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Frequency Analysis: Graphical Approach for Log-Pearson Type III
CVEEN 4410 Engineering Hydrology
CVEEN 4410: Engineering Hydrology
Frequency Analysis (continued)
Topic and Goal: Use frequency analysis of historical data to forecast hydrologic events
Apply graphical frequency analysis for data that fit the Log-
Pearson Type 3 Distribution
Specific Objectives
Be able to:
Frequency
Analysis
Objectives
Graphical
Frequency
Analysis: Log-Pearson
Type 3
Distribution
Frequency Analysis: Graphical Approach for Log-Pearson Type III
CVEEN 4410 Engineering Hydrology
Assume you’ve acquired a dataset, analyzed it
using histograms, and you decide that it probably
fits a Log-Pearson type III distribution.
The next step, then, is to evaluate the fit for a LP3
distribution. If the fit is indeed “good,” then
proceed with forecasting estimates.
Frequency
Analysis
Objectives
How do you
evaluate
which
probability
distribtuion
applies to your
data?
Graphical
Frequency
Analysis: Log-Pearson
Type 3
Distribution
Frequency Analysis: Graphical Approach for Log-Pearson Type III
CVEEN 4410 Engineering Hydrology
“Recipe” for LP3 frequency curve analysis:
1. Create a series that consists of the logarithms Yi of the
annual maximum flood series
2. Compute the sample mean, standard deviation, Sy, and
standardized skew, Cs, of the logarithms of Step 1
3. For selected values of the exceedance probability (P) obtain
values of the standardized variate K from Table 3-4 (round
the skew to the nearest tenth)
4. Determine the values of the LP3 curve for the exceedance
probabilities selected in Step 3 using the equation
in which Y is the logarithmic value of the LP3 curve
5. Take the antilogarithm of the Yi values and use them to
plot the LP3 frequency curve (if necessary)
Frequency
Analysis
Objectives
How do you
evaluate
which
probability
distribtuion
applies to your
data?
Graphical
Frequency
Analysis: Log-Pearson
Type 3
Distribution
Frequency Analysis: Graphical Approach for Log-Pearson Type III
CVEEN 4410 Engineering Hydrology
4
Frequency
Analysis
Objectives
How do you
evaluate
which
probability
distribtuion
applies to your
data?
Graphical
Frequency
Analysis: Log-Pearson
Type 3
Distribution
Problem Statement:
Check it these data in
spreadsheet file
“wildcat_creek_runoff_data.xls”
posted on website.
First order of business: calculate the
skew of the log data.
Frequency Analysis: Graphical Approach for Log-Pearson Type III
CVEEN 4410 Engineering Hydrology
5
Frequency
Analysis
Objectives
How do you
evaluate
which
probability
distribtuion
applies to your
data?
Graphical
Frequency
Analysis: Log-Pearson
Type 3
Distribution
Problem Statement:
First order of business: calculate the
skew of the log data.
Frequency Analysis: Graphical Approach for Log-Pearson Type III
CVEEN 4410 Engineering Hydrology
6
Frequency
Analysis
Objectives
How do you
evaluate
which
probability
distribtuion
applies to your
data?
Graphical
Frequency
Analysis: Log-Pearson
Type 3
Distribution
Problem Statement:
First order of business: calculate the
skew of the log data.
Frequency Analysis: Graphical Approach for Log-Pearson Type III
CVEEN 4410 Engineering Hydrology Frequency
Analysis
Objectives
How do you
evaluate
which
probability
distribtuion
applies to your
data?
Graphical
Frequency
Analysis: Log-Pearson
Type 3
Distribution
Next step: for selected
values of the exceedance
probability (P) obtain
values of the standardized
variate K from Table 3-4
(round the skew to the
nearest tenth).
Frequency Analysis: Graphical Approach for Log-Pearson Type III
CVEEN 4410 Engineering Hydrology
K
Exceedance
Probability
Frequency
Analysis
Objectives
How do you
evaluate
which
probability
distribtuion
applies to your
data?
Graphical
Frequency
Analysis: Log-Pearson
Type 3
Distribution
Frequency Analysis: Graphical Approach for Log-Pearson Type III
CVEEN 4410 Engineering Hydrology
K
Exceedance
Probability
Frequency
Analysis
Objectives
How do you
evaluate
which
probability
distribtuion
applies to your
data?
Graphical
Frequency
Analysis: Log-Pearson
Type 3
Distribution
For all LP3 graphical
problems, plot the LP3
frequency curve using
abscissa (x-axis) values
of 99% exceedance
probability, 50%, 20%,
10%, 4%, 2%, 1% and
0.5%.
Recall that exceedance
probability corresponds
to the top x-axis of
regular probability
paper. Now, how about the
ordinate (y-axis)
values for your
frequency curve?
Frequency Analysis: Graphical Approach for Log-Pearson Type III
CVEEN 4410 Engineering Hydrology
K
Exceedance
Probability
Frequency
Analysis
Objectives
How do you
evaluate
which
probability
distribtuion
applies to your
data?
Graphical
Frequency
Analysis: Log-Pearson
Type 3
Distribution
Now, how about the
ordinate (y-axis)
values for your
frequency curve?
Well, you have a series of
frequency factors, K.
Recall that
Y =Y +KSywhere Y is your random
variable (or in this case,
the log of your RV).
Thus, you may calculate
values of Y for your
ordinate axis,
corresponding to each of
the exceedance
probabilities (99, 50, 20,
10, 4, 2, 1, 0.5).
Frequency Analysis: Graphical Approach for Log-Pearson Type III
CVEEN 4410 Engineering Hydrology
K
Exceedance
Probability
Frequency
Analysis
Objectives
How do you
evaluate
which
probability
distribtuion
applies to your
data?
Graphical
Frequency
Analysis: Log-Pearson
Type 3
Distribution
Y =Y +KSy
Thus, you may calculate
values of Y for your
ordinate axis,
corresponding to each of
the exceedance
probabilities (99, 50, 20,
10, 4, 2, 1, 0.5).
But first, let’s scale that y-
axis, using:
Frequency Analysis: Graphical Approach for Log-Pearson Type III
CVEEN 4410 Engineering Hydrology Frequency
Analysis
Objectives
How do you
evaluate
which
probability
distribtuion
applies to your
data?
Graphical
Frequency
Analysis: Log-Pearson
Type 3
Distribution
1.4
1.3
1.2
1.1
1.0
0.9
0.8
0.7
Frequency Analysis: Graphical Approach for Log-Pearson Type III
CVEEN 4410 Engineering Hydrology
K
Exceedance
Probability
Frequency
Analysis
Objectives
How do you
evaluate
which
probability
distribtuion
applies to your
data?
Graphical
Frequency
Analysis: Log-Pearson
Type 3
Distribution
Y =Y +KSy
Thus, you may calculate
values of Y for your
ordinate axis,
corresponding to each of
the exceedance
probabilities (99, 50, 20,
10, 4, 2, 1, 0.5).
The first value, for 99%
exceedance P, then, is:
Y = 1.109 + (-2.544)(0.144)
= 0.743
or, to plot, use
X (or P) = 99%, Y = 0.743
Now, to calculate Y, recall:
Frequency Analysis: Graphical Approach for Log-Pearson Type III
CVEEN 4410 Engineering Hydrology Frequency
Analysis
Objectives
How do you
evaluate
which
probability
distribtuion
applies to your
data?
Graphical
Frequency
Analysis: Log-Pearson
Type 3
Distribution
1.4
1.3
1.2
1.1
1.0
0.9
0.8
0.7
Frequency Analysis: Graphical Approach for Log-Pearson Type III
CVEEN 4410 Engineering Hydrology
K
Exceedance
Probability
Frequency
Analysis
Objectives
How do you
evaluate
which
probability
distribtuion
applies to your
data?
Graphical
Frequency
Analysis: Log-Pearson
Type 3
Distribution
Y =Y +KSy
The second value, for 50%
exceedance P, then, is:
Y = 1.109 + (0.05)(0.144)
= 1.116
or, to plot, use
X (or P) = 50%, Y = 1.116
Using:
with:
Frequency Analysis: Graphical Approach for Log-Pearson Type III
CVEEN 4410 Engineering Hydrology Frequency
Analysis
Objectives
How do you
evaluate
which
probability
distribtuion
applies to your
data?
Graphical
Frequency
Analysis: Log-Pearson
Type 3
Distribution
1.4
1.3
1.2
1.1
1.0
0.9
0.8
0.7
Frequency Analysis: Graphical Approach for Log-Pearson Type III
CVEEN 4410 Engineering Hydrology
K
Exceedance
Probability
Frequency
Analysis
Objectives
How do you
evaluate
which
probability
distribtuion
applies to your
data?
Graphical
Frequency
Analysis: Log-Pearson
Type 3
Distribution
Y =Y +KSy
The third value, for 20%
exceedance P, then, is:
Y = 1.109 + (0.853)(0.144)
= 1.232
or, to plot, use
X (or P) = 20%, Y = 1.232
and so on....
Using:
with:
Frequency Analysis: Graphical Approach for Log-Pearson Type III
CVEEN 4410 Engineering Hydrology Frequency
Analysis
Objectives
How do you
evaluate
which
probability
distribtuion
applies to your
data?
Graphical
Frequency
Analysis: Log-Pearson
Type 3
Distribution
1.4
1.3
1.2
1.1
1.0
0.9
0.8
0.7
Frequency Analysis: Graphical Approach for Log-Pearson Type III
CVEEN 4410 Engineering Hydrology Frequency
Analysis
Objectives
How do you
evaluate
which
probability
distribtuion
applies to your
data?
Graphical
Frequency
Analysis: Log-Pearson
Type 3
Distribution
Frequency Analysis: Graphical Approach for Log-Pearson Type III
CVEEN 4410 Engineering Hydrology Frequency
Analysis
Objectives
How do you
evaluate
which
probability
distribtuion
applies to your
data?
Graphical
Frequency
Analysis: Log-Pearson
Type 3
Distribution
Now, we need to plot the data, to
check fit against the frequency
curve!
Frequency Analysis: Graphical Approach for Log-Pearson Type III
CVEEN 4410 Engineering Hydrology Frequency
Analysis
Objectives
How do you
evaluate
which
probability
distribtuion
applies to your
data?
Graphical
Frequency
Analysis: Log-Pearson
Type 3
Distribution
Frequency Analysis: Graphical Approach for Log-Pearson Type III
CVEEN 4410 Engineering Hydrology Frequency
Analysis
Objectives
How do you
evaluate
which
probability
distribtuion
applies to your
data?
Graphical
Frequency
Analysis: Log-Pearson
Type 3
Distribution
1.4
1.3
1.2
1.1
1.0
0.9
0.8
0.7
Frequency Analysis: Graphical Approach for Log-Pearson Type III
CVEEN 4410 Engineering Hydrology
Exceedance
Non-Exceedance
Frequency
Analysis
Objectives
How do you
evaluate
which
probability
distribtuion
applies to your
data?
Graphical
Frequency
Analysis: Log-Pearson
Type 3
Distribution
1.4
1.3
1.2
1.1
1.0
0.9
0.8
0.7
Frequency Analysis: Graphical Approach for Log-Pearson Type III
CVEEN 4410 Engineering Hydrology
Exceedance
Non-Exceedance
Frequency
Analysis
Objectives
How do you
evaluate
which
probability
distribtuion
applies to your
data?
Graphical
Frequency
Analysis: Log-Pearson
Type 3
Distribution
1.4
1.3
1.2
1.1
1.0
0.9
0.8
0.7
Frequency Analysis: Graphical Approach for Log-Pearson Type III
CVEEN 4410 Engineering Hydrology
Exceedance
Non-Exceedance
Frequency
Analysis
Objectives
How do you
evaluate
which
probability
distribtuion
applies to your
data?
Graphical
Frequency
Analysis: Log-Pearson
Type 3
Distribution
1.4
1.3
1.2
1.1
1.0
0.9
0.8
0.7
Frequency Analysis: Graphical Approach for Log-Pearson Type III
CVEEN 4410 Engineering Hydrology Frequency
Analysis
Objectives
How do you
evaluate
which
probability
distribtuion
applies to your
data?
Graphical
Frequency
Analysis: Log-Pearson
Type 3
Distribution
Next time (after midterm exam):
we’ll start with “routing methods,”
including channel routing and
reservoir routing. You’re
encouraged to read sections 4.1
and 4.2 in your text before next
week.