LOG-Person Type III Distribution
Transcript of LOG-Person Type III Distribution
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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.
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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
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Mean:
Alternatively,
=IS THE STANDARD DEVIATION
n
Xx
loglog
1
/)log()(log 22
log
n
nxxx
X
XKXX
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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
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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
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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
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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
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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
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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
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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.
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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(
)!(!
!
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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
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