Characterization and Short-term prediction of Droughts

over India using Copula-based Approaches

Poulomi Ganguli

Postdoctoral Research Associate, Northeastern University

The work reported here is based on Ganguli’s dissertation at IIT Bombay in Mumbai, India under

supervision of Prof. M. Janga Reddy in Civil Engineering

Sustainability & Data Sciences Laboratory

Northeastern University, Boston, MA

Date: July 26, 2013

Drought & Water Stress

~1/5th of the world’s population currently experiencing water stress

Water scarcity projected to increase by 2050

(Vörösmarty et al. 2010; Arnell et al. 2011)

Water scarcity can impact both water and food security

Properties

Deficient rainfall compared to a regional average (Druyan 1996)

Multi-attribute: Severity, Duration, Peak, spatial Extent

Recurring feature of climate variability (or change)

Extended period (season, a year, or more) and larger spatial extentcompared to most other climate or weather extremes

Impacts are non-structural & difficult to quantify (Obasi 1994).

Global Water Scarcity

Source: World Water Development

Report 4: World Water Assessment

Programme (WWAP), March 2012

Little or no water scarcity

> 75% of river flows are withdrawn

> 60% of river flows are withdrawn

< 25% of river flows are

withdrawn, but malnutrition exists

Not estimated

Classification of Major Drought Types and Monitoring

Meteorological Hydrological Agricultural Socio-economic

(Precipitation deficit ) (Reduction in surface &

sub-surface water flow)

(soil moisture & plant

behavior due to

insufficient water

content)

(Water-supply

shortages – failure of

water management

practices)Indices

• Percent of Normal

precipitation (PNP; Werick

et al. 1994)

• Effective Drought Index

(EDI; Byun & Wilhite 1999)

• Deciles (Gibs & Maher 1967)

• Standardized Precipitation

Index (SPI; Mckee et al. 1993)

Indices

• Standardized Runoff Index

(SRI; Shukla & Wood 2007)

• Palmer Hydrological Drought

Index (PHDI; Palmer 1965)

• Surface Water Supply Index

(SWSI; Shafer & Dezman 1982)

• Groundwater Resource Index

(GRI; Mendicino et al. 2008)

Indices

• Crop Moisture Index (CMI;

Palmer 1968)

• Soil Moisture Deficit Index

(SMDI; Narasimhan & Srinivasan

2005)

• Soil Moisture Percentile

(SMP; Andreadis et al. 2005)

• Normalized Difference

Vegetation Index (NDVI;

Rouse et al. 1973)

Indices

• Falkenmark Indicator

(Falkenmark 1989)

• Relative Water Demand

(RWD; Vörösmarty et al. 2000)

• Water Poverty Index (WPI;

Sullivan et al. 2003)

• Water Stress Indicator

(WSI; Smakhtin et al. 2005)

• Watershed Sustainability

Index (WSI; Chavez & Alipaz

2007)Our Focus

Annual Temperature

Annual PDSI

Annual Precipitation

Annual Soil Moisture

Palmer model derived PET

Palmer model derived ET

CCSM4 ETSta

nd

ard

ize

d V

alu

e

Year

• PDSI was originally designed as a surrogate metric for soil moisture and runoff

• Currently, soil moisture and runoff may be directly available

• PDSI may exaggerate future drought severity

• Soil moisture projections may show relatively reduced drying and drought frequency

Is a transition to semi-permanent drought

conditions Imminent in US Great Plains ? Hoerling et al. (2012)

Abrupt decline in PDSI

decline in soil moisture values

Departure in CCSM4 ET is consistent

with decline in soil moisture

Increasing drought under global warming in observations and models

Dai, 2012, Nature Climate Change

Pre

cip

ita

tio

nP

DS

I%

Dry

Are

a

PDSI_Th

PDSI_PM

Little change in global drought over past 60 years

Sheffield et al., 2012, Nature

“major issue with forcing data for

calculation of PDSI” – Trenberth 2012

Indices do matter!

Year

Sc_PDSI_PM all forcing

Sc_PDSI_PM without obs. surface

warming

Dependence among drought attributes convey important information but

cannot be well handled by traditional multivariate modeling

ti te

t

SP

I

Time Interval (months)

,

1

D

i i t

t

S SPIDrought severity

0

Non drought

duration (Dn)Drought duration (D)

Threshold level

Drought Events

-2.00

-1.00

1.00

2.00

2

3.00

-3.00Peak (P)

1

t

SPIthr 20 percentile

(SPI) Droughts: Attributes and Indices

Intensity, ,

1_

1. 1,2,...,

gridN

t i t thr i t

idr grids

I SPI SPI SPI t nN

1

Spatial Extent,

1

1

.

, 1,2,...,

grid

grid

N

i t thr i

it N

i

i

SPI SPI A

A t n

A

1

Copulas

Copulas are mapping functions that capture the rank-invariant dependencestructure among random variables, which is obtained by joining marginaldistribution of any form.

Marginal process Copula functionJoint Distribution

function

F1(x1)

F2(x2)

Fd(xd)

Copulas

C (u1, u2, …, ud)

Joint CDF

F (X1, X2, …, Xd)

Construction of Copula (Favre et al. 2004)Joint density of Gumbel-Hougaard copula (http://rgm3.lab.nig.ac.jp/RGM/)

Different Copula Class Employed & their Significance in Hydrologic Literature

Archimedean Elliptical

Bivariate Case

AMH Clayton Frank BB1

Extreme value

Gumbel - Hougaard Galambos

Plackett

Student’s t

1u2u

2

3u

1

Trivariate Case

Fully Nested Archimedean (FNA) Student’s t

Michele & Salvadori

(2003); Favre et al.

(2004)

Chowdhary et al.

(2011)

Genest & Favre (2007) Shiau (2006)

Kao & Govindaraju

(2008)

Grimaldi & Serinaldi (2006)

Genest et al. (2007)

Hierarchical Structure of FNA

Clayton Gumbel - Hougaard Frank

Understanding Droughts over India: Challenges & Opportunities

Disaster prone regions in India. Source:

http://www.ldeo.columbia.edu/chrr/research/pr

ofiles/india

Extreme drought years in India. Source: Jayaraman (2003)

• Possible failures of southwest monsoon

• Drought prone areas in India lie inarid (19.6 %),semi – arid (37%) andsub-humid (21%) regions

(CMP, 2010)

• Finer resolution and multivariate patterns important

• Climate variability and change impacts

• Projection uncertainties

Droughts over India: Applications and Case Studies

Evaluate the potential of copulas for droughts

Multivariate frequency of droughts

Drought risk assessment

• Severity-Duration-Frequency (S-D-F)

• Intensity-Area-Frequency (I-A-F) curves

Identification of critical drought regions

Climate Variability & Droughts

• Groundwater drought studies

• ENSO and droughts

Climate Change & Droughts

• Ensemble drought prediction model (laggeddrought indices & large scale climate indices)

Location Map of Study Areas

Trend Analysis of SPI-6 time series using Mann-Kendall Test with correction for ties and Autocorrelation

• Long-term trend – 110 yrs (1896 – 2005)

• trend in June (J), July (J) & June – Sept (JJAS) period for W. Rajasthan

• trend in June for Saurashtra – Kutch; Aug. and JJA S period for Marathwada

• No substantial drying trend was observed

• Short term trends for time windows: 36 yrs (1896 – 1931), 35 yrs (1932 – 1966) & 39 yrs (1967 – 2005)

• For 1932 – 1966 in June for W. Rajasthan & Sauarashtra – Kutch

• No trend at other time windows

Trivariate drought property – Severity (S), duration (D) & Peak (P) - Student’s t copula

Computation of Joint Return Period: Primary - OR (∪) & AND (∩) & secondary ( )

•

•

•

•

Shift towards Near

Normal/wetter

Condition

No. of drought

occurrence during

this period

T

; ;DSP DS DSP SP DSP DPT T T T T T

DSP ST T

DSP ST T

DSP DSP DSPT T T

Sub-critical (F (X) < t)

Super-critical (F (X) > t)

critical layer

Concept of Secondary Return Period

Conditional probabilities of drought severity given duration (months) and peak exceeding 50th percentile threshold level

W. Rajasthan Saurashtra & Kutch Marathwada

• Severity-Duration-Frequency (S-D-F) relationship provides regional drought characteristics

• Series of curves at different return periods are plotted keeping duration on horizontal axis &severity as vertical axis.

• Can be obtained from Copula-based conditional return period

| |

| |

,,

1 | 1

S D

S D d S D d

S D S D d D

C F s F dT C

F s d C F d

where N = total length of SPI series (years); n = no. of drought eventsN

n

Duration (months)

Se

ve

rity

5-yr

10-yr

50-yr

S-D-F diagram

0 4 8 120

5

10

15

20

25

30

Duration (months)

Severity

(a)

2-year 5-year 10-year 25-year 50-year 100-year

0 4 8 120

2

4

6

8

10

12

14

Duration (months)

Severity

(b)

0 4 8 120

2

4

6

8

10

12

14

Duration (months)

Severity

(c)

S-D-F curves for (a) Western Rajasthan (b) Saurashtra and Kutch (c) Marathwada during 1900 – 2008 obtained using conditional

distribution of Gumbel-Hougaard copula; The best copula was selected using goodness-of-fit test (Genest et al. 2009) based on

parametric bootstrap approach and upper tail dependence test

Trend analysis of gridded (0.5° × 0.5°) SPI-6 during 1971-2005 in western Rajasthan during (a) June (b) July (c)August & (d) September and (e) total monsoon (June to September) period.

(a) (b)

(c) (d) (e)

0 20 40 60 80 1000

50

100

150

200

250

300

350

Areal Threshold

Num

ber

of

month

s u

nder

dro

ught

(a)

0 100 200 300 4000

5

10

15

20

25

30

35

Number of months under drought

Fre

quency

(c)

0 20 40 60 80 1000

50

100

150

200

(b)

PAUD

Fre

quency

7 % of total area =

14, 004 Km2

240 months

(a) Number of months under drought at different percentage areal threshold (b) histogram of percentage areaunder drought (PAUD) (c) histogram of number of months under drought.

Drought intensity and PAUD values for 30 drought events identified during study period (1971-2005).

1971 1974 1977 1980 1983 1986 1989 1992 1995 1998 2001 2004 20070

0.5

1

1.5

2

2.5

Inte

nsity

Year

1971 1974 1977 1980 1983 1986 1989 1992 1995 1998 2001 2004 20070

20

40

60

80

100

PA

UD

Intensity PAUD

Kendall’s = 0.34;Spearman’s = 0.47

(p-value < 0.0001)

0 10 20 30 40 50 60 70 80 90 1001

1.5

2

2.5

Percentage Area under Drought

Inte

nsity

2-year

5-year

10-year

20-year

25-year

50-year

Drought Intensity – Area – Frequency (I-A-F) curves at different return periods using Gumbel-

Hougaard copula and copula-based conditional distribution

-2 0 2 4-6

-4

-2

0

2

4

MEI Index

SP

I-6

Spearman's = -0.14

(a)

-4 -2 0 2 4-3

-2

-1

0

1

2

3

MEI Index

SP

I-6

Spearman's = -0.30

(b)

-2 -1 0 1 2-6

-4

-2

0

2

4

MEI Index

SP

I-6

Spearman's = -0.14

(c)

Scatter plots of SPI-6 and MEI time series at each ENSO phase during 1896-2005 : (a) El Niño (b) La Niña and(c) Neutral states. The p-values of correlation as shown in Fig. are all less than 0.0001.

Climate State Drought characteristics Mean Max

Without ENSO Severity 4.0 30.68

Duration (months) 2.8 18

Peak 1.4 4.32

El Niño

Severity 5.48 30.68

Duration (months) 3.6 18

Peak 1.47 4.32

La Niña

Severity 1.91 6.05

Duration (months) 1.80 6.00

Peak 1.15 1.89

Table: Drought characteristics without & with accounting ENSO phases

ENSO state Percentile levels Drought property quantiles at different

percentiles

S D (months) P

50th 1.87 2 1.09 0.021

El Niño 75th 7.36 6 1.90 0.088

90th 12.76 7 2.49 0.332

95th 25.91 10 3.31 0.600

50th 1.19 1 0.96 0.098

La Niña 75th 2.79 2 1.50 0.515

90th 4.08 4 1.78 0.487

95th 5.22 5 1.84 0.575

50th 1.58 1 1.32 0.115

Neutral 75th 4.73 4 1.59 0.146

90th 8.22 5 2.56 0.488

95th 14.76 7 2.99 0.618

|SP D dP

Table: Conditional probability of drought severity & peak given duration exceedingcertain threshold for different ENSO state drought conditions

Conditional probability of drought severity & peak given duration d’ = 90th percentile threshold level at differentENSO state drought conditions

020

0

50

0.5

1

Severity

El Nino

PeakP

(S

s

, P

p

| d' 1

= 2

)

020

02

40

0.5

1

SeverityPeak

P (

S

s,

P

p| d' 2

= 6

)

020

02

40

0.5

1

SeverityPeak

P (

S

s,

P

p| d' 2

= 7

)0

200

24

0

0.5

1

Severity

La Nina

Peak

P (

S

s,

P

p| d' 1

= 1

)

020

02

40

0.5

1

SeverityPeak

P (

S

s,

P

p| d' 2

= 2

)

020

02

40

0.5

1

SeverityPeak

P (

S

s,

P

p| d' 2

= 4

)

020

02

40

0.5

1

Severity

Neutral

PeakP (

S

s,

P

p| d' 1

=

1)

020

02

40

0.5

1

SeverityPeak

P (

S

s,

P

p| d' 2

= 4

)

020

02

40

0.5

1

SeverityPeak

P (

S

s,

P

p| d' 2

= 5

)

(a)

(b)

(c)

El Niño La Niña Neutral

Risk Assessment of Groundwater Drought in Manjara Basin Aquifer, India

Location map of the study area

Association between ENSO-index, JJAS precipitation &DGWT is analyzed using rank correlation

Kendall’s between

• DGWT & precipitation: -0.60

• Precipitation & ENSO index: -0.37

• DGWT & ENSO index: 0.24

2 different bivariate copulas: AMH & Frank families

Observed vs. 1000 simulated samples fitted with Frank Archimedean Copulas for different hydro-climatological variables at Khed

Observed

Simulated

(a)

(b)

(c)Conditional distributions of (a) DGWT for given precipitation

intensity; (b) precipitation for given ENSO index; (c) DGWT for

given ENSO index

Deterministic prediction of SPI-6 using Support Vector Regression (SVR)

Drought prediction models at 1, 2 & 3-months lead

• Models without seasonal partition

• Combined seasonal: Feb-May (FMAM), June-September (JJAS) & October – January (ONDJ)

Generate random samples from selected copulas using inverse conditional distribution

• Ensembles of predicted SPI is retrieved from inverse probability integral transform

SPI6(t + l) = f (SPI6 (t – z1), MEI (t – z2), AMO (t – z3), IOD (t – z4), RF (t – z5))

1

Yy F v

0

0.1

0.2

0.3

0.4

0.5

0.6

1 2 3

CR

PS

Lead Times (months)

without Combined Seasonal

0

0.2

0.4

0.6

0.8

1 2 3

NSS

Lead Times (months)

without Combined Seasonal

Performance comparison of ensemble drought prediction models at different lead times

Feb-95 Jun-95 Oct-95 Feb-96 Jun-96 Oct-96 Feb-97 Jun-97 Oct-97 Feb-98 Jun-98 Oct-98 Feb-99 Jun-99 Oct-99-4

-2

0

2

4

Months

SP

I-6

Observed SPI SVR-Predicted SPI SVR-Copula Ensemble Mean

Feb-00 Jun-00 Oct-00 Feb-01 Jun-01 Oct-01 Feb-02 Jun-02 Oct-02 Feb-03 Jun-03 Oct-03 Feb-04 Jun-04 Oct-04

-4

-2

0

2

4

Months

SP

I-6

Time series comparison of 3-month lead predicted SPI using SVR & ensemble generated using SVR-copulaapproach. Box-plots shows uncertainty associated with drought prediction

Discussions

Long-term trend indicates shift towards near normal/wetter condition for westernRajasthan, Saurashtra – Kutch & Marathwada region

Short term drying trend was observed for Western Rajasthan & Saurashtra - Kutch during 1932- 1966 time frame

Severity-Duration-Frequency (S-D-F) relationship showed drought in Western Rajasthan is themost severe as compared to other regions

Regional trend analysis of SPI-6 indicate central part is the most drought affected

Combined seasonal model gives better prediction estimate as compared to the model withoutseasonal partition

Relevant Publications

Ganguli, P; Reddy, M.J. (2013). Ensemble prediction of regional droughts using climate inputs and SVM-copula approach. Accepted for

publication in Hydrological Processes.

Ganguli, P; Reddy, M.J. (2013). Analysis of ENSO based climate variability in modulating drought risks over Western Rajasthan in India.

Journal of Earth System Sciences. 1: 253 – 269.

Reddy, M.J. Ganguli, P. (2013). Spatio-temporal analysis and derivation of copula-based intensity-area-frequency curves for droughts in

western Rajasthan (India). Stochastic Environmental Research and Risk Assessment. Doi: 10.1007/s00477-013-0732-z

Ganguli, P.; Reddy, M.J. (2013). Evaluation of trends and multivariate frequency analysis of droughts in three meteorological subdivisions

of western India. Doi: 10.1002/joc.3742

Reddy, M.J.; Ganguli, P (2012).Risk Assessment of hydro -climatic variabiliy on groundwater levels in the Manjara Basin aquifer in India

using Archimedean copulas. Journal of Hydrologic Engineering. 17(12): 1345 – 1357.

Reddy, M.J.; Ganguli, P. (2012). Application of copulas for derivation of drought severity-duration-frequency curves. Hydrological

Processes. 26(11): 1672-1685.

Thank You…

Appendices

Cu

mu

lati

ve P

rob

ab

ilit

y

Aggregated Precipitation

SPI

Fig. Computation of SPI (Source: Pai et al. 2011)

More application is found in southwest Asia drought studies, due to limitedinput data requirements and simplicity in computation.

Family Parameter Space

Archimedean Class

AMH NA

Clayton NA

Frank NA

BB1 NA

Extreme value class

Galambos NA

Gumbel-Hougaard

Table: Expressions for bivariate Archimedean and Extreme value class of copula families and their associated properties (Nelsen, 2006; Joe 1997)

1 2

1 21 1 1

u u

u u

1 2,C u u A w t

1

1 2 1u u 0,1

1t

1 21 11log 1

1

u ue e

e

, 1ln

1

te

e

1,11 1

lnt

t

12

2 21 1

11

1 21 1 1u u1

2

0, ;

1,

21 1t

1

1 2 1 2expu u u u 0,1

1 1w w

1

1 2exp u u 1, 1

1w w ln t

1 1

1 1 2 2log , logu u u u

Copula Parameter Estimation

Method of moments based on rank-based dependence measures: Kendall’s or Spearman’s

Maximum Pseudo-likelihood method (MPL)

Student’s t copula: Two Step estimation procedure

,1 ,2

,1 ,2

1 1

ln ln , ln , 1,...,1 1

n ni i

i i

i i

R RL c U U c i n

n nU

ˆ argmax ln LU

,ˆ ˆsin

2i j ijr

,1 ,2 ,2, 1

ˆ ˆarg max log , ; ,n

i i i j

i

c U U r

To avoid trapping at local optimal solution using gradient-based search technique a real-coded genetic algorithm (R-GA) is applied to get optimal copula parameter.

Goodness-of-fit test

Empirical Process

Where,

2

1 2 1 2 1 2, , , , 0,1nn C u u C u u u uu =

1 2 ,1 1 ,2 2 1 2

1 1

1 1ˆ ˆ, ,1 1

n ni i

n i i

i i

R SC u ,u U u U u u u

n n n nI I

2

22

1 2 1 2 1 2 ,1 ,2 ,1 ,20,1

1

, , , , ,n

e

n n n n i i i i

i

S n C u u C u u dC u u C U U C U U

*

1

1 Nk e

val n n

k

p S SN

I

nC u Empirical copula

ˆC u parametric copula

(a) For bivariate case

*

2* * * * * *

,1 ,2 ,1 ,2

1

U , U U , Ukn

nk k k k k k

n n i i i i

i

S C Cwhere

TDC Captures the concordance between extreme values in lower leftquadrant tail & upper right quadrant tails of the variables.

where

Non-Parametric TDC

CFG estimator

LOG estimator:

1 1

1 2 , 1 ,lim 2 lim 2 1

1 1U U C

u u

u C u u C u u

u u

,C u C u u

21 1, 2, 1, 2,

1 1 1 1ˆ 2 2exp log log log logmax ,

nCFG

U

i i i i iu un u u

ˆlog 1 , 1ˆ 2 , 1,..., 1

log 1

n

LOG

U

k kC

n nk n

k

n

Tail Dependence Coefficient (TDC) Test

Copula Class Copula Family Parameter (s) llmax p-value

Archimedean Clayton = 2.36 35.9 0.41 0.004

Frank = 14.95 79.85 0.05 0.00

BB1 = 0.95, = 1.11 40.14 0.21 0.00

Extreme value GH = 4.37 89.65 0.04 0.004

Galambos = 3.67 89.47 0.04 0.002

Plackett Plackett = 59.94 78.36 0.07 0.00

Elliptical Student’s t = 3.02, r = 0.958 70.75 0.03 0.002

Table: Estimated parameters & performance of different copula Families at westernRajasthan during 1900 - 2008

e

nS

ˆ

ˆ

1ˆ

2ˆ

ˆ

ˆ

ˆ

ˆ

Table: TDC estimates of different copulas

ˆ ˆ0.834; 0.874CFG LOG

Obs Obsfor empirically transformed observed data

ˆU

ˆCFG

U

ˆLOG

U

ˆˆ CFG

Uˆˆ CFG

Uˆˆ LOG

Uˆˆ LOG

U

Copula Families

Clayton 0 0.528 0.029 0.160 0.092

Frank 0 0.747 0.018 0.691 0.066

BB1 0.631 0.678 0.010 0.644 0.047

Gumbel-Hougaard 0.828 0.827 0.008 0.826 0.031

Galambos 0.568 0.701 0.010 0.700 0.037

Plackett 0 0.772 0.015 0.680 0.120

Student’s- t 0.784 0.857 0.011 0.806 0.036

Drought Prediction

Deterministic Prediction of Drought using Support Vector Regression (SVR)

T

iˆ , + , , , , 1,..., , ,n n

i i iy f b R y R D y i n R y Rx w w x x x x

Solving a set of linear equations with objective function

2

, ,1

1min , ,

2 2

n

iw b

i

l b2

w w

Equality constraintsT

i i+iy bw x

optimization problem is solved by employing Lagrange’s multipliers

i i

1

x x , x ,SVn

i

i

f k b

2

2, exp , 0

2

i j

i j

x xk x x

*

,

1

1 1,

2

N

obs obs t

t

CRPS F x E X X E X xN

Continuous Rank Probability Score (CRPS) (Gneiting & Raftery, 2007)