Dr. Ardhasena SopaheluwakanCenter for Climate, Agro- and Marine-Climate, BMKGBMKG

Characterization of precipitation time series using copulas: the case of the Jakarta Observatory and proposal for YMC

Badan Meteorologi Klimatologi dan Geofisika

BMKG

Contents

Idea: characterizing wet events using bivariate distribution approach:severity and intensity

Data for this study

1880 1900 1920 1940 1960 1980 20000

100

200

300

400

500

600

700

800

Year

Rai

nfal

l [m

m]

Time series for this study:

Monthly data from 1864 2013 [150 years]

BMKG

HomogeneityBMKG

Homogeneity testing was carried out using the Penalized Maximal F-test methodwith RHtestV4 from Wang et.al.

The testing finds no inhomogeneity of the (log-transformed) monthly precipitationvalues.

1880 1900 1920 1940 1960 1980 20000

1

2

3

4

5

6

Year

Log

Rai

nfal

l [m

m]

SPI category & characteristics of wet monthsBMKG

SPI value range Category

SPI > 2 Extremely wet

1.5 < SPI 1

Intensity of a wet event:Severity/Duration

In this study we are interested to study the characteristics of wet months, with SPI > 1, for two periods:Period1: 1864 1963, andPeriod2: 1964 2013

CopulasBMKG

Copula function

Joint Distribution

Function

CopulasJoint CDFJoint CDF

Marginal process

COUPLING of Severity Duration Intensity:

Copulas are mapping functions that capture the dependence structureamong random variables, by joining marginal distributions

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Case study of JakartaBMKG

Considers the wet episodes from SPI3 distributions of the months DJF

Characterize severity, duration, intensity, return period, (conditional) probabilities for two different periods: 1864-1963 and 1964-2013

Scatterplot and histogram of duration, severity and intensityBMKG

1 2 3 4 5

1

1.5

2

2.5

Duration

Inte

nsity

1 2 3 4 5

0

2

4

6

8

10

Duration

Sev

erity

1 1.5 2 2.5

0

2

4

6

8

10

Intensity

Sev

erity

Period: 1864 - 1963

Scatterplot and histogram of duration, severity and intensityBMKG

Period: 1964 - 2013

1 2 3 4 5 60.8

1

1.2

1.4

1.6

1.8

2

2.2

2.4

Duration

Inte

nsity

1 1.5 2 2.5

0

2

4

6

8

10

12

Intensity

Sev

erity

1 2 3 4 5 6

0

2

4

6

8

10

12

Duration

Sev

erity

Marginals fittingBMKG

Marginal Distribution

MLEs Parameters Log-likelihood

Exponential -133.05

Weibull a = 2.405; b = 1.960 -111.53

Gamma a = 3.421; b = 0.619 -110.18

Log-normal -109.16

Duration 1864 - 1963

1.5 2 2.5 3 3.5 4 4.50

0.1

0.2

0.3

0.4

0.5

Duration

Den

sity

1.5 2 2.5 3 3.5 4 4.5 5 5.50

0.1

0.2

0.3

0.4

0.5

0.6

Duration

Den

sity

Marginal Distribution

MLEs Parameters Log-likelihood

Exponential -87.26

Weibull a = 2.475; b = 1.846 -75.26

Gamma a = 3.247; b = 0.672 -75.50

Log-normal -72.16

Duration 1964 - 2013

Marginals fittingBMKG

Marginal Distribution

MLEs Parameters Log-likelihood

Exponential -161.05

Weibull a = 3.439; b = 1.586 -149.92

Gamma a = 2.392; b = 1.279 -148.00

Log-normal -145.59

Severity 1864 - 1963

2 3 4 5 6 7 8 90

0.1

0.2

0.3

0.4

0.5

Severity

Den

sity

Marginal Distribution

MLEs Parameters Log-likelihood

Exponential -69.04

Weibull a = 3.439; b = 1.586 -149.92

Gamma a = 3.581; b = 1.631 -63.63

Log-normal -61.75

Severity 1964 - 2013

2 3 4 5 6 7 8 9 100

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

Severity

Den

sity

Marginals fittingBMKG

Marginal Distribution

MLEs Parameters Log-likelihood

Exponential -166.27

Weibull a = 1.514; b = 4.655 -33.18

Gamma a = 23.796; b = 0.058 -18.77

Log-normal -15.93

Intensity 1864 - 1963

Marginal Distribution

MLEs Parameters Log-likelihood

Exponential -41.69

Weibull a = 1.460; b = 5.540 -2.66

Gamma a = 31.401; b = 0.043 -0.38

Log-normal -0.83

Intensity 1964 - 2013

1.2 1.4 1.6 1.8 2 2.20

0.05

0.1

0.15

0.2

0.25

0.3

Intensity

Den

sity

1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.90

0.05

0.1

0.15

0.2

0.25

0.3

0.35

Intensity

Den

sity

Return periodBMKG

Shiau, 2006

BMKG

3

5

5

10

10

20

20

5050

50

100100

100

Severity

Dur

atio

n (m

onth

)Return Periods T(D>=d and S>=s)

0 5 100

1

2

3

4

5

6

7

8

3

5

5

10

10

2020

20

5050

50

100100

100100

Severity

Dur

atio

n (m

onth

)

Return Periods T(D>=d and S>=s)

0 5 100

1

2

3

4

5

6

7

8

Period: 1864 - 1963 Period: 1964 - 2013

Using Gumbelcopula

1 2 3 4 5 6 70

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Severity

Con

ditio

nal p

roba

bilit

y

Conditional distribution of severity, given wetness duration exceeding d

Conditional probabilityBMKG

d = 1 d = 2 d = 3

For the same value ofprobability, and a givenduration, severity ofwet events areincreasing for theperiod of 1964 2013compared to the period1864 1963.

Black: 1864 1963Blue: 1963 - 2013

Badan Meteorologi Klimatologi dan Geofisika

BMKG

Conclusions and remarks

Copula based approach makes it possible to create joint (bivariate)distribution from predetermined marginals.

Characterization of wet events in terms of their joint distribution waspresented. The characterization was done for two different periods, 1864 1963 and 1964 2013.

Result shows that for the latter period, for a fix probability of occurrence of aprescribed wet event duration, the severity increases, compared to the formerperiod.

This means that in the latter period, the climate in DJF is wetter.

Badan Meteorologi Klimatologi dan Geofisika

BMKG

Proposal for YMC research activity

Using bivariate distributional approach to see the impact of MJO (through theMJO index) to in-situ daily precipitation (anomaly).

Previous studies by numerous authors using satellite data, using traditionalcorrelation approach.

Model the joint distribution of MJO index precipitation and see upper taildependence (UTD) in the joint distribution. Intuitively UTD means:

That with large values of MJO index also large values of rainfall (anomalies)are expected: when, where and under what conditions,

Badan Meteorologi Klimatologi dan Geofisika

BMKG

Proposal for YMC research activity

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Badan Meteorologi Klimatologi dan Geofisika

Thank you

ardhasena(at)bmkg.go.idardhasena(at)gmail.com

BMKG