Consistency of the Empirical Distributions of Navigation Positioning System Errors with Theoretical Distributions-Comparative Analysis of the DGPS and EGNOS Systems in the Years 2006 and 2014.

Positioning systems are used to determine position coordinates in navigation (air, land and marine). The accuracy of an object's position is described by the position error and a statistical analysis can determine its measures, which usually include: Root Mean Square (RMS), twice the Distance Root Mean Square (2DRMS), Circular Error Probable (CEP) and Spherical Probable Error (SEP). It is commonly assumed in navigation that position errors are random and that their distribution are consistent with the normal distribution. This assumption is based on the popularity of the Gauss distribution in science, the simplicity of calculating RMS values for 68% and 95% probabilities, as well as the intuitive perception of randomness in the statistics which this distribution reflects. It should be noted, however, that the necessary conditions for a random variable to be normally distributed include the independence of measurements and identical conditions of their realisation, which is not the case in the iterative method of determining successive positions, the filtration of coordinates or the dependence of the position error on meteorological conditions. In the preface to this publication, examples are provided which indicate that position errors in some navigation systems may not be consistent with the normal distribution. The subsequent section describes basic statistical tests for assessing the fit between the empirical and theoretical distributions (Anderson-Darling, chi-square and Kolmogorov-Smirnov). Next, statistical tests of the position error distributions of very long Differential Global Positioning System (DGPS) and European Geostationary Navigation Overlay Service (EGNOS) campaigns from different years (2006 and 2014) were performed with the number of measurements per session being 900'000 fixes. In addition, the paper discusses selected statistical distributions that fit the empirical measurement results better than the normal distribution. Research has shown that normal distribution is not the optimal statistical distribution to describe position errors of navigation systems. The distributions that describe navigation positioning system errors more accurately include: beta, gamma, logistic and lognormal distributions.

Statistical Distribution Analysis of Navigation Positioning System Errors-Issue of the Empirical Sample Size.

Positioning systems are used to determine position coordinates in navigation (air, land, and marine). Statistical analysis of their accuracy assumes that the position errors (latitude-δφ and longitude-δλ) are random and that their distributions are consistent with the normal distribution. However, in practice, these errors do not appear in a random way, since the position determination in navigation systems is done with an iterative method. It causes so-called "Position Random Walk", similar to the term "Random Walk" known from statistics. It results in the empirical distribution of δφ and δλ being inconsistent with the normal distribution, even for samples of up to several thousand measurements. This phenomenon results in a significant overestimation of the accuracy of position determination calculated from such a short series of measurements, causing these tests to lose their representativeness. This paper attempts to determine the length of a measurement session (number of measurements) that is representative of the positioning system. This will be a measurement session of such a length that the position error statistics (δφ and δλ) represented by the standard deviation values are close to the real values and the calculated mean values (φ¯ and λ¯) are also close to the real values. Special attention will also be paid to the selection of an appropriate (statistically reliable) number of measurements to be tested statistically to verify the hypothesis that the δφ and δλ distributions are consistent with the normal distribution. Empirical measurement data are taken from different positioning systems: Global Positioning System (GPS) (168'286 fixes), Differential Global Positioning System (DGPS) (864'000 fixes), European Geostationary Navigation Overlay Service (EGNOS) (928'492 fixes), and Decca Navigator system (4052 fixes). The analyses showed that all researched positioning systems (GPS, DGPS, EGNOS and Decca Navigator) are characterized by the Position Random Walk (PRW), which resulted in that the empirical distribution of δφ and δλ being inconsistent with the normal distribution. The size of the PRW depends on the nominal accuracy of position determination by the system. It was found that measurement sessions consisting of 1000 fixes (for the GPS system) overestimate the accuracy analysis results by 109.1% and cannot be considered representative. Furthermore, when analyzing the results of long measurement campaigns (GPS and DGPS), it was found that the representative length of the measurement session differs for each positioning system and should be determined for each of them individually.