Transcript DES601-Module03

Basic Hydrology &
Hydraulics: DES 601
Module 3
Flood 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.
1yr.
5 - year ARI =
= 0.2 = 20% AEP
5yr.
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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)
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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)
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Example – Beargrass Creek
Time-series: (YYYY,Peak Q)
• Illustrates
concepts
related
to
probability,
magnitude, and the underlying mechanics of
assessing such behavior.
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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)
Rank i
Cum. Freqi =
N +1
• Plot Magnitude and Cum.Freq
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Cumulative Relative Frequency
QPEAK
Example – Beargrass Creek
• So at this step, we have an “empirical” probabilitydischarge 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)
AEP
1
x -m
F(x) = (1+ erf (
))
2
2s
Magnitude
Error function
(like a key on a calculator
e.g. log(), ln(), etc.)
Distribution Parameters
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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.
AEP
1
x -m
F(x) = (1+ erf (
))
2
2s
Magnitude
• 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)
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Summary
• Probability and Magnitude
Frequency Curve
are
Related
via
a
• 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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