sensors Article 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 Mariusz Specht Department of Transport and Logistics, Gdynia Maritime University, Morska 81-87, 81-225 Gdynia, Poland; [email protected] Abstract: 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 Citation: Specht, M. Consistency of assessing the fit between the empirical and theoretical distributions (Anderson-Darling, chi-square the Empirical Distributions of and Kolmogorov-Smirnov). Next, statistical tests of the position error distributions of very long Navigation Positioning System Errors Differential Global Positioning System (DGPS) and European Geostationary Navigation Overlay with Theoretical Distributions— Service (EGNOS) campaigns from different years (2006 and 2014) were performed with the number Comparative Analysis of the DGPS of measurements per session being 900’000 fixes. In addition, the paper discusses selected statistical and EGNOS Systems in the Years 2006 distributions that fit the empirical measurement results better than the normal distribution. Research and 2014. Sensors 2021, 21, 31. has shown that normal distribution is not the optimal statistical distribution to describe position https://dx.doi.org/10.3390/s21010031 errors of navigation systems. The distributions that describe navigation positioning system errors more accurately include: beta, gamma, logistic and lognormal distributions. Received: 17 November 2020 Accepted: 21 December 2020 Keywords: statistical distribution; navigation positioning system; position error; Differential Global Published: 23 December 2020 Positioning System (DGPS); European Geostationary Navigation Overlay Service (EGNOS) Publisher’s Note: MDPI stays neu- tral with regard to jurisdictional claims in published maps and institutional affiliations. 1. Introduction The assumption that a position line error in navigation has a normal distribution is commonplace for book authors [1–3], as well as in monographs, regulations and standards directly related to the statistical analysis of position errors [4,5]. It should be noted, however, Copyright: © 2020 by the author. Li- that several scientific publications draw attention to existing differences between the censee MDPI, Basel, Switzerland. This article is an open access article distributed empirical and theoretical distributions. Global Positioning System (GPS) is the basic under the terms and conditions of the positioning system used in navigation. Its operational characteristics are periodically Creative Commons Attribution (CC BY) described in several technical standards, of which Global Positioning System Standard license (https://creativecommons.org/ Positioning Service Signal Specification versions have already been issued five times; licenses/by/4.0/). in 1993, 1995, 2001, 2008, and 2020. The first version of this document [6] states expressly Sensors 2021, 21, 31. https://dx.doi.org/10.3390/s21010031 https://www.mdpi.com/journal/sensors Sensors 2021, 21, x FOR PEER REVIEW 2 of 21 Sensors 2021, 21, 31 2 of 23 cally described in several technical standards, of which Global Positioning System Standard Positioning Service Signal Specification versions have already been issued five times; in 1993, 1995, 2001, 2008, and 2020. The first version of this document [6] states thatexpressly the empirical that the error empirical distributions error distributions are overlaid are with overlaid Gauss with distributions, Gauss distributions, as a basis for as a comparisonbasis for comparison with theoretical with theoretical expectations expect (Figureations1). The(Figure theoretical 1). The distributionstheoretical distribu- were generatedtions were using generated the means using and the standardmeans and deviations standard of deviations the empirical of the datasets. empirical The datasets. error distributionsThe error distributions are based uponare based measured upon datameas fromured thedata GPS from Control the GPS Segment Control monitor Segment stationsmonitor recorded stations forrecorded three months. for three months. (a) (b) (c) FigureFigure 1. 1.Comparison Comparison of of empirical empirical data data of of the the GPS GPS position position error error in in North North (a (),a), East East (b ()b and) and vertical vertical (c()c axes) axes with with the the theoretical theoretical normal normal distribution. distribution. Own Own study study based based on: on: [ 6[6].] TheThe presented presented differences differences in in local local axes axes must must result result in in significant significant differences differences in in the the fit fit ofof 2D 2D position position error error with with the the chi-square chi-square distribution. distribution. Therefore, Therefore, in in the the same same document, document, FigureFigure2 presents2 presents the the empirical empirical (64 (64 m) m) and and theoretical theoretical (83 (83 m) m) values values of of twice twice the the Distance Distance RootRoot Mean Mean Square Square (2DRMS) (2DRMS) measure. measure. It It should should be be stressed stressed that that since since the the estimation estimation error error waswas as as high high as as 19 19 m, m, 95% 95% of of the th measurementse measurements will will be smallerbe smaller than than this. this. In thisIn this situation, situation, it isit difficult is difficult to support to support the use the of use normal of normal distribution distribution for the calculation for the calculation of the basic of quantitythe basic describingquantity describing the system the positioning system positioning accuracy accuracy (2DRMS). (2DRMS). Similar conclusions Similar conclusions concerning con- the inconsistency of the statistical distributions of Differential Global Positioning System (DGPS) and GPS position errors are raised by the Frank van Diggelen, but with much smaller discrepancies [5]. Sensors 2021, 21, x FOR PEER REVIEW 3 of 21 Sensors 2021, 21, 31 cerning the inconsistency of the statistical distributions of Differential Global Positioning3 of 23 System (DGPS) and GPS position errors are raised by the Frank van Diggelen, but with much smaller discrepancies [5]. FigureFigure 2.2. NominalNominal GPSGPS SPSSPS horizontalhorizontal errorerror distribution.distribution. OwnOwn studystudy basedbased on:on: [[6].6]. In the authors’ research on the accuracy of various navigation positioning systems, two In the authors’ research on the accuracy of various navigation positioning systems, measures of accuracy were also often compared: 2DRMS and R95. The latter is an empirical two measures of accuracy were also often compared: 2DRMS and R95. The latter is an quantity calculated by sorting errors from the smallest to the largest. This value is higher empirical quantity calculated by sorting errors from the smallest to the largest. This value than 95% of the measurements made. Please note that if the empirical 2DRMS statistics is higher than 95% of the measurements made. Please note that if the empirical 2DRMS fit the chi-square distribution, these values should be almost identical. The author’s statistics fit the chi-square distribution, these values should be almost identical. The au- research into the accuracy of various Global Navigation Satellite Systems (GNSS), such thor’s research into the accuracy of various Global Navigation Satellite Systems (GNSS), as DGPS and European Geostationary Navigation Overlay Service (EGNOS) [7], GNSS such as DGPS and European Geostationary Navigation Overlay Service (EGNOS) [7], geodetic networks, as well as multi-GNSS solutions [8,9], has repeatedly shown significant GNSS geodetic networks, as well as multi-GNSS solutions [8,9], has repeatedly shown discrepancies between 2DRMS and R95 measures. significant discrepancies between 2DRMS and R95 measures. In order to quantify the discrepancy between the 2DRMS and R95 measures, let us In order to quantify the discrepancy between the 2DRMS and R95 measures, let us analyse the results of the position accuracy tests of six different mobile phones working inanalyse parallel, the whichresults were of the conducted position