Asynchronous and time-differenced RTK for ocean applications using the BeiDou short message service

Abstract

The ocean real-time kinematic (ORTK) technique was proposed to realize high-precision GNSS positioning for long baselines with differential corrections transmitted through the BeiDou short message service, where the positioning accuracy is from subdecimeter to centimeter and the convergence time is several minutes without integer ambiguity resolution. In this paper, the ORTK method is improved based on the dual-frequency global positioning system and triple-frequency BeiDou navigation satellite system. The convergence time is shortened to a few seconds using the following modifications/innovations. First, the uncombined corrections are used instead of the ionosphere-free corrections. A more efficient encoding strategy is proposed to compress the uncombined corrections. Second, the ultrarapid precise ephemeris provided by the Tongji BeiDou Analysis Center is assimilated in the corrections to mitigate the orbit errors, and a long-baseline ionosphere-weighted model is applied to estimate the ionospheric delays. Third, an asynchronous and time-differenced positioning filter is employed to address the time-delays and time-correlations of the corrections. Finally, the float solutions in our earlier study are upgraded to fixed solutions by partially fixing the ambiguities. Experiments on a 320 km baseline indicate that the ambiguities can be correctly fixed within 10 s and that the positioning accuracy is approximately 1.1 cm for both horizontal components and 5 cm for the vertical component.

Appendix

Following the idea in Petovello et al. (2009), a Kalman filter solution with time-correlated observations is derived in this Appendix. A new observation vector can be formed by differencing the observation vectors of two consecutive epochs as

$$ \begin{aligned} {\varvec{z}}_{k} = &\, {\varvec{l}}_{k} - {\varvec{S}}_{k,k - 1} {\varvec{l}}_{k - 1} \\ = & \,{\varvec{A}}_{k} {\varvec{\xi}}_{k} - {\varvec{S}}_{k,k - 1} {\varvec{A}}_{k - 1} {\varvec{\xi}}_{k - 1} + {\varvec{\eta}}_{k} - {\varvec{S}}_{k,k - 1} {\varvec{\eta}}_{k - 1} \\ \end{aligned} $$

(24)

Applying state transition Eqs. (18) and (19), we have a new observation equation as

$$ \begin{array}{*{20}c} {{\varvec{z}}_{k} = {\varvec{H}}_{k} {\varvec{\xi}}_{k} + {\varvec{v}}_{k} } \\ \end{array} $$

(25)

where

$$ \begin{array}{*{20}c} {{\varvec{H}}_{k} = {\varvec{A}}_{k} - {\varvec{S}}_{k,k - 1} {\varvec{A}}_{k - 1} } \\ \end{array} $$

(26)

$$ \begin{array}{*{20}c} {{\varvec{v}}_{k} = {\varvec{S}}_{k,k - 1} {\varvec{A}}_{k - 1} {\mathbf{w}}_{k} + {\varvec{\eta}}_{k} - {\varvec{S}}_{k,k - 1} {\varvec{\eta}}_{k - 1} } \\ \end{array} $$

(27)

The variance of the new observation noise vector ${\varvec{v}}_{k}$ is

$$ \begin{array}{*{20}c} {{\varvec{Q}}_{{{\varvec{v}}_{k} }} = {\varvec{S}}_{k,k - 1} {\varvec{A}}_{k - 1} {\varvec{Q}}_{{w_{k} }} {\varvec{A}}_{k - 1}^{{\text{T}}} {\varvec{S}}_{k,k - 1}^{{\text{T}}} + {\varvec{Q}}_{{{\varvec{\eta}}_{k} }} + {\varvec{S}}_{k,k - 1} {\varvec{Q}}_{{{\varvec{\eta}}_{k - 1} }} {\varvec{S}}_{k,k - 1}^{{\text{T}}} } \\ \end{array} $$

(28)

Here, ${\varvec{v}}_{k}$ is time-uncorrelated since ${\mathbf{w}}_{k}$, ${\varvec{\eta}}_{k}$ and ${\varvec{\eta}}_{k - 1}$ are all independent of each other and time-uncorrelated. However, ${\varvec{v}}_{k}$ is correlated to the process noise vector ${\mathbf{w}}_{k}$ with a covariance matrix as

$$ \begin{array}{*{20}c} {{\varvec{C}}_{k} = {\varvec{Q}}_{{w_{k} }} {\varvec{A}}_{k - 1}^{{\text{T}}} {\varvec{S}}_{k,k - 1}^{{\text{T}}} } \\ \end{array} $$

(29)

In such a case, a standard Kalman filter is not applicable, and the filter solution should be instead resolved as (Brown and Hwang 1992)

$$ \begin{array}{*{20}c} {\bar{\boldsymbol{\xi }}_{k} = \hat{\boldsymbol{\xi}}_{k - 1} + {\mathbf{w}}_{k} } \\ \end{array} $$

(30a)

$$ \begin{array}{*{20}c} {{\varvec{Q}}_{{\bar{\boldsymbol{\xi }}_{k} }} = {\varvec{Q}}_{{\hat{\boldsymbol{\xi }}_{k - 1} }} + {\varvec{Q}}_{{{\mathbf{w}}_{k} }} } \\ \end{array} $$

(30b)

$$ \begin{array}{*{20}c} {\hat{\boldsymbol{\xi }}_{k} = \bar{\boldsymbol{\xi }}_{k} + {\varvec{J}}_{k} \left( {{\varvec{z}}_{k} - {\varvec{H}}_{k} \bar{\boldsymbol{\xi }}_{k} } \right)} \\ \end{array} $$

(30c)

$$ \begin{array}{*{20}c} {{\varvec{Q}}_{{\hat{\boldsymbol{\xi }}_{k} }} = \left( {{\varvec{I}} - {\varvec{J}}_{k} {\varvec{H}}_{k} } \right){\varvec{Q}}_{{\bar{\boldsymbol{\xi }}_{k} }} - {\varvec{J}}_{k} {\varvec{C}}_{k}^{{\text{T}}} } \\ \end{array} $$

(30d)

where

$$ \begin{array}{*{20}c} {{\varvec{J}}_{k} = \left( {{\varvec{Q}}_{{\bar{\boldsymbol{\xi }}_{k} }} {\varvec{H}}_{k}^{{\text{T}}} + {\varvec{C}}_{k} } \right)\left( {{\varvec{H}}_{k} {\varvec{Q}}_{{\bar{\boldsymbol{\xi }}_{k} }} {\varvec{H}}_{k}^{{\text{T}}} + {\varvec{Q}}_{{{\varvec{v}}_{k} }} + {\varvec{H}}_{k} {\varvec{C}}_{k} + {\varvec{C}}_{k}^{{\text{T}}} {\varvec{H}}_{k}^{{\text{T}}} } \right)^{ - 1} } \\ \end{array} $$

(31)