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