(3) The above approximation is particularly useful

refers to estimating the uncertainty for ordered

for extrapolating the frequency curve for stages

observations. However, the theory can be extended if

corresponding to the channel floodplain elevation.

an equivalent years of record is available for an

Under these circumstances, the frequency curve is

estimated frequency curve. For example, assume that

relatively flat and the sampling uncertainty in the

the information used to develop a flow-frequency

frequency curve actually decreases with increasing

curve from hypothetical events is worth 20 years of

deviation from the mean stage. The above approx-

equivalent record length. Plotting positions for the

imation captures this effect because the standard

20 years can be calculated and the corresponding *Y*j

deviation is inversely proportional to the local value of

determined from the hypothetical frequency curve.

the probability density function. The behavior of the

The *Y*j can then be used as previously described to

standard deviation in this circumstance is very

calculate confidence limits.

different than that for unregulated streamflow

frequency curves where the standard deviation

increases with distance from the mean flow.

(1) The asymptotic approximation used for the

(4) Although general, the above approximation is

uncertainty distribution differs depending on whether

too conservative for application to hypothetical and

or not stage or flow is the focus. In the case of stage,

regulated frequency curves which are approximately

the uncertainty is determined based on the result that

analytic for extreme probabilities, or for stage-

the binomial distribution is asymptotically normal for

frequency curves which do not have broad overbank

large *n*. Under this assumption, the variance of a

areas. Consequently, as an alternative approach, the

normal uncertainty distribution for quantile *X *is given

order statistic approximation of uncertainty is

as

extended for these flow-frequency curves using the

following asymptotic approximation to the uncertainty

in a normal distribution (Kottegoda 1980, p 233)

2

(5)

2S

2

2

(6)

% *Z*p

'

2*n*

where

where

distribution

(7)

frequency curve of interest

where

Note that the probability density function is the inverse

of the slope of the nonexceedance frequency curve.

(2) A suggested guideline for applying this

asymptotic approximation given by Dixon and Massey

(1957) is that *np *$ 5. Given this criterion, the method

probability *p*

could be applied at most to the 0.25-nonexceedance

probability or about a 2-year event for *n *= 20 years.

Equation 6 is used exclusively for hypothetical or

regulated frequency curves and is used to constrain the

Given this limited range, the order statistic approach

variance estimated by Equation 5.

is used to estimate uncertainty for the range of the

data, and the asymptotic approximation is used to

extrapolate the results beyond the range of data.

4

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