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<br />6 <br /> <br />TECHNIQUES OF WATER-RESOURCES INVESTIGATIONS <br /> <br />. <br /> <br />values of 0,. Matalos (1963) describes the math- <br />ematical fitting process without use of a table <br />of frequency factors. The computer program <br />entitled "Revised Flood Statistics" is available <br />for fitting a Pearson Type III curve to data. <br />An example of fitting a Pearson Type III <br />curve by use of computed parameters and a <br />table of frequency factors is given with the <br />example of graphical fitting under the section <br />of that name. <br /> <br />Type I extreme-value distribution <br />(Gumbel) <br /> <br />This is a 2-parameter distribution having a <br />constant skew of 1.139. The parameters are <br /> <br />U=X-YN/O< <br /> <br />and <br /> <br />I/o< =S/UN, <br /> <br />Mean flood is 606,200 cfs=X <br /> <br />Standard deviation, S=..jC2:,X'-NX')/N-l <br />=175,200. <br /> <br />From table 2 for N=89, <br /> <br />YN=.558 <br />UN= 1.20, <br /> <br />then <br /> <br />1/0< = S/UN= 175,200/1.20 = 146,000, <br />U=X-YN/O< <br />=606,200- (0.558) (146,000) =524,700. <br />The equation of the straight line is <br /> <br />and <br /> <br />X=u+Y/o<=524,700+ 146,000y. <br /> <br />The relation y=ln T may be used to define <br />plotting points for large recurrence intervals. <br /> <br />y=ln T=2.303 log T. <br /> <br />where u is the mode, I/o< is a scale parameter, For T=50 years, y=3.91, <br />X and S are the sample mean and standard <br />deviation respectively, and UN and UN are func- and <br /> <br />tions of N, the number of items in the sample. X=524,700+ 146,000(3.91) = 1,096,000 cfs. . <br />Values of UN and UN for N from 8 to 1,000 are <br />tabulated by Gumbel (1958, p. 228). Part of Table 2.-Means and standard deviatians af reduced <br />Gumbel's table is given in table 2. exlremes <br />The mean and standard deviation of the [Em...... from. men. compIe.. table by Gumbel] <br />sample are computed, UN and UN are read from <br />the table, u and I/o< are computed from the <br />above formulos, and the straight line <br /> <br />X=U+Y/o< is determined. <br /> <br />This straight line is plotted on Powell- <br />Gumbel probability paper using the "reduced <br />variate y" scale. The Geological Survey Form <br />9-179a, flood data plot, does not have a re- <br />duced variate scale (but the recurrence interval <br />is related to the reduced variate). On 9-179a <br />plot the mean, X, at the 2.33-year recurrence <br />interval and use the approximate relation <br />y=ln T to locate another point on the straight <br />line. <br />Following is a sample computation for annual <br />floods on Columbia River near The Dalles <br />, <br />Oreg., for 1858-1946. See U.S. Geological Sur- <br />vey Water-Supply Paper 1080, page 337, for <br />data. <br /> <br />N 'N ON <br />10 0.4952 0.9497 <br />15 .5128 1.021 <br />20 .5236 1.063 <br />25 .5309 1.091 <br />30 .5362 1.112 <br />35 .5403 1.128 <br />40 .5436 1.141 <br />45 .5463 1.152 <br />50 .5485 1.161 <br />60 .5521 1.175 <br />70 .5548 1.185 <br />80 .5569 1.194 <br />90 .5586 1.201 <br />100 .5600 1.206 <br />200 .5672 1.236 <br />500 .5724 1.259 <br />1,000 .5745 1.269 <br /> <br />The line is defined on the graph of figure 5 <br />by the points <br /> <br />X =606,200 cfs at 2.33 years <br /> <br />and <br /> <br />1,096,000 cfs at 50 years. <br />The plotted points for the period 1858-1948 . <br />