Geoscience Reference
In-Depth Information
that a better fit to the data is provided by the
Gringorten formula (equation 6.4):
()
=
−
−
( )
bX a
e
FX
e
(6.5)
r
N
−
+
044
012
.
0 5772
.
()
=
(6.6)
=−
(6.4)
aQ
FX
.
b
π
As illustrated in the worked example, the difference
between these two formulae is not great and often
the use of either one is down to personal preference.
(6.7)
b
=
6
σ
With knowledge of
F
(
X
) you can find
P
(
X
) and the
average recurrence interval (
T
(
X
)) for a certain size
of flow:
X
. The formulae above can be rearranged to
give you the size of flow that might be expected for
a given average recurrence interval (equation 6.8):
Extrapolating beyond your data set
The probabilities derived from the Weibull and
Gringorten formulae give a good description of the
flood frequency within the measured stream record
but do not provide enough data when you need to
extrapolate beyond a known time series. This is a
common hydrological problem: we need to make
an estimate on the size of a flood within an average
recurrence interval of fifty years but only have
twenty-five years of streamflow record. In order to
do this you need to fit a distribution to your data.
There are several different ways of doing this, the
method described here uses the method of moments
based on the Gumbel distribution. Other dis-
tributions that are used by hydrologists include the
Log-Pearson Type III and log normal. The choice of
distribution is often based on personal preference
and regional bias (i.e. the distribution that seems to
fit flow regimes for a particular region).
=−
()
TX
TX
1
(6.8)
Xa
b
lnln
()
−
1
In the formula above ln represents the natural
logarithm. To find the flow for a fifty-year average
recurrence interval you must find the natural log-
arithm of (50/49) and then the natural logarithm of
this result.
Using this method it is possible to find the
resultant flow for a given average recurrence interval
that is beyond the length of your time series. The
further away from the length of your time series you
move the more error is likely to be involved in the
estimate. As a general rule of thumb it is considered
reasonable to extrapolate up to twice the length of
your streamflow record, but you should not go
beyond this.
Method of moments
If you assume that the data fits a Gumbel distri-
bution then you can use the method of moments to
calculate
F
(
X
) values. Moments are statistical
descriptors of a data set. The first moment of a data
set is the mean; the second moment the standard
deviation; the third moment skewness; the fourth
kurtosis. To use the formulae below (equations 6.5-
6.7) you must first find the mean (
Q
¯
)
and standard
deviation
(
σ
Q
) of your annual maximum data series.
The symbol
e
in the equations 6.5-6.7 is the base
number for natural logarithms or the exponential
number (
≈
2.7183).
Low flow frequency analysis
Where frequency analysis is concerned with low
flows rather than floods, the data required are an
annual minimum series. The same problem is found
as for annual maximum series: which annual year to
use when you have to assume that the annual
minimum flows are independent of each other. At
mid-latitudes in the northern hemisphere the
calendar year is the most sensible, as you would
expect the lowest flows to be in the summer months
(i.e. the middle of the year of record). Elsewhere an
