A method for updating GNSS satellite ultra-rapid clock offsets and orbits with the aid of a covariance intersection algorithm

Abstract

Global navigation satellite system (GNSS) ultra-rapid clock offsets and orbit products are essential for near-real-time and real-time uses. To meet the requirements of accuracy and timeliness in high-accuracy applications, the issuing rates of ultra-rapid products are increased to six or three hours. However, there is an appreciable fluctuation of positioning residuals during the period of updating of products. To improve the performance of GNSS rapid services including the GPS, GLONASS, GALILEO and BeiDou, this paper proposes a method for updating satellite ultra-rapid clock offsets and orbits based on the covariance intersection algorithm, where kernel tricks are used to model the series in the position domain. Moreover, the parameter characteristics of the clock and orbit and the unknown inter-series correlation are considered in the model of satellite ultra-rapid products. Meanwhile, a sparse strategy is used in solving the model coefficients; i.e., the least absolute shrinkage and selection operator (LASSO) strategy. Several experimental schemes show that 1) jumps and gross errors in the GNSS ultra-rapid clock offset affect the modeling and services and should be detected and repaired before high-accuracy applications; 2) the updating of clock offsets and orbits reduces the steadiness of product services, while position residual fluctuations are introduced using GNSS precise point positioning solutions; 3) improved clock offset and orbit series can be obtained using the covariance intersection algorithm and kernel tricks; 4) the LASSO strategy can automatically and effectively choose and estimate the model coefficients of clock offset and orbit series; and 5) the proposed method can smooth the short-term arcs of GNSS products and positioning services by 29.8–99.5% for Multi-GNSS Experiment stations compared with original series. It is thus concluded that the proposed updating strategy is meaningful for improving GNSS satellite ultra-rapid products.

Appendix

We set two groups of observations as y₁ and y₂. The matrix expression is

$$\left[ {\begin{array}{*{20}c} {{\mathbf{y}}_{1} } \\ {{\mathbf{y}}_{2} } \\ \end{array} } \right] = \left[ {\begin{array}{*{20}c} {{\mathbf{F}}_{1} } \\ {{\mathbf{F}}_{2} } \\ \end{array} } \right]{\mathbf{x}} + \left[ {\begin{array}{*{20}c} {{\mathbf{e}}_{1} } \\ {{\mathbf{e}}_{2} } \\ \end{array} } \right],$$

(19)

where e₁ and e₂ are the model errors and F₁ and F₂ are the observation functions. The co-factor matrix is thus written as

$${\text{cov}} \left[ {\begin{array}{*{20}c} {{\mathbf{e}}_{1} } \\ {{\mathbf{e}}_{2} } \\ \end{array} } \right] = \left[ {\begin{array}{*{20}c} {{\mathbf{Q}}_{11} } & {{\mathbf{Q}}_{12} } \\ {{\mathbf{Q}}_{21} } & {{\mathbf{Q}}_{22} } \\ \end{array} } \right] \le \left[ {\begin{array}{*{20}c} {\frac{1}{\omega }{\mathbf{Q}}_{11} } & 0 \\ 0 & {\frac{1}{1 - \omega }{\mathbf{Q}}_{22} } \\ \end{array} } \right].$$

(20)

The solution to x is obtained from Eqs. (19) and (20) as

$${\hat{\mathbf{x}}} = {\hat{\mathbf{Q}}}_{{{\mathbf{xx}}}} \left[ {\omega {\mathbf{F}}_{1}^{{\text{T}}} {\mathbf{Q}}_{11}^{ - 1} {\mathbf{y}}_{1} + (1 - \omega ){\mathbf{F}}_{2}^{{\text{T}}} {\mathbf{Q}}_{22}^{ - 1} {\mathbf{y}}_{2} } \right],$$

(21)

where ${\hat{\mathbf{Q}}}_{xx}$ is expressed as

$${\hat{\mathbf{Q}}}_{{{\mathbf{xx}}}} = \left[ {\omega {\mathbf{F}}_{1}^{{\text{T}}} {\mathbf{Q}}_{11} {\mathbf{F}}_{1} + (1 - \omega ){\mathbf{F}}_{2}^{{\text{T}}} {\mathbf{Q}}_{22} {\mathbf{F}}_{2} } \right]^{ - 1} .$$

(22)

Moreover, we obtain the covariance of ${\hat{\mathbf{x}}}$ as

$$\begin{aligned} {\text{cov}} \left[ {{\hat{\mathbf{x}}}} \right] &= {\hat{\mathbf{Q}}}_{{{\mathbf{xx}}}} {\text{cov}} \left[ {\omega {\mathbf{F}}_{1}^{{\text{T}}} {\mathbf{Q}}_{11}^{ - 1} {\mathbf{y}}_{1} + (1 - \omega ){\mathbf{F}}_{2}^{{\text{T}}} {\mathbf{Q}}_{22}^{ - 1} {\mathbf{y}}_{2} } \right]{\hat{\mathbf{Q}}}_{{{\mathbf{xx}}}} \hfill \\& = {\hat{\mathbf{Q}}}_{{{\mathbf{xx}}}} {\text{cov}} \left\{ {\left[ {\omega {\mathbf{F}}_{1}^{{\text{T}}} {\mathbf{Q}}_{11}^{ - 1} (1 - \omega ){\mathbf{F}}_{2}^{{\text{T}}} {\mathbf{Q}}_{22}^{ - 1} } \right]\left[ {\begin{array}{*{20}c} {{\mathbf{y}}_{1} } \\ {{\mathbf{y}}_{2} } \\ \end{array} } \right]} \right\}{\hat{\mathbf{Q}}}_{{{\mathbf{xx}}}} \hfill \\& = {\hat{\mathbf{Q}}}_{{{\mathbf{xx}}}} \left[ {\omega {\mathbf{F}}_{1}^{{\text{T}}} {\mathbf{Q}}_{11}^{ - 1} (1 - \omega ){\mathbf{F}}_{2}^{{\text{T}}} {\mathbf{Q}}_{22}^{ - 1} } \right]\left[ {\begin{array}{*{20}c} {{\mathbf{Q}}_{11} } &{{\mathbf{Q}}_{12} } \\ {{\mathbf{Q}}_{21} } &{{\mathbf{Q}}_{22} } \\ \end{array} } \right]\left[ {\begin{array}{*{20}c} {\omega {\mathbf{Q}}_{11}^{ - 1} {\mathbf{F}}_{1} } \\ {(1 - \omega ){\mathbf{Q}}_{22}^{ - 1} {\mathbf{F}}_{2} } \\ \end{array} } \right]{\hat{\mathbf{Q}}}_{{{\mathbf{xx}}}} . \hfill \\ \end{aligned}$$

(23)

We can further modify Eq. (23) as

$$\begin{gathered} {\text{cov}} \left[ {{\hat{\mathbf{x}}}} \right] \le {\hat{\mathbf{Q}}}_{{{\mathbf{xx}}}} \left[ {\omega {\mathbf{F}}_{{1}}^{{\text{T}}} {\mathbf{Q}}_{11}^{ - 1} (1 - \omega ){\mathbf{F}}_{{2}}^{{\text{T}}} {\mathbf{Q}}_{22}^{ - 1} } \right]\left[ {\begin{array}{*{20}c} {\frac{1}{\omega }{\mathbf{Q}}_{11} } & 0 \\ 0 & {\frac{1}{1 - \omega }{\mathbf{Q}}_{22} } \\ \end{array} } \right]\left[ {\begin{array}{*{20}c} {\omega {\mathbf{Q}}_{11}^{ - 1} {\mathbf{F}}_{1} } \\ \quad {(1 - \omega ){\mathbf{Q}}_{22}^{ - 1} {\mathbf{F}}_{2} } \\ \end{array} } \right]{\hat{\mathbf{Q}}}_{{{\mathbf{xx}}}} . \hfill \\ \qquad = {\hat{\mathbf{Q}}}_{{{\mathbf{xx}}}} \left[ {\omega {\mathbf{F}}_{1}^{{\text{T}}} {\mathbf{Q}}_{11}^{ - 1} {\mathbf{Q}}_{11} {\mathbf{Q}}_{11}^{ - 1} {\mathbf{F}}_{1} + (1 - \omega ){\mathbf{F}}_{{2}}^{{\text{T}}} {\mathbf{Q}}_{22}^{ - 1} {\mathbf{Q}}_{22} {\mathbf{Q}}_{22}^{ - 1} {\mathbf{F}}_{2} } \right]{\hat{\mathbf{Q}}}_{{{\mathbf{xx}}}} \hfill \\ \qquad= {\hat{\mathbf{Q}}}_{{{\mathbf{xx}}}} \left[ {\omega {\mathbf{F}}_{1}^{{\text{T}}} {\mathbf{Q}}_{11}^{ - 1} {\mathbf{F}}_{1} + (1 - \omega ){\mathbf{F}}_{{2}}^{{\text{T}}} {\mathbf{Q}}_{22}^{ - 1} {\mathbf{F}}_{2} } \right]{\hat{\mathbf{Q}}}_{{{\mathbf{xx}}}} = {\hat{\mathbf{Q}}}_{{{\mathbf{xx}}}} {\hat{\mathbf{Q}}}_{{{\mathbf{xx}}}}^{{ - {\mathbf{1}}}} {\hat{\mathbf{Q}}}_{{{\mathbf{xx}}}} = {\hat{\mathbf{Q}}}_{{{\mathbf{xx}}}} \hfill \\ \end{gathered}$$

(24)