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