1 INTRODUCTION

The Global Navigation Satellite System (GNSS)

become an essential enabler of technology and socio-

economic applications (systems and services) [5, 6].

Based on a dedicated technology infrastructure and the

fulfilled presumptions of the GNSS PNT operations,

the GNSS offers the Positioning, Navigation, and

Timing (PNT) service of the unprecedented quality [5],

even in scenarios of exceptional conditions (such as:

space weather/geomagnetic/ionospheric storms,

outstanding multipath conditions, severe

meteorological events) of the GNSS positioning

environment [15]. As with the all technology systems,

the nature of the system design, operations

shortcomings, and the consequent vulnerabilities

render the GNSS PNT performance susceptible to a

number of natural ionospheric and tropospheric delay,

multipath effects), operational (inaccurate ephemeris

data) and artificial (spoofing, jamming) effects [8, 9].

Development of the resilient GNSS PNT has become

the research subject of the prime importance, as rising

number of the GNSS PNT applications require robust,

available, affordable and stable PNT Quality of Service

An Assessment of Long-term Spatial Agnosticism

of GNSS Positioning Degradation Risks Due

to Ionospheric Conditions

N. Sikirica

, I. Hedji

, M. Mikša

, D. Brčić

, E. Ciriković

& R. Filjar

1,3

University of Aplied Sciences Hrvatsko zagorje Krapina, Krapina, Croatia

Virovitica University of Applied Sciences, Virovitica, Croatia

University of Rijeka, Rijeka, Croatia

ABSTRACT: The Global Navigations Satellite Systems (GNSS) have been evolved into an essential infrastructure

of modern civilisation, a public goods, and enabler of rapidly growing number of technology and socio-economic

applications. However, GNSS applications often lack fundamental details on GNSS Positioning, Navigation, and

Timing (PNT services performance to define and determine their Quality of Service (QoS). The lack of alignment

with the core GNSS PNT deprives GNSS applications of assessing the risks of the GNSS PNT utilisation, thus

leaving GNSS applications unable to prepare alternatives and mitigate the causes of GNSS PNT performance

disruptions. Here we contributed to solution of the problem with the introduction and long-term performance

assessment of the risk model of ionospheric-caused GNSS positioning degradation. Called the Probability of

Occurrence (PoO), our team defined the risk model of GNSS positioning degradation caused by ionospheric

conditions based on the long term observations of occurrences of degraded GNSS positioning performance. In

the process of the GNSS risk model validation, the long-term PoO risk models are developed using the annual

2014 stationary GNSS horizontal positioning error observations derived from the GNSS pseudoranges collected

at the International GNSS Service (IGS) reference stations situated in polar (Iqaluit, Canada) and sub-equatorial

regions (Darwin, Australia). Two GNSS risk models are compared for similarity using statistical methods of

Hausdorff distance and Cramér–von Mises statistical test. Research results show that two GNSS risk models are

spatially agnostic, since no significant difference in two long-term GNSS risk models is found. The research results

supports the conclusion of generality of the PoO GNSS risk model, and its ability to serve GNSS applications

developers, operators, and users in determination of the QoS of particular GNSS applications.

http://www.transnav.eu

the International Journal

on Marine Navigation

and Safety of Sea Transportation

Volume 19

Number 1

March 2025

DOI: 10.12716/1001.19.01.11

(QoS) that should propagate into the QoS of targeted

GNSS applications, rendering them of the same nature

[2, 6, 8]. Developers, operators, and users of the GNSS-

based applications are particularly concerned with the

lack of integration between the GNSS PNT QoS and the

QoS of the particular GNSS application, which may

have its own set of the PNT performance requirements

[5]. The risk of the GNSS PNT performance not meeting

the needs and requirements of the targeted GNSS

application has become the prime issue and potential

obstacle in the market acceptance of the GNSS PNT as

the fundamental enabler and driver of a fast range of

the PNT applications [5]. Currently, developers,

operators, and users cannot assess such a risk in an

easy, formal, and convincing manner. Furthermore, the

characterisation of such a risk for GNSS application in

terms of long-term behaviour is still in its infancy [12].

Recently, we introduced a novel GNSS performance

degradation risk assessment model, which allows

developers, operators, and users of a GNSS application

to assess the risk of the GNSS PNT QoS failure to meet

the requirements and needs of the targeted GNSS

applications [12]. The GNSS PNT risk assessment

mode, dubbed the Probability-of-Occurrence (PoO)

model allows the GNSS application’s evaluators to

assess the probability that the GNSS PNT performance

will not meet the exact GNSS application requirement

for positioning accuracy in the observed scenario of the

GNSS usage [12]. The PoO model was demonstrated as

successful in the case of the risk of GNSS PNT

degradation due to ionospheric effects, the single

principal natural cause of the GNSS positioning

performance degradation, among the others in the core

GNSS PNT error budget [12, 15, 16].

Here a research is presented as an application the

introduced PoO model in evaluation of the long-term

spatial characterisation of the risk of GNSS PNT

degradation due to ionospheric effect. The research

hypothesises that the long-term GNSS PNT

degradation risk due to ionospheric effects should be

considered spatially agnostic, in sense that it does not

variate significantly worldwide. In that terms, the

generalised, global nature of models of the GNSS PNT

degradation risks due to ionospheric effects will be

confirmed, which will render them applicable for

GNSS applications regardless of the intended

geographical area of deployment.

The manuscript presenting the research reads, as

follows. This Section introduces the reader to the

problem, and outlines the motivation and the research

approach taken. Section 2 describes the methodology

and material (data, observations, models) used in the

research. Section 3 outlines research results. Section 4

discusses the research results in terms of the problem

solution, and summarises the research contributions,

their shortcomings, and plans for the research

continuation in the future.

2 METHOD AND MATERIAL

The GNSS PNT performance suffers from the

ionospheric conditions, as the single major cause of

degrading effects. This research addresses the issues

affecting the mass-market of the single-frequency

commercial-grade GNSS receivers, such as those found

in smartphones, utilising the essential GNSS position

estimation algorithm. Such devices comprise a

majority of GNSS receivers used currently and

projected to remain as such in the near future. The

ionospheric effects cause a delay in the satellite signal

propagation, and consequently in the pseudorange

measurement Δρi [m] from the i-th satellite, which is

determined by frequency of satellite carrier wave f

[Hz], and the vertical ionospheric profile N(h)

[electrons/m

] with h [m] being the height above the

mean sea level, as shown in (1), while the slant factor F

reflects the effects of the actual angle at which a satellite

signal enters the ionosphere [4, 15].

(1)

The vector of GNSS ranging errors Δρ, related to the

set of satellites used in a single-frequency single

position estimate determination, is mapped onto GNSS

positioning error vector Δx, comprising the three-

dimensional error components, northing, easting, and

vertical, as given in (2) [15].

( )

x G G G





−

 = 

(2)

At the same time, the GNSS positioning error vector

is determined for every GNSS positioning estimate as

a difference between the estimated position and the

true position of a GNSS receiver. Error determination

is performed per components of the positioning vector,

as desscribed in more details in [12]. The true position

of an IGS GNSS receiver is specified in the IGS network

description [11].

The horizontal GNSS positioning error, xhor in [m],

is determined using the northing, xnorthing in [m], and

easting, xeasting in [m], components of the GNSS

positioning error vector Δx, as given in (2).

hor northing easting

x x x=+

(2)

Horizontal GNSS error is used for development of

the Probability-of-Ocurrence risk model of GNSS PNT

degradation due to ionospheric effects, with the GNSS

pseudoranges used in the position and positioning

error estimation retaining the effects of the ionospheric

conditions only. Determination of the horizontal GNSS

errors may be performed using a bespoke software, by

collection of empirical data, or by post-processing of

archive GNSS pseudorange observations by a GNSS

Software Defined Radio (SDR) receiver, such as

RTKLIB [17] and gLAB Tool Suite (Research group of

Astronomy and Geomatics [14], configured to work as

a post-processing tool.

Using the GNSS pseudoranges observed using a

stationary GNSS receiver, with observations

uncorrected for ionospheric effects, over the

sufficiently long period, a database of horizontal GNSS

positioning errors will serve as a reasonable sample of

population of various scenarios of ionospheric

disturbances and their effects on GNSS PNT

performance. A suitably selected period of data

collection will represent sufficiently well not only the

depth of various ionospheric events, but the frequency

of their occurrences, in real situation manner. The

horizontal GNSS errors database established in a such

manner may serve in development of the PoO risk

model development, as described previously in [12].

We defined the PoO model as a Complementary

Cumulative Distribution Function (CCDF) of an

annual (year-long) set of horizontal GPS positioning

errors, taken every 30 s, throughout the year [12]. Thus,

the PoO model may be understood as an application of

the CCDF, a statistical method, on a suitably crafted

context statistically defining the GNSS PNT

degradation risk.

Here we argue that PoO models developed in the

manner described above for stationary receivers set in

different geographical regions may serve for

assessment of the long-term spatial agnosticism of

GNSS positioning degradation risks due to ionospheric

conditions. The aim of the research is therefore to

determine if two PoOs developed for geographically

separated test sites show similarity or statistically

significant difference. The risk of the GNSS PNT

utilisation results from the statistically independent

probability of the harmful event and the damage, as its

consequence. The contribution of this research is

focused on the former, while the GNSS application

developers, operators, and users may find the latter

either in references, such as [5], or in their own

specifications of QoS.

The comparison of two PoOs may be performed in

various ways. Our team selects three statistical

methods for comparing two PoO curves, including: (i)

Dynamic Time Warping (DTW) method [7, 18], (ii) the

Hausdorff distance [10], and (iii) the Cramér–von

Mises statistical test [1, 3].

The Dynamic Time Warping (DTW) method

calculates the distance between two given vectors

representing points on two respective PoOs in

consideration, resulting with two new vectors showing

the best possible matches between data points on PoO

curves [7, 18]. The matching may be presented in a

graphical form, with the data points matched in the

one-to-one manner [7]. Vertical and horizontal lines

mean that a single point on the one axis

matches/represents more than one point on the another

[7, 18].

The Hausdorff distance of two sets of points is

defined as the largest of all the distances from a point

in one set to the closest point in the other set [10].

The Cramér–von Mises statistical test essentially aims

at determination of the goodness of fit of an theoretical

cumulative distribution function and a given empirical

one [1, 3]. Additionally, the test may serve the purpose

of assessing fit between the two given empirical

cumulative distribution functions [1, 3]. This research

utilises Cramér–von Mises statistical test for

determination of the similarity/fit between the two

given PoO models.

The proof of similarity between the PoOs concerned

will then serve as the proof of that the GNSS

positioning degradation risk due to ionospheric

conditions is spatially agnostic. The presented

methodology may be summarised with the generalised

algorithm depicted in Figure 1.

Figure 1. Research methodology

The research presented use two sets of the GPS

pseudorange observations taken at the International

GNSS Service (IGS) [11] reference stations at Iqaluit,

Canada and Darwin, Australia, depicted in Figure 2,

every 30 s throughout the year 2014 to determine the

PoOs. The IGS reference stations shall comply to the

IGS standard on the reference station set-up,

configuration, and operation. Interested parties are

advised to find details on the IGS internet-site [11]. The

authors select a single GNSS system (GPS) to present

the case of the proposed PoO and its characterisation

in a clearer terms, without the need to complicate the

research procedure with a concern of multi-GNSS

positioning. Future research will consider the multi-

GNSS usage effects on the PoO model.

The selection of the IGS reference stations for this

research is motivated by the aim of the research to

cover the long-term case of extremes of the ionospheric

conditions in polar and sub-equatorial regions,

respectively. Since both regions extend harsh

ionospheric conditions, caused by different processes,

the aim of presented research is to show that those

produce the statistically similar PoO outcomes.

Figure 2. Positions of the two IGS stationary reference

stations at Iqaluit, Canada and Darwin, Australia,

respectively.

The GPS position estimates are obtained from the

respective GPS observations using the RTKLIB GNSS

SDR [17], configured as a commercial-grade single-

frequency GNSS receiver, with configuration shown in

Figure 3.

Figure 3. Configuration of a commercial-grade single-

frequency RTKLIB GNSS SDR receiver

A tailored software for horizontal GNSS

positioning errors determination, and the PoO model

development and validation is developed for the

purpose of this research in the open-source R

environment for statistical computing (R project team,

2025).

3 RESEARCH RESULTS

Two datasets, with the respected number of horizontal

GPS positioning errors per testing site, are used in the

presented research for the PoO model determination,

as follows: Darwin, Australia – 1,042,933 instances,

Iqaluit, Canada – 1,028,597 instances. The respected

PoO models are developed to determine the risk of

GNSS PNT degradation due to ionospheric effects for

Darwin, Australia, and Iqaluit, Canada. The PoO

models are depicted graphically in Figure 4, together

with the results of the DTW analysis. The DTW

analysis shows a reasonably good match between the

points of two PoO curves in consideration. Visible

difference in the assessed risks of surpassing the

horizontal GPS error threshold may be found in the

range [2 m, 6 m].

Figure 4. The PoOs of Darwin, Australia and Iqaluit, Canada,

with the graphical presentation of DWT analysis

The DWT diagram shows the reasonable

correspondence between the two PoO curves, as

depicted in Figure 5.

Figure 5. The DWT diagram of the PoOs of Darwin,

Australia, and Iqaluit, Canada

Results of statistical analysis of the two PoOs are

summarised in Table 1.

Table 1. Results of statistical test examining similarity of the

two PoOs in consideration

value of

statistic

p-value,

with α = 0.05

Hausdorff distance

16.13018

Cramér–von Mises statistical test

1.340

0.114

Determined Hausdorff distance for the two PoOs

compared shows a maximal discrepancy of 16.13%,

which may be considered a proof of a good similarity

between the models examined. The Cramér–von Mises

statistical test examines the null-hypothesis of two set

of the PoO points having the same cumulative

distribution functions. The statistical tests returns the

p-value = 0.114, which suggest retaining the null-

hypothesis, with the statistical significance α = 0.05.

The three tetsing methods deployed prove the two

PoOs develop using data from distant geographical

region have the spatially agnostic long-term risk of the

GPS positioning performance degradation due to

ionospheric conditions and disturbances.

4 CONCLUSION

Presented research aims at evaluation of spatial

characterisation of the long-term risk of the GNSS

positioning performance degradation due to

ionospheric conditions and disturbances. Therefore,

the research does not attempt to address either the real-

time PNT performance improvement, including the

improvement of the GNSS positioning accuracy, or

nowcast/forecast/correct in real time the disturbing

natural and artificial effects to the GNSS PNT

performance.

Two Probability-of-Occurrence (PoO) risk models of

GPS positioning performance degradation due to

ionospheric conditions are developed for

geographically separated experimental IGS data

collection sites at Darwin, Australia (sub-eqatorial

region) and Iqaluit, Canada (polar region). The

selection of experimental sites is driven by the fact of

both regions being exposed to extreme ionospheric

conditions, but of different respected natures. Two

datasets of horizontal GPS positioning errors during

Year 2014 contain more than one million observations

each, thus covering all scenarios of ionospheric

disturbances by their intensity as well as the frequency

of appearance. The universality of the proposed PoO

model in terms of spatial agnosticism, i. e. spatially

independent performance is not immediately obvious,

thus needed to be scrutinised objectively. The two PoO

models are assessed for their similarity and potential

spatial agnosticism using three statistical analysis

methods: (i) Dynamic Time Warping (DTW) method,

(ii) the Hausdorff distance, and (iii) the Cramér–von

Mises statistical test. All three tests return results in

favour of significant similarity between the two PoOs

under consideration, with the Cramér–von Mises

statistical test of the null-hypothesis of the same

cumulative distribution function returning the p-value

= 0.114, at statistical significance of α = 0.05. The tests

present the objective justification for retaining the null-

hypothesis of the research that the long-term risk of

GNSS positioning performance degradation due to

ionospheric conditions is spatially agnostic.

The Solar activity, the main driver of ionospheric

conditions, extends a cyclic dynamics, with the

identified periods of 11, 22, and 84 years. The research

presented in the manuscript addresses the year 2014,

which has been selected for its vigorous solar activity,

characteristic for a year at the times of the Solar

maximum. Considering the GNSS PNT utilisation risk

assessment, it makes a case close to the worst one,

rendering it suitable for the evaluation of the near-

worst-case risk assessment scenario. Further to this, a

consideration of a ful-year scenario covers all four

seasons, reflecting the actual seasonal dynamics. The

authors believe that the experimental set-up will

suffice in demonstration of the proposed methodology.

Future research will take into account a wider period

of the two-times the base period of Solar activity, until

the limitations of experimental IGS data availability are

reached.

The presented research contributes to

understanding of the risk of GNSS Positioning,

Navigation, and Timing (PNT) utilisation in fulfilment

of the Quality of Service (QoS) of a targeted GNSS-

based application. Furthermore, the PoO model appear

as a valuable tool for GNSS application developers,

operators, and users, which may allow them to easily

and transparently assess the risk of the GNSS

positioning performance degradation due to

ionospheric conditions. Massive datasets arranged for

the presented research remain an invaluable starting

point for future research on the nature of GNSS PNT

performance variability, and for further validation of

the introduced PoO GNSS utilisation risk model.

Finally, the approach and methodology developed for

this research may remain applicable for the risk

assessment of GNSS PNT degradation casued by other

positioning environment adversarial effects.

REFERENCE

[1] Anderson, T W. (1962). On the Distribution of the Two-

Sample Cramer–von Mises Criterion. Annals of

Mathematical Statistics, 33 (3), 1148–1159.

doi:10.1214/aoms/1177704477

[2] Bonenberg, L, Motella, B, Fortuny Guasch, J. (2023).

Assessing Alternative Positioning, Navigation and

Timing Technologies for Potential Deployment in the EU,

EUR 31450 EN. Publications Office of the European

Union, Luxembourg. ISBN 978-92-68-01163-8,

doi:10.2760/596229, JRC132737

[3] Cramér, H. (1928). On the Composition of Elementary

Errors. Scandinavian Actuarial Journal, 1928 (1), 13–74.

doi: 10.1080/03461238.1928.10416862

[4] Davies, K. (1990). Ionospheric Radio. Peter Peregrinus

Ltd. London, UK.

[5] EUSPA. (2025). EUSPA GNSS User Needs and

Requirements Library. European Agency for Spaace

Programme (EUSPA). Prague, Czechia. Available at:

https://www.gsc-europa.eu/electronic-library/gnss-

market-and-user-reports#userneeds

[6] Flytkjær, R et al. (2023). The economic impact on the UK

of a disruption to GNSS, final report. London Economics.

London, UK. Available at:

https://londoneconomics.co.uk/blog/publication/the-

economic-impact-on-the-uk-of-a-disruption-to-gnss-

2021/

[7] Giorgino, T. (2009). Computing and Visualizing Dynamic

Time Warping Alignments in R: The dtw Package.

Journal of Statistical Software, 31(7), 1–24.

https://doi.org/10.18637/jss.v031.i07

[8] Goward, D. (2016). Prioritizing dangers to the United

States from Threats to GPS: Ranking Risks and Porposed

Mitigations. Resilient Navigation and Timing

Foundation. Alexandria, VA. Available at:

https://rntfnd.org/wp-content/uploads/12-7-Prioritizing-

Dangers-to-US-fm-Threats-to-GPS-RNTFoundation.pdf

[9] GPS Spoofing Workgroup. (2024). GPS Spoofing: Final

Report of the GPS Spoofing Workgroup. OPSGROUP.

Available at: https://ops.group/blog/gps-spoofing-final-

report/

[10] Henrikson, J T. (1999). Completeness and Total

Boundedness of the Hausdorff Metric. MIT

Undergrduate Journal of Mathematics. Available at:

https://www.semanticscholar.org/paper/Completeness-

and-Total-Boundedness-of-the-Hausdorff-

Henrikson/91b0de0eb189355e27a26e3848d41c818c95350

[11] IGS. (2025). International GNSS Service database of

GNSS observations, maintained by NASA. Available at,

with the required free registration:

https://cddis.nasa.gov/archive/gnss/data/daily/

[12] Malić, E, Sikirica, N, Špoljar, D, Filjar, R. (2023). A

method and a model for risk assessment of GNSS

utilisation with a proof-of-principle demonstration for

polar GNSS maritime applications. TransNav, 17(1), 43-

50. doi: 10.12716/1001.17.01.03

[13] R project team. (2025). The R: A language and

environment for statistical computing. R Foundation for

Statistical Computing. Vienna, Austria. Available

through: https://www.r-project.org/

[14] Research group of Astronomy and Geomatics (gAGE).

(2024). gLAB Tool Suite. gAGE, Universitat Politecnica de

Catalunya (UPC). Barcelona, Spain. Available at:

https://gage.upc.edu/en/learning-materials/software-

tools/glab-tool-suite

[15] Sanz Subirana, J., Juan Zornoza, J. M., and Hernandez-

Pajares, M. (2013). GNSS Data Processing – Volume I:

Fundamentals and Algorithms. ESA Communications.

Available at: http://bit.ly/1tDzJIQ

[16] Smith, J et al. (2024). Mapping the ionosphere with

millions of phones. Nature, 635, 365-369. doi:

https://doi.org/10.1038/s41586-024-08520-8

[17] Takasu, T, rtklibexplorer. (2024.). RTKLIB Software-

Defined Radio GNSS Receiver, revision demo5 b34k.

Available at:

https://github.com/rtklibexplorer/RTKLIB/releases

[18] Wang, X et al. (2010). Experimental comparison of

representation methods and distance measures for time

series data. Data Mining and Knowledge Discovery, 2010,

1–35. doi: https://doi.org/10.48550/arXiv.1012.2789