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On reliable data-driven partial GNSS ambiguity resolution

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Abstract

In high-precision global navigation satellite system applications, it is often not possible to simultaneously meet the requirements for fast and reliable integer ambiguity resolution. For a given reliability constraint in form of a user-defined, tolerable probability of an incorrect ambiguity estimate, resolving a subset of ambiguities instead of the full set can be beneficial. We discuss a fixed failure rate implementation of a data-driven, likelihood-ratio-based partial ambiguity resolution technique. A key problem in this context is the efficient determination of a scalar that is a model-dependent threshold value. This problem is approached via a conservative functional approximation of the threshold value. The only input parameter of the function is the integer least-squares failure rate of the system model under consideration. Numerically simulated single and combined system GPS/Galileo single baseline cases with single- and dual-frequency measurements are used to analyze the impact of the approximation. The results indicate that the conservative description hardly affects the performance of the algorithm, while the predefined failure rate is not exceeded. Moreover, it is shown that the presented data-driven partial ambiguity resolution approach clearly outperforms a purely model-driven scheme based on the bootstrapping failure rate.

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Acknowledgments

This work was initiated during the author’s stay as a visiting researcher at the GNSS Research Centre at Curtin University, Perth, Australia. The discussions with Prof. Peter Teunissen and his helpful suggestions for this work are greatly appreciated. In particular, the idea of developing functional threshold descriptions comes from him.

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Correspondence to Andreas Brack.

Appendix

Appendix

Proof of Theorem

The failure rate is decreasing with increasing threshold value; the success rate is neither decreasing nor increasing with increasing threshold value.

For any realization of \({\hat{\varvec{a}}}\), the log-likelihood-ratios are \(l_{i} \left( {{\hat{\varvec{a}}}} \right)\) with i = 1, …, n. Given a certain value of \(l_{\text{th}}\) and the corresponding \(P_{\text{f}}\) and \(P_{\text{s}}\):

Assume \(l^{\prime}_{\text{th}} > l_{\text{th}}\): For any realization of \({\hat{\varvec{a}}}\), \({\mathcal{I}}(l^{\prime}_{\text{th}} ) \subseteq {\mathcal{I}}(l_{\text{th}} )\) (20), where \({\mathcal{I}}(l_{\text{th}} )\) denotes the index set \({\mathcal{I}}\) for the specific \({\hat{\varvec{a}}}\) corresponding to \(l_{\text{th}}\).

  • If \({\varvec{z}}^{{ \star }}_{{({\mathcal{I}}(l_{\text{th}} ))}} = {\varvec{a}}_{{({\mathcal{I}}(l_{\text{th}} ))}}\) (success), it can now happen that \({\mathcal{I}}(l^{\prime}_{\text{th}} ) = \emptyset\) (no success, no failure) or \({\varvec{z}}^{{ \star }}_{{({\mathcal{I}}(l^{\prime}_{\text{th}} ))}} = {\varvec{a}}_{{({\mathcal{I}}(l^{\prime}_{\text{th}} ))}}\) (success).

  • If \({\varvec{z}}^{{ \star }}_{{({\mathcal{I}}(l_{\text{th}} ))}} \ne {\varvec{a}}_{{({\mathcal{I}}(l_{\text{th}} ))}}\) (failure), it can now happen that \({\mathcal{I}}(l^{\prime}_{\text{th}} ) = \emptyset\) (no success, no failure) or \({\varvec{z}}^{{ \star }}_{{({\mathcal{I}}(l^{\prime}_{\text{th}} ))}} = {\varvec{a}}_{{({\mathcal{I}}(l^{\prime}_{\text{th}} ))}}\) (success) or \({\varvec{z}}^{{ \star }}_{{({\mathcal{I}}(l^{\prime}_{\text{th}} ))}} \ne {\varvec{a}}_{{({\mathcal{I}}(l^{\prime}_{\text{th}} ))}}\) (failure).

  • If \({\mathcal{I}}(l_{\text{th}} ) = \emptyset\), then \({\mathcal{I}}(l^{\prime}_{\text{th}} ) = \emptyset\).

Integration over all possible realizations of \({\hat{\varvec{a}}}\) leads to the following conclusions:

  • Increasing \(l_{\text{th}}\) to \(l^{\prime}_{\text{th}}\) decreases the probability of the event failure, i.e., \(P^{\prime}_{\text{f}} < P_{\text{f}}\), where \(P^{\prime}_{\text{f}}\) corresponds to \(l^{\prime}_{\text{th}}\).

  • Increasing \(l_{\text{th}}\) to \(l^{\prime}_{\text{th}}\) does not allow for a statement on \(P^{\prime}_{\text{s}}\), i.e., \(P^{\prime}_{\text{s}}\) may be smaller, equal, or greater than \(P_{\text{s}}\), where \(P^{\prime}_{\text{s}}\) corresponds to \(l^{\prime}_{\text{th}}\). □

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Brack, A. On reliable data-driven partial GNSS ambiguity resolution. GPS Solut 19, 411–422 (2015). https://doi.org/10.1007/s10291-014-0401-9

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