Single-Epoch GNSS Array Integrity: An Analytical Study

Abstract

In this contribution we analyze the integrity of the GNSS array model through the so-called uniformly most powerful invariant (UMPI) test-statistics and their corresponding minimal detectable biases (MDBs). The model considered is characterized by multiple receivers/satellites with known coordinates where the multi-frequency carrier-phase and pseudo-range observables are subject to atmospheric (ionospheric and tropospheric) delays, receiver and satellite clock biases, as well as instrumental delays. Highlighting the role played by the model’s misclosures, analytical multivariate expressions of a few leading test-statistics together with their MDBs are studied that are further accompanied by numerical results of the three GNSSs GPS, Galileo and BeiDou.

Appendix

Proof of (3)

The model’s misclosures, forming the condition equations, can be formulated through pre-multiplying the corresponding observation vector by an orthogonal complement basis matrix of the design matrix (Teunissen 2000). In case of the single-epoch ambiguity-float scenario, the carrier-phase observations are all reserved to determine the DD ambiguities, thus leaving the code observations to contribute to the redundancy of the model. Given the observations Eq. (1), the code-only design matrix A, together with its orthogonal complement basis matrix B, can therefore be expressed as (per baseline)

$$ \displaystyle\begin{array}{rcl} \begin{array}{l} A\mapsto [\begin{array}{ccc} e_{f} \otimes D_{s}^{T}g&,&\mu \otimes I_{s-1} \end{array} ] \Rightarrow \\ \quad \quad B^{T}\mapsto \left [\begin{array}{c} (D_{f}^{T}\mu )^{\perp T}D_{f}^{T} \otimes c_{d\vert \tau }^{2}g^{T}D_{s}(D_{s}^{T}W_{s}^{-1}D_{s})^{-1} \\ \mu ^{\perp T} \otimes (D_{s}^{T}g)^{\perp T} \end{array} \right ] \end{array} & &{}\end{array}$$

(22)

from which (3) follows. That the misclosures M ₁ and M ₂ are mutually uncorrelated follows from the identities $D_{s}^{T}\bar{g} = D_{s}^{T}g$, and $(D_{s}^{T}g)^{\perp T}D_{s}^{T}g = 0$. □ 

Proof of Theorem 

Equation (6) is indeed another expression of the UMPI test-statistic T _q presented in Teunissen (2000). In terms of the model’s misclosures M, T _q and its MDB-squared | | ∇ | | ² read

$$\displaystyle\begin{array}{rcl} T_{q} = \frac{1} {q}M^{T}Q_{ MM}^{-1}P_{ C_{M}}M& &{}\end{array}$$

(23)

$$\displaystyle\begin{array}{rcl} \vert \vert \nabla \vert \vert ^{2} = \frac{\nu _{q,\alpha,\gamma }} {d_{Y }^{T}C_{M}^{T}Q_{MM}^{-1}C_{M}d_{Y }}& &{}\end{array}$$

(24)

To complete the proof, we thus need to show

$$\displaystyle{ \begin{array}{l} \mathit{tr}(Q_{\mathit{MM}}^{-1}P_{C_{M}}\mathit{MM}^{T}) = M^{T}Q_{\mathit{MM}}^{-1}P_{C_{M}}M, \\ \mathit{tr}(P_{C_{M}}) = q\end{array} }$$

(25)

The first expression follows from the trace-property tr(UV ) = tr(VU) for any matrices U and V of an appropriate size, and the fact that the trace of a scalar is equal to the scalar itself. The second expression follows from the equality between the trace of a projector and its rank, that is

$$\displaystyle{ \mathit{tr}(P_{C_{M}}) = \mathit{rank}(P_{C_{M}}) = q, }$$

(26)

since rank(C _M ) = q. □ 

Proof of (12)

In case of the atmosphere-fixed scenario, no differential atmospheric delays are to be estimated, i.e. μ = 0 and g = 0. This yields μ ^⊥ = I _f and $(D_{s}^{T}g)^{\perp } = I_{s-1}$. According to (3), the frequency-difference misclosures M ₁ vanishes, and the vectorized version of the atmosphere-free misclosures M ₂ takes the following form

$$\displaystyle{ M_{\tilde{p}} = [D_{n}^{T} \otimes I_{ f} \otimes D_{s}^{T}]\mathit{vec}[\tilde{P}] }$$

(27)

with the variance matrix (cf. (2))

$$\displaystyle{ Q_{M_{\tilde{p}}M_{\tilde{p}}} = D_{n}^{T}D_{ n} \otimes Q_{p} \otimes D_{s}^{T}W_{ s}^{-1}D_{ s} }$$

(28)

Upon choosing the array-detector structure (11), matrix C _M of $M_{\tilde{p}}$, introduced in Theorem 1, reads then

$$\displaystyle{ C_{M_{\tilde{p}}} = D_{n}^{T}D_{ n} \otimes I_{f} \otimes D_{s}^{T}D_{ s} }$$

(29)

Similar expressions are formulated for the carrier-phase observations $\tilde{\varPhi }$, in case the ambiguities are fixed to their integers. The structures of $M_{\tilde{\phi }}$, $Q_{M_{\tilde{\phi }}M_{\tilde{\phi }}}$ and $C_{M_{\tilde{p}}}$ are thus identical to those of $\tilde{P}$. Substituting $M = [M_{\tilde{p}}^{T},M_{\tilde{\phi }}^{T}]^{T}$,

$$\displaystyle\begin{array}{rcl} Q_{\mathit{MM}} = \left [\begin{array}{cc} Q_{M_{\tilde{p}}M_{\tilde{p}}} & 0 \\ 0 &Q_{M_{\tilde{\phi }}M_{\tilde{\phi }}} \end{array} \right ],\quad C_{M} = \left [\begin{array}{cc} C_{M_{\tilde{p}}} & 0 \\ 0 &C_{M_{\tilde{\phi }}} \end{array} \right ],& &{}\end{array}$$

(30)

an application of Theorem 1 gives (cf. (6))

$$\displaystyle\begin{array}{rcl} T_{q} = \frac{\mathit{tr}\{Q_{M_{\tilde{p}}M_{\tilde{p}}}^{-1}P_{C_{M_{\tilde{p}}}}M_{\tilde{p}}M_{\tilde{p}}^{T}\} + \mathit{tr}\{Q_{M_{\tilde{\phi }}M_{\tilde{\phi }}}^{-1}P_{C_{M_{\tilde{\phi }}}}M_{\tilde{\phi }}M_{\tilde{\phi }}^{T}\}} {\mathit{tr}\{P_{C_{M_{\tilde{p}}}}\} + tr\{P_{C_{M_{\tilde{\phi }}}}\}} & &{}\end{array}$$

(31)

The proof follows then from

$$\displaystyle{ P_{C_{M_{\tilde{\phi }}}} = P_{C_{M_{\tilde{p}}}} = I_{n-1} \otimes I_{f} \otimes I_{s-1}, }$$

(32)

and

$$\displaystyle\begin{array}{rcl} \begin{array}{l} \mathit{tr}\{Q_{M_{\tilde{p}}M_{\tilde{p}}}^{-1}M_{\tilde{p}}M_{\tilde{p}}^{T}\} = \mathit{tr}\{[Q_{p}^{-1} \otimes W_{s}P_{e_{s}}^{\perp }]\ \tilde{P}P_{D_{n}}\tilde{P}^{T}\}, \\ \mathit{tr}\{Q_{M_{\tilde{\phi }}M_{\tilde{\phi }}}^{-1}M_{\tilde{\phi }}M_{\tilde{\phi }}^{T}\} = \mathit{tr}\{[Q_{\phi }^{-1} \otimes W_{s}P_{e_{s}}^{\perp }]\ \tilde{\varPhi }P_{D_{n}}\tilde{\varPhi }^{T}\}\end{array} & &{}\end{array}$$

(33)

with the projectors $P_{e_{s}}^{\perp } = W_{s}^{-1}D_{s}(D_{s}^{T}W_{s}^{-1}D_{s})^{-1}D_{s}^{T}$, and $P_{D_{n}} = D_{n}(D_{n}^{T}D_{n})^{-1}D_{n}^{T}$.

The proof of(14), (17), (19)and(21)goes along the same lines as the proof of(12). □