# Ambiguity-Resolved Model Tests for Carrier-Phase GNSS
## 4 The Ambiguity-Resolved Tests
### AR Identification
Using the representation \(\underline{T}(\underline{\hat{a}}_{0},\underline{\hat{a}})=||\underline{\hat {c}}(\underline{\hat{a}})||^{2}+||\underline{\hat{a}}_{0}-\underline{\hat{a} }||^{2}_{Q_{\text{dr}}\phi_{0}}-||\underline{\hat{a}}||^{2}_{Q_{\text{dr}}\phi _{0}}\) and neglecting the difference between \(\underline{\hat{a}}_{0}\) and \(\underline{\hat{a}}\), we obtain  
\[\underline{T}(\underline{\hat{a}},\underline{\hat{a}})=||\underline{\hat{c}} (\underline{\hat{a}})||^{2}_{Q_{\text{dr}}(a)\setminus\{\text{dr}\}} \tag{17}\]  
Note that this statistic also follows directly from the AK test statistic when \(a\) in (9) is replaced by the integer estimator \(\underline{\hat{a}}\). Therefore, it can be seen as a direct ambiguity-resolved generalization of the AK test, having the advantage that now only the integer estimation under \(\mathcal{H}_{a}\) needs to be performed. The PDF of \(\underline{\hat{x}}=\underline{\hat{c}}(\underline{\hat{a}})\) was already given in [20], Equation (36), as \(f_{\ell}(x)=\sum_{z\in\mathbb{Z}^{n}}f_{\ell(z)}(x)\text{P}[\underline{\hat{a} }=z]\), with \(\underline{\hat{c}}(z)\sim\mathcal{N}_{q}(c+\hat{C}^{+}A(a-z),Q_{\text{$\ell$}(a )\setminus\{\text{dr}\}})\). This result can be used to determine the PDF of (17). **Lemma 2** (Distribution of \(\underline{T}(\underline{\hat{a}},\underline{\hat{a}})\)): _The PDF of the test statistic (17) is given as_  
\[f_{\Upsilon(\underline{\hat{a}},\underline{\hat{a}})}(x)=\sum_{z\in\mathbb{Z}^{ n}}f_{\chi^{2}(q,\lambda_{z})}(x)\text{P}[\underline{\hat{a}}=z] \tag{18}\]with noncentrality parameter \(\lambda_{z}=||c+\bar{C}^{+}A(a-z)||_{Q_{(\bar{a})}(a)}^{2}\)._  
Proof. : See Appendix A. Note that if the ambiguity success rate goes to one, \(\mathsf{P}[\underline{a}=z]\uparrow 1\), the PDF (18) indeed converges to that of the AK test (9). Also note that, as before, we could have used the PDF of \(\underline{\xi}=\underline{\xi}(\underline{a})\) directly, by means of the HD test: reject \(\mathcal{H}_{0}\) if \(f_{\xi}(x)<\mu_{a}\), to test for the significance of the bias \(c\). Such a test was proposed for outlier testing in [18]. However, as mentioned earlier, such would require dealing with the infinite sum for _each_ individual execution of the test, while at the same time, its GNSS performance will generally be similar to that of (17). Furthermore, for the test statistic \(T(\underline{a},\underline{a})\), we have its exact distribution available (cf. (18)). The fact that it also consists of an infinite sum is not an issue, since its usage for critical value and power computations can usually be performed off-line at the designing stage, see [19]. Although the ambiguity success rates are not required to be as large as needed for ambiguity-fixed GNSS baseline estimation [1], it is clear from the above considerations that they still play a driving role in the performance of the ambiguity-resolved tests. Since it may happen that _full_ ambiguity success rates become drastically reduced when switching from \(\mathcal{H}_{0}\) to \(\mathcal{H}_{a}\), particularly when the dimension of \(c\) increases, _partial_ ambiguity resolution (PAR) is a strategy to keep the reduction in ambiguity success rate at bay. This is illustrated for ionospheric testing in the example of Figure 4. It shows how PAR can obtain large \(\mathcal{H}_{a}\) success rates, thus enabling the conditions of Theorem 1, thereby significantly improving the power compared to that of AF testing.