Vector model of danger

Reliability Engineering and System Safety 37 (1992) 65-71

Vector model of danger* Andrzej H. Szymanek Technical University of Radom, Department of Transport, ul. Malczewskiego 20, 26-600 Radom, Poland (Received 1 February 1991; accepted 1 May 1991)

In this paper a mathematical model is described in which the danger to any real object is interpreted as a vector in a given vector space. This model also provides mathematical interpretations of danger speed as well as danger field, notions which can broaden the range of safety science. The model presented is intended to give a solid base of formal language to system safety theory. This paper includes an example of the vector interpretation of danger, and the possibility of mathematical description of danger in various human activities and natural systems.

. Safety theory methods. It is necessary that these methods model, evaluate and forecast the danger to various real objects more completely than non-classical mathematics methods. First of all one refers here to fuzzy set theory, game theory and catastrophy theory. System analysis methods should be developed as well. The nature of accidents requires a probabilistic approach to safety analysis. The possibilities of employing probabilistic methods in modelling and safety system analysis have not been exhausted; in particular one is referring here to stochastic processes theory and its applications.

1 INTRODUCTION In the nineties one may expect theories concerning the safety of real objects. These theories will be based on the results of examining the safety of various technical and social activities as well as natural systems. Systems safety theory must cover, among other things, the following three fields: 1. Safety philosophy. It is necessary to establish general principles allowing the interpretation of the nature of danger. It is not possible to recognize the nature of danger directly, since danger is a partially subjective category: from the human point of view it is as big as can be accepted. This principle allows the description of danger by analysis of the risk of dangerous events which are characteristic for the object analyzed. Important here is the problem of risk valuation: 'How safe is safe enough?', l 2. The formal language of safety theory. It is necessary to strictly and objectively interpret the basic notions used so far, intuitively or on the basis of empirical premises. Among such basic and widely used notions are danger, risk, conflict, accident, safety. It is also necessary to introduce new notions, which describe safety problems better.

One should aim to make compatible theories properly interpreting the 'nature' of danger in various classes of human activity and natural systems. It is obvious that theoretical models must be identified and verified by truthful empirical data. The problem is to register such dangerous events as: incidents, conflicts, accidents, faulty and potentially dangerous human behavior, etc. Valuation and forecast of any real object safety should be preceded by objective interpretation of the basic notions of a safety field. This requirement is the basis of this paper. The author shows here some of the considerations on creating a formal safety theory language of real objects. As 'real objects' we mean simple material objects as well as systems. First of all we mean human technical activity systems, social systems and natural systems. Among real objects we include all phenomena and processes which generate danger. In this paper a vector interpretation of danger in objects of these types has been proposed. Such interpretation is close to intuition and allows us to

* An earlier draft of this paper was presented on the III Sympozjum Bezpieczenstwa Systemow 3rd Symposium on Safety Systems, held in Kiekrz, Poland, 16-19 October 1990, (Polish language only).

Reliability Engineering and System Safety 0951-8320/92/$05.00 © 1992 Elsevier Science Publishers Ltd, England. 65

66

Andrzej H. Szymanek

treat 'danger' as a mathematical category. Hence, formal interpretation of new notions broadening the range of safety science is possible. Some of the results come from Ref. 2.

3 D A N G E R RELATION

On set fl we define the relation p = p(toi, toj): /~

: to,ptoj ~ Us(to,) -> Up(%)

(2)

mi, mje.~

2 D A N G E R POTENTIAL A N D PROTECTIVE POTENTIAL

Consider a set fl of arbitrary objects. To each real object to e Q two kinds of potential can be assigned; Us(to)--danger potential (negative potential): object's imminent ability to destructively influence other objects; Up(to)--protective potential (positive potential): object's imminent ability to compensate for the destructive influence of another object (or itself). Examples of Us and Up potentials are shown in Table 1. Set Q can be formally written down as follows: Q = {to: if the potential exists: Us(to) and (or) Up(to)}

Relation p is called here 'a relation of object wj endangered by object wi'. This relation is of fundamental importance to the interpretation of any real object danger. The difficulty in making use of relation p in real object safety analysis consists in: --determining which physical, chemical, psychophysiological and other quantities (parameters) may be considered to be potentials Us,

up; ----estimating and comparing values of potentials U~,

Up. The inequality of formula (2) is not, of course, the only possible definition of the danger relation p. For instance, in probabilistic interpretation, perhaps more adequate, the danger relation may be written as follows:

(1) /~

Each real object to e fl can be:

: to,p% ~ @(Us(toi) -> Ue(toj)) > 0

--only a source of danger to = tos: when only the Us(to) potential can be assigned to it; ---only an endangered object to = top: when only the Us(to) potential can be assigned to it; wboth a source of danger and an endangered object to = tosP: when both UN(to) and Up(to) potentials can be assigned to it at the same time.

where ~(UN(toi)I> Up(toj))> 0 means that inequality Us(toi)/> Up(%) is fulfilled with certain probability ~. Making use of the probabilistic danger relation, the probability described by formula (3) should be estimated; potentials Us, Up denote random variables or stochastic processes, if their time dependence is taken into account.

Table 1. Examples of Us and Ur potentials for various objects

Source of danger

Potential Us

Object of danger

Potential Up

Sea

Sea aggressiveness Earth seismicity

Ship

Ship's stability

Building

Nuclear power plant

Radiation

Man

Load

Weight of load

Structural component

Poison

Poison's toxicity

Man

Industry

Ability to emit pollution

Ecosystem

Human operator

Dangerous

Human operator

Building's stability Organism's resisting power Structure's mechanical resistance Organism's resisting power Ecosystem's, resisting power Human reliability

Earth's Crust

(3)

mi, mi~Q

errors

Vector model of danger

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4 DANGER CLASSIFICATION

6 STATE OF DANGER

Based on existing considerations, separable danger classifications in real objects (systems) have been worked out.

A danger relation is time-dependent and describes changes of potentials Us, Up in relation to each other. Any system SD state in that case has an interpretation of 'state of danger' (emergency) existing between objects to;, toj e f~. One needs to emphasize that the general term 'danger' does not need to be used. As seen from the above considerations, it may be replaced by a precise term 'state of danger'. Since a terminological tradition does exist, we will consider these terms as synonyms.

I classification criterion: algebraic properties of relation p: --self-dangers (relation p reflexiveness); --bilateral dangers, mutual (relation symmetry); --direct and indirect dangers (relation transitivity).

p p

Generally the danger relation is antireflexive, antisymmetric but not transitive. However it's possible to determine conditions concerning potentials Us, Up for which this relation is an equivalence relation on set f~. II classification criterion: nature of potentials Us,

Up: --material dangers: (road, sea, other); --energetistic dangers (heat, radioactive, other); --information dangers (telecommunicational, radiotechnical; appearing in computer network, other); --immaterial dangers (psychical, social, political); -----economical dangers. III classification criterion: relation p duration:

7

DANGER AS A VECTOR?

Let us consider a Cartesian coordinate system UN, Up. In this coordinate system we find any point knowing the values of potentials Us = UN(to), Up = Up(to). Note that a good location characteristic of this point is its position vector. Note as well that the state of danger can be described by values of potentials UN(toi), Up(toj). The state of danger existing between objects to, toj can be formally described as a vector. This vector we will call the 'danger vector', denoted as d. The danger vector is a vector of a point position in the coordinate system Us(to), Up(tO), and the inequality described by formula (2) is also true. In symbolic notation the definition of the danger vector is as follows: d = [Us(to,), Up(toj)]

--dangers of short duration (traffic conflicts); --long-lasting dangers (ecological dangers); ---cyclic dangers (natural calamities). IV classification criterion: degree of system complexity of danger source and endangered object: --'system-system' dangers (ecological dangers); --'system-single object' dangers (e.g. dangers of pedestrian in road traffic); --'single object-system' dangers (e.g. danger of water poisoning in water supply system by terrorist).

for

UN(toi) >-- Up(toj)

The idea of interpreting the state of danger as a vector from the potential space UN(to), Ue(to) is shown in Fig. 1. The set of all vectors d we denote as Ud. It is a subset of set U of all vectors u, i.e. position vectors of any point in potentials space Us(to), Up(to). In the probabilistic description of danger, formula (5) must be completed with probabilities ~[Us(to/)],

~[Up(to,)]. The reasoning behind formula (5) corresponds well with intuition, since in describing danger it is not

5 CONCEPTION MODEL OF DANGER SYSTEM

'Up c-

Consider a set Q of any real objects and relation p. Using a general system definition3 one can, based on f~ and p, define system St,, which is called here conventionally the 'danger system' of object toj endangered by object to;. Thus the danger system defined on set fl of real objects is an ordered pair: SD = (f~, p(to;, to/))

for any

to;, toj e f~

[3. >

~(~

UN(Wi ), Up(Wj)

0~ fn

(4)

The danger relation p is here a system-formative relation.

(5)

0

I Danger potential

UN

Fig. 1. Interpretation of 'danger vector'.

68

Andrze] H. Szymanek

enough simply to give its value. It is necessary as well to give 'direction' and 'sense' of danger: those are features of vectors not scalars. $ DANGER TRAJECTORY

Formula (5) gives a vector interpretation of the state of danger, i.e. the state of system So at any given moment. We are interested in how to describe changes of state of system SD, and in the limited time interval [0, T]. We introduce following mapping:

u, = [0, r]--~ U

(6)

Here u, = u(t) is a vector function of the real variable t e [0, T]. In accordance with formula (6), to each number t e [0, T] is assigned the vector u e U, with the following coordinates:

u = [Ur~(toi; t), U~,(toj; t)] for t ~ [0, T] (7) Note that each value of vector function (6) corresponds to a point in three-dimensional space, of which two dimensions are the potentials Us(toi), Up(toj), whereas the third dimension is time t. In geometrical interpretation the vector u envelope curve for t ~ [0, T] is the space curve. We call this curve 'a trajectory of danger system states'. We will denote it as u(t). Curve u(t) may have two characteristic runs; --if Us(toi; t) < Up(toj; t), for each t e [0, T]: then function u(t) will be called the 'safety trajectory' and we will denote it as s(t), --if Us(to~; t) -> Up(toj; t), for each t ~ [0, T]: then function u(t) will be called the 'danger trajectory' and we will denote it as d(t). Of special interest are those cases when trajectory is partially trajectory s(t) and trajectory d(t) (see Fig. 2).

u(t)

catast

/~s(t)K/~

s(t)

>e

~

n

0

catastrophe

Dangerpotential

The appearance of danger relation p(t), for t ~ [0, T] does not imply the failure or damage of the endangered object. Between the appearance of the danger relation and the appearance of the state traditionally referred to as an 'accident' a certain continuum of intermediate states of danger (emergencies), i.e. a 'danger continuum' exists. The problem arises as to how to divide the danger continuum into a state of danger classes. This is a problem of danger scaling. In hitherto existing literature these states are known as: incidents, conflicts, near-accidents. The scaling of states of danger should be carried out according to two criteria:

10 DANGER SPEED

d(t) C_

9 THE PROBLEM OF D A N G E R SCALING

--the criterion of state of danger probability or other measure characterizing the possibility of appearance of danger, e.g. fuzzy measure; --the criterion of the state of danger severity; this problem is connected with the notion of risk.

Up

C+

Between trajectory u(t) and axis t the threedimensional surface can be spread. This surface is the geometrical representation of mutual changes in potentials UN(to; t) and Up(to; t) in the time interval [0, T]. A certain trajectory u(t), for t ~ [0, T] can be assigned to each line located on this surface. In Fig. 2 an example of such a trajectory is shown with arrows. At point C_ there is a transition from the state where Us(to;; t) < Up(toj; t) to the state where UN(toi; t) --> Up(to# t). This transition point may be interpreted as the initial moment of the accident process. At point C÷ there is the inverse transition. This transition point may be interpreted as the moment of elimination of danger. In mathematical terms transitions C_ and C÷ are of a structural nature, meaning that they can be interpreted as catastrophes. Using the methods of differential geometry one can work out the equation of the 'danger surface' on which all possible object trajectories u(t) are located (see Fig. 2).

UNt-

Fig. 2. Interpretation of 'danger trajectory' and 'danger surface'.

Mathematical formalism introduced until now provides the possibility of defining and interpreting new danger characteristics. This is in the case when the time derivative of the vector function n(t), t ~ [0, T] is defined. Such a derivative indicates the timedependent state of danger and this is why it can be interpreted as 'danger speed'. Danger speed can be described by a new vector function v,: d

v,=v(t)=~n(t),

for

t~[0, T]

(8)

Vector model of danger 1 Model (hypothesis) of potentials scalarfield

Danger speed is the vector

[d

d

69

]

v= dttUs(to`.;t)'-dtUP(to/;t) , for t e [ 0 , T l (9) The geometrical interpretation of instantaneous danger speed is a vector tangent to a trajectory u(t) at the point t e [0, T]. The sense of a vector v shows whether a danger is increasing or decreasing. However, this vector length is a measure of the danger increase or speed decrease. Danger speed is the Vth danger classification criterion (see Section 4), and can indicate --increasing dangers ----decreasing dangers --periodic dangers

11 DANGER FIELD Up to now we have interpreted the state of danger as a vector in two-dimensional vector space U. The next step in creating our theory will be to interpret geometrical space around the danger source. Let us consider the three-dimensional geometrical space M =M(X, Y, Z) around object toi with destructive potential Urn(to`.). Let object to,. be located in the middle m =m(0, 0, 0) of the cartesian coordinate system. Then we can assume that there is interaction of object to`.via its potential Us(to`.) in each point m of space M. If we place any object to/ with defensive potential Ul,(to/) in any point m of space M, then potentials Us(to`.) and Up(to/) will give the danger relation described by formula (2) or (3). Then there exists a state of danger (emergency) between objects to`. and to/. The above argument enables two propositions to be submitted.

Proposition 1: 'on danger transmission' At each point m of three-dimensional geometrical space M around object to~ with negative potential Us(to`.) there is a destructive interaction of the said object. Potential Us(to`.) is 'transmitted' to each point in space M.

Proposition 2: 'on danger compensation' If in any point of three-dimensional geometrical space M around object to; with negative potential Us(to`.), an object toi with positive potential Up(to/) is placed, then the potential Us(to`.) is partially or completely compensated. In that case two alternative danger hypotheses are sensible. They are presented here as mathematical models.

Any point m of three-dimensional geometrical space M around object to`. with negative potential UN(to,.) can be unequivocally characterized by assigning to it a non-negative value AU, i.e. the difference between the negative potential Us(to`.) and the positive potential Up(to/) of a certain object to/located in point m e M. This hypothesis is described by the following mapping: AU:M---> AU(M) (10) In the above formula, AU(M) is a scalar function of a point in space M; the value of AU(M) is in this case the 'scalar field of the difference of danger source potential and endangered object potential'. The scalar field AU(M) can also be described by the scalar function of the vector variable, i.e. the position vector r of points of space M in relation to the origin of the coordinates (i.e. in relation to the danger source):

AU:M---} AU(r)

(11)

In the above hypothesis the notion of potential difference AU plays the key role. From the above considerations it is known that potentials Un(to~), Up(to/) define the vector d (see formula (5)). So the problem of danger 'transmission' around the danger source can be as well described using vector notation. This is the idea of the second hypothesis.

2. Danger vector field model (hypothesis) Any point m in the three-dimensional geometrical space M around object to`. with negative potential UN(to`.) can be unequivocally characterized by assigning vector w to it. The length of this vector is proportional to AU. This hypothesis is shown as the following mapping: w:M-~w(n)

(12)

Quantity w(M) is the vector function of points of space M; the quantity w(M) is then the 'vector danger field'. The source of this field is object to`. with potential UN(toi). The vector field w(M) can also be described by a vector function of the vector variable, i.e. a function of position vector r of points in space M in relation to the origin of coordinates (i.e. in relation to the danger source):

w: r---~w(r)

(13)

Describing the above hypothesis more explicity, one can say: in each point m of geometrical space M around a danger source, vector w can be 'originated'. The length of this vector is proportional to potential difference A U characterizing the danger source and the endangered object. Vector w is located on a straight line connecting danger source and endangered object (point m). The sense of vector w is from the danger source. Space M 'filled' with vectors w is the vector danger field.

Andrzej H. Szymanek

70

We will present here a general formula describing the vector danger field. In order to do this we will assume that vectors w are not only proportional to AU; two more assumptions are made, which can be empirically verified: --the potential difference A U depends upon distance of object coj from danger source toi; this dependency can be described by introducing function f(r), where r is the length of vector r; --the potential difference A U depends indirectly upon 'coefficient of proportionality' n, which characterizes physical properties of the space M regarding transmission of the danger potential

UN( i).

following assumptions: ----energy E(m;t) is interpreted as potential Us; --the building stability S(m;t) is interpreted as potential Up.

2. State of danger: due to formula (5), this is the vector d. d = rE(m; t), S(m; t)], where: E(m; t) >- S(m; t), for t e [0, T]

3. Danger trajectory; this is the envelope curve of vectors described by the formula u = rE(m; t), S(m; t)]

In connexion with the above, the danger field is described as follows: (14)

w = A U(at) of(r) o ~

In the above formula symbol ~ is the versor of vector r. Vectors w are directed from the field source. So the following dependency occurs: r

versor: E = -

(15)

r

The vector danger field can be described by the following general formula:

w = r -1 o A U(~r) of(r) or

(16)

(21)

for

t e [0, T]

(22)

4. Danger speed: this is the vector function vt described by the general formula (8). Vector v of instantaneous danger speed will be computed by time-differentiating the vector u coordinates:

d

v=

E(m; t), -dt S(m; t)

]

for

t e [0, T] (23)

In the case when the building stability S(M;t) is treated as constant, formula (23) shows that the instantaneous danger speed is the time-derivative of the earthquake energy; it can be written as the vector

v=

[d~ E ( m ; t ) , O ]

for

tE[0, T]

(24)

The potential difference AU can be written as follows:

AU = :t o Us(toi; 0; t) - Up(toj; m(r); t) for m e M , and t~[0, T]

(17)

Finally, the vector danger field around object to; can be presented in the following form: W----r -1 of(r) o [~o Us(toi; O; t) - Up(a~j;m(r); t)]or formeM,

and

tE[0, T]

(18)

w = r -1 of(r) o [:to E(0; t) - S(m; t)] or

13 EXAMPLE: VECTOR DANGER FIELD AROUND AN EARTHQUAKE EPICENTRE

for m E M

An interpretation of the danger to buildings during earthquake has been made. 1. Danger relation: building endangering in time t occurs when the instantaneous value of shock strength E(m; t) surpasses building construction stability S(m; t); this can be written as follows:

E(m,t)>-S(m,t)

for t~[0, T]

(19)

In probabilistic interpretation this is:

~(E(m, t) >- S(m, t)) > 0 for t ~ [0, t]

5. Danger field; there is an intrinsic vector danger field around the epicentre. It has unequivocal physical interpretation. It corresponds to the area in which the earthquake energy 'comes out'. Mathematically, the danger field is a vector function of building position in relation to the conventional origin of coordinates, i.e. in relation to the epicentre. In accordance with formula (18), the danger field around the epicentre has the following mathematical form:

(20)

The formulas above result from eqn (2) and the

and

t e [ 0 , T]

(25)

Coefficient n can be interpreted as the lithosphere damping coefficient; only a part of energy E(0;t) released at the epicentre reaches the building (point m). Stability S(m;t) is equivalent to the minimal earthquake energy which will cause building damage; let it be denoted conventionally as Emi,. In accordance with the above assumptions, the danger field in the space around the epicentre can be written formally as the vector field:

w=r-2°[:t°E(O;t)-Emin]°r

for

t e [ 0 , T]

(26)

For instance, in the case of a flat area, the danger field

Vector model of danger is circular:

X y2 o [g Wx - - X 2 ..[_ - X2 ~ ..[_re Wy --

o E(O; t) -- Emin] 1

° [~roE(0; t) - Emin]

J

for

t • [ 0 , T] (27)

x, y are coordinates of the building in relation to the epicentre.

14 CONCLUSIONS The model of the vector danger field allows the evaluation of the quantitative negative influence of an analyzed object upon other objects located a certain distance away. This model enables dangers to various real objects to he described. Vector interpretation of danger should give new possibilities for mathematical system safety modelling. Vector algebra and vector analysis can be used. 'Danger sum' and 'danger product' can also be defined and analyzed. Using probabilistic interpretation of the danger relation, new problems can be formulated. Problems of mathematical formalization of other

71

basic notions such as risk, conflict and safety also need to be studied. The next stage of formalization of danger science should be creating danger valuation and forecast models of various real objects. The model presented here may first find use in safety analysis of real objects, where potentials UN, Up interpretation is unequivocal. This includes dangerous natural phenomena and processes such as earthquakes, volcano eruptions, hurricanes and aggressive sea environments. In civilization, the interpretation of object potentials Urn, Up is unequivocal if the danger 'carrier' is energy in its various forms.

REFERENCES 1. Jaeger, T. A., Das Risikoproblem in der Technik. Schweizer Archiv fiir angewandte Wissenschaft und Technik, XXXVI (7), (1970). 2. Szymanek, A. H., Wektorowy model zagroSenia obiektu. Paper presented at Materiaty III Sympozjum Bezpieczehstwa System6w, ed, K. Wa~yfista-Fiok, Instytut Techniczny wojsk Lotniczych Warsaw, Poland, 16-19 October, 1990. 3. Hall, A. D. & Fagen, R. E., Definition of system. In General Systems, vol. 1. 1956.