Geoscience Reference
In-Depth Information
which has a lower bound at
x
=
a
. The corresponding density function is
a
1
/
b
b
x
−
1
−
1
/
b
f
(
x
)
=
(13.86)
In the practical application of this function, the parameters
a
and
b
can be derived
simply by least squares linear regression of the logs of the observed values
X
against the
logs of their return periods
T
r
, in accordance with Equation (13.84).
The power distribution has been found useful in the description of numerous phe-
nomena, such as fragmentation, earthquakes, volcanic eruptions, mineral deposits, and
land forms, among others. In hydrology, the power distribution probably found its ear-
liest application in the description of rainfall intensities. Equation (3.3), whose origins
go back at least to the work of Meyer (1917), is in the form of (13.84). (See also
Figure 3.16.) A noteworthy feature in this particular application of Equation (13.84) is
that its coefficient
a
is also a power function of the duration
D
of the rainfall event,
for values of
D
in excess of 2 h. In a different context, namely in the description of
capillary retention of water in soils, the form of Equations (8.14), (8.15) and (8.16), after
substitution of (8.5), suggests a power distribution and fractal features of the smaller
pores; this is illustrated for a sand in the example of Figure 8.20, indicating a straight
line for large values of the capillary suction
H
.
More recently, the power distribution has also been used to describe flow maxima.
Turcotte (1994) and Malamud
et al
. (1996) have presented cases where it provided a
better fit with flood data than the generalized log-gamma distribution. However, the
distribution appears to be more applicable to partial duration flow data than to annual
flow maxima. Partial duration flow series contain all the data above a given pre-defined
base, whereas annual flow series contain only the peak discharge rates observed during
each year of the record. The main disadvantage of an annual series is that in some years
a number of events may be larger than the annual event in other years. In the analysis of
very large events this is rarely a problem, because the two types of data series tend to be
nearly the same for events with return periods in excess of about three time units, years in
this case. Hence, in the estimation of the parameters
a
and
b
in the power distribution for
annual peak flows, it is advisable to use only the data whose return period is larger than
3 y, or whose probability of non-exceedance is larger than 0.67. This is illustrated below
in Example 13.9. The performance of the power distribution, to describe annual peak
flows in comparison with other distributions, is also illustrated Figures 13.9, 13.10, 13.12
and 13.14. It can be seen that it tends to produce smaller values of the non-exceedance
probability and of the return period, and therefore will usually lead to more conservative
design values.
Example 13.9. Power law distribution applied to annual peak flows
The Sheepscot River, at North Whitefield, Maine, drains a basin which is subject to
strong maritime influence. The gaging station is located at 31 m above sea level at
44
◦
13
23
N and 69
◦
35
38
W; the upstream drainage area covers 376 km
2
, and the cor-
rected (Korzoun
et al
., 1977) long term average annual precipitation is around 1300 mm.
