Theoretical analysis of the multi-GNSS contribution to partial ambiguity estimation and R-ratio test-based ambiguity validation

Abstract

The fast and high-precision positioning with multiple Global Navigation Satellite Systems (multi-GNSS) has been challenging for decades. Although the single-frequency single system (SF-SS), satellite selection for multi-GNSS, and multi-GNSS-based partial ambiguity resolution (PAR) can achieve rapid positioning, the varying theoretical bases of them result in different fixed reliability of ambiguities. Hence, we provide the theory analyzing the ambiguity resolution capabilities of the named systems. By adding satellite observations, the equations giving the variance–covariance matrix variation of the original float parameters are derived. Then, the relationship between the ambiguity dilution of precision (ADOP) values of the original ambiguity vector (OAV) before and after adding observations is obtained. This is followed by the analyses of the changing trends in the OAV’s probability density function, integer least-squares pull-in region, and the R-ratio test-based integer aperture pull-in region. In terms of precision, ADOP, and R-ratio test-based fixed reliability of ambiguities, the analyses indicate that the multi-GNSS can improve the partial ambiguity estimation and validation. Besides, compared to satellite selection and SF-SS, the PAR is optimal. The BeiDou Navigation Satellite System (BDS) and the Global Positioning System (GPS)-based single-epoch positioning experiments showed that both BDS B1 and B1-based PAR outperform GPS L1 and L1-based PAR in terms of ADOP and R-ratio test-based fixed reliability. The ADOP of the former is smaller than 0.14, and both the R-ratio test-based acceptance and success rates are up to 99.64%. Finally, the false alarm, failure, and detection rates are reduced to 0.34%, 0.0%, and 0.02%, respectively.

Appendix A: The proof of positive definiteness of matrices A 1 and A 2 in (10)

For the symmetrical positive definite matrix ${\varvec Q}_{{{{\hat{b}\hat{b}}}}}$, there is a 3-order real invertible matrix ${\varvec S}_{{\text{1}}}$ making $\varvec{Q}_{{\hat{b} \hat{b}}} = \varvec{S}_{1}^{ {T}} \cdot \varvec{S}_{1}$ (Horn and Johnson 1999). Hence, considering the positive definite matrix ${\varvec P}_{{\text{2}}}^{{{\text{ - 1}}}}$, $\forall {\varvec{x}} \ne {\bf{0}}$, it holds that $\varvec{x}^{T} \left( {\varvec{P}_{2}^{{ - 1}} + \varvec{A}_{3} \varvec{Q}_{{\hat{b}\hat{b}}} \varvec{A}_{3}^{T} } \right)\varvec{x}\, > \;0$. Thus, $\varvec{P}_{2}^{{ - 1}} \; + \;\varvec{A}_{3} \varvec{Q}_{{\hat{b}\hat{b}}} \varvec{A}_{3}^{T}$ is a symmetrical positive definite matrix. Similarly, there is a real invertible matrix S₂ making $\left( {\varvec{P}_{2}^{{ - 1}} + \varvec{A}_{3} \varvec{Q}_{{\hat{b}\hat{b}}} \varvec{A}_{3}^{T} } \right)^{{ - 1}} = \varvec{S}_{2}^{T} \cdot \varvec{S}_{2}$ and $\varvec{A}_{1} = \left( {\varvec{S}_{2} \varvec{A}_{3} \varvec{Q}_{{\hat{b}\hat{b}}} } \right)^{T} \cdot \varvec{S}_{2} \varvec{A}_{3} \varvec{Q}_{{\hat{b}\hat{b}}}$, and it holds that rank$\left( {\varvec{S}_{2} \varvec{A}_{3} \varvec{Q}_{{\hat{b}\hat{b}}} } \right)$ = rank (A₃) and rank (A₃)≤ 3. When rank (A₃) = 3, A₁ is a positive definite matrix, and for rank (A₃) < 3, A₁ is a positive semidefinite matrix.

Hence, there is a 3-order real matrix S₃ making $\varvec{A}_{1} = \varvec{S}_{3}^{T} \cdot \varvec{S}_{3}$ and $\varvec{A}_{2} = \left( {\varvec{S}_{3} \varvec{B}_{1}^{T} \varvec{B}_{{1,\;\lambda }}^{{ - 1}} } \right)^{T} \cdot \varvec{S}_{3} \varvec{B}_{1}^{T} \varvec{B}_{{1,\;\lambda }}^{{ - 1}}$. For the 3 x n-order matrix ${\varvec{S}}_{3} {\varvec{B}}_{1}^{T} {\varvec{B}}_{{1,\;\lambda }}^{{ - 1}}$, it holds that rank $\left( {\varvec{S}_{3} \varvec{B}_{1}^{T} \varvec{B}_{{1,\;\lambda }}^{{ - 1}} } \right) \le \min ~\left\{ {rank\left( {\varvec{S}_{3} } \right),\;rank\left( {\varvec{B}_{1}^{T} } \right)} \right\} < n$. Therefore, A₂ is a positive semidefinite matrix.

End of Proof.