**3. Results**

The research aims were formulated in the form of questions:


Research on distributions of life times began with the analysis of GPS position error results. A recording session of 168 286 fixes was studied. This can be considered statistically representative, according to the analyses conducted in [37]. Figure 7 presents the graph of the position error as a function of time. The figure also shows two accuracy values (1 m and 2 m), which correspond to the requirements of road transport for vehicle identification and hydrography for special order.

**Figure 7.** GPS position error as a function of time (168 286 fixes) and two decision thresholds corresponding to the requirements of road transport for vehicle identification and hydrography for special order.

To automate the process of determining life and failure times, the position error values needed to be assigned to one of the operational statuses: 1 (life status, when the temporary position error is below a pre-set limit value) or 0 (failure status, in the opposite case) at any point in time. For this purpose, a calculation sheet was prepared in Mathcad 15 (Figure 8).

**Figure 8.** Mathcad worksheet for determining life and failure times based on the GPS 2013 measurement campaign.

In the sheet shown in Figure 8, the calculations began with importing measurement data consisting of position errors. The position error limit was then determined arbitrarily. In the worksheet, it is 1m. Next, two vectors were created consisting of life (vector\_11) and failure (vector\_10) times.

Both vectors were saved as text files and then uploaded to the EasyFit software, where they were analysed for fit to typical statistical distributions. These distributions included: beta, Cauchy, chi-square, exponential, gamma, Laplace, logistic, lognormal, normal, Pareto, Rayleigh, Student's and Weibull.

For GPS position errors, calculations were carried out for two variants, 1 m and 2 m, corresponding to example navigation applications such as from road transport for vehicle identification (1 m) and hydrography for special order (2 m). In P-P plots, the empirical probability distribution function is plotted against the theoretical distribution. The observations are first sorted in descending order. The *i*-th observation is then plotted on one axis as *<sup>i</sup> <sup>n</sup>* (i.e., the value of the observed cumulative distribution) and the other axis as *F(xi)*, where *F(xi)* is the value of the theoretical probability distribution function for

respective observation *xi*. If the theoretical cumulative distribution is a good approximation of the empirical distribution, then the points on the diagram should be close to the diagonal. The research began with an analysis of GPS failure times. The results of P-P plot analyses are presented in Table 1. Moreover, the list of theoretical distributions with the best (top) and worst fit can be seen next to each of the graphs.

The analyses show that the assumption (very common in technology) that the life and failure times follow exponential distribution is questionable. It is clear that, for distribution function, values below 0.9 (both empirical and theoretical) are poorly fitted. On the contrary, above this value the fit is very good, which may suggest that the failure (*λ*) and renewal (*μ*) rates can be calculated very reliably on its basis. It has also been stressed that it is not an exponential distribution but a lognormal distribution that provides a very good approximation of the life and failure time statistics. Furthermore, analyses performed on two different decision thresholds (1 m and 2 m) produced similar conclusions regarding the mismatch between the empirical and theoretical (exponential) distributions for distribution function values below 0.9.

Therefore, it is reasonable to conduct identical analyses for other navigation positioning systems such as DGPS and EGNOS. These analyses have been performed for a very large sample of 900 000 fixes. Table 2 presents the PDFs of life and failure times, together with the best-fit distributions for the DGPS system. Moreover, Table 2 shows the P-P plots for DGPS and EGNOS systems.

The distribution of DGPS system life and failure times is similar to that of GPS, although empirical data fit theoretical exponential distribution much better. On the contrary, for EGNOS it is clear that the empirical distributions of life and failure times deviate significantly from the theoretical distributions. However, as in the case of GPS and DGPS systems, for values above 0.9, the fit is very good.

**Table 2.** Statistical analysis of life and failure times for empirical DGPS and EGNOS position errors (1 m).

The following conclusions can be drawn from the empirical studies and theoretical analyses carried out:


The next research stage was to determine the failure and renewal rates of the renewal process. Both of these values made it possible to determine the course of the availability function calculated for an arbitrary position error.

Finally, to present the position error distribution functions of the GPS, DGPS and EGNOS systems, repeated calculations of the probability value corresponding to a given position error value ware performed. It has been assumed that to determine the distribution function, the position error value was increased from 0, every 0.1 m, up to 4 m. The course of distribution functions calculated based on the reliability model is presented in Figure 9.

**Figure 9.** Position error distribution functions of the GPS, DGPS and EGNOS systems calculated using the reliability model.

Please note that the DGPS and EGNOS systems achieved positioning accuracies below 1m(*p* = 0.95) on a very large and probably representative sample. Therefore, it is concluded that both systems can be successfully used in navigation applications requiring positioning accuracies of 1 m (*p* = 0.95), even though the official system characteristics given in [57] are 3m(*p* = 0.95) horizontally and4m(*p* = 0.95) vertically.

It needs to be particularly emphasised here that it is not the purpose of this article to establish the actual value of the position error of the studied systems, including the EGNOS system. This paper aims to propose a new reliability model which will allow, as an alternative to the classical method using the 2DRMS value, one to determine the navigation system positioning accuracy and the corresponding probability.

#### **4. Discussion**

The presented reliability model requires an assessment of its accuracy. This can only be carried out in relation to calculations performed directly on empirical (real) values. The most reliable method of determining the position error value larger than 95% of the population of the remaining errors is to calculate it by sorting the errors from the smallest to the largest. This method of error determination is used in several publications [30,32,33]. There is no doubt that with a very large number of measurements this method produces the most reliable results because it does not assume any statistical distribution of empirical position errors. Therefore, to assess which of the two models (classical or reliability) is closer to the true value, the R95 value obtained by the method of sorting the position errors was used.

There are two curves presented in Figure 10. The first curve is the empirical distribution function of the sorted position errors (red) from the GPS system, which takes the value of 2.039 m (*p* = 0.95). This is an R95 value that can be considered close to true due to a large number of measurements (168 286 fixes). The distribution function calculated from the proposed reliability model (green) reaches a value of 2.044 m (*p* = 0.95). In contrast, the 2DRMS value calculated in the classical way (blue) takes the value of 2.240 m (*p* = 0.95).

**Figure 10.** Comparison of three methods for calculating position error values larger than 95% of the population of remaining errors. Analysis of results for the GPS system.

From Figure 10, it follows that a much better approximation of the R95 value was obtained by applying the reliability model than by using the 2DRMS measure.

To verify the accuracy of the reliability method on systems other than GPS, an identical analysis was performed for DGPS and EGNOS systems. It needs to be emphasised that, because the calculations used a very large sample of 900 000 fixes, the results can be considered reliable and representative (Figure 11).

**Figure 11.** Comparison of three methods for calculating position error value larger than 95% of the population of the remaining errors. Analysis of results for: (**a**) DGPS; (**b**) EGNOS.

Empirical distribution function graphs (obtained by position error sorting) and those obtained based on the reliability model, for DGPS and EGNOS systems, prove that the reliability model provided a better approximation of the true value than the commonly used 2DRMS measure. For the DGPS system, the value considered to be the true R95 was 0.748 m. Calculation using the reliability model yielded 0.756 m (*p* = 0.95), while the 2DRMS measure was 0.885 m. The same is true for the EGNOS system. The value of the R95 was 0.854 m and the value calculated using the reliability model was 0.856 m, whereas the value of the 2DRMS was 0.901 m. Please note that many authors [29] have already noted the overestimation of the actual values by the 2DRMS measure, which is confirmed by the results presented in Figures 10 and 11. Therefore, it may be concluded

that the proposed reliability model calculates the R95 value much more accurately than the classical model.

#### **5. Conclusions**

This paper proposes a new method, an alternative to the classical solution based on the 2DRMS measure, for determining navigation system positioning accuracy, which, in its essence, is based on the reliability model. The random variables are life and failure times in the positioning process, and not, as was the case in the classical model, the position errors. This method can be successfully used in assessing the suitability of a positioning system for a specific navigation application. It allows for the calculation of the system's position error with a probability of 95% more accurately than using the classical approach. The method was applied to determine the positioning accuracy of modern navigation systems: GPS (168 286 fixes), DGPS (900 000 fixes) and EGNOS (900 000 fixes). Although empirical distributions of life and failure times differ from the theoretical exponential distribution (for distribution functions with a probability below 0.9), the method provides high accuracy of the final results. An additional advantage of this method lies in the rather simple calculation algorithm.

What was a surprising result of this research was that the lognormal distribution presented a very good fit to the empirical data on life and failure times of all three systems (GPS, DGPS and EGNOS). This requires additional analysis in future research.

Tests conducted on very large measurement samples have proven that the proposed method provides a much more precise determination of positioning accuracy in navigation systems compared to the 2DRMS measure. Thanks to its application, it is possible to determine the position error distribution of the navigation system more precisely, as well as to indicate applications that can be used by this system, ensuring the safety of the navigation process.

It should be noted that the proposed method is not limited only to navigation positioning systems. With minor modifications, it can be successfully applied to the applications listed in Figure 1, which all differ in positioning accuracy and availability. The R95 measure, which was used to compare two models (classical or reliability), is a narrow scope of application. It is intended to determine the position error value for a strictly defined confidence level of 95%.

**Funding:** This research was funded from the statutory activities of Gdynia Maritime University, grant number WN/2021/PZ/05.

**Institutional Review Board Statement:** Not applicable.

**Informed Consent Statement:** Not applicable.

**Data Availability Statement:** Not applicable.

**Conflicts of Interest:** The author declares no conflict of interest.

#### **References**

