Risk Analysis in Water Quality Problems.
Souza, Raimundo1
Chagas, Patrícia2
1,2
Departamento de Engenharia Hidráulica e Ambiental Centro de Tecnologia - UCF
Campus do Pici, P. O. Box 6018 60451 - 970, Fortaleza, Ceará
Cornell University
1
School of Civil and Environmental Engineering 466 Hollister Hall
Ithaca, NY, 14850
Emails: rs237@cornell.edu; pfchagas@yahoo.com
Abstract
Engineering Risk Analysis has been a very strong methodology in order to measures the uncertainties presents in all Engineering processes. In the field of Water Quality, the presence of this uncertainty comes from the different sources. Data base, approximation theory, numerical methods for the solutions of the differential equations are some of these sources. This work applies the methods of probabilities to evaluate the uncertainties present in the water pollution analysis processes. The results have shown that even that the method of probability is a very strong method, there are some restrictions to use the method.
Introduction
With the development of Technology and increasing of the population, the water quality pollution has become an important topic to be worried by scientist around the world. This fact has brought a very strong development to new theories, in order to get a better understanding about the processes that involves the water quality aspects. Today, the water quality problems are so important as the water quantity problems. In fact, a modern way to study water problem, concerning with water resources management, should involve the integration of the water quality and water quantity together as an unique problem.
On the other hand, the solution for this class of problem, usually comes with a very strong set of uncertainty. These uncertainties are incorporated in the processes of solution by different ways. Data set, approximations solution in the numerical methods, are some sources that can bring uncertainties to the final solution. These uncertainties must be understood and must be measured, so that, a better interpretation of the results could be done.
Engineering risk analysis is a strong way that is being developed in the present day, to manage and bring a better understanding to these uncertainties.
Following Ganoulis, (1994), Engineering Risk and Reliability analysis provides a general methodology for the assessment of the safety of engineering projects. Yet, following
Ganoulis, (1994), risk and reliability assessment of water pollution is a useful tool to quantify uncertainties and to evaluate their consequences on water resources.
This material presents a general methodology of engineering risk analysis applied to water pollution problem, in order to establish a better understanding of the uncertainty that comes from any solution method in the process of analysis of these classes of problems.
The results of this analysis has shown that it is important an efficient set of hydrologic data associated with a very powerful method of solution, so that it will be possible to have a better control of the water pollution problem.
Engineering Risk Analysis Theory
It is extremely difficult to define risk in a single set of word. The reason for that concerns with the high level of confusion surrounding the aspects of this subject. In general risk could be established in qualitative aspect as in quantitative aspect. The latter one is usually called Engineering Risk Analysis. It is important to observer that the qualitative aspect of risk brings one idea about failure or success of some defined event. In such way risk comes relative to hazard and safeguards, where hazard is defined as a source of damage or injury.
Thus, it is possible to say that risk could be expressed by the symbolic relationship
) , (h s R
risk = (1)
Where, h means hazard and s means safeguard. For example, given a safeguard, the large is the hazard the bigger will be the risk. On the other hand, for a given hazard, the bigger is the safeguard, the smaller will be the risk. However, it should be point out that the relationship (1), just establishes the idea of the behavior of the function R(h,s). It cannot establish any quantification of Risk.
The quantification of risk involves look for answers for three basic questions.
• What can happens?
• How often failures is expected?
• What is the likely consequence?
As Ganoulis, (1994), pointed out, the research to answer the first two questions involves the establishment of the part of the uncertainty analysis of the systems. For example, the answer for the first question is given by writing scenarios describing what might go wrong and in which way this could happen. In order to get answer for the second question it is important to introduce uncertainty aspect into the method of analysis. Such way can be done considering the all variables of the problem as stochastic ones. Thus, the answer for these questions can be investigated through some stochastic method available. Actually there are two methods that can be used in order to quantify risk. The first one is the probabilistic method where all set of variables in the problem, is defined as random variables. The other one is the Fuzzy Method, where all set of variable is considered as fuzzy set. In this research it was used the probabilistic method.
In order to formulate the engineering risk analysis it is important to define a scenario that can be considered as a reference. To do so, let suppose that the capacity C0 of any system to resist any external load E0 could be defined as a Random Variable. In other word, the pair (C0, E0) could be considered as a pair of random variable. This means that all uncertainty that comes in consequences of the estimation of (C0, E0) are quantified by probabilistic methods.
Therefore, the way to calculate the risk of failure of any environmental system cold be done by the application of the probabilistic theory over the set of random variable (C0, E0).
Suppose the system of random variable (C0, E0), has the probability distribution function (C, E) and probability density function (c, e) related with other through the equation
∫
∞ −= x
dx x c x
C( ) ( ) (2)
∫
∞ −= x
dx x e x
E( ) ( ) (3)
Where,
C(x) is defined as being the probability distribution function or cumulative distribution function of the set random variable C0;
E(x) is defined the probability distribution function of the set of random variable E0; c(x) and e(x) are respectively the probability density function of the set of random variable (C0, E0).
It is important to note that, by the probability theory, the cumulative distribution function and the probability density function have the following properties;
0 ) (x ≥
C and
∫
∞∞ −
=1 ) (x dx
c (4)
0 ) (x ≥
e and
∫
∞∞ −
=1 ) (x dx
e (5)
The equations (2) and (3) have the mean of measures of probability. For example, in the equation (1) C(X) is the probability that the random variable X does not exceed the random variable x.
After defining the probability distribution function and the probability density function,
∫
∫
∞= > =
c
ec
f P E C e c dc de
p
0 0
] ) , ( [ ]
[ ψ (6)
Where, ψec(e,c) is the joint probability density function of the two continuous random variables E0 and C0.
It is important to note that
ψec(e,c) ≥0 and
∫ ∫
( , ) =1 ∞∞ −
∞
∞ −
dedc c e ec
ψ . (7)
Consequently, the probability of a success will be estimated by ;
f
s P
P =1− (8)
Where, Ps will be defined as P[C>E].
Applying some probabilistic properties over the equation (5), and suppose that E and C are independent random variables, equation (5) can be transformed to;
de dc c e
p
c
c e
f ( )[ ( ) ]
0
0
∫
∫
∞= ψ ψ (9)
In another words, if one can get the probability density function of the two continuous random variables C0 and E0, it is possible to quantify the risk of failure of any environmental system with capacity C and load E, through the equation (9).
On the other hand, the application of the probability methods to quantify risk has some restrictions, considering that this methods needs the definition of the probability density function of each random variables. This mission, sometimes, is not so easy. That restriction has brought the development of new techniques, in order to solve this problem. The new technique that is in the very beginning but has brought good results in water resources, is the Fuzzy Theory. In this new technique one does not need so much data in order to establish some risk analysis. However, as it was said before, this methodology is in very beginning.
Application to Pollution Problem
In order to apply this theory in the Pollution Problems, it is important define some random variables that should be managed. To do so, let us suppose a body of water with capacity or some limit level of concentration C. This body of water is receiving some load of pollutant P. As it was pointed out before, C and P are random variables.
Therefore, probabilistic method of analysis can be applied in order to give some measurement of the risk of pollution of that body of water.
Suppose yet that both C and P are positive random variable with probabilistic density functions fc and fp. In such way, the risk R can be calculated through the equation;
dp dc c f p f C P P R c c p( )[ ( ) ] ] [ 0 0
∫
∫
∞ = >= (10)
where, C and P are independent random variables.
Thus R is the risk that the concentration P comes to be bigger than the capacity C. The opposite of R will be defined as the probability that C be always bigger than P. This situation is called reliability.
The reliability is thus;
R
r =1− (11) For example, let suppose that both C and P are normal distribution with probability density function defined by;
c k c c c e k
f = − ’ and p p kpp
e k
f = − ’ (12)
Therefore the risk can be calculated by;
dc dp e k e k C P P
R kc
p c p k p c p ] [ ] [ ’ ’ 0 0 − − ∞
∫
∫
= >= (13)
Solving this equation, the risk will be;
) ( ’ ’ ’ p c p p c k k k k k R +
= (14)
It is important to observe that, in this simple case, kc and kp depend on the expected values Ec and Ep and the variance σp2 and σc2. However, if a most complicated distribution is found for C and P, the probability density function of the random variables must be calculated and, in such case, the mission could not be so easy.
A different way to contours this difficult is to apply the Fuzzy Set Theory. This theory permits to analyze the uncertainty inherent in data, mathematical model, parameters, and boundary conditions, with a few information. In this theory, one needs just a few data to represent them as a fuzzy number. However, the Fuzzy Set Theory, that has been applied
in all field of science, has its application in the Environmental Engineering in the very beginning , needing more development in its principles.
The probability theory, that is more available for this purpose, has brought good results to the risk analysis. However, its application becomes very limited if the set of data is not sufficient to get a probability density function. This is a very common situation concerning with the water pollution problem. In this case, the probability theory cannot be applied, in order to analyze uncertainty inherent to the Environmental Engineering Problem.
Conclusions
After the analysis of the probability theory to evaluate and quantify risk concerting with pollution in a body of water, one can conclude that the theory is extremely useful to system with a consistent set of data. On the other hand, for systems without a consistent set of data, the chance to get good results, on this mission, will be very scarce. The reason for that is that the probability theory needs, in its application, the probability density function for all set of random variables that should be analyze, in order to measure the risk of any environmental system. This cannot be done with an inconsistent data set. In this case the Fuzzy Set Theory could be a better way to this kind of study.
References
Ganoulis, J., 1994, “Engineering Risk Analysis of Water Pollution: Probabilities and Fuzzy Sets”, VCH.
Ganoulis, J, 1991, “ Water Resources Engineering Risk Assessment”. NATO, ASI Series,Vol. 29, Heidelberg – Springer Verlag.
Stakhiv, E, 1986, “ Risk Analysis Considerations of Dam Safety, In Engineering Reliability and Risk in Water Resources”. L. Duckstein and E. J. Plate, editors.
Bagtzouglou, A. C., A. F. B Tompson and D. E. doudherty,1991, Probabilistic Simulation for Reliable Solute Source Identification in Heterogeneous Porous Medium”. In: Ganoulis (ed.), Water Resources Engineering Risk Assessment, NATO, ASI Series, Vol. 29, Heidelberg, Springer - Verlag.
Bardossy, A., I. Bogardi and L. Dudkstein, 1990, “Fuzzy Regression in Hydrology. Water Resource Res. 26(7).
The author Dr. Souza would like to thank the Government of Brazil that, through the CAPES, has supported His sabbatical license at the Cornell University.
