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Event Tree AnalysisBest Practices in Dam and Levee Safety Risk AnalysisPart A – Risk Analysis BasicsChapter A-5July 2019
Objectives
•Define event tree terminology and rules•Demonstrate common applications
2
Outline of Topics
•Structure•Terminology•Calculations•Construction
3
Key Concepts• Event Tree Analysis is an inductive modeling technique that uses
Boolean logic to evaluate a sequence of events• Frequently used concepts and techniques include
• Conditional – Probability depends on an event that has occurred• Intersection – Used to multiply probabilities• Mutually Exclusive – Used to sum probabilities• Partitioning – Used to discretize continuous functions• Consistent Percentile – Used to combine uncertainties
4
Event Tree Analysis• A model for estimating risk• Depicted by an event tree• Used to decompose and
discretize a complex sequence of events
• Improves understanding of potential failure modes
• Alternative models• Fault tree analysis• Stochastic simulation
5
Initiating EventForward Looking (Inductive) Logic
• Chronological• Causal Chain
Consequences
Example• Verbal PFM description
• In a given year, an earthquake occurs with a peak horizontal acceleration between 0.6g and 0.8g. The ground motion triggers foundation liquefaction which causes instability of the upstream embankment slope. The resulting slope failure lowers the crest of the dam to a level below the reservoir pool. Overtopping of the lowered crest ensues causing erosion and breach of the dam.
• Key events• Earthquake occurs with PHA between 0.6g and 0.8g• Foundation liquefaction is triggered• Upstream slope instability lowers the crest• Overtopping erodes the lowered crest• Breach occurs
6
Possible Event Tree
7
Earthquake PHA0.6g – 0.8g
Liquefaction
No Liquefaction
Slope Instability
No Slope Instability
Overtopping
No Overtopping
Breach
No Breach
Terminology
Flood
Stage< 1520
Stage1520-1550
Stage> 1550
Non Breach
Non Breach
Non Breach
Monolith Slides
Monolith Slides
Spillway Erodes
Spillway Erodes
Life Loss
Life Loss
Life Loss
Life Loss
Life Loss
Life Loss
Life Loss
0.99
0.009
0.001
1.00
0.02
0.05
0.93
0.04
0.1
0.86
100
40
5
170
50
15
Initiating Event
Node
Probability
Pathway
Consequences
Rules and Math• Branches must be mutually exclusive
• Only one outcome can occur• Probabilities across branches can be summed
• Probabilities must be conditional• Probability of an event depends on all events
along pathways to the left• Probabilities along pathways can be multiplied
• Branches must be collectively exhaustive• The sum of probabilities across all branches
must equal one
9
Stage1520-1550
Non Breach
Monolith Slides
Spillway Erodes0.009
0.02
0.05
0.93
Monolith Slides
Spillway Erodes
Non Breach
Single Tree Format
Flood
Stage< 1520
Stage1520-1550
Stage> 1550
Non Breach
Non Breach
Non Breach
Monolith Slides
Monolith Slides
Spillway Erodes
Spillway Erodes
Life Loss
Life Loss
Life Loss
Life Loss
Life Loss
Life Loss
Life Loss
0.99
0.009
0.001
1.00
0.02
0.05
0.93
0.04
0.1
0.86
100
40
5
170
50
15
0.009 * 0.02 = 0.00018
0.009 * 0.05 = 0.00045
0.009 * 0.93 = 0.00837
0.001 * 0.04 = 0.00004
0.001 * 0.1 = 0.0001
0.001 * 0.86 = 0.00086
0.99 * 1.0 = 0.99
Separate Potential Failure Mode Trees
Flood
Stage< 1520
Stage1520-1550
Stage> 1550
Monolith Slides Life Loss
Monolith Slides Life Loss
Monolith Slides Life Loss
Flood
Stage< 1520
Stage1520-1550
Stage> 1550
Spillway Erodes Life Loss
Spillway Erodes Life Loss
Spillway Erodes Life Loss
Non breach event tree not shown
Calculating APF
Flood Stage1520-1550
Stage> 1550
Non Breach
Non Breach
Monolith Slides
Monolith Slides
Spillway Erodes
Spillway Erodes
Life Loss
Life Loss
Life Loss
Life Loss
Life Loss
Life Loss
0.009
0.001
0.02
0.05
0.93
0.04
0.1
0.86
100
40
5
170
50
15
0.00018
0.00045
0.00837
0.00004
0.0001
0.00086
APF(Monolith Sliding) = 0.00018 + 0.00004 = 0.00022
P(Event A) = Sum of end branch p values for all pathways that contain Event A
Calculating ALL
Flood Stage1520-1550
Stage> 1550
Non Breach
Non Breach
Monolith Slides
Monolith Slides
Spillway Erodes
Spillway Erodes
Life Loss
Life Loss
Life Loss
Life Loss
Life Loss
Life Loss
0.009
0.001
0.02
0.05
0.93
0.04
0.1
0.86
100
40
5
170
50
15
0.00018
0.00045
0.00837
0.00004
0.0001
0.00086
ALL(Monolith Sliding) = 0.00018 (100) + 0.00004 (170) = 0.0248
E(C | Event A) = Sum of end branch p*c values for all pathways that contain Event A
Partitioning
14
• Tree branches are discrete• Input functions are continuous• Analogous to Simpson’s rule for integration• Numerical precision
• Number of partitions (more is better)• Location of partitions (capture shape changes)
• Can generate intervals manually or automatically• Intervals can be regular or irregular spacing
Example
15
Flood
Stage= 1500
Stage= 1507
Stage= 1537
Stage= 1580
Stage= 16001480
1500
1520
1540
1560
1580
1600
1620
0.00010.0010.010.11
Peak
Res
ervo
ir St
age
Annual Chance Exceedance
Continuous
Discrete Approximation
1 - 0.5 = 0.5
Exceedance interval
Non-ExceedanceInterval Partition Probability
Partition Stage0.5 – 0.1 = 0.4
0.1 – 0.01 = 0.09
0.01 – 0.001 = 0.009
0.001 – 0 = 0.001
∑ (area under the curve)= 1
These partitions are mutually exclusive
1 2 3 4 5
1
2
3
4
5
Avoid Double Counting
16
Flood
Stage> 1500
Stage> 1507
Stage> 1537
Stage> 1580
Stage> 1600
1480
1500
1520
1540
1560
1580
1600
1620
0.00010.0010.010.11
Peak
Res
ervo
ir St
age
Annual Chance Exceedance
Continuous
Discrete Approximation
1
0.5
0.1
0.01
0.001
∑ > 1, not goodDo Not Use Exceedance ProbabilitiesThese partitions are not mutually exclusive
1
12
3
4
5 2
3
4
5
System Response Curves
17
Flood
Stage= 1500
Stage= 1507
Stage= 1537
Stage= 1580
Stage= 1600
0.5
0.4
0.09
0.009
0.001
Breach
Breach
Breach
Breach
Breach
3E-5
8E-5
0.003
0.00001
0.0001
0.001
0.01
0.1
1
1500 1520 1540 1560 1580 1600
Prob
abili
ty o
f Bre
ach
Peak Stage
0.09
0.25
Variable Transformation• Peak stage is typically used as the independent variable to
combine the hazard, system response, and consequence functions• Peak stage defined as a function of AEP• SRP and consequences defined as a function of peak stage
• Other variables might be• More convenient – Probability of failure as a function of overtopping depth• Better indicator – Consequences as a function of peak outflow
• Event tree calculations can be set up to perform and apply these transformations
• Overtopping depth defined as stage minus top of levee• Peak outflow defined as function of flood AEP
18
Monte Carlo Analysis• Branch probability estimates and consequences can be modeled
with uncertainty• Monte Carlo analysis can be used to combine these uncertainties
to obtain the uncertainty distribution for APF and ALL
19
Distribution of Sums and Products• Because event tree math is additive and multiplicative
• The mean AFP and mean ALL can be estimated by using the means of the input distributions
• Can become problematic in other models with operations that are not strictly additive or multiplicative
• Use the mean of the output distribution from a Monte Carlo simulation
• The distribution of AFP and ALL will typically trend toward a normal or log normal distribution because of the central limit theorem
20
0
0.002
0.004
0.006
1 2 3 4
Syst
em R
espo
nse
Prob
aibl
ityFlood Loading Interval
Median (50th Percentile)
15th and 85th Percentile
Curve Sampling
21
• Independent sampling of each load partition can generate physically impossible samples
30%Flood interval 1
Flood interval 2
Flood interval 3
Flood interval 4 0.3
0.9
0.2
0.4
90% 20%
40%Sampled SRP curve cannot decrease with increasing load
0
0.002
0.004
0.006
1 2 3 4
Syst
em R
espo
nse
Prob
aibl
ityFlood Loading Interval
Median (50th Percentile)
15th and 85th Percentile
Sampled System Response (70th Percentile)
Consistent Percentile Sampling
22
• Sample a single percentile and apply to all loading partitions
70%Flood interval 1
Flood interval 2
Flood interval 3
Flood interval 4
0.7
70%
70%
70%
“Risk taking is inherently failure prone. Otherwise, it would be called sure thing taking.”
-Jim McMahon
Exercise
Flood Stage1520-1550
Stage> 1550
Non Breach
Non Breach
Slope instability
Slope Instability
Internal Erosion
Internal Erosion
Life Loss
Life Loss
Life Loss
Life Loss
Life Loss
Life Loss
0.009
0.001
0.02
0.07
0.91
0.08
0.14
0.78
30
60
5
80
140
15Calculate APF for slope instabilityCalculate ALL for slope instability
Stage< 1520
0.99
Solution
Flood Stage1520-1550
Stage> 1550
Non Breach
Non Breach
Slope instability
Slope Instability
Internal Erosion
Internal Erosion
Life Loss
Life Loss
Life Loss
Life Loss
Life Loss
Life Loss
0.009
0.001
0.02
0.07
0.91
0.08
0.14
0.78
30
60
5
80
140
15
Calculate APF for slope instability = 0.00018 + 0.00008 = 0.00026Calculate ALL for slope instability = (0.00018 * 30) + (0.00008 * 80) = 0.0118
Stage< 1520
0.99 0.009 * 0.02 = 0.00018
0.001 * 0.08 = 0.00008
�Event Tree AnalysisObjectivesOutline of TopicsKey ConceptsEvent Tree AnalysisExamplePossible Event TreeTerminologyRules and MathSingle Tree FormatSeparate Potential Failure Mode TreesCalculating APFCalculating ALLPartitioningExampleAvoid Double CountingSystem Response CurvesVariable TransformationMonte Carlo AnalysisDistribution of Sums and ProductsCurve SamplingConsistent Percentile SamplingSlide Number 23ExerciseSolution