LOG-Person Type III Distribution

LOG-PEARSON TYPE III DISTRIBUTION

The recommend procedure for

use of the log-pearson

distribution is to convert the

data series to logarithms and

compute.

Where:

= is a flood of specified probability

=Is the mean of flood series

=Is the number of years of record

=Is s frequency factor define by specific

distribution

3log

3232

))(2)(1(

)log(2)(loglog3)(log

xnnn

xxxnxnG

xXx logloglog

xKXX

X

X

n

K

Mean:

Alternatively,

=IS THE STANDARD DEVIATION

n

Xx

loglog

1

/)log()(log 22

log

n

nxxx

X

XKXX

Also,

Standard Deviation:

Skew Coefficient:

Where:

The value of X for any probability level is computed

from modified,

1

)log(log 2

log

n

XXX

3log

3

))(2)(1(

)log(log

Xnn

XXnG

XKXX

XKXX logloglog

The probability density function of type III is,

Where: where: is the third moment about the

mean= is the variance is gamma function is the base of

napierian logarithms

32

23

1

0

2

3

2/0

)1(

2

14

)1()(

ce

c

a

n

ca

c

ea

XX

c

cXc

3G6

2e

EXTREME-VALUE TYPE I DISTRIBUTION

Fisher and Tippett found that the distribution

of the maximum(or minimum) values selected

from n samples approached a limiting form as

the size of the samples increased. When the

initial distributions within the samples are

exponential, the type I distribution is given by

.

yee1

Where:

-is the probability of a given flow being equaled or

exceeded

-is the base of napierian logarithm

-is the reduced variate or is the function of probability

Where:

-is the mean of the data series

-is the standard deviation

e

y

XyXX )45.07797.0(

yee1

X

X

Return Period, years

Probability Reduced Variate, y

K

1.58 0.63 0.000 -0.450

2.00 0.50 0.367 -0.164

2.33 0.43 0.579 0.001

5 0.20 1.500 0.719

10 0.10 2.250 1.300

20 0.05 2.970 1.870

50 0.02 3.902 2.590100 0.01 4.600 3.140

200 0.005 5.296 3.680

400 0.0025 6.000 4.230

The table shows the values of K for various return

period. So when using Gringorten plotting position from

this equation or , no

correction for record length is considered necessary.

Two or more computed value of X define a straight line

on extreme- value probability paper.

1

44.0

n

m44.0

12.0

m

nTr

)1ln(ln y

Gumbel was the first to suggest the use of the

extreme-value distribution for floods, and the

distribution is commonly referred to as the

Gumbel distribution. Gumbel’s argument for the

use of this distribution was that each year of

record constituted a simple with n=365 and the

annual flood was the maximum value from the

sample. Hence, it could be assumed that the

flood case conformed to the conditions specified

by Fisher and Tippett.

SELECTION OF DESIGN FREQUENCY

There are situation when one is concerned with the

probability of a flood occurring during specified

interval of future time. For example, what flood

probabilities exist during the construction period of

a dam? The probability that the flood with an

average probability of occurrence will be

exceeded exactly k times during an N-year period is

given by the binomial distribution

kJ

kkNk kNk

NJ

)1(

)!(!

!

The probability of one or more

exceedances in N years is found by taking

k=0 and noting that the probability of

exceedance is one minus the probability of

nonexceedance, norMOREJ )1(11

Thanks for l istening!!!

God bless you all

REPORTED BY:

BRYANBONG C. ZAILON