Consider some minefield which at time contains m mines as sketched in fig. 2.3, where each mine has been assigned a number
0 t =
{1,2,..., }
k ∈ m
Fig. 2.3. Minefield containing m = 10 mines.
1 2
3 4
5
6 7 8
10 9
As minefield accidents by nature are random events, the central quantity in a risk assessment of the above minefield is the probability distribution , where z denotes the number of accidents in the minefield during a future observation period of a certain length.
In what follows, an observation which starts at time t and ends at time will be denoted as indicated in fig. 2.4. The time unit in fig. 2.4 is arbitrary, but as accident statistics in so-called Landmine Impact Surveys typically report the number of casualties observed during a two-year period, we will assume that | ( = 2 years for all t.
( ) p z
t+1 ( )t
∆
) t |
m
∆
Fig. 2.4. Time axis
t =−1 t =0 t =1 t =2
time t
∆(0) ∆(1)
Now, let denote the number of minefield accidents which might occur during in the minefield from fig. 2.3. To calculate we will by way of introduction look at mine no. 1 from fig. 2.3. During mine no. 1 will either detonate or not. To record this event, let denote the binary random variable which takes the value 1 if mine no. 1 is set off and 0 otherwise.
0 {1,2,..., } Z ∈
∆(0) p z( )0
∆(0)
10
Z
To calculate , that is, the probability of mine no. 1 being set off during , it is valuable to consider the sequence of events which is a prerequisite for a detonation: Firstly, during there has to be a “contact” between mine no. 1 and a person, a vehicle, etc.
Secondly, to detonate during the “contact”, mine no. 1 has to be exposed to a pressure which is equal to or exceeds a certain threshold value.
1
( )0
p z ∆(0)
∆(0)
The very simplified account given above covers up certain difficulties. First of all, the notion a “contact” is ill-defined, as the triggering of a mine not necessarily implies a physical contact between the mine and say a person. Secondly, to set off a mine the triggering pressure has to be exerted at the right part of the mine or at the right part of the ground above a buried mine.
To overcome the above difficulties and to keep our model considerations simple, we will assume that every mine can be characterized by an individual contact zone, that is, a surface in 3D-space with the following properties:
1) To set off the mine, a pressure equal to or exceeding a certain threshold pressure (TP) has to be exerted within the boundary of the contact zone.
2) The threshold pressure is constant over the contact zone.
Examples of contact zones for different types of mines are sketched in fig. 2.5 below.
Depending on whether the mine is located on the surface of the ground or buried, the contact zone may or may not coincide with parts of the casing of the mine.
Fig. 2.5 Contact zones of mines. The red coloured areas denote the contact zones of mines of various designs.
Fig. 2.5.a Fig. 2.5.b Fig. 2.5.c Fig. 2.5.d
tripwire
Fig. 2.5.e
Fig. 2.5.f
surface level
tripwire
mine
The introduction of contact zones allows us to clarify the “contact” concept: Whenever a person, a vehicle, etc., touches the contact zone of a mine, we will refer to the event as a
“contact”.
The idealized model of a uniform threshold pressure can be sketched as in fig. 2.6 below.
Fig. 2.6. Probability of detonation. denotes the probability of detonation given a pressure P is exerted on the contact zone of a mine. The value “TP” denotes the threshold pressure of the mine.
det(P) p
TP PHpressureL 1
pdetHPL
It should be noted that not all mines fit into the idealized model sketched in fig. 2.6. We will however ignore cases such as the PFM-1 anti-personnel mine which can be triggered by the accumulated effect of successive contacts due to its pressure fuzed liquid explosive.
The magnitude of the threshold pressure of a mine will in general depend on factors such as
• type of mine (AP-mine, AT-mine)
• fuzing mechanism
• condition of mine (ageing, corrosion)
• vertical position of mine.
Whether the threshold pressure of a mine is reached during a random contact will in general depend on the kind of activity during the contact (walking, driving, ...). In addition, for a given type of activity the pressure exerted on a mine will presumably vary from contact to contact due to its stochastic nature. To incorporate this variability into our model we will assign the minefield from fig. 2.3 a probability distribution which denotes the probability of observing a contact pressure of magnitude CP during a contact with a randomly selected mine. The contact pressure is here defined as the maximum pressure exerted on a randomly selected mine during a contact.
( p CP)
It follows from the considerations above that mine no. 1 subsequent a random contact only will detonate with a certain probability which can be calculated as φ1
(2.01)
1
1 ( )
TP
p CP dCP φ
∞
=
∫
,where in equation (2.01) denotes the threshold pressure of mine no. 1. The parameter will be denoted the conditioned probability of detonation of mine no. 1.
TP1
φ1
After having introduced these facilitating concepts, a closed expression for can be obtained in the following way: Let denote the random variable which counts the number of times the contact zone of mine no. 1 is struck during the period . The probability of mine no. 1 not being set off can be written as
01
( ) p z X1
∆(0)
01 1 (2.02)
1 1
2
1 1
1 1
0
( 0) ( 0)
( 1)(1 )
( 2)(1 ) ...
( )(1 ).i
i
p Z p X
p X p X
p X i φ φ
∞ φ
=
= = = +
= − +
= − +
=
∑
= −If X1 follows a Poisson distribution with intensity , that is λ1
1 1 11
1
( ) ,
!
x
p x e x
λ λ
= − (2.03) where E[X1]=λ1, it follows that
1
1 1
1 1
0
0
( 0) (1
! .
i
i i
p Z e
i e
λ
λ φ
λ φ
∞ −
=
−
= = −
=
∑
1)1
Z0
0
(2.04)
Consequently p z( )01 takes the form
(2.05)
1 1
1 1
1 1 0
0 1
0
1 if
( ) if 0.
e z
p z e z
λ φ λ φ
−
−
⎧⎪ − =
=⎪⎪⎨⎪⎪⎪⎩ =
If the stochastic variables furthermore are independent, it follows that the distribution of can be calculated as
1 2
0, 0,.., 0m
Z Z Z
0 1
m k
k
Z
=
=
∑
0 (2.06)
1
( ) m ( ),k
k
p z p z
=
=
∑
where p z( )0k is given as
0 0 (2.07)
0
1 if
( ) if 0,
k k
k k
k k
k
e z
p z e z
λ φ λ φ
−
−
⎧⎪ − =
=⎪⎪⎨⎪⎪⎪⎩ = 1
0
and the sum denoted by Σ in equation (2.06) includes all vectors for which . In spite of the simple structure of equation (2.07) the model embedded in this equation reflects the combined action of several factors, that is,
1 2
0 0 0
( , ,...,z z zm)
1 2
0 0 ... 0m
z +z + z =z
• the types, conditions and vertical locations of the mines present (reflected byTPk )
• the activities taking place in the mined area (reflected through p CP( ))
• the intensities of the activities taken place in the mined area (reflected by ). λk The utility of the model may be questioned as neither m nor the true values of the parameters will be known in the general case. We might however have some, albeit incomplete information at hand which makes it possible to make a qualified guess at their true values by means of probability distributions , and . From these distributions can be calculated numerically.
{{ , }}φ λk k
( )
p m p( )φ p( )λ ( )0
p z
In the present chapter we will follow a slightly different course. That is, by introducing two additional assumptions the stochastic variable from (2.06) can be turned into a binomially distributed variable. Apart from its simple analytical structure the binomial model demands as input only two parameters to calculate .
Z0
( )0
p z Table 2.1. Applied notation in minefield model.
Factor Represents Factor Represents
t time TPk Threshold pressure of mine no. k.
( )t
∆ Observation period [t ; t+1] CP Contact Pressure
m Number of mines p CP( ) Probability of CP during contact.
Z t Number of accidents in ∆( )t φk The probability of detonation of mine no. k given a random contact.
( )t
p z Probability of observing accidents in .
zt
( )t
∆
Xk Number of random contacts with mine no. k during ∆( )t
k
Zt 0-1 variable. Indicates whether mine no. k has been set off in ∆( )t .
λk The expected value of Xk