Statistical techniques for investigating rainfall variability at monthly and annual time scale E. Ferrari E. Ferrari Dipartimento di Difesa del Suolo, Università della Calabria Rende (CS) – ITALY International Workshop on: Evaluation des changements globaux sur les regimes Evaluation des changements globaux sur les regimes hydrologique et les ressources an eau hydrologique et les ressources an eau Université Mohamed V-Agdal, Faculté des Sciences Rabat (Morocco) 10 - 11 Décembre 2009
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Statistical techniques for investigating rainfall variability at monthly and annual time scale
International Workshop on: Evaluation des changements globaux sur les regimes hydrologique et les ressources an eau Université Mohamed V-Agdal, Faculté des Sciences Rabat (Morocco) 10 - 11 Décembre 2009. - PowerPoint PPT Presentation
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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)
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
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
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
• 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