ETL 1110-2-537
31 Oct 97
d. Equivalent years of record. The discussion
(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 Yj
deviation is inversely proportional to the local value of
determined from the hypothetical frequency curve.
the probability density function. The behavior of the
The Yj 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
e. Asymptotic approximations.
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)
p(1&p)
2
SX '
(5)
n fX ( x )2
S2
2S
2
2
(6)
Sx
% Zp
'
n
2n
where
where
SX = standard deviation of the uncertainty
distribution
S = standard deviation of frequency curve
p = exceedance probability
Zp = normalized deviate computed as
n = record length
X&M
Zp '
(7)
fX(x) = probability density function for the
S
frequency curve of interest
where
Note that the probability density function is the inverse
of the slope of the nonexceedance frequency curve.
M = mean of the frequency curve
(2) A suggested guideline for applying this
X = flow corresponding to exceedance
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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