1 october 2009

Regional Frequency Analysis (RFA)

Cong Mai Van

Ferdinand Diermanse

1 october 2009

Problem definition

10-3

10-2

10-1

100

0

100

200

300

400

500

600

700

800

annual probability of exceedance [year-1]

rain

fall

inte

nsity

[m

m/h

r]

HKO; duration: 1 dayGEV: k= -0.08 sig= 59.91 mu= 159.87

data

GEV fit95% bounds

1 october 2009

Trading space for time

Available time series are “by definition” too short for extreme value analysis

consequence: large uncertainties

combining data from different stations (trading space for time) can reduce the uncertainties

1 october 2009

RFA

Principle: pooling data by using information from neighbouring locations,

which are considered from homogeneous regions

The main stages:

1. screening of data;

2. identification of homogeneous regions;

3. test for discordant stations

4. choice of a regional frequency distribution;

5. estimation of the regional frequency distribution.

1 october 2009

How to combine?

Popular method: the Index Flood Method: distributions in all locations are

assumed to be multiples of the “average” distribution (the regional growth curve) -> shape is the same for al stations

f

regional growthcurve Qr(f)

Q

Qi(f)= µiQr(f),

1 october 2009

derivation of regional growth curve

normalise data: divide data by mean (µ) -> new mean = 1

derive fits of normalised data for each station

regional growth factor = “mean” of all fits e.g.:

o mean of the parameters of the distribution functions

o for each frequency: mean of the quantiles (mean [Qi(f)])

1 october 2009

L-moments approach in RFA

described by Hosking and Wallis , 1997 (the “bible of RFA”)

L-moments (linear moments) are alternative estimates of the classic statistical moments (mean, standard deviation, skewness and curtosis)

found to be “superior” in estimating parameters of distribution functions in many applications

1 october 2009

coefficients of L-mean for n=20

0 2 4 6 8 10 12 14 16 18 200

0.2

0.4

0.6

0.8

1

sorted value

coef

fcie

nt

lmoment: 1

*1/20

1/n*(X1+X2+ … + Xn)

1 october 2009

Concept of L-variance: example with 2 observations

measure for “spread”

X2 – X1

1 october 2009

coefficients of L-variance for n=50

0 5 10 15 20 25 30 35 40 45 50

-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

sorted value

coef

fcie

nt

lmoment: 2

*1/50

1 october 2009

Concept of L-skewness: example with 3 observations

measure for “spread”

X3 -2X2 + X1

1 october 2009

coefficients of L-skewness for n=50

0 5 10 15 20 25 30 35 40 45 50-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

1.2

sorted value

coef

fcie

nt

lmoment: 3

*1/50

1 october 2009

coefficients of L-kurtosis for n=50

0 5 10 15 20 25 30 35 40 45 50

-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

sorted value

coef

fcie

nt

lmoment: 4

*1/50

1 october 2009

selection of distribution function based on L-moments

Skewness and Kurtosis provide information about the shape

-0.1 0 0.1 0.2 0.3 0.40

0.05

0.1

0.15

0.2

0.25

glog

gev

logn3

pearson-III

gpd

L-skewness

L-ku

rtos

is

L-moment ratio diagram

stations

region

1 october 2009

Identification of discordant stations

-0.1 0 0.1 0.2 0.3 0.40

0.05

0.1

0.15

0.2

0.25

L-skewness

L-ku

rtos

is

discordant stations

stations

discordant stations

Wilks’ discordancy test

1 october 2009

Region homogeneity test

simulate a homogeneous region with same L-moments as the observed region

sample large number of simulated series in all stations

derive measure of heterogenity for each sampled set of simulated series.

compare observed meaure of heterogeneity with measures of simulated series

1 october 2009

example

0.01 0.015 0.02 0.025 0.03 0.0350

20

40

60

80

100

120

140

V

prob

abili

ty d

ensi

tyobserved and simulated values of heterogenity measure V

PDF

observed value

1 october 2009

Original fits (lines are crossing, large diversity)

10-3

10-2

10-1

100

0

100

200

300

400

500

600

700

probability of exceedance]

rain

fall

inte

nsity

[m

m/h

r]

fits before application of RFA

1 october 2009

RFA fits (no crossing lines, smaller diversity)

10-3

10-2

10-1

100

50

100

150

200

250

300

350

400

450

probability of exceedance]

rain

fall

inte

nsity

[m

m/h

r]

fits after application of RFA

1 october 2009

compare original and RFA fit

10-3

10-2

10-1

100

0

50

100

150

200

250

300

350

400

450

probability of exceedance]

rain

fall

inte

nsity

[m

m/h

r]

fitted gev-distributions for station: station_1

data

initial fitRFA fit

1 october 2009

Wave recording locations along the Dutch coast

Station Abbrev.

1. Eierlandse Gat ELD

2. Euro Platform EUR

3. K13A Platform K13

4. Lichteleiland Goeree LEG

5. Noordwijk Meetpost MPN

6. Scheur West SCW

7. Schiermonnikoog Noord SON

8. Schouwenbank SWB

9. Ijmuiden Munitie Stortplaats YM6

Application 1: Dutch coast North sea

1 october 2009

Station N D(I)

SCW 23 1.96

MPN 24 2.43

SWB 47 1.35

LEG 33 0.9

ELD 52 0.11

EUR 67 0.45

K13 72 0.17

SON 58 1.23

YM6 61 0.42

Wave data Dutch Coast

Discordance tests of the datasets

1 october 2009

Homogeneity tests

Wave data Dutch Coast

1 october 2009

Goodness of fit to find a “bestfit”

Distribution

Wave data

L-Kurt. Z Value

GEN. LOGISTIC 0.212 6.21

GEN. EXTREME VALUE 0.178 4.23

GEN. NORMAL 0.165 3.48

PEARSON TYPE III 0.141 2.1

GEN. PARETO 0.097 -.52 *

1 october 2009

RFA of waves data, Dutch coast

conventional fitconventional fit

1 october 2009

Application 2: Vietnam coast

Station Abbrev.

PhuLe NamDinh Phule

VanUc ThaiBinh Vanuc

DoSon HaiPhong Doson

CuaCam HaiPhong Cuacam

HonDau HaiPhong Hondau

VanLy NamDinh Namdinh

BinhMinh NinhBinh Ninhbinh

BaLat-SongHong Balat

AnPhu HaiPhong Anphu

1 october 2009

Discordance & homogeneity test

Station N D(I)

Phule 108 0.46

Vanuc 80 0.25

Doson 100 2.32

Cuacam 91 0.99

Hondau 100 1.89

Namdinh 99 1.04

Ninhbinh 87 0.31

Balat 60 0.04

Anphu 100 1.71

1 october 2009

RFA of storm surge data, Vietnam coast

0.9 1 1.1 1.2 1.3 1.4 1.510

-3

10-2

10-1

100

Extreme Water Levels (Normalized) [-]

Exc

eeda

nce

freq

uenc

y (N

orm

aliz

ed)

[-]

Normalized 9 South China Sea locations along Vietnamese Coasts

0.8 0.9 1 1.1 1.2 1.3 1.4 1.510

-3

10-2

10-1

100

Extreme Water Levels (Normalized) [-]

Exc

eeda

nce

freq

uenc

y (N

orm

aliz

ed)

[-]

Regional Frequency Distribution fitted all 9 sites

0.8 0.9 1 1.1 1.2 1.3 1.4 1.510

-4

10-3

10-2

10-1

100

Extreme Water Levels (Normalized) [-]

Exc

eeda

nce

freq

uenc

y (N

orm

aliz

ed)

[-]

Regional Frequency Distribution fitted to 8 sites

300 350 400 450 500 55010

-4

10-3

10-2

10-1

100

101

Extreme Water Levels (cm)

Exc

eeda

nce

freq

uenc

y (p

er y

ear

Exceedance RF curve for Phule station, Nam Dinh coast

PhuLeVanuc

Doson

Cuacam

HondauNamdinh

Ninhbinh

BalatAnphu

RFA fitted Wakeby

RFA fitted GPD

RFA fitted Wakeby

RFA fitted GPD

RFA fitted GDP at-site

conventional fit

1 october 2009

Discussions

•The GPD appears to be the optimal regional fit for the POT extreme sea datasets.

•Uncertainty of the quantile estimates with RFA for both application cases is found smaller than conventional data fitting methods

•Differences between the at-site quantile estimates and the regional quantile estimates can be quite high (up to ~1.0m for the extreme extrapolations of 10.000 years). It is better to rely on the regional quantile estimates for decision making, as suggestion in Hosking and Wallis (1997).

•A convex curvature is presented in the normalized regional growth curves for both wave and storm surge data. This would lead to a regional upper limit of the extreme value for waves and surges, which is more physical relevant.

•Adding more sites to the existing databases for each data type may result in more accurate predictions of the extreme quantiles.