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Table 20.2 Division of important uncertainties for levee example onto those related to natural variability and those related to knowledge uncertainty Naturally varying (aleatory) conditions Extreme storm in upstream basin Peak ood discharge ( ood-frequency) Peak river stage for a given discharge Flood duration given peak ood Spatial variation of soil properties Knowledge (epistemic) uncertainties Average strength of the levee ll Presence of aw zones in the levee Existence of permeable lenses in foundation Critical duration of overtopping to cause failure
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Levee overtops No
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S > 48 Yes Q > 200 No S < 48 Yes S= No Q > 200 S = 48 Yes S=0
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Figure 20.17 Alternative simple event trees for levee overtopping: Top showing dichotomous representation of outcome: bottom showing continuous representation of outcome.
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An event tree is built from a set of events, each represented by a node. The branches emanating from an event node represent possible outcomes. Since these are mutually exclusive and collectively exhaustive, the outcome follows exactly one branch. Figure 20.17 shows the initiating event of a ood discharge greater than some value. The subsequent event, river stage, is summarized in the upper tree by a dichotomous outcome. In this case, the corresponding river stage is either greater than some value, or less than that value (by convention, the upper branch at each event usually corresponds to the associated event being true ). In the lower tree, the event is summarized by a continuous outcome between lower and upper bounds. A probability distribution is associated with each event node. When the outcomes are de ned as discrete, the probabilities are represented in a probability mass function (pmf). When the outcomes are continuous, the probabilities are represented in a probability density function (pdf). Calculations in the rst case are made by multiplying branch
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probabilities by their corresponding consequences, and in the latter case by integrating the consequences over the pdf. An event tree quickly becomes large. Expanding a simple chain of events leading from an initiating event of high rainfall to strength failure of a levee could pass through at least the following ve events, each of which might have many possible outcomes: (1) an extreme storm generates rainfall of a given amount; (2) discharge in the river exceeds some value; (3) stage rises above some critical elevation; (4) high water lasts for a critical length of time; and nally (5) the levee fails due to high loads and rising pore pressures. Simplifying to dichotomous events yields the tree of Figure 20.18, which is already large. A more realistic model might include consideration of whether soft zones ( aws) exist within the embankment, how high the pore pressures rise, whether a levee experiencing structural instability might yet retain the ood ow, and so on. One problem with any of the simple logic tools used to model risks associated with system performance, such as in uence diagrams or event trees, is that the tools are static analyses that decompose events into individual realizations. This simpli es the analysis, but may lose important features. Since the tree grows combinatorially with the number of events, care must be taken to keep event chains as short as can be reasonably accommodated. Consider an issue suggested by Zielinsky (2001) of the height and duration of overtopping. Each is important for failure, and they act jointly. Shallow overtopping, that is, not much water volume passing over the levee, may cause failure, but only if the condition lasts for a long time; if the overtopping is brief, the levee may withstand potential erosion. On the other hand, deep overtopping even if brief may cause failure. Figure 20.19 shows two cases of overtopping. On the left, the overtopping is shallow but lasts for 20 hours; on the right, the overtopping is deep but lasts a much shorter time, perhaps three or four hours. Both are cases of overtopping, but the levee may respond differently to each. A simple in uence diagram or event tree may not capture this combined dependence. Events may need to be de ned by depth-duration pairs in order to model the behavior of the levee. For convenience, event trees are usually separated by initiating event (Figure 20.20). Thus, in a dam safety study it is common to see three, four, or even more event trees, associated respectively with extreme storms, earthquakes, aws in design or construction,
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