Basic Hydrology & Hydraulics: DES 601

Module 3Flood Frequency

Probability and Discharge

• Discharge is the flow rate (cubic feet per second) in a conduit (stream, pipe, overland, etc.)

• Probability is the chance of observing a particular value of discharge or greater in a given period, typically a year.

• These exceedance probabilities are sometimes expressed for stage (depth), or hydraulic structure capacity.

• Chapter 4, HDM

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Annual Exceedance Probabiltiy

• In TxDOT HDM, the preferred terminology is Annual Exceedence Probability (AEP)• In other contexts recurrence intervals are used

interchangeably• 1-percent chance, 0.01 chance, and 100-year

recurrence interval all represent the same “amount” of probability.• In recent years, the use of T-year designation is

discouraged because it is easy to misinterpret!.

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Annual return interval• An annual return interval is an alternative way to

express the AEP.

• The abbreviation is ARI.

• The ARI is the average number of periods (years) between periods containing one or more events (discharges) exceeding a prescribed magnitude.

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Annual Exceedance Probabiltiy

• Probability of observing 20,000 cfs or greater in any year is 50% (0.5) (2-year).

Exceedance

Non-exccedance

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Annual Exceedance Probabiltiy

• Probability of observing 150,000 cfs or greater in any year is 1% (0.01) (100-year)

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Estimating Probability• Subjective assessment – probability you will be bored in the next 10

minutes (hard to judge, depends on my “entertainment value”, time of day, how well you slept, interest, etc.)

• Fault-tree analysis – probability that a system (computer) will fail but linking the failure probabilities of individual components (transistors, capacitors, etc.)

• Historical outcome analysis – estimate probability on past system behavior (this is the method used in hydrology most of the time)

Reference: Engineering Statistics Handbook; Section 1.3.3.22; NIST – US Commerce Department.

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Estimating Probability

• Historical outcome analysis – estimate probability on past system behavior (this is the method used in hydrology most of the time)

• Time-series – e.g. annual peak discharge versus time

• No anticipation the peak comes on the same day each year

• Anticipate that the annual peaks are sort of caused by similar, random, processes

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Estimating Probability

• Time-series – e.g. annual peak discharge versus time

• Appeal to the concept of “relative frequency” as a model to explain the time-series behavior.

• Each year is a roll of “dice”, we record the result, and use the result to postulate the long-term average, anticipated behavior

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Probability plots

• The probability plot is a graphical technique for assessing whether or not a data set follows a given distribution such as the normal or Weibull.

• The distribution is the model of the observations, hence it is kind of important to be comfortable we are choosing the most appropriate model from our tool kit.

• Perfect agreement is impossible! If the model exactly fits, we probably made an error (i.e. plotted model vs. model, instead of data vs. model)

Reference: Engineering Statistics Handbook; Section 1.3.3.22; NIST – US Commerce Department.

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Example – Beargrass Creek• Illustrates concepts related to probability,

magnitude, and the underlying mechanics of assessing such behavior.

Tim

e-se

ries

: (Y

YY

Y,P

eak

Q)

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Example – Beargrass Creek• Generally, rank series (small to big, big to small –

analyst preference).• Assign a relative frequency to each year assuming

each year is a dice roll (independent, identically distributed)• Typical “relative frequency” is the Weibull plotting

position (there are others, next module)

• Plot Magnitude and Cum.Freq

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Cum

ulat

ive

Rel

ativ

e F

requ

ency

QP

EA

K

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Example – Beargrass Creek• So at this step, we have an “empirical” probability-

discharge plot.• Sometimes can use as-is, but usually we fit a

distribution model to the plot, and make inferences FROM THE MODEL!

• As an illustration, we can fit a normal distribution to the time series (next slide)

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Normal Distribution using the Time-Series Mean and Variance as fitting parameters

Fit is not all that great

Point here is to illustrate how AEP models are constructed from observations.

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Example – Beargrass Creek

• Assume we “like” this fit, then one can interpolate/extrapolate from the distribution model (and dispense with underlying data)

Error function (like a key on a calculator

e.g. log(), ln(), etc.)

AEP

Distribution Parameters

Magnitude

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Example – Beargrass Creek

• Naturally we would prefer to supply a “F” and recover the “x” directly – not always possible, but in a lot of cases it is.

• More importantly, is when we extrapolate – the participant should observe the 1% chance value is NOT contained in the observation record.

• To estimate from the model, we simply find the value “x” that makes “F” equal 0.01 (about 3920 cfs in this example)

AEP

Magnitude

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Summary

• Probability and Magnitude are Related via a Frequency Curve

• The probability is called the Annual Exceedance Probability (AEP) or Annual Recurrence Interval (ARI). AEP is the preferred terminology

• Historical observations are examined to construct “models” of the probability and discharge relationship

• These models are used to extrapolate/interpolate to recover magnitudes at prescribed values of AEP

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