500-year flood - can it be reliably estimated? - ltrrkatie/kt/floods-usgs/nsf...t he method adopted...

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500-Year Flood – Can It Be Reliably Estimated?

By Joseph D. Countryman PE; D. WRE

President MBK Engineers, 2450 Alhambra Boulevard, Sacramento, California,95817, PH (916) 456-4400; FAX (916) 456-0253; email:[email protected]

AbstractThe “500-Year” flood has been proposed for a national urban flood standard

(Galloway 2005). This recommendation overlooks the problems associated withestimating the “500-Year” flood. The approved pdf (IACWD 1982) for making theseestimates in the United States is the LP III pdf. The approved methodology provideserratic and highly unreliable estimates of the “500-Year” flood in several Californiawatersheds. A case study of the American River flood estimates is presented todemonstrate the issues. A comparison of flood flow estimates for several probabilitydistribution functions and the associated Confidence Intervals is presented. The paperfinds that extrapolation of pdfs should be avoided and the use of “ConfidenceIntervals” for project formulation and levee certification should be abandoned. It isrecommended to use physically based design floods rather than statistically-baseddesign floods and/or Risk Analysis.

BackgroundThe federal government, through the National Flood Insurance Program

(NFIP), has established a defacto minimum flood control standard for urban areas of100-year flood protection. A 100-year flood has a 1 percent chance of beingexceeded in any given year. It is now understood that a very high residual risk offlooding resides with a 100-year level of flood protection. Concern that urban areascould be subject to an unreasonably high residual flood risk has resulted inrecommendations for changing the current 100-year minimum standard. Arecommendation that the Standard Project Flood (SPF), or 500-year flood, beconsidered as the new NFIP standard for urban areas was recently presented tocongress (Galloway 2005).

Reasonableness of a Flow Frequency Standard.The SPF is not defined by an exceedence frequency, but by a storm and runoff

analysis. It is “the most severe flood producing rainfall depth-area durationrelationship and isohyetal pattern of any storm that is reasonably characteristic of the

World Environmental and Water Resources Congress 2007: Restoring Our Natural Habitat © 2007 ASCE

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region” (USACE 1965). The SPF was the Corps designated design standard for urbanareas. The Corps of Engineers (Corps) is no longer relying on the SPF as thepreferred methodology for establishing an urban design standard. The Corps iscurrently requiring a Risk Analysis approach to project formulation (USACE 2005).This methodology relies on statistically derived frequency curves and theiruncertainty bounds (Confidence Intervals) to design an optimized flood controlproject. This raises the following questions for the hydrologists and engineers that aretasked with developing the frequency curves that are the very essence of the RiskAnalysis methodology:

1. Can the Log Pearson III, pdf, the current federally approved pdf (IACWD1982, pg 3), or any other pdf be reliably used to estimate the 500-year flood?2. Are the Confidence Intervals for the selected pdf reliable?

Theoretically, a 100-year, 500-year, or 1250-year design flood definition isstraight forward and easy to understand. Our ability to define these floods with anydegree of certainty is in question. The remainder of this paper will analyze the 500-year flood estimates for the American River in California

Watershed DescriptionThe American River watershed (see Figure 1) drains the west side of the

Sierra Nevada mountain range and is located just east of Sacramento, California. Thewatershed includes 1888 square miles of drainage. The longest channel length is 170miles and the maximum elevation of the basin is greater than 8,000 feet. The recordflood for the basin occurred in 1997 and had a peak flow of approximately 290,000cfs.

Can the 500-year Flood be Reliably Estimated?An Interagency Advisory Committee on Water Data was established to

recommend procedures for developing 100-year flood estimates. This advisorycommittee developed guidelines for estimating the 100-year flood and published theirfinding in Bulletin #17B (IACWD 1982). Bulletin #17B provided detailedinformation on estimating the 100-year flood. The adopted method was fitting theLog Pearson Type III probability distribution function (LP III WRC) to historic data.There are numerous problems with developing a design flood from a frequencyanalysis. Vit Klemes has identified a major issue for the statistical approach asdescribed below:

“ …from a hydrological point of view, very extreme floods and their causes tend tobe outliers by definition, i.e., very little, if any, information about their likelihood iscontained in the frequencies of relative small floods of which the bulk of a typicalflood sample is composed. Extrapolating distribution models fitted to these samplesis tantamount to extrapolating the small flood dynamics beyond the range it canphysically function.” (Klemes 2000, pg 155)

The following quote from Bulletin #17B illustrates the concern:


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“The accuracy of flood probability estimates based upon statistical analysis of flooddata deteriorates for probabilities more rare than those directly defined by the periodof systematic record. This is partly … because the basic underlying distribution offlood data is not known exactly.” (IACWD 1982, pg 19)

American River Flood EstimatesAn example of the condition articulated above is shown on Figure 2. The LP

III pdf was fit to 102 years of American River flow data. Two different methods offitting the LP III pdf to the data set were used. The method adopted by Bulletin #17Bis the LP III WRC pdf on Figure 2. The LP III BOB (LP III MM, shown on Figure 2)fitting technique is described by Bernard Bobee and Fahim Ashkar (Bobee & Ashkar1991, pgs 96-98). The differences between the estimates increase dramatically forfloods greater than the 20-year flood, and are nearly 70 percent different for the1,000-year flood. These differences are based only on the fitting technique for the LPIII pdf. Since it is impossible to know the true underlying pdf of any given data set (ifone actually exists!), flood frequency estimates were made using various pdfsaccepted for use in hydrology (Bobee & Ashkar 1991). This analysis used the 102years of American River flood data and the computing capability of the HYFRANcomputer program (Univ. of Quebec 2002). A summary of these results is providedon Figure 3. All of the pdfs studied have similar estimates for the 10-year flood. Thepercentage differences dramatically increase for floods larger than the 100-year flood.The LP III WRC 500-year flood estimate of 331,000 cfs would be greater than the10,000-year flood estimate based on the Exponential pdf.

The Corps of Engineers, through the Hydrologic Engineering Center,contracted with MGS Engineering Consultants to develop a stochastic flood modelfor the purpose of estimating American River flood flow frequencies (USACE-HEC2005). The NRC had recommended this procedure as a way of extrapolating thefrequency curve beyond the historic data set (NRC 1999, pg 66). The MGS reportfound that the computed stochastic frequency curve values were lower than the LP IIIWRC frequency curve values. The stochastic model frequency values were adjustedsuch that the 100-year MGS flood estimate matched the LP III WRC 100-year floodestimate (USACE-HEC 2005, pg 30). For this paper the MGS stochastic frequencycurve was adjusted to the 10-year flood rather than the 100-year flood, to minimizethe extrapolation differences between the various pdf estimates. Table 1 comparesthe estimated 3-day flood flows utilizing various methods.

Table 1 American River Flood EstimatesFlow values in 3-Day-CFS

LP III WRC Exponential MGS Stochastic MGS10-year adj

10-year 73,000 74,500 67,000 73,00050-year 151,000 96,000 89,000 97,000100-year 196,000 148,000 150,000 163,000500-year 331,000 199,000 223,000 243,000


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The data presented in Figure 3 and Table 1 demonstrates the inherentinstability in extrapolating pdfs beyond the information available in the historicrecord. Figure 3 shows the LP III WRC quantile estimate increases much faster thanfor the other pdfs displayed and the LP III WRC extrapolation is unbounded on theupper end. This raises the question, “Where will the precipitation come from togenerate this unbounded estimate?”

The Probable Maximum Flood (PMF) is defined as follows: “…that flooddischarge that would result from the combination of the most severe and criticalmeteorological and hydrologic conditions considered reasonably possible in a region”(USACE 2001, pg. 1). During the 1960s and 1970s, it was considered standardpractice at the Corps of Engineers (based on personal experience) to adjust frequencycurve extrapolations such that the PMF would not have an annual exceedanceprobability greater than 1 in 10,000 (10,000-Year flood). The PMF for the AmericanRiver has a volume of 430,000 3-day-cfs (USACE 2001, Chart 20). The AmericanRiver PMF would have an annual exceedance probability of 1 in 1,500 based on theLP III WRC pdf. All of the other pdfs in Figure 3 show the PMF has an annualexceedance probability of less than 1 in 10,000.

It is evident that selection of a pdf should include an evaluation of the physicalhydrology of a drainage basin. If this is not done, very significant extrapolationerrors may be introduced and project formulation and optimization procedures thatrely on these extrapolations will not achieve the goal of project cost efficiency andsafety. The LP III WRC pdf extrapolation for the American River does not includethese evaluations and appears to be significantly overestimating floods greater thanthe 100-year flood. A much larger question must be addressed, “Can theextrapolation of any pdf, beyond the data set used to “curve fit” the pdf, providereliable information unless it is shown to be consistent with the physical watershedand meteorologic conditions in the region?” The answer to this question is selfevident; do not extrapolate the pdfs without physical verification/justification.

Are the Confidence Intervals Reliable?

Once flow frequency curves are adopted based on a given pdf, it is commonpractice in flood frequency analysis to calculate Confidence Intervals for thefrequency curve. An example of this is shown on Figure 4. The fundamentalscientific basis for the Confidence Intervals has not been clearly established.Mathematically, it can be shown that if the pdf for the population of data is known,then an approximation of the sampling error of the population can be estimated. Thisis the essence of Confidence Intervals. In flood hydrology, only one sample isavailable and the true underlying pdf of the population of flood flows is unknown.There is uncertainty that the annual maximum flows are a homogeneous populationbecause the small annual floods do not have meteorologic commonality with the largefloods. Further it is common to assume that there is no upper boundary of possibleflows. All of these conditions make the calculation of Confidence Intervals


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speculative, at best, and extremely misleading at worst. It would take 10,000 years ofdata (assuming stationarity of the meteorologic and hydrologic factors governingflood flows) to prove this bold statement. The proof should be on those that useConfidence Intervals in project optimization calculations or for levee certificationguidelines to show that the Confidence Intervals have validity.

American River Confidence IntervalsFigure 5 displays the 500-year flood estimates and the 90 percent Confidence

Intervals for eleven (11) different pdfs and/or fitting techniques. All pdf curve fittingis based on the identical 102 years of recorded flows. The HYFRAN (Univ. ofQuebec 2002) computer program was used to perform all of the calculationssummarized in Figures 5 & 6 with the exception of the LP III WRC pdf. Thecomputer program HEC-SSP (USACE-HEC 2006) was used for the LP III WRCcalculations. The “90 percent Confidence Intervals” range from a low of 62,000 cfsfor Pearson III pdf to 497,000 cfs for the GEV MM pdf and to 250,000 cfs for theapproved LP III WRC pdf. This information raises serious questions about theefficacy of the Confidence Interval calculations. All calculations are based on thesame relatively long record, yet enormous differences in the Confidence Intervals areevident. This data points to the conclusion that the Confidence Intervals are notprimarily the result of record length. In fact, if it is assumed that the LP III WRC pdfstatistics were based on 500 years of record instead of 102 years of record, the “90percent Confidence Interval” would change to 106,000 cfs. This would still be 170percent larger than the “90 percent Confidence Interval” of the Pearson III pdf basedon 102 years of record.

Another major defect in the Confidence Intervals displayed on Figure 5 is thatthe estimates show the upper bound exceeding the PMF. This Figure shows that theGen Gamma ML, LP III WRC, GEV MWM and the Log-Nor ML pdfs all have upperbound estimates of the “90percent Confidence Interval” greater than the PMF. Thisresult is not supported by the physical hydrologic and meteorologic conditions in theregion. What physical factors in the atmosphere, or in the watershed, exist that couldconceivably result in the PMF becoming the 500-year flood? Is our 102 year samplethis unrepresentative? If the climate changed, or the watershed changed in a verydramatic and currently inexplicable way, the stationarity of the data set would becalled into question. No, we must imagine a situation that would not change thecurrent physical watershed or climatic conditions and still result in the PMFbecoming the 500-year flood. The proponents of the Confidence Interval calculationsmust explain this seeming flaw in the statistical result.

The absolute magnitude of the Confidence Intervals appears to beunreasonably large. Figure 6 displays the 500-year flood Confidence Intervals for thevarious pdfs as a percentage of the “Best Estimate” of the 500-year flood. CommonSense dictates that the “90 percent Confidence Interval” for the 500-year floodestimate should not be larger than the Best Estimate of the 500-year flood! Wheneverthe Confidence Interval is greater than 50 percent of the Best Estimate, it is likely thata serious problem exists. Yet with Risk Analysis in project formulation, this has not


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been considered. A distressing aspect of this is that the Corps of Engineers, as part oftheir adoption of Risk Analysis methodology, has embraced Confidence Intervals forlevee certification (Davis 2006). The Corps will no longer certify a levee unless itincludes an analysis of the Confidence Intervals. As can be seen from the datapresented herein, certification of a levee to a 500-year flood standard may wellrequire the ability of the levee to pass the PMF or larger flood. The data presentedshows that the calculated Confidence Intervals can be extreme and not consistent withthe hydrology or meteorology of a watershed. The use of Confidence Intervals forproject formulation or for levee certification is not advised.

ConclusionThe use of historic data to estimate floods larger than those that have been

experienced is the essence of stochastic hydrology. The fitting of mathematicalfunctions (pdfs) to the ordered historic annual maximum flood flow data set for agiven watershed is the accepted procedure. The pdfs (the LP III WRC pdf is theadopted pdf for the United States) and the parameters of the pdfs that are used toadjust the mathematical functions to fit the available data are not related to physicalfeatures of the landscape or to hydrologic or meteorologic conditions in a region. Theaccepted procedures provide reasonably consistent estimates of annual flood flowexceedance probabilities for floods that are smaller than the maximum flood in thehistoric data set. As shown in this paper and as supported by numerous authoritiescited herein, the extrapolation of the pdfs beyond the data set is problematic andpotentially unreliable. Therefore, any estimate of the 500-year flood should includehydrologic and meteorologic support for the extrapolation. Common sense items suchas the PMF representing a maximum value for the pdf and Confidence Intervalextrapolations should be utilized. A wiser and more scientifically justifiable pathwould be to develop design floods that are based on hydrologic and meteorologicconditions in the area of a proposed project. Risk Analysis software requires thefrequency curves to be extrapolated in order to purportedly allow optimization of aproject formulation process. As with all software, if the hydrologic input representedby the frequency curve extrapolation is unreliable, then the optimization that is theobject of the analysis is also unreliable. It is the responsibility of the hydrologist topoint out this inconvenient reality!

The utilization of Confidence Intervals is currently being promoted by theCorps of Engineers as part of their adoption of Risk Analysis in project formulationand in levee certification. The fact that the Confidence Intervals are dependent uponthe assumption that the adopted pdf represents the true population of flood events fora given watershed is often overlooked. There is no evidence that the pdf that iscalibrated to a limited number of data points accurately represents the full populationof flood events. The use of Confidence Intervals should be restricted to situations towhich their efficacy can be demonstrated.


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Figure 1 Location Map

Figure 2 American River Curve Fitting Techniques for LP III pdf


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Figure 3 American River pdf Comparison

Figure 4 American River Confidence Interval for LP III WRC pdf


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Figure 5 American River 500-year Flood – 90% Confidence Interval

Figure 6 500-year Flood Confidence Interval as % Best Estimate Flood


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References

Bobée B. and Ashkar, F., (1991). The Gamma Family and Derived DistributionsApplied in Hydrology, Water Resources Publications, Colorado.

Davis, Darryl, P.E., D.WRE, (December 2006), Policy letter – USACE LeveeCertification Policy and Risk Analysis.

Gallaway, G.E., (2005). “Statement to Committee on Transportation andInfrastructure, U.S. House of Representatives.”

Interagency Advisory Committee on Water Data, Hydrology Subcommittee(IACWD) (1982). Guidelines for Determining Flood Flow Frequency, Bulletin#17B.

Klemes, Vit, (2000). Common Sense and Other Heresies: Selected Papers inHydrology and Water Resources Engineering, Canadian Water ResourcesAssociation, Cambridge, Ontario.

National Research Council (NRC), (1999). Improving American River FloodFrequency Analysis, National Academy Press, Washington, D.C.

Univ. of Quebec, (2002). Chair in Statistical Hydrology, INRS-ETE. SoftwareHYFRAN for hydrologic frequency analysis, version 1.1

USACE, (1965). Engineer Manual 1140-2-1411.

USACE (2001). American River Basin, California, Folsom Dam and Lake, RevisedPMF Study.

USACE, (2005). Engineer Regulation 1105-2-101. Risk Analysis for Flood DamageReduction Studies.

USACE-HEC, (2005). Stochastic Modeling of Extreme Floods on the AmericanRiver at Folsom Dam.

USACE-HEC, (2006). Software: HEC-SSP ver. 1.0 Beta.


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