# Ambiguity-Resolved Model Tests for Carrier-Phase GNSS
## 4 The Ambiguity-Resolved Tests
### AR Integer Testing  
The distribution of Theorem 1 considers the case when the integer ambiguity success rate under \(\mathcal{H}_{a}\) approaches one, i.e., the case when the integer estimated ambiguity may be considered as known. We now consider the other extreme and assume the ambiguity vector under \(\mathcal{H}_{a}\) to be an unknown real-valued parameter instead of integer-valued. The test statistic relevant for this case follows if instead of \(\underline{a}\), the ambiguity-float estimator \(\underline{a}\) is used. We obtain, instead of (16), the test statistic  
Figure 4: PAR ionospheric power function compared with AK and AF counterparts for single-epoch, triple-frequency Galileo model with undifferenced pseudorange standard deviation \(\sigma_{p}=20\) cm. PAR allowed the \(\mathcal{H}_{a}\) ambiguity success rate to increase from 38.4% to 99.6%.  
\[\underline{T}(\underline{\hat{a}}_{0},\underline{\hat{a}})=||\underline{\hat{z}}||^ {2}_{Q_{2\ell}}+||\underline{\hat{z}}_{0}||^{2}_{Q_{0\ell}q_{0}} \tag{19}\]  
As \(\underline{\hat{c}}\) and \(\underline{\hat{c}}_{0}\) are independent, the PDF of the input \([\underline{\hat{z}}^{T},\underline{\hat{c}}_{0}]^{T}\) follows, similarly to (14), as \(f_{\varepsilon,\underline{\hat{c}}_{0}}(\gamma,\epsilon)=f_{\varepsilon}( \gamma)f_{\underline{\hat{c}}_{0}}(\epsilon)\), with \(\underline{\hat{c}}\sim\mathcal{N}_{q}(c,Q_{2\ell})\). With \(a\) assumed as real-valued under \(\mathcal{H}_{a}\), \(a\in\mathbb{R}^{n}\), but integer-valued under \(\mathcal{H}_{0}\), \(a\in\mathbb{Z}^{n}\), the statistic (19) is the test statistic to use for _integer testing_ of the ambiguities. When \(c=0\), it simplifies to \(\underline{T}(\underline{\hat{a}}_{0},\underline{\hat{a}})=||\underline{\hat{ c}}_{0}||^{2}_{Q_{0\ell}q_{0}}=||\underline{\hat{a}}_{0}-\underline{\hat{a}}_{0}||^ {2}_{Q_{0\ell}q_{0}}\).  
Being able to statistically test for the integerness of estimable ambiguities is important when linking GNSS models. Dependent on how different GNSS systems are combined or on how well receiver hardware delays can be calibrated, real-valued biases may become lumped with the carrier-phase ambiguities, thereby removing the integerness of the estimable ambiguities. For instance, when signals of different systems are integrated, inter-system biases in both the carrier-phase and code data may occur [21; 22]. Also, in the case of low-cost, mass-market GNSS receivers, the stability of the hardware delays may not be such that integerness of the carrier-phase ambiguities is guaranteed. For instance, it is demonstrated in [14; 15] that smartphone phase observations may indeed be contaminated by receiver effects that destroy the integer property of the ambiguities. Therefore, being able to statistically test for such occurrences becomes important. This is also true in case of opportunistic PNT when one tries to use carrier-phase signals from terrestrial transmitters or LEO communication satellites for precise positioning [23; 24].  
As an example illustration, we consider the single-epoch, dual-frequency, two-satellite GNSS geometry-free model [1]. The ambiguity bias vector under \(\mathcal{H}_{a}\), propagated into the ambiguity residual \(\underline{\hat{c}}_{0}=\underline{\hat{a}}_{0}-\underline{\hat{a}}_{0}\), is denoted as \(\delta=[\delta_{1},\delta_{2}]^{T}\). This bias vector is, like the ambiguity residual \(\underline{\hat{c}}_{0}\), confined to the origin-centered pull-in region of the integer ambiguity estimator. Hence, the domain of the power function \(F(\delta)=\mathsf{P}[||\underline{\hat{a}}_{0}-\underline{\hat{a}}_{0}||^{2} _{Q_{0\ell}q_{0}}>\tau_{a}|\mathcal{H}_{a}]\) is confined to this pull-in region as well, which, in case of ILS estimation, is a hexagon. Figure 5 shows, for when \(\underline{\hat{a}}_{0}\) is chosen as the ILS estimator, the contour aligns with the color bar of the power function \(F(\delta)\). It clearly shows how the power increases as the ambiguity bias vector moves away from the origin towards the edges of the ILS pull-in region.