Streamflow measurement

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CE461 Hydrology
Unit 9 Peak Discharge Frequency Analysis
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Probability Functions

• Probability density functions may be derived
  using two approaches
• Inductive Approach (Nonlinear Curve Fit)
• By fitting a function to data
• Deductive Approach (Parametric Model)
• By fitting data to a known function
• Eg. Normal distribution

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Hydrologic Probability Functions

• In hydrology, the following four deductive models
  are frequently used
• Normal Distribution
• Log normal Distribution
• Extreme Value Type I (Gumbel) Distribution
• Log Pearson III Distribution

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Normal Distribution
F(z)
0
z
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Normal Distribution Tables
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Basic Statistics

• Each analytical deductive model can be fit
  using basic statistics from the underlying
  hydrologic data
• Mean
• Variance
• Skew

m X Xi / N
s2 S2 (Xi - X)2 / N - 1
gs Cs N (Xi - X)3 / (N - 1)
(N-2) S3
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Log-Normal Distribution
The lognormal distribution is obtained by simply
using the normal distribution with X log
X Mean of X mean of the logs of X Variance of X
variance of the logs of X
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Extreme Value Type I (Gumbel) Distribution
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Log-Pearson III


e
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Hydrologic Probability Functions

• Deductive hydrologic probability functions are
  usually calibrated using one of two methods
• Graphical Approach
• Analytic Approach

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Graphical Method

• Obtain n years of average annual flows
• Rank the flows from largest to smallest
• Assign each flow a rank r r 1 for largest, rn
  for smallest
• Assign each flow a probability using
• Plot the flows vs probability on a selected
  probability graph

P(Q) r/(n1)
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Graphical Method (Cont)
Probability
Each probability function has its own type of
graph paper Fit best straight line through the
data on each graph Select function which yields
the lowest fitting error.
Normal
Log normal
Log Pearson III
Gumbel
Q
Return Interval
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Graphical Method (Cont)
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Analytical Method

• In using an analytical approach, we work with the
  inverse form of the hydrologic probability
  function. In general, the equation may be
  expressed as

Standard deviation of the random variable
Y Y KTSY
Frequency factor
Random variable
Mean of the random variable
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1a. Normal Method

• Y Q
• Y mean of the Ys
• SY standard deviation of the Ys
• KT f(T the return frequency)

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1b. Log-Normal Method

• Y log Q or ln Q
• Y mean of the Ys
• SY standard deviation of the Ys
• KT f(T the return frequency)

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1c. Gumbel Method

• Y Q
• Y mean of the Ys
• SY standard deviation of the Ys
• KT f(T the return frequency,
• N number of years )

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1d. Log Pearson III Method

• Y log Q or ln Q
• Y mean of the Ys
• SY standard deviation of the Ys
• KT f(T the return frequency, Cs skew
  coefficient)
• Note when the number of years of historical Qs
  is less than 100, the skew coefficient must be
  adjusted using skews from nearby stations

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Water Resources Council Method for Log Pearson III
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Map Skew Cm
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Application

• Hydrologic probability functions P( ) can be
  used in two different applications
• Design Application Given Tq find q
• Analysis Application Given q find Tq

P(Q gt q) 1/Tq 1/100 0.1
Tq 1/ P(Q gt q)
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Normal Method (Appl)

• Find Q which has a return interval of T
• Determine Q and Sq from Qs
• Determine KT from table or equation
• Solve for Q Q KTSq
• Determine the return interval for a discharge of
  Q
• Determine Q and Sq from Qs
• Solve for KT Q - Q / Sq
• Solve for T by interpolating from table or
  equation

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Lognormal Method (Appl)

• Find Q which has a return interval of T
• Determine Y and SY from Qs using either Y
  log Q or Y ln Q
• Determine KT from table or equation
• Solve for Y Y KTSY
• Solve for Q using Q 10Y or eY
• Determine the return interval for a discharge of
  Q
• Determine Y and SY from Qs using either Y
  log Q or Y ln Q
• Solve for KT Y - Y / SY
• Solve for T by interpolation from table or by
  equation

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Gumbel Method (Appl)

• Find Q which has a return interval of T
• Determine Q and Sq from Qs
• Determine KT given T and N using table
• Solve for Q Q KTSq
• Determine the return interval for a discharge of
  Q
• Determine Q and Sq from Qs
• Solve for KT Q - Q / Sq
• Solve for T by given KT and N by interpolating
  from table

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Extreme Value Type I Probability Tables
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Log Pearson III Method (Appl)

• Find Q which has a return interval of T
• Determine Y and SY from Qs using either Y
  log Q or Y ln Q
• Determine skew coefficient Cs
• Determine KT using
• Solve for Y Y KTSY
• Solve for Q using Q 10Y or eY
• Determine the return interval for a discharge of
  Q
• Determine Y and SY from Qs using either Y
  log Q or Y ln Q
• Determine skew coefficient Cs
• Solve for KT Y - Y / SY
• Solve for T by interpolating from Table of KT and
  Cs

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Log Pearson III Probability Tables