## 6 Conclusions  
This work proposed an LEO-SIN orbit extrapolation method based on MSEC and MS-FARIMA, focusing on solving the problems of low accuracy and low timelessness in existing ephemeris prediction algorithms.  
Figure 13: DFO estimation errors based on the proposed method: (**a**) UE position 1, (**b**) UE position 2, and (**c**) UE position 3.  
This method decomposed the extrapolation error based on the satellite ephemeris into three categories: RE, TE, and PE. The orbit extrapolation scheme that combines MSEC and MS-FARIMA achieved the elimination of the RE and TE.  
This method reduced the ephemeris extrapolation error by 33.5% compared to existing methods, which also contributes to the enhanced performance of DFO estimation in MPDTSC. Future work could incorporate artificial intelligence techniques such as deep learning to further improve the adaptability and accuracy of the orbit extrapolation model.  
This work proposed an LEO-SIN orbit extrapolation method based on MSEC and MS-FARIMA. The conclusions are as follows:  
* We are the first to propose separating the broadcast ephemeris extrapolation errors into three types: RE, TE, and PE. We focused on improving the extrapolation accuracy by diminishing and compensating for the RE and TE. We also derived the lower bound of the orbit extrapolation error based on single-satellite ephemeris information;
* This work first proposed a novel and accurate ephemeris extrapolation framework that involves the ephemerides of multi-satellite and multi-stream data TSFs;
* An MSEC orbit extrapolation error elimination algorithm that analyzes the spatial and temporal associations between corresponding and different satellites at different times was designed. This algorithm utilized multiple sets of ephemeris information from MSEC to diminish the RE. The focus was primarily on improving the extrapolation accuracy by addressing the RE and TE;
* This work is the first to design an MS-FARIMA algorithm, which supports data from the same attribute information at different times for different objects. The reasons for the improved accuracy of the ephemeris extrapolation error based on MSEC and multi-stream information were also explored;
* The orbit extrapolation scheme that combined MSEC and MS-FARIMA achieved higher accuracy than the single-satellite, single-stream scheme;
* This work proposed a DFO estimation and compensation algorithm for the LEO-SIN based on the employment of the above-optimized ephemeris. It theoretically derives the impacts of different ephemeris errors on DFO estimation, enhancing the DFO estimation accuracy.  
Conceptualization, W.L., K.W. and Z.H.; methodology, W.L., J.Y. and Z.H.; validation, W.L., J.Y. and K.W.; formal analysis, [PERSON], [PERSON] and Z.H.; investigation, [PERSON], J.Y. and Y.L. ([PERSON]); resources, [PERSON], [PERSON] ([PERSON]) and H.K.; writing--original draft preparation, [PERSON], J.Y. and Z.H.; writing--review and editing, [PERSON] and J.Y.; supervision, [PERSON], T.W., Z.D. and Z.L.; project administration, [PERSON], Z.D. and J.Z.; funding acquisition, [PERSON], K.W. and Z.D. All authors have read and agreed to the published version of the manuscript.  
This work was supported by the Natural Science Foundation of Chongqing Province (CSTB2023 NSCQLMX0008) and the National Key R&D Program of China (2022 YFB2902605).  
The data are contained within the article.  
Author [PERSON] was employed by the company China Satellite Network Exploration Co., Ltd. The remaining authors declare that the re-search was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.