Instantaneous BeiDou+GPS RTK positioning with high cut-off elevation angles

Abstract

As the Chinese BeiDou Navigation Satellite System (BDS) has become operational in the Asia-Pacific region, it is of importance to better understand as well as demonstrate the capabilities that a combination of BeiDou with GPS brings to positioning. In this contribution, a formal and empirical analysis is given of the single-epoch RTK positioning capabilities of such a combined system. This will be done for the single- and dual-frequency case, and in comparison with the BDS- and GPS-only performances. It will be shown that with the combined system, when more satellites are available, much larger than the customary cut-off elevations can be used. This is important, as such measurement set-up will significantly increase the GNSS applicability in constrained environments, such as e.g. in urban canyons or when low-elevation multipath is present.

Appendix

Proof of Theorem (Combined-system ADOP) The ambiguity variance matrix of the combined system can be expressed in the single-system variance matrices as

$$\begin{aligned} \begin{array}{l} Q_{\hat{a}\hat{a}}= \left[ \begin{array}{ll} Q_{\hat{a}_{G}\hat{a}_{G}} &{} 0\\ 0 &{} Q_{\hat{a}_{B}\hat{a}_{B}} \end{array} \right] - \left[ \begin{array}{c} Q_{\hat{a}_{G}\hat{b}_{G}} \\ Q_{\hat{a}_{B}\hat{b}_{B}} \end{array} \right] [Q_{\hat{b}_{G}\hat{b}_{G}}\\ \qquad \qquad +\,Q_{\hat{b}_{B}\hat{b}_{B}} ]^{-1} \left[ \begin{array}{c} Q_{\hat{a}_{G}\hat{b}_{G}} \\ Q_{\hat{a}_{B}\hat{b}_{B}} \end{array} \right] ^{T} \end{array} \end{aligned}$$

Upon taking the determinant, we get

$$\begin{aligned} \begin{array}{l} |Q_{\hat{a}\hat{a}}|=|Q_{\hat{a}_{G}\hat{a}_{G}}||Q_{\hat{a}_{B}\hat{a}_{B}}| |I-[Q_{\hat{b}_{G}\hat{a}_{G}}Q_{\hat{a}_{G}\hat{a}_{G}}^{-1}Q_{\hat{a}_{G}\hat{b}_{G}}\\ \qquad \qquad \,+\,Q_{\hat{b}_{B}\hat{a}_{B}}Q_{\hat{a}_{B}\hat{a}_{B}}^{-1}Q_{\hat{a}_{B}\hat{b}_{B}}][Q_{\hat{b}_{G}\hat{b}_{G}}+Q_{\hat{b}_{B}\hat{b}_{B}} ]^{-1}| \end{array} \end{aligned}$$

where we made use of the determinant property $|I_{m}-BC|=|I_{n}-CB|$ for $m \times n$ matrices $B$ and $C^{T}$. With the use of

$$\begin{aligned} Q_{\check{b}_{*}\check{b}_{*}}=Q_{\hat{b}_{*}\hat{b}_{*}}-Q_{\hat{b}_{*}\hat{a}_{*}}Q_{\hat{a}_{*}\hat{a}_{*}}^{-1}Q_{\hat{a}_{*}\hat{b}_{*}} \end{aligned}$$

we can then finally write

$$\begin{aligned} |Q_{\hat{a}\hat{a}}|=|Q_{\hat{a}_{G}\hat{a}_{G}}||Q_{\hat{a}_{B}\hat{a}_{B}}| |\frac{Q_{\check{b}_{G}\check{b}_{G}}+Q_{\check{b}_{B}\check{b}_{B}}}{Q_{\hat{b}_{G}\hat{b}_{G}}+Q_{\hat{b}_{B}\hat{b}_{B}}}| \end{aligned}$$

(18)

from which the expression (12) for the combined ADOP follows. $\square$