1 INTRODUCTION

One of the important duties in port activities and

shipping is the prevention of oil release from port

installations and ships and the spread of oil spills that

often have dangerous consequences for port and sea

water areas (Bogalecka & Kołowrocki 2018,

Dąbrowska & Kołowrocki 2019A, NOAA). Thus, as

the first step, there is a need for methods of oil spill

domain movement modelling based on determination

of the oil spill central point drift curve determination

and the oil spill domain probable placement at any

moment after the accident that could be the tools for

increasing the shipping safety and effective port and

sea environment protection. Even if, the real trajectory

of the oil spill central point and the oil spill domain

movement are different from those determined by the

proposed methods, they can be useful in the port and

sea environment protection.

The oil spill central point drift trend, the oil spill

domain shape and its random position distribution

fixed for different hydro-meteorological conditions

allow us to construct the model of determination of

the area in which, with the in advance fixed

probability, the oil spill domain is placed (Dąbrowska

& Kołowrocki 2019A). This way, the area determined

for oil spill allow us to mark the domain where the

actions of mitigating the oil release consequences

should be performed. This approach is proposed to

make oil releases at the sea prevention and mitigation

actions more effective.

The general model of the oil spill domain

determination based on the probabilistic approach

may be practically applied in the oil spill

consequences mitigation actions at the sea after its

unknown parameters’ statistical identification.

Statistical experiments should be performed

according to the methods of the model unknown

parameters estimation. Thus, the methods of

evaluation of unknown parameters of the oil spill

central point drift curve and the joint density function

should be proposed. Moreover, the procedures of

their practical evaluations should be done as well

(Dąbrowska & Kołowrocki 2019A).

Probabilistic Approach to Determination of Oil Spill

Domains at Port and Sea Water Areas

E. Dąbrowska

& K. Kołowrocki

Gdynia Maritime University, Gdynia, Poland

ABSTRACT: A new method of oil spill domains’ determination, based on a probabilistic approach, is

recommended. A semi-Markov model of the process of changing hydro-

meteorological conditions is

constructed. To describe the oil spill domain central point position a two-dimensional stochastic process is

used. Parametric equations of oil spill domain central point drift trend curve for different kinds of hydro-

meteorological conditions are determined. The general model of oil spill domain determination for various

hydro-meteorological conditions is proposed. Moreover, approximate expected stochastic prediction of the oil

spill domain movement in constant and changing hydro-meteorological conditions is proposed.

http://www.transnav.eu

the

International Journal

on Marine Navigation

and Safety of Sea Transportation

Volume 14

Number 1

March 2020

DOI:

10.12716/1001.14.01.05

2 MODELLING PROCESS OF CHANGING

HYDRO-METEOROLOGICAL CONDITIONS AT

OIL SPILL AREA

We denote by A(t) the process of changing hydro-

meteorological conditions at the sea water areas

where the oil spill happened and distinguish m its

states from the set A = {1,2,...,m} in which it may stay

at the moment t, t ∈ <0,T>, where T > 0.

Further, we assume a semi-Markov model of the

process A(t) and denote by

ij its conditional sojourn

time in the state i while its next transition will be done

to the state j, where i, j = 1,2,...,m, i ≠ j (Dąbrowska &

Kołowrocki 2019A). Under these assumptions, the

process of changing hydro-meteorological conditions

A(t) is completely described by the following

parameters (Dąbrowska & Kołowrocki 2019A):

− the vector of probabilities of its initial states at the

moment t = 0

[p(0)] = [p

1(0), p2(0),..., pm(0)]; (1)

− the matrix of probabilities of its transitions

between the particular states

] =





















222

, (2)

where p

ii = 0 for i = 1,2,...,m;

− the matrix of distribution functions of its

conditional sojourn times θij at the particular states

(t)] =













)

()()(

)()()(

2221

1211

tWtW

tWtWtW

mmmm







, (3)

where W

(t) = 0 for i = 1,2,...,m;

− the expected values (mean values) of its

conditional sojourn times θij at the particular states

= E[

∫

∞

tdW

(t), i, j = 1,2,...,m, i ≠ j. (4)

Having the above parameters of the process

of changing hydro-meteorological conditions A(t),

t ∈ <0,T>, T > 0, this process following characteristics

can be determined (Dąbrowska & Kołowrocki 2019A):

− the distribution functions of the unconditional

sojourn time θi of the process of changing hydro-

meteorological conditions at the particular states i,

i = 1,2,...,m,

i(t) =

∑

j 1

(t), i = 1,2,...,m; (6)

− the mean values of the unconditional sojourn time

θi of the process of changing hydro-meteorological

conditions at the particular states i, i = 1,2,...,m,

i = E[

i] =

∑

ij], i = 1,2,...,m. (7)

3 MODELLING TREND OF OIL SPILL CENTRAL

POINT DRIFT

First, for each fixed state k, k ∈ {1,2,…,m}, of the

process A(t) and time t ∈ <0,T>, where T is time we

are going to model the behaviour of the oil spill

domain

),(tD

we define the central point of this oil

spill domain as a point

)),(),(( tytx

t ∈ <0,T>,

k ∈ {1,2,…,m}, on the plane Oxy that is the centre

of the smallest circle, with the radius

),(tr

t ∈ <0,T>, k ∈ {1,2,…,m}, covering this domain (Figure

1). Thus, for the fixed oil spill domain

),(tD

have

() ()

() ,

xt xt

)()(

)

(

tyty

t ∈ <0,T>, k ∈ {1,2,…,m}, (8)

where the

))

(),((

tyt

and

))(

),((

222

ytx

are the most distant points of the oil

spill domain

),(tD

t ∈ <0,T>, k ∈ {1,2,…,m}, and

the radius

),(tr

called the radius of the oil spill

domain

),(

is given by

,)]()

([)]()([

)(

tyt

ytxtxtr

−+−=

t ∈ <0,T>, k ∈ {1,2,…,m}. (9)

(t), y

(t))

k k

(t), y

(t))

k k

C(x

(t), y

(

t))

(t)

Figure 1. Interpretation of central point of oil spill

definition.

Further, for each fixed state k, k = 1,2,…,m, of the

process A(t) and time t, t ∈ <0,T>, we define a two-

dimensional stochastic process

)),(),(( tYtX

t ∈ <0,T>,

such that

)

(

: <0,T> → R

where

),(tX

)(tY

respectively are an abscissa and

an ordinate of the plane Oxy point, in which the oil

spill central point is placed at the moment t while the

process A(t), t ∈ <0,T>, is at the state k. We set

deterministically the central point of oil spill domain

in the area in which an accident has happened and an

oil release was placed in the water as the origin O(0,0)

of the co-ordinate system Oxy. The value of a

parameter t at the moment of accident we assume

equal to 0. It means that the process

)),(),(( tYtX

a random two-dimensional co-ordinate (a random

position) of the oil spill central point after the time t

from the accident moment and that at the accident

moment t = 0 the oil spill central point is at the point

O(0,0), i.e.

).0,0())0(),0(( =

After some time, the central point of the oil spill

starts its drift along a curve called a drift curve. In

further analysis, we assume that processes

)),(),(( tYtX

t ∈ <0,T>, k ∈ {1,2,…,m},

are two-dimensional normal processes

)),(),(),(),(),(( ttttmtmN

σσρ

with varying in time expected values

)],([)(

tXEtm

)],

([)

( tYE

standard deviations

),(t

)(t

, t ∈ <0,T>, k ∈ {1,2,…,m},

and correlation coefficients

),(t

i.e. with the joint density functions

),( yx

))((1)(

)(2

ttt

ρσπσ

−

))((

[

)))((1(2

exp{

tmx

σρ

−

)()(

))())(((

)(2

tmytmx

σσ

−−

−

]

))((

tmy

−

,),(

Ryx ∈

t ∈ <0,T>, k ∈ {1,2,…,m}. (10)

where

),(tX

)

(

respectively are an abscissa and

an ordinate of the plane Oxy point, in which the oil

spill central point is placed at the moment t while the

process A(t), t ∈ <0,T>, is at the state k. We set

deterministically the central point of oil spill domain

in the area in which an accident has happened and an

oil release was placed in the water as the origin O(0,0)

of the co-ordinate system Oxy. The value of a

parameter t at the moment of accident we assume

equal to 0. It means that the process

)),

(),(( tYtX

a random two-dimensional co-ordinate (a random

position) of the oil spill central point after the time t

from the accident moment and that at the accident

moment t = 0 the oil spill central point is at the point

O(0,0), i.e.

).0,

0())

0(),

0(( =

Thus, the points

))(),(( tmtm

, t ∈ <0,T>, k ∈ {1,2,…,m},

create a curve K

called an oil spill central point drift

trend (Figure 2) which may be described in the

parametric form











>∈<=

.,0

),(

)(

Ttty

txx

(11)

(t), m

(t))

k k

Figure 2. Oil spill central point drift trend.

4 MODELLING OIL SPILL DOMAIN

4.1 Probabilistic approach

We are interested in finding the search domain D

(t), t

∈ <0,T>, k ∈ {1,2,…,m}, such that the central point of

oil spill domain is placed in it with a fixed probability

p. More exactly, we are looking for c such that

(

))())(),(((

)(

pdxdy

xtDtYtXP

kkk

∫∫

=∈

t ∈ <0,T>, k ∈ {1,2,…,m}, (12)

where

)

(

))

(

))

((

[

))(

(

)

,{(

−

–

)(

)

(

))())(

(

)

σσ

−

]

))((

tmy

−

}

≤

, t ∈ <0,T>,

k ∈ {1,2,…,m}, (13)

(t)

Figure 3. Domain D

(t) of integration bounded by an ellipse.

is the domain bounded by an ellipse being the

projection on the plane 0xy (Figure 4) of the curve

rising as the result of intersection (Figure 3) of the

density function surface

},),(),,(:),,{(

Ryxyxzzyx

∈==

ϕπ

(14)

and the plane

:),

,{(

zyx

exp[

))((

1)()(2

ttt

−

ρσπσ

(x,y) ∈ R

}, t ∈ <0,T>, k ∈ {1,2,…,m}. (15)

Since

))())(),((( tDtYtXP

kkk

∈

exp[1

c−−

t ∈

<0,T>,

k ∈ {1,2,…,m}, (16)

then for a fixed probability p, the equality

)))(),(((

kkk

DtYtXPp ∈=

t ∈ <0,T>, k ∈ {1,2,…,m}, (17)

holds if

).1ln(2

pc −−=

(18)

Thus, the domain in which at the moment t the

central point of oil spill is placed with the fixed

probability p is given by

)(tD

))((

[

))((1

:),{(

tmx

σρ

−

–

)()(

))())(((

)(2

tmytmx

σσ

−−

]

))((

))

(

−

≤ – 2ln(1 – p)}, t ∈ <0,T>, k ∈ {1,2,…,m}. (19)

Considering the above and the assumed in Section

3 definition of the central point of oil spill, for each

fixed state k, k ∈ {1,2,…,m}, of the process A(t) and

time t ∈ <0,T>, we define the oil spill domain

)

))(

(

))

((

[

))((1

:),

{(

σρ

−

–

)()(

))

())(((

)(2

tmytmx

σσ

−−

]

))((

tmy

−

≤ – 2ln(1 – p)},

t ∈ <0,T>, k ∈ {1,2,…,m}, (20)

where

),()()( trtt

σσ

()(

)( tr

σσ

t ∈ <0,T>, k ∈ {1,2,…,m}, (21)

and

),(tr

t ∈ <0,T>, k ∈ {1,2,…,m}, (22)

is the radius of the oil spill domain

),(tD

t ∈ <0,T>, k ∈ {1,2,…,m}.

(t)

y – m

(t) = ρ

(t) (x – m

(t))

k k k

(t)

(t) +

cρ

(t)

k k

(t), m

(t))

k k

(t) – cσ

(t)

k k

(t) + cσ

(t)

k k

(t) – cρ

(t)

k k

Figure 4. Domain D

(t) covering oil spill central point with probability p.

(t)

y – m

(t) = ρ

(t) (x – m

(t))

k k k

(t)

(t) + cρ

(t)

k k k

(t), m

(t))

k k

(t) – cσ

(t)

k k

(t) + cσ

(t)

k k

(t) – cρ

(t)

k k k

(t)

(t) + cσ

(t)

(t) – cσ

(t)

k k k

(t) + cρ

(t)

(t) –

cρ

(t)

k k k

Figure 5. Oil spill domain

).(tD

The graph of the oil spill domain

)(t

is given

in Figure 5.

To find the oil spill domain

)(tD

determined

by (20)-(22) and presented in Figure 5, the statistical

methods of its general model unknown parameters

estimation are proposed in (Dąbrowska & Kołowrocki

2019A). These methods are presented in the form of

algorithms giving successive steps which should be

done to evaluate these unknown model parameters

on the base of statistical data coming from

experiments performed at the sea.

4.2 Oil spill domain for fixed hydro-meteorological

conditions

We suppose that the process A(t) for all t ∈ <0,T>, is at

the fixed state k, k ∈ {1,2,…,m}. Assuming a time step

∆t and a number of steps s, s ≥ 1, such that

(s – 1)∆t < M

k ≤ s∆t,

,Tts ≤

∆

(23)

where

k = E[θk], k ∈ {1,2,…,m}, (24)

are the expected value of the process A(t), t ∈ <0,T>,

sojourn times θ

k, k = 1,2...,m, at the state k determined

in Section 2, after multiple applying sequentially the

procedure from Section 4.1, for

,,,2,1 tsttt ∆∆∆= 

(25)

we receive the following sequence of oil spill domains

(Figure 6)

).(.,..),2(),( tsDtDtD

kkk

∆∆∆

(26)

Hence, the oil spill domain

k ∈ {1,2,…,m}, is

described by the sum of determined domains of the

sequence (26)

),(...)2()1()(

tsDtDtDtiDD

kkk

∆∪∪∆∪∆=∆=



,,...,2,1 mk =

(27)

and illustrated in Figure 6.

(0), m

(0))

k k

2∆t

s∆t

(2∆t)

(s∆t)

...

Figure 6. Oil spill domain for fixed hydro-meteorological

conditions.

Remark 1. The oil spill domain

defined by (27)

and illustrated in Figure 6 is determined for constant

radius

,)(

rtr =

t ∈ <0,T>, k ∈ {1,2,…,m}. If the

radius is not constant, we define the sequence of

domains (Dąbrowska & Kołowrocki 2019A)

...

)

()()(

∪∆∪∆=∆=∆

tDtDtaDtb



( tbD

∆∪

,...,2,

1 sb

k ∈ {1,2,…,m},

where

(:)( t

aDtaD

∆=∆

,,...,2,1 ba =

,...,2

b =

,...,2,

1 mk =

defined by (20) with the following substitutions:

),(:)( tamtm

∆=

(0), m

(0))

k k

1∆t 2∆t

s∆t

(1∆t)

Figure 7. Oil spill domain at the time 1∆t.

(0), m

(0))

k k

s∆t

(1∆t)

(2∆t)

1∆t

2∆t

Figure 8. Oil spill domain at the time 2∆t.

(0), m

(0))

k k

s∆t

(2∆t)

(3∆t)

3∆t

(1∆t)

1∆t

∆t

Figure 9. Oil spill domain at the time 3∆t.

(0), m

(0))

k k

s∆t

(2∆t)

(s∆t)

(1∆t)

((s – 1)∆t)

1∆t 2∆t

...

Figure 10. Oil spill domain at the time s∆t.

),(:)

( tamtm

∆=

),()()(:)( tbrtbtbt

∆+∆=∆=

σσσ

),()()(:)( t

brtbtbt

∆+∆=∆=

σσσ

,,...,2,1 ba =

,,...,2,1 s

b =

k ∈ {1,2,…,m}.

This oil spill domain movement is illustrated in

Figures 7-10.

4.3 Oil spill domain in varying hydro-meteorological

conditions

We assume that the process of changing hydro-

meteorological conditions in succession takes the

states k

1, k2, ..., kn+1, ki ∈ {1,2,...,m}, i = 1,2,...,n+1. For a

fixed step of time ∆t, after multiple applying

sequentially the procedure from Section 4.1:

− for

t ∆

∆∆



(28)

at the process A(t) state k

− for

,,,)2(,)1(

211

tststst ∆∆+∆+= 

(29)

at the process A(t) state k

− for

,,,

(,)1(

tststst

nnn

∆∆

∆+=

−−



(30)

at the process A(t) state k

we receive the following sequence of oil spill domains

(Figure 11):

),(

.,..),2(),1(

sDtDtD

∆

∆∆

(30)

),(.,..),)2((),)1((

tsDtsDtsD

kkk

∆∆+∆+

(31)

),(

.,.

.),)2

((),)

1((

Dts

∆∆+

∆+

−−

(32)

where s

i, i = 1,2,...,n, are such that

i-1)∆t <

∑

j 1

+1 ≤ si∆t, i = 1,2,...,n,

Tts

≤∆

(33)

and

+1 = E[θk

+1], j = 1,2...,n, (34)

are the expected value of the process A(t), t ∈ <0,T>,

conditional sojourn times θ

j+1

, j = 1,2...,n at the states

j, upon the next state is kj+1, j = 1,2...,n, ki ∈ {1,2,...,m},

i = 1,2...,n, determined in Section 2.

Hence, the oil spill domain

,...,

1 n

kkk

k1, k2, ..., kn

∈ {1,2,...,m}, is described by the sum of determined

domains of the sequences (30)-(32), given by

 

kkk

tjsDD

,...,

))((

−

∆+=

)](...)

2()1([

111

tsDtDt

kkk

∆∪∪∆∪∆=

(

) (

)

)](

...)2()

1([

Dts

DtsD

∆∪∪∆+

∪∆+∪

( ) ( )

)](...)2()1([

tsDtsDtsD

∆∪∪∆+∪∆+∪

−−

for k1, k2, ..., kn ∈ {1,2,...,m},

(35)

Remark 2. The oil spill domain

kkk

,...,

defined by

(35) and illustrated in Figure 11 is determined for

constant radiuses

,)(

rtr =

t ∈ <0,T>,

i ∈ {1,2,...,m}, i = 1,2,...,n. If the radiuses are not

constant, we define the sequence of domains for each

sate k

i, ki ∈ {1,2,...,m}, i = 1,2,...,n, in a way similar to

that described in Remark 1 in Section 4.2, i.e. we

define the sequence of domains

 

kkk

tasDtb

1 1

,...,

)()(

= =

−

∆+=∆D

( ) ( )

)](...21[

tsDtDtD

∆∪∪∆∪∆=

( ) ( )

)](...)2()1([

tsDtsDtsD

kkk

∆∪∪∆+∪∆+∪

( ) ( )

)](...)2()1([

tsDtsDtsD

∆∪∪∆+∪∆+∪

−−

for

,...,2

−

ki ∈ {1,2,...,m}, i = 1,2,...,n,

where

),(:)(

tasDtasD

∆+=∆+

−−

,,...,2,1

ba =

,,...,2,1

1−

−=

iii

ssb

ki ∈ {1,2,...,m},

i = 1,2,...,n,

defined by (20) with the following substitutions:

),(:)(

tasmtm

∆+=

−

),(:)(

tasmtm

∆+=

−

)(:)(

tbst

∆+=

−

σσ

),()(

tbsrtbs

∆+

∑

+∆+=

−

)(:)(

tbst

∆+=

−

σσ

),()(

tbsrtbs

∆+

∑

+∆+=

−

,,...,2,1

ba =

,,...,

−

iii

ki ∈ {1,2,...,m},

i = 1,2,...,n.

(0), m

(0))

k k

2∆t

∆t

(2∆t)

∆t)

1∆

∆t (

+ 1)

∆t

(1∆t)

∆t)

((s

+ 1)∆t)

Figure 11. Oil spill domain for changing hydro-

meteorological conditions.

The oil spill domain movement in this case can be

illustrated in a similar way (a bit more complicated)

to that given in Figures 7-10.

5 CONCLUSIONS

The improvement of the methods of the oil spill

domains determination is the main real possibility of

the identifying the pollution size and the reduction of

time of its consequences elimination. Therefore, it

seems to be necessary to start with the new and

effective methods of the oil spill domains at port and

sea waters determination in constant and changing

hydro-meteorological conditions. The most important

criterion of new methods should be the time of the oil

spill consequences minimising. One of the essential

factors that could ensure these criteria fulfilment is

the accuracy of methods of the oil spill domain

determination. Those methods should be the basic

parts of the general problem of different kinds of

pollution identification, their consequences reduction

and elimination at the port and sea water areas to

elaborate a complete information system assisting

people and objects in the protection against the

hazardous contamination of the environment. One of

the new efficient methods of more precise

determination of the oil spill domains determination

could be a probabilistic approach to this problem

presented in this paper and preliminarily in

(Dąbrowska & Kołowrocki 2019A).

The oil spill domains determined for different

hydro-meteorological conditions can be also done for

other kind of spills, dangerous for the environment.

The proposed probabilistic approach to oil spill

domains determination would surely improve the

efficiency of people activities in the environment

protection. A weak point of the method is the time

and cost of the experiments necessary to perform at

the port and sea water areas in order to identify

statistically particular components of the proposed

models (Dąbrowska & Kołowrocki 2019A). Especially

experiments needed to evaluate drift trends and

parameters of the central point of oil spill position

distributions can consume much time and be costly as

they have to be done for different kind of spills and

different hydro-meteorological conditions in various

areas. A strong and positive point of the method is the

fact that the experiments for the fixed port and sea

water areas and fixed hydro-meteorological

conditions have to be done only once and the

identified models may be used for all environment

protection actions at these regions and also

transferred for other regions with similar hydro-

meteorological conditions.

The proposed stochastic approach can be

supplemented by the Monte Carlo simulation

approach (Dąbrowska 2019) to the spill oil domain

movement investigation proposed in (Dąbrowska &

Kołowrocki 2019A, 2019B). These two approaches are

the authors’ primary original approaches to the oil

spill domain determination which are intended to be

significantly developed with the close considering the

contents of publications cited in references below.

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terminal. Proc. International Conference on Industrial

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