interpolation. The recommendation herein is to

determine the uncertainty in this region based on a

The inverse of the slope of the estimated nonexceed-

weighted average of the uncertainty calculated for the

ance frequency curve is used to compute the density

largest observed event and smallest hypothetical event.

function. Correspondingly, the order statistic and

The weight is inversely proportional to the distance

asymptotic approximation estimates are matched for

from the respective events plotted on probability paper.

quantiles less than *Y*l as

needs to be constrained by the characteristics of the

field conditions. This is also true of uncertainty distri-

(11)

2

butions. Typical situations where the field conditions

need to be considered are in applications to regulated

frequency curves. Regulated frequency curves should

Equation 5 can now be used to extrapolate the

approach the inflow frequency curves as the effect of

estimates of uncertainty using the computed values of

the reservoir diminishes. Certainly, the uncertainty

distributions should also become equivalent as these

equivalent record lengths from Equation 6 for hypo-

two frequency curves become equivalent.

thetical or regulated frequency curves.

distribution for a stage-frequency curve on the

Sacramento River will be used to demonstrate the

(1) Estimation of frequency curves involves

procedure outlined previously. The frequency curve

various kinds of information. Ideally, a sufficient

developed for the location is shown in Figure 2 from

number of gauged observations are available to

data listed in Table 2. Also shown in Table 2 is the

estimate the frequency curve. This is rarely the case.

mean and standard deviation for the frequency curve

If gauged observations are not available, rainfall-

which was computed using trapezoidal rule

runoff analysis can be used to develop a synthetic

frequency curve. The equivalent years of record are

If one assumes that the frequency curve was estimated

determined for this frequency curve from Table 1

using information that is worth 20 years of record,

reproduced here from Table 4-5 of EM 1110-2-1619.

ordered values are obtained from the frequency curve

The order statistic methodology can be used to esti-

using the Weibull plotting position as shown in

mate the uncertainty in the frequency curve given the

Table 3. The uncertainty distribution will be

equivalent years of record as described in paragraph

computed by following the steps outlined in paragraph

1-5*d*.

1-6*a*.

(2) In some cases, a mixture of gauged observa-

(1) Step 1: The uncertainty distribution for each

tions and hypothetical events is used to estimate the

ordered value was computed using the incomplete beta

frequency curve. The gauge record length is only

function via Equation 2.

sufficient to estimate the frequency curve for relatively

high-frequency events. The hypothetical events are

(2) Step 2: The mean and standard deviation of

used to extend the frequency curve. The confidence

the uncertainty distribution at each ordered value was

and corresponding uncertainty in the frequency curve

computed via Equations 6 and 7 and are shown in

differs for the region of the curve defined by the

Table 4. Note again that these moments are only

observed or hypothetical events.

computed for uncertainty distributions that can be

defined for exceedance probabilities minimally

(3) In this case, the uncertainty in the region of the

ranging from 0.1 to 0.9.

frequency curve determined by the observed events is

calculated using the observed record length. Likewise,

(3) Step 3: Inspection of Table 4 indicates that the

the uncertainty in the region influenced by the hypo-

maximum standard deviations are *S*l = 2.35 and *S*m

thetical events is determined by an appropriate equiv-

= 1.54 corresponding to stages *Y*l = 12.33 and *Y*m

alent years of record. A transition region exists where

= 21.29. The standard deviation of the normal distri-

the frequency curve is defined between the largest

bution approximation to the uncertainty distribution

observed event and the smallest hypothetical event by

6

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