FLOOD00764

• County: Statewide
• Community: Nationwide
• Title: Use of Extreme Value Theory in Estimating Flood Peaks from Mixed Populations
• Date: 2/1/1980
• Prepared By: Utah State University Water Research Laboratory
• Floodplain - Doc Type: Educational/Technical/Reference Information

Thus

F(x)

{1 x>b
= exp{- (b:Xr-(~,xr}X>b' c>O, 0'>0

(12)

The bound parameter b was taken to be the
same for both componenls. Numerically b is
the most difficult of the three paramet~rs to
estimate and the one to which the distribu-
t ion is least sensitive.

A least squares estimation technique
reported in Canfield and Borgman (1975) was
improved and used to estimale the parameters
of Equat~on 12. Let h(i), i = 1,2, ..., n
be the Ith order statistic of n annual
maximum flood flows. Estimates of the
parameters in Equal ion 12 are taken to be
those values which minimize,

n
.=" [E(lnF(h(.)))-ln
i=l :1

2
(F(h(i)))J Wi

(13)

where Wi is a weIght factor such that

var

(In F(\i)))
(In F(h(i)))

(14)

W =
i

var

and E(.) is the expected value operalor.
The variance .of ln F(h(i)) is defined by

2i var (In F(h(i))) = E[ln F(h(i))
2
- E(ln F(h(i)))J .

(15)

The values of
F (h ( i))) ar e
computed USIng
trapezoid rule.

EOn F(h(i))) and var
nonparametric and may
numerical integration by

On
be
the

E [In F(h(i))J

(i_l)ni(n_i) !1\n F(h(i)) [F(h(i)) Ji-I
o

'[I-F(h(i))Jn-idF(h(i))

(16)

2
E[{E[ln F(h(i))J-ln F(h(i))} J
[ 2 2
E (In F(h(i))) J - (E[ln F(h(i)) J}

11
nl 2 i-I
(i-l)!(n-i)! [In F(h(i))J [F(h(i))J
o
In-i
'[I-F(h(i)) dF(h(i)}

- (ci-l) 7~n-l) !11 In F(h(i))
. [1-F(h(O) t-idF~h(i)r

[F(h(i))]i-l

Lindgren (1976), page 218, gives the density
function of the ith order statistic and, page
113, the expectat ion of a funet ion of a
random variable. For convenIence let,

ELi = E [In F(h(i))]
2
ELSQi E[{E[ln F(h(i))]-ln F(h(i))} ]

Yi = b - h(i)

0.' (0.1,0.2) = (a,a')

8' = (81,82) = (c~ ' (C,\a')

From this information, Equation 13 can be
rewrItten as
n 0.1 a.2 2
ljJ = L {ELi +81Yi + 82Yi } Wi
i=l

al a. 2 2
E {~ ELi + SlYi ~ +82Yi ~}
1=1

(17)

A FORTRAN program for computat ion of ELi
and ELSQi are found in A~pendix B. Esti-
mat ion ot a, a', c and c is accomplished
by estimating a. and 6 and then solvIng for a,
a', c and c' respectIvely.

In order to minimize Equat ion
appropr iale part ial der ivat ives of 1/1
evaluated and set equal to zero.
~ ,n 0'.') n 20.1
-""--,s = I. W~ELi "+91 I: W.Y.
1 i=1.L 1=1 :1 1

17,
are

n
+62 l:
1=1

0.1+02
W1Y1

o

(18)

i!t = ~
382 1=1

n a. +0'.
EWy12+8
1=1 i i 2

n 2il2
E WiY1
1=1

"2
WiELiYi +61

o

(19)

l!:L_ n "I n u1+a2
L WiELiYi InYi+82 L WiYi In Yi
'"I i=l i=1
n 2"1
+ 81 E WtYi In Yi 0 (20)
1=1

n
.3!L = L
3a2 1=1

n al+a2
E WiYi
i=1

In Y.
,

"2
WiELiYi In Y1 + 61

+ 82

n 202
L WiYi
i=1

In Y.
,

o

(21)

Solving Equation 18 for 82 yields,

'2

n
- E W
i=1 i

2"1
WiYi

"1 n
ELiYi - 61 r
i=1
n a1+u2
E Wi Yi
i=1

(22)

16