2

formulas have been proposed in the past. However,

estimating the expected annual damage presumes that

(9)

the expected value of the frequency curve (the

% (*Y*i&*M*)2)

expected probability estimate) is preserved in the

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

function.

(3) The standard deviation computed in step (2) is

set equal to that of the normal distribution to compute

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

(1) Compute the order statistic confidence limits

deviation is equated to that computed in step (2). Let

for the computed plotting positions corresponding to

the appropriate record length. A natural selection of

corresponding maximum variances *S*m and *S*l 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-

hypothetical or regulated frequency curves,

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 *Y*m and *Y*l .

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

for an equivalent record length for extrapolation. For

uncertainty distributions with exceedance probabilities

quantile values greater than *Y*m this becomes

defined minimally between 0.9 and 0.1. The mean

and standard deviation for the uncertainty distribution

(10)

computed with Equation 2 are computed based on a

2

probability weighted basis using trapezoidal rule

integration as

where

(8)

5