WATER & SOIL - These are not the droids you are looking for.

P=45
O= 15
x4
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'=
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o
t.25 2'.O ¡b zs sb loo rooo T
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o.f o.3 0.5 0.7 0.9 0.95 o.9986
YH
F(x)
Figure 3.2 The plot of the df in Fig. 3.1 as a function of its ¡educed variate,
F(x) =l-P(X>x)
34
where P(
ote the probability of
non-exce
respectively.
Allied with
dence is the notion of
recurrence interval or return period, which is the reciprocal
of the probability of exceedence in a time unit. For instance,
if a flood peak is exceeded on average 20 times in
every 100 years, it has a return period of 5 years, or a probability
of exceedence in any one year of 0.20. Thus
P(X>x) = I T
which, from Equation
F(x) = I
3.4, gives
_l
T
whereT = returnperiod.
The curvatu
35
36
and the reliability of the extrapolation when the fitted frequency
line is straight rather than curved. A straight line
for the df can be obtained by rescaling the F(x) axiJ with a
on the F(x) axis, using the relationship between y and F(x)
and Equations 3.3-3.6. In this way different probability
papers, e.g., Normal and Gumbel, can be constructed to
produce straight line plots of their corresponding distribution
functions. Figure 3.2 illustrates the use of a reduced
yariate (y¡) for the Normal distribution shown in Figure
3.1. In accordance with convention, Figure 3.1 has been realigned
in Figure 3.2, so that the variate scale is now on the
ordinate and the probability and reduced variate scales are
on the abscissa; future mention of frequency curves is with
reference to this type of plot. The actual application of a reduced
variate is explained fully in section 3.1.3 with reference
to the Gumbel distribution.
typical ofthat Despite the use of a reduced variate, a straight line will
for a Normal
are plotted to not give a good fit when the data sample does not conform
natural scales.
curvature for, to the assumed distribution. However, it may still be possible
to obtain a good fit with a straight line by first trans-
when fitting ã
sferred to as a
relative frequency or simply a frequency distribution) to a forming the data sample. The most common transformation
is the changing of each sample item to its logarithm.
data sample, it is easier to visually assess the goodness-of-fit
Water & soil technical publication no. 20 (1982)
l4