Developing Hydrologic Hazard Curves & Flood Hydrographs for Dam Safety Risk Assessment, Exercises of Hydrology

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Guidelines for Evaluating

Hydrologic Hazards

U.S. Department of the Interior Bureau of Reclamation June 2006

MISSION S TATEMENTS

The mission of the Department of the Interior is to protect and provide access to our Nation’s natural and cultural heritage and honor our trust responsibilities to Indian tribes and our commitments to island communities.

The mission of the Bureau of Reclamation is to manage, develop, and protect water and related resources in an environmentally and economically sound manner in the interest of the American public.

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Contents

• Executive Summary S- Page
• 1. Introduction...............................................................................................................................
• 2. Background ...............................................................................................................................
  • 2.1 General
  • 2.2 Public Protection Guidelines
  • Dam Safety Program Processes..............................................................................................................
• 3. Process........................................................................................................................................
  • 3.1 Data Sources...................................................................................................................................
  • 3.2 Flood Frequency Extrapolation
  • 3.3 Flood Peak and Volume Relationships...........................................................................................
• 4. Analysis Techniques..................................................................................................................
  • 4.1 Flood Frequency Analysis with Historical/Paleoflood Data
    • 4.1.1 Historical and Paleoflood Data
    • 4.1.2 Mixed-Population Graphical Approach............................................................................
    • 4.1.3 Expected Moments Algorithm..........................................................................................
    • 4.1.4 FLDFRQ3
  • 4.2 Hydrograph Scaling and Volumes................................................................................................
  • 4.3 GRADEX Method
  • 4.4 Australian Rainfall-Runoff Method
    • 4.4.1 Approach............................................................................................................................
    • 4.4.2 Calibration
    • 4.4.3 Strengths and Limitations
  • 4.5 Stochastic Event-Based Precipitation Runoff Modeling with the SEFM.....................................
  • 4.6 Stochastic Rainfall-Runoff Modeling With CASC2D
  • 4.7 PMF Analysis Technique
• 5. Characterization of Hydrologic Hazards..............................................................................
  • 5.1 Integration of the PMF into Hydrologic Hazard Evaluations.......................................................
  • 5.2 Characterization of Hydrologic Risk for the CFR........................................................................
  • 5.3 Detailed Hydrologic Studies.........................................................................................................
• 6. Case Studies.............................................................................................................................
  • 6.1 Los Banos Dam - Hydrograph Scaling 6.1.1 Los Banos Hydrologic Hazard Curves Using Flood Frequency Analysis and
  • 6.2 A.R. Bowman Dam - Hydrograph Scaling 6.2.1 A.R. Bowman Hydrologic Hazard Curves Using Flood Frequency Analysis and
    • 6.2.2 A.R. Bowman Hydrologic Hazard Estimates Based on a Stochastic Event Flood Model
    • 6.2.3 A.R. Bowman Hydrologic Hazard Estimates Using Bayesian Statistical Estimation
      • Implications 6.2.4 Combined Hydrologic Hazard Estimates for Risk Analysis and Dam Safety
  • 6.3 Fresno Dam - Hydrograph Scaling 6.3.1 Fresno Dam Hydrologic Hazard Curves Using Flood Frequency Analysis and
    • 6.3.2 Fresno Dam Hydrologic Hazard Analysis Using the GRADEX Method.........................
• 7. Summary..................................................................................................................................
• 8. Bibliography ............................................................................................................................

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Executive Summary

The purpose of this document is to establish guidelines for generating hydrologic hazard information for use in evaluating hydrologic risk at dams. This information is intended to be used for risk analysis and prioritization of further work at Bureau of Reclamation (Reclamation) dams and other U.S. Department of the Interior facilities. Hydrologic hazard information consists of a flood frequency analysis and frequency flood hydrographs for a full range of Annual Exceedance Probabilities (AEP) necessary for decision making.

Reclamation has developed an approach toward developing hydrologic hazard curves for use in evaluating dam safety issues. The procedure relies on extracting information from existing studies to the fullest extent possible. The procedures and analysis techniques defined in these guidelines allow for the possibility, and even plausibility, that peak discharge and volume estimates may exceed the probable maximum flood (PMF). This is a function of the uncertainty and inconsistency among and between analysis techniques. Therefore, in these cases, the PMF is believed to represent the upper limit to hydrologic risk.

The procedure for developing hydrologic hazard curves considers the dam safety decision criteria, potential dam failure mode and dam characteristics, available hydrologic data, possible analysis techniques, resources available for analysis, and tolerable level of uncertainty. Dam safety decision criteria determine the probabilistic range of floods needed to address hydrologic issues. The potential dam failure mode and dam characteristics impact the type of hydrologic information needed to assess the problem. The specific elements selected to be incorporated in an analysis of hydrologic hazards should consider the tolerable level of uncertainty. To reduce the uncertainty in the estimates, additional data collection and use of more sophisticated solution techniques may be required.

Reclamation currently uses a combination of seven hydrologic methods to develop hydrologic hazard curves. These general techniques include:

• Flood frequency analysis with historical/paleoflood data
• Hydrograph scaling and volumes
• The GRADEX Method
• The Australian Rainfall-Runoff Method
• Stochastic event-based precipitation runoff modeling with stochastic event flood model
• Stochastic rainfall-runoff modeling with CASC2D
• The PMF

It is believed that increasing the level of effort and sophistication of analysis technique increases the level of confidence associated with the results.

The amount of effort expended on analyzing a hydrologic hazard depends on the nature of the problem and the potential cost of the solution. A staged approach toward evaluating a hydrologic safety issue is recommended. Initially, very little effort is expended to determine the

Guidelines for Evaluating Hydrologic Hazards

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magnitude of the hydrologic hazard. Reclamation attempts to make use of all the available studies for the site of interest. Often, the PMF and initial flood frequency studies are the only hydrologic studies available before the start of a probabilistic investigation. When other hydrologic studies have been performed, available data will be used to decrease uncertainty in results as well as provide an overall assessment of hydrologic risk.

Dam safety evaluations usually begin by characterizing hydrologic risk for the Comprehensive Facility Review (CFR) process. If detailed studies have been conducted for the site of interest, they are summarized, consolidated, and presented to the risk assessment team. About two-thirds of Reclamation’s dams can safely accommodate the PMF; when the PMF is selected as the inflow design flood, no additional work may be required unless other hydraulic issues need evaluation. Additional hydrologic work begins with a flood frequency analysis developed for peak flows and volumes and hydrograph scaling. It is believed that this type of information is sufficient to address hydrologic issues and make dam safety decisions at about 80 percent of the remaining dams. For the sites that still have potential safety problems, more sophisticated solution techniques than the initial flood frequency analysis and hydrograph scaling may be required.

When planning more detailed studies, the goal is to achieve a balance between the amount of hydrologic analysis needed to address the issues and the level of effort required to conduct the study. As the studies get more detailed, the results should become more precise and contain less uncertainty.

When multiple methods are used, alternative hazard curves are developed by weighting results from the individual analyses. A team of hydrologists evaluates the alternatives and selects the one most representative for the site for use in the risk assessment. Selection of the final hydrologic hazard curve depends on the experience of the hydrologists and the assumptions that went into each analysis.

Three case studies, Los Banos, Fresno, and A.R. Bowman Dams, are presented in these guidelines to illustrate the variety of methods available. These sites were chosen to demonstrate the use of flood frequency analysis and hydrograph scaling to characterize the flood hazard and more detailed followup studies, where available. The A.R. Bowman example shows how multiple studies were combined into a single flood hazard curve for use in risk assessment.

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Risk assessment methods provide techniques to organize and plan the data collection and technical studies necessary to evaluate dam safety issues at a site. The risk assessment process allows the risk assessment team to consider the possible adverse outcomes to a given loading condition and compute the risk associated with each possible outcome. The process involves identifying all of the possible loading conditions, dam responses, exposure conditions, and consequences. The overall risk from the dam is the accumulation of the risks associated with each of these factors (Bureau of Reclamation, 1999).

When evaluating hydrologic hazards, a systematic means of developing flood hazard relationships is needed for risk-based assessments to determine hydrologic adequacy for Reclamation dams. The nature of the potential failure mode and characteristics of the dam and reservoir dictate the type of hydrologic information needed. For some sites, only a peak- discharge frequency analysis may be required, while at other sites, flood volumes and hydrographs may be required. The goal of any hydrologic analysis is to provide the hydrologic information needed to make dam safety decisions at the least possible cost.

2.2 Public Protection Guidelines

Guidance for providing adequate and consistent levels of public protection in the evaluation and modification of existing dams and the design of new structures are described in the Guidelines for Achieving Public Protection in Dam Safety Decisionmaking , (Bureau of Reclamation, 2003a). The reader may refer to the guidelines for a complete description of the assessment measures used by Reclamation in making dam safety decisions.

Determining an appropriate level of public protection involves assessing the existing risks, determining the need for risk reduction, and, where needed, evaluating specific alternatives to reduce risk. Because the total needs for the agency’s financial and human resources generally exceed the available resources, the Public Protection Guidelines were prepared to assist Reclamation staff in presenting public safety information to decisionmakers for prioritizing among projects and allocating limited resources.

Reclamation’s Public Protection Guidelines consist of two assessment measures of risk that are considered in the decision process for a dam: (1) the probability of dam failure and (2) the life loss consequences resulting from unintentional reservoir release. The annual probability of failure guideline considers the accumulation of risks from Reclamation’s total inventory of dams. The life loss guideline deals with agency public trust responsibilities.

Dam Safety Program Processes

Hydrologic hazard information is generally required during four stages of the dam safety program process. These four stages include the Comprehensive Facility Review (CFR), Issue Evaluation (IE), Corrective Action Study (CAS), and Final Design (FD). Most projects do not progress through each stage of the process because the process is intended to address dam safety deficiencies, and many projects either have no deficiencies or the safety issues can be resolved without a need for structural modifications. The remainder of this section of the guidelines will

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briefly describe the four stages of the dam safety program process that require hydrologic hazard information. For more detailed information about the dam safety process, the reader should review the references cited.

The CFR provides a mechanism for early detection of developing and/or existing dam safety issues. The CFR is performed every 6 years and consists of a state-of-the-art review of the dam and its performance, previous studies/analyses (including hydrology), construction practices, downstream consequences, risk, and dam safety decisions (Bureau of Reclamation, 1998). The CFR is used to identify risks at individual dams and to prioritize further work. Once hydrologic hazard information is developed for the CFR, the Dam Safety Office determines whether or not additional hydrologic studies are required to make decisions during subsequent stages of the process.

The IE stage is used to confirm problems identified previously. Data collection and/or analysis activities are focused on addressing specific dam safety issues and updating risk estimates. At the conclusion of the IE, the decision makers determine whether or not actions are required to reduce risk at the dam (Bureau of Reclamation, 2003b).

A CAS formulates and evaluates risk reduction alternatives. Data is collected and analyzed to the extent necessary to develop the details of identified alternatives, to estimate project costs, and to provide sufficient information to allow decision makers to select and justify the proper course of action. The baseline risk analysis is updated to show the risk reduction potential of each of the developed alternatives (Bureau of Reclamation, 2003b).

During the FD stage, the conceptual design is transformed into the final design. Additional data collection and analysis are used to improve the design, reduce and refine project costs, and finalize design drawings and specifications (Bureau of Reclamation, 2003b).

3. Process

The elements selected for incorporation in an analysis of hydrologic hazards must consider the potential dam failure mode and dam characteristics, available hydrologic data, possible analysis techniques, resources available for analysis, and tolerable level of uncertainty. The potential dam failure mode and dam characteristics impact the type of hydrologic information needed to assess the problem. Some problems may require only a peak-discharge frequency curve, while others may need complete hydrographs. The available data, possible analysis techniques, resources available, and needs of the decision makers influence the selection of elements to be included in developing hydrologic hazard curves.

The process that follows provides a systematic approach for estimating hydrologic hazard curves that can be used for dam safety decisionmaking. It recognizes that additional studies do not always lead to better decisions. Therefore, the process relies on using existing data and previous analyses as much as possible to produce hydrologic information suitable for dam safety decisionmaking at the least possible cost.

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from regional precipitation, regional streamflow, and regional paleoflood sources should provide the highest assurance of credible characterization of low AEP floods. The information that follows was developed in a workshop sponsored by Reclamation and documented in Bureau of Reclamation, 1999.

For Reclamation dam safety risk assessments, flood estimates are needed for AEPs of 1 in 10,000 and possibly ranging down to as low as 1 in 100,000,000. Developing credible estimates at these low AEPs generally requires combining data from multiple sources and a regional approach. Table 3-1 lists the different types of data that can be used as a basis for flood frequency estimates and the typical and optimal ranges of credible extrapolation for AEP (Bureau of Reclamation, 1999). In general, the optimal ranges are based on the best combination(s) of data envisioned in the western U.S. in the foreseeable future. Typical ranges are based on the combination(s) of data that are commonly available and analyzed for most sites.

Table 3-1.—Data types and extrapolation ranges for flood frequency analysis (Bureau of Reclamation, 1999)

Type of data used for flood frequency analysis

Range of credible extrapolation for annual exceedance probability Typical Optimal At-site streamflow data 1 in 100 1 in 200 Regional streamflow data 1 in 500 1 in 1, At-site streamflow and at-site paleoflood data 1 in 4,000 1 in 10, Regional precipitation data 1 in 2,000 1 in 10, Regional streamflow and regional paleoflood data 1 in 15,000 1 in 40, Combinations of regional data sets and extrapolation 1 in 40,000 1 in 100,

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, and extrapolation beyond twice the length of record, or to about 1 in 200 AEP, is generally not recommended (Interagency Advisory Committee on Water Data [IACWD], 1982). Likewise, for regional streamflow data the optimal range of credible extrapolation is established at up to 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 estimating extreme floods for dam safety risk assessment. This climatic constraint indicates that an optimal range for extrapolation from paleoflood data, when combined with at-site gaged data, for a single stream should be up to about 1 in 10,000 AEP. For regional precipitation data, a similar range 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

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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 for an AEP range up to about 1 in 100,000 under optimal conditions.

In many situations, credible extrapolation ranges may be less than optimal. Typical ranges 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 range of credible extrapolation might be up to 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 range of credible extrapolation might be up to an AEP of 1 in 15,000 based on regional streamflow and regional paleoflood data.

The information presented in table 3-1 is intended as a guide; each situation is different and should be assessed individually. The ranges of extrapolation should be determined by evaluating the lengths of records, number of stations in a hydrologically homogeneous region, degree of correlation between stations, and other data characteristics that 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 amount of data and flood experience for any site or region constrain the range of the floods to which AEPs can be assigned based solely on data. In general, the scientific range 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 PMF for a site. However, there is a need in dam safety risk assessment to determine the probability of occurrence of very large floods with very small AEPs. The lack of an ideal data set does not absolve the hydrologist from extending the flood frequency relationship to cover the full range of AEPs needed for risk assessment. Therefore, a systematic approach is provided for estimating hydrologic hazard curves that can be used for dam safety decisionmaking.

Floods can be categorized, according to the Australian Rainfall and Runoff: A Guide to Flood Estimation (Nathan and Weinmann, 2001), as large, rare, and extreme. These flood categories are shown in figure 3-1. Large floods generally encompass events for which direct observations and measurements are available. Rare floods represent events located in the region between direct observations and the credible range of extrapolation from the data. Extreme floods generally have very small AEPs, which are beyond the credible range of extrapolation but are still needed for dam safety risk assessments. Occasionally, Reclamation has an interest in floods with an AEP as low as 1 in 10^8.

Extreme floods border on the unknowable. Uncertainty is very large and unquantifiable. Since data cannot support flood estimates in this AEP range, hydrologists and engineers must rely on our knowledge and understanding of hydrologic processes to estimate extreme floods. Oftentimes, these floods may result from unforeseen and unusual combinations of hydrologic parameters generally not represented in the flood history at a particular location. One potential upper bound to the largest flood at a particular site of interest is the PMF.

Guidelines for Evaluating Hydrologic Hazards

1.00E+

1.00E+

1.00E+

1.00E+

Annual Exceedance Probability (%)

Peak Discharge (ft

3 /s)

1.00E+

1.00E+

1.00E+

Volume (acre ft)

99.0 95

0 84.070.0^50

0 30.0 16 .0 5. 10.0 1.0 0.30.1 0.01 0. 0.0001 1 (

-6^ ) 1 (

2.5 -5^ )

Peak Discharge 1-Day Volume 3-Day Volume 5-Day Volume 7-Day Volume 15-Day Volume

Figure 3-2.—Example hydrologic hazard curve.

4. Analysis Techniques

The main probabilistic and engineering hydrology methods that are currently being used, applied, and under investigation by the Flood Hydrology Group are summarized in this section of the report. There are seven general techniques:

• Flood frequency analysis with historical/paleoflood data
• Hydrograph scaling and volumes
• The GRADEX Method
• The Australian Rainfall-Runoff Method
• Stochastic event-based precipitation runoff modeling with the stochastic event flood model (SEFM)
• Stochastic rainfall-runoff modeling with CASC2D

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• The PMF

Other models and approaches are briefly noted by reference in each section. General sources of models and approaches for estimating extreme floods are listed in Maidment (1993), Singh (1995), and Bureau of Reclamation (1999). Methods to calculate extreme floods and associated probabilities have recently been revised and published in the United Kingdom (Institute of Hydrology, 1999) and Australia (Nathan and Weinmann, 2001).

4.1 Flood Frequency Analysis with Historical/Paleoflood Data

There are three main techniques that Reclamation currently uses to develop a peak-flow frequency curve and integrate streamflow (gage) data, historical data, and paleoflood data. The first is a mixed-population graphical approach (England et al., 2001). The two other techniques are statistical models that use gage, historical, and paleoflood data. The Expected Moments Algorithm (EMA) (England, 1999) uses moments to estimate the parameters of a log-Pearson Type III (LP-III) distribution and is consistent with Bulletin 17B (IACWD, 1982). A Bayesian maximum likelihood approach is used by FLDFRQ3 (O’Connell, 1999) to estimate a peak-flow frequency curve with historical and paleoflood data and uncertainties. All three techniques have been used for estimating flood peaks at various Reclamation dams.

4.1.1 Historical and Paleoflood Data

Many different kinds of historical and paleoflood data can be used for flood frequency analysis. Historical flood data are typically extreme floods that have occurred and were described in some qualitative or quantitative fashion before establishing a stream gaging station. The typical information that is available for historical floods is the date of occurrence and the height of the water surface (Thomson et al., 1964). In many cases, people physically mark, on a relatively permanent surface, the approximate high-water mark of a flood (Thomson et al., 1964; Leese, 1973; Natural Environment Research Council, 1975; Sutcliffe, 1987; Fanok and Wohl, 1997).

Paleoflood hydrology is the study of past or ancient floods that occurred before the time of human observation or direct measurement by modern hydrologic procedures (Baker, 1987). The basic types of paleoflood indicators that are useful for flood frequency analysis are paleostage indicators and botanical evidence (Wohl and Enzel, 1995; Baker, 2000). Recent investigations, techniques, and analyses for collecting and using paleoflood data are discussed in House et al. (2002). Fluvial geomorphic evidence includes erosional and/or depositional features that are used to infer paleostages or non-inundation levels. The fluvial geomorphic evidence used in paleoflood and flood frequency studies that represents paleostage indicators includes: silt lines, scour lines, slackwater deposits, boulder and gravel bars, and modified geomorphic surfaces (Costa, 1978; Baker, 1987; Kochel and Ritter, 1987; Jarrett and Costa, 1988; Salas et al., 1994; Jarrett and England, 2002; Levish, 2002). Botanical evidence consists of vegetation that records evidence of a flood (or several floods) or indicates stability of a geomorphic surface for some time period. Botanical evidence of floods includes: corrosion scars, adventitious sprouts, tree age, and tree-ring anomalies (Hupp, 1987).

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e' = 3 e = 1

k = number of floods exceeding Qo = e + e' = 4

historical period h systematic (gage) record s

Qt

t total record length n = h + s

Peak Discharge

Water Year

discharge threshold Qo

Figure 4-1.—Example of peak discharge time series with historical period and discharge threshold Q

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o. The shaded area represents floods of unknown magnitude less than^ Qo.

4.1.2 Mixed-Population Graphical Approach

A mixed-population graphical peak-discharge frequency approach has been developed by Reclamation (England et al., 2001). The graphical approach is an at-site frequency method and the frequency curve is constructed in two distinct parts: (1) standard hydrologic statistical methods are used to define a frequency curve for return periods less than and including the 100-year return period (e.g., IACWD, 1982; Ries and Crouse, 2002) and (2) graphical methods are used for estimates greater than the 100-year return period. Peak discharge estimates from gaging stations are used to define the first part of the curve and at-site paleoflood data are used to define the second part of the curve. The first part is estimated assuming an LP-III distribution. One of three at-site techniques and associated computer programs is typically used to estimate the parameters of the LP-III distribution, calculate quantiles, and estimate confidence intervals: (1) the Bulletin 17B Method (IACWD, 1982) and FREQY (Carson, 1989); (2) expected moments methods (Cohn et al., 1997) and EMA (England, 1999); or (3) Bayesian maximum likelihood and FLDFRQ3 (O’Connell, 1999). Historical information is included in the at-site frequency analysis when it is available. Historical data can be used to adjust a so-called “high outlier” using FREQY, EMA, or FLDFRQ3. Low outliers can be adjusted using IACWD (1982) methods. The second portion of the frequency curve is estimated assuming a 2-parameter log- Normal (LN-2) distribution. It is defined between the 100-year and the available paleoflood data return periods, and extrapolated beyond the paleoflood data using this LN-2 distribution. Two points are typically used to estimate this portion of the flood-frequency curve: (1) the LP-III

Guidelines for Evaluating Hydrologic Hazards

model 100-year peak discharge estimate and (2) the midpoint in time and discharge of the paleoflood data. Logarithms (base 10) of the peak flows and standard Normal variates of return periods are used to estimate the LN-2 parameters using least squares (England, 2000). The LN- distribution was found to reasonably represent daily standardized precipitation in the western United States (Lane, 1997).

The mixed-population graphical approach is used to estimate flood hazard curves. The approach has been developed so that one can estimate an extreme flood frequency curve at any location in the western United States with a minimal amount of effort using existing streamflow data and some site-specific paleoflood data. There are two main assumptions of this graphical approach for estimating extreme flood probabilities: the upper portion of the frequency curve is appropriately defined by the 100-year peak discharge and paleoflood data and the extrapolation of this portion of the curve using a LN-2 model is appropriate. An example peak-flow frequency curve using the graphical approach is shown in figure 4-2. The approach has been reviewed by Kuczera (2000). Kuczera pointed out the major weaknesses were the use of an envelope curve, lack of confidence intervals, and extrapolation. Kuczera recommended that regional growth curves be used to compliment the use of envelope curves.

1 10 100 1000 10000

5000

6000

7000

8000 9000 1000010000

20000

30000

40000

50000

60000

70000

80000

(^10000010000090000)

200000

300000 General Storm PMF Peak Discharge 233,700 ft^3 /s

Spillway capacity 53,000 ft^3 /s at El. 3565 (top of dam)

This flood frequency relationship is based on available streamflow data and preliminary paleoflood data. This information presented is only suitable for use in a CFR Baseline Risk Analysis. These curves should not be extrapolated. Refer to the report for a discussion of uncertainties.

Preliminary Regional Paleoflood Estimates 70,000 - 165,000 ft^3 /s in 200 to 10,000 years

Preliminary Range of Regional Peak Discharge Observations 70,000 - 165,000 ft^3 /s

Observed Peaks middle estimate upper and lower estimates

Peak Discharge (ft

3 /s)

Return Period (years)

Figure 4-2.—Example application of mixed-population graphical flood frequency curve using peak discharges on the South Fork Flathead River near Hungry Horse, Montana.

4.1.3 Expected Moments Algorithm

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The EMA (Lane, 1995; Lane and Cohn, 1996; Cohn et al., 1997, 2001) is a new moments-based parameter estimation procedure that was designed to incorporate many different types of systematic, historical, and paleoflood data into flood frequency analysis. EMA assumes the