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Figure 20.18 Simple event tree for structural/geotechnical strength instability of levee.
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Figure 20.19 Height and duration of overtopping interact to affect levee performance.
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Failure Stringers exist Duration >week 50% 100% of crest Duration <week Geotechnical Failure No Failure Stringers do not exist Piping occurs No Failure
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Figure 20.20 Hydrologic, piping, and strength failure parts of the levee failure event trees.
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mechanical or electrical dysfunction of an appurtenant structure, and possibly more exotic things. Note that this tree combines states of nature ( soft soil ll exists ) with system states of the levee ( high pore pressure in levee ). That is, it does not separate aleatory and epistemic uncertainties into a logic tree and event tree but combines them. The dashed vertical line indicates separate event trees leading to distinct failure modes but arising from the same initiating event. The triangle at a terminating end of a path through the
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EVENT TREE ANALYSIS
Table 20.3 A comparison of in uence diagrams and decision trees (Marshall and Oliver 1995) Model features In uence diagrams Modeling Timing Conditional independence Size Data Variable type Asymmetry Modeling usefulness Shows timing of all decisions and uncertain events. Shows dependence among uncertain events and decisions. Number of nodes grows linearly with the number of variables. Identi es dependencies of variables without need for data. Both continuous and discrete decisions and probabilities. Scenarios with different event sequences not distinguished. Most useful in initial stages of modeling; captures interaction between decision maker and analyst. Shows timing of all decisions and uncertain events. Dependence among uncertain events and decisions not shown. Number of terminal nodes and paths grows exponentially with the number of variables. Decision, probability and result data shown explicitly. Both continuous and discrete decisions and probabilities. Shows asymmetric structure of problem. Useful in depicting detailed uncertain event outcomes and decisions, and model solution; dif cult to display large problems. Event or decision trees
Solution process Bayes theorem Solution method Indicated by arc reversal, but calculation not shown. Reduction by a set of reduction operations possible using advanced methods. Indicated by node reversal (a separate event tree may be used as an aid for calculation). Uses simple rollback algorithm.
tree indicates a failure consequence. Despite the common logical content of in uence diagrams and event or decision trees, the practical uses of the two differ, as summarized in Table 20.3.
20.3.2 Consequences
The consequences that result from a dam incident or failure depend on a large number of factors beyond the behavior of the dam itself. For example, consequences depend on safety functions such as barriers, safety systems, operating procedures, operator actions, and so on, and on how they respond to an initiating event or an indication of dysfunction of the dam proper. Safety functions may include systems that respond automatically to an initiating event or to a failure, alarms that alert operators or other cognizant personnel that an event or failure has occurred, prede ned operating procedures that follow an alarm, or barriers or other containment facilities intended to limit the effects of an initiating event or failure. Other considerations that in uence the consequences resulting from initiating event or failure include time of day, meteorological conditions, downstream warning systems, and emergency response procedures to protect life and property.
BRANCH PROBABILITIES
Event Tree Consequence Tree Logic Tree
States of Nature
System Events Consequences
Figure 20.21 Consequence tree summarizing various events that may occur upon a particular outcome (leaf) of the event tree.
Typically, these many considerations are not of the dam system itself or are not part of the uncertainties related to limited knowledge about natural processes or the functioning of the dam. Thus they are separated into a consequence model or, more typically, into a separate consequence tree.
20.3.3 Consequence trees
As in probabilistic risk analyses (PRA) of nuclear power plants (McCormick 1981), a separate event tree is often used to model consequences of a failure; for example to provide an estimate of the consequences conditional on loss of pool or some other damage state of the dam (Figure 20.21). This sub-tree incorporates exposure cases and other downstream activities or events, such as the effectiveness of evacuations or other risk mitigation procedures. Usually, this downstream sub-tree can be treated independently of the event tree for the dam itself, but it is often no less complicated. Event trees are evaluated by generating a set of probability-consequence pairs for each mutually exclusive end node in the tree. In principle, consequences could be expressed in utilities (Keeney and Raiffa 1976), rather than direct physical units (e.g. dollars or lives lost), but usually they are not. Multiplying probability by consequence to obtain an expected consequence implicitly treats high-probability low-consequence outcomes as equivalent to low-probability high-consequence outcomes, as long as the product is the same. For government projects this is usually taken to be reasonable, given governments asset positions and the multiplicity of projects (Morgan and Henrion 1990). For a private owner, such risk neutrality may not be reasonable.