Geoscience Reference
In-Depth Information
function. The term “return period” stems from the fact that, whenever the observations
are made at regular intervals in time, the number of observations is a time expressed
in the same units. It should be emphasized that the return period represents the average
number of observations. This does not mean that the event will occur once every
T
r
number of observations. Thus a 100 y flood need not occur every 100 y. In fact, it may
very well occur next year, or again, it may not occur at all for another 1000 y, although
that is unlikely.
In practical applications, the return period as given by Equation (13.15) is normally
used to characterize phenomena whose severity increases with their magnitude
x
.For
example, a 500 y flood is more severe and causes more damage than a 100 y event. When,
on the other hand, the severity of the event decreases with its numerical magnitude, the
return period should be defined as
1
F
(
x
)
T
r
(
x
)
=
(13.18)
This will ensure that, for instance in the case of a drought, lower flows and lower rainfall
amounts, as measures of drought severity, will be characterized by longer return periods.
The theoretical significance of Equation (13.18) is,
mutatis mutandis
, the same as (13.15).
13.2.4
Empirical probability plots
It is often useful to plot the data from a hydrologic measurement record to gain a general
idea of their statistical characteristics. The (empirical) probability plot, also known as
the frequency plot or frequency curve, and sometimes as the quantile plot, is a common
tool for this purpose. This plot is a graphical representation of the probability (of non-
exceedance or exceedance) of the individual data in the record against their respective
magnitudes, i.e.
F
F
(
x
).
For a data record, consisting of
n
items, normally measured at regular intervals, the
procedure is as follows. (i) Tabulate the
n
data,
X
m
, ranked in increasing magnitude,
so that
X
1
≤
=
X
n
; (ii) assign an order number to each item, in
accordance with its respective subscript, namely 1
X
2
≤··· ≤
X
m
≤··· ≤
n
; (iii) estimate the
(empirical) probability
P
m
, that the magnitude of the item will not be exceeded, for each
item by means of a suitable
plotting position
formula. This plotting position
P
m
is used as
an estimate of the value of the unknown probability
F
(
x
,
2
,...,
m
,...,
X
m
) for the observed event
X
m
. In the past numerous plotting position formulae have been suggested. Cunnane
(1978) has presented a critical review of the history and properties of some of the more
common ones.
=
Plotting position
Probably the oldest and intuitively the simplest is
P
m
=
n
. The problem with this
expression is that the largest item on record is assigned a probability of one, that is
certainty; in other words, with this formula it is assumed that the magnitude of the largest
will never be exceeded in the future. Because this is impossible, the use of this plotting
position is tantamount to discarding the largest item in the record. This difficulty may be
m
/
