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<br />(1) For-y=O <br /> <br />r= a[(1-c)t+b] <br />(t+b)1 +c <br /> <br />.................... (10) <br /> <br />(2) For -y = 1 <br /> <br />a[(1- c)(td - t) + b] <br />r = 0 :s t :s td.... (11) <br />[(td' t) + b] 1 + c <br />(3) For 0 < -y < 1 <br /> <br />a[(1 . c)(td . t/'Y) + b] <br />r = [(td _ th) + b] 1 + c 0 :s t :s -ytd.. (12) <br /> <br />r= <br /> <br />a[(1.cXt-oyld)/(I--y)+b] <br />[(t --ytd)/(1 - -y)+ b] I +c <br />-ytd:S t:S td....,.... - . . . . . . . . (13) <br /> <br />Hyetograph equations for negative b <br /> <br />In this case, the value of c cannot exceed unity, <br />Moreover, because of the nature of Eq. 9, a small <br />portion of hyetograph for all three types must be <br />given a constant intensity, (a/bC)[(1 - c)/(1 + c)]C in <br />order to avoid the breakdown when t :s b. <br />Hyetograph equations for the three types are derived <br />and listed as follows: <br /> <br />(1) For -y= 0 <br /> <br />a (1 _c~c <br />r = if T"+Z) <br /> <br />t:S 2b .. .. .. .. (14) <br />r:c <br /> <br />a[(1 - c) t . b] <br />(t _ b)l + c <br /> <br />(2) For -y = 1 <br /> <br />2b <br />-..... (15) <br />1 - c <br /> <br />r = <br /> <br />t2: <br /> <br />a[(1 - c)(td - t) - b] <br />[(td - t) - b] 1 + c <br />2b <br />o < t < td - -1 ................. (16) <br />- - -c <br /> <br />r <br /> <br />( )c <br />a 1 - c 2b <br />r = if ~ td - r:c :s t :s td.. (17) <br /> <br />(3) For 0 < -y < 1 <br /> <br />a[(l - c)(td - th) - b] <br />r = <br />[(td-t/'Y).bjl+C <br /> <br />O:S t :s -ytd. .B:r ................. (18) <br />1 - c <br /> <br />r = b~ ( : ~ ~)C <br /> <br />-ytd. 2br:s < 2b(1 --y) (19) <br />1 _ c t - rtd + I _ c . . . . . . . <br /> <br />a[(1-c)(t.rtd)/(1.r)-b] <br />r = ] +c <br />[(t - rtd)/(1 . r) - b] <br /> <br />2b(1 - r) < (20) <br />rtd + 1 _ c - t:S td...""...,... <br /> <br />For examining the validity of Eqs. 10 through <br />20, substituting the equation or equations for each <br />case into Eq, 8 and performing the integration over <br />the respective integration limits as specified gives <br />exactly ravtd' However, for negative b, if Eq. 8 is <br />satisfied, there is an apparent discontinuity in r, for <br />example, at t = 2b/(1 - c) in the case of r = 0 with r = <br />(a/bC)[(1 . c)/(1 + c)] 1 +c obtained by substituting t <br />= 2b/(1 - c) into Eq. 15. For application, the values of <br />the parameters characterizing the hyetograph <br />equations such as a, b, c, td, and 'Y need to be <br />evaluated. <br /> <br />Evaluation of Hyetograph Parameters <br /> <br />PaJametric values, a, b, and c <br /> <br />Taking logarithm of both sides of Eq. 9 yields <br /> <br />lograv=loga-clog(td t b)....,.,.... (21) <br /> <br />Preul and Papadakis (1973) determined the a band <br />c values by plotting data points (raJ, tj) for] ='1, 2, <br />..., n with various assumed values of b on log-log <br />paper until a straight line was established. Because <br />Eq. 21 is linear in log rav and log (t :tb) for a given <br />value of b, the determination of the a, b, and c values <br />can be accomplished in a systematic way by using the <br />method of least squares and an optimization <br />technique similar to the method of steepest descent <br />for optimizing an unconstrained problem. The <br />optimization problem formulated herein is <br />tantamount to the one to fmd the a, b, and c values <br />for minimizing the expression <br /> <br />n ' <br />F(a,b,c) = ~ [log r J -log a <br />j = 1 av <br /> <br />'2 ) <br />+ c10g(tdJ t b)] .......,.,.... (22 <br /> <br />8 <br />