provides a quantitative definition of risk (Kaplan and Garrick, 1981; ASME/ANS, 2009; IPET 2009) in probabilistic terms:

image

where v = frequency of occurrence or exceedance, and is a measure of the aleatory uncertainty (the randomness of events); ρ = probability as a measure of the confidence to which an estimate of v is the true value, or the epistemic uncertainty in the estimate of v. For purposes of a flood risk analysis, the consequences of flooding are measured in terms of the economic damages. Typically, results of risk analysis would be expressed in terms of a frequency distribution on economic damages (dollars). This is denoted

image

where n() = frequency of occurrence per year; and c = consequences measured in dollars.

Another risk metric is the expected annual losses, which can be estimated by

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The estimate of flood consequences and their frequency of occurrence are uncertain, that is v(c) cannot be determined with certainty; there is epistemic in estimating the frequency and magnitude of floods, uncertainty in the performance of flood protection systems, and the magnitude of consequences. There is therefore a distribution on the estimate of v(c). A formal analysis of the uncertainties in the components of a risk analysis (h, v, and C) produces a probability distribution on the estimate of v(c), the estimate of risk. This probability distribution that quantifies the uncertainty in the estimate of v(c) can be thought of as a distribution from which confidence intervals on the estimate of risk can be derived.3 Kaplan and Garrick (1981) provide a quantitative definition of risk that includes the quantification of aleatory and epistemic uncertainties and the estimate of consequences:

image

or, in terms of the risk example above, equation (I-5) can be written,

image

where, v = frequency of occurrence or exceedence, as is a measure of the aleatory uncertainty; C = a consequence metric (e.g., economic impact, public safety); ρ = probability as a measure of epistemic uncertainty in the estimate; and where,

image

The frequency of occurrence is a measure of the aleatory uncertainty of events or outcomes (consequences), their randomness or stochastic character. An example of aleatory uncertainty is the occurrence of flood events in a watershed, or the performance of a levee for a given flood level. Equation (I-6) defines a discrete probability mass function on the frequency distribution of consequences, where probability is a measure of epistemic uncertainty in the estimate of v.

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3 The notion of confidence intervals is a useful concept for purposes of illustrating the measure of uncertainty in the risk analysis results. In fact, however the assessment of uncertainties is not same as a statistical analysis of a dataset from which confidence intervals on parameter estimates are derived.



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