Article Improved Short-Term Clock Prediction Method for Real-Time Positioning Yifei Lv, Zhiqiang Dai *, Qile Zhao *, Sheng Yang, Jinning Zhou and Jingnan Liu GNSS Research Center, Wuhan University, 129 Luoyu Road, Wuhan 430079, China; [email protected] (Y.L.); [email protected] (S.Y.); [email protected] (J.Z.); [email protected] (J.L.) * Correspondence: [email protected] (Z.D.); [email protected] (Q.Z.); Tel.: +86-181-6410-1559 (Z.D.); +86-186-2770-0081 (Q.Z.) Academic Editor: Fabio Dell’Acqua Received: 5 April 2017; Accepted: 1 June 2017; Published: 6 June 2017 Abstract: The application of real-time precise point positioning (PPP) requires real-time precise orbit and clock products that should be predicted within a short time to compensate for the communication delay or data gap. Unlike orbit correction, clock correction is difficult to model and predict. The widely used linear model hardly fits long periodic trends with a small data set and exhibits significant accuracy degradation in real-time prediction when a large data set is used. This study proposes a new prediction model for maintaining short-term satellite clocks to meet the high-precision requirements of real-time clocks and provide clock extrapolation without interrupting the real-time data stream. Fast Fourier transform (FFT) is used to analyze the linear prediction residuals of real-time clocks. The periodic terms obtained through FFT are adopted in the sliding window prediction to achieve a significant improvement in short-term prediction accuracy. This study also analyzes and compares the accuracy of short-term forecasts (less than 3 h) by using different length observations. Experimental results obtained from International GNSS Service (IGS) final products and our own real-time clocks show that the 3-h prediction accuracy is better than 0.85 ns. The new model can replace IGS ultra-rapid products in the application of real-time PPP. It is also found that there is a positive correlation between the prediction accuracy and the short-term stability of on-board clocks. Compared with the accuracy of the traditional linear model, the accuracy of the static PPP using the new model of the 2-h prediction clock in N, E, and U directions is improved by about 50%. Furthermore, the static PPP accuracy of 2-h clock products is better than 0.1 m. When an interruption occurs in the real-time model, the accuracy of the kinematic PPP solution using 1-h clock prediction product is better than 0.2 m, without significant accuracy degradation. This model is of practical significance because it solves the problems of interruption and delay in data broadcast in real-time clock estimation and can meet the requirements of real-time PPP. Keywords: real-time precise point positioning; real-time clock estimation; short-term prediction 1. Introduction Precise point positioning (PPP) has been vigorously developed since its introduction. High-precision orbit and clock products are required by PPP technology [1]. The Global Navigation Satellite System (GNSS) ranging measurement in substance is equal to the time measurement between the satellite clock and receiver clock and usually, the users fix satellite clock corrections and estimate receiver clock corrections to eliminate the error of the clock. In short, the accuracy of the satellite clock is directly related to the accuracy of PPP. However, a 12-day to 18-day delay occurs before International GNSS Service (IGS) final orbit and clock products are released, and even IGS rapid products are released after 17 h to 41 h [2,3]. Sensors 2017, 17, 1308; doi:10.3390/s17061308 www.mdpi.com/journal/sensors Sensors 2017, 17, 1308 2 of 14 Therefore, such products can only be utilized by users in post PPP applications. Real-time PPP has gradually become a research focus in satellite navigation and positioning, but IGS final and rapid products cannot meet real-time demands. IGS also provides ultra-rapid products (IGU-P). The precision of IGU-predicted orbit products is about 5 cm, which is equivalent to that of IGS final products and satisfies post PPP applications. By contrast, the official accuracy of IGU-P clock products is only 3 ns. Besides, the sampling rate of the clock has a significant impact on the PPP solution in kinematic PPP [4]. Apparently, 15-min interval IGU-P clocks cannot be used for high- precision PPP. To promote real-time PPP application, IGS set up a real-time working group in 2002 to start the prototype research, and the IGS Real Time Service (RTS) was officially launched in 2013 to provide service for real-time applications [2]. The accuracy of real-time clocks is about 0.3 ns, the broadcast delay is 25 s, and the sampling interval is 5 s [5]. This can be regarded as a reference for real-time clocks estimation. Apart from the integrated products provided by IGS, the analysis center of IGS also provides users their own products. Nevertheless, several problems are encountered in real-time clock applications. First, a delay occurs in real-time observation receiving and product broadcasting. Second, the processing of real-time clock correction estimation software results in a delay. Third, the data stream often loses information or encounters interruption. Table 1 shows the loss rate of real-time clock products in each analysis center between days 349 and 358 in 2015 (except for day 357). The sampling interval is 5 s. In the most ideal case, nearly 10% of the epoch is lost, and this loss seriously affects the accuracy and stability of real-time PPP. Table 1. Loss rate of real-time clock products in each analysis center. Federal Agency Natural Centre National European Space GeoForschungsZ for Cartography Resources d’Etudes Day Agency (ESA) entrum (GFZ) IGS (%) and Geodesy Canada (NRCan) Spatiales (CNES) (%) Potsdam (%) (BKG) (%) (%) (%) 349 43.7 37.9 53.3 44.8 50.8 49.1 350 23.3 23.1 14.2 28.0 35.1 28.3 351 37.5 28.4 30.7 44.4 51.8 33.2 352 10.3 6.7 7.9 6.0 7.5 6.9 353 7.9 4.8 5.5 4.6 5.7 5.5 354 17.3 15.5 16.1 63.4 15.4 15.9 355 13.8 13.2 14.1 47.9 17.0 15.2 356 20.9 12.3 24.2 81.8 21.0 10.5 358 15.6 15.8 17.5 3.3 15.7 15.8 The current Global Positioning System (GPS) constellation can be grouped into three types. Block IIR and Block IIR-M satellites carry rubidium clocks, and the newest generation of Block IIF is equipped with an improving cesium and rubidium atomic frequency standard (AFS). Previous studies have shown that AFS operated on Block IIF is more stable than other types [6,7], and rubidium AFS is more stable than cesium AFS [7,8]. The type of satellite and atomic clock on 1 May 2016 is shown in Table 2. Moreover, the behavior of GPS on-board clocks has been fully discussed by researchers. Periodic signals related to orbital dynamics have been detected in all clock types [9]. Knowing these variations in the satellite clock is important for improving the prediction model of clock. Table 2. The type of satellite and atomic clock on 1 May 2016. Satellite Type Clock Type Pseudo Random Noise (PRN) Block IIR Rb 2, 11, 13, 14, 16, 18, 19, 20, 21, 22, 23, 28 Block IIR-M Rb 5, 7, 12, 15, 17, 29, 31 Rb 1, 3, 6, 9, 10, 25, 26, 27, 30, 32 Block IIF Cs 8, 24 Most studies and models focused on medium- and long-term forecast analyses, such as one day or several days, the purpose of which is to provide products similar to IGU ultra-fast products to Sensors 2017, 17, 1308 3 of 14 realize real-time PPP with ultra-fast orbit products. This case is obviously different from the short-term clock prediction required for real-time clock difference calculation. Therefore, the introduction of short-term real-time clock correction prediction is important to guarantee the broadcast stability of real-time clock correction and improve real-time PPP accuracy. Many methods, such as polynomial, gray, autoregressive integrated moving average (ARIMA), and neural network models, can be used for clock error prediction [10–12]. The linear or quadratic polynomial model with periodic terms is used as the model of IGU-P products. Huang corrected the original IGU-P clock model, and the accuracy of the improved model is better than that of IGU-P [13]. Several studies focused on short-term real-time clock correction prediction. Lou proposed a polynomial model with periodic items and clock jumps [14]. Song used a polynomial model with periodic items to model the single-difference sequence of clock correction, which was obtained by conducting first-order difference processing on adjacent clock data. The accuracy of this method using 300 s to predict 60 s is better than 0.3 ns [15], which is insufficient for real-time PPP. This study focuses on short-term clock correction prediction and proposes a new model to improve prediction precision. The prediction residuals between linear sliding forecast and real-time clock product are analyzed and modeled through fast Fourier transform (FFT) [16]. Periodic signals can be eliminated by applying the additional residual model in the subsequent sliding forecast. The experimental results show that with the proposed model, the static PPP achieves precision better than 0.1 m and the kinematic PPP achieves precision better than 0.2 m.