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<br />PATTERN SKEWNESS IN ACTUAL STORMS
<br />
<br />The computation of the storm pattern skew-
<br />ness, 'Y, can proceed by means of either Eqs. 12 and
<br />13 (for positive b) or Eqs, 18, 19, and 20 (for nega-
<br />tive b) with the help of an optimization technique,
<br />However, the a, b, and c values in these equations for
<br />each actual hyetograph to be analyzed must be deter-
<br />mined first before the 'Y value can be computed. A
<br />computation procedure similar to the one used in the
<br />formulation of the rate-duration relationship (Eq, 6)
<br />can be set up for each hyetograph to determine the a,
<br />b, and c values.
<br />
<br />The values of a, b, and c for an actual
<br />hyetograph are readily determined by first arranging
<br />the hyetograph in the order of intensity in a way
<br />similar to the formulation of Eq. 6 and then
<br />computing the a, b, and c values by means of the least
<br />squares of the expression shown in Eq, 22. On
<br />substitution of the a, b, and c values just obtained
<br />from Eq. 22 into Eqs. 12 and 13 (for positive b) or
<br />Eqs. 18, 19, and 20 (for negative b), the -y value is
<br />determined by minimizing the following expression.
<br />For positive b,
<br />
<br />n-y
<br />F (-y) = 1;
<br />j=l
<br />
<br />[. a[O - c)(td - tj/-y) + b1] 2
<br />rL ((td-t/-y)+bll+c
<br />
<br />n
<br />+ 1;
<br />j=n-y+l
<br />
<br />[. a[o-c)(tL-ytd)!(I--Y)+b1] 2
<br />rL . I +c
<br />[(tL-ytd)!(1--y)+bl
<br />
<br />...............,....... (61)
<br />
<br />and for negative b,
<br />
<br />F(-y)=
<br />
<br />n-yl
<br />1;
<br />j=1
<br />
<br />[, a[(I-C)(td-tj/-Y)-b1] 2
<br />rL 1+
<br />[(td- t/-y)- bl c
<br />
<br />n-y2
<br />+ 1;
<br />j= n-yl+ 1
<br />
<br />[rj _!..- 6~)C] 2
<br />bC \1 + c
<br />
<br />n
<br />+, 1;
<br />J=n-y2 + 1
<br />
<br />[, a[(.l - c)(tj - -ytd)!(1 - -y) - b]] 2
<br />rJ -, 1 + c
<br />[(tL 1td)!(1 -1) - bl
<br />
<br />'" "'"...,.",...",. (62)
<br />
<br />in which I1y is the number of measured data points
<br />before the peak in the case of positive b; n is the total
<br />number of measured data points within td; and fl..y].
<br />and ~ are respectively the numbers of measured
<br />data pomts before and after the constant rate around
<br />the peak zone as postulated in Eq, 19 for negative b,
<br />An optimization technique similar to that for
<br />minimizing the objective function in Eq, 22 can be
<br />used to determine the 1 value, Note that in the
<br />optimization process the numbers of measured data
<br />points before and after the peak, I1y for positive b
<br />(and n11 and n12 for negative b), vary depending on
<br />the location of the peak assumed in the hyetograph,
<br />11 is expected that the best-fitted hyetograph does
<br />not necessarily have the theoretical peak fall within
<br />the duration of the highest intensity in the actual
<br />hyetograph,
<br />
<br />The optimization technique described above
<br />was developed primarily for evaluating the pattern
<br />skewness ( -y value) in actual storms. In application of
<br />the preceding method, however, one must be aware
<br />of all the assumptions made in the optimization
<br />process. The most questionable approach in the
<br />method is, of course, related to the suitability of the
<br />equations and optimization criterion developed in
<br />order for the synthetic hyetograph to best fit the
<br />recorded hyetograph. For example, if the actual
<br />storm under study is double- or triple-peaked or,
<br />sometimes even more complicated, multiple-peaked,
<br />the hyetograph equations (1.e" Eqs. 12 and 13 for
<br />positive band Eqs. 18 through 20 for negative b)
<br />which were derived based on the assumption of a
<br />single-peak storm do not seem to be accurate enough
<br />to describe the actual hyetographs, as will be seen
<br />later from given examples. The numbers of measured
<br />data points such as n, n1 ' nn ' and nn in Eqs. 55
<br />and 56 could also become another source of errors.
<br />Since the accuracy of the result depends greatly on a
<br />number of data points used in the curve-fitting
<br />process, as a general rule in this simplified
<br />"univariaten optimization technique, the more data
<br />
<br />29
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