"FORECASTING OF THE RAINFALL AND THE DISCHARGE OF THE NAMORONA RIVER IN VOHIPARARA AND FFT ANALYSES OF THESE DATA

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1 "FORECASTING OF THE RAINFALL AND THE DISCHARGE OF THE NAMORONA RIVER IN VOHIPARARA AND FFT ANALYSES OF THESE DATA Andry Tefinanahary RABENJA, Adolphe RATIARISON and Jean Marc RABEHARISOA Suspensions Rheology Laboraory, Deparemen of Physics, Universiy of Anananarivo, Madagascar I - Inroducion: The ime series predicion corresponds regularly o he analysis of observaions occasional in he ime. In his aricle, he monhly ime sep has been chosen for discharge and rainfall daa. In his aricle, he combinaion of he mehods self regressive and moving average has been used in order o foresee he evoluion of he rainfall and he discharge. The firs, ARMA(p,q) model, he second, he ARIMA(p,d,q) model inegraes he daa, and he SARIMA(p,d,q)(P,D,Q) process ha akes in accoun he seasonal variaions[4]. The crieria of Nash-Sucliff has been used o value he efficiency of he models used [6]. The frequenial analysis by Fas Fourier Transform (FFT) permis o deec he hidden periods in he daily daa of rain and debi. II - Mehodology: II-1 Saion of survey: Our saion of survey is he draining basin of Vohiparara. Localized a 21 14' Laiude Souh ' Longiude Eas. This saion conrols a draining basin of 445km2. The middle aliude of his basin is of 1250m. The hydroelecric saion of Vohiparara has been implaned in 1929 for he survey of he debis of he Namorona River. The scale of ORSTOM has been insalled in 1951, downsream he firs, o 400m upsream of he firs fall. Sormy in 1960, because of a repairing of he road, i has been reinsalled in Ocober 1960, hen in January 1970 afer he passage of he JANE cyclone. The period of survey spreads of 1952 à1989 [3]. II-2 Daa: The rainfall daa in millimeer, and of discharge in cubic meer per second have been goen in he meeorological saion, a he hydrological service of Ampandrianomby in Anananarivo. The daa spread on abou hiry years. 1/12

2 II-3 ARMA(p,q), ARIMA(p,d,q), and SARIMA(p,d,q)(P,D,Q) models: The ARMA model is a mixure of he self regressive models and moving average. A process ( ) is an ARMA(p,q) process if i is saionary, ha means heir whie noises ε andε k are independen, for all k, for all, as: = Φ Φ + ε + θ ε θ ε, for all. 1 1 p p 1 1 q q These processes are saionary under some condiions, and can be wriing as: Φ ( L) = θ ( L) ε Where Φ ( L) = II Φ1L... Φ pl θ ( L) = II + θ L θ L And 1 p p p II is he marix ideniy, L represens he delay operaor, o he sense where L = 1, and wih he convenion L p ο p 1 = L L, eiher: Lp p =, he serie (Y), asy = L p is hen he serie ( ) rearded of p periods. In he same way, a non saionary process is inegraed of order 1, differeniaed unil i becomes a saionary process; ( ) non saionary will be said inegraed of order 1 if he process ( Y ) defines by: Y = 1 = (1 L) is saionary. The process () can be wrien like his: Π ( L) = Φ( L)(1 L) d = θ ( L) ε Will be he ARIMA(p,d,q process), ε is a whie noise. For he real daa, d will be noed d = 1, 2 or 3 (o he maximum). I means ha (Y) defines like difference of d order of he process (). Π( L) is he produc of delay operaors L. SARIMA models i is a model ARMA inegraed and aking ino accoun he seasonal par of he daa [4]. II-3-1 Esimae of he models parameers: For he model of p and q order, i for 1 o p and j for 1 o q, he parameers [ ] i Φ and θ j will be deermine by he maximum verisimiliude mehod. The probabiliy of appariion of he elemens is raised if he verisimiliudes funcion V ( Φ, θ, σ ) is maximum, ha means: V ( Φ, θ, σ ) = P( Φ, θ, σ ) 2/12

3 The variables valuers Φ and θ ha would reurn he probable forecasing values areϕ and ϑ as : p( ϕ, ϑ) p( φ, θ ) Wha comes back o maximize is logarihm, eiher: ln( p( φ, θ, σ )) φ φ = ϕ The verisimiliude valuers * and ln( p( φ, θ, σ )) (ln( p( φ, θ, σ )) =0, θ = ϑ = 0, and σ = σ * θ σ * ε, as well as he previous values of * * and ε are he soluions of hese equaions, he iniial values and. ε are supposed known. Then, ln(v) is maximal if T φ, θ ε * * = 1 ε,, is absoluely minimal. All hese models of forecasings, o know ARMA(p,q),ARIMA(p,d,q) and SARIMA(p,d,q)(P,D,Q), are based on he DURBIN-LEVINSON algorihm [4]: ˆ T + h = α k + 1 k k = 1 The α k parameers are choosen in order o minimize he predicion error. II-4 Fas Fourier Transform: The Direc Fourier Transform (TFD) is a powerful reamen ool of he signal by frequenial analysis, The algorihm Fas Fourier Transform (FFT) reduced he number of necessary operaions o calculae a TFD in an O(n order n log2), ha is a considerable progress. The FFT permis he calculaion of he specer of a periodic signal and o deduc hidden periods of i. For avecor x, he FFT is like his : II-5 Crieria of Nash-Sucliff: The crieria of Nash-Sucliff permis o assess a hydrological model, i is defined like follows: E = 1 ( Q ( Q o o Q Q o m ) 2 ) 2 Where Q o is he observed discharge a ime, Q m he modelled discharge a ime, and he average of he observed discharge. The value of E varies of o 1, he bes model is he one having a value of E close o 1, ha means reliable o 100% [7]. 3/12

4 III-Resuls: III-1 Resuls of he discharges forecasing: III-1-1 ARMA(2,12) forecasing of he monhly discharge of 1952 o 1979: The monhly daa of discharge spread from January 1952 o December The forecasing wih he ARMA (2,12) mehod had begun he January 1980 and have been finished in December debi(m3/s) monh ARMA real debi Face1: Comparison of he ARMA(2,12) forecasing and he real value of he monhly discharge. For he ARMA forecasing of he monhly discharge of he year 1980, he monhs of January, June, Sepember, Ocober, presens a deviaion. 4/12

5 III-1-2 ARIMA(2,1,12) forecasing of he monhly discharge of 1952 o 1979: The forecasing wih he ARIMA model has been made on he same period ha wih ARMA. 40 debi(m3/s) ARIMA real debi monh Face 2: Comparison of he ARIMA(2, 1, 12) forecasing wih he real value of discharge. A he ime of he use of he ARIMA(2, 1, 12) forecasing, ha means when he process was inegrae once, a reducion of difference beween he forecasing and he real value of monhly discharge has been observed. III-1-3 SARIMA(2,1,12)(1,1,1) forecasing of he monhly discharge of 1952 o 1979 The forecasing wih he SARIMA mehod has been made on he same period ha he wo previous mehods. debi(m3/s) 35,00 30,00 25,00 20,00 15,00 10,00 5,00 0, monh SARIMA real debi Face 3: Comparison of he SARIMA (2, 1, 12) (1, 1, 1) forecasing wih he real value of discharge. 5/12

6 While inroducing he seasonal componens, an improvemen has been noiced for he forecasing. III-2 Resuls of rainfall forecasing: III-2-1 ARMA(24,12) forecasing : rainfall (mm) ARMA real value monh Face 4: Values forecas by ARMA(24,12) and values real of he monhly rains. A gap is noiced ener he forecasing wih he ARMA(24,12) model and he real values, his disance is consan safe for he firs and he las monh of forecasing. III-2-2 ARIMA(24,1,12) forecasing: rainfall(mm) monh ARIMA real value Face 5: Values foreseen by ARIMA (24,1,12) and values real of he monhly rains. The ARIMA(24,1,12) model esimaes he values of rain beer. On his Face, he difference beween he ARIMA(24,1,12) forecasing and he real value is less imporan han he one observed wih he model ARMA. 6/12

7 III-2-3 SARIMA(24,1,12)(1,1,1) forecasing: rainfall(mm) SARIMA real value monh Face 6: Values forecasing by SARIMA(24,1,12)(1,1,1) and real values of rains. On his Face, he difference beween he forecasing values and he real values are minimal III-3 Resuls of FFT applied o he daa of monhly discharge of 1952 o 1980: The specers of fourier are calculaed by he FFT algorihm and are represened on he face according o he periods. Period = Period = Face 7: daily debi specer according o he periods. On his face, he periods ha correspond o he maximum of he specers are 181,65 and 371,95 days, he period 181,65 corresponds o December and 371,95 corresponds o December 31, /12

8 For he period equal o 180,91 days, here are he discharge superior or equal o 10m3/s: years mohs days Discharges(m 3 /s) 1952 december 23 10, december 14 12, december 1 22, november november 22 19, november 10 17, ocober 31 42, ocober 13 19, ocober 11 29, ocober 5 37, sepember sepember 12 14, sepember 4 12, augus 20 14, augus 9 16, augus 6 14, july 24 19, july 14 36,8 Table 1 : discharge for he period 180,91 For he period 388,95 days, he debis superior or equal o 20m3/s are raised: years mohs days Discharges(m 3 /s) 1954 february 6 24, sepember 14 38, ocober december 26 36,6 Table 2 : discharge for he period 388,95 8/12

9 On hese ables 1 and 2, we noe ha he debis superior or equal o 20m3/s appears on February, July, Augus, Sepember, Ocober, November, December. III-4 FFT daily rainfall : The specers of fourier of he daily daa of rain have been calculaed by he FFT. Period = Period = Face8: FFT of he daily daa of rain according o he periods. On his face, he periods ha correspond o he maximum of he specers are 181,65 and 371,95 monhs, he period 181,65 corresponds o December and 11,597 corresponds o December 31, /12

10 For he period equal o 181,65 days, here are he rains of heighs supérior or equal o 20 mm: years monhs Days Rainfalls(mm) 1952 december 13 98, december 14 21, december 5 22, november 29 33, november 3 53, ocober 16 22, ocober 6 31, sepember augus 18 24, augus 5 63, july 13 25, july 12 22,7 Table 3 : rainfall for he period 181,65. In his able he heighs of rains ha correspond o he period days are superior or equal o 20 mm For he period equal o 371,95 days, one raised rains superior or equal o 20 mm: years monhs days Rainfalls(mm) 1953 december 31 73, december , december 25 66, december 27 42, december 25 43, january 7 20,7 Table 4 : rainfall for he period 371,95. The values of rains had for he period 371,95 are he imporans values of December and January rains. 10/12

11 IV - DISCUSSION and CONCLUSION: In his survey, ARMA(2,12),ARIMA(2,1,12), and SARIMA(2,1,12)(1,1,1) are he models use for he forecasing of he monhly discharge. Wih he model ARMA (2, 12), he misakes of forecasing are more imporan. I is due o he non inegraion of he daa. Wih he ARIMA model (2, 1, 12) ha inegraes he daa once, an improvemen has been goen, however i remains again a difference wih he real values. While using he es of Nash Sucliff, we can noed ha i is he SARIMA(2,1,12)(1,1,1) model wih coefficien of Nash Sucliff 0.54 eiher a reliabiliy of 54% ha foresees bes he monhly discharge, in conclusion, he bes model of forecasing is he one ha firs inegraes he daa, and ha akes in accoun he seasonal flucuaions, wha is he case of he SARIMA(2,1,12 model) (1,1,1). For he forecasing of monhly rainfall daa, he p parameers and q used s are no he same ha hose of he monhly debis. I is he fac ha he used parameers change according o he daa o foresee. The choice of hese parameers is made while using he maximum verisimiliude mehod. The models used for his monhly rain forecasing are: ARMA(24,12), ARIMA(24,1,12) and SARIMA(24,1,12)(1,1,1). The daa of rainfall presen a seasonal par of he daa ha divides he year in wo: of November o April, he season of he srong rains and May o Ocober, he dry season. Thus, i is he SARIMA model ha esimaes bes he real values of he monhly rains. The frequenial analysis " Fas Fourier Transform" gives he periods of repeiions of rains and debis; The values of rains or debis ha correspond o hese periods are maximum values. In conclusion, he SARIMA model is he more adaped for he forecasing of he daa of he rainfall and he monhly discharge and he FFT permis o deermine he dae o which rains or debis of imporan values occurs.. 11/12

12 BIBLIOGRAPHY: [1]: Hydrological direcion of France overseas, 1954, auhor: overseas of scienific and echnical research office(orstom), page ; Ediion Paris 1993 and CD-ROM version [2] Elemens of surface hydrology, auhor: J. P. Laborde, Docor of science in hydrology. Ediion [3] Sreams and river of Madagascar, auhor: Joël Danloux, Luc Ferry, Pierre Chaperon. I.R.D ediion Paris 1993, version CD-ROM [4] Time series, heories and applicaions. Volume 1, Inroducion o he heory of he processes in discree imes, Model ARIMA and mehod of Box& Jenkins, auhors: Arhur Charpenier, DESS Mahémaique of he decision Ediion [5] O. Sievers, K. Fraedrich, C.C. Raible, Wheaher and Forecasing 15, 623 (2000). [6] Course of hydrology, auhor,: André Musy. hp://hydran.unibe.ch/hades/hades_fr.hm [7] Forecasing of he river fluxes by concepual model, Par 1, auhor: Nash, J.E and Sucliff, Journal of hydrology page: /12