**1. Introduction**

The main goal behind positioning systems is to provide air [1–5], land [6–10] and marine navigation [11–15] applications with such accuracy that the process is carried out safely. Such applications may include the following: a ship entering a port, driving a car along a route, landing an aircraft at an airport, stopping a tram on a platform, etc. The decision to qualify a positioning system as safe for a given navigation application is made based on a comparison of the position error characterizing the system with the minimum navigation requirements for a specific application specified in normative documents, such as radio navigation plans [16–19] and other recommendations or regulations [20–24]. These requirements most often include the following: positioning accuracy, availability, continuity, fix rate, integrity, operation range, reliability [25].

There is no doubt that, for decades of navigation development, it is the positioning accuracy that has been, and still is, the decisive factor for the use of a system for a specific navigation task, while the availability level of a specific position error is related to the threat that a positioning failure may pose to the safety of an object. Hence, air navigation requires top positioning availability. Figure 1 presents a synthesis of the requirements for different navigation applications in terms of the maximum permissible position error and its availability. The requirements found in various normative documents were used for the analysis.

**Citation:** Specht, M. Determination of Navigation System Positioning Accuracy Using the Reliability Method Based on Real Measurements. *Remote Sens.* **2021**, *13*, 4424. https:// doi.org/10.3390/rs13214424

Academic Editor: Serdjo Kos

Received: 30 September 2021 Accepted: 2 November 2021 Published: 3 November 2021

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**Copyright:** © 2021 by the author. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/).

**Figure 1.** Requirements for navigation positioning systems. Own study based on [16–24,26].

Figure 1 shows that each of the navigation applications should use a system with adequate positioning accuracy. This means, among other things, that a system with a positioning accuracy of 100 m cannot be used in hydrography as it requires a system with a positioning accuracy of less than 20 m for order 2, 5 m for orders 1a/1b, 2 m for special order and 1 m for exclusive order.

The navigation system positioning accuracy refers to the overlap between the statistics of the measured position coordinates and their real values or values assumed to be real (most often the average value from latitude (*ϕ*) and longitude (λ) measurements) [19]. A position accuracy measure is its error, which can be evaluated with reference to any dimension: plane (2D) or space (3D). In navigation, a position error is defined as the radius of the circle or sphere within which a certain proportion of position determinations should statistically fall. It is commonly accepted in navigation that position error statistics refer to 95% of the population, as the safety of the navigation process depends largely on the navigation system positioning accuracy. The Twice the Distance Root Mean Square (2DRMS(2D)) is used as a position accuracy measure. Its determination starts with the calculation of the DRMS(2D), which is the square root from the sum of the squares of the standard deviations of the position coordinates relative to *ϕ* and *λ*, as per the following relationship:

$$DRMS(2D) = \sqrt{\left(s\_{\varphi}\right)^2 + \left(s\_{\lambda}\right)^2},\tag{1}$$

where: *sϕ*—standard deviation of the geodetic (geographic) latitude; *sλ*—standard deviation of the geodetic (geographic) longitude.

The probability of the DRMS(2D) lies in the 63.2–68.3% range and depends on the relationship between the standard deviations. For *s<sup>ϕ</sup>* = *sλ*, *p* = 0.63, while for the relation *s<sup>ϕ</sup>* = 10 ·*sλ*, *p* = 0.68.

To provide greater statistical reliability of the DRMS(2D) in navigation, the 2DRMS(2D) measure is commonly used, taking the following form:

$$2DRMS(2D) = 2\sqrt{\left(s\_{\varphi}\right)^2 + \left(s\_{\lambda}\right)^2}.\tag{2}$$

In navigation literature, the 2DRMS corresponds to a probability lying in the range 95.4–98.2% and is related to the relationship between the standard deviations determined with the two coordinates. Figure 2 shows the geometric interpretation of the position error.

**Figure 2.** Geometric interpretation of the concept of navigation system position error in 2D plane using DRMS and 2DRMS values.

Taking the 2DRMS value as the primary position accuracy measure by a navigation system is based on the assumption that *ϕ* and *λ* errors are normally distributed [27,28]. However, this belief has been questioned in several publications. The most important standard describing Global Positioning System (GPS) accuracy characteristics [28] states that the difference between the empirical value (64 m) and the theoretical value (83 m), as determined by the 2DRMS measure, was as much as 19 m. Similar conclusions concerning the inconsistency of the statistical distributions of Differential Global Positioning System (DGPS) and GPS position errors were raised by Frank van Diggelen, but with much smaller discrepancies [29].

The author's research, conducted on various navigation positioning systems, has repeatedly confirmed the existence of such discrepancies. They related to systems such as DGPS and the European Geostationary Navigation Overlay Service (EGNOS) [30], the Global Navigation Satellite System (GNSS) and geodetic networks and multi-GNSS solutions [31–33].

Questioning normal distribution as a model for navigation positioning system errors has prompted the search for other methods to determine the value of the position error with 95% probability, as commonly used in navigation [27,28]. One of the methods based on reliability theory has already been proposed in [25]. This method allows the navigation system positioning availability to be determined for a specific (given) value of the position error based on life and failure times, and not based on measurement errors. The positioning system is fit when the position error does not exceed the allowable error. The failure period is defined as the logical negation of the fit period. A comparison of both methods (classical and reliability) is presented in Figure 3.

**Figure 3.** Comparison of the classical and reliability methods for assessing the positioning system's ability to meet the accuracy requirements for a navigation application.

In the classical approach, for a fixed probability (usually amounting to 95%), the position error (value of the 2DRMS) is calculated, whereas in the reliability method an acceptable error value is determined first, and only then is its probability is calculated. In the first method, the position error is a random variable and in the second method the random variable is the life or failure time. In the first method, 1D errors are assumed to be normally distributed and 2D errors are assumed to follow a chi-square distribution, while in the second method, exponential distributions of life and failure times are assumed.

The characteristics of the classical (based on the normal distribution) approach are as follows:

• The calculations are based on simple Root Mean Square (RMS) determination relationships;


The reliability approach has the following characteristics:


This paper aims to compare both methods and to evaluate the positioning system in terms of its fitness for a specific navigation application, based on real measurement data from the positioning system (long sessions). Therefore, the scientific purpose of this article are as follows:


To assess which of the methods allows for more precise determination of the statistical value of the position errors, the measured position errors were sorted from the smallest to the largest, and based on this the value of the error that is greater exactly than 95% of the error population will be determined. In the navigation literature [34], this value is referred to as the R95 measure.

This is the third article in a series of monothematic publications "Research on empirical (actual) statistical distributions of navigation system position errors" [35–37]. The main scientific aim of this series is to answer the question of what statistical distributions follow the position errors of navigation systems such as GPS, Global Navigation Satellite System (GLONASS), BeiDou Navigation Satellite System (BDS), Galileo, DGPS, EGNOS and others. It must be emphasised that the purpose of both this paper and the whole series of publications is not to analyse the causes of Position Random Walk (PRW), such as ionospheric and tropospheric effects, multipath, noise, etc. This article rather analyses the statistical distributions of 1D and 2D position errors resulting from PRW. The causes might be very complex and probably deserve a separate series of publications.
