1 INTRODUCTION

The probabilistic approach to determination of oil

spill domains at port and sea water areas is proposed

in (Dąbrowska & Kołowrocki 2019B). In this paper,

the stochastic approach is supplemented by the Monte

Carlo simulation approach (Dąbrowska 2019, Law &

Kelton 2000, Zio & Marseguerra 2002) to the oil spill

domain movement in changing hydro-meteorological

conditions (Dąbrowska & Kołowrocki 2019A, 2019B).

First, the model of the process of changing hydro-

meteorological conditions is defined and its

parameters are introduced. The identification

methods of the unknown parameters of process of

changing hydro-meteorological conditions is

described in (Dąbrowska & Kołowrocki 2019A). Next,

a bit modified probabilistic model of oil spill domains,

considered in (Dąbrowska & Kołowrocki 2019B) is

introduced. After that, Monte Carlo simulation

approach general procedure is created and applied to

generating the process of changing hydro-

meteorological conditions at oil spill area and to the

prediction of oil spill domain in varying hydro-

meteorological conditions.

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 & Soszyńska-

Budny 2018, 2019A). Under these assumptions, the

process of changing hydro-meteorological conditions

A(t) is completely described by the following

parameters (Dąbrowska & Soszyńska-Budny 2018,

Kołowrocki & Soszyńska-Budny 2011):

− the vector of probabilities of its initial states at the

moment t = 0

[p(0)] = [p

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

Monte Carlo Simulation Approach to Determination of

Oil Spill Domains at Port and Sea Water Areas

E. Dąbrowska & K. Kołowrocki

Gdynia Maritime University, Gdynia, Poland

ABSTRACT: Monte Carlo simulation method of oil spill domains determination based on the probabilistic

approach to the solution of this problem is proposed. A semi-Markov model of the process of changing hydro-

meteorological conditions is constructed and its parameters are defined. The general stochastic model of oil

spill domain movement for various hydro-meteorological conditions is described. Monte Carlo simulation

procedure is created and applied to generating the process of changing hydro-meteorological conditions and

the prediction of the oil spill domain movement impacted by these changes conditions.

http://www.transnav.eu

the

International Journal

on Marine Navigation

and Safe

ty of Sea Transportation

Volume 14

Number 1

March 2020

DOI:

10.12716/1001.14.01.06

− the matrix of probabilities of its transitions

between the particular states

] =













mmmm











22221

11211

, (2)

where

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

− the matrix of distribution functions of its

conditional sojourn times θij at the particular states

(t)] =













)()()(

22221

11211

tWtWtW

mmmm







, (3)

where

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

3 MODELLING OIL SPILL DOMAIN IN VARYING

HYDRO-METEOROLOGICAL CONDITIONS

We assume that the process of changing hydro-

meteorological conditions A(t) in succession takes the

states

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 in

(Dąbrowska & Kołowrocki 2019B):

− for

,,,2,1

tsttt ∆∆∆= 

(4)

− at the process A(t) state k

− for

,,,)2(,)1(

211

tststst ∆∆+∆+= 

(5)

− at the process A(t) state k

…

− for

,,,)2(,)1(

tststst

∆∆+∆+=

−−



(6)

− at the process A(t) state k

we receive the following sequence of oil spill

domains:

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

111

tsDtDtD

kkk

∆∆∆

(7)

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

tsDtsDtsD

kkk

∆∆+∆+

(8)

…

(.,

.),

((

),)

1((

∆

∆+

−

(9)

where s

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

i-1)∆t <

∑

≤ si∆t, i = 1,2,...,n,

,Tts

≤∆

(10)

and

1+jj

j = 1,2...,n, (11)

are the realizations 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, introduced in Section 2 and in (Dąbrowska

& Kołowrocki 2019B).

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 (7)-(9), given by

 

kkk

tjsDD

,...,

))((

−

∆+=

)]

(...)

2()1

([

tsDtDtD

kkk

∆∪∪∆∪∆=

( )

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

tsDtsDtsD

kkk

∆∪∪∆+∪∆+∪

…

( )

)]

(...)2()1([

tsDtsDtsD

∆∪∪∆+∪∆+∪

−−

for k1, k2, ..., kn ∈ {1,2,...,m},

0s =

(12)

The oil spill domain

kkk

,...,

defined by (12) is

determined for constant radiuses

)(

r =

t ∈ <0,T>, ki ∈ {1,2,...,m}, i = 1,2,...,n.

If the radiuses are not constant, we define the

sequence of domains for each state k

i, ki ∈ {1,2,...,m},

i = 1,2,...,n, in a way similar to that described in

Remark 1 in (Dąbrowska & Kołowrocki 2019B), i.e. we

define the sequence of domains

 

kkk

tasDtb

1 1

,...,

)()(

= =

−

∆+=∆D

( ) ( )

)](...21[

tsDtDtD

∆∪∪∆∪∆=

( ) ( )

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

tsDtsDtsD

kkk

∆∪∪∆+∪∆+∪

…

(

) (

)

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

tsDtsDtsD

∆∪∪∆+∪∆+∪

−−

for

,...,2

1−

−

iii

ki ∈ {1,2,...,m},

i = 1,2,...,n, (13)

where

),(:)(

tasDtasD

∆+=∆+

−−

,,...,2,1

ba =

,,...,2,1

1−

−=

iii

ssb

ki ∈ {1,2,...,m},

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

with the following, modified slightly in comparison

that defined in (Dąbrowska & Kołowrocki 2019B),

substitutions:

),()(:)(

tamtsmtm

∆+∆=

−

()(:)(

tamtsmtm

∆+∆=

−

)(:)(

tast

∆+=

−

σσ

),()(

tbrtas

∆

∑

+∆+=

−

)(:)(

tast

∆+=

−

σσ

),()(

tbrtas

∆

∑

+∆+=

−

where

)

(

=∆

,0)

(

=∆tsm

for

i = 1,2,...,bi, bi = 1,2,..., si – si-1, ki ∈ {1,2,...,m}, i = 1,2,...,n,

0 =0.

4 MONTE CARLO SIMULATION PREDICTION OF

THE OIL SPILL DOMAIN

4.1 Generating process of changing hydro-meteorological

conditions at oil spill area

We denote by k

i = ki(q), i ∈ {1,2,...,m}, the realization of

the process' A(t) initial state at the moment t = 0.

Further, we select this initial state by generating

realizations from the distribution defined by the

vector [p(0)]

m, according to the formula

i(q) = kξ,

∑

pξ–1(0) ≤ q <

∑

pξ(0),

ψ ∈ {1,2,...,m}, (14)

where q is a randomly generated number from the

uniform distribution on the interval

〈

0,1) and p(0) for

ξ = 0 equals 0.

After selecting the initial state k

i, i ∈ {1,2,...,m}, we

can fix the next operation state of the process of

changing hydro-meteorological conditions at oil spill

area. We denote by k

j = kj(g), j ∈ {1,2,...,m}, i ≠ j, the

sequence of the realizations of the operation process'

consecutive states generated from the distribution

defined by the matrix [p

ij]m

m. Those realizations are

generated for a fixed i, i ∈ {1,2,...,m}, according to the

formula

j(g) = kξ,

∑

pi ξ–1 ≤ g <

∑

pi ξ,

ψ ∈ {1,2,...,m}, ψ ≠ i, (15)

where g is a randomly generated number from the

uniform distribution on the interval

〈

0,1) and p

i 0 = 0.

We can use several methods generating draws

from a given probability distribution, e.g. an inverse

transform method, a Box-Muller transform method,

Marsaglia and Tsang’s rejection sampling method

(Dąbrowska 2019). The inverse transform method

(also known as inversion sampling method) is

convenient if it is possible to determine the inverse

distribution function (Grabski & Jaźwiński 2009). This

section will consider only this one sampling method,

but the other methods are discussed in (Law & Kelton

2000, Rao & Naikan 2016, Zio & Marseguerra 2002).

We denote by

)

(

i,j ∈ {1,2,…,m}, i ≠ j,

= 1,2,…,n, the realization of the conditional sojourn

times θ

ij of the process A(t), t ∈ <0,T>, generated from

the distribution function W

ij(t), where

denotes the

subsequent number of the sojourn times realizations

and n is the number of those sojourn time realizations

during the experiment time T. Thus, using the inverse

transform method, the realization

)(

is generated

from

ij =

1−

(h), i,j ∈ {1,2,…,m}, i ≠ j, (16)

where

1−

(h) is the inverse function of the

conditional distribution function W

ij(t) and h is a

randomly generated number from the interval

〈

0,1);

Having the realizations

)(

, i,j ∈ {1,2,…,m}, i ≠ j,

= 1,2,…,n, of the process A(t), it is possible to

determine approximately the entire sojourn time as

the sum of all sojourn time realizations during the

experiment time T, applying the formula

n =

∑

)(

, i,j ∈ {1,2,...,m}, n = 1,2,... . (17)

The exemplary realisation of the process A(t) and

the entire sojourn time is presented in a figure below.

(3)

(2)

state

(4)

(5)

(1)

…

(6)

T t

Figure 1. Exemplary realization of the process A(t).

4.2 General procedure of Monte Carlo simulation

application to determine the oil spill domain in

varying hydro-meteorological conditions

The procedure of generating and estimating the

parameters of the process of changing hydro-

meteorological conditions at oil spill area

characteristics is formed as follows.

First, we have to draw a randomly generated

number g from the uniform distribution on the

interval

〈

0,1). Next, we can select the initial state k

i ∈ {1,2,...,m}, according to (14). Further, we draw

another randomly generated number g from the

uniform distribution on the interval

〈

0,1). For the

fixed i, i ∈ {1,2,...,m}, we select the next state k

j ∈ {1,2,...,m}, j ≠ i, according to (15). Subsequently, we

draw a randomly generated number h from the

uniform distribution on the interval

〈

0,1). For the

fixed i and j, we generate a realization t

ij of the

conditional sojourn time θ

ij from a given probability

distribution, according to (16). Then, we compare the

realization t

ij of the conditional sojourn time with the

experiment time T. If the realisation t

ij of the

conditional sojourn time is less than the experiment

time T, we draw the sequence of domains

)(

,...,,

kkk

∆D

for

,,...,

2,1

1−

−=

ki ∈ {1,2,...,m}, i = 1,2,...,n,

using formula (13).

As the realisation t

ij is the first one, we put

= 1

and consequently

1 =

)1(

Further, we substitute i

= j and repeat drawing

another randomly generated numbers g and h

(selecting the states k

j and generating another

realization

)(

= 2, of the conditional sojourn time.

Having the realizations

)(

, i,j ∈ {1,2,…,m}, i ≠ j,

= 1,2, of the process A(t), we calculate the entire

sojourn time

n, n = 1,2,... , applying the formula (17),

i.e. we have

2 =

)1(

)2(

Further, we compare it with time T. If the sum

2 is

less than the experiment time T, we draw the

sequence of domains using formula (13).

We repeat the procedure above until the sum

n of

all generated realizations

)(

= 1,2, …, n, reach a

fixed experiment time T. Consequently, we calculate

the entire sojourn time

n, according to (17) and draw

the sequence of domains using formula (13).

Finally, we put together all the sequences of

domains draw before and we get the oil spill domain

movement (Figure 2). In the interval 〈0,

1) the number

of ellipses is s

1 – s0 = s1, in the next intervals 〈

2 –

2),

…, 〈

n –

n–1,

n) the number of ellipses are respectively

2 – s1, s3 – s2, …, sn – sn–1, where si, I = 1,2, …, n, are

defined by (10).

The general Monte Carlo simulation flowchart for

generating and determination of a process of

changing hydro-meteorological conditions at oil spill

area is illustrated in Figure 3.

4.3 Monte Carlo simulation prediction of the oil spill

domain in varying hydro-meteorological conditions

Using the procedures of the process of changing

hydro-meteorological conditions at oil spill area

prediction described in Sections 4.1-4.2 and the

modified method of the domain of oil spill

determination presented in Section 4.3 in (Dąbrowska

& Kołowrocki 2019B) the Monte Carlo simulation oil

spill domain prediction can be done.

The modified method of the domain of oil spill

determination presented in Section 4.3 in (Dąbrowska

& Kołowrocki 2019B) depends on changing the

procedure (4)-(12) by replacing the conditions (10)-

(12) by conditions:

The s

i, i = 1,2,...,n, existing in (4)-(9), according to

(10), are such that

i – 1)∆t <

∑

j 1

j+1

= si∆t, i = 1,2,...,n,

,Tts

≤∆

(18)

and

j+1

, j = 1,2...,n – 1, (19)

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

conditional sojourn times

j+1

, j = 1,2...,n – 1

at the states k

j, upon the next state is kj+1, j = 1,2...,n – 1,

j, kj+1, ∈ {1,2,...,m}, j = 1,2...,n – 1, defined in

Section 4.1.

(0),

(0))

(2∆t)

∆t)

1∆t

2∆t

∆t = 5∆t

2)∆t s

∆t = 9∆t (s

+ 2)∆t s

∆t = 15∆t

∆t

1)∆t

3)∆t (s

1)∆t (s

5)∆t

= s

∆t

= 18∆t

(1∆t)

∆t)

((s

+ 1)∆t)

((s

+ 2)∆t)

((s

+ 1)∆t)

(1)

(2)

(3)































−=

tty











)(

tyy

txx

((s

+ 3)∆t)

∆t)

((s

+ 2)∆t)

((s

+ 5)∆t)

∆t)

(4)

Figure 2. Oil spill domain for changing hydro-meteorological conditions.

Set:

• T := const., T > 0;

• ∆t := const., ∆t > 0;

•

)(: qkk

)(: gkk

i,j ∈ {1,2,…,m}, i ≠ j;

•

)(:

)(

(

−

i,j ∈ {1,2,…,m}, i ≠ j,

= 1,2,…,n;

•

∑

)(

, i,j ∈ {1,2,…,m}, n = 1,2,... .

Fix the initial state k

Start

Generate

Set:

0:)(

)(

=ht

1:=

0:=

Fix state k

Generate

Yes

: +=

νν

Fix the realisation

)(

of conditional sojourn time

≥ T

Calculate the entire

sojourn time

Draw the sequence

of domains at the

hydro-

meteorological

state k

Stop

Draw the oil spill

domain movement

Figure 3. General Monte Carlo flowchart for prediction of

the oil spill domain in varying hydro-meteorological

conditions.

5 CONCLUSIONS

The proposed Monte Carlo simulation approach

allows us for the determination of oil spill domains at

port and sea water areas in changing hydro-

meteorological conditions. It is a new supplementary

method to the probabilistic methods of the oil spill

domains determination in the varying hydro-

meteorological states presented in (Dąbrowska &

Kołowrocki 2019A, 2019B), (Kim et.al. 2013) and

(Chen, Li & Li 2007). The comparison of results of

these methods’ applications in real conditions should

lead to the selection one of them with the best

accuracy and develop in the future research.

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