international institute for geo-information science and earth observation (itc) isl 2004 riskcity...

International Institute for Geo-Information Science and Earth Observation (ITC)

ISL 2004

RiskCity

Introduction to Frequency Analysis of hazardous events


Extreme Events“Man can believe the impossible. But man can never believe the improbable.”

- Oscar Wilde

Linda O. Mearns NCAR/ICTP

“It seems that the rivers know the [extreme value] theory. It only remains to convince the engineers of the validity of this analysis.”

–E. J. Gumbel


ISL 2004

Objective of FA exercise• The objective of this exercise is to practice different

methods of frequency analysis for floods and for earthquakes and to gain insight in magnitude – frequency relationship.

• Keep in mind that the methods presented in this exercise are just a selection of all existing methods.

• In this exercise ILWIS is not being used.


ISL 2004

M-F relationship• Magnitude-frequency relationship is a relationship where events

with a smaller magnitude happen more often than events with large magnitudes.

• Magnitude is related to the amount of energy released during the hazardous event, or refers to the size of the hazard.

• Frequency is the (temporal) probability that a hazardous event with a given magnitude occurs in a certain area in a given period of time.


ISL 2004

M-F relationship• The M – F relationship does hold true for many different

disaster types, but as can be seen in the table, not for all:

Disaster type Occurrence possible M - F relationship

Hydro-meteorological LightningHailstormTornadoIntense rainstormFloodCyclone/ HurricaneSnow avalancheDrought

Seasonal (part of the year)Seasonal (storm period)Seasonal (“tornado season”)Seasonal (rainfall period)Seasonal (rainfall period)Seasonal (cyclone season)Seasonal (winter)Seasonal (dry period)

RandomPoisson , gammaNegative binomialPoisson, Gumbelgamma, log-normal, GumbelIrregularPoisson, gammaBinomial , gamma

Environmental Forest fireCrop diseaseDesertificationTechnological

Seasonal (dry period)Seasonal (growing season)ProgressiveContinuous

RandomIrregularProgressiveIrregular

Geological EarthquakeLandslideTsunamiSubsidenceVolcanic eruptionCoastal erosion

ContinuousSeasonal (rainfall period)ContinuousContinuousIntermittent (magmachamber)Seasonal (storm period)

Log-normalPoissonRandomSudden or progressiveIrregularExponential , gamma


Flooding

• Return period/exceeding probability

• Extreme value distribution by Gumbel method

• Intensity-duration-frequency relationships


What is the return period of a rain event over a 100 mm/day ?

Maximum daily preciptation (mm)

1930 1940 1950 1960 19700 111.8 26.6 47.4 18.38

1 116 24.6 111.6 19.48

2 112 56.8 21.6 117

3 15.42 23.2 8.2 66.2

4 24.6 17.12 11.44 50.4

5 77 44 9.9 30 60.4

6 358 35.8 3.46 19.58 28.2

7 34.4 92 50.6 140 109

8 50.8 13.94 116.6 88.6 25.4

9 9.88 41.2 20.2 30.4

Between 1935 and 1978: 9 events

4

1

1

16

3

6

5

58 Intervals, ranging from 1 to 16 years

The sum of the intervals =4 + 1 + 1 + 16 +3 + 6 + 5 + 5 = 41

Average = 41/8 = 5.1 years

Annual exceedence probability of a rain event over 100 mm/day = 100 / 5.1 = 19.5 %

FloodingReturn period/exceeding probability


Flooding

Q100 has a greater probability of occurring during the next 100 yrs (63%) than during the next 5 years ( 5%)

For the average annual risk!

Return period/exceeding probability


Year Rainfall Year Rainfall Year Rainfall Year Rainfall Year Rainfall1917 615 1927 785 1937 592 1947 541 1957 6551918 655 1928 732 1938 549 1948 632 1958 7441919 709 1929 615 1939 627 1949 564 1959 4521920 699 1930 940 1940 612 1950 612 1960 9401921 498 1931 765 1941 630 1951 688 1961 5691922 701 1932 732 1942 488 1952 597 1962 5001923 653 1933 546 1943 513 1953 544 1963 5691924 726 1934 526 1944 620 1954 721 1964 5031925 770 1935 701 1945 551 1955 538 1965 8081926 602 1936 612 1946 699 1956 622 1966 853

Flooding

Annualrainfall

450 - 500 500 - 550 550 - 600 600 - 650 650 - 700

Frequency 3 9 6 11 6

Annualrainfall

700 - 750 750 - 800 800 - 850 850 - 900 900 -950

Frequency 8 3 1 1 2

Frequency Analysis


Flooding

0

2

4

6

10

8

12

500 550 600 650 700 750 800 850 900450 950

Rainfall classes (mm)

Fre

qu

enc

yFrequency Analysis


-2 +2 +3+1-1-3 X

0.683

0.954

0.997

P(R < 642) = 0.5

P(R > 642) = 0.5

P(532 < R < 752) = 0.683

Mean: 642 mm

Standard deviation: 110 mm

P(422 < R < 862) = 0.954

P(312 < R < 972) = 0.997

FloodingFrequency Analysis


Flooding

Unfortunately, a large amount of events is right skewed;

• Magnitude of events are absolutely limited at the lower end and not at the upper end. The infrequent events of high magnitude cause the characteristic right-skew

• The closer the mean to the absolute lower limit, the more skewed the distribution become

• The longer the period of record, the greater the probability of infrequent events of high magnitude, the greater the skewness


Flooding

0

2

4

6

10

8

12

500 550 600 650 700 750 800 850 900450 950

Rainfall classes (mm)

Fre

qu

en

cy

• The shorter the time interval of recording, the greater the probability of recording infrequent events of high magnitude, the greater the skewness

• Other physical principles tend to produce skewed frequency distributions: e.g. drainage basin size versus size of high intensity thunderstorms.


FloodingSolution to right-skewness:

Use Extreme Value transform (other names: Double exponential transform or Gumbel transformation):

1. Rank the values from the smallest to the largest value

2. Calculate the cumulative probabilities: P=R/(N+1)*100%

3. Plot the values against the cumulative probability on probability paper and draw a straight line (best fit) through the points

4. From the line, estimate the standard deviation and mean

5. Estimate all other required probabilities versus values


FloodingExtreme value distribution by Gumbel method


FloodingIntensity-duration-frequency relationships

IDF curves are calculated for a certain station and it cannot be extrapolated to other areas.

Intensity

Duration


Per day

0.5436360 (every 4 minutes….)

3600 (every 24 seconds….)

Earthquakes


Earthquakes


Earthquakes• The Gutenberg-Richter Relation

log N(M) = a – bM

Gutenberg-Richter plots are made for various data sets all over the world, and most of them end up having a b value very close to 1, usually slightly less.

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