Risk-based decisions require different types of information than standards-based decisions. Traditional sources of information used for estimating probabilities of extreme floods include gaged streamflow records, indirect discharge measurements, and precipitation records. Generally these data sources have records that are less than 100 years in length. This framework for flood characterization for risk assessments uses the length of the data record and other characteristics of the data to determine the credible extrapolation limits used in the flood frequency analysis. Because risk assessments require estimation of floods with annual exceedance probabilities of 1 in 10,000, or less, emphasis is placed on developing probabilistic estimates using regional hydrometeorological data and paleoflood information. The uncertainties associated with descriptions of flood flow exceedance probabilities are likely to be substantial and an important attribute to convey into the risk assessment.

No single approach is capable of providing estimates of extreme floods over the full range of annual exceedance probabilities required for risk assessment. Therefore, results from a number of approaches need to be combined to yield a composite flood characterization; this means several methods and sources of data are needed. The application of several independent methods applicable to the same range of annual exceedance probabilities will increase the credibility and resulting confidence in the results.

The U.S. Bureau of Reclamation is now making extensive use of quantitative risk assessment in support of dam safety decision making (Von Thun and Smart, 1996). An important input to Dam Safety Risk Assessment is the development of probabilistic extreme flood estimates. This shifts the focus for dam safety flood evaluation from routing a single 'maximum' event (i.e. the probable maximum flood, PMF) to consideration of the entire range of plausible inflow flood events, and ultimately to the magnitude-frequency relationship of maximum reservoir stages.

For floods, the risk assessment process involves selecting a spillway evaluation flood (SEF) based on the probability of dam failure and the severity of the incremental consequences of dam failure. Past practice also examined consequences, but without formal consideration of probability of failure; if consequences were judged to be large, the SEF was chosen as the PMF.

Reclamation has identified the need for a review of its present procedures for developing probabilistic extreme flood estimates and their associated uncertainties for use in dam safety risk assessment. Where practical, Reclamation would like to develop improved procedures. The overall objective is to develop a practical, robust, consistent, and credible framework for developing probabilistic extreme flood estimates for Dam Safety Risk Assessment. The desired outcome is a robust framework in which components can be improved in the future as the state-of-the-art develops.

The framework was developed by inviting a group of approximately 20 professionals with extensive experience in the theoretical and practical aspects of physical, paleo-, and statistical flood hydrology and hydrometeorology to participate in a one-week workshop held at Utah State University in June 1997. Participants from North America, Australia, and the United Kingdom reviewed current Reclamation practice, and evaluated various advances in developing probabilistic extreme flood estimates for their potential role in the needed framework. A smaller group met in Denver to develop the details of the framework. This paper summarizes the findings of these groups.

Present Reclamation risk assessment practice uses a staged approach for conducting risk assessments (USBR, 1997a). Project schedule and budget constraints are considered in determining the type of flood assessment prepared at each stage. While each risk assessment is unique, the following stages are generally used in Reclamation risk assessments:

1. Screening Level Risk Assessment: An evaluation of risk that includes definition of load probabilities and consequences for all load classes (flood, earthquake, and static). Structure failure probabilities and associated uncertainties are also considered in a global sense, but detailed event trees are not usually prepared. An emphasis at this stage is to maximize the use of available information, without conducting new analyses or collecting additional data. The intent is to identify areas where risks are potentially high and to determine the need for further evaluations and data collection. Results of these evaluations are used to determine Reclamations risk profile and to 'screen' out dam safety issues where additional funding and effort appears to have little potential for reducing dam safety risks.

2. Scoping Level Risk Assessment: A more detailed evaluation of risks is performed for the dam safety issues identified in a screening level risk assessment. This level of risk assessment typically involves more detailed treatment of event trees, load probabilities, structural response, and consequences. The intent is to invest sufficient effort so that the risk assessment team understands the major contributors to risk to enable formulation of risk reduction strategies and to determine the need for additional analyses and investigations.

3. Decision Level Risk Assessment: At this level, more detailed evaluation of risks is performed to provide decision makers with the information necessary to reach a dam safety decision for a structure. The decision may be related to continuing project operations, correcting dam safety deficiencies, selecting among risk reduction alternatives, or determining the need for interim actions to reduce risk while long term plans are developed. The intent is to provide decision makers with sufficient pertinent risk information such that the risk reduction objective can be effectively considered along with other Reclamation objectives. At this level of risk assessment, detailed loading information, structural response analyses, and consequence evaluations are developed for all significant issues. This type of risk assessment focuses on reducing uncertainties in the risk estimates and evaluating risk reduction actions.

The proposed framework for developing probabilistic extreme flood estimates for risk assessment uses the length of record and other characteristics of the data to determine the extrapolation limits for flood frequency analysis. Traditional sources of information used for flood hazard analyses include streamflow and precipitation records. Generally, these data sources have records that are less than 100 years in length, although in some cases these records can be extended to about 150 years using historical information. Regional precipitation and streamflow data can create pooled data sets from short periods of observation, and paleoflood data can extend records of floods to periods of up to several thousand years.

Many different types of streamflow information are used in developing probabilistic extreme flood estimates for risk assessment. Streamflow data are used in flood hazard assessment as input for frequency studies or as the basis for developing flood hydrographs. The usual source of these data is the streamflow records collected and maintained by the U.S. Geological Survey. However, similar data are collected and archived by many other Federal and State government agencies and some non-government organizations. Streamflow records consist of data collected at established gaging stations and indirect measurements of streamflow at other sites. Streamflow data can include estimates of peak discharge, as well as average or mean discharge for various time periods. Most streamflow measurements on U.S. streams began after 1900 with only a few records dating back that far. Most often, streamflow records at a single site range in length from about 20 to 60 years. Completeness of the data set may vary from station to station.

Precipitation and weather data used in hydrologic models can include rainfall, snowfall, snow water equivalent, temperature, solar radiation, and wind speed and direction from individual weather stations, as well as remote sensing information and radar information for broader regions. Data types available from various sources vary greatly in record length and quality throughout the United States. Some of these types of data (i.e., snowfall, snow water equivalent, solar radiation, and wind) are limited to record lengths of less than about 30 years; basic rainfall and temperature data are available for some stations for up to 150 years, but in most cases are limited to less than 100 years.

Historical data can provide a means for extending the length of record for many types of data, in particular for observations of the most extreme events. These data are most commonly used to extend streamflow records of peak discharge prior to organized stream gaging. Historical observations can provide information for other types of data such as weather patterns and the frequency of extreme storm events, or changes in land use or vegetation that may be significant to runoff modeling calculations. However, as with any type of historical data, the accuracy and validity of the observations must be carefully assessed and compared to the other types of data used in the analysis.

Paleoflood hydrology is the study of past or ancient flood events which occurred prior to the time of human observation or direct measurement by modern hydrological procedures (Baker, 1987). Unlike historical data, paleoflood data do not involve direct human observation of the flood events. Instead, the paleoflood investigator studies geomorphic and stratigraphic records (various indicators) of past floods, as well as the evidence of past floods and streamflow derived from historical, archeological, dendrochronologic, or other sources. The advantage of paleoflood data is that it is often possible to develop records that are 10 to 100 times longer than conventional or historical records from other data sources in the western United States. In addition, the paleoflood record is a long-term measure of the tendency of a river to produce large floods. In many cases, paleoflood studies can provide a long-term perspective, which can put exceptional annual peak discharge estimates in context and assist in reconciliation of conflicting historical records.

Paleoflood data generally include records of the largest floods, or commonly the limits on the stages of the largest floods over long time periods. This information can be converted to peak discharges using a hydraulic flow model. Generally, paleoflood data consist of two independent components. One component is a peak discharge estimate; the second is a time period or age over which the peak discharge estimate applies. Paleoflood studies can provide estimates of peak discharge for specific floods in the past, or they can provide exceedance and non-exceedance bounds for extended time periods. Each of these differing types of paleoflood data must be appropriately treated in flood frequency analyses.

The primary basis for a limit on credible extrapolation of extreme flood estimates derives from the characteristics of the data and the record length used in the analysis. The data used in the analysis provide the only basis for verification of the analysis or modeling results, and as such, extensions beyond the data cannot be verified. Different risk assessments require flood estimates for different ranges of annual exceedance probability (AEP), and therefore analysis procedures and data sources should be selected to meet project requirements. The greatest gains to be made in providing credible estimates of extreme floods can be achieved by combining regional data from multiple sources. Thus, analysis approaches that pool data and information from regional precipitation, regional streamflow, and regional paleoflood sources should provide the highest assurance of credible characterization of low AEP floods.

For many Reclamation dam safety risk assessments, flood estimates are needed for AEPs of 1 in 10,000 and ranging down to 1 in 100,000, or even lower. Developing credible estimates at these low AEPs generally require combining data from multiple sources and a regional approach. Table 1 lists the different types of data which can be used as a basis for flood frequency estimates, and the typical and optimal limits of credible extrapolation for AEP, based on workshop discussions or subsequent communications. The limits presented in the table represent a general group consensus; however, opinions differed amongst workshop participants. In general, the optimal limits are based on the best combination(s) of data envisioned in the western U.S. in the foreseeable future. Typical limits are based on the combination(s) of data which would be commonly available and analyzed for most sites.

Many factors can affect the equivalent independent record length for the optimal case. For example, gaged streamflow records in the western United States only rarely exceed 100 years in length, and extrapolation beyond twice the length of record, or to about 1 in 200 AEP, is generally not recommended (IACWD, 1982). Likewise, for regional streamflow data the optimal limit of credible extrapolation is established at 1 in 1,000 AEP by considering the number of stations in the region, lengths of record, and degree of independence of these data (Hosking and Wallis, 1997). For paleoflood data, only in the Holocene epoch, or the past 10,000 years, is climate judged to be sufficiently like that of the present climate, for these types of records to have meaning in estimates of extreme floods for dam safety risk assessment. This climatic constraint indicates that an optimal limit for extrapolation from paleoflood data, when combined with at-site gaged data, for a single stream should be about 1 in 10,000 AEP. For regional precipitation data, a similar limit is imposed because of the difficulty in collecting sufficient station-years of clearly independent precipitation records in the orographically complex regions of the western United States. Combined data sets of regional gaged and regional paleoflood data can be extended to smaller AEPs, perhaps to about 1 in 40,000, in regions with abundant paleoflood data. Analysis approaches that combine all types of data are judged to be capable of providing credible estimates to an AEP limit of about 1 in 100,000 under optimal conditions.

In many situations, credible extrapolation limits may be less than optimal. Typical limits would need to reflect the practical constraints on the equivalent independent record length that apply for a particular location. For example, many at-site streamflow record lengths are shorter than 100 years. If in a typical situation the record length is only 50 years, then the limit of credible extrapolation might be an AEP of about 1 in 100. Similarly, many paleoflood records do not extend to 10,000 years, and extensive regional paleoflood data sets do not currently exist. Using a record length of about 4,000 years, a typical limit of credible extrapolation might be an AEP of 1 in 15,000 based on regional streamflow and regional paleoflood data.

The information presented in Table 1 is intended as a guide; each situation is different and should be assessed individually. The limits of extrapolation should be determined by evaluating the length of record, number of stations in a hydrologically homogeneous region, degree of correlation between stations, and other data characteristics which may affect the accuracy of the data.

Ideally, one would like to construct the flood frequency distribution for all floods that could conceivably occur. However, the limits of data and flood experience for any site or region place practical limits on the range of the floods to which AEPs can be assigned. There does not appear to be sufficient data to justify computation of AEPs less than 1 in 100,000. In general, the scientific limit to which the flood frequency relationship can be credibly extended, based upon any characteristics of the data and the record length, will fall short of the probable maximum flood (PMF) for a site. PMF estimates provide a useful reference to past practice and can be compared with extreme floods characterized for risk assessment. However, the workshop participants concluded that there is limited scientific basis for assigning an AEP to the PMF. For precipitation data, similar limitations apply to extrapolations that approach values described by probable maximum precipitation.

At Site Flood Frequency Analysis

Frequency analysis is an information problem: if one had a sufficiently long record of flood flows, or perhaps rainfall for a basin, then a frequency distribution for a site could be determined with good precision, so long as change over time due to anthropogenic or natural processes did not alter the distribution of floods. In most situations available data are insufficient to precisely define the annual exceedance probability of large floods. This forces hydrologists to use practical knowledge of the physical processes involved, and efficient and robust statistical techniques, to develop their estimates (Stedinger et al., 1993).

Fitting a distribution to data sets allows both a compact and smoothed representation of the frequency distribution revealed by the available data, and a systematic procedure for extrapolation to frequencies beyond the range of the data set. Given a family of distributions, one can estimate the parameters of that distribution so that required quantiles and expectations can be calculated with the "fitted" model. Appropriate choices for distribution functions can be based upon examination of the data using probability plots, the physical origins of the data, previous experience, or prescriptive guidelines.

Several general approaches are available for estimating the parameters of a distribution. A simple approach is the method of moments, which uses the available sample to compute estimators of the distribution's parameters. The Federal guidelines published in Bulletin 17B (IACWD, 1982) recommend fitting a Pearson type 3 distribution to the common base 10 logarithms of the peak discharges. It uses at-site data to estimate the sample mean and variance of the logarithms of the flood flows, and a combination of at-site and regional information to estimate skewness.

Another method that may be used to estimate the parameters of a distribution for at-site frequency analysis is the Expected Moments Algorithm (EMA). EMA (Cohn et al., 1997) is a moments-based estimation procedure and is identical to the existing Bulletin 17B (IAWCD, 1982) approach when no high or low outliers are present. The EMA method was developed to utilize historical and paleoflood information in a censored data framework. This approach explicitly acknowledges the number of known and unknown values above and below a threshold, similar to a maximum-likelihood approach. Three types of at-site flood information are used: systematic stream gage records; information about the magnitudes of historical floods; and knowledge of the number of years in the historical period when no large flood occurred.

Still another method, which has strong statistical motivation, is the method of maximum likelihood. Maximum likelihood estimators (MLEs) have very good statistical properties in large samples, and experience has shown that they generally do well with records available in hydrology. In many cases MLEs cannot be reduced to simple formulas, so estimates must be calculated using numerical methods (Stedinger et al., 1988; O'Connell, 1997).

L-moments are another way to summarize the statistical properties of hydrologic data. Sample estimators of L-moments are linear combinations (and hence the name L-moments) of the ranked observations, and thus do not involve squaring or cubing the observed values as do the product-moment estimators. As a result L-moment estimators of the dimensionless coefficients of variation and skewness are almost unbiased and have very nearly a normal distribution (Hosking and Wallis, 1997).

In hydrology, sufficient information is seldom available at a site to adequately determine the frequency of rare events using frequency analysis. This is certainly the case for the extremely rare events which are of interest in dam safety risk assessment. The National Research Council (1988) has proposed several general strategies, including substituting space for time for estimating extreme floods. One substitutes space for time by using hydrologic information at different locations in a region to compensate for short records at a single site.

Three approaches (Cudworth, 1989) have been considered for regional flood frequency analysis: (1) average parameter approach; (2) index flood approach; and (3) specific frequency approach. With the average parameter approach, some parameters are assigned average values based upon regional analyses, such as the log-space skew or standard deviation. Other parameters are estimated using at-site data, or regression on physiographic basin characteristics, perhaps the real or log-space mean. The index flood method is a special case of the average parameter approach. The specific frequency approach employs regression relationships between drainage basin characteristics and particular quantiles of a flood frequency distribution.

Index Flood Method. The index flood procedure is a simple regionalization technique with a long history in hydrology and flood frequency analysis (Dalrymple, 1960). It uses data sets from several sites in an effort to construct more reliable flood-quantile estimators. A similar regionalization approach in precipitation frequency analysis is the station-year method, which combines precipitation data from several sites without adjustment to obtain a large composite record to support frequency analyses. The concept underlying the index flood method is that the distributions of floods at different sites in a "region" are the same except for a scale or index-flood parameter which reflects the size, rainfall and runoff characteristics of each watershed. Generally the mean is employed as the index flood (Hosking and Wallis, 1997).

Average Shape Parameter. As at-site records increase in length, procedures that estimate two parameters, with at-site data to be used with a regional shape parameter, have been shown to perform better that index flood methods in many cases (Stedinger and Lu, 1995). For record lengths of even 100 years, 2-parameter estimators with a good estimate of the third shape parameter, are generally more accurate than are 3-parameter estimators (Lu and Stedinger, 1992; Stedinger and Lu, 1995). However, whether or not it is better to also regionalize the coefficient of variation depends upon the heterogeneity of the regions and the coefficients of variability of the flows. In regions with high coefficients of variation (and high coefficients of skewness) index flood methods are more attractive.

Regional Regression. Regional analysis can be used to derive equations to predict the values of various hydrologic statistics (including means, standard deviations, quantiles, and normalized regional flood quantiles) as a function of physiographic characteristics and other parameters. Stedinger and Tasker (1985, 1986a, 1986b) developed a specialized Generalized Least Squares (GLS) regression methodology to address the regionalization of hydrologic statistics. Advantages of the GLS procedure include more efficient parameter estimates when some sites have short records, an unbiased model-error estimator, and a better description of the relationship between hydrologic data and information for hydrologic network analysis and design.

Precipitation-runoff modeling is typically used as an event-based method for determining extreme floods. A single set of hydrometeorological parameters and watershed characteristics are used to simulate a design flood event. The major inputs to a design event-based precipitation-runoff model are: (1) climate data (rainfall, snowfall, and other variables needed to predict snowmelt); (2) losses (infiltration/interception); (3) physical watershed characteristics for runoff and routing simulations (drainage areas, watershed and channel slopes, lag times, antecedent moisture, etc.); (4) precipitation-runoff transformation function; and (5) runoff conveyance/routing mechanisms. Model output includes runoff hydrographs at user-specified locations, maximum peak discharges, and total runoff volumes. Examples of this type of model include HEC-1 (USACE, 1990) and RORB (Laurenson and Mein, 1995).

In the stochastic approach, hydrologic model inputs are treated as random variables. Monte Carlo sampling procedures are used to allow the input variables to vary in accordance with their observed distributions, including the observed dependencies among some climatic and hydrologic parameters. The use of the stochastic approach with regional precipitation information allows the estimation of flood magnitude-frequency curves for flood peak discharge, flood runoff volume, and reservoir level. An example of this type of model is discussed by Barker et al. (1997).

This method combines the work of atmospheric modelers and regional precipitation analysis to derive a precipitation magnitude-frequency curve (Chin et al., 1997). The atmospheric model is used to generate storms over the watershed, and the findings from the regional analysis are used to estimate the annual exceedance probability of point and areal precipitation generated by the model. Using distributed precipitation-runoff modeling, snowpack and other antecedent conditions can be combined to estimate a simulated flood frequency curve using a Monte Carlo approach.

The data generation and continuous simulation modeling approach is based on Monte Carlo generation of long and detailed sequences of hydrometeorological variables, including precipitation, air temperature, and wind speed and direction. In order to represent spatial differences across the watershed adequately, it is necessary to generate hydrometeorological variables for several sites concurrently. Hydrological models of watershed behavior and hydraulic models of confluences, wave effects and reservoir outlets are used to simulate the reservoir water level continuously. An estimated magnitude-frequency relationship of maximum reservoir stages is input to the risk assessment (Calver and Lamb, 1996).

No single approach is capable of providing the needed characterization of extreme floods over the full range of annual exceedance probabilities that may be required for risk assessment. In particular, characterization of floods with AEPs less than 1 in 10,000 can be expected to require that results from a number of approaches, based on multiple data sources, need to be combined to yield a composite flood frequency description. The application of several independent methods and types of data applicable to the same range of annual exceedance probabilities will increase the credibility and resulting confidence in the results.

Table 2 lists various methodologies that were considered for characterizing extreme floods to support dam safety risk assessment. A flood frequency analysis must be combined with each of these methodologies to assign annual exceedance probabilities to the floods.

The framework developed for Reclamation does not propose a specific methodology for rigorously combining information from these differing data sources and methodologies in an overall statistical framework. In some cases the information may be combined statistically, and in other cases one set of results may be used as a bound on the frequency distribution obtained by analysis of other data. Clearly, this process will require a measure of judgement. Regardless of the approach taken for combining results, it should incorporate sound physical and scientific reasoning for weighting or combining results.

All floods characterized for the risk assessment process should display the uncertainties resulting from the analysis. As the risk assessment moves from the screening and scoping levels to the decision level, uncertainty should be reduced and better quantified so that appropriate information is included in the dam safety decision-making process.