The ratio test for future GNSS ambiguity resolution

Abstract

The performance of the popular ambiguity ratio test is analyzed. Based on experimental and simulated data, it is demonstrated that the current usage of the ratio test with fixed critical value is not sustainable in light of the enhanced variability that future global navigation satellite system (GNSS) ambiguity resolution will bring. As its replacement, the model-driven ratio test with fixed failure rate is proposed. The characteristics of this fixed-failure rate ratio test are described, and a performance analysis is given. The relation between its critical value and various GNSS model parameters is also studied. Finally, a procedure is presented for the creation of fixed failure rate look-up tables for the critical values of the ratio test.

Appendix

Here, we will work with the reciprocal of the ratio test, which means that $\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{a}$ is accepted if $\frac{{q({\mathbf{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{a} }})}}{{q({\mathbf{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{a}^{\prime}}})}} \le \frac{1}{c} = \mu$, since the threshold value $\mu$ is then limited as $0 \le \mu \le 1$.

Procedure for determining the threshold value $\mu$ based on simulations

• 1. For a given model, calculate P _f,ILS , for example, use the upper bound from integer bootstrapping (Teunissen 1999). If P _f,ILS $\le$ P _f , set $\mu = 1$, otherwise continue with step 2.

• 2. Using Monte Carlo simulations, generate N samples of normally distributed float ambiguities: ${\hat{\mathbf{a}}}_{{\mathbf{i}}} \sim N(0,{\mathbf{Q}}_{{{\mathbf{\hat{a}\hat{a}}}}} ),{\mkern 1mu} \begin{array}{*{20}c} {} & {i = 1, \ldots ,N} \\ \end{array}$.

• 3. Determine the ILS solutions (best and second-best candidates) ${\mathbf{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{a} }}_{{\mathbf{i}}} {\mathbf{, \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{a}^{\prime}}}_{i} , \, R_{i} = \frac{{q({\mathbf{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{a} }}_{{\mathbf{i}}} )}}{{q({\mathbf{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{a}^{\prime}}}_{i} )}}$.

• 4. The simulation-based failure rate as function of $\mu$ is given by ${\mathbf{P}}_{{\mathbf{f}}} (\mu ) = \frac{{N_{f} }}{N}$ with $N_{f} = \mathop {\mathop \sum \limits^{N} }\limits_{i = 1} \omega (R_{i} ,{\mathbf{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{a} }}_{{\mathbf{i}}} )$ where $\omega (R_{i} ,\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{a}_{i} )$ equals 1 if $R_{i} \le \mu \;,\;{\mathbf{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{a} }}_{{\mathbf{i}}} \ne 0$, and 0 otherwise.

Choose $\mu_{\hbox{min} } = (\hbox{min} (R_{i} ) - 10^{ - 16} )$ since this results in $N_{f} = 0{\mkern 1mu} \Rightarrow {\mkern 1mu} {\mathbf{P}}_{{\mathbf{f}}} (\mu_{\hbox{min} } ) = 0$, $\mu_{\hbox{max} } = \hbox{max} (R_{i} )$ which results in ${\mathbf{P}}_{{\mathbf{f}}} (\mu_{\hbox{max} } ) = {\mathbf{P}}_{{{\mathbf{f,ILS}}}}$.

• 5. Use a root finding method to find $\mu \in [\mu_{\hbox{min} } ,\mu_{\hbox{max} } ]{\text{ so that }}{\mathbf{P}}_{{\mathbf{f}}} (\mu ) - {\mathbf{P}}_{{\mathbf{f}}} = 0$.

This will give the solution since ${\mathbf{P}}_{{\mathbf{f}}} (\mu_{\hbox{min} } ) - {\mathbf{P}}_{{\mathbf{f}}} < 0$ and ${\mathbf{P}}_{{\mathbf{f}}} (\mu_{\hbox{max} } ) - {\mathbf{P}}_{{\mathbf{f}}} > 0,$

and the failure rate is monotonically increasing for increasing $\mu$.

Procedure for creating the look-up table

• 1. Generate many different models based on varying satellite geometry (system, time and location), number of frequencies, number of epochs, measurement noise, baseline length and the availability and accuracy of atmosphere corrections.

• 2. For each model, use the simulation approach described above to find the threshold value for the given fixed failure rate P _f.

• 3. Select all models with the same number of ambiguities n.

• 4. Plot the ILS failure rate P _f,ILS versus the threshold value $\mu$ determined with the simulations for all models with the same n (See example in Fig. 9. Note that here $c = 1/\mu$ is shown).

• 5. In order to be conservative, one can manually select the points creating the blue solid line as in Fig. 9, which ensures that with the corresponding threshold value for a given ILS failure rate, this value $\mu = 1/c$ is somewhat lower than required, that is, more conservative.

• 6. The solid blue line can then be used to determine the appropriate threshold values $\mu$ for the ILS failure rates in the table, and these can be stored in the column corresponding to dimension n.