Geoscience Reference
In-Depth Information
that you must choose the class intervals for the
histogram carefully so that the probability density
function is an accurate representation of the data.
Too large an interval and the distribution may
be shaped incorrectly, too small and holes in the
distribution will appear.
One way of avoiding the difficulties of choosing
the best class interval is to use a rank order dis-
tribution. This is often referred to as a plotting
position formula.
25
20
15
10
5
0
Annual maximum flow (X) (m
3
/s)
Figure 6.11
Frequency distribution of the Wye annual
maximum series.
The Weibull formula
through the histogram. This is the
probability density
function
which represents the smoothed version of
your frequency histogram.
In flood frequency analysis there are three
interrelated terms of interest. These terms are inter-
related mathematically, as described in equation
6.2 in the text below.
The first step in the method is to rank your annual
maximum series data from low to high. In doing
this you are assuming that each data point (i.e. the
maximum flood event for a particular year) is
independent of any others. This means that the year
that the flood occurred in becomes irrelevant.
Taking the rank value, the next step is to calculate
the
F
(
X
) term using equation 6.3. In this case
r
refers to the rank of an individual flood event (
X
)
within the data series and
N
is the total number of
data points (i.e. the number of years of record):
1
The probability of excedence:
P
(
X
). This is the
probability that a flow (
Q
) is greater than, or
equal to a value
X
. The probability is normally
expressed as a unitary percentage (i.e. on a scale
between 0 and 1).
2
The relative frequency:
F
(
X
). This is the prob-
ability of the flow (
Q
) being less than a value
X
.
This is also expressed as a unitary percentage.
r
()
=
(6.3)
FX
N
+
1
3
The average recurrence interval:
T
(
X
). This
is sometimes referred to as the return period,
although this is misleading.
T
(
X
) is a statistical
term meaning the chance of excedence once every
T
years over a long record. This should not be
interpreted as meaning that is exactly how many
years are likely between certain size floods.
In applying this formula there are two important
points to note:
1
The value of
F
(
X
) can never reach 1 (i.e. it is
asymptotic towards the value 1).
2
If you rank your data from high to low (i.e. the
other way around) then you will be calculating
the
P
(
X
) value rather than
F
(
X
). This is easily
rectified by using the formula linking the two.
()
=−
()
()
=
PX
1
FX
(6.2)
1
1
TX
=
−
()
()
PX
1
FX
The worked example on pp. 113-114 gives the
F
(
X
),
P
(
X
) and
T
(
X
) for a small catchment in
mid-Wales (Table 6.3).
The Weibull formula is simple to use and
effective but is not always the best description of an
annual maximum series data. Some users suggest
It is possible to read the values of
F
(
X
) from a
cumulative probability curve; this provides the
simplest method of carrying out flood frequency
analyses. One difficulty with using this method is
