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Discharge Stage >Y >X <X <Y >Y <Y {<X,<Y } {>X,<Y } {<X,>Y } Outcome {>X,>Y }
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Simple event tree for discharge and stage of a river reach.
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probabilities at the second branching are the conditional probabilities of discharge for known stage. The joint probabilities of the right-hand side outcomes are the same in both cases. The event tree is simply a convenient way of consistently representing the relationships among events and conditional probabilities. Viewing event trees as statements of joint probability presumes no causality from parent node to child, and no temporal ordering. Events can logically appear in the tree in arbitrary chronology. Causality is irrelevant. If an event tree is viewed as a logical statement about information and beliefs, then again both events and condition variables co-exist in the tree. This is the common approach in most geotechnical studies. The uncertainties associated with events and condition variables can be convolved, or they may be separated into two distinct parts: an event tree to summarize occurrences in time or space, along with a logic tree to summarize the state of knowledge about the dam and its environment. An event tree is thus a representation of what we know, not a model of a particular dam (Figure 20.8). Evolving practice in seismic hazard, nuclear safety, and some other disciplines is to separate aleatory random variables from epistemic uncertainties. Two trees are then created: an event tree and a logic tree. The event tree contains only aleatory events; the logic tree contains only epistemic uncertainties. The logic tree is a representation of the possible realizations of the states of nature controlling probabilities within the event tree.
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System 2 Success State (S2) Failure State (F2) Success State (S2) Failure State (F2)
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Leaf a4 , b 1 , g 1 States of Nature g a b Initiating Event (I )
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Figure 20.8 Logic tree describing xed-but-unknown conditions (states) of nature, as conditioning point for event tree of system.
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EVENT TREE ANALYSIS
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The event tree structure appends to individual leaves of the logic tree, so that all calculations within the event tree are conditioned on the realized states of nature in the logic tree. This simultaneously accounts for dependencies anywhere in the event tree due to the common realization of states of nature in the logic tree. The nal combined tree need be no more complex than before, since the number of combinations of all uncertain events, both aleatory and epistemic, remains the same.
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20.2 In uence Diagrams
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An in uence diagram graphs relationships among initiating events, states of nature, conditions of the system, and consequences. Creating an in uence diagram is, in essence, creating the basic model of the risk analysis from which event trees, reliability calculations, and other models arise. There is not a rigid set of steps or a recipe for creating an in uence diagram, but only a structured procedure to help maintain logical consistency. As in all modeling, the in uence diagram should represent the logic of the system, in uences upon the system, and uncertainties affecting system performance. To the extent possible, it should do so parsimoniously; it should seek an ef cient representation. Obviously, this, too, is an art, requiring interpretation and judgment. The enterprise of creating an in uence diagram seeks a relatively simple representation with comparatively few parameters, which, despite its simplicity, closely represents the behavior of the dam. An in uence diagram provides a visual model of the structure of a risk analysis problem. This includes the timing of events, relationships among risky outcomes, and uncertain events, quantities, or variables. The intent of an in uence diagram is to help visualize how system components, uncertainties, and outcomes are interrelated; and especially to support the development of a systems risk model of the components, uncertainties, and outcomes. The in uence diagram involves no mathematical model; it deals only with relationships among entities. Two important advantages of in uence diagrams over event or decision trees make them useful in the early stages of risk analysis. First, even a complicated problem can sometimes be represented in a simple in uence diagram. An in uence diagram represents a single uncertain quantity by a single node, and does not lead to the combinatory explosion of branches associated with event trees and fault trees. Secondly, in uence diagrams explicitly show interrelationships among events. In an event tree there is usually no immediately obvious way to track the causal dependence of one uncertain quantity on another.