Lucht An example: dual versus triple frequency GPS even with a high enough success-rate, fixing to the wrong integer ambiguities is still possible when one or more observations are erroneous. A success-rate close enough to one does therefore not release one from the obligation of performing statistical tests for model validation. It does however make it much more easier to perform such tests. The higher the success-rate, the sooner one is allowed to apply the classical theory of statistical hypothesis testing. Figure 3: By taking the integral of the probability density function (at left) over the pull-in region for each integer vector, the probability is obtained that this vector will result as the integer least-squares solution. The probabilities are given at right for the integer vectors between -1 and 1. The integral over the area for the correct integer vector, in this case (0,0), gives the success-rate. It is about 0.85 in this example. The success-rate can also serve as a tool in analyzing the benefits of three frequency GPS or as a design-tool in choosing the three frequencies forthe planned European 'Galileo', see [2]. As such an example, we will compare the performance of the current dual-frequency GPS with the future triple-frequency GPS, also referred to as 'modernized GPS'. This comparison will be based on the so-called geometry- free model. This model is the simplest possible GNSS model that still allows the estimation of integer carrier phase ambiguities. In its most basic form the model consists of the double-differenced pseudo range and carrier phase observations of two receivers to two satellites, parameterized in terms of an unknown double- differenced satellite-receiver range, unknown ambiguities and an unknown ionospheric delay. The ionospheric delay is included so as to make the model applicable for long baselines. We will first study the dual-frequency success rate in its dependence on a varying second frequency, whilst the first frequency is kept fixed to the GPS LI frequency. This is shown in figure 4, at left. First note that the success rate fails to exceed the very small value of 0.025 within the frequency range shown. This stipulates the poor performance of instantaneous dual-frequency ambiguity resolution for long baselines. Hence, for these cases one can not expect dual-frequency ambiguity resolution to be successful. The figure also shows that the success rate reaches its minimum when the two frequencies coincide and that the success rate gets larger when the frequency separation gets larger. This contradicts the popular belief that ambiguity resolution would benefit from choosing the frequencies close together. It is of course still true that frequencies with little separation would allow one to construct a wide-lane with a corresponding very large wavelength. However, as the figure shows this turns out to be counterproductive as far as the overall success rate is concerned. In fact, as the figure shows, the success rate will be identical to zero when the two frequencies coincide. This is understandable when one recognizes that a nonzero frequency separation is needed per se in order to be able to estimate the ionospheric delays. When the two frequencies coincide, the ionospheric delay becomes non-estimable and the variance-covariance matrix 107

Digitale Tijdschriftenarchief Stichting De Hollandse Cirkel en Geo Informatie Nederland

Lustrumboek Snellius | 2000 | | pagina 118