Characterization and Short-term prediction of … and Short-term prediction of Droughts over India...
Transcript of Characterization and Short-term prediction of … and Short-term prediction of Droughts over India...
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 75% of river flows are withdrawn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 60% of river flows are withdrawn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withdrawn, but malnutrition exists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)