The Hydrologic Frequency Analysis Work Group is a work group of the
Hydrology Subcommittee of the Advisory Committee on Water Information
(ACWI). The Terms of Reference of this work group were approved by the
Hydrology Subcommittee on October 12, 1999 and are available on the ACWI
web page. The work group was formed to provide guidance on issues
related to hydrologic frequency analysis and replaced the Bulletin 17B
Work Group that had existed since 1989. The Hydrologic Frequency
Analysis Work Group is open to individuals from public and private
organizations. The current members of the work group are also given on
the ACWI web page. The initial objectives of the work group are to
 Develop a set of frequently asked questions and answers on the use
of Bulletin 17B guidelines,
 Prepare a position paper that provides guidance on determining the
most appropriate methodology for flood frequency
analysis for ungaged watersheds, and
 Prepare a position paper on methodologies for flood frequency
analysis for gaged streams whose upstream flows are
regulated by detention structures.
In response to the second objective above, the work group has
prepared a paper that provides some guidance on evaluating flood
frequency estimates for ungaged watersheds. This paper, entitled
"Evaluation of Flood Frequency Estimates for Ungaged Watersheds" is
provided below for informational purposes. This is not a guideline or
standard but a possible approach for evaluating the reasonableness of
flood frequency estimates for ungaged watersheds.
Any comments on this paper should be provided by email to Will
Thomas, Vice Chairman of the Hydrologic Frequency Analysis Work Group, at wthomas@mbakercorp.com.
Evaluation of Flood Frequency Estimates for Ungaged Watersheds
Wilbert O. Thomas, Jr.
Michael Baker, Jr.
Michael M. Grimm
Federal Emergency Management Agency
Richard H. McCuen
University of Maryland
Introduction
Two approaches for estimating the magnitude and frequency of flood
discharges for ungaged watersheds are those methods based on statistical
(regression) analysis of data collected at gaging stations and
deterministic rainfallrunoff models that use rainfall input and
algorithms to convert rainfall excess to flood discharges. Flood
Insurance Guidelines and Specifications for Study Contractors (Federal
Emergency Management Agency (FEMA, 1995) is an example of guidelines
that describe the use of both regional regression equations and
rainfallrunoff models for estimating flood discharges for flood
insurance studies and map revisions. FEMA recommends the use of the
most recent regional regression equations published by the
U. S. Geological Survey (USGS), if these equations are applicable for
the studied streams. Where regional regression equations are not
applicable due to flow regulation, flood detention storage, rapid
watershed development, or other unique basin characteristics, FEMA
recommends the use of a rainfallrunoff model.
This paper describes an approach for evaluating flood discharges from
regression equations and rainfallrunoff models and judging the
reasonableness of discharges using a measure of uncertainty such as the
standard error. Example cases from flood insurance studies are
described to illustrate how uncertainty is used in the selection of a
final discharge estimate. This approach could be used for other
analyses such as the design of bridges and culverts for ungaged
watersheds where frequency estimates are available from both regression
equations and rainfallrunoff models.
Objective of Review Procedures
The estimation of flood discharges for floodplain management is just
one example of the need for flood frequency analyses for ungaged
watersheds. In an effort to expedite the processing of flood insurance
studies, FEMA recommends the review of flood discharges prior to their
use in hydraulic and mapping analyses. The 1percent annual chance
(base flood) discharge is used by FEMA to define the Special Flood
Hazard Areas, those areas inundated by the base flood, on Flood
Insurance Rate Maps (FIRMs). The intent of the hydrologic review is to
obtain agreement on the base flood discharge prior to the hydraulic
analysis to avoid revisions to the hydraulic and mapping analyses
because of subsequent hydrologic revisions. This approach to review is
similar to analyses for design of hydraulic structures (bridges,
culverts, levees, dams, etc.) where the hydrologic analysis is completed
and reviewed prior to the design and construction of the hydraulic
structure.
Often flood frequency estimates are available from previous studies
or are developed for the ongoing study by different methods for
comparison purposes. The objective of the following review procedures
is to determine which estimates are reasonable and can be used in
floodplain management. These procedures are intended as general
technical guidance for judging the reasonableness of flood discharges
and not as a set of rules to be followed strictly. The review
procedures provide a framework for evaluating flood discharges and
provide some quantitative guidance for selecting a flood estimate.
Engineering judgement is needed in the application of these review
procedures as this is not a "cookbook" approach. There is no intent to
imply that the examples provided are inclusive of all situations or that
other input data, such as rainfall data, should not be investigated.
The intent of the review procedures is not to identify a best method
for a region or best method under given watershed conditions but to
identify a reasonable estimate for a given application. It is assumed
that different methods can provide reasonable estimates and that no one
method is universally superior.
Most hydrologic analyses have the potential for being used for flood
insurance studies even though they were undertaken for other purposes.
For example, hydrologic analyses performed for the design of a new
bridge or culvert are often later submitted to FEMA by the State
Department of Transportation to revise the FIRM through the Letter of
Map Revision process. Hydrologic analyses for flood insurance studies
are used to illustrate the review procedures.
Description of Hydrologic Methods
The review procedures primarily involve the comparison of flood
frequency estimates from rainfallrunoff models to those from regression
equations and gaging station data. Rainfallrunoff models used for
design purposes and floodplain management are usually based on a
singleevent design storm with the assumption that the rainfall
frequency equals runoff frequency. This approach also assumes that the
design rainfall events have uniform spatial distribution over the
watershed and a specified temporal distribution. These models are
typically not calibrated to observed flood data. Given these
assumptions and those involving antecedent moisture and infiltration
rates, there is uncertainty in characterizing the frequency of flood
discharges from rainfallrunoff models based on the designevent
approach.
However, estimates from other methods such as continuous simulation
models are sometimes used in design of hydraulic structures and in
floodplain management (Brown and Steffen, 1997). Continuoussimulation
rainfallrunoff models account for changes in soil moisture between
storm events and use historical rainfall and other climatic data to
estimate peak flows. Frequency analyses are then performed on the
simulated peak flows to determine design discharges such as the
1percent chance flood. These models are often assumed to be more
accurate than other methods for estimating the magnitude and frequency
of design flood discharges because they are calibrated to observed data,
estimate antecedent moisture conditions from observed data, and rely on
the temporal and spatial distributions of historical rainfall. However,
the flood data used for calibration often lack a major flood. Thomas
(1987) has shown that frequency curves generated from a continuous
rainfallrunoff model used by USGS (Dawdy and others, 1972) for
extending flood records on small watersheds tend to exhibit less
variance than frequency curves based on observed flood data. Conclusive
evidence of greater accuracy of continuous simulation models,
particularly for extreme floods, has not been reported. Additionally,
continuoussimulation models are not commonly used because of their
significant data requirements and the time and effort involved in their
calibration. However, it should be noted that recent improvements in
development of data bases and GIS technology enable the user to apply
continuous simulation models with considerable less effort than in the
past.
Regression equations are developed by relating flood discharges at
gaging stations to watershed and climatic characteristics using
leastsquares regression techniques. If the regression equations are
applicable to a given stream, then reasonable estimates of the magnitude
and frequency of flood discharges should be obtained. A pilot test was
conducted to compare procedures for estimating flood discharges for
natural watersheds in the Midwest and Northwest USA (U.S. Water
Resources Council (USWRC, 1981)). Analyses for these two regions
indicated that regression equations provided more unbiased and
reproducible estimates of flood discharges than rainfallrunoff models
such as HEC1 and TR20. Even though regression equations are
calibrated to gaging station data, they may provide biased flood
estimates if they are based on outdated gaging station data or do not
include explanatory variables unique to the watershed of interest.
Accuracy of Discharge Estimates
In addition to the flood estimates themselves, the accuracy or
uncertainty of the estimates is considered in making decisions about
reasonable estimates. Uncertainty exists in all methods and, therefore,
it is advisable to compare all estimates and use the accuracy of each
estimate in deciding the best discharge estimate to use. When comparing
discharge estimates computed using different methods (e.g.,
rainfallrunoff models and regional regression equations), the various
estimates are considered reasonable if they are within a predefined
error band. The standard error is recommended as a predefined error band
for judging the reasonableness of flood discharges since this measure of
uncertainty is easy to compute, is frequently used, is often reported in
the literature and is better understood by engineers and
hydrologists.
The standard error of flood discharges from gaging station data can
be determined using procedures described by Kite (1988). The standard
error of gaging station estimates can also be estimated using 84percent
onesided confidence limits as described in Bulletin 17B (IACWD, 1982).
The approach by Kite (1988) is favored since this approach considers the
uncertainty in the skew coefficient while the Bulletin 17B approach does
not. The standard errors of estimate or prediction of the USGS
regression equations are given in regional flood frequency reports
(e.g., Dillow, 1996).
The standard error of rainfallrunoff model estimates is not usually
known, although the USWRC report (1981) suggested that it is larger then
the standard error of regression estimates. This is due, in part, to
the fact that rainfallrunoff models based on a singleevent design
storm are not usually calibrated to regional data. Confidence limits or
standard errors of flood discharges from rainfallrunoff models can be
estimated if an equivalent years of record is assumed for the flood
discharges as described by the USACE (1996) as part of riskbased
analyses. However, there is no established practice of estimating the
uncertainty of flood estimates from rainfallrunoff models by this or
any other procedure.
The standard error of the flood discharge is not the only factor in
determining significant differences for floodplain mapping. The change
in elevation of the base flood is also very important as discussed in
FEMA 37 (FEMA, 1995). Standard errors of flood discharges from
regression equations and limited gaging station data often exceed 40
percent. Since flood depths are proportional to the approximate square
root power of discharge, this implies that a 40 percent change in
discharge translates to about a 20 percent change in depth (or
elevation). If flood depths exceed 5 feet, then a 20 percent change is
about plus or minus one foot which is usually considered significant in
the National Flood Insurance Program. Base flood depths in the main
channel usually exceed 10 feet even for small streams so plus or minus
one standard error in the base flood discharge is likely to transform to
a significant change (on the order of 2 feet) in watersurface
elevation.
The review procedures described herein are often applied to the flood
discharges prior to the hydraulic analyses or determination of
watersurface elevations. As described earlier, the motivation for the
review procedures from a FEMA perspective was to obtain consensus on
flood discharges prior to hydraulic analyses. However, the change in
base flood elevations resulting from a standard error change in the base
flood discharge can quite likely be determined from prior (effective)
hydraulic analyses.
It is possible that the base flood discharges may be statistically
insignificant, yet there is a significant change in the watersurface
elevations. Under these conditions, the decision about the appropriate
elevation to use should be based on hydraulic considerations such as the
best modeling approach or the most current hydraulic data.
Hydrologic Analysis Based on a RainfallRunoff
Model
Flood discharges are updated for flood insurance studies for several
reasons such as the availability of a more physicallybased
rainfallrunoff model, updated regression equations, or changing
landuse or hydraulic conditions. The proposed base flood discharges
from a rainfallrunoff model should be compared to flood discharges at
gaging stations with watershed characteristics within the range of those
for the studied stream(s), to base flood discharges from USGS regression
equations (if they are applicable), to the effective discharges used for
previous flood insurance studies in that community, and to discharges
computed from other available hydrologic analyses. Flood frequency
estimates for the gaging stations used in this evaluation should be made
in accordance with the methodology presented in Bulletin 17B,
Guidelines For Determining Flood Flow Frequency (Interagency
Advisory Committee on Water Data (IACWD), 1982). If the watershed under
study is urbanized, then the regional regression equations should be
adjusted for urbanization using procedures such as those described by
Sauer and others (1983) and Jennings and others (1994). The urban
equations developed by Sauer and others (1983) are applicable nationwide
and were based on observed and modeling data through 1978. It may be
time to update these equations using more recent data and current
statistical procedures.
The regression equations are considered applicable for evaluating
rainfallrunoff model estimates if the watershed, climatic, and
urbanization characteristics for the studied streams are within the
range of those of the gaging stations used to develop the equations and
regulation by flood detention structures does not significantly effect
flow rates. The applicability of the regression equations can be
determined from a plot of the explanatory variables, as illustrated in
Figure 1, for data for the Piedmont Region in Maryland. The Piedmont
Region is that area between the Appalachian Mountains of western
Maryland and the Fall Line that runs from Washington, DC, through
Baltimore to the northern extremes of the Chesapeake Bay.
Figure 1
As illustrated in Figure 1, an ungaged watershed with a drainage area
of 0.5 square miles and a forest cover of 70 percent is outside the
cloud of the data and is, therefore, an extrapolation of the regression
equations. Note that the drainage area and forest cover are
individually within the limits of the data, but the combination of a
small watershed with high forest cover is not represented in the data
set.
The gaging stations used in the evaluation of rainfallrunoff model
estimates should also have watershed characteristics that are within the
range of the characteristics of the studied streams. Base flood
discharges for gaging stations can be obtained from recently published
USGS regional flood reports. It may be appropriate to update the flood
frequency estimates for the gaging stations using Bulletin 17B
guidelines (IACWD, 1982). Decisions on whether to update the station
frequency curves are dependent upon factors such as the existing length
of record, the time since the analyses were last updated, and whether
major floods have occurred since the last update. The gaging station
and regression estimates are used to judge the reasonableness of the
rainfallrunoff model estimates.
The base flood discharges from rainfallrunoff models, gaging station
data, regression equations and previous (effective) flood insurance
studies are plotted against drainage area on logarithmic paper to
determine if the proposed rainfallrunoff model discharges are
reasonable. The error bars of plus or minus one standard error should be
shown about the gaging station or regression estimates. The review
procedures are illustrated using data submitted for flood insurance
studies for two communities.
Application in Lake County, California
The first example is for a study of selected streams in Lake County
in California. The proposed discharges were estimated using the
U.S. Army Corps of Engineers (USACE) HEC1 model (USACE, 1990). Two of
the studied streams, Adobe and Highland Creeks, have gaging stations
upstream of floodcontrol reservoirs. The reaches of these streams that
are to be mapped are downstream of the reservoirs. USGS regression
equations documented in Waanenen and Crippen (1977) were applied to the
unregulated (upstream) reaches of the studied streams.
The effective base flood discharges used in previous flood insurance
studies are compared in Figure 2 with
discharges from the HEC1 model, gaging station data, and USGS
regression estimates. The effective base flood discharges either are
for the studied streams or for other streams in the county with similar
drainage areas. The gaged data are based on 24 years of record each for
Adobe Creek (6.36 square miles) and Highland Creek (11.9 square miles).
The vertical bars about the gaged data represent plus and minus one
standard error computed by methods given in Kite (1988), i.e., 27
percent for Adobe Creek and 30 percent for Highland Creek. The vertical
bars for the USGS regression estimates represent plus one standard error
of estimate (66 percent from Waananen and Crippen, 1977). Only the plus
standard errors are shown for the USGS regression equations because the
HEC1 discharges for the unregulated stream reaches are greater than
those for the regression equations.
Figure 2 near here.
For the unregulated reaches of the studied streams, the proposed
HEC1 discharges are within one standard error of the USGS regression
estimates. The same is generally true for the gaging station data
except one of the HEC1 discharges is situated slightly below the
onestandarderror bound for the Highland Creek gaged data. If most of
the HEC1 estimates are within the standard error bound of the gaging
station and regression estimates, then logic dictates that the HEC1
estimates are reasonable.
The three proposed discharges clearly outside the onestandarderror
bound are for regulated reaches of Adobe and Highland Creeks. The
regulated estimates are shown in Figure 2 to evaluate if the base flood
discharges for the regulated reaches are less than those for the
unregulated reaches for a comparable drainage area. Given the
comparison in Figure 2, FEMA concluded that
the proposed HEC1 base flood discharges are reasonable for use in the
hydraulic analysis. The conclusion implies that the differences in the
unregulated discharges from the HEC1 model and gaging station and
regression estimates are not significantly different. Therefore, the
proposed HEC1 discharges were used for floodplain mapping.
In the Lake County, California example, the HEC1 model or some
deterministic model is needed since two of the streams, Adobe and
Highland Creeks, are regulated by reservoirs. The example described
above was just part of the review process to judge the reasonableness of
the HEC1 inflow peak discharges to the reservoirs. Additional review
considerations were the reasonableness of the starting reservoir
elevations for the base flood routings and the shape and volume of the
inflow hydrographs.
Application in St. Francis County, Arkansas
The second example is in Forrest City in St. Francis County,
Arkansas. The proposed discharges were estimated using a HEC1 model
(USACE, 1990) and balanced design storms based on rainfall data from
U.S. Weather Bureau (USWB) TP40 (USWB, 1961), rainfall losses
calculated using the Natural Resources Conservation Service (NRCS)
runoffcurvenumber method, kinematicwave calculations for routing the
rainfall excess to the main collector channels, and normaldepthstorage
routing.
The HEC1 discharges were compared to gaging station data and
regression equations developed by Hodge and Tasker (1995). Forrest City
lies in two hydrologic regions as defined by Hodge and Tasker (1995):
Region C represented by Crowleys Ridge where the channel slopes are
steep and Region D which is the remains of the old alluvial floodplain
of the Mississippi River where channel slopes are flat. Regression
estimates were determined for the studied streams by weighting the
regression estimates for Regions C and D proportional to the drainage
area in each region.
Figure 3 compares the HEC1 discharges, the
weighted regression estimates, and gaging station data in Regions C and
D. As illustrated in Figure 3, the HEC1 base
flood discharges over predict in comparison with the areaweighted
estimates from the USGS regression equations and with gaging station
data even within Region C (region of steep channel slopes). In fact,
the HEC1 discharges are generally greater than the USGS weighted
regression estimates plus one standard error of prediction. The
weighted standard error of prediction varies with watershed
characteristics and was estimated using a computer program provided by
Hodge and Tasker (1995). The average standard error of prediction for
the studied streams is 45 percent and the vertical bars extending from
the weighted regression estimates represents plus 45 percent. Only the
plus standard errors are shown since the HEC1 estimates are greater
than the regression estimates.
Figure 3
The HEC1 base flood discharges were considered to be too high for the
following hydrologic reasons: use of saturated antecedent moisture
conditions, inappropriate application of kinematic wave routing
computations, and runoffcurve numbers that are higher than those used
in previous studies in the region. On the basis of the comparisons
given in Figure 3, FEMA concluded that the
proposed HEC1 base flood discharges were inappropriate. This
conclusion is supported by the HEC1 discharges being outside the
standard error bounds of the weighted regression estimates and high in
comparison to gaging station data. Since the USGS regression equations
are applicable to the studied streams and a flood hydrograph is not
needed, FEMA's recommendation was to use the regression equations for
the flood insurance study. An alternative approach to using the USGS
regression equations is to revise the HEC1 model so that the model base
flood discharges fall within one standard error of prediction of the
weighted regression estimates. The use of the USGS regression equations
was considered more cost effective than revising the HEC1 model.
Hydrologic Analysis Based on Regional Regression
Equations
Regional regression equations are frequently used in estimating base
flood discharges for flood insurance studies. As with rainfallrunoff
models, the regional regression equations should be evaluated before
using the base flood estimates. Regression estimates should be compared
to the effective discharges for the community, to base flood discharges
from other regression equations published by USGS and other agencies
that are applicable for the region, and to base flood discharges at
gaging stations in the vicinity of the community. In general, the
proposed regression estimates should be based on the most recent
equations published by the USGS. If the most current USGS regression
equations are not used, then reasons should be given as to why the other
equations are more appropriate.
An example where an earlier version of the USGS regression equations
may be more appropriate is in southern Arizona. Regression equations
developed by Eychaner (1984) are based on drainage area, channel slope,
and basin shape. More recent equations published for southern Arizona
by Thomas and others (1994) are based on only drainage area.
Evaluations of flood insurance studies in southern Arizona have
indicated that the regression equations developed by Eychaner (1984)
provide more accurate estimates of base flood discharges than Thomas and
others (1994) for long, narrow, watersheds with flat channel slopes.
As noted earlier, the regression equations should be adjusted for
urbanization, if appropriate. Procedures for making these urban
adjustments are described in Sauer and others (1983) and Jennings and
others (1994). If the urbanization factors for the studied streams are
outside the range of the regression equations or if the watershed is
undergoing rapid landuse change, then the effects of urbanization
should be evaluated using a rainfallrunoff model. The base flood
discharges for gaging stations used to evaluate the regression estimates
should be determined as described above under the section on
rainfallrunoff models.
The base flood estimates from the above sources should be plotted
against drainage area on logarithmic paper similar to the examples in
Figures 23. The proposed discharges are considered reasonable if the
regression equations are applicable, were applied correctly, and are
consistent with the gaging station data used in the evaluation.
USGS regression equations for some states were last updated in the
mid to late 1970's. These regression equations may not be indicative
of the current flood discharges if major floods have occurred since
publication of the regression equations. An example of this is
California where some of the regional equations developed by Waananen
and Crippen (1977) do not reflect major floods that occurred in
different parts of the State in 1980, 1983, 1986, 1995 and 1997. The
USGS is in the process of updating the 1977 regression equations. As
described above, an approach for evaluating if regression estimates are
reasonable is to compare these estimates to updated gaging station data
in the region.
Future Research Needs
Procedures are well documented for determining the standard error of
gaging station and regression estimates. Confidence limits can be
estimated for flood discharges from rainfallrunoff models but
assumptions about the equivalent years of record are needed (USACE,
1996). Future research should be oriented to determining the accuracy
of flood discharges estimated from single (design) event rainfallrunoff
models since the use of these models is prevalent in hydraulic design
and floodplain management. If the accuracy of flood discharges from
rainfallrunoff models could be objectively determined, then the
feasibility of weighting these estimates with regression estimates could
be evaluated.
Additional research is needed to determine the most appropriate
criteria for distinguishing between flood estimates based on different
hydrologic methods. One standard error was chosen because it is often
available and more understood by engineers and hydrologists. Additional
statistical criteria as well as economic and hydrologic criteria should
be evaluated in judging the reasonableness of flood discharges for
ungaged watersheds.
Summary
Procedures for evaluating flood discharges based on rainfallrunoff
models and regional regression equations were described. Two examples
of using the standard error (or standard error of prediction) of flood
discharges to judge the reasonableness of flood discharges were
presented. In the first example, the proposed HEC1 base flood
discharges were rejected for hydrologic reasons and were shown to be
reasonable (within one standard error) in comparison to USGS regression
equations and gaging station data. In the second example, the proposed
HEC1 base flood discharges were shown to over predict (outside one
standard error of prediction) in comparison to USGS regression equations
and gaging station data. The USGS regression equations were recommended
for use because this was a more costeffective approach than revising
the HEC1 model.
The review procedures described in this paper are considered an
approach for determining reasonable estimates for flood discharges for
ungaged watersheds. The procedures are predicated on the assumption
that flood estimates that differ by less than one standard error (of
estimate or prediction) are not significantly different. The choice of
one standard error represents a statistical criterion and is used
because this uncertainty measure is often available. Other statistical
criteria, such as 50percent confidence limits (75percent onesided
limits) or 90percent confidence limits (95percent onesided limits),
could be adopted and the review procedures proposed herein could still
be used. As more experience is obtained with the use of these review
procedures, it may be worthwhile to revise the statistical criteria.
These statistical criteria could be replaced by economic or hydrologic
criteria, if such values could be agreed upon, were readily available,
and were properly verified.
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