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more rapidly with increasing size of the drainage area as the severity of the flood
increases.
As an aside, if Equation (13.92) is valid, it follows that, unless b T is independent of
T r and a T is a power function of T r , the underlying assumption of the power distribution
(13.82) cannot be generally valid. For example, substitution of (13.92) into (13.82), yields
the following criterion for the validity of the power distribution
a 100
a 10
a 50
a 5
A b 100 b 10
A b 50 b 5
=
(13.93)
This shows that in this case the ratio K 10 in (13.82) depends on the drainage area; in other
words, the equality in (13.93) needed for the validity of the power distribution requires a
certain size A of the basin, so that it is practically never satisfied. For instance, in the case
of the typical values obtained for Maine (see Figure 13.16), for A = 100 km 2 , the left-hand
side of (13.93) is 1.67, and the right-hand side is 1.82; for A = 1000 km 2 , the left-hand side
of (13.93) is 1.54 and the right-hand side is 1.65.
Theoretical distribution functions with regionalized moments
The underlying assumption of this approach is that the moments in a hydrologically homo-
geneous region depend on known or measurable basin and climate characteristics. Thus,
once the moments can be estimated for an ungaged basin within the region on the basis of
these characteristics, it becomes possible to calculate the parameters of the selected prob-
ability distribution function. In principle, because several moments, namely the mean, the
variance and the skew coefficient can be related to basin characteristics, the method is less
restrictive than the index-flood method, which makes use of only the first moment. For two-
parameter distributions the skew is not needed, and for three-parameter distributions, as it
tends to be unreliable, a regional value can be assumed. The method has not been widely
applied. For example, in their application to the Klamath Mountains in northern California,
Cruff and Rantz (1965) found that the sample mean M and the sample standard deviation
S are both related to the catchment area A and to the mean annual basinwide precipitation,
respectively, by power functions similar to Equation (13.91).
REFERENCES
Abramowitz, M. and Stegun, I. A. (editors). (1964). Handbook of Mathematical Functions, Appl. Math.
Ser. 55 . Washington, DC: National Bureau of Standards.
Bailey, J. F., Patterson, J. L. and Paulhus, J. L. H. (1975). Hurricane Agnes rainfall and floods,
June-July 1972 . Geol. Survey Prof. Paper 924, Washington, DC: US Department of the Interior.
Benson, M. A. (1950). Use of historical data in flood-frequency analysis. Trans. Amer. Geophys. Un. ,
31 , 419-424.
(1962a). Evolution of methods for evaluating the occurrence of floods , Geol. Survey Water-Supply
Paper 1580-A. Washington, DC: US Department of the Interior.
(1962b). Factors influencing the occurrence of floods in a humid region of diverse terrain , Geol.
Survey Water-Supply Paper 1580-B. Washington, DC: US Department of the Interior.
(1968). Uniform flood-frequency estimating methods for Federal agencies. Water Resour. Res. , 4 ,
891-908.
Bloschl, G. and Sivapalan, M. (1997). Process controls on regional flood frequency: coefficient of
variation and basin scale. Water Resour. Res. , 33 , 2967-2980.
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