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Last modified
1/26/2010 10:06:41 AM
Creation date
10/5/2006 3:55:55 AM
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Floodplain Documents
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Statewide
Title
Techniques of Water-Resources Investigations of the US Geological Survey Frequency Curves
Date
1/3/1997
Prepared By
USGS
Floodplain - Doc Type
Educational/Technical/Reference Information
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<br />8 <br /> <br />TECHNIQUES OF WATER-RESOURCES INVESTIGATIONS <br /> <br />usually results. It is difficult to define such a <br />curve by the few observations; it is customary, <br />therefore, to use a graph sheet having the ab- <br />scissa graduated in such a way that a particular <br />theoretical frequency curve will plot as a <br />straight line. Such graph sheets are available <br />for the normal, lognormal, and Gumbel Type <br />I distributions. It is possible to prepare such <br />a scale for any two-parameter distribution. <br />Although sets of data of the same type may <br />not appear to lie on straight lines on a parti- <br />cular plotting paper, the lines of good fit usually <br />are only slightly curved in one direction. Such <br />lines may be more confidently defined from <br />the plotted points than sharply curved lines. <br />An additional advantage of the probability <br />graph appears when a straight line is a reason- <br />able interpretation of the plotted points; then <br />the straight line is a frequency curve of the <br />theoretical type on which the plotting paper <br />is based. A discussion of normal-probability <br />paper is given by Dixon and Massey (1957, <br />p. 55-57). It should be clearly understood that <br />a frequency curve is not necessarily normal <br />just because the points are plotted on normal- <br />probability paper (or has a Gumbel distribu- <br />tion because the points are plotted on Gumbel <br />probability paper); only when the frequency <br />curve is a straight line is this true. <br />The mean of a normal distribution corre- <br />sponds to the 0.5 probability or to the 2-year <br />recurrence interval. But a curved line on <br />normal-probability paper represents a skewed <br />distribution whose mean is not at 2-year re- <br />currence interval. The effect of skew on the <br />relation of mean to recurrence interval is easily <br />demonstrated by use of the Gumbel Type I <br />distribution which has a fixed positive skew. <br />As used for flood analyses, the mean occurs at <br />2.33-year recurrence interval. But if the same <br />Gumbel distribution is used to represent the <br />frequency of floods less than, the positions of <br />the mean and median are reversed, and the <br />mean plots at about 1.59 years. This effect is <br />shown by figures 3 and 4. The discharge cor- <br />responding to the 2.33-year recurrence interval <br />as obtained from a curved line on Gumbel <br />probability paper is not the mean. It can, how- <br />ever, be used as a characteristic discharge as <br />could the 2-year value or any other near the <br />central part of the distribution. <br /> <br />. <br /> <br />Example of graphical fitting <br /> <br />The annual discharges for the years 1915-50 <br />inclusive in table 3, column 2, can be used to <br />define a frequency curve. The curve can be <br />cumulated from the high end or from the low <br />end, depending on whether the data are arrayed <br />from the high end or from the low end. Both <br />arrays are given in table 3. <br /> <br />Water <br />y"" <br /> <br />Table 3.-Computation of plotting position <br /> <br />Order num. Plotting <br />Q ber, m: high* position <br />est as No.1 (n+l)!m <br /> <br />19Ui 264 <br />16 374 <br />17 332 <br />18 SOW <br />19 359 <br />1920 333 <br />21 ... <br />22 417 <br />23 346 <br />24 320 <br />1926 271 <br />26 214 <br />27 530 <br />28 304 <br />'" 271 <br />1930 271 <br />31 304 <br />32 400 <br />33 327 <br />34 4Ui <br />1935 402 <br />36 362 <br />37 320 <br />38 272 <br />3D 2.. <br />1940 279 <br />41 303 <br />42 310 <br />43 276 <br /><< 317 <br />l~ 3M <br />.. 38' <br />47 359 <br />48 ... <br />49 406 <br />1960 570 <br /> <br />" <br />11 <br />,. <br />I6 <br />13 <br />18 <br />. <br />, <br />17 <br />21 <br />31 <br />" <br />2 <br />.. <br />.. <br />.. <br />2' <br />. <br />2() <br />. <br />8 <br />12 <br />22 <br />8. <br />" <br />28 <br />2' <br />" <br />'" <br />23 <br />16 <br />10 <br />" <br />. <br />, <br /> <br />1.09 <br />3.37 <br />1.95 <br />2.31 <br />2.86 <br />2.06 <br />11.2 <br />7.40 <br />2.18 <br />1.76 <br />1.19 <br />1.03 <br />lS,li <br />1.48 <br />1.16 <br />1.12 <br />1.48 <br />4.11 <br />1.86 <br />6.16 <br />4.62 <br />3.09 <br />1.68 <br />1.23 <br />1.06 <br />1.32 <br />1.37 <br />1.54 <br />1.28 <br />1.61 <br />2,47 <br />3.70 <br />2.65 <br />.... <br />6.30 <br />87.0 <br /> <br />Order num. <br />ber m: low. <br />est as No. 1 <br /> <br />. <br />2. <br />18 <br />21 <br />" <br />,. <br />" <br />32 <br />2(} <br />" <br />. <br />1 <br />" <br />12 <br />, <br />. <br />11 <br />28 <br />17 <br />31 <br />2ll <br />" <br />16 <br />, <br />2 <br />, <br />10 <br />13 <br />8 <br />" <br />22 <br />2' <br />23 <br />33 <br />.. <br />.. <br /> <br />PloUing <br />position <br />(n+I)/mi <br /> <br />11.2 <br />1.43 <br />2.06 <br />1.16 <br />1.64 <br />1." <br />1.09 <br />1.16 <br />1.86 <br />2.31 <br />6.16 <br />37,0 <br />1.96 <br />3.09 <br />7.40 <br />.." <br />3.37 <br />1.32 <br />2.18 <br />1,19 <br />1.28 <br />1.48 <br />2.47 <br />'.30 <br />18.6 <br />4.11 <br />3.70 <br />2.8.1 <br />'.62 <br />2.", <br />I." <br />1.37 <br />1.61 <br />1.12 <br />1.23 <br />1.03 <br /> <br />. <br /> <br />Arraying 20 or 25 items, that is, arrangmg <br />them in order of magnitude, and assigning <br />order numbers, can be done readily by obser- <br />vation. For a larger number of items, various <br />schemes may be used. One method is to write <br />each item and its year of occurrence on a card, <br />then arrange the cards in order of magnitude, <br />number the cards, and transfer the order num- <br />bers to the table of items. Another method <br />utilizes transparent plastic strips, one for each <br />period of record used. Each strip has a cali- <br />brated length equal to the abscissa scale on <br />Geological Survey Form 9-179a or 9-179b. On <br />the strip are marked the plotting positions for <br /> <br />. <br />
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