Geoscience Reference
In-Depth Information
avoided by the formula
P
m
=
n
, but this yields certainty for the smallest item in
the record. Hazen (1930) proposed an intermediate position, namely
P
m
=
(
m
−
1)
/
n
.
An obvious feature of Hazen's choice is that with Equation (13.15) it produces a return
period
T
r
=
(
m
−
1
/
2)
/
2
n
for the largest item in the record; in other words, the resulting return
period is twice as long as the period over which the data have been recorded. This is
unacceptable, according to Gumbel (1958), because the estimated return period of the
largest event should approach the length of the period of record, as
n
becomes large.
These difficulties are not encountered in the Weibull formula
m
P
m
=
(13.19)
n
+
1
Gumbel (1958), and many after him, have recommended Equation (13.19), by noting
that beside the avoidance of difficulties for
m
0 or 1, so that all data can be plotted, it
also has the following advantages: (i) it is independent of the distribution function
F
(
x
);
(ii) the return period of the largest (or smallest, as the case may be) observation approaches
the number of observations
n
; (iii) all observations are equally spaced on the frequency
scale, which means that the difference between the plotting positions of the
m
th and
the (
m
=
1)th observation is a function only of
n
; (iv) it is intuitively simple and can be
readily implemented. Its main advantage, however, is that it can be theoretically justified
as the mean of the probability of the
m
th smallest observation; this can be proved as
follows.
+
Derivation of Weibull plotting position
Consider again a sample of
n
observations, after they have been ranked in increasing
magnitude, so that
X
1
<
X
n
. The probability distribution of
the
m
th smallest observation by itself is given by Equation (13.3), or
X
2
<
···
<
X
m
<
···
<
X
m
F
(
X
m
)
=
f
(
x
)
dx
(13.20)
−∞
and the density of this observation by itself is
f
(
X
m
)
(13.21)
However,
X
m
does not occur by itself, but in conjunction with (
n
−
1) other observations;
among these remaining (
n
−
1) observations, [(
n
−
1)
−
(
m
−
1)] observations exceed
1) do not. The probability of this occurrence is given by the binomial
distribution (see Section 13.3.2 below), and equals
X
m
and (
m
−
(
n
−
1)!
1)!
F
m
−
1
F
m
)
n
−
m
P
{
(
m
−
1)successes
,
(
n
−
m
)failures
}=
(1
−
m
(
n
−
m
)!(
m
−
(13.22)
in which the symbol
F
m
=
F
(
X
m
) is introduced for conciseness of notation, and
“success” refers to non-exceedance. The probabilities of each observation are inde-
pendent. Moreover, the
m
th smallest observation,
X
m
, can occupy
n
different places in
the sequence of the remaining (
n
−
1) observations, namely in front of all of them and
