Statistical techniques for investigating rainfall

variability at monthly and annual time scale

E. FerrariE. Ferrari

Dipartimento di Difesa del Suolo, Università della CalabriaRende (CS) – ITALY

Statistical techniques for investigating rainfall

variability at monthly and annual time scale

E. FerrariE. Ferrari

Dipartimento di Difesa del Suolo, Università della CalabriaRende (CS) – ITALY

International Workshop on: Evaluation des changements globaux sur les regimes Evaluation des changements globaux sur les regimes

hydrologique et les ressources an eauhydrologique et les ressources an eau

Université Mohamed V-Agdal, Faculté des SciencesRabat (Morocco)

10 - 11 Décembre 2009

Basic detailManagement of water resources and analysis of water balance need investigations about variability of longer time aggregation rainfalls, such as MONTHLY AND ANNUAL RAINFALLS.

Objectives of the work

Review of some statistical techniques forReview of some statistical techniques for:

Trend Trend detectiondetection (usually monthly and annual rainfalls)

Seasonality Seasonality analysisanalysis (usually monthly rainfalls)

Analysis of Analysis of impactimpact of of rainfall changes rainfall changes (annual (annual rainfall)rainfall)

Application on drainage basins of Southern Application on drainage basins of Southern ItalyItaly

TECHNIQUES for SHIFT and TREND detectionTREND detection

• MANN-KENDALL test

• SPEARMAN rank correlation test

• Linear regression analysis

• MANN-WHITNEY test (step-change)

TECHNIQUES for SEASONALITY analysisSEASONALITY analysis

• FOURIER analysis

• one-way analysis of variance

• lag1 and lag12 month-to-month correlation versus season

• procedures based on monthly mean / SD / skewness

TECHNIQUE for evaluating IMPACT of RAINFALL IMPACT of RAINFALL CHANGECHANGE

Simple rainfall change scenarios Variability of water resources potentiality

1st Case study

2nd Case study

3rd Case study

H0: no trend vs H1: trend

H0: no trend vs H1: trend

Time series: x1, x2, …, xn

01

00

01

Sifm

Sifm

Sifm

SPEARMAN’s rank correlation Test

If H0 holds, the statistic

D D N ( E(D) , Var(D) ) N ( E(D) , Var(D) )

The test statistic ZSN(0,1)

Trend is significant ifTrend is significant if:

1nn

ixR61D 2

n

1i

2i

1n

1DVar

0DE

DV

DZS

11 D

α/21S ZZ

MANN-KENDALL Test

If n8 and H0 holds, the statistic S S N( E(S) , Var(S) ) N( E(S) , Var(S) )

The test statistic ZMKN(0,1)

Trend is significant ifTrend is significant if:

SV

mSZMK

1n

1i

n

1ijij xxsgnS

tn

1iiii 52t1tt52n1nn

181

SVar

0SE

α/21KM ZZ

01

00

01

Sifm

Sifm

Sifm

1) TECHNIQUES for TREND detection

Non-Non-parametric parametric

teststests

1) TECHNIQUES for TREND detection

LINEAR REGRESSION analysis

A parametric approach for tend detection may concern the linear regression analysis, expressed as:

Y= β0+β1X+ε.

Assuming as null hypothesis that no trend occurred in data series (ββ11=0=0), for each series the confidence interval of the

slope parameter β1 is:

where sxy is the covariance and tn-1,1-α/2 is the 100(1-α/2) percentage point of a Student’s t distribution with n-2 degree of freedom.

Trend is significant if the Confidence Interval of the Trend is significant if the Confidence Interval of the slope parameter slope parameter ββ11 does not contain the 0 valuedoes not contain the 0 value, ,

that is the population value of slope parameter that is the population value of slope parameter when no trend occurred.

2n

1ii

xy2α2,1n1

xx

stb

Fourier Fourier analysianalysi

ss 1/31/3

ji,HMonthly Monthly rainfallsrainfalls

λHY ji,ji, Power transformation ~ Normal distribution

1

μ

Y

υ1

Zjj Y

ji,

Y

Deseasonalization - standardization

a

jj

n

1kkk0YY j

6πk

senbj6πk

cosaa21~

a

jj

n

1kkk0YY j

6πk

senbj6πk

cosaa21~

12

1jY0 j

m61

a

12

1jYk kj

6π

cosm61

aj

12

1jYk kj

6π

senm61

bj

12

1jY0 j

v61

a

12

1jYk kj

6π

cosv61

aj

12

1jYk kj

6π

senv61

bj

meanmean

coefficient coefficient of of variationvariation

ja,Y2α1Yja,Y2α1Y NszmNszmjjjj

jYˆ

trigonometric interpolation

CI of coefficient of variationcoefficient of variation

jY

Montecarlo techniques

2) TECHNIQUES for SEASONALITY analysis

Seasonal variability Seasonal variability (FOURIER analysis)

Fourier cofficients

a0(μ), ai

(μ), bi(μ)

a0(ν), ai

(ν), bi(ν)

CI of the meanmean

For prefixed and significance level least numberleast number of of harmonicsharmonics

jY

jY~

jY~

Further tests

3N1N2N1N6N(0,N~g Z1, Skewness coefficientSkewness coefficient

Kurtosis coefficientKurtosis coefficient g2,Z CI evaluated through Montecarlo method

Once removed seasonality Test on Test on PROCESS RANDOMNESSPROCESS RANDOMNESS (possible residual correlation structure)

Anderson TestAnderson TestIf Z is a stricly stationary and independent normally distributed process, the sample autocorrelation coefficients rZ,k

kn1kn

zkn

1r

kn1kn

zkn

12α1kZ,2α1

CI of rZ,k

nk,1

Zkn1,

Z

nk,1Z

kn1,Z

kn1,Z

kZ, ssmmp

r

Analysis of reduced variate Analysis of reduced variate ZZ

2) TECHNIQUES for SEASONALITY analysis

Fourier Fourier analysianalysi

ss 2/32/3Time series: x1, x2, …, xn

If the values of rZ,k belong to the Ics the hypothesis that the process is purely random cannot be rejected

Z~N(0,1)Z~N(0,1) (normal probabilistic plot)

α1N,N δ

N1N1

D

if

1k

2

α1α1 2δ1)π(2k

21

expδ2π

α1where 1- is evaluated by:

The hypothesis that they come from the same statistical universe, at

significance level , has to be refused.

Comparisons between different periods (decades):Comparisons between different periods (decades):

Period X: Sample z1’, z2’,…, zn’

Period Y: Control sample z1”, z2”, …, zn” (H0: stationary

period)

N.B. Control sample is excluded from data used for calibration of the model

zzsupD NNz

N,N

N

1nn

1

N

zzfor1

zzzforNn

zzfor0

N

1nn

1

N

zzfor1

zzzforNn

zzfor0

Two sample Kolmogorov-Smirnov test

2) TECHNIQUES for SEASONALITY analysis

Use of the Use of the modelmodel

Fourier Fourier analysianalysi

ss 3/33/3

Procedure

• Fitting of a probability distribution to data (areal annual rainfall) observed in the stationary period 1916-80.

• Hypothesis on the variability of parameters of the probability distribution hypothesized for the 30-year transitional period 1981-2010 due to rainfall change ( estimation of parameters for 3 different models assessed for the transitory period).

• Simulation of annual raifalls over the next 30-year period (2010-39) for each model hypothized in the transitory period.

• Probability estimation of the maximum cumulated deficit of water resources potentiality for n-year temporal windows ( Monte-Carlo techniquesMonte-Carlo techniques).

3) ANALYSIS OF IMPACT OF RAINFALL CHANGE3) ANALYSIS OF IMPACT OF RAINFALL CHANGE

Probabilistic evaluation of variability of water resources potentiality depending on occurrence of rainfall change scenarios

SIMN Stazione pluviometrica SIMN Stazione pluviometrica680 Mezzana di Lucania 1240 Acquaformosa930 Villapiana scalo 1250 Fagnano Castello940 Francavilla Marittima 1260 S. Marco Argentano950 S. Lorenzo Bellizzi 1280 Spezzano Albanese scalo960 Civita 1290 Caselle970 Cassano allo Ionio 1296 Macchia Albanese976 Sibari 1300 S. Giorgio Albanese980 Piane Crati 1310 Schiavonea984 Serra Pedace 1320 S. Giacomo d'Acri990 Trenta 1340 Staggi

1000 Domanico 1350 Difesella1010 Cosenza 1360 Longobucco1020 Cerisano 1370 Bocchigliero1030 S. Pietro in Guarano 1480 Quaresima c.c.1040 Rende 1490 Lorica c.c.1050 Rose 1494 Rovale c.c.1060 Montalto Uffugo 1500 Nocelle1070 Laghitello c.c. 1510 Sculca1080 S. Martino di Finita 1520 Monteoliveto c.c.1090 Camigliatello Silano 2990 Parenti1100 Cecita ex Acquacalda 3000 Rogliano1110 Pinutello 3030 Aiello Calabro1120 Acri 3040 Amantea1130 Torano scalo 3050 Fiumefreddo Bruzio1140 Tarsia 3060 Paola1150 S. Sofia d'Epiro 3070 Cristiano c.c.1160 S. Agata c.c. 3080 Guardia Piemontese1170 Morano Calabro 3090 Cetraro super.1180 Castrovillari 3100 Belvedere Maritt. scalo1184 Piano Campolongo 3110 Cirella1190 Firmo 3124 Verbicaro scalo1200 S. Agata d'Esaro 3160 Campotenese c.c.1210 Malvito 3170 Mormanno1220 Roggiano Gravina 3180 Papasidero1230 San Sosti 3190 Orsomarso

ROSSO stazione esterna al bacinoNERO stazione interna al bacino

Cosenza

Rainfall data baseRainfall data base: : monthly rainfalls observed in 70 rain gauges

(1916–2008)

ITALYITALY

CALABRIACALABRIA

Crati RiverBasin (~2500

km2)

Crati RiverBasin (~2500

km2)

All examined area (~5000 km2)

All examined area (~5000 km2)

Application on basins of Southern Italy

47

54

54

60

49

60

63

66

49

54

48

57

58

63

42

63

50

53

68

67

60

51

68

61

65

49

Years

-1.98

0.57

-3.06

-1.08

-5.22

-1.03

-2.80

0

-2.01

-1.65

-0.27

-0.67

-1.34

-0.21

0.61

0.52

-5.29

-2.65

0.75

-4.03

-2.69

-4.09

-2.29

-2.44

-4.21

-2.42

ZMK

-2.16

0.50

-2.92

-1.11

-5.19

-0.93

-2.58

0.05

-1.99

-1.63

-0.38

-0.63

-1.44

-0.12

0.53

0.44

-4.89

-2.49

0.76

-4.03

-2.62

-4.05

-2.33

-2.44

-4.41

-2.77

ZS

-3.48

-0.70

-6.80

-3.02

-17.18

-2.30

-2.37

+0.22

-5.10

-4.20

-1.25

-1.54

-3.19

+0.47

+2.55

+1.54

-25.03

-8.49

+1.74

-6.61

-4.20

-14.23

-3.40

-6.44

-6.02

-4.87

Lin.Regr.

980

1220

1230

1240

1260

1290

1020

1030

1040

1050

1060

1070

1080

1090

1110

990

1000

1010

1200

1190

1180

1170

1150

1140

1130

1120

Code

SIGNIFICANT TREND

Blue: NO Red: YES

26 rain gauges internal to Crati basin

26 rain gauges internal to Crati basin

MANN-KENDALL ZZMKMKN(0,1)N(0,1)

SV

mSZMK

1n

1i

n

1ijij xxsgnS

SPEARMAN’s Rank Correlation ZZSSN(0,1)N(0,1)

1nn

ixR61D 2

n

1i

2i

DV

DZS

Linear regression H0:

{{ββ11=0=0}}

Y= β0+β1X+ε 2n

1ii

xy2α2,1n1

xx

stb

11stst example exampleRESULTS RESULTS from TREND ANALYSISTREND ANALYSIS of annual rainfalls

na(μ) a0

(μ) a1(μ) b1

(μ) a2(μ) b2

(μ)

22 18,00 4.08 1.74 0.02 -1.04

λ=1/2

97,5%mean2,5%

2

4

6

8

10

12

14

Month

Mean of H1/22 harmonics 1 harmonic

Monthly rainfalls of Crati basin

G F M A M J J A S O N D

Best number of harmonics

Normalizing value

22ndnd example exampleRESULTS RESULTS from FOURIER ANALYSIS of monthly rainfalls

Model calibrationModel calibration (monthly mean)(monthly mean)

na(ν) a0

(ν) a1(ν) b1

(ν) a2(ν) b2

(ν)

22 0.694 -0.11

6

-0.075 0.033

0.087

λ=1/2

97,5%

2,5%mean

0.1

0.2

0.3

0.4

0.5

0.6

0.7

G F M A M J J A S O N DMonth

Cv of H1/2

2 harmonics 1 harmonic

Monthly rainfalls of Crati basin

Best number of harmonics

Normalizing value

22ndnd example exampleRESULTS RESULTS from FOURIER ANALYSIS of monthly rainfalls

Model calibrationModel calibration (monthly Cv)(monthly Cv)

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0 6 12 18 24

Lag k (months)

Auto

corr

ela

tion c

oeffi

cients

r(k)

Confidence interval

(α=0.05)

22ndnd example exampleRESULTS RESULTS from FOURIER ANALYSIS of monthly rainfalls Verification of the modelVerification of the model (Anderson test for serial correlation)(Anderson test for serial correlation)

Power transformation

(λ=0.5)

Removal of

periodicity

Presence of correlation of process {Z} can be

rejected+

22ndnd example exampleRESULTS RESULTS from FOURIER ANALYSIS of monthly rainfalls Verification of the modelVerification of the model (Z~Standardized normal distribution) (Z~Standardized normal distribution)

Test on coefficient of skewness [-0.196 < -0.051-0.051 < 0.196]

Test on coefficient of Kurtosis [2.65 < 2.742.74 < 3.42]

-2.5 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 2.50.01

0.02

0.05

0.10

0.25

0.50

0.75

0.90

0.95

0.98

0.99

Reduced variable ZZ

Prob

abili

tyPP

-2.5 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 2.50.01

0.02

0.05

0.10

0.25

0.50

0.75

0.90

0.95

0.98

0.99

-2.5 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 2.50.01

0.02

0.05

0.10

0.25

0.50

0.75

0.90

0.95

0.98

0.99

Reduced variable ZZ

Prob

abili

tyPP

N=600 sample values(12x50 years)

-2

-1

0

1

2

-2 -1 0 1 2expected values of Z

obse

rved v

alu

es

of

Z

Years 1991-2000 Years 1981-1990 Years 1971-1980

Confidence interval

22ndnd example exampleRESULTS RESULTS from FOURIER ANALYSIS of monthly rainfalls Verification of the modelVerification of the model (Analysis of decades 1981-90 and 1991-(Analysis of decades 1981-90 and 1991-

2000) 2000)

Statistic DN’,N” (critical value 0.175)

Decade 1991-00

1981-90

1971-80

1961-70

1951-60

1931-40

1921-30

1991-00

--- 0.133 0.192 0.250 0.242 0.225 0.183

1981-90

No --- 0.167 0.192 0.242 0.200 0.142

1971-80

Yes No --- 0.092 0.125 0.067 0.092

1961-70

Yes Yes No --- 0.142 0.092 0.133

1951-60

Yes Yes No No --- 0.100 0.158

1931-40

Yes Yes No No No --- 0.117

1921-30

Yes No No No No No ---

Decade X Sample z1’, z2’,…, zN’ (N’=120) Decade Y Sample z1”, z2”, …, zN”

(N”=120)

DN’,N”

Are statistical variations between paired decades significant ?

22ndnd example exampleRESULTS RESULTS from FOURIER ANALYSIS of monthly rainfalls Verification of the modelVerification of the model (Two-sample Kolmogorov-Smirnov test (Two-sample Kolmogorov-Smirnov test

α=0.05) )

Time period after 1980

994.3869.4………851.51132.5

20082007………19821981ti

6564………21j=i-65Model for transitional period with discrete

parameter

1980t;tH ii )(* mmh

Areal annual rainfalls – drainage basin of Crati River

500

750

1000

1250

1500

1750

1910 1920 1930 1940 1950 1960 1970 1980 1990 2000 2010years

h (mm)

Time period up to 1980Model for stationary period with discrete

parameter

1980t;H i

Known observations:

1420.51243.9………1065.1899.7

19801979………19171916ti

6564………21i

)(mmh

Known observations:

33rdrd example exampleRESULTS RESULTS from ANALYSIS of IMPACT of RAINFALL CHANGE

Models for Models for stationarystationary (1921-80) and and transitionaltransitional periodsperiods (1981-2010)

Mean values of models for stationary period (1921-1980) and transitional period

(1981-2010)

0

500

1000

1500

1910 1920 1930 1940 1950 1960 1970 1980 1990 2000 2010

Years

)mm(h Stationary periodStationary period Transitional period

STATIONARY MODELSTATIONARY MODELProbabilistic distribution Trans-Normal (T-N)

2αθ

02

1αθ

00H

1β

h

2θ

1exp

β

h

θ1Φβ2π

αhf

0θ0,β0,α0;h 0

Probability density function

Φ() Standard normal distribution

• β0 scale parameter

• α,θ form parameters

If θ0 Log-normal distr. If α=1 Box-Cox distr.

Likelihood function

n

1i

2αθ

0

i

2

n

1i

1αθi

nθαn0

2n

n

i0 1β

h

2θ

1exph

θ1Φβ2π

αhθ,βα,L

Values of parameters

data: mm899.7h1 mm1065.1h2 mm1243.9h64 mm1420.5h65

it holds: 0.201θ ˆ5.482α ˆ mm1173β0 ˆ…

T-N distribution fitted to data of period 1916-1980 (stationary model with discrete parameter - white noise)

600 800 1000 1200 1400 1600 18000.0050.01

0.02

0.05

0.10

0.25

0.50

0.75

0.90

0.95

0.98

0.99 0.995

Areal annual rainfalls in Crati River basin (mm)

Normal probability plot

MODEL adopted for transitional period 1981-2010MODEL adopted for transitional period 1981-2010

Probability distribution

normaltranshfnormal-trans hfHH

θ,tηβα,;hfθβ,α,;hf i0HH

variation of scale parameter (β) unchanged parameters (α, θ)

HtηtH ii

mean:

variance:

HEtηtHE ii

HVartηtHVar i2

i

HCtHC ViV

HγtHγ 1i1

Variation coefficient:(unchanged)

skewness:(unchanged)

Statistics

1. Discontinuos 1. Discontinuos model with model with constant constant

parameterparameter

Likelihood function (constrained)

1980tfa

1980tf1tη

i

ii or

or

1980tfaβ

1980tfβtβ

i0

i0i or

or

m

1j

2θα

0

j

2

m

1j

1-θα

j

mθαm

02m

m

j 1βa

h

θ2

1exph

θ1Φβa2π

αhaL

ˆˆˆˆ

ˆˆ ˆˆˆˆ

ˆm sample dimension

sample

Estimation of parameter

mm1132.5h1

mm851.5h2

mm869.4h27

mm994.3h28

0.8305a ˆ

mm974.2βatβ 0i ˆˆˆ…

Discontinuos model with Discontinuos model with constant parameterconstant parameter

Sca

le p

ara

mete

r

Years1910 1920 1930 1940 1950 1960 1970 1980 1990 2000 2010

500

1000

1500

0

m1,2,...,j,h j

2. Continuous 2. Continuous model with model with linear linear

parameterparameter

m dimension of series

samplem1,2,...,j,h j

mm1132.5h1

mm851.5h2

mm869.4h27

mm994.3h28

…

Continuous model with Continuous model with linear parameterlinear parameter

Sca

le p

ara

mete

r

Years1910 1920 1930 1940 1950 1960 1970 1980 1990 2000 2010

500

1000

1500

0

1980tfbj1

1980tf1tη

i

ii or

or

1980tfbj1β

1980tfβtβ

i0

i0i or

or

1980t65ij i

m

1j

2θα

0

j

2

m

1j

1-θα

jmθαm

02m

m

j 1bj1β

h

θ2

1exph

θ1Φbj1β2π

αhbL

ˆˆˆˆ

ˆˆ ˆˆˆˆ

ˆ

0.009710b ˆ

mmj0.00971011173jb1βtβ 0i ˆˆˆ

Likelihood function (constrained)

Estimation of parameter

mm1132.5h1

mm851.5h2

mm869.4h27

mm994.3h28

…

Continuous model with Continuous model with rational parameterrational parameter

Sca

le p

ara

mete

r

Years1910 1920 1930 1940 1950 1960 1970 1980 1990 2000 2010

500

1000

1500

0

1980tf

cj1

11980tf1

tηi

i

i or

or

1980tf

cj1

β1980tfβ

tβi

0

i0

i or

or

1980t65ij i

m

1j

2θα

0

j

2

m

1j

1-θα

jmθαm

02m

m

j 1cj1β

h

θ2

1exph

θ1Φcj1β2π

αhcL

ˆˆˆˆ

ˆˆ ˆˆˆˆ

ˆ

0.01273c ˆ

mmj0.012731

1173

jc1

βtβ 0

i

ˆ

ˆˆ

3. Continuous model 3. Continuous model with with rational rational parameterparameter

m dimension of series

sample

Likelihood function (constrained)

Estimation of parameter

m1,2,...,j,h j

30-year simulation period30-year simulation period (2010-2039)

Sca

le p

ara

mete

r

Years1980 1990 2000 2010 2020 2030 2039

500

1000

1500

0

250

750

1250

1750

30-year period analysed through

Monte-Carlo simulation

30-year period used for parameters

calibration

In the simulation period 2010-2039 the maximum deficit of water resources potentiality has been evaluated (extrapolation of the different models through Monte Carlo simulation)

Discontinuous model with constant

parameter

Continuous model with

linear parameter

Continuous model with

rational parameter

Transitional period Simulation period

Maximum deficit of water resources Maximum deficit of water resources potentialitypotentiality

K forecasting time span (30 years) p time span used for deficit evaluation (1, 2, 3, 4, 5 years) ti , t0 index of the year (t0+K ≤ 1980)μH expected value of annual rainfall in stationary period

0,max0,max0,maxD dDPdQmax0,

Steps of the procedure Evaluation of QQDD00,max,max(d(d00,,maxmax))

Evaluation of QQDmaxDmax(d(dmaxmax) ) under the hypothesis of:1) discontinuous constant

model2) continuous linear model 3) continuous rational modelMonte Carlo

techniques

Steps of the procedure Evaluation of QQDD00,max,max(d(d00,,maxmax))

Evaluation of QQDmaxDmax(d(dmaxmax) ) under the hypothesis of:1) discontinuous constant

model2) continuous linear model 3) continuous rational modelMonte Carlo

techniques

H

1p

0r

rtK2010t2010

Kmax pμ

H

1100pD

ii

min

H

1p

0rKttt

Kmax0, pμ

H

1100pDi

min00

Exceedence probability

maxmaxmaxD dDPdQmax

Results (period=2 years)

Variazione della probabilità di deficit della potenzialità idrica(biennale)

0

0,2

0,4

0,6

0,8

1

10 20 30 40 50 60

Deficit della potenzialità idrica (%)

Pro

ba

bili

tà d

i su

pe

ram

en

to

t < 1981 - stazionario

t > 2010 - costante

t > 2010 - lineare

t > 2010 - razionale

P=0.5

25% 37%

39% 40%

Variation of deficit probability of water resources potentiality2-year period2-year period

Deficit of water resources potentiality (%)

stationaryconstantlinearrational

With reference to the next 30-year period, the deficit of water resources potentiality cumulated in a period of 2 years2 years may turn from 25% to about 40%.

Exce

eden

cepr

obab

ility

Variazione della probabilità di deficit della potenzialità idrica(quinquennale)

0

0,2

0,4

0,6

0,8

1

5 10 15 20 25 30 35 40 45 50

Deficit della potenzialità idrica (%)

Pro

ba

bili

tà d

i su

pe

ram

en

to

t < 1981 - stazionario

t > 2010 - costante

t > 2010 - lineare

t > 2010 - razionale

P=0.5

13% 28%

31%31.5%

Results (period= 5 years)

Variation of deficit probability of water resources potentiality5-year period5-year period

Deficit of water resources potentiality (%)

stationaryconstantlinearrational

With reference to the next 30-year period, the deficit of water resources potentiality cumulated in a period of 5 5 yearsyears may turn from 13% to about 30%.

Exce

eden

cepr

obab

ility

RAINFALL VARIABILITY ANALYSESRAINFALL VARIABILITY ANALYSES in Southern Italy

1) TREND ANALYSIS1) TREND ANALYSIS of ANNUAL and MONTHLY RAINFALLSANNUAL and MONTHLY RAINFALLS

Nonparametric / Parametric tests Nonparametric / Parametric tests

• Mann-Kendall / Spearman tests / Linear regression analysis

decreasing trend for most part of rain gauges

2) SEASONALITY ANALYSIS2) SEASONALITY ANALYSIS of MONTHLY RAINFALLSMONTHLY RAINFALLS

Interpretation through Truncated Fourier seriesInterpretation through Truncated Fourier series

• 2 harmonics for mean and coefficient of variation

• decreasing values for decades 1981-90 and 1991-003) IMPACT3) IMPACT of RAINFALL CHANGERAINFALL CHANGE

• TN distribution for stationary period• hypotheses on models for transitional period• Monte Carlo simulation Increase of probability concerning the deficit of water resources potentiality for 30-year future period

CONCLUSIONCONCLUSIONSS