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<br />REGIONAL ANALYSES OF STREAMFLOW CHARACTERISTICS
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<br />equation can be applied to ungaged basins.
<br />Under these conditions the average is likely
<br />to be biased. On the other hand, if the records
<br />are independent and the chance variation is
<br />extremely large, the regression analysis
<br />should produce a good answer, but the quality
<br />of the results may not be recognized because
<br />of the large standard error. Thus the success
<br />of a regionalization procedure by regression
<br />analysis cannot be measured in terms of the
<br />standard error of regression alone.
<br />However, for a given set of data, the regres-
<br />sion equation with the smallest practical
<br />standard error should be used. Improvement
<br />of the regression equation not only reduces the
<br />standard error but reduces the portion of the
<br />standard error that is due to differences in
<br />basin characteristics.
<br />A regional regression having a large stand-
<br />ard error may provide a good answer if most
<br />of that standard error is due to chance varia-
<br />tion. But since we have no way of knowing
<br />how well the regression describes the real dif-
<br />ferences among basins, we usually conclude
<br />that a relation with a large standard error has
<br />much room for improvement.
<br />A very small standard error of regression
<br />indicates little chance variation among the
<br />records used. The practice of reducing the
<br />residual variation to near zero by assigning
<br />various coefficients to subareas of the total
<br />area represented by the regression must be
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<br />based on the assumption that the residual
<br />variation is largely due to unexplained differ-
<br />ences in basin characteristics, and thus that
<br />the chance variation is small. This assumption
<br />does not seem justified. More likely, the major
<br />part of the residual variation is due to chance.
<br />Benson (1962a,b, 1964) discusses and shows
<br />examples of the multiple-regression method of
<br />regional analysis of flood peaks. The following
<br />example outlines the procedure.
<br />Table 1 lists the 2-, 25-, and 50-year floods,
<br />the drainage area, and the mean annual basin
<br />precipitation for gaging stations in Snohom-
<br />ish River basin, Washington (Collings, 1971).
<br />A graphical regression using these data is
<br />shown in figure 1 and the gage sites are shown
<br />in figure 2. See Riggs (1968a) for method of
<br />making a graphical multiple regression.
<br />This graphical step is preliminary and may
<br />be bypassed in an analysis, but it takes little
<br />time and usually clearly indicates the suitabil-
<br />ity (or lack of suitability) of the model to be
<br />used in the mathematical fitting. In figure 1
<br />the plotted points indicate the statistical sig-
<br />nificance of both independent variables.
<br />Standard error of the graphical regression
<br />can be esti mated.
<br />Not all graphical regressions are as clear
<br />cut as that of figure 1. Consequently the re-
<br />gression is usually determined by mathemati-
<br />cal fitting, preferably by digital computer.
<br />The computer program produces the standard
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<br />Table I.-Data from Snohomish River basin, Washington
<br />
<br />Station
<br />
<br />Annual flood peak (cfs)
<br />at indicated recurrence interval (years)
<br />
<br />Drainage
<br />am
<br />(sq mi)
<br />
<br />Mean
<br />annual
<br />precipitation
<br />(in.)
<br />
<br />2
<br />
<br />1330. S. F. Skykomish_ --- ---I
<br />1335. Troublesome__u______ _n
<br />1345. Skykomish_n____._ ______
<br />1350. Wallace______u__________
<br />1375. Sultanuu________________
<br />1410. Woods___________________
<br />1415. M. F. Snoqualmie_________
<br />1420. N. F. Snoqualmie__________
<br />1440. S. F. Snoqualmie_ _ _ __ __ ___
<br />1445. Snoqualmie_ _ _ _ __ _ _ _ _ _ ____
<br />1460. Patterson_ _ _ __ __ __ _ _ __ _ _ __
<br />1407. Griffinnn________________
<br />1475. N. F. ToIL_______________
<br />1480. S. F. Toltm______________
<br />1485. ToIL____________________
<br />1490. Snoqualmie_____n________
<br />1525. Pilchuckn_n_______n__n
<br />1530. L. Pilchuck_____________
<br />
<br />22,600
<br />920
<br />36,100
<br />1,990
<br />16,700
<br />1,210
<br />12,500
<br />7,440
<br />4,190
<br />26,500
<br />201
<br />393
<br />5,000
<br />3,450
<br />7,780
<br />28,200
<br />5,080
<br />281
<br />
<br />.
<br />
<br />25 50
<br />54,400 63,300
<br />2,760 -
<br />87,800 102,000
<br />3,570 4,000
<br />35,200 39,600
<br />2,300 2,580
<br />27,100 -
<br />16,600 19,100
<br />8,080 -
<br />63,500 -
<br />309 -
<br />944 1,120
<br />9,540 -
<br />6,700 -
<br />16,100 17,900
<br />59,400 67,400
<br />9,120 10,200
<br />627 -
<br />
<br />355
<br />10.6
<br />535
<br />19.0
<br />74.5
<br />56.4
<br />169
<br />64.0
<br />81.7
<br />375
<br />15.5
<br />17.1
<br />39.2
<br />19.7
<br />81.4
<br />603
<br />54.5
<br />17.0
<br />
<br />116
<br />176
<br />119
<br />141
<br />151
<br />59
<br />127
<br />139
<br />112
<br />118
<br />47
<br />65
<br />112
<br />123
<br />105
<br />102
<br />114
<br />53
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