Endpoint-based optimal fractal interpolation for predicting BDS-3 system time offsets

Abstract

In order to improve the interoperability within the Global Navigation Satellite System (GNSS), the International Committee on Global Navigation Satellite Systems published a joint statement in December 2019 that stated that all GNSS providers agree to monitor and broadcast the time offsets between each system timescale and the Universal Time Coordinated (UTC) or the rapid realization of UTC (UTCr). This commitment requires the study of precise prediction models for system time offsets. The prediction model of system time offsets is different from that of the atomic clock because of the control of the system timescale. The offsets between the system time of the Beidou Satellite Navigation System-3 (BDS-3) and the National Time Service Center (NTSC), called [UTC(NTSC)-BDT], have two main periods of 12 h and 24 h, according to the Fast Fourier Transform analysis. The rescaled range (R/S) analysis demonstrates that it has long memory, making it a fractal time series with a memory period of about 10.4 h. While using the fractal interpolation method to predict the [UTC(NTSC)-BDT] series, we found that the prediction error reaches its minimum value if adding disturbance on the estimated endpoint of the forecasted interval. After verifying the correlation between the estimated endpoints with the minimal interpolation error and minimal prediction error and proving the existence and uniqueness of the estimated endpoint with the minimal interpolation error, we established the endpoint-based optimal fractal interpolation prediction method. The experimental results indicate that the average prediction accuracy of the proposed prediction model is improved by 57.90% and 39.26% compared to that of a quadratic model and standard fractal prediction model, respectively. The accuracy analysis results of numerical tests indicate that the proposed prediction model can restrain the divergence of prediction error. Finally, we transform the [UTC(NTSC)-BDT] into [UTCr -BDT] using the [UTCr-UTC(NTSC)] published by the Bureau International des Poids et Mesures (BIPM) to meet the requirement of GNSS interoperability. The prediction accuracies of daily [UTCr-BDT] using the proposed prediction model are no more than 1.5 ns with uncertainty about 6 ns.

Appendix: Specific calculation process of fractal interpolation bias

The main purpose of this appendix is to provide the specific computing process of parameter $\beta_{j}$. This will help us ensure the existence and uniqueness of the estimated endpoint disturbance for minimal interpolation error.

First, we will simplify the parameters defined by equation (16). Since the interpolation series $\{ t_{i} \}$ ($i = 0,1, \cdots ,N$) is always an arithmetic progression, we have:

$$\left\{ \begin{gathered} a_{i} = \frac{{t_{i} - t_{i - 1} }}{{t_{N} - t_{0} }} = \frac{1}{N} \hfill \\ a_{N + 1} = \frac{{t_{N + 1} - t_{N} }}{{t_{N + 1} - t_{0} }} = \frac{1}{N + 1} \hfill \\ \end{gathered} \right.$$

(40)

Then, the sums of $e_{i}$, $c_{i}$ and $f_{i}$ are calculated as follows:

$$\begin{aligned} & \sum\limits_{i = 1}^{N} {e_{i} } = \sum\limits_{i = 1}^{N} {\frac{{t_{N} t_{i - 1} - t_{0} t_{i} }}{{t_{N} - t_{0} }}} = \frac{1}{{t_{N} - t_{0} }}\sum\limits_{i = 1}^{N} {\left( {t_{N} t_{i - 1} - t_{0} t_{i} } \right)} \\ & \quad = \frac{1}{{t_{N} - t_{0} }}\left( {\sum\limits_{i = 1}^{N} {t_{N} t_{i - 1} } - \sum\limits_{i = 1}^{N} {t_{0} t_{i} } } \right) = \frac{1}{{t_{N} - t_{0} }}\left( {t_{N} \sum\limits_{i = 1}^{N} {t_{i - 1} } - t_{0} \sum\limits_{i = 1}^{N} {t_{i} } } \right) \\ & \quad = \frac{1}{{t_{N} - t_{0} }}\left[ {t_{N} \left( {\sum\limits_{i = 0}^{N} {t_{i} } - t_{N} } \right) - t_{0} \left( {\sum\limits_{i = 0}^{N} {t_{i} } - t_{0} } \right)} \right] \\ & \quad = \frac{1}{{t_{N} - t_{0} }}\left[ {\left( {t_{N} - t_{0} } \right)\sum\limits_{i = 0}^{N} {t_{i} } + t_{0}^{2} - t_{N}^{2} } \right] = \sum\limits_{i = 0}^{N} {t_{i} } - \left( {t_{N} + t_{0} } \right) \\ & \quad = \left( {t_{N} + t_{0} } \right)\frac{N + 1}{2} - \left( {t_{N} + t_{0} } \right) = \left( {t_{N} + t_{0} } \right)\frac{N - 1}{2} \\ \end{aligned}$$

(41)

$$\begin{aligned} \sum\limits_{i = 1}^{N} {c_{i} } & = \sum\limits_{i = 1}^{N} {\frac{{x_{i} - x_{i - 1} - \mu_{i} (x_{N} - x_{0} )}}{{t_{N} - t_{0} }}} = \sum\limits_{i = 1}^{N} {\frac{{x_{i} - x_{i - 1} }}{{t_{N} - t_{0} }}} - \frac{{(x_{N} - x_{0} )}}{{t_{N} - t_{0} }}\sum\limits_{i = 1}^{N} {\mu_{i} } \\ & = \frac{{x_{N} - x_{0} }}{{t_{N} - t_{0} }} - \frac{{x_{N} - x_{0} }}{{t_{N} - t_{0} }}\sum\limits_{i = 1}^{N} {\mu_{i} } = \frac{{x_{N} - x_{0} }}{{t_{N} - t_{0} }}\left( {1 - \sum\limits_{i = 1}^{N} {\mu_{i} } } \right) \\ \end{aligned}$$

(42)

$$\begin{aligned} & \sum\limits_{i = 1}^{N} {f_{i} } = \sum\limits_{i = 1}^{N} {\frac{{t_{N} x_{i - 1} - t_{0} x_{i} - \mu_{i} (t_{N} x_{0} - t_{0} x_{N} )}}{{t_{N} - t_{0} }}} \\ & \quad = \frac{1}{{t_{N} - t_{0} }}\left( {t_{N} \sum\limits_{i = 1}^{N} {x_{i - 1} } - t_{0} \sum\limits_{i = 1}^{N} {x_{i} } - (t_{N} x_{0} - t_{0} x_{N} )\sum\limits_{i = 1}^{N} {\mu_{i} } } \right) \\ & \quad = \frac{1}{{t_{N} - t_{0} }}\left( {t_{N} \sum\limits_{i = 0}^{N} {x_{i} } - t_{N} x_{N} - t_{0} \sum\limits_{i = 0}^{N} {x_{i} } + t_{0} x_{0} - (t_{N} x_{0} - t_{0} x_{N} )\sum\limits_{i = 1}^{N} {\mu_{i} } } \right) \\ & \quad = \sum\limits_{i = 0}^{N} {x_{i} } + \frac{1}{{t_{N} - t_{0} }}\left( {t_{0} x_{0} - t_{N} x_{N} - (t_{N} x_{0} - t_{0} x_{N} )\sum\limits_{i = 1}^{N} {\mu_{i} } } \right) \\ & \quad = \sum\limits_{i = 0}^{N} {x_{i} } - \left( {\frac{{x_{N} \left( {t_{N} - t_{0} } \right) + t_{0} \left( {x_{N} - x_{0} } \right)}}{{t_{N} - t_{0} }} + \frac{{x_{0} \left( {t_{N} - t_{0} } \right) - t_{0} \left( {x_{N} - x_{0} } \right)}}{{t_{N} - t_{0} }}\sum\limits_{i = 1}^{N} {\mu_{i} } } \right) \\ & \quad = \sum\limits_{i = 0}^{N} {x_{i} } - x_{N} - t_{0} \frac{{x_{N} - x_{0} }}{{t_{N} - t_{0} }} - \left( {x_{0} - t_{0} \frac{{x_{N} - x_{0} }}{{t_{N} - t_{0} }}} \right)\sum\limits_{i = 1}^{N} {\mu_{i} } \\ & \quad = \sum\limits_{i = 0}^{N} {x_{i} } - x_{N} + x_{0} - x_{0} - t_{0} \frac{{x_{N} - x_{0} }}{{t_{N} - t_{0} }} - \left( {x_{0} - t_{0} \frac{{x_{N} - x_{0} }}{{t_{N} - t_{0} }}} \right)\sum\limits_{i = 1}^{N} {\mu_{i} } \\ & \quad = \sum\limits_{i = 0}^{N} {x_{i} } + \left( {x_{0} - t_{0} \frac{{x_{N} - x_{0} }}{{t_{N} - t_{0} }}} \right)\left( {1 - \sum\limits_{i = 1}^{N} {\mu_{i} } } \right) - x_{N} - x_{0} \\ & \quad = \sum\limits_{i = 1}^{N - 1} {x_{i} } + \left( {\frac{{t_{N} x_{0} - t_{0} x_{N} }}{{t_{N} - t_{0} }}} \right)\left( {1 - \sum\limits_{i = 1}^{N} {\mu_{i} } } \right) \\ \end{aligned}$$

(43)

For the parameter $e_{N + 1}$, we have:

$$\begin{aligned} e_{N + 1} & = \frac{{t_{N + 1} t_{N} - t_{0} t_{N + 1} }}{{t_{N + 1} - t_{0} }} = \frac{{t_{N + 1} t_{N} - t_{N} t_{0} + t_{N} t_{0} - t_{0} t_{N + 1} }}{{t_{N + 1} - t_{0} }} \\ & = \frac{{t_{N} \left( {t_{N + 1} - t_{0} } \right) + t_{0} \left( {t_{N} - t_{N + 1} } \right)}}{{t_{N + 1} - t_{0} }} = t_{N} - \frac{{t_{0} }}{N + 1} \\ \end{aligned}$$

(44)

According to equation (31), the expression of $\beta_{j}$ contains two parts:

$$\begin{aligned} \beta_{j} & = \underbrace {{\mu_{j} \left( {(1 - \lambda )(x_{N} - \mu_{N + 1} x_{0} ) + \mu_{N + 1} x^{*} } \right)}}_{{{\text{Part}}\;{\text{I}}}} \\ & \quad { + }\underbrace {{\phi_{j} \left( {L_{N + 1} (t^{*} )} \right) - \frac{1}{N}\sum\limits_{i = 1}^{N} {F_{j} \left( {L_{i} (t^{*} ),F_{i} (t^{*} ,x^{*} )} \right)} }}_{{{\text{Part}}{\kern 1pt} {\text{II}}}} \\ \end{aligned}$$

(45)

Since Part I of $\beta_{j}$ is already simplified, we will focus on the last part. According to (15):

$$\begin{aligned} & \phi_{j} \left( {L_{N + 1} (t^{*} )} \right) - \frac{1}{N}\sum\limits_{i = 1}^{N} {F_{j} \left( {L_{i} (t^{*} ),F_{i} (t^{*} ,x^{*} )} \right)} \\ & \quad = \phi_{j} \left( {L_{N + 1} (t^{*} )} \right) - \frac{1}{N}\sum\limits_{i = 1}^{N} {\left( {\phi_{j} \left( {L_{i} (t^{*} )} \right) + \mu_{j} F_{i} (t^{*} ,x^{*} )} \right)} \\ & \quad = \frac{1}{N}\sum\limits_{i = 1}^{N} {\phi_{j} \left( {L_{N + 1} (t^{*} )} \right)} - \frac{1}{N}\sum\limits_{i = 1}^{N} {\phi_{j} \left( {L_{i} (t^{*} )} \right)} - \frac{1}{N}\sum\limits_{i = 1}^{N} {\mu_{j} F_{i} (t^{*} ,x^{*} )} \\ & \quad = \frac{1}{N}\sum\limits_{i = 1}^{N} {\left[ {\phi_{j} \left( {L_{N + 1} (t^{*} )} \right) - \phi_{j} \left( {L_{i} (t^{*} )} \right)} \right]} - \frac{{\mu_{j} }}{N}\sum\limits_{i = 1}^{N} {F_{i} (t^{*} ,x^{*} )} \\ \end{aligned}$$

(46)

For the first segment of (46), using (40), (41) and (44), we obtain:

$$\begin{aligned} & \frac{1}{N}\sum\limits_{i = 1}^{N} {\left[ {\phi_{j} \left( {L_{N + 1} (t^{*} )} \right) - \phi_{j} \left( {L_{i} (t^{*} )} \right)} \right]} \\ & \quad = \frac{1}{N}\sum\limits_{i = 1}^{N} {\left( {c_{j} L_{N + 1} (t^{*} ) + f_{j} - c_{j} L_{i} (t^{*} ) - f_{j} } \right)} = \frac{{c_{j} }}{N}\sum\limits_{i = 1}^{N} {\left( {L_{N + 1} (t^{*} ) - L_{i} (t^{*} )} \right)} \\ & \quad = \frac{{c_{j} }}{N}\sum\limits_{i = 1}^{N} {\left( {a_{N + 1} t^{*} + e_{N + 1} - a_{i} t^{*} - e_{i} } \right)} \\ & \quad = \frac{{c_{j} }}{N}\left[ {t^{*} \left( {\sum\limits_{i = 1}^{N} {a_{N + 1} } - \sum\limits_{i = 1}^{N} {a_{i} } } \right) + \sum\limits_{i = 1}^{N} {e_{N + 1} } - \sum\limits_{i = 1}^{N} {e_{i} } } \right] \\ & \quad = \frac{{c_{j} }}{N}\left[ {t^{*} \left( {\frac{N}{N + 1} - 1} \right) + N\left( {t_{N} - \frac{{t_{0} }}{N + 1}} \right) - \left( {t_{N} + t_{0} } \right)\frac{{\left( {N - 1} \right)}}{2}} \right] \\ & \quad = \frac{{c_{j} }}{N}\left( { - \frac{{t^{*} }}{N + 1} + Nt_{N} - \frac{{Nt_{0} }}{N + 1} - \frac{{\left( {N - 1} \right)t_{N} }}{2} - \frac{{\left( {N - 1} \right)t_{0} }}{2}} \right) \\ & \quad = \frac{{c_{j} }}{N}\left( { - \frac{{t^{*} }}{N + 1} + \frac{N + 1}{2}t_{N} - \frac{{Nt_{0} }}{N + 1} - \frac{{\left( {N + 1} \right)t_{0} }}{2} + t_{0} } \right) \\ & \quad = \frac{{c_{j} }}{N}\left( {\frac{{t_{0} - t^{*} }}{N + 1} + \frac{N + 1}{2}\left( {t_{N} - t_{0} } \right)} \right) \\ \end{aligned}$$

(47)

For the second segment of (46), using (42) and (43), we obtain:

$$\begin{aligned} & \sum\limits_{i = 1}^{N} {F_{i} (t^{*} ,x^{*} )} = \sum\limits_{i = 1}^{N} {c_{i} t^{*} } + \sum\limits_{i = 1}^{N} {\mu_{i} x^{*} } + \sum\limits_{i = 1}^{N} {f_{i} } \\ & \quad = t^{*} \sum\limits_{i = 1}^{N} {c_{i} } + x^{*} \sum\limits_{i = 1}^{N} {\mu_{i} } + \sum\limits_{i = 1}^{N} {f_{i} } \\ & \quad = t^{*} \frac{{x_{N} - x_{0} }}{{t_{N} - t_{0} }}\left( {1 - \sum\limits_{i = 1}^{N} {\mu_{i} } } \right) + x^{*} \sum\limits_{i = 1}^{N} {\mu_{i} } + \sum\limits_{i = 1}^{N - 1} {x_{i} } + \left( {\frac{{t_{N} x_{0} - t_{0} x_{N} }}{{t_{N} - t_{0} }}} \right)\left( {1 - \sum\limits_{i = 1}^{N} {\mu_{i} } } \right) \\ & \quad = \left( {\frac{{t^{*} \left( {x_{N} - x_{0} } \right) + t_{N} x_{0} - t_{0} x_{N} }}{{t_{N} - t_{0} }} - x^{*} } \right)\left( {1 - \sum\limits_{i = 1}^{N} {\mu_{i} } } \right) + x^{*} + \sum\limits_{i = 1}^{N - 1} {x_{i} } \\ & \quad = \left( {\frac{{x_{N} \left( {t^{*} - t_{0} } \right) + x_{0} \left( {t_{N} - t^{*} } \right)}}{{t_{N} - t_{0} }} - x^{*} } \right)\left( {1 - \sum\limits_{i = 1}^{N} {\mu_{i} } } \right) + x^{*} + \sum\limits_{i = 1}^{N - 1} {x_{i} } \\ & \quad = \left( {\eta x_{N} + \left( {1 - \eta } \right)x_{0} - x^{*} } \right)\left( {1 - \sum\limits_{i = 1}^{N} {\mu_{i} } } \right) + x^{*} + \sum\limits_{i = 1}^{N - 1} {x_{i} } \\ \end{aligned}$$

(48)

where $\eta = {{\left( {t^{*} - t_{0} } \right)} \mathord{\left/ {\vphantom {{\left( {t^{*} - t_{0} } \right)} {\left( {t_{N} - t_{0} } \right)}}} \right. \kern-\nulldelimiterspace} {\left( {t_{N} - t_{0} } \right)}}$. Substituting results (46), (47) and (48) into (45), the specific expression of $\beta_{j}$ is given as:

$$\begin{aligned} \beta_{j} & = \mu_{j} \left( {(1 - \lambda )(x_{N} - \mu_{N + 1} x_{0} ) + \mu_{N + 1} x^{*} } \right) \\ & \quad { + }\phi_{j} \left( {L_{N + 1} (t^{*} )} \right) - \frac{1}{N}\sum\limits_{i = 1}^{N} {F_{j} \left( {L_{i} (t^{*} ),F_{i} (t^{*} ,x^{*} )} \right)} \\ & = \mu_{j} \left( {(1 - \lambda )(x_{N} - \mu_{N + 1} x_{0} ) + \mu_{N + 1} x^{*} } \right) \\ & \quad + \frac{{c_{j} }}{N}\left( {\frac{{t_{0} - t^{*} }}{N + 1} + \frac{N + 1}{2}\left( {t_{N} - t_{0} } \right)} \right) \\ & \quad - \frac{{\mu_{j} }}{N}\left[ {\left( {\eta x_{N} + \left( {1 - \eta } \right)x_{0} - x^{*} } \right)\left( {1 - \sum\limits_{i = 1}^{N} {\mu_{i} } } \right) + x^{*} + \sum\limits_{i = 1}^{N - 1} {x_{i} } } \right] \\ \end{aligned}$$

(49)

It is obvious that parameter $\beta_{j}$ does not contain $\tilde{x}_{N + 1}$. This completes the calculation of the fractal interpolation bias $\Delta \Gamma$.