30]. [PERSON] (2019) realized 1-h update multi-GNSS ultra-rapid orbit determination by multi-threading and changed observation intervals [31]. The accuracy of predicted orbit was improved and the kinematic PPP solutions could achieve a shorter time to first fix and higher positioning accuracy. To promote the accuracy of the predicted multi-GNSS orbit, we proposed a new approach for multi-GNSS hourly ultra-rapid orbit determination. This approach improves the efficiency of multi-GNSS ultra-rapid orbit solution without changing the original observation data or processing interval, and retains the continuity and consistency of the original parameters to be estimated using the multi-process and multi-thread method based on Message Passing Interface (MPI)/Open Multi Processing (OpenMP). MPI is a message passing library specification which provides a powerful and portable way for expressing parallel programs. OpenMP is a set of compiler directives as well as an API for programs that provides support for parallel programming in shared-memory environments. Therefore, MPI/OpenMP can achieve higher accuracy for multi-GNSS ultra-rapid orbit as the initial orbit is precise, but the length of extrapolation time is shortened significantly. To realize that, first and foremost, the construction of the normal equation for initial value of satellite orbit is divided into multi-arc parallel processing by the multi-process method based on MPI, and each process achieves consistency through processing the arc data and interacting matrix. In addition, the multi-thread method is used in parameter elimination and orbit extrapolation to further improve the efficiency. In this approach, the GPS+GLONASS+Galileo+BDS observations are processed together in one common procedure with the double-differenced network model. The promotion of computational efficiency, accuracy and stability is validated based on the data from MGEX and IGS stations.