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.