Branch probabilities and the bases for their calculation in .NET framework

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Table 20.5 Branch probabilities and the bases for their calculation
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Event node 0.001 0.1 0.7 0.2 0.25 0.75 0.15 0.5 0.2 0.1 0.5 0.2 Poisson process Seepage model Slope failure model Slope failure model Seepage model ditto Rating curve Rating curve Stochastic process
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Conditioning event Event branch Pr Method
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Natural variability
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Limited knowledge
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Extreme storm
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Water height
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Extreme storm
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Flood frequency curve Rating curve
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Extreme storm
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Water level above crest 50 99% of crest
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Extreme storm
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Duration
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less than 50% of crest More than a week
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Bottom conditions, debris/ice, etc. Bottom conditions, debris/ice, etc. Bottom conditions, debris/ice, etc. ditto ditto
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Distribution type, mean, variance, & skew Statistical regression parameters Statistical regression parameters Statistical regression parameters ditto
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Less than a week
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Stringers
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None
Flood frequency curve Flood frequency curve Poisson process
Sand boils
Stringers exist
Ave number per reach and size Spatial variability of permeability
Failure
Sand boils
Soft soil
None
High pore pressure
Duration
Spatial variability of soil parameters Ave number per reach and size Spatial variability of permeability Spatial variability of soil parameters
GT Failure
Soft soil, pore pressure
Chance exploration detected Soil parameters and spatial model parameters Statistics of soil properties, model uncertainty Chance exploration detected Soil parameters and spatial model parameters Statistics of soil properties, model uncertainty
EVENT TREE ANALYSIS
Conditional probability given earlier events in tree.
BRANCH PROBABILITIES
The reason for this result is that the existence of the soft lenses makes both events simultaneously more likely, while the non-existence makes both simultaneously less likely. Since the two events are no longer independent, their marginal probabilities can no longer be simply summed to give the probability that either one or the other occurs. The correlation due to a common dependence reduces the risk to the structure in this case, but the outcome could be the reverse in other circumstances. Branch probabilities associated with separate event nodes can be correlated through of any of the following: 1. Causal dependence, meaning that one event physically causes another. For example, liquefaction-induced settlement may directly lead to overtopping of an embankment, thus the liquefaction event and overtopping event would not be independent of one another. If the liquefaction settlement occurs, the probability of overtopping might be greatly enhanced. 2. Probabilistic correlation, meaning that two uncertainties may share a common dependence on a third uncertainty, as in the case of the low-density soil lenses in the example above. Whether the low-density soil lenses exist or not simultaneously changes the probability of liquefaction cracking and of overtopping.
Extreme storm = 0.001
Water level above crest p = 0.1 < 50 % of crest p = 0.99 Piping p = 0.5
Overtopping
1.0 E-4 -
Failure p = 0.2 No Failure
2.6 E-6 -
>week p = 0.25 50% 100% of crest p = 0.7
Stringers p = 0.15
No piping p = 0.5 None p = 0.85
<week p = 0.75 Geotechnical High pore Failure pressure No Failure Low pore pressure 1.0 E-7 -
>week p = 0.25 50% 100% of crest p = 0.2
Soft soil fill Good soil fill
<week p = 0.75
Figure 20.22 Event tree for levee failure during extreme storm, with estimated branch probabilities.
EVENT TREE ANALYSIS
3. Spatial or temporal autocorrelation, meaning that two uncertainties depend on the spatial or temporal realization of some third uncertainty which itself exhibits stochastic dependence in space or time. The performances of two sections of a long levee may depend on soil engineering properties in the naturally occurring valley bottom, which, when modeled as a stochastic (aleatory) process, exhibit a long wave length of correlation in space; thus adjacent sections will exhibit similar settlements or factors of safety against slope instability. 4. Statistical correlation, meaning that two uncertainties are simultaneously estimated from a xed set of data and therefore in uenced by a common sampling variability error. In soil mechanics, a common and almost always overlooked statistical correlation is that between soil cohesion, c, and soil friction angle, , which, being regression parameters of the Mohr-Coulomb model, are negatively correlated, given a nite number of test data.
20.5 Levee Example Revisited
Figure 20.22 shows the event tree for levee failure during an extreme ood that was developed earlier, but now with branch probabilities. Table 20.5 shows the justi cation for each probability estimate, the estimation approach, and the sources of uncertainty due to natural variation and limited knowledge.
Expert Opinion
Many important uncertainties in risk analysis are not amenable to quantitative estimation from data. In some cases there are no data at all, only the judgment of experts. These uncertainties have traditionally been treated using expert opinion. The tacit knowledge of experts is based on intuition, unenumerated past experience, subjective theory, and other important but qualitative beliefs. How should we interpret expert testimony and include it in quantitative risk analysis How do experts estimate probabilities associated with qualitative judgment Can expert opinion be quanti ed in a way that is repeatable, and if so, how Is there any way to test the external validity of expert opinion This chapter surveys the growing eld of expert elicitation of subjective probabilities, and summarizes the emerging understanding of the psychology of probability assignment.