Fractal behavior of BDS-2 satellite clock offsets and its application to real-time clock offsets prediction

The satellite clocks of the BeiDou Navigation Satellite System (BDS) have characteristic differences compared to those of the global positioning system and Galileo. Therefore, the satellite clock offsets prediction method of BDS is different from those of these two systems. The basis for establishing a more appropriate prediction model is to clarify the statistical characteristics of the BDS satellite clock offsets, which can be reflected by fitting residuals of precise clock errors. Fractal behavior is generally not considered in existing studies. In this study, the rescaled range analysis method is improved by using the moving-average method in order to verify the fractal behavior of the BDS clock offsets. The computation results of the Hurst exponent show that the BDS clock offsets are fractal series with long-term memory, and the memory spans are obtained by V-statistic calculation. The quadratic polynomial fitting residuals of BDS clock offsets are fitted by using the periodic model and fractal interpolation model, where the latter approach has a higher accuracy. In the predictive modeling process, the quadratic polynomial forecasting model is improved by using the memory span, so that the forecasting model not only inherits the overall clock offsets trend but also considers the local memory trend. The fractal interpolation prediction model of the clock offsets is established by using the extension method of the affine iteration function system. The experimental results show that the average prediction accuracy of the fractal model in 3, 6, 12, and 24 h can reach 1.4890, 2.0222, 3.1609, and 4.9278 ns, which is 57.74%, 50.20%, 52.66%, and 49.42% higher than the products from the China iGMAS ultra-rapid prediction, respectively.