Geoscience Reference
In-Depth Information
13.4.4
The generalized log-gamma distribution
A random variable is said to be described by the log-gamma distribution, also called the
log-Pearson Type III distribution, when its logarithms obey a three-parameter gamma
distribution. A common way (see Benson, 1968) of determining the probability with this
distribution consists of the application of Table 13.2 with the first three moments of the
logarithms of the data, as calculated with Equation (13.13).
In the late 1960s this distribution was recommended by a Federal interagency group for
adoption by all government agencies in the United States for flood frequency analysis;
this recommendation was arrived at (Benson, 1968; Thomas, 1985) to foster greater
consistency and uniformity in planning for flood-plain management and water-resources
development. It is now still widely used in the United States for this purpose. Further
details and recommendations by this interagency group for the practical application of
this distribution can be found in Bulletin 17B (Interagency Advisory Committee on
Water Data, 1982). Beside the standard application of the method, Bulletin 17B also
contains suggestions on plotting historical data (i.e. dating from prior to the period
of record), and regionalization with data from hydrologically similar watersheds. A
more comprehensive treatment of this distribution was presented by Bobee and Ashkar
(1991).
As noted earlier, the skew coefficient tends to exhibit greater variability between
samples than the mean and the variance. This may be overcome by regionalization
(see Section 13.5.2 below). Several techniques have been used in the past to obtain a
regional value, in place of the locally calculated value, if the data record is short. These
include the construction of a map with iso-lines obtained by interpolation of the values
computed at the existing gaging stations in the region; another possibility is the derivation
of a regression relationship between the available skew values in the region and basin
characteristics; finally, as a third possible approach, the skew may simply be taken as
the average of all available skew values from the records in the region with long records;
the average can also be weighted by multiplying each available value by the number of
years of record at that gaging station divided by the average number of years of record
of all stations in the region. Hardison (1974) has presented regional values of the skew
coefficient for the annual peak flow rates in rivers in the United States, but with the
availability of additional data since then, these results have gradually become obsolete.
Tasker and Stedinger (1986) have further improved the estimation procedure of regional
skew values.
Example 13.7. Log-gamma distribution applied to annual peak flows
In Council Creek, near Stillwater, Oklahoma, the flowrates have been measured from
1934 until 1993, but some peak flow information is also available for 1912; the data have
been published by the US Geological Survey (see also http://waterdata.usgs.gov/nwis).
The station is located at an elevation of 252 m above sea level, at 36
◦
06
58
N and
96
◦
52
03 W in the Prairie region of North America and has a drainage area of 80.3 km
2
;
the corrected average annual precipitation was estimated (Korzoun
et al
., 1977) to be
