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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.

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Statistical techniques for investigating rainfall variability at monthly and annual time scale

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  1. 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 Statistical techniques for investigating rainfall variability at monthly and annual time scale E. Ferrari Dipartimento di Difesa del Suolo, Università della Calabria Rende (CS) – ITALY

  2. Basic detail Management of water resources and analysis of water balance needinvestigations about variability of longer time aggregation rainfalls, such as MONTHLY AND ANNUAL RAINFALLS. Objectives of the work  Review of some statistical techniques for: Trend detection(usually monthly and annual rainfalls) Seasonality analysis(usually monthly rainfalls) Analysis of impact of rainfall changes (annual rainfall)  Application on drainage basins of Southern Italy

  3. TECHNIQUES for SHIFT and TREND detection • MANN-KENDALL test • SPEARMAN rank correlation test • Linear regressionanalysis • MANN-WHITNEY test (step-change) 1st Case study • TECHNIQUES for SEASONALITY analysis • FOURIER analysis • one-way analysis of variance • lag1 and lag12 month-to-month correlationversus season • procedures based on monthly mean / SD / skewness 2nd Case study TECHNIQUE for evaluating IMPACT of RAINFALL CHANGE Simple rainfall change scenarios  Variability of water resources potentiality 3rd Case study

  4. SPEARMAN’s rank correlation Test If H0 holds, the statistic D  N ( E(D) , Var(D) ) The test statistic ZSN(0,1) Trend is significant if: MANN-KENDALL Test If n8 and H0 holds, the statistic S  N( E(S) , Var(S) ) The test statistic ZMKN(0,1) Trend is significant if: Non-parametric tests 1) TECHNIQUES for TREND detection Time series: x1, x2, …, xn H0: no trend vs H1: trend

  5. 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 (β1=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 slope parameter β1does not contain the 0 value, that is the population value of slope parameter when no trend occurred.

  6. Fourier analysis 1/3 2) TECHNIQUES for SEASONALITY analysis Seasonal variability (FOURIER analysis) Monthly rainfalls Power transformation ~ Normal distribution Deseasonalization - standardization mean coefficient of variation Fourier cofficients a0(μ), ai(μ), bi(μ) a0(ν), ai(ν), bi(ν) trigonometric interpolation For prefixed and significance level  least numberof harmonics CI of the mean CI of coefficient of variation  Montecarlo techniques

  7. Fourier analysis 2/3 2) TECHNIQUES for SEASONALITY analysis Analysis of reduced variate Z Time series: x1, x2, …, xn Once removed seasonality  Test on PROCESS RANDOMNESS(possible residual correlation structure) Anderson Test If Z is a stricly stationary and independent normally distributed process, the sample autocorrelation coefficients rZ,k If the values of rZ,k belong to the Ics the hypothesis that the process is purely random cannot be rejected CI of rZ,k Further tests Z~N(0,1) (normal probabilistic plot) Skewness coefficient g2,Z CI evaluated through Montecarlo method Kurtosis coefficient

  8. Fourier analysis 3/3 2) TECHNIQUES for SEASONALITY analysis Use of the model Comparisons between different periods (decades): Period X: Sample z1’, z2’,…, zn’ Period Y: Control samplez1”, z2”, …, zn” (H0: stationary period) N.B. Control sample is excluded from data used for calibration of the model Two sample Kolmogorov-Smirnov test The hypothesis that they come from the same statistical universe, at significance level , has to be refused. where 1-is evaluated by:

  9. 3) ANALYSIS OF IMPACT OF RAINFALL CHANGE Probabilistic evaluation of variability of water resources potentiality depending on occurrence of rainfall change scenarios • Procedure • Fitting of a probability distribution to data (areal annual rainfall)observedin the stationary period1916-80. • Hypothesis on the variability of parameters of the probability distributionhypothesized for the 30-year transitionalperiod1981-2010 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 techniques).

  10. Application on basins of Southern Italy Rainfall data base: monthly rainfalls observed in 70 rain gauges (1916–2008) Cosenza All examined area (~5000 km2) Crati River Basin (~2500 km2) ITALY CALABRIA

  11. Code Years ZMK ZS Lin.Regr. 980 49 -2.42 -2.77 -4.87 990 65 -4.21 -4.41 -6.02 1000 61 -2.44 -2.44 -6.44 1010 68 -2.29 -2.33 -3.40 1020 51 -4.09 -4.05 -14.23 1030 60 -2.69 -2.62 -4.20 1040 67 -4.03 -4.03 -6.61 1050 68 0.75 0.76 +1.74 1060 53 -2.65 -2.49 -8.49 1070 50 -5.29 -4.89 -25.03 1080 63 0.52 0.44 +1.54 1090 42 0.61 0.53 +2.55 1110 63 -0.21 -0.12 +0.47 1120 58 -1.34 -1.44 -3.19 1130 57 -0.67 -0.63 -1.54 1140 48 -0.27 -0.38 -1.25 1150 54 -1.65 -1.63 -4.20 1170 49 -2.01 -1.99 -5.10 1180 66 0 0.05 +0.22 1190 63 -2.80 -2.58 -2.37 1200 60 -1.03 -0.93 -2.30 1220 49 -5.22 -5.19 -17.18 1230 60 -1.08 -1.11 -3.02 1240 54 -3.06 -2.92 -6.80 1260 54 0.57 0.50 -0.70 1290 47 -1.98 -2.16 -3.48 SIGNIFICANT TREND Blue: NORed:YES 1st example RESULTS from TREND ANALYSIS of annual rainfalls 26 rain gauges internal to Crati basin MANN-KENDALLZMKN(0,1) SPEARMAN’s Rank CorrelationZSN(0,1) Linear regression H0:{β1=0} Y= β0+β1X+ε

  12. 2 harmonics 1 harmonic Mean of H1/2 14 12 10 8 6 97,5% mean 4 2,5% 2 G F M A M J J A S O N D Monthly rainfalls of Crati basin Month 2nd example RESULTS from FOURIER ANALYSIS of monthly rainfalls Model calibration(monthly mean) Normalizing value Best number of harmonics λ=1/2

  13. 2 harmonics 1 harmonic Cv of H1/2 0.7 97,5% mean 0.6 2,5% 0.5 0.4 0.3 0.2 0.1 G F M A M J J A S O N D Month Monthly rainfalls of Crati basin 2nd example RESULTS from FOURIER ANALYSIS of monthly rainfalls Model calibration(monthly Cv) Normalizing value Best number of harmonics λ=1/2

  14. r(k) Confidence interval 0.3 0.2 0.1 Autocorrelation coefficients 0 0 6 12 18 24 -0.1 Lag k (months) -0.2 (α=0.05) -0.3 2nd example RESULTS from FOURIER ANALYSIS of monthly rainfalls Verification of the model(Anderson test for serial correlation) Power transformation (λ=0.5) + Removal of periodicity Presence of correlation of process {Z} can be rejected

  15. 2nd example RESULTS from FOURIER ANALYSIS of monthly rainfalls Verification of the model(Z~Standardized normal distribution) Test on coefficient of skewness[-0.196 < -0.051< 0.196] Test on coefficient of Kurtosis[2.65 < 2.74 < 3.42] N=600 sample values (12x50 years)

  16. 2nd example RESULTS from FOURIER ANALYSIS of monthly rainfalls Verification of the model(Analysis of decades 1981-90 and 1991-2000) Years 1991-2000 Years 1981-1990 Years 1971-1980 Confidence interval 2 observed values of Z 1 0 -2 -1 0 1 2 expected values of Z -1 -2

  17. 2nd example RESULTS from FOURIER ANALYSIS of monthly rainfalls Verification of the model(Two-sample Kolmogorov-Smirnov test α=0.05) 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 ?

  18. Areal annual rainfalls – drainage basin of Crati River h (mm) 1750 1500 1250 1000 750 500 1910 1920 1930 1940 1950 1960 1970 1980 1990 2000 2010 years 3rd example RESULTS from ANALYSIS of IMPACT of RAINFALL CHANGE Models for stationary(1921-80)and transitional periods(1981-2010) Time period up to 1980 Known observations: Model for stationary period with discrete parameter i 1 2 … … … 64 65 ti 1916 1917 … … … 1979 1980 899.7 1065.1 … … … 1243.9 1420.5 Known observations: Time period after 1980 j=i-65 1 2 … … … 64 65 Model for transitional period with discrete parameter ti 1981 1982 … … … 2007 2008 1132.5 851.5 … … … 869.4 994.3

  19. Transitional period Stationary period 1500 1000 500 0 1910 1920 1930 1940 1950 1960 1970 1980 1990 2000 2010 Years Mean values of models forstationary period (1921-1980)andtransitional period (1981-2010)

  20. Φ() Standard normal distribution • β0 scale parameter • α,θ form parameters • If θ0 Log-normal distr. • If α=1 Box-Cox distr. Values of parameters … data: it holds: STATIONARY MODEL Probabilistic distribution Trans-Normal (T-N) Probability density function Likelihood function

  21. Normal probability plot 0.995 0.99 0.98 0.95 0.90 0.75 0.50 0.25 0.10 0.05 0.02 0.01 0.005 600 800 1000 1200 1400 1600 1800 Areal annual rainfalls in Crati River basin (mm) T-N distribution fitted to data of period 1916-1980 (stationary model with discrete parameter - white noise)

  22. Statistics Variation coefficient: (unchanged) skewness: (unchanged) mean: variance: MODEL adopted for transitional period 1981-2010 Probability distribution • variation of scale parameter (β) • unchanged parameters (α, θ)

  23. Discontinuos model with constant parameter 1500 Scale parameter 1000 500 0 1910 1920 1930 1940 1950 1960 1970 1980 1990 2000 2010 Years 1. Discontinuos model with constant parameter Likelihood function (constrained) m sample dimension sample Estimation of parameter …

  24. Continuous model with linear parameter 1500 Scale parameter 1000 500 0 1910 1920 1930 1940 1950 1960 1970 1980 1990 2000 2010 Years 2. Continuous model with linear parameter Likelihood function (constrained) m dimension of series sample Estimation of parameter …

  25. Continuous model with rational parameter 1500 1000 Scale parameter 500 0 1910 1920 1930 1940 1950 1960 1970 1980 1990 2000 2010 Years 3. Continuous model with rational parameter Likelihood function (constrained) m dimension of series sample Estimation of parameter …

  26. 30-year simulation period(2010-2039) Discontinuous model with constant parameter 1750 30-year period used for parameters calibration 30-year period analysed through Monte-Carlo simulation 1500 1250 Continuous model with rational parameter 1000 Scale parameter 750 500 Continuous model with linear parameter 250 Simulation period Transitional period 0 1980 2039 2010 1990 2000 2020 2030 Years 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)

  27. Maximum deficit of water resources potentiality 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 Steps of the procedure Evaluation of QD0,max(d0,max) Evaluation ofQDmax(dmax) under the hypothesis of: 1) discontinuous constant model 2) continuous linear model 3) continuous rational model Monte Carlo techniques Exceedence probability

  28. Results(period=2 years) Variation of deficit probability of water resources potentiality 2-year period stationary constant linear rational Deficit of water resources potentiality (%) With reference to the next 30-year period, the deficit of water resources potentiality cumulated in a period of 2 years may turn from 25% to about 40%.

  29. Results(period= 5 years) Variation of deficit probability of water resources potentiality 5-year period stationary constant linear rational P=0.5 13% 28% 31% 31.5% Deficit of water resources potentiality (%) With reference to the next 30-year period, the deficit of water resources potentiality cumulated in a period of 5 years may turn from 13% to about 30%.

  30. CONCLUSIONS RAINFALL VARIABILITY ANALYSES in Southern Italy • 1) TREND ANALYSIS of ANNUAL and MONTHLY RAINFALLS • Nonparametric / Parametric tests • Mann-Kendall / Spearman tests / Linear regression analysis •  decreasing trend for most part of rain gauges • 2) SEASONALITY ANALYSISof MONTHLY RAINFALLS • Interpretation through Truncated Fourier series • 2 harmonics for mean and coefficient of variation • decreasing values for decades 1981-90 and 1991-00 • 3) IMPACTof RAINFALL CHANGE • TN distribution for stationary period • hypotheses on models for transitional period • Monte Carlo simulation  Increase of probability concerning thedeficit of water resources potentiality for 30-year future period

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