A Neural Network approximation of Fixed Failure-rate Ratio Test for PPP Ambiguity Resolution

The correct resolution of integer carrier-phase ambiguities is a key element for improving user precise positioning solutions, especially during the convergence. However, wrongly fixed ambiguities might deteriorate the solution, so posing a potential threat for safety-critical applications. Within the framework of the Least-squares AMBiguity Decorrelation Adjustment (LAMBDA) method, a Fixed Failure-rate Ratio Test (FFRT) has been proposed to generate ratio-test critical values according to a tolerable failure rate.

For a given failure rate, FFRT thresholds’ computation via Monte Carlo (MC) simulations is generally not computationally efficient. At the same time, with the recent LAMBDA 4.0 toolbox implementation, fitting functions introduced by Hou et al. (2016) were integrated for computing these critical values and therefore controlling the failure rate to prevent unnecessary false alarms. Still, Lookup Tables (LT) represent a conservative approach rather than a close approximation to critical values for a given model strength.

In this contribution we leverage the latest developments in Machine Learning (ML), thus focusing on a Neural Network (NN) function approximation. The latter one considers the components of the ambiguity variance-covariance matrix as input and provides the FFRT critical value for a given failure rate. In this way, it is possible to provide a very accurate approximation (close to MC-based results) with an efficiency comparable to LT approach in use by the latest LAMBDA 4.0 implementation.

For the NN training, several GNSS scenarios are synthetically generated based on actual satellite orbits. Hence, we produce datasets for Precise Point Positioning with Ambiguity Resolution (PPP-AR) user models in an uncombined form, based on a Kalman Filter. It is numerically shown how considering the ‘conditional variances’ as inputs for the NN is sufficient for an approximation of FFRT thresholds, which are otherwise too conservative when using the LT approach developed by Hou et al. (2016). The NN results are therefore assessed based on independent datasets not involved during the training stage.

These three approaches, i.e. MC, NN, LT, are ultimately compared in PPP-AR processing using real data from 30 well-distributed stations from the IGS global network, based on the use of CODE Final products. It is further shown how adoption of fixed critical values for the ratio test, like 2 or 3, often leads to a very conservative ambiguity validation, i.e. returning float solutions when RT is rejected. On the other hand, a properly defined FFRT estimator allows improving user convergence time, as well as enabling more advanced ambiguity validation strategies for PPP-AR, as discussed in this work.

This research has been funded by the ESA’s Navigation Innovation and Support Program (NAVISP) Element 1 programme [https://navisp.esa.int/project/details/307/show].