ETL 1110-2-537
31 Oct 97
f. Plotting position. Numerous plotting position
n
S ' E 0.5((Yi&1&M)2
2
formulas have been proposed in the past. However,
i'2
estimating the expected annual damage presumes that
(9)
IBp(i,n&i%1)&IBp(i&1,n&i)
the expected value of the frequency curve (the
% (Yi&M)2)
IBp(n,1) & IB(1,n)
process. Consequently, the Weibull plotting position
is preferred, because it is the expected estimate of the
where M and S are estimates of the sample mean and
exceedance or nonexceedance probability.
standard deviation of the uncertainty distribution
computed for any quantile using the incomplete beta
1-6. Application to Estimated Frequency
function.
Curves
a. Methodology. The order statistic approach
(3) The standard deviation computed in step (2) is
provides estimates of uncertainty for a limited range.
uncertainty in the frequency curve. In other words, the
An approach is suggested herein for utilizing the order
standard deviation of the distributions computed with
statistic estimates to obtain the uncertainty distribution
the incomplete beta function is set equal to that of the
for the range of frequency curve probabilities of
normal distribution to obtain an approximate uncer-
interest. The approach taken is to fit a normal distri-
tainty distribution. A limitation might be placed on
bution to the confidence limits obtained with the order
the use of the standard deviations computed in the
statistic approach. The normal distribution was
previous step in estimating the equivalent normal
selected because it matches the asymptotic approxi-
distribution. In general, a reasonable expectation is
mation used to extrapolate uncertainty estimates.
that the uncertainty should be nondecreasing once a
Furthermore, the normal distribution is convenient for
maximum value has been reached. Two of these
use with a Monte Carlo simulation. The general steps
maxima will occur near the extreme ends of the
involved in estimating the equivalent normal distri-
plotted points. Consequently, the maximum values of
bution are as follows:
the standard deviation computed should define the
range where the normal distribution standard
deviation is equated to that computed in step (2). Let
for the computed plotting positions corresponding to
Ym and Yl be the ordered observations where the
the appropriate record length. A natural selection of
corresponding maximum variances Sm and Sl have
the location and number of points would be at the
been determined.
ordered observations.
(4) The calculation of the normal approximation
(2) Calculate the mean and standard deviation of
in step (3) is only useful for computing the
the uncertainty distributions developed from the
uncertainty distribution for the range of quantiles (e.g.,
incomplete beta function at each of the points selected
stages or flows) used in step (2). The asymptotic
in step (1). The calculation is approximate because a
approximations for either stage, Equation 5, or
full range of probabilities for the uncertainty distri-
bution is not obtained from the order statistic
Equation 6, are matched at this point to the order
approach. The calculation of uncertainty is limited by
statistic estimates by equating variances at Ym and Yl .
the range of observations or record length as was
For example, consider equating variance for a stage-
described previously (Figure 1). Consequently, the
frequency curve. This is done by solving Equation 5
mean and standard deviation are computed only for
uncertainty distributions with exceedance probabilities
quantile values greater than Ym this becomes
defined minimally between 0.9 and 0.1. The mean
and standard deviation for the uncertainty distribution
p(1&p)
nm '
(10)
computed with Equation 2 are computed based on a
2
Sm fY ( Ym )2
where
IBp( i, n&i%1 ) & IBp ( i&1, n&i ))
n
M ' E 0.5 (Y(i&1) % Yi)
(8)
p = exceedance probability for Ym
IBp ( n, 1 ) & IBp ( 1, n )
i'2
5