Nominal and extreme error performance of the UNB3 tropospheric delay model
Canada and the United States have been co-operating for many years to develop approvals for Global Positioning System (GPS) use in aviation. Under a standing agreement between the two nations, the University of New Brunswick (UNB) has been approached to produce an improved tropospheric delay model to be used in airborne Wide Area Augmentation System (WAAS) receivers and to estimate the bounds of the tropospheric delay contribution to the WAAS error budget. Previous research work at UNB resulted in a proposed tropospheric propagation delay model for use in aircraft receivers operating in the GPS-WAAS navigation system. This model, denoted UNB3, does not require real-time input of meteorological parameters, relying instead on a look-up table to model the latitude and seasonal trend of atmospheric parameters. Such a concept is susceptible however, to both rare, anomalous conditions and local conditions that are significantly different from the average conditions at the same latitude and season. This study has investigated the performance of the UNB3 model over an extremely wide range of atmospheric conditions. A 10-year radiosonde data set has been used as a “truth” source against which to compare the model. Because of the extremely large nature of this data set, and the requisite time required to process it, the error analysis has been confined to the error experienced in the zenith direction only. This component is expected to experience the largest error, whereas the mapping function component is known to be affected less significantly. The nominal zenith error of the UNB3 model is −2 ± 5 cm (1σ), a level of performance that compares very well with that determined by our previous study. This confirms that only a comparatively small amount of data is needed to study the nominal error performance of any tropospheric delay model. Of course the actual error varies widely for any one time and place, but we have found from our extensive data set that only 0.0075% of the residual zenith errors equal or exceed a range of ±20 cm. Those errors that do equal or exceed the ±20 cm range can be used to predict future “extremes”. The extreme negative errors are limited by the maximum value of the wet zenith delay (~27 cm), assuming an unusually dry, tropical atmosphere and negligible error in the hydrostatic component. Fitting an extreme value distribution to the 10 values representing the largest negative errors in each year, suggests that the worst case error is a few centimetres smaller. The largest positive error is harder to determine. While probably not completely unlimited, it has not been possible to determine an analytical maximum. Empirically however, the extreme value distribution fitted to the 10 values representing the largest positive errors in each year, suggests a maximum on the order of 55 cm or more occurring at least once during an average of 25 years. The direct impact of an unmodelled tropospheric error on the GPS vertical position determination is reduced by the dependence on the satellite geometry. Suitable elevation angle weighting schemes reduce the impact further, provided that a majority of satellites are not congregating near the same low elevation angle. This fact, when combined with the very rare rate of occurrence, means that unmodelled tropospheric errors are almost certainly negligible for WAAS users when using the UNB3 model.