Uncertainty and Reliability Analysis in Water Resources Engineering

Yeou-Koung Tung, Professor

Department of Civil & Structural Engineering, Hong Kong University of Science & Technology, Clear Water Bay, Kowloon, Hong Kong UNCERTAINTIES ENGINEERING IN WAT ER RESOURCES Natu ral uncertainty is associated with the inherent random ness of natural processes such as the occurrence of precipitation and floo d events. The occurrence of hydrological events often display va riations in tim e and in space. Their occurrences and intensities could not be predicted precisely in advance. Due to th e fact that a model is only an abstraction of the reality, which generally involves certain degrees of simplifications and idealizations. Model uncertainty reflects the inability of a model or design technique to represent precisely the system's true physical behavior. Parameter uncertainties resulting from the inability to quantify accurately the model inputs and param eters. Parameter u ncertainty could also be caused by change in operational conditions of hydraulic structures, inherent variability of inputs and parameters in time and in space, and lack of sufficient amou nts of data. Data uncertainties include (1) measurement errors, (2) inconsistency and non-homo geneity o f data, (3) d ata handling and transcription errors, and (4) inadequate represen tation of data sample due to time and space limitations. Operational uncertainties include those associated with construction, manufacture, deterioration, maintenance, and human. The magnitude of this type of uncertain ty is largely dependent on the workmanship and quality control during the construction and manufacturing. Progress ive deterioration due to lack of proper maintenance could result in changes in resistance coefficients and structural capacity reduction. The purpose of this article is to briefly summari ze the state-of-the-art of uncertainty and reliability analyses procedures in water resources engineering. For more detailed descriptions of the va rious techniques and applications can be found in the two references at the end of this article. IM P L I C A T I O N S O F U N C E R T A I N T Y A ND PURP OSES OF UN CER TAIN TY A NAL YSIS In water resou rces engin eering de sign and analysis, the decisions on the layout, capacity, and operation of the system largely depend on the system response under some anticipated design cond itions. W hen some of the 13

Water resources engin eering design an d analysis d eal with the occurrence of water in various parts of a hydrosystem and its effects on environmental, ecological, and socioeconomical settings. Du e to the extreme complex nature of the phy sical, chem ical, biological, and socioeconomical processes involved, tremendous efforts have been devoted by researchers attempting to have a better understanding of the pro cesses. One beneficial product of these research efforts is the development of a model which describes the interrelationships and interactions of the components involved in the processes. Herein, the term m odel is used in a very loo se man ner, referrin g to any structural or nonstructural ways of transforming inputs to produce some forms of outputs. In water resources enginee ring, mo st mode ls are structural which take the forms of mathematical equations, tables, graphs, or c o m puter programs. The model is a useful tool for engineers to assess the system performance under various scenarios based on which efficient designs or effective management schemes can be formulated. Desp ite numerous research efforts made to further our understanding of various processes in hydrosystems, there is still much more that are beyond our firm grasp. Therefo re, uncerta inties exist du e to our lac k of perfe ct knowledge concerning the phenomena and processes involve d in prob lem def inition and resolution . In general, uncertainty due to inherent random ness of physical processes cannot be eliminated. On the other hand, uncertainties such as those associated with lack of complete knowled ge about the p rocess, mode ls, parameters, data, and etc. could be reduced through research, data collection, and careful manufacturing. In wate r resources engineering, uncertainties involved can be divided into four basic categories: hydrologic, hydraulic, structural, and eco nomic . More sp ecifically, in wate r resources engineering analyses and designs uncertainties could arise from the various sources including natural u n c e r t a i n t i e s , m o d e l u n c e rt a in t ie s , p a ra m ete r uncertainties, data uncertainties, an d opera tional uncertainties.

components in a hydrosystem are subject to uncertainty, the system responses under the design conditions cannot be assessed w ith certainty. Therefore, the conventional determ inistic design practice is inappro priate because it is unable to account for possible variation of system responses. In fact, the issues involved in the design and analysis of hydrosystems un der unc ertainty are multidimen sional. An engineer has to conside r various c riteria including, but not lim ited to, cost of the system, probab ility of failure, and consequence of failure so that a prope r design c an be m ade for th e system . In water resources engineering design and modeling, the design quantity and system output are functions of several system parameters not all of them can be quantified with absolute accuracy . The task o f uncertain ty analysis is to determine the uncertainty featu res of the sy stem ou tputs as a function of un certainties in th e system mode l itself and the stochastic variables involved. It provides a formal and systema tic framew ork to qu antify the u ncertainty associated with the system output. Furthermore, it offers the designer useful insights regarding the contribution of each stochastic variable to the overall uncertainty of the system outputs. Such knowledge is essential to identify the 'important' parameters to which more attention sh ould be given to have a better assessment of their values and, accordingly, to reduce the overall uncertainty of the system outpu ts. MEASURES OF UNCERTAINTY Several expressions have been used to describe the degree of uncertainty of a parameter, a function, a model, or a system. In gener al, the unce rtainty associated with the latter three is a result of combined effect of the uncertainties of the con tributing param eters. The most complete and ideal description of unce rtainty is the probability density function (PDF) of the quantity subject to uncertainty. However, in most practical problems such a probability function cannot be derived or found precisely. Another measure of the uncertain ty of a qu antity is to express it in terms of a reliability domain such as the confidence interval. A confidence interval is a numerical interval that wou ld capture the quan tity subject to uncertain ty with a specified probabilistic confidence. Nevertheless, the use of confidence intervals has a few drawback s: (1) the parameter population may not be norm ally distributed as assumed in the conventional procedures and this problem is particularly important 14

when the sample s ize is small; (2) n o mean s is available to directly comb ine the co nfidence intervals of individual contributing random compo nents to give the o verall confidence interval of the system. A useful alternative to quan tify the level o f uncertain ty is to use the statistical m omen ts associated with a qu antity subject to uncertainty. In particular, the variance and standard deviation which measure the dispersion of a stochastic variable are commonly used. AN OVERVIEW O F UNC ERT AINT Y ANA LYSIS TECHNIQUES Several techniques can be applied to conduct uncertain ty analysis of water re sources e ngineer ing prob lems. E ach technique has different levels of mathematical comp lexity and data requirem ents. Broadly sp eaking, those techniques can be classified into two categories: analytical approaches and approximated approa ches. The selection of an appropriate technique to be used depends on the nature of the problem at hand including availability of inform ation, resources constraints, model complexity, and type and accuracy of results de sired. Analytical Techniques This section briefly describes several analytical methods that allow an analytical derivation of the exact PDF and/or statistical moments of a model as a function of several stochastic variables. Several useful analytical techniques for uncertainty analysis including derived distribution technique and various integ ral transform techn iques. Although these analytical techniques are rather restrictive in practical applications d ue to the comp lexity of most models, they are, neverthe less, powerful tools for deriving comp lete inform ation about a stochastic pro cess, including its distribution, in some situations. The analytical techniques described herein are straightforward. However, the success of implementing these procedures largely depends on the functional relation, forms of the PDFs in volved , and ana lyst's mathe matical sk ill. Deriv ed Distribution Technique- This derived distributio n method is also known as the transformation of variables technique. Example applications of this technique can be found in modeling the distribution of pollu tant decay process and rainfall-runoff modeling. Fourier Transform Technique - The Fourier transform of the PDF of a stochastic v ariable X results in the so-called the character istic function . The characteristic function of

a stochastic variable always exists and two distribution functions are identical if and only if the corresponding characteristic functions are identical. Therefore, given a character istic function of a stochastic variable, its PDF can be uniquely determined through the inverse Fourier transform. Also, the sta tistical mom ent of the sto chastic variable X can be obtained by using the character istic function. Fourie r transform is particularly useful when stochastic variables are independent and linearly related. In such case s, the convolution property of the Fourier transform can be applied to derive the characteristic function of the resu lting stocha stic variable. Laplace and Ex ponen tial Transform Techniques - The Laplace and exponential transforms of the PDF of a stochastic variable lead to the moment generating function. Similar to the characteristic function, statistical moments of a stochastic variable X can be derived from its moment generating function. There are two deficiencies associate d with the moment generating functions: (1) the moment gene rating function of a stochastic variable may not always exist, and (2) the correspondence between a PDF and moment generating function may not necessarily be unique. However, the existence and unique conditions are generally satisfied in most situations. Fourier and exponential transforms are frequen tly used in uncertainty analysis of a model that involves exponentiation of stochastic variables. Examples of their applications can be found in probabilistic cash flow analysis an d proba bilistic mod eling of pollutant decay. Mellin Transform Technique - When the functional relation of a model satisfies the product form and the stochastic variables are independent and non-negative, the exact moments for model output of any order can be derived analytically by the M ellin transfor m. The Mellin transform is particularly attractive in uncertain ty analysis of hydrologic and hydraulic problems because many mode ls and the involved parameters satisfy the above two conditions. Similar to the convolution property of the Laplace and Fourier transforms, the Mellin transform of the convolution of the PDFs associated with independent stochastic variables in a produ ct form is simply equal to the product of the Mellin transfo rms of individu al PDFs. Applications of the M ellin transfor m can b e found in econo mic benefit-cost analysis, and hydrology and hydraulics. One cau tion abou t the use of th e Mellin transform is that under some combinations of distribution and functional form, the resulting transform may not be defined. This could occur especially when quotients or variables with negative exponents are involved. 15

Estimations of Probabilities and Quantiles Using Mom ents - Althou gh it is gene rally difficult to analytically derive the PDF from the results of the integral transform techniques described above and the approximation techniques in the next section, it is, however, rather straightforward to obtain or estimate the statistical mom ents of the sto chastic variable one is interested in. Based on the computed statistical mom ents, one is able to estimate the distribution and quantile of the stochastic variable . One po ssibility is to base on the asy mptotic expansion about the norm al distribution for calculating the values of CDF and qu antile, and th e other is to base on the maxim um en tropy co ncept. Approximation Techniques Most of the models or design procedures used in water resources engineering are nonlinear and highly complex. This basically prohibits any attempt to derive the probability distribution or the statistical moments of model output analytically. As a practical alternative, engineers frequently resort to methods that yield approximations to the statistical properties of uncertain model output. In this section, several methods that are useful for uncertainty analysis are briefly described. First-order variance estimation (FOVE) method - The method, also called the variance propagation method, estimates uncertainty features associated with a model output based on the statistical properties of model's stochastic variables. The basic idea of the method is to approximate a model by the first-order Taylor series expansion. Commonly, the FOVE method takes the expansion point at the means of the stochastic variables. Consider a hydraulic or hydrologic design quantity W whic h is related to N stochastic va riables X=(X 1, X 2, ..., X N) as W = g(X 1, X 2, ..., X N) The mean of W, by the FOVE m ethod, can be estimated as E[W] . g(: 1, : 2, ..., : N) in which : i is the mean of the i-th stochastic variable. When all stochastic variables are independent, the variance of the desig n quan tity W can be approximated as Var[W] .s 12 F 12 + s22 F 22 + ... +s N2 F N2 in which si is the first-order sensitivity coefficient of the i-th stochastic variable and F i represents the corresponding standard devia tion. From the above equation, the ratio,

s i2F i2/Var[W], indicates the propor tion of ov erall uncertain ty in the design quantity contributed by the uncertainty associated with the stochastic variable X i. In general, E[g(X)] g(:) unless g(X) is a linear function of X. Improvemen t of the accuracy can be made by incorporating higher-order terms in the Taylor expansion. However, the inclusion of the higher-order terms rap idly increases not only the mathematical complication b ut also the required information. The method can be ex pande d to include the second-order term to improve estimation of the mean to account fo r the presence of model nonlinearity and correlation between stochastic variables. The method does not re quire knowledge of the PDF of stochastic variables which simplifies the analysis. However, this advantage is also the disadvantage of the method because it is insensitive to the distributions of stochastic variables on th e uncertainty ana lysis. The FOVE method is simple and straightforward. The computational effort associated with th e metho d largely depends on the ways how the sensitivity coefficients are calculated. For simple analytical functions the computation of derivatives are trivial task s. However, for functions that are complex and/or implicit in the form of computer programs, or charts/ figures, the task of computing the derivatives could become cumbersome or difficult. In such cases probabilistic point estimation techniques can be viable alternatives. There are many applications of the FOVE method in the literature. Example applications of the method can be found in open channel flow, groundwater flow, water quality modeling, benefit-cost analysis, grave l pit migratio n analysis, storm sewer design, culverts, and bridges. Probab ilistic Point Estimation (PE) Methods - Unlike the FOVE metho ds, prob abilistic PE methods quantify the model uncertainty by performing model evaluations without computing the model sensitivity. The methods generally is simpler and more flexible especially when a model is either complex or non-analytical in the forms of tables, figure, or computer programs. Several types of PE methods have been developed and app lied to unc ertainty analysis and each has its advantages and disadvantages. It has been shown that the FOVE method is a special case of the probabilistic PE methods when the uncertainty of stochastic v ariables are small. Rosenblueth in 1975 developed a method for handling stochastic variables that are symmetric and the method is later extende d to treat no n-sym metric stochastic variables 16

in 1981. T he basic idea of Rosenblueth's PE method is to approx imate the original PDF or PMF of the stochastic variable by assumin g that the en tire proba bility mass is concentrated at two points. The four unknow ns, namely, the locations of the two poin ts and the corresponding probab ility masses, are determined in such a manner that the first three m omen ts of the orig inal stocha stic variable are preserved. Fo r problem s involvin g N stoch astic variables, the two points for each variable are com puted and permu tated to produce a total of 2N possible points of evaluation in the parameter space based on which the statistical moments of the model outputs are computed. The potential drawback of Rosen blueth's PE metho d is its practical application due to explosive nature of the computation requirement. For moderate or large N, the number of required model evaluations could be too numerous to be imp lemente d practica lly, even on the computer. Example applications of Rose nblueths PE method for unce rtainty ana lysis can be found in groundwater flow model, dissolved oxygen deficit mo del, and brid ge pier sco uring m odel. To circum vent the sh ortcom ing in computation, Harr developed an alternative PE method that reduces the 2N model evaluations required by Rosenblueth's method down to 2N. Harrs method utilizes the first two m omen ts (that is, the mean and covariance) of the involved stochastic variables. The method is appropriate for treating stochastic variables that are normal. The theoretical basis of Harr's PE method is built on the orthogonal transformation using eigenvalue-eigenvector decomposition which maps correlated stochastic variables from their original space to a new domain in which they become uncorre lated. Hen ce, the analysis is greatly simplified. Harrs PE method h as been a pplied to uncertain ty analysis of a gravel p it migration mode l, regional equations for unit hydrograph param eters, groundwater flow models, and parameter estimation of a distributed hydrod ynam ic mod el. Recently, Li proposed a computationally practical PE method that allows incorporation of the first four mom ents of correlated stoch astic variab les. In fact, Ros enbl ueth 's solutions are a special case of Lis solution. Lis method requires (N 2+3N+2)/2 evaluations of the mode l. W hen the polynomial order of the model is four or less, Lis method yields the exact expected value of the mode l. Among the three probabilistic PE algorithms described above, Harrs method is the most attractive from the comp utational viewpoint. However, the method cannot

incorpo rate additiona l distribution al information of the stochastic variables other than the first two m oments. Such distributio nal information could have important effects on the results of uncertainty analysis. To incorpo rate the i n fo r m ation about the marginal distributions of involv ed stocha stic variables, a transformation between non-normal parameter space and a multivariate standard normal space has been incorporated into Harrs method. The resulting method preserves the computational efficiency of Harrs PE method while extends its c apability to hand le multiva riate non-n ormal sto chastic var iables.

this, various variance reduction techniques have been developed. Applica tions of Monte Carlo simulation in water resources engineering are abundant. Examples can be found in groundwater, benefit-co st analysis, w ater quality mode l, pier-scou ring pred iction, and open ch annel. Resampling Techniques - Note that M onte Ca rlo simulations are conducted under the condition that the probab ility distribution and the associated population parameters are known for the stochastic variables involved in the system. The observed data are not directly utilized in the simulation. Unlike the Monte Carlo simulation approach, resampling techniques reproduce random data exclusively on the basis of observed ones. The two resampling techniques that are frequently used are jackknife method and bootstrap method. REL IABIL ITY AN ALY SIS In many water resource engineering problems, uncertainties in data and in the ory, inclu ding de sign and analysis procedures, warrant a probabilistic treatment of the problems. The failur e associate d with a hydr aulic structure is the result of the combined effect from inherent randomness of extern al load an d variou s uncertain ties involved in the analysis, design, construction, and operation al proced ures desc ribed pre viously. Failure of an engineering system occurs when the load (external forces or demands) on the system exceeds the resistance (strength, capacity, or supply) of the system. In hydrau lic and hydrologic analyses, the resistance and load are frequently functions of a num ber of stoc hastic variables. Without considering the time-dependence of the load and resistance, static reliability model is generally applied to evaluate the system performance subject to a single wo rst load eve nt. However, a hydraulic structure is expected to serve its designed function over an expected period of t ime. In such circumstances, time-dependent reliability models are used to incorporate the effects of service duration, random ness of occurrence of loads, and possible change of resistance characteristics over time. In reliability analysis, the load and resistance functions are often comb ined to esta blish a performance function, W (X), which divides the system state into a safe (satisfactory) set defined by W(X)$0 and a failure (unsatisfactory) set defined by W(X)<0. The boundary separating the safe set and failure set is a surface defined 17

Mon te-Carlo Simulati on - Simulation is a process of replicating the real world based on a set of assumptions and conceived mode ls of reality. Because the purpose of a simulation model is to duplica te reality, it is a useful tool for evaluating the effect o f different designs o n system performance. The M onte Ca rlo procedure is a numerical simulation to reprod uce sto chastic variables preserving the specified distributional pro perties. Several books have been written for gene rating un ivariate random numbers. A number of computer program s are available in the pub lic dom ain. The challeng e of Mo nte Carlo simulation lies in generating multivariate random variates. Compared with univariate random variate generators, algorithms for multivariate random variates are much more restricted to a few joint distributions such as multivariate normal, multivariate lognormal, mult ivariate gamm a, and few others. If the multivar iate stochastic variables involved are correlated with a mixture of marginal distributions, the joint PDF is d ifficult to formulate. Rather th an preser ving the f ull multiva riate features, practical multivariate Monte Carlo simulation procedures for problems involving mixtures of nonnormal stochastic variables have be en deve loped to preserve the marg inal distributio ns and co rrelation of involved stoch astic variables. In uncertainty analysis, the implemen tation of bruta lforce type of sim ulation is straightforward but can be very comp utationally intensive. Furthermore, because the Mon te Carlo s im ulation is a sampling procedure, the results obtained inevitably involve sampling errors which decrease as the sam ple size increase s. Increasing sample size for achiev ing high er precis ion generally means an increase in computer time for generating random variates and data processing. Therefore, the issue lies on using the minimum possible computation to gain the maximum possible ac curacy f or the qu antity under e stimation. For

by W (X)=0 which is called th e failure surf ace or lim itstate surface. T he com monly used safety factor and safety margin are the special cases of the performance function. Alternatively, the reliability index, defined as the ratio of the mean to the standard deviation of the performance function, is another frequently used reliability indicator. Computa tion of Reliability The computation of reliability requires knowledge of probab ility distributions of load an d resistance, or the performance function , W. This comp utation of reliability is called load -resistance in terference . Direct Integration Method - The method of direct integration requires the PDFs of the load and resistance or the performance fu nction be k nown or derive d. This information is seldom available in practice, especially for the joint PDF , because of the com plexity of hydrolo gic and hydraulic models used in design. Explicit solution of direct integration can be obtained for only a few PDF s. For most PDFs numerical integration may be necessary. When using numerical integration, difficulty may be encountered when o ne deals w ith a multiv ariate problem. Mean-Value First-Order S eco n d-M o men t (M FOSM ) Method - The M FOSM metho d for reliab ility analysis employs the FOV E meth od to estim ate the mean and standard deviation of the per forma nce fun ction W (X) from which the reliability index is computed. Several studies have shown that reliability is not g reatly influenced by the choice of distribution for the performance function and the a ssump tion of a normal distribution is quite satisfacto ry, excep t in the tail portion of a distribution. The M FOSM method has been used widely in various h ydraulic structural and facility designs such as storm se wers, culv erts, levees, f lood plains, and open chan nel hydraulics. The application s of the M FOSM method is simple and straightforward. How ever, it posse sses certain weakness es in addition to the difficulties with accu rate estimation of extrem e failure pro babilities as m entioned above. These w eaknesse s include: (1 ) Inappr opriate choice of the expa nsion po int; (2) Inab ility to hand le distributions with large skew co efficient; (3) G enerally poor estimation of the mean and v ariance o f highly nonlinear functions; (4) Sensitivity of the computed failure pro bability to the formulation of the performance function W; (5) In ability to incorp orate ava ilable information on probab ility distributions. The general ru le of thumb is not to rely on the result of the MFOSM method if any of the following conditions exist: (a) high 18

accuracy requirem ents for the estimated reliability or risk; (b) high nonlinea rity of the p erform ance fun ction; many skewed random variables are involved in the performance function. Advanced First-Order Second-Moment (AFOSM) Method - The m ain thrust o f the AFO SM m ethod is to reduce the error of the MFOSM m ethod associated with the nonlinearity and non-invariability of the performance function, while keeping the advantages and simplicity of the first-order approximation. The expansion point in the AFOSM method is located on the failure surface defined by the limit-state equation. Among all the possible values of x that fall on the limitstate surface one is more concerned with the combination of stochastic variables that wo uld yield the lowe st reliability or highest risk. The point on the failure surface with the lowest reliability is the one having the shortest distance to the point wher e the me ans of the stochastic variables are located. This point is called the design point or the mo st probab le failure po int. With the mean and standard deviation of the performance function computed at the design point, the AFOSM reliability index can be determined. At the design point, the sensitivities of the failure probability with respect to each of the stocha stic variable can be computed easily. Methods for treating non-normal and cor related stoc hastic variables have been develop ed for the AFOS M m ethod. Due to the nature of nonlinear optimization, the algorithm AFOSM does not necessarily converge to the true design point associated with the minimum reliability index. Therefore, different initial trial points be used and the smallest reliability index be selected to compute the reliability. Time -to-Failu re Ana lysis Any system w ill fail eventually; it is just a matter of time. Due to the presence of many uncertainties that affect the operation of a physical system, the time that the system fails to satisfactorily perform its intend ed function is a random variable. Instead of co nsidering detailed interactions of resistance and load ing ove r time, a syste m or its componen ts can be treated as a black box or a lumpedparameter system and their performances are observed over time. This reduces the reliability analysis to a onedimensional problem involving time as the only random variable. The term 'time' could be used in a more general sense. In some situations other physical scale me asures, such as distance or length, may be appropriate for system perform ance ev aluation.

Failure and Repair Characteristics - The time-to-failure analysis is particularly suitable for a ssessing the reliability of systems and/or components which are repairable. For a system that is repairable after its failure, the time period it would take to have it repaired back to the operational state is uncertain. Therefore, the time-to-repair (TTR) is also a rand om va riable. For a repairab le system o r comp onent, its service life can be extende d indefin itely if repair work can restore the system as if it was new . Intuitively, the probability of a repairable system available for service is greater than that of a non -repairab le system. The failure den sity functio n is the probability distribution that governs the time occurrence of failure and it serves as the comm on thread in the reliability assessme nts in timeto-failure analysis. Among them, the exponential distribution perhaps is the most widely u sed. Besid es its mathematical simplicity, the exponential distribution has been found, b oth phe nome nologic ally and e mpirica lly, to adequ ately describe the time-to-failure distribution for c o m ponents, equipment, and systems involvin g comp onents w ith mixtu res of life distrib utions. In general, th e failure rate for many systems or compo nents has a bathtub shape in that three distinct life periods, namely , early life (or in fant mortality) period, useful life period, and wear-out life period are identified. A commonly used reliability measure of system performance is the mea n-time-to -failure (M TTF) w hich is the expected time-to-failure. For repairable water reso urces systems, such as pipe networks, pump stations, storm runoff drainage structures, failed components within the system can be repaired or replaced so that the system can be put back into service. The time required to have th e failed syste m repa ired is uncertain and, conseq uently, the total time req uired to restore the system from its failu re to operational state is a random variable. Like the time-to -failure, the ra ndom time-to-re pair (TTR) has the repair density function describing the random characteristics of the time required to repair a failed system when f ailure occ urs at time z ero. The repair probab ility is the probability that the failed system can be restored within a given time period and it is sometimes used for measuring the maintainability. The mean-timeto-repair (MTTR) is the expected value of time-to -repair of a failed system which measures the elapsed time required to perform the maintenance operation. 19

The MTTF is a proper measure of the mean life span of a non-rep airable system. For a repairable system, a more representative indicator for the fail-repair cycle is the mean-time-between-failure (MTBF) which is the sum of MTTF and MTTR. Availab ility and Un availability - A repairable system experiences a repetition of repair-to-failure and failure-torepair processes during its service life. Hence, the probab ility that a system is in operating condition at any given time t for a repairable syst em is different than that of a non-repairable system. The term availability is generally used for repairable systems to indicate the probab ility that the system is in operating condition at any given time t. On the o ther hand , reliability is app ropriate for non-rep airable systems indicating the probability that the system has been continu ously in its operating state starting from time zero up to tim e t. Availab ility can also be interpreted as the percentage of time that the system is in operating condition within a specified time period. On the other hand, unavaila bility is the percentage o f time that the system is n ot available for the intended service in a specified time period, given it is operational at time zero.

SYSTEM RELIABILITY

Most systems involve many sub-systems and comp onents whose performances affect the performance of the system as a whole. The reliability of the entire system is affected not only the reliability of individual sub-systems and compo nents, but also the interaction and configuration of the subsystem s and com ponen ts. Furth ermore, water resources systems involve multiple failure modes, that is, there are several potential modes of failure in which the occurrence of any or a combination of such failure modes constitute the system failure. Due to the fact that different failure modes might be defined over the same stochastic variables space, the failure modes are generally correlated. For a complex system involv ing many sub-systems, comp onents and con tributing sto chastic var iables, it is generally difficult, if not imp ossible, to directly assess the reliability of the system. In dealing with a complex system, the general approach is to reduce the system configuration, based on its component arrangement or modes of operation, to a simpler system for which the analysis can be performed easily. However, this goal may not be achieved for all cases necessitating the development of a special procedure. Some of the

potentially useful techniques for water resources system reliability evaluation are briefly described below. State Enumeration Method - The metho d lists all possible mutua lly exclusive states of the system components that define the state of the entire system. In general, for a system containing M components in which each can be classified into N op erating states, th ere will b e N M possible states for the entire system. Once all th e possible system states are enu merate d, the states that resu lt in successful system operation are identified and the probab ility of the occurrence of each successfu l state is computed. The last step is to sum all of the successful state proba bilities which yield the sy stem reliab ility. Path Enumeration Method - A path is defined as a set of comp onents or modes of operation which lead to a cer tain state of the system. In system reliability analysis, the system states of interest are those of failed state and operational state. The tie-set analysis an d cut-set an alysis are the two w ell-known tech niques. The cut-set is defined as a set of system components or modes of operation which, when failed, causes failure of the system. Cut-set analysis is powerful for evaluating system reliability for two reasons: (1 ) it can be ea sily programmed on digital computers for fast and efficient solutions of any general system configu ration, espe cially in the form of a network, and (2) the cut-sets ar e directly related to the modes of system failure. The cut-set method utilizes the m inimum cut-set for calculating the system failure pro bability. A minim um cu t-set impli es that all components of the cut-set must be in the failure state to cause system failure. Therefo re, the com ponen ts or modes of operation involved in the minimum cut-set are effectively connected in parallel and each minimum cut-set is connected in series. As the complem ent of a cut-set, a tie-set is a minim al path of the system in which system components or modes of operation are arranged in series. Consequently, a tie-set fails if any of its components or modes of operation fail. The main disadvantage of the tie-set method is that failure modes are not directly identified. Direct identification of failure mode s is sometimes essential if a limited amount of a resource is available to place emphasis on a few dominan t failure modes.

decompo se the entire system into simple series and/or parallel subsystems for which the reliability or failure probab ility of subsystems can be easily evaluated. Then, the reliability of the entire system is obtained by combining those of the sub-systems using conditional probability rule. Fault Tree A nalysis - Conceptua lly, fault-tree an alysis traces from a sy stem failu re backward, searching for possib le causes of the failure. A fault tree is a logical diagram represen ting the consequence of component failures (basic or primary failures) on system failure (top failure or top ev ent). The f ault tree consists of event sequen ces that lead to system failure. RISK-BASED DESIGN OF WATER RESO URCES S Y ST E M S Reliability analysis can be applied to design of various hydrau lic structures w ith or without consider ing risk co sts which are the cos ts associated with the failure of hydrau lic structures or systems. The risk-based lea st cost design of hydraulic structures promises to be, potentially, the most significant a pplication of reliability an alysis. The risk-based design of water resources engineering structures integrates the procedures of economic, uncertainty, and reliab ility analyses in the design practice. Engineers using a risk -based d esign procedure consider trade-offs among various factors such as risk, econ omics, and other performance measures in hydraulic structure design. When risk-based design is embedded into an optimization framework, the combined procedure is called optima l risk-based design. Because the cost associated with the failure of a hyd raulic structure cannot b e predicte d from year to year, a practical way to quan tify it is to use an e xpecte d value on the annual basis. The total annual expected cost is the sum of the annual installation cost and annual expected damage cost. In general, as the structural size increases, the annual installation cost increases while the annual expected damage cost associa ted with fa ilure decreases. The optimal risk-based design determines the optimal structural size, configuration, and operation such that the annual total expected cost is minimized. In the optim al risk-base d design s of hyd raulic structures, the thrust of the exercise is to evaluate annual expected damage cost as the function of the PDFs of loading and 20

Conditional Probability Approach - The ap proach starts with a selection of key components and modes of operation whose states (opera tional or faile d) wou ld

resistance, damage function, and the types of unce rtainty considered. The conventional risk-based hydraulic design considers only the inherent hydrolo gic unce rtainty du e to the random occurrence of loads. It does n ot consider hydrau lic and eco nomic uncertain ties. Also, the probab ility distribution of the load to the water resources system is assumed known which is generally not the case in reality. However, the evaluation of annual expected cost can be made by further incorporating the uncertainties in hydra ulics, hydrological model and parameters. To obtain an accurate estimation of annual expected damage associated with structural failure would require the consider ation of all u ncertainties, if such can be practically done. Otherwise, the annual expected damage would, in most c ases, be und erestimate d, leading to inaccura te optima l design. REFERENCES Yen, B.C. an d Tun g, Y.K., Reliability a nd Un certainty Analyses in Hydraulic Designs, the American Society of Civil Engineers, New York, NY, 1993. Tung, Y.K., "U ncertainty and reliab ility analysis ," Chapter 7, In: Water Resources Handbook, edited by L.W. Mays. McGraw-Hill Book Company. 1996.

Yeou-K oung Tung is presently a professor at the Hong Kong University of Science and Technology. He is also a Professor of Statistics and a statistical h ydrolog ist with the Wyoming Water Research Center at the University of Wyoming where he has been since 1985 . Prior to that time he was an Assistan t Professor of Civil Engineering at the University of Nevada at Reno. He obtained the B.S. (1976) in civil engineering from Tamkang College in the Repu blic of China, and the M.S. (1978) and Ph.D. (1980) from the Univ ersity of Texas at Austin. His teaching has been in a wide range of topics including hydrology, water resources, operation s research, a nd prob ability and statistics. He has published extensively in the areas of application of operations research and probability and statistics in solving hydrologic and water resources problems. In 1987 he received the American Society of Civil Engineers Collingwood Prize. He has been very active with the committees in the American Society of Civil Engineers

21