1 INTRODUCTION

Satellite navigation has mature to become a public

goods, an essential component of the national

infrastructure, and the enabling technology for

growing number of technical and socio-economic

systems [1, 2, 3]. Global Navigation Satellite System

(GNSS) comprises several global satellite navigation

systems, including Global Positioning System (GPS),

operated by the US, which provide the Positioning,

Navigation, and Timing (PNT) services in an

interoperable and co-operative manner [4, 5, 6].

Matured enough to be one of the cornerstones of

modern society, GNSS still keeps traditions that

constraint developments and utilisations [6, 7].

Among them is the system-centric view of GNSS

positioning performance assessment and declaration,

which does not concern with the potential application

and its requirements [2, 6]. Consequentially, GNSS

application developers, operators, and users cannot

either assess systematically the effects and risks the

GNSS positioning performance degradation renders

to GNSS application Quality of Service, or develop the

GNSS application QoS resilient to potential short-term

GNSS performance degradations or outages [2, 3, 6, 8,

9, 10]. The problem becomes more emphasised in the

wide range of disciplines utilising GNSS, including

the maritime sector [3, 6, 8, 9, 10].

A Method and a Model for Risk Assessment of GNSS

tilisation with a Proof-of-Principle Demonstration for

olar GNSS Maritime Applications

. Malić

, N. Sikrica

, D. Špoljar

& R. Filjar

2,3

Independent researcher, Split, Croatia

University of Applied Sciences Hrvatsko zagorje Krapina, Krapina, Croatia

University of Rijeka, Rijeka, Croatia

ABSTRACT: The GNSS positioning performance is commonly defined and described in terms unspecified to

particular GNSS-based application. The approach causes difficulties to GNSS application developers, operators,

and users, rendering the impact assessment of GNSS performance on the GNSS application Quality of Service

(QoS) particularly difficult. Here the Probability of Occurrence (PoO) Model is introduced, which allows for a

risk assessment of the probability for the GNSS positioning accuracy failure to meet the requirements of the

particular GNSS-based application. The proposed PoO Model development procedure requires a large set of

position estimation errors observations, which shall cover a range of classes of positioning environment (space

weather, troposphere, multi-path etc.) disturbances affecting GNSS positioning accuracy. As result, the PoO

Model becomes a tool that returns the probability of failure in meeting the positioning accuracy requirements of

the GNSS applications considered, thus providing the input for a GNSS deployment risk assessment. The

proposed PoO Model and its development procedure are demonstrated in the case of polar region positioning

environment, with raw GNSS pseudorange observations taken at the International GNSS Service (IGS) Network

reference station Iqualuit, Canada are used for the PoO Model development. The PoO Model proof-of-principle

is then used to estimate the pro

bability of the unmet required positioning accuracy for a number of polar

maritime navigation applications. Manuscript concludes with a discussion of the PoO Model benefits and

shortcomings, a summary of contribution, and intentions for the future research.

http://www.transnav.eu

the

International Journal

on Marine Navigation

and Safety of Sea Transportation

Volume 17

Number 1

March 2023

DOI: 10.12716/1001.17.01.

Disciplines, such as air or maritime transport and

traffic, attempt to assist GNSS applications with

standards of required GNSS performance [11, 12, 13,

14]. An excellent review of current standards with

direct amendments proposals from expert panels has

been performed by European Agency for Space

Programme (EUSPA), which conducts bi-annual re-

assessment of GNSS user needs and requirements per

disciplines, based on expert panel opinions [8]. Still,

the current state-of-the-art does not overcome the gap

between the GNSS operator’s expression of GNSS

performance, which naturally considers it without

knowledge of the actual positioning conditions

around GNSS users, and the GNSS application’s

needs to assess the risk of GNSS positioning failure in

actual conditions of usage [6, 7, 8, 11, 13].

Here we propose the solution for the growing

problem by introduction of the method and the

model/index for the risk assessment of GNSS

utilisation for the particular GNSS application with

known requirement for GNSS position accuracy.

Called the Probability of Occurrence (PoO), the index

is capable of advising GNSS application developer,

operator, and user on the risk of GNSS not meeting

the required positioning accuracy level at the certain

conditions of usage. Such a knowledge may assist

GNSS application developers, operators, and users in

making the objective inference on the need for

suitable alternative when the GNSS PNT

underperform in relation to GNSS application, thus

rendering GNSS application resilient. The proposed

method and model are demonstrated in this

manuscript on the case of risk assessment of the GPS

PNT ionospheric effects for single-frequency

commercial-grade GPS positioning in the Arctic

region.

This manuscript is structured, as follows. This

chapter introduces the problem and outlines the

research hypothesis. Chapter 2 describes (2.1) the

theoretical foundations and the proposal for PoO

model and the methodology for its development, (2.2)

the experimental observations needed for the PoO

model development, (2.3) a proof-of-principle case

study the PoO model development, and (2.4)

demonstrates and discusses PoO model utilisation for

risk assessment for a particular maritime

task/application. Chapter 3 concludes the manuscript

with the outline of contributions and findings, and

proposals for future research.

2 GPS UTILISATION RISK PROBABILITY-OF-

OCCURANCE (POO) METHOD AND MODEL

The Probability of Occurrence (PoO) model is

proposed in this research as a single GPS application-

centric model for the GPS utilisation risk estimation in

defined GPS positioning conditions and utilisation

scenario. This Section outlines the method and

describes the material required for the PoO model

development and deployment for the GPS utilisation

risk assessment.

2.1 Method

The proposed GPS/GNSS Probability of Occurrence

(PoO) risk index is defined based on empirical

identification of the GPS/GNSS positioning

degradation [2, 3, 4, 6, 9, 10, 15] risk in characteristic

positioning environments [3, 6, 9, 10, 16, 17, 18], and

in accordance to a fundamental statistical principles

[19, 20], as follows.

Let X be a statistical random variable, and x its

value. Experimental observations of a variable may be

considered values x of a statistical variable X.

Statistical distribution of a variable X serve as its

statistical model, characterised with two essential

functions: the cumulative distribution function (CDF)

and the probability density function (PDF) [19, 20].

The probability density function f

X(x) of statistical

variable X is defined as the function that returns the

probability of X acquiring the value exactly equal to x

[19, 20], as shown in (1).

( ) ( )

[ ]

, : 0,1

F x PX x F= = →

(1)

The cumulative distribution function (CDF) F

X(x) is

defined as the function that returns the probability

that X will acquire the value less then, or equal to x

[19, 20], as expressed in (2). Two essential functions of

a statistical variable X are mutually related [19, 20].

( ) ( )

[ ]

( )

, : 0,1

F x P X x F f x dx

−∞

= ≤ →=

∫



(2)

The complementary cumulative distribution

function (CCDF), or tail distribution, is derived from

the cumulative distribution function, and defined as

expressed in (3).

( ) (

) ( )

Fx PXx Fx

= >=−

(3)

The CCDF outcome may be interpreted as the

probability of X exceeding (being larger than) x.

Introduction of CCDF may serve as a numerical

indicator of probability of risk that an observed

variable exceeds a critical value, established in the

domain of interest.

Statistical distribution functions may be estimated

analytically using various statistical methods [19, 20],

implemented within either stand-alone software

packages, such as CumFreq [21], or in programming

environments, such as the open-source R environment

for statistical computing [22]. Software libraries, such

as the R-based fitdistrplus [23], allow for estimation of

experimental PDF and CDF, fitting them to theoretical

statistical distributions [19, 20]. Furthermore,

statistical tests, such as Kolmogorov-Smirnov and

Shappiro-Wilk, may be uitilised to confirm

compliance of an experimental statistical distribution

to the particular theoretical one, thus completing the

identification of the experimental statistical

distribution [19, 20]. Once estimated and identified,

the experimental CDF may be used for derivation of

the CCDF for the process/data pool in question.

The CCDF approach is applied in the research

presented for definition of the Probability of

Occurence model development method, which utilises

a massive data set of GPS positioning error

observations, as a statistical variable. The PoO model

emerges as the CCDF of the observed GPS positioning

errors over a long time in the specified conditions of

positioning environment and GPS application

requirements for the particular scenario of utilisation.

The PoO model application for risk estimation

requires understanding and specification of means of

utilisation in an GPS application-oriented sense, as

well as the targeted effect as the source of the risk. The

PoO model is derived from a large data set of

experimental GPS position observations in

positioning conditions and the particular scenario of

usage, of interest for a particular GPS-based

application.

The GPS positioning environment effects that may

cause GPS positioning degradation may include [3, 4,

5, 6, 24, 25]: (i) GPS ionospheric delay, (ii) GPS

ionospheric scintillation, (iii) GPS multipath in a

specified class of terrain (forest, mountain, urban,

semi-urban, rural, ocean), (iv) GPS tropospheric delay

etc.

The GPS utilisation scenario should be described

by specification of user equipment (for example:

commercial-grade single-frequency GPS receiver, as

found in most smartphones still), utilisation of SBAS

(for example: EGNOS) or other advanced positioning

techniques (such as Real-Time Kinematics, RTK)

instead of essential GPS, expected utilisation

environment (indoors vs outdoors, urban, rural,

mountains, forrest, maritime, air, vehicle etc.) [3, 4, 5,

6, 24, 25].

Once the targeted risk effects is defined, and GPS

positioning environment and GPS application

utilisation scenario specified, a massive set of GPS

positioning error observations should be obtained,

either by long-term data collection using individual

equipment, or by utilisation of trusted data. Section

2.2 should be consulted for more details.

With the massive GPS positioning error data set is

at hand, the CCDF should be estimated, using the

experimental PDF and CDF estimation methods. The

experimental CCDF then serves as the Probability of

Occurrence (PoO) model. Te PoO model returns the

probability of GPS positioning failure for a given GPS

positioning error threshold. With the GPS positioning

error threshold set as a maximum acceptable

positioning error of the GPS application in

consideration, the PoO will return the probability of

occurrence the event in which the GPS positioning

process will fail to meet the requested positioning

accuracy threshold for particular GPS application.

The PoO model development method is developed

in a formal manner to be implemented easily within a

programming environment of a choice. The lack of

‘ground truth’ complicates the validation of the PoO

model performance. An alternative cross-validation

approach [20] is taken for this research, thus allowing

for objective assessment of the PoO model correctness.

2.2 Material

Material required for the PoO model development

comprises GPS positioning errors taken over a long

time with the specified satellite navigation

receiver/position estimation method in the

geographical, terrain, and positioning environment

(space weather, ionospheric, tropospheric, multipath,

satellite visibility) conditions [6]. The concept of

material collection aims at creation of a large GPS

positioning error observations as a statistical sample

that represent correctly the population of GPS

positioning errors which can occur in the real-time

GPS-based application usage. It is essential that the

representation (sample) resembles frequency of GPS

positioning degradation at different levels as best as

possible [19, 20].

The user GPS equipment specifications, and GPS

application requirements should be defined by a risk

assessor in collaboration with interested parties (GPS

application developers, operators, and users) [2, 6, 7].

Positioning environment conditions and GPS

positioning error datasets may be acquired either

through a tailored long-term field campaigns, or

obtain from databases operated by trusted third

parties.

As an example, in the case of GPS ionospheric

effects risk assessment, the 24 hours-a-day GNSS raw

pseudorange observations at reference stations across

the world have been continuously collected, with

exposed ionospheric effects and mitigated all the

others, for the exact purpose of evaluation of the

ionospheric effects on GNSS positioning performance.

Available from the International GNSS Service (IGS)

database [26], the GNSS raw pseudorange observation

may be processed with a suitably configured GNSS

Software-Defined Radio in the post-processing mode

[2, 6] to allow for generation of GNSS positioning

estimates and estimation of GNSS positioning error [4,

5, 6, 25, 27], as required for the PoO model

development. Other sources of GNSS raw

pseudorange observations include the internet-based

open-access databases, such as Sonel [28], and the

EUREF Permanent GNSS Network [29]. The GPS

positioning environment is systematically described

from the perspective of space weather, geomagnetic,

and ionospheric conditions with a number of the

internet-based open-access archives of global data and

wide range of indices [3, 6, 9, 30], provided by

different national and international organisations and

agencies. The US NASA maintains a well-structured

observation-rich OMNIWeb repository [31] with a

Graphical User Interface (GUI) aimed at selection of

data [32]. The International Service of Geomagnetic

Indices [33] allows for a free access to archived

derived values of geomagnetic descriptors. The

Intermagnet [34] network offers a free access to

structured global observations of geomagnetic

conditions, as observed with a geographically spread

network maintained to serve scientists, researchers,

and engineers. Alternative third-party sources of

observations are also available, with the APIs

supporting the computer-based data access.

2.3 A case study of practical PoO model development

The research presented is established a method for

PoO model development, and defined the required

inputs. The case scenario of the PoO model

development for the risk assessment of the single-

frequency commercial-grade GPS utilisation in

maritime sector in polar regions is developed, and

presented as a proof-of-principle demonstration.

This research is driven by the rising interest in

sailing in polar region, a geographical area with

known exposure to dynamic space weather,

geomagnetic, and ionospheric conditions known to

produce considerable degradation of the GPS

positioning performance [3, 4, 10, 11]. Consequently, a

case scenario is established, resembling market

conditions with prevailing share of the single-

frequency commercial-grade unassisted GPS

receivers. The common GPS receiver is assumed to be

utilising just the standard correction models provided

by GPS operator, thus correcting the ionospheric,

tropospheric, and satellite clock effects in the

standardised manner [6, 27]. A suitably configured

Software-Defined Radio GNSS receiver is used to

produce the GPS position, based on the raw GPS

pseudorange observations taken at the reference

station in the polar region. The position of the

reference station was determined by precise geodetic

methods. The GPS positioning residuals xresidual(t)

are used as the GPS positioning error estimates xe(t)

at the time instant of GPS positioning t, with xGPS(t)

denoting the vector of GPS position components

estimates, and xref(t) denoting the true position vector

of GPS receiver, as determined by a precise geodetic

method as given with (4).

(

) (

)

residual e GPS ref

x txtx txt= = −

(4)

The GPS observations used in the proof-of-

principle PoO model development were taken at the

IGS [26] reference station Iqualuit, Canada, using a

stationary GNSS receiver collecting continuously the

raw GNSS pseudoranges 24-hours-a day at 30 s

sampling interval. Observations taken throughout

2014 are selected as a representative sample of the

population. Total of 1 028 713 GPS position estimates

ate derived from the massive data set of raw GPS

pseudorange observation, after those were fed into

RTKLIB [35], a, SDR GNSS receiver, to produce GPS

positioning, and GPS positioning error estimates.

Space weather, geomagnetic, and ionospheric

disturbances at various scales occurred in that year,

with their frequency of occurrence resembling the

long-standing pattern, as confirmed with the

examination of the Dst index [3, 9, 36, 37] of

geomagnetic storms/disturbances in 2014, as depicted

in Figure 1.

Figure 1. Time series of Dst, geomagnetic disturbance index,

throughout 2014.

Dst data set is obtained from [32, 33], for the

purpose of identification of particular classes of the

GPS positioning environment: (i) (relatively) quiet

geomagnetic condition, (ii) positive (first) phase of the

geomagnetic storm, (iii) negative (deep through, and

recovery) phase of the geomagnetic storm [36, 37].

Such classification allows for establishing three cross-

validation [20] scenarios for the PoO model

performance validation.

The raw GPS pseudorange observations are then

fed into the RTKLIB GNSS SDR receiver, set in the

post-processing mode and with configuration as a

single-frequency commercial-grade sole-GPS receiver

[27, 35], as outlined in Figure 2.

Figure 2. Configuration of RTKLIB as a single-frequency

sole-GPS satellite navigation receiver

The RTKLIB receiver returns the GPS positioning

estimates, which are then transformed into GPS

positioning errors using the method (4). The method

is applied in the tailored software developed for the

purpose of this research in the R environment for

statistical computing. The GPS positioning error

statistical analysis and the PoO model development

are performed with the R-based software developed

under this research.

The PoO model in the presented proof-of-principle

is targeting the horizontal GPS positioning accuracy,

derived from the positioning accuracy of horizontal

components of position, as the index referred to in

GPS-based application requirements for maritime [7].

Before the PoO model development takes place, the

obtained components of GPS positioning error vector

are examined for their statistical properties, as

outlined with box-plot diagrams in Figure 3.

Figure 3. Box-plot of GPS positioning error vector

components

Figure 3 reveals a number of outliers, caused by

extensive ionospheric storms in the region, which

were affecting the GPS positioning quality with the

resulting risks for GPS-based applications.

A cross-validation method is developed for the

PoO model validation, based on classification of GPS

positioning error vector into one of the three classes of

geomagnetic conditions [36, 37]: (i) quiet conditions,

(ii) positive phase of geomagnetic storm, (iii) negative

phase of geomagnetic storm. Essential statistical

properties for classes (ii) and (iii), and the total sample

are given in Table 1.

Table 1. Statistical properties of the total set of horizontal

GPS positioning errors, and the two subsets related to

geomagnetic storm development

________________________________________________

GPS No. of Mean Median Variance

horizontal observations

error [m] in 2014

________________________________________________

Total 1019769 1.9549 1.6300 4.89951

Dst > 30 nT 24 2.0077 1.5496 1.629124

Dst < -40 nT 222 1.8110 1.6709 1.328492

________________________________________________

The box-plots of horizontal GPS positioning errors

remain balanced in regard to the Dst value ranges

during quiet geomagnetic conditions. However, it is

evident from Figures 4, and 5, respectively, that the

median of GPS positioning error rises in regard to the

absolute value of Dst during disturbed geomagnetic

conditions, thus justifying the classification approach.

Figure 4. Box-plots of GPS positioning errors per ranges of

Dst values, positive phase of geomagnetic storm

Figure 5. Box-plots of GPS positioning errors per ranges of

Dst values, negative phase of geomagnetic storm

Cross-validation approach is justified further with

the comparison of horizontal GPS positioning error

data sets for geomagnetic disturbance classes (ii) and

(iii). Related statistical tests [19, 20] reveal no

similarities in two subsets of the original observations.

Table 2. Statistical tests results of comparison of GPS

positioning error subsets during phases of geomagnetic

storms

________________________________________________

Compared t-test, p-value, F-test, p-value,

with H0: means H0: variances

General are equal are equal

________________________________________________

Dst > 30 nT 0.8411 0.00227

Dst < -40 nT 0.06421 < 2.2e-16

________________________________________________

Finally, the method outlined in Section 2 is applied

to yield three PoO models, for the whole sample

(total) of horizontal GPS positioning errors, and for

the subsets relating to geomagnetic conditions classes

(ii), and (iii). The method application results in three

PoO models, respective to geomagnetic conditions, as

depicted in Figure 6.

Figure 6. Three PoO models, for the whole set of horizontal

GPS positioning errors (total, in red), for the positive phase

of geomagnetic storm (Dst > 30 nT, in blue) subset, and for

the negative phase of geomagnetic storm (Dst < -40 nT, in

green) subset

In consideration of the PoO model performance,

the negative phase PoO fits well the total PoO, thus

confirming the total PoO model success. The reason

may be found in the fact that statistical properties of

the horizontal GPS positioning errors during the

negative phase of the geomagnetic storms are clearly

identified within the total pool of horizontal GPS

positioning errors. A relatively small subset of

horizontal GPS positioning errors observed during the

positive phase of geomagnetic storm creates a unique

statistical pattern, resulting in a slightly different PoO.

Difference is particularly visible in the 1 m – 6 m

range of horizontal GPS positioning errors. However,

the size of related sample may lead to inference of

neglecting the observed difference in PoOs.

2.4 PoO model application on particular GPS application

The total PoO model, expressed either in graphical or

analytical form may be utilised to asses the risk of

GPS utilisation, in the presented positioning

environment conditions and the scenario of

utilisation, in a specific GPS application defined by its

requirements for horizontal GPS positioning accuracy

[7, 38].

Let us assume the PoO is expressed in the

analytical form, as given in (5).

( )

risk

P f requested horizontal accuracy=

(5)

A specific GPS-based application should define its

request for the highest acceptable horizontal GPS

positioning error, and use it as the value for requested

horizontal accuracy. Applied to the PoO model (5),

the GPS-based application receives the probability of

horizontal GPS positioning accuracy not meeting its

request, and may consider potential alternatives for

periods of degraded GPS positioning performance.

Determination of the PoO/risk for a particular

requested horizontal GPS accuracy may be performed

using the analytical model, or graphically, as shown

in Figure 7.

Figure 7. Graphical determination of PoO/risk of horizontal

GPS positioning accuracy not meeting the requested level,

based on the PoO curve

Considering the proof-of-principle demonstration

scenario, a maritime GPS-based application requiring

the positioning accuracy of 5 m may find the

probability of approx 4% that the required accuracy

level will not be met. Reference [7] states such a

request may be set for the navigation in port

operations. GPS application operator and user may

consider implementation and operation of a

redundant positioning system in a confined area

(port) to overcome the risk, or the utilisation of

integrated navigation (for example, GPS+INS), for the

period of degraded GPS positioning performance.

The proposed method, and PoO model

demonstrated in the proof-of-principle scenario, may

be generalised towards any positioning indicator

requested, as well as to utilisation multiple GNSS

position estimation.

3 CONCLUSIONS

The lack of objective and systematic risk assessment of

GPS/GNSS utilisation renders raising number of

GPS/GNSS-based applications uncertain of potential

derogation of their Quality of Service (QoS), or even

failure to deliver, due to potentially unacceptable

GPS/GNSS positioning performance degradation.

The research presented addresses the problem

with the proposal for a method for development of

Probability of Occurrence model, a statistical model,

for risk assessment of GPS utilisation in particular

positioning environment and for a specific GPS-based

application that extends its requirements for GPS

positioning accuracy.

Contributions of the presented research

summarise, as follows:

1. A method of the Probability of Occurrence (PoO)

model of GPS utilisation.

2. Assemblage of a year-long massive database of GPS

positioning errors at Iqualuit, Canada in Arctic

region, presented in the open-source access

manner.

3. A cross-validation method for the PoO validation,

based on geomagnetic conditions classification.

4. A method of the PoO utilisation for the GPS

utilisation risk assessment for the specified GPS-

based application.

5. An R-based software for PoO model development

and validation.

The proposed method and the PoO model are

demonstrated in the case scenario of a sole-GPS,

single-frequency commercial-grade GPS positioning

in the Arctic polar region. As the result, the PoO

model is developed based on the experimental data

taken in the real and statistically representative

conditions. The PoO model utilisation for risk

assessment is demonstrated in the case of a maritime

GPS application. The research will continue with

improvement and advancement of developed R-based

software that will allow for specification of

positioning environment and GPS-based application

utilisation scenario, as well as with generalisation of

the PoO method development.

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