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<br />Conditional Probability Adjustment <br /> <br />After the peak-streamflow frequency curve parameters have been determined, the historically- <br />weighted frequency curve can be tabulated. If no low outliers, zero flows, or below-gage-base peaks are <br />present, this process is simply a matter of looking up the ~earson Type III standardized ordinates, kg, p for <br />the desired skew coefficient (g) and probability (p) and cqmputing the logarithmic frequency curve <br />ordinates by the formula: <br /> <br />10gQp = 1\1 + S ~kg, p) <br /> <br />(14) <br /> <br />When peaks below the flood base are present, how~ver, the above calculation determines a <br />conditional frequency curve Q describing only those peaks above the base. To account for the fraction of <br />the population below the flood base, the following argumeht is used: the probability that an annual peak will <br />, , <br />exceed a streamflow x (above the flood base) is the produft of the probability that the peak will exceed the <br />base at all, times the conditional probability that it will ex~eed x, given that it exceeds the base, The first of <br />these factors is just the probability PO; the second factor!s the probability on the conditional frequency <br />curve at streamflow x. Thus the unconditional curve, Q* ; assigns a probability Po (p) to the streamflow <br />having probability p on the above-base curve, Conversely"an exceedance probability p on the unconditional <br />curve Q* corresponds to the probability piP 0 on the original above-base curve Q. Thus the ordinates of <br />the unconditional curve can be computed directly by the formula: <br /> <br />10gQ*p = 1\1+S(k~,(p/po)) <br /> <br />1\1, S , and G are the logarithmic mean, standard deviation and skew coefficient of the above-base <br /> <br />distribution, <br /> <br />(15) <br /> <br />Because this distribution does not have the PearsoniType III shape, it is only used as an intermediate <br />step in constructing an equivalent Pearson Type III curve. first, the three points Q* 0.50, Q* 0.10 and Q* 0.01 <br />are computed using the above formula, Then a logarithmic Pearson Type III curve is passed through these <br />three points; its mean, standard deviation, and skew coef*cient, 1\1', S', and G', are found by solving the <br />three simultaneous equations: <br /> <br />1\1' + S'(k- ) = 10gQ*p (forp' = 0.50, 0.10, and 0.01) <br />G',p <br /> <br />(16) <br /> <br />An exact solution requires a laborious interpolation in th~ Pearson Type III tabl:s; ~e Bulle:in 17B <br />guidelines present a direct formula based on a linear app~oximation. Note that M', S', and G' represent <br />the contributions of all the observed peaks, those below the base as well as those above, whereas 1\1 , S , <br />and G did not. The resulting unconditional frequency C\1fve, when floods below the base have been <br />detected, then is: <br /> <br />10gQp = 1\1' + S'(k-, ) <br />, G ,p <br /> <br />(17) <br /> <br />PEAKFQ <br /> <br />9 <br /> <br />DRAFT. 1/30/98 <br />