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<br />kFx
<br />r =- ......, ,.,.... (5)
<br />av tl
<br />
<br />in which the exponent, n, need not vary with
<br />frequency, but only vary with geographical location,
<br />as may the exponent, x.
<br />
<br />A more general formula, which incorporates the
<br />characteristics of a hyperbolic-type formula, Eq. 1,
<br />with those of a parabolic-type formula, Eq. 3, was
<br />first developed by Sherman (1931). It had the form:
<br />
<br />a
<br />r = . . . . . . , . . . . . , (6)
<br />av (t +bjD
<br />. d
<br />
<br />in which a, b, and c are parameters, the values of
<br />which depend on meteorological localities. The value
<br />of a may be evaluated by using the same expression as
<br />Cin Eq_ 4.
<br />
<br />Another formula similar to Eq. 6 was proposed
<br />by Kiefer and Chu (1957).
<br />
<br />a
<br />r= .............(7)
<br />av (tdc+b)
<br />
<br />From Eqs. 6 and 7, two different types of design
<br />storm patterns could be formulated. There seems no
<br />apparent advantage of using one equation over the
<br />other as far as the derivation of the synthetic
<br />hyetograph equation i. concerned. However, there is
<br />certainly a slight advantage of using Eq. 6 over Eq. 7
<br />in the case of computing a, b, and c values because
<br />Eq. 6, but not Eq. 7, can readily be transformed into
<br />a linear form in unknowns on log-log scale.
<br />
<br />Hathaway's (1945) study of Yarnell's (1935)
<br />rainfall intensity-frequency data for a large number of
<br />precipitation stations indicated that there were fairly
<br />consistent relationships between the average intensity
<br />of rainfall for a period of 1 hour and the average rates
<br />of comparable frequency for shorter and longer
<br />intervals, regardless of the geographical location of
<br />the stations or frequency of 1 hoUr rainfall. These
<br />relations were referred to as the Ustandard" inten-
<br />sity-duration curves (Williams, 1950), the use of
<br />which might eliminate the need for the formulas
<br />mentioned above. The similarity between standard
<br />intensity-duration curves for various localities was
<br />explained by Williams (1950) as the fact that high-
<br />intensity short-duration storms from which most of
<br />the data were obtained are of the convective type,
<br />accompanied by electrical phenomena. Since the
<br />physical laws governing the rainfall-producing charac.
<br />teristics of such storms afe the same everywhere, the
<br />only variable is the frequency with which various
<br />intensities occur.
<br />
<br />Design Storm Temporal Patterns
<br />
<br />The customary rainfall-duration curve or for-
<br />mula for any specific frequency gives only the average
<br />rainfall intensity for any particular duration and
<br />applies no information as to how the intensity varies
<br />from the average during the duration period. How.
<br />ever, the rainfall intensity practically always does
<br />vary in nature. Nature never allows a rain to fall at
<br />one rate for the whole duration. The importance of
<br />the role played by the variation in the rainfall
<br />intensity in the design of urban drainage facility was
<br />recognized by Bernard as early as in 1936 in his
<br />discussion of Horner's (1936) paper. SOImd design
<br />demands the assumption of limiting conditions, and
<br />therefore, rainfall is arranged in critical order. This
<br />special temporal arrangement of rainfall intensities
<br />tends to modify the conception of the storm as
<br />having a desired frequency, Breihan (1940) deter-
<br />mined the probable rates at which the rainfall
<br />actually occurred.
<br />
<br />Hicks (1944) derived the storm temporal
<br />pattern by plotting the major storms of the Los
<br />Angeles station of the U.s, Weather Bureau as mass
<br />curves with the center of the most intense S~minute
<br />duration at a common point (90-minute elapsed time
<br />for design purposes), The result was a storm pattern
<br />between "medium" and "delayed" types.
<br />
<br />Kiefer and Chu (1957) developed a method for
<br />determining a storm temporal pattern based on the
<br />rainfall intensity.duration-frequency formula, Eq. 7.
<br />They applied the method to a Chicago urban area for
<br />storm sewer design. Later the method using Eq. 6 was
<br />applied by Bandyopadhyay (1972) and Preul and
<br />Papadakis (1973) to an urban drainage area in Guhati,
<br />India, and Cincinnati, Ohio, respectively. In all of
<br />these previous investigations, different time-
<br />coordinate systems were utilized for the description
<br />of temporal patterns before and after the peak of a
<br />storm. From the engineering point of view, however,
<br />expressions of the storm pattern equations in two
<br />different time-coordinate systems present a big prob-
<br />lem because they cannot be readily solved on an
<br />electronic computer.
<br />
<br />Three types of synthetic storm patterns could
<br />be formulated from the rainfall intensity-duration-
<br />frequency formula. The storm patterns were called
<br />(1) a completely advanced (initial burst) type; (2) a
<br />completely delayed (fmal burst) type; (3) an inter-
<br />mediate type. In application, the evaluation of the
<br />storm parameters such as a, b, and c in Eq. 6 or 7 as
<br />well as the rainfall duration, td, and the skewness, "y,
<br />of the time distribution appearing in the pattern
<br />equation becomes a major task.
<br />
<br />4
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