Regionalization of Rainfall

Dr. V. V. SrinivasAssociate ProfessorAssociate Professor

Department of Civil Engineering, IISc Bangalore 560 012, India

Overview of PresentationOverview of Presentation

• Introduction

• Approaches to regionalization of precipitation• Approaches to regionalization of precipitation

• Issues and concerns

• Methodology proposed for regionalization, and frequency Analysis

• Case studies

Introduction

Unpredicted rains and consequent floods/droughts can cause large scaledevastation (e.g., loss of life, failure of infrastructure)

Infrastructure includes structures that support a society (e.g., water supplyand sewage networks, flood control structures, roads, railways, buildings,power grids, telecommunications (Internet, telephone lines, broadcasting))

Boys fishing in flood waterBangaloreH R d B lA house in JP Nagar, Bangalore Bangalore Hosur Road, Bangalore

Underpass: Railway Station and KSRTC Bus Bangalore

Flood-submerged road nearAhmadabadv

Flood-submerged road nearAhmadabadMumbai

UP Floods: Man carrying his livelihood on his shoulders

dademocrazy.blogspot.com/

Amritsar

Mumbai

MumbaiUttar Pradesh

Flooded Airports

California

O’Hare, Chicago

Flooded CA Airporthttp://paganofamily.us/ images/matt/rain.jpg

ILL EFFECTS OF DROUGHTS

7

Introduction

The need to alleviate devastation provided impetus to develop models foraccurate prediction of extreme hydrometeorological events and theiraccurate prediction of extreme hydrometeorological events and theircharacteristics

Challenge stems from paucity of at-site data to calibrate the model parameters

An appealing solution - Regional analysisRegion: A set of sites depicting similar hydrological behavior

Regionalization: Process of identifying regions

Regional Frequency Analysis: Frequency analysis based on Regional information

IntroductionIntroduction

Typical Applications

Agricultural planning

Design of drainage systems and other civil infrastructure across waterDesign of drainage systems and other civil infrastructure across water bodies

Estimating frequency and areal extent of meteorological droughts.

There is enough demand for precipitation quantile estimates[Hydrometeorological design studies center (HDSC) of NOAA’sNational weather service, USA]

Approaches to Regionalization of PrecipitationApproaches to Regionalization of Precipitation

Elementary linkage analysis (e.g., Adelekan, 1998) and its variation (Gadgil et al.,1993)

Approaches to Regionalization of PrecipitationApproaches to Regionalization of Precipitation

Spatial correlation analysis (e.g., Sumner and Bonell, 1988)

Common factor analysis (e.g., Bärring, 1987)

Empirical orthogonal function analysis (e.g., Bedi and Bindra, 1980; Kulkarni et al.,1992)

Principal component analysis (PCA) (e g Singh and Singh 1996; Iyengar andPrincipal component analysis (PCA) (e.g., Singh and Singh, 1996; Iyengar andBasak, 1994)

Cluster analysis (e.g., Easterling, 1989; Guttman, 1993; Obregón and Nobre, 2006;Venkatesh and Jose 2007)Venkatesh and Jose, 2007)

PCA in association with cluster analysis (e.g., Baeriswyl and Rebetez, 1997;Dinpashoh et al., 2004; Satyanarayana and Srinivas, 2008).

10

Elementary linkage analysis (McQuitty, 1957)y g y ( y, )

Area is partitioned into regions such that each station in a region is more highlycorrelated with another station in the same region, than with stations in anyother regionother region.

Steps(1) Identify a pair of stations having highest cross-correlation to form a cluster.

(2) Specify a correlation threshold

(3) From the remaining stations assign those stations to the cluster whose(3) From the remaining stations, assign those stations to the cluster whosecorrelation with any of the stations in the cluster is greater than the specifiedcorrelation-threshold .

(4) Repeat step (3) till no more stations can be assigned to the cluster.

Elementary linkage analysis related approach (Gadgil et al., 1993)y g y pp ( g , )

Steps(1) Identify a pair of stations having least cross-correlation (in precipitation

data) to form central stationsdata) to form central stations.

(2) Form a region around a central station, by grouping it with a station that issignificantly correlated with the central station.

(3) From the remaining stations, assign those stations to the region whosecorrelation with any of the stations in the region is greater than the specifiedcorrelation-thresholdcorrelation-threshold .

(4) Chose new central stations from the stations that are not assigned to any ofthe existing regions, such that it is as poorly correlated as possible to theexisting central stations.

(5) Repeat steps (3) and (4) for the new central stations.

(6) Repeat steps (5) and (6) till all the stations are allocated to regions.

Spatial correlation analysis

Steps(1) Cross-correlation between rainfall at different stations in the study area is considered as a function of direction and distance between the stations.considered as a function of direction and distance between the stations.

(2) A two-dimensional plot between cross-correlation and distance between the corresponding stations, called correlogram, is prepared and patterns are di ddiscerned.

(3) Smooth appearance of the correlogram indicates a homogeneous rainfall region, while irregularities in the correlogram indicate heterogeneous regiong , g g g g

Common factor analysisy

Spatial patterns of common factors are used to delineate regions ofhomogeneous precipitation.

Principal Component (PC) analysis (more often referred to asempirical orthogonal function analysis)

Inter-station correlation/covariance of precipitation in the study region isdescribed using minimal number of extracted components that areorthogonal to each other.

Spatial patterns of leading PCs are analyzed to delineate coherent regions of homogeneous precipitation variability.

This analysis could involve either plotting the unrotated and/or rotated PCloadings on the map of the study region (e.g., Ehrendorfer, 1987), orrepresenting stations as points in two-dimensional planes of leading PCs(e.g., Gadgil and Joshi, 1983).(e.g., Gadgil and Joshi, 1983).

Issues & concernsIssues & concerns

Statistics computed from precipitation data are used for regionalization ofprecipitation

Issues & concernsIssues & concerns

p p

mean of annual/seasonal/monthly/daily precipitation

mean number of wet days

ratio of minimum to maximum average two-month precipitation

parameters of hydrologic distributions fitted to precipitation data

C i h h i l i f i li i i l dConcerns with the conventional practice of regionalization include:

Requirement of adequate number of sites having sufficiently long andcontemporaneous precipitation records to form meaningful regions

Lack of specific strategy to validate the delineated regions for homogeneity andto predict quantiles at ungauged sites

15

There is a need to develop effective methodology for regionalization of precipitation

Methodology proposed for Regionalization Methodology proposed for Regionalization and Frequency Analysisand Frequency Analysisq y yq y y

Selection of Attributes

Preparation offeature vectors

Formation ofclustersclusters

Selection of optimumnumber of clusters

Testing the homogeneity of clusters

Are the clustersHomogeneous?

Adjustment ofHeterogeneous

clustersNO

YES

Prediction of quantiles of predictands

Clustering Clustering Algorithms available in literatureAlgorithms available in literatureClustering Clustering Algorithms available in literatureAlgorithms available in literature

The clustering algorithms available in literature can be broadly classified as

Partitional [e.g., K-means and K-medoids algorithms, fuzzy c-means clustering]

Hierarchical [e.g., single-linkage, complete-linkage and Ward’s algorithms.]

Density-based [e.g., DBSCAN (Density-Based Spatial Clustering of

Applications with Noise) and OPTICS (Ordering Points To Identify the Clustering

Structure]

Grid-based [e.g., STING (a STatistical INformation Grid approach) and CLIQUE

(Clustering In QUEst).].

Model-based and [e.g., COBWEB ]

Boundary-detecting. [e.g., SVC (Support Vector Clustering) ]

Methodology proposed for Regionalization, Methodology proposed for Regionalization, and Frequency Analysisand Frequency Analysis

Selection of Attributes

q y yq y y

Preparation offeature vectors

Formation ofclusters

Selection of optimumnumber of clusters

Testing the homogeneity of clusters

Are the clustersHomogeneous?

Adjustment ofHeterogeneous

clustersNO

YES

Prediction of quantiles of predictands 18

Definition of Regional HomogeneityDefinition of Regional Homogeneity

A region is said to be homogeneous if data at all sites in that region followA region is said to be homogeneous if data at all sites in that region followsame distribution.

Degree of homogeneity is decide by comparing the between-site dispersionof the sample L-moment ratios for the sites in a region, with dispersion thatwould be expected for a homogeneous region.

19

Methods for parameter estimation

Method of Moments

Method of Maximum Likelihood

Probability Weighted Moments

L-Moments

LL-Moments

LH-Moments

L-moments

22

Advantages of L-momentsg

Lower bias in estimation of higher order moments

Ability to characterize a wide range of distributionsAbility to characterize a wide range of distributionsL-moments exist whenever the mean of a distribution exists.L-moments can be computed even when some higher conventional moments fail to exist.

R b t t th f tli i th d tRobustness to the presence of outliers in the dataL-moments are less affected than conventional moments by outliers in the dataFor estimation of higher order moments, conventional moments give greater weight to theextreme tails of the distributionL t it ti ti f hi h d t b li bi ti f l tL-moments permit estimation of higher order moments by linear combinations of elementsof an ordered sample.

Boundedness of L-moment ratiosL-moment ratios vary between 1 and -1, allowing easy interpretation of statisticsConventional moments are unbounded.

Ease in identifying population distributionL-moments framework offers higher probability of discrimination between samples from different distributions

Hosking & Wallis (1997)

Hosking & Wallis (1997)

1 n

Wang (1996) derived the following direct sample estimators of the L-moments

111

1 n

j:nnj

ˆ xC

λ=

= ∑

( )12 1 1

12

1 12

nj n j

j:nnj

ˆ C C xC

λ − −= −∑122 jC =

( )1 13 2 1 1 2

1 1 23

nj j n j n j

j:nˆ C C C C xλ − − − −= − +∑( )3 2 1 1 2

133 j:nnjC =∑

( )1 1 11 1 nj j j j j jˆ ∑( )1 1 1

4 3 2 1 1 2 314

1 1 3 34

j j n j j n j n jj:nn

j

ˆ C C C C C C xC

λ − − − − − −

== − + −∑

( )1 1 11 1 3 3n

j j n j j n j n jˆ C C C C C C xλ − − − − − −= +∑

h d d l h ( 1) l l h l d ( ) l

( )4 3 2 1 1 2 314

3 34

j j j j j jj:nn

jC C C C C C x

Cλ

== − + −∑

In the ordered sample, there are (j - 1) values less than or equal to j:nx , and (n - j) values

greater than or equal to j:nx . In any combination of four values drawn from the sample (which

includes x ) the following possibilities exist:includes j:nx ), the following possibilities exist:

1. For j:nx to be the largest, the other three values have to come from the (j - 1) smaller

1jvalues, and there are a total of 13

j C− such combinations

2. For j:nx to be the smallest, the other three values have to come from the (n - j) larger

values, and there are a total of 3n jC− such combinations.

3. For j:nx to be the second smallest, one value has to come from (j - 1) smaller values and

there are a total of 11

j C− such combinations, and other two values have to come from (n - j)

larger values, and there are a total of 2n jC− such combinations.

( )1 1 14 3 2 1 1 2 3

14

1 1 3 34

nj j n j j n j n j

j:nnj

ˆ C C C C C C xC

λ − − − − − −

== − + −∑

4. For j:nx to be the third smallest, two values have to come from (j - 1) smaller values and

1jthere are a total of 12

j C− such combinations, and other one value has to come from (n - j)

larger values, and there are a total of 1n jC− such combinations.

Equation for kurtosis is derived by substituting 4 4:X , 3 4:X , 2 4:X and 1 4:X in Equation (3.9) by 1

3j C−

j:nx , 12

j C− × 1n jC− j:nx , 1

1j C− × 2

n jC−j:nx , and 3

n jC− j:nx respectively (for j

=1 2 n) and dividing by nC (i e the number of all possible combinations of drawing four=1,2,…,n), and dividing by 4C (i.e., the number of all possible combinations of drawing fourvalues from the sample) to compute the expected value.

LL--momentsmoments

11

:

1 1n

r j n

n jb n x

−

− − −⎛ ⎞ ⎛ ⎞= ⎜ ⎟ ⎜ ⎟

⎝ ⎠ ⎝ ⎠∑Sample Probability weighted moment,

10 :

1

n

j nj

b n x−= ∑

:1

r j nj rr r= +

⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

∑

( )( )

11 :

2

11

n

j nj

jb n x

n− −

=−∑ ( )( )

( )( )1

2 :3

1 21 2

n

j nj

j jb n x

n n−

=

− −=

− −∑1j= ( )2 1j n= ( )( )3 1 2j n n=

l b=

: th smallest observation in an ordered sample of size representsj nx j n

Sample L-moments are defined as:

1 0

2 1 0

3 2 1 0

26 6

r

l bl b bl b b b

== −= − +

( ) ( )

*1 ,

0

0,1,2,..., 1

1 !

r r k kk

r k

in general l p b

r n

r k

+=

−

=

= −

+

∑2 1

3 3 2

Sample Coefficient of L variation ( CV), Sample L skewness, = Sample L kurtosis

L t l lt l l

t l l

− − =−

( ) ( )( ) ( )

*, 2

1 !

! !r k

r kwhere p

k r k

− +=

−4 4 2

2

Sample L kurtosis, In general, = r r

t l lt l l− =

30

Method of Moments Karl Pearson (1902)

Data values are each assigned ahypothetical mass equal to theirrelative frequency of occurrence(1/n)

It is imagined that the system ofmasses is rotated about origin

Best values of parameters of afrequency distribution as those forwhich moments of the probabilitydensity function about the origin areequal to the corresponding moments ofthe sample data.

Method of Maximum Likelihood Fisher (1922)

Best values of parameters of a frequency distribution as those which maximize thelikelihood or joint probability of occurrence of the observed sample.

Issues:

The method is computationally intensive than the method of moments.

In the case of multi-parameter probability distributions, problems of convergence in

the solution of maximum likelihood iterative procedure could arise.

Given: Sample of independent and identically distributed observations x1, x2, …, xn

Divide the sample into intervals of equal length ‘dx’

Probability density for X=xi is f(xi)

Probability that random variable will occur in the interval including xi is f(xi) dx

The Joint probability of occurrence of observations

n ⎤⎡= [f(x1) dx] [f(x2) dx]…[f(xn) dx] =

Since the vale of dx is fixed, maximizing is equivalent is maximizing

( )ni

n

idxxf ⎥

⎦

⎤⎢⎣

⎡∏=

)(1

( )ni

n

idxxf ⎥

⎦

⎤⎢⎣

⎡∏ )(

1⎥⎦

⎤⎢⎣

⎡∏ )(

1i

n

ixf

which is called Likelihood function, L.

i⎥⎦

⎢⎣ =1

⎥⎦

⎢⎣ =1i

Issues:

The method is computationally intensive than method of moments.

In the case of multiparameter probability distributions problems of convergence inIn the case of multiparameter probability distributions, problems of convergence inthe solution of maximum likelihood iterative procedure could arise.

Methodology proposed for Regionalization Methodology proposed for Regionalization and Frequency Analysisand Frequency Analysis

Selection of Attributes

q y yq y y

Preparation offeature vectors

Formation ofclusters

Selection of optimumnumber of clusters

Testing the homogeneity of clusters

Are the clustersHomogeneous?

Adjustment ofHeterogeneous

clustersNO

YES

Prediction of quantiles of predictands 35

Definition of Regional HomogeneityDefinition of Regional Homogeneity

A region is said to be homogeneous if data at all sites in that region followA region is said to be homogeneous if data at all sites in that region followsame distribution.

Degree of homogeneity is decide by comparing the between-site dispersionof the sample L-moment ratios for the sites in a region, with dispersion thatwould be expected for a homogeneous region.

36

HETEROGENEITY MEASURES (Hosking and Wallis, 1997)

VALIDATION

L CV t it i ( )iL-CV at site i :

L-skewness at site i :

( )it

( )3

itL skewness at site i :

L-kurtosis at site i :

3t

( )4it

Regional average L-CV: ( )

1 1

N NR i

i ii i

t n t n= =

=∑ ∑

Regional average L-skewness: ( )3 3

1 1

N NR i

i ii i

t n t n= =

= ∑ ∑N N

Regional average L-kurtosis: ( )4 4

1 1

N NR i

i ii i

t n t n= =

= ∑ ∑

Weighted standard deviation of at-site sample L-CVs:

VALIDATIONHETEROGENEITY MEASURES (Contd…)

Weighted standard deviation of at-site sample L-CVs:

2/12)( )(V

⎭⎬⎫

⎩⎨⎧

−= ∑∑N

iN Ri

i nttn

Weighted average distance from the site i to the group weighted mean in the 2 dimensional space of L CV and L skewness:

11 ⎭⎩ == ii

{ } ∑∑ −+−=N

ii

N

i

RiRii nttttn

11

2/123

)(3

2)(2 )()(V

in the 2-dimensional space of L-CV and L-skewness:

== ii 11

Weighted average distance from the site i to the group weighted mean in the 2-dimensional space of L-skewness and L-Kurtosis:

{ } ∑∑ −+−=N

ii

N

i

RiRii nttttn

11

2/124

)(4

23

)(33 )()(V

in the 2 dimensional space of L skewness and L Kurtosis:

== ii 11

HETEROGENEITY MEASURES (Contd…)

Vμ

Vμ

: Mean of V values from Monte-Carlo (MC) simulations

: Mean of V2 values from MC simulations2Vμ

3Vμ

: Mean of V2 values from MC simulations

: Mean of V3 values from MC simulations

Vσ

Vσ

: Standard deviation (SD) of V values from MC simulations

: SD of V values from MC simulations2Vσ

3Vσ

: SD of V2 values from MC simulations

: SD of V3 values from MC simulations

MC simulations: 50039

HETEROGENEITY MEASURES (Hosking and Wallis, 1997)

VALIDATION

Heterogeneity measure (HM) based L-CV: V

V1 σ

)μ(VH

−=

2V22

)μ(VH

−=HM based on L-CV and L-Skewness:

Vσ

2V2 σ

H

3V3 )μ(V −HM based on L-Skewness and L-Kurtosis:

3

3

V

V33 σ

)μ(H =

Acceptably homogeneous : H < 1

Possibly homogeneous/heterogeneous : 1≤ H < 2

Definitely heterogeneous : H ≥ 2

Methodology proposed for Regionalization Methodology proposed for Regionalization and Frequency Analysisand Frequency Analysis

Selection of Attributes

q y yq y y

Preparation offeature vectors

Formation ofclusters

Selection of optimumnumber of clusters

Testing the homogeneity of clusters

Are the clustersHomogeneous?

Adjustment ofHeterogeneous

clustersNO

YES

Prediction of quantiles of predictands 41

Estimation of precipitation quantiles

The regional quantile estimate at site i for T-year recurrence interval isdetermined by using index flood method (Dalrymple, 1960) as,

is scaling factor (e.g., mean, median) for site I

ˆ ˆ ˆ( ) ( )i i kQ T q Tμ=

ˆiμ

is a dimensionless quantile function known as regional growth curveˆkq

42

Selection of Attributes

Preparation offeature vectors

Formation ofclusters

Selection of optimum

Testing the homogeneity of clusters

Selection of optimumnumber of clusters

of clusters

Are the clustersHomogeneous?

Adjustment ofHeterogeneous

l tNO

Homogeneous? clusters

YES

Prediction of quantiles f di dof predictands

43

Adjustment of Heterogeneous ClustersHosking and Wallis (1997):

eliminating one or more sites from the data set

dividing a region to form two or more new regions

allowing a site to be shared by two or more regions

i it f i t th imoving one or more sites from a region to other regions

dissolving regions by transferring their sites to other regions

merging a region with one or more regionsg g g g

merging two or more regions and redefining groups

obtaining more data and redefining regions

Adjustment of Heterogeneous ClustersAdjustment of Heterogeneous Clusters

The primary option that is considered for adjusting a heterogeneous cluster is elimination of one or more sites that are grossly discordant

ith t t th it ithi th l t Th l di d twith respect to other sites within the cluster. The grossly discordant sites are identified using the discordancy measure given by Equation

11 T 11 ( ) ( )3i k i iD N −= − −u u S u u

where is discordancy measure for ith site in a cluster k having sites.iD kNwhere is discordancy measure for ith site in a cluster k having sites.

is a vector containing L- moment ratios (LMR) of predictand at site i

i th i ht d f th LMR d S i i t i

i k

T( ) ( )( )

i 3 4 i iit t t⎡ ⎤⎢ ⎥⎣ ⎦

=u

u is the unweighted group average of the LMR and S is a covariance matrix

kNi∑ u T( )( )

kN∑S

u

1i

i

kN=∑

=u

uT

1( )( )i i

i== − −∑S u u u u

45

Att ib t S t d

For regionalization:

Attributes Suggested

Large scale atmospheric variables influencing precipitation in the study area, instead of statistics computed from observed precipitation

For regionalization:

Location parameters (Latitude, longitude, elevation)

Seasonality measures

For validation:

Statistics computed from observed precipitation

47

Delineation of Homogeneous Summer Monsoon rainfall regions

Data Considered

NCEP-NCAR reanalysis data: to delineate homogeneous rainfall regions

2.5 degree resolution

M thl l f 1951 t 2004Monthly values from 1951 to 2004

S No Variable name Pressure levels in mb

Pressure Variables:

15 Variables

S.No. Variable name Pressure levels in mb

1 Air temperature 925, 700, 500, 200

2 Geopotential height 925 500 2002 Geopotential height 925, 500, 200

3 Specific humidity 925, 850

4 Zonal wind (U-wind) 925, 200( ) ,

5 Meridional wind (V-wind) 925, 200

S.No. Variable name

49

Surface Variables: 1 Surface pressure

2 Precipitable water 49

Data Considered

Digital Elevation Model (DEM): SRTM-DEM [3 arc second (~90 m) resolution]

IMD gridded daily rainfall data (1×1 degree) for validating regions

• IMD data was prepared using 54years of daily data (1951-2004) at2140 gauging stations

50

Seasonality of RainfallSeasonality of Rainfall

The julian day identified in year, ( ) can be represented as a vector of unit magnitude, and direction (in radians) (Burn, 1997, Ouarda et. al., 2006).

υ dυ

2 0 2d πθ θ≤ ≤

υθ

The directional mean

0 2D

dNυ υ υθ θ π= ≤ ≤ is number of days in a yearDN

⎛ ⎞ 1 1n n1tan ; 0y x

xθ − ⎛ ⎞= ≠⎜ ⎟

⎝ ⎠( ) ( )

1 1

1 1cos ; sini in n

i ii i

x yn nυ υ

υ υ

θ θ= =

= =∑ ∑

NMean Julian day

A 30-day long sliding window is used to determine the Julian day depicting middle day of a 30-day maximum precipitation in each year

2DNMD θπ

=

a 30 day maximum precipitation in each year.

51

Feature vector formation – Hard & Fuzzy Clusteringy g

Climate data are re-gridded to one degree resolution.

25 one degree grid points surrounding target grid box are considered

Th 4500 l (15 i blThus 4500 values (15 variables × 25 grid points × 12 months) are obtained for each of the 294 IMD grid boxes.

4 PCs from 4500 values, latitude, longitude, elevation of grid box and seasonality of rainfall are considered asrainfall are considered as attributes.

52

Feature vector formation – Dynamic Fuzzy Clustering

Climate data is considered as time varying

y y g

Each month of summer monsoon is considered as time step

Total time steps = 54 years X 4 months (monsoon months) = 216y ( )

Principal components, extracted at each time step from the climate data, are considered as time varying attributes

Latitude, Longitude, Average elevation of IMDgrid box are included as additional attributes

53

Validity indices considered

2

1 1

1( ) ( )c N

PC ikk i

V uN = =

= ∑ ∑UFuzzy partition coefficient(maximum value refers to optimum partition)

1 1

1( ) ( )c N

PE ik ikk i

V u log uN = =

⎡ ⎤= − ∑ ∑⎢ ⎥⎣ ⎦U

Fuzzy partition entropy(minimum value refers to optimum partition)

( ) 1( ) 11

PCFPI

c VVc

× −= −

−UU

( )V U

Fuzziness performance index(minimum value refers to optimum partition)

( )( )( )

PENCE

VVlog c

=UU

2( )c N

u μ∑ ∑ v x

Normalized classification entropy(minimum value refers to optimum partition)

1 12

( )( : )

min

ik k ik i

XB,ml kl ,l k

uV ,

N

μ

= =

≠

−∑ ∑=

−

v xU V X

v vExtended Xie-Beni index

(minimum value refers to optimum partition)

54

Clusters in Optimal Partition by DFCM

O ti titi i id tifi d bOptimum partition is identified by visual interpretation, previousknowledge about the number of regions, and observing heterogeneity g , g g yvalues

Clusters : 22Fuzzifier value : 1.2

55

294 IMD grid boxes

New Regions formed using DFCM and Large Scale Atmospheric Variablesand Large Scale Atmospheric Variables

Characteristics of the SMR Regions Formed by Parthasarathy et al., 1995Characteristics of the SMR Regions Formed by Parthasarathy et al., 1995

AHO : 00PHE : 00

Number of Heterogeneity

PHE : 00HET : 05

Regionname

Number ofIMD gridpoints

Heterogeneity measures Region type

Peninsular 49 23.28 5.93 0.26West Central 86 10 89 0 64 1 33

Definitely HeterogeneousWest Central 86 10.89 0.64 -1.33

Northwest 69 20.96 5.87 -1.08Central northeast 59 4.32 -0.73 -1.90

Northeast 36 4.44 -0.91 1.06 57

Characteristics of the SMR Regions Formed by Parthasarathy et al., 1995Characteristics of the SMR Regions Formed by Parthasarathy et al., 1995After Adjustments to Improve HomogeneityAfter Adjustments to Improve Homogeneity

No of Heterogeneity

Region name

No. ofIMD grid

points

g ymeasures grid points

eliminatedH1 H2 H3

Peninsular 27 0.75 -0.34 1.35 22West Central 62 0.80 -1.17 -2.03 24

Northwest 40 0.84 -0.86 -1.90 29CentralCentral

Northeast 45 0.74 -0.86 -1.47 14

Northeast 32 0.45 -1.30 -1.06 04

Delineation of Homogeneous annual rainfall regions

294 out of 357 IMD grid boxes covering the study area were considered.The discarded 63 grid boxes (which are shown with cross symbols) are inHimalayan mountainous regionHimalayan mountainous region.

(1) Fifteen large scale atmospheric variables

(2) Location descriptors (latitude, longitude and altitude))

(3) Seasonality: The Julian day depicting middle day of a 30 day maximum rainfall30-day maximum rainfall period in each year was used to define seasonality of rainfall.

Heterogeneity of the region in the presence of cross-correlation

20 122 ( 1)H H N+

Heterogeneity of the region in the presence of cross correlation(Castellarin et al., 2008)

20 122 ( -1)adj cH H . Nρ= + ×

2ρ Average of squares of cross-correlations of concurrent rainfall for the sites in a cluster

cN Number of sites in the cluster1000

1PC

2PCs

10001PC

2PCs

100

3PCs

4PCs

4PCs only

mbe

r of s

ites

100

3PCs

4PCs

4PCs only

mbe

r of s

ites

Num

Num

10-22610141822

Hadj value

10-22610141822

Hadj value

Figure: Soft clusters (plausible homogeneous annual rainfall regions) in optimum partition identified using fuzzy cluster analysis

Homogeneous regions of Annual rainfall

Ref: Satyanarayana and Srinivas (2011)

1000Soft clusters

1000Soft clusters

100

Hard clusters

r of s

ites

100

Hard clusters

r of s

ites

100

Num

ber100

Num

ber

1000

Soft regions

1000

Soft regions

10-226101418

Hadj value

10-226101418

Hadj value

100

IMD sub-divisions

er o

f site

s

100

IMD sub-divisions

er o

f site

sN

umbe

Num

be

10-22610141822263034

Hadj value

10-22610141822263034

Hadj value

Figure . Meteorological sub-divisions of India (Guhathakurta, 2008)

SMR: GLO distribution is suitable to fit pooled information of regions inNortheast and Central Northeast India, whereas GNO is a default option

f fto fit pooled data in the rest of the country

Annual Rainfall: Wakeby distribution was found suitable for Northeastand Central Northeast India, GNO distribution was found suitable forand Central Northeast India, GNO distribution was found suitable forTamil Nadu and North Uttar Pradesh, and GEV distribution (followed byGNO and PE3 distributions) was found suitable for Gujarat andRajasthan. For the rest of India, GLO distribution was found suitable.

PublicationsPublications

Satyanarayana, P., Srinivas, V. V. (2008) Regional frequency analysis of precipitationusing large-scale atmospheric variables, Journal of Geophysical Research -Atmospheres, American Geophysical Union, Vol. 113, D24110, December 2008,doi:10.1029/2008JD010412.

Satyanarayana, P., and Srinivas, V. V. (2011) Regionalization of precipitation in datasparse areas using large scale atmospheric variables - A fuzzy clustering approach.Journal of Hydrology, Elsevier, Netherlands, Vol. 405, Issues 3-4, pp.462-473

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