Geoscience Reference
In-Depth Information
distribution that best fits the observed frequency distribution for the data during
the period for which observations are available. Once this has been done, estimates
of the probabilities of extreme occurrences or the probability of exceeding limits
can be made graphically or algebraically.
Over the years, the selection of an appropriate probability distribution to use in
such design problems has been the subject of much debate among engineering
hydrologists and civil engineers. Five broad groupings of possible distributions are
used:
(1)
Conventional frequency distributions (e.g., normal, Poisson, gamma);
(2)
Grumbel distributions (there are numerous);
(3)
Pearson distributions (particularly Types I and III);
(4)
Extremal distributions (Types I, II, III); and
(5)
Transformal distributions (e.g., logarithmic and polynomial transforms).
However, the selection of a particular probability distribution is as much an art
as a science, and is a decision that is aided by experience but influenced by
personal preference. Hydrometeorological understanding has little to offer
to aid such a detailed choice. For this reason, extended discussion of the
appropriateness of particular assumed probability distributions for particular
tasks is beyond the scope of a text such as this, which is concerned with
understanding hydrometeorological and hydroclimatological phenomena and
processes.
Nonetheless, an example of the general approach used is appropriate, and a
thirty year time series of annual rainfall for Musoma, Tanzania is used for this
purpose, see Table 13.3. The first step is to rank the data in either ascending or
descending order on the basis of the magnitude of the annual rainfall. This is done
in ascending order from 1 to 30 in Table 13.3. On the basis of this set of observa-
tions, the return period,
T
, for an event with the magnitude corresponding to rank
m
is given by:
( )
n
+
(13.11)
T
=
m
where
n
is the number of years in the data series, in this case 30. Alternatively, the
probability,
P
, of an event of rank
m
being equaled or exceeded is:
m
P
=
(13.12)
( )
n
+
The use of (
n
1) in these equations implies that the period for which data are
available is merely a sample and neither the highest ranked sample value, nor the
lowest, represents the true highest or lowest values in the population as a whole.
It is possible to assign a notional probability for each value of annual precipitation
from Equation (13.10), and these can be plotted to describe the sampled probability
distribution. In Fig. 13.13 such a plot is made on log-normal graph paper for the
data in Table 13.3. It is approximately a straight line, implying the probabilities are
+
