Risk Analysis in Water Quality Problems. Souza, Raimundo 1 Chagas, Patrícia 2 1,2 Departamento de Engenharia Hidráulica e Ambiental

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1 Risk Analysis in Water Quality Problems. Downloaded from aselibrary.org by Uf - Universidade Federal Do Ceara on 1/29/14. Coyright ASCE. For ersonal use only; all rights reserved. Souza, Raimundo 1 Chagas, Patríia 2 1,2 Deartamento de Engenharia Hidráulia e Ambiental Centro de Tenologia - UCF Camus do Pii, P. O. Box , Fortaleza, Ceará Cornell University 1 Shool of Civil and Environmental Engineering 466 Hollister Hall Ithaa, NY, s: rs237@ornell.edu; fhagas@yahoo.om Abstrat Engineering Risk Analysis has been a very strong methodology in order to measures the unertainties resents in all Engineering roesses. In the field of Water Quality, the resene of this unertainty omes from the different soures. Data base, aroximation theory, numerial methods for the solutions of the differential equations are some of these soures. This work alies the methods of robabilities to evaluate the unertainties resent in the water ollution analysis roesses. The results have shown that even that the method of robability is a very strong method, there are some restritions to use the method. Introdution With the develoment of Tehnology and inreasing of the oulation, the water quality ollution has beome an imortant toi to be worried by sientist around the world. This fat has brought a very strong develoment to new theories, in order to get a better understanding about the roesses that involves the water quality asets. Today, the water quality roblems are so imortant as the water quantity roblems. In fat, a modern way to study water roblem, onerning with water resoures management, should involve the integration of the water quality and water quantity together as an unique roblem. On the other hand, the solution for this lass of roblem, usually omes with a very strong set of unertainty. These unertainties are inororated in the roesses of solution by different ways. Data set, aroximations solution in the numerial methods, are some soures that an bring unertainties to the final solution. These unertainties must be understood and must be measured, so that, a better interretation of the results ould be done. Engineering risk analysis is a strong way that is being develoed in the resent day, to manage and bring a better understanding to these unertainties. Following Ganoulis, (1994), Engineering Risk and Reliability analysis rovides a general methodology for the assessment of the safety of engineering rojets. Yet, following 1 Coyright ASCE 24 World Water Congress 21 Bridging the Ga

2 Downloaded from aselibrary.org by Uf - Universidade Federal Do Ceara on 1/29/14. Coyright ASCE. For ersonal use only; all rights reserved. Ganoulis, (1994), risk and reliability assessment of water ollution is a useful tool to quantify unertainties and to evaluate their onsequenes on water resoures. This material resents a general methodology of engineering risk analysis alied to water ollution roblem, in order to establish a better understanding of the unertainty that omes from any solution method in the roess of analysis of these lasses of roblems. The results of this analysis has shown that it is imortant an effiient set of hydrologi data assoiated with a very owerful method of solution, so that it will be ossible to have a better ontrol of the water ollution roblem. Engineering Risk Analysis Theory It is extremely diffiult to define risk in a single set of word. The reason for that onerns with the high level of onfusion surrounding the asets of this subjet. In general risk ould be established in qualitative aset as in quantitative aset. The latter one is usually alled Engineering Risk Analysis. It is imortant to observer that the qualitative aset of risk brings one idea about failure or suess of some defined event. In suh way risk omes relative to hazard and safeguards, where hazard is defined as a soure of damage or injury. Thus, it is ossible to say that risk ould be exressed by the symboli relationshi risk = R( h, s) (1) Where, h means hazard and s means safeguard. For examle, 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 oint out that the relationshi (1), just establishes the idea of the behavior of the funtion R(h,s). It annot establish any quantifiation of Risk. The quantifiation of risk involves look for answers for three basi questions. What an haens? How often failures is exeted? What is the likely onsequene? As Ganoulis, (1994), ointed out, the researh to answer the first two questions involves the establishment of the art of the unertainty analysis of the systems. For examle, the answer for the first question is given by writing senarios desribing what might go wrong and in whih way this ould haen. In order to get answer for the seond question it is imortant to introdue unertainty aset into the method of analysis. Suh way an be done onsidering the all variables of the roblem as stohasti ones. Thus, the answer for these questions an be investigated through some stohasti method available. Atually there are two methods that an be used in order to quantify risk. The first one is the robabilisti method where all set of variables in the roblem, is defined as random variables. The other one is the Fuzzy Method, where all set of variable is onsidered as fuzzy set. In this researh it was used the robabilisti method. 2 Coyright ASCE 24 World Water Congress 21 Bridging the Ga

3 In order to formulate the engineering risk analysis it is imortant to define a senario that an be onsidered as a referene. To do so, let suose that the aaity C of any system to resist any external load E ould be defined as a Random Variable. In other word, the air (C, E ) ould be onsidered as a air of random variable. This means that all unertainty that omes in onsequenes of the estimation of (C, E ) are quantified by robabilisti methods. Downloaded from aselibrary.org by Uf - Universidade Federal Do Ceara on 1/29/14. Coyright ASCE. For ersonal use only; all rights reserved. Therefore, the way to alulate the risk of failure of any environmental system old be done by the aliation of the robabilisti theory over the set of random variable (C, E ). Suose the system of random variable (C, E ), has the robability distribution funtion (C, E) and robability density funtion (, e) related with other through the equation Where, x C ( x) = ( x) dx (2) x E ( x) = e( x) dx (3) C(x) is defined as being the robability distribution funtion or umulative distribution funtion of the set random variable C ; E(x) is defined the robability distribution funtion of the set of random variable E ; (x) and e(x) are resetively the robability density funtion of the set of random variable (C, E ). It is imortant to note that, by the robability theory, the umulative distribution funtion and the robability density funtion have the following roerties; C ( x) and ( x) dx = 1 (4) e ( x) and e ( x) dx = 1 (5) The equations (2) and (3) have the mean of measures of robability. For examle, in the equation (1) C(X) is the robability that the random variable X does not exeed the random variable x. After defining the robability distribution funtion and the robability density funtion, risk an be quantified by the exression; 3 Coyright ASCE 24 World Water Congress 21 Bridging the Ga

4 f = P[ E > C] = [ ψ ( e, ) d] de (6) e Downloaded from aselibrary.org by Uf - Universidade Federal Do Ceara on 1/29/14. Coyright ASCE. For ersonal use only; all rights reserved. Where, ψ e (e,) is the joint robability density funtion of the two ontinuous random variables E and C. It is imortant to note that ψ e (e,) and ψ ( e, ) ded = 1. (7) Consequently, the robability of a suess will be estimated by ; P s P f e = 1 (8) Where, P s will be defined as P[C>E]. Alying some robabilisti roerties over the equation (5), and suose that E and C are indeendent random variables, equation (5) an be transformed to; f e ] = ψ ( e)[ ψ ( ) d de (9) In another words, if one an get the robability density funtion of the two ontinuous random variables C and E, it is ossible to quantify the risk of failure of any environmental system with aaity C and load E, through the equation (9). On the other hand, the aliation of the robability methods to quantify risk has some restritions, onsidering that this methods needs the definition of the robability density funtion of eah random variables. This mission, sometimes, is not so easy. That restrition has brought the develoment of new tehniques, in order to solve this roblem. The new tehnique that is in the very beginning but has brought good results in water resoures, is the Fuzzy Theory. In this new tehnique one does not need so muh data in order to establish some risk analysis. However, as it was said before, this methodology is in very beginning. Aliation to Pollution Problem In order to aly this theory in the Pollution Problems, it is imortant define some random variables that should be managed. To do so, let us suose a body of water with aaity or some limit level of onentration C. This body of water is reeiving some load of ollutant P. As it was ointed out before, C and P are random variables. 4 Coyright ASCE 24 World Water Congress 21 Bridging the Ga

5 Therefore, robabilisti method of analysis an be alied in order to give some measurement of the risk of ollution of that body of water. Suose yet that both C and P are ositive random variable with robabilisti density funtions f and f. In suh way, the risk R an be alulated through the equation; Downloaded from aselibrary.org by Uf - Universidade Federal Do Ceara on 1/29/14. Coyright ASCE. For ersonal use only; all rights reserved. ] R = P[ P > C] = f ( )[ f ( ) d d (1) where, C and P are indeendent random variables. Thus R is the risk that the onentration P omes to be bigger than the aaity C. The oosite of R will be defined as the robability that C be always bigger than P. This situation is alled reliability. The reliability is thus; r = 1 R (11) For examle, let suose that both C and P are normal distribution with robability density funtion defined by; f k = ke and Therefore the risk an be alulated by; f k = k e (12) k k R = P[ P > C] = k e [ k e d] d (13) Solving this equation, the risk will be; kk R = (14) k ( k + k ) It is imortant to observe that, in this simle ase, k and k deend on the exeted values E and E and the variane σ 2 and σ 2. However, if a most omliated distribution is found for C and P, the robability density funtion of the random variables must be alulated and, in suh ase, the mission ould not be so easy. A different way to ontours this diffiult is to aly the Fuzzy Set Theory. This theory ermits to analyze the unertainty inherent in data, mathematial model, arameters, and boundary onditions, with a few information. In this theory, one needs just a few data to reresent them as a fuzzy number. However, the Fuzzy Set Theory, that has been alied 5 Coyright ASCE 24 World Water Congress 21 Bridging the Ga

6 in all field of siene, has its aliation in the Environmental Engineering in the very beginning, needing more develoment in its riniles. Downloaded from aselibrary.org by Uf - Universidade Federal Do Ceara on 1/29/14. Coyright ASCE. For ersonal use only; all rights reserved. The robability theory, that is more available for this urose, has brought good results to the risk analysis. However, its aliation beomes very limited if the set of data is not suffiient to get a robability density funtion. This is a very ommon situation onerning with the water ollution roblem. In this ase, the robability theory annot be alied, in order to analyze unertainty inherent to the Environmental Engineering Problem. Conlusions After the analysis of the robability theory to evaluate and quantify risk onerting with ollution in a body of water, one an onlude that the theory is extremely useful to system with a onsistent set of data. On the other hand, for systems without a onsistent set of data, the hane to get good results, on this mission, will be very sare. The reason for that is that the robability theory needs, in its aliation, the robability density funtion for all set of random variables that should be analyze, in order to measure the risk of any environmental system. This annot be done with an inonsistent data set. In this ase the Fuzzy Set Theory ould be a better way to this kind of study. Referenes Ganoulis, J., 1994, Engineering Risk Analysis of Water Pollution: Probabilities and Fuzzy Sets, VCH. Ganoulis, J, 1991, Water Resoures Engineering Risk Assessment. NATO, ASI Series,Vol. 29, Heidelberg Sringer Verlag. Stakhiv, E, 1986, Risk Analysis Considerations of Dam Safety, In Engineering Reliability and Risk in Water Resoures. L. Dukstein and E. J. Plate, editors. Bagtzouglou, A. C., A. F. B Tomson and D. E. doudherty,1991, Probabilisti Simulation for Reliable Solute Soure Identifiation in Heterogeneous Porous Medium. In: Ganoulis (ed.), Water Resoures Engineering Risk Assessment, NATO, ASI Series, Vol. 29, Heidelberg, Sringer - Verlag. Bardossy, A., I. Bogardi and L. Dudkstein, 199, Fuzzy Regression in Hydrology. Water Resoure Res. 26(7). The author Dr. Souza would like to thank the Government of Brazil that, through the CAPES, has suorted His sabbatial liense at the Cornell University. 6 Coyright ASCE 24 World Water Congress 21 Bridging the Ga