ETL 1110-2-537
31 Oct 97
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 Yl as
needs to be constrained by the characteristics of the
p(1&p)
nl '
field conditions. This is also true of uncertainty distri-
(11)
2
Sl fY ( Yl )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
nm and nl . A similar procedure is used to obtain
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.
d. Example 1. The estimation of the uncertainty
b. Scenarios.
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
1-5d.
1-6a.
(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 Sl = 2.35 and Sm
thetical events is determined by an appropriate equiv-
= 1.54 corresponding to stages Yl = 12.33 and Ym
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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