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<br />EXTREME VALUE APPLICATION - HOMOGENEOUS DATA
<br />
<br />Given the need to supplement empirical
<br />fit with theoretical considerations, the
<br />}urpose of this section is to evaluate
<br />extreme value theory as a tool in identifying
<br />a distribution for annual floods. It should
<br />be understood that in all likelihood no
<br />single distribution is correct for all flood
<br />series. For example, rIver basins wIth
<br />large carry-over storage or streams which
<br />flow only intermittently may violate the
<br />assumptions of extreme value theory. In the
<br />first case, flood peaks depend on flows In
<br />the previous year; and in the second, a data
<br />set with large numbers of zero flows IS not
<br />really an extreme value SItuatIon.
<br />
<br />However, if the theory can be shown
<br />to apply in more normal SItuatIons, the
<br />hypotheses of the theory are sufficiently
<br />general to expect it to be widely applicable.
<br />In this sect ion a theoretical distribution
<br />IS selected by matching physical character-
<br />istics of stream flow with the mathematical
<br />characterIstics of the various extreme value
<br />forms. Applicability is examined by trying
<br />to fit the data for selected stations
<br />with long periods of record from around the
<br />world (Table 2) used in the study of Bobee
<br />and Robitaille (1977). (See Appendix H.) The
<br />same measures of goodness-of-fit is used in
<br />order to compare these results with those ob-
<br />tained from the distribution of their study.
<br />
<br />Table 2.
<br />
<br />Extreme Value Distributions
<br />
<br />Before proceedIng, some basic elements
<br />of extreme value theory need to be reviewed.
<br />Extreme value random variables are defined as
<br />follows. Let xI,_x2, ..., Xn be asample
<br />of independent, Identically distrIbuled,
<br />conlinuous random variables. Let
<br />
<br />Zn =:: max(x1.x2.
<br />
<br />x )
<br />n
<br />
<br />(2)
<br />
<br />...,
<br />
<br />and
<br />
<br />Yn min(x1.x2, ..., xn)
<br />
<br />(3)
<br />
<br />Extreme value theory IS concerned with the
<br />asymptotIC distributIon of sequences (Zn
<br />bn}fan and (Yn - bn1)fan' as n = 1,2,....
<br />The norming values an, bn, an', bn' are
<br />dIctated by the theory. The interesting
<br />result of the theory is that if an asymptot ic
<br />distribution eXIsts, there are only three
<br />types for Zn and three types for Yo- The
<br />mathematical characteristics for the random
<br />variables Xl which determine the resulting
<br />dIstributIon tor Zn and Yo are given by
<br />Gnedenko (1943). These results are difficult
<br />to use because the distribution function must
<br />be known. A less mathemat ieal but more
<br />workable approach is suggested here.
<br />
<br />Selected stations exhibiting homogeneous sources.
<br />
<br />Station Country River Location Drainage Record Missing Years Years of
<br /> Area. Km2 Record
<br />bB24 Mali Senegal Bakel 218,000 1903-1966 64
<br />HE60 USA Susquehanna Harrisburg. PA 62,400 1891-1967 1906.1922.1927 70
<br /> 1935,1938,1951
<br />1B06 India Krishna Vij ayawada 251,355 1901-1960 60
<br />BF40 Czech. Decin Elbe 51,104 1851-1968 1857,1863,1866.1873 108
<br /> 1874,1879,1884,1898
<br /> 1918,1921
<br />BE38 Germany Hofkirchen Danube 47,495 1901-1968 68
<br />BF19 Norway Gloma Langnes 40,170 1902-1968 1964 66
<br />CF25 USSR Neman Smalininkai 81.200 1812-1969 1944.1945,1946 155
<br />mE19 Canada Hope Fraser 203,000 1912-1970 59
<br />JE792 Canada Headingley Assinibione 162,000 1914-1970 57
<br />IF 00 Canada Medicine Hat S.Saskatchewan 58,400 1913-1970 58
<br />KF62 Canada Saskatoon S.Saskatchewan 139,500 1912-1970 59
<br />KF53 Canada Prince Albert N. Saskatchewan 119,500 59
<br />hE88a Canada Amos Hurricana 3,680 1915-1969 1932,1933 53
<br />JF50a Canada Slave Falls Winnipeg 126,000 1908-1970 1909,1911-1912,1917 50
<br /> Power Plant 1922-1926,1931,1934
<br /> 1939-1942,1949,1958
<br /> 1961.1962.1964,1965
<br /> 1967
<br />
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