Extreme Flood Frequency 

Analysis: 

Concepts, Philosophy and Strategies

Jery R. Stedinger

Cornell University and USGS

Member HFAWG

sabbaticals USGS, US ACE

with V. Griffis, A. Veilleux, E. Martins, T. Cohn
Challenge

Estimate frequency of rare floods 

(exceedance probabilities of 1 in 100, 

or 

1 in 10,000) using limited data provided 

by streamflow gauge at the site ...

plus other sources of information.
What Data Do We Use?

Record at streamflow gauge at the site.

Historical data at the site

Paleoflood data at the site

Regional analysis of gaged data, 

historical and paleoflood Information
Bulletin 17B

• Uniform flood frequency techniques used 

by US Federal agencies 

• Bulletin not updated in 20+ years

– despite significant amount of research

– additional 30 years of data for skew map

– better statistical procedures for censored data
Bulletin 17C

• Revision on the way

• Provides a tune up addressing

– Use of historical information

– Potentially Influential Low Floods (PILFs)

– Censored & interval data

– Confidence intervals & uncertainty analyses

• Also developing better regional skew
Regional Skew Estimation

Regional skew Gg

from B17 

skew map

-0.5 < Gg <+0.7

 

Map SE = 0.55

 MSE[Gg] = 0.302

Effective record

length = 17 yrs
Regional Skew Southeast

1976 (MSE = 0.302)

2,972 sites nationally

2009 (MSE = 0.14) revised 1/09

342 independent watersheds

γ = 0 as good as 40 

years of record

Maps claims to be as 

good as 17 years
Historial & Paleoflood

data at a site:

Ways to extend the record.
Sources of Historical Information

Long-term Residences’ Memory

Written accounts, markers, pictures

Geomorphologic evidence 

slack-water deposits

scour lines

high-water marks

undisturbed areas
Paleoflood Data

(Source: Jarrett 1991, modified from Baker 1987)

Paleoflood stage 

Non-exceedence level 

(bound) 

Threshold level
Historical and Gauged Record

 

h years s years
 

Value of Historical Information

i.e. Bulletin 17B
 

S = 20

84

70

50

84 - 20 = 64 additional yrs

70 - 20 = 50 additional yrs

50 - 20 = 30 additional yrs

Value of Historical Information
 

S = 20

Value of Historical Information

i.e. Bulletin 17B
 

S = 20

40

24

40 - 20 = 20 additional yrs

24 - 20 = 4 additional yrs

Value of Historical Information
Effective record length

 

i.e. Bulletin 17B
Average Gains when Estimating 100-year Flood by 

fitting 2-Parameter Lognormal Distribution 

(s = 20 years; h = 100 years)

 

i.e. Bulletin 17B
Average Gains when using MLE's 

to Estimate 1, 2, and 3 Lognormal Parameters

 
Observation:

Much larger Average Gains when 

estimating 3 parameters.

Shape parameter is hard to pin down 

with short records, and their estimators 

are sensitive to extreme observations.
Generalized Extreme Value 

(GEV) distribution
GEV Prob. Density Function large x
Index Flood & Regionalization

Substituting Space for Time.
Principles for better estimates of 

extreme flood probabilities

U.S. National Research Council report (1988)

• Substitute space for time using "regionalization."

• Introduce additional structure into models based 

upon physical insight and hydrologic experience.

• Focus on extremes or "tails" of flood distribution, so 

less important and common floods do not distort fitted 

distribution for extreme events.
Hosking and Wallis 

(1997)

Development of

L-moments for regional 

flood frequency 

analysis.
Index Flood

procedure

combines many 

flood records

Quantile

Estimator

Flood Frequency Analysis 

Paradigm 
New Paradigm for Flood 

Frequency Analysis
Credibility of Estimators

• At site record: 50 - 100 years

• With historical info: 100-200 years

• Regional 20 records: 20*50 = 1,000 years

• At-site & paleo. info: 1,000 years

• At-site & regional paleofloods 10,000 years

See USBR(1999, 2004) for similar analysis

of limits of credibility of empirical estimators
Empirical Analysis Assumes

• Observed records contains representatives of 

critical events that represent LARGE & RARE 

floods at a site.

• Nature is “smooth” and “continuous” so that 

fitting a mathematically nice GEV or LP3 

distribution is reasonable for extrapolation. 
Estimating Flood Risk

B17B

Empirical

Use observed

Flood Data

PFHA (?)

Imagine

Construct using

regional flood,

rainfall, climate, 

storm-track, & 

orographic data

T = 100 to 1000 comfortably T = 10,000 to 1 million
Addressing Uncertainty

• Consider

Natural Variability (Aleatory Uncertainty) versus 

Knowledge Uncertainty (Epistemic Uncertainty-Limited 

data to estimate parameters)?

• Do they combined to get total risk? 

• Do we give nominal risk with uncertainty bounds 

reflecting epistemic uncertainty?

• Flood risk estimates in US (following Bulletin 17B) 

include only aleatory uncertainty. 
MCMC for LP3 Quantiles with

only systematic information, s = 100
Ilustration of Uncertainty

Drew samples of size n = 100

From 2-parameter lognormal distribution.

Fit 2-parameter lognormal distribution AND

Fit 3-parameter LP3 (with log moments)

Estimated events exceeded with probability of 

1 in 100 through 1 in a million.

Report ratio of exceedance probability of “design floods”

to anticipated exceedance probability.

and Coefficient of Variation of “design flood” estimators.
Coefficient of Variation of Design 

Flood Estimators

1 in 100 1000 10,000 100k Million

 Nominal Exceedance Probability p

Consider 2-p Lognormal, and 3-p LP3 distributions
Error in Exceedance Probability

Ratio exceedance probability of “design flood” to p

1 in 100 1000 10,000 100k Million

 Nominal Exceedance Probability p

2-p Lognormal

3-p LP3 distribution
New Paradigm for Flood 

Frequency Analysis

Reflecting

Uncertainty in

Model parameters
Historical advice
“We must accomplish a task 

more difficult to many minds 

than daring to know. 

We must dare to be ignorant.”

Concluding two sentences:

Karl Pearson

The Grammar of Science, 1892
“In view of the magnitude and ubiquity of the hydroclimatic 

change apparently now under way, however, we assert 

that stationarity is dead and should no longer serve as a 

central, default assumption in water-resource risk 

assessment and planning. Finding a suitable successor is 

crucial for human adaptation to changing climate.”

Science, Feb. 2008
Clmate Change – So What?

(1) Gaged records tell us what common floods 

look like and where we start. 

(2) Then we need to estimate the change.

(3) But how to determine where we go from here?
Climate Change & Risk Projection

• Imagine 50-100 year record of floods 

• Plus paleoflood record illustrating largest 

floods over 1000-2000 years.

• Fit a line to estimate 1-in-100,000 risk ?

• Paleoflood record likely includes periods of 

greater & lesser risk, so risk extrapolation 

OVER estimates actual risk in our time frame.
Summary

• Flood records for last century do not reveal reliably flood 

risk of 1 in 10,000 or 1 in a million.

• At-site Historical and Paleoflood information helps.

• Regionalization of recent & paleoflood records helps

• Extrapolation more uncertain with 3-parameter LP3.

• Climate change confuses things

– Public loss of confidence in analysis

– Extrapolating beyond paleoflood data likely conservative
Issues for Discussion

• Records of real floods may be insufficient for our 

purposes: analysis instead needs to imagine.

• Uncertainty can be large part of computed total risk.

– Are we eady to combine variability & knowledge 

uncertainty into a total risk?

– Can we do the computation appropriately using 

honest informative Bayesian priors ? 
Thank you for your attention.

Jery Stedinger
Donald Rumsfeld, Secretary of Defense 

under President George W. Bush from 2001 to 2006

said at Department of Defense news briefing, February 2002: 

"Reports that say that something hasn't happened are always 

interesting to me, because as we know, 

there are known knowns;

there are things we know we know. We also know there are 

known unknowns; that is to say 

we know there are some things we do not know. 

But there are also unknown unknowns

-- the ones we don't know we don't know. 

And if one looks throughout the history of our country and other 

free countries, it is the latter category 

that tend to be the difficult ones."
Generalized Extreme Value 

(GEV) distribution

Gumbel's Type I, II & III Extreme Value distr.:

F(x) = exp{ – [ 1 – (κ/α)(x-ξ)]1/κ } for κ ≠ 0

μX = ξ + (α/κ) [1 – Γ(1+κ)]

Var[ X ] = (α/κ)2 [Γ(1+2 κ) –- Γ2(1+κ)]

E[ (X-μX)3 ] = (α/κ)3 [ – Γ(1+3κ) + 3 Γ(1+κ) Γ(1+2 κ) – 2Γ3(1+κ) ]

κ = shape; α = scale, ξ = location.

Mostly 

-0.25 < κ ≤ 0