INTRODUCTION. 2. Characteristic curve of rain intensities. 1. Material and methods

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1 Determination of dates of eginning and end of the rainy season in the northern part of Madagascar from 1979 to IZANDJI OWOWA Landry Régis Martial*ˡ, RABEHARISOA Jean Marc*, RAKOTOVAO Niry Arinavalona*, RAMIHARIJAFY Rodolphe*, RATIARISON A Adolphe* *University of Antananarivo - Doctoral School of Physics and Applications - Physics of the Gloe and Environment - Dynamics of the Atmosphere Climate and Oceans. Astract This study aims to determine the dates of eginning and end of the rainy seasons, their durations, for the period. We used: the method of six degree polynomial, the anomalous accumulation, and the optimal least squares estimator. The results show that on average, the dates of eginning of the seasons move from East to West. In most cases, the dates of eginning and end, and durations of seasons of rains, show a strong polynomial trend of sixth degree. Keywords: six degree polynomial; anomalous accumulation; optimal least-squares estimator; analysis; ackground; oservation. INTRODUCTION Water cycle is one of the major components of the climate, and its implications on rainfall patterns are important. In tropical and sutropical zone, particularly in Madagascar, rainfall variaility is a major factor of vulneraility of societies. Indeed, the economy remains very largely ased on products of the agricultural sector (example: vanilla in the Northern part, product whose Madagascar is the 1st world producer). Moreover, the capacity of adaptation of population to the climatic risks is still limited too much. This study was undertaken to determine the dates of eginning and end of seasons of rains y implementing three methods.. Characteristic curve of rain intensities Characteristic curve of rain intensities is otained, using an even ( pair ) polynomial function of the sixth degree y filtering the precipitation values, in the purpose to determinate the dates of the eginning and the end of the rainy season For the daily gloal averages in the whole Northern part of Madagascar, ranging etween 4 and 54 East and etween 11 and 15 South, over the 33 seasons etween 1979 and 01, we otained (following equation): (1) 1. Material and methods In this study we used three methods: the method of polynomial of the sixth degree, the method of Liemann (anomalous accumulation) and the optimal estimator of least-squares. We used precipitations data from the European Center of Medium Range Weather forecasting (ECMWF) with the format netcdf. Calculations were made using MATLAB 015 software and tales were made under EXCEL. ˡ author : IZANDJI OWOWA Landry Régis Martial*ˡ Doctorant : izandji@yahoo.fr, RABEHARISOA Jean Marc* Maître de Conférence, RAKOTOVAO Niry Arinavalona* Maître de Conférence, RAMIHARIJAFY Rodolphe*, Docteur en Physique et Application, RATIARISON A Adolphe* Professeur Titulaire Figure 1: Average seasonal variations of the daily rains and the corresponding values, and the characteristic curve of precipitation intensities represented y the values filtered y a polynomial of the sixth degree over 33 seasons in the whole of the Northern part of Madagascar. In theory, the two minima on each side of the maximum, respectively represent the date of the eginning (efore the maximum), and the completion date (after maximum) of the season of rain. 1

2 But, if the minima are not on the positive part of the y axis which represents the values of precipitation, we will take then as minimum, the point of intersection of the values filtered curve with the x axis. Ref [1] 3. Method of polynomial of the sixth degree This method, as its name indicates it, uses an even ( pair ) polynomial function of the sixth degree for the determination of the dates of the eginning and the end of the rainy season. This method was explained to the preceding section. We have elow the general form of the polynomial of the sixth degree. () 4. Method of Liemann : Anomalous Accumulation (AA) The index called Anomalous Accumulation (AA) is given y : (3) The eginning and the end of the rain season then are respectively determined y the date of the minimum and the maximum on the curve of Anomalous Accumulation Ref []. Figure : Anomalous accumulation of the daily average of annual precipitations over the period of on the Northern part of Madagascar A quick analysis makes it possile to note that indeed, the method of Liemann (anomalous accumulation) has a tendency to reduce the real duration of the season, as shown in the following figure. Figure 3 : Comparison etween the methods of Liemann and the polynomial of the sixth degree There are 5 months and 01 day of difference over the duration of the season etween the two methods, which is very considerale. Therefore we need to find a way to make a choice. 5. Analysis of optimal least-squares: Data assimilation concepts and methods. Ref [3] The data assimilation indicates the various methods making it possile to take into account, in the initialization of a forecasting digital model, certain oservation data arrived to this center after the model launched its calculations (forecasts in real time). A comparison is made then, with regular times intervals, etween values produced y the new oservations on the one hand, and corresponding values predicted y the model on this term on the other hand. From the differences etween the rough outline (draft) of evolution provided y the model and the new oservations, it then ecomes possile to rectify the walk of the model, y periodically modifying its forecasting results at determined moments, and these results, once modified, ecome new initial data for the future forecasts. 1) Data assimilation analysis In the presentation of this method of analysis, we will refer sometimes at the true model state. It is an expression referring to the est state possile to represent y the model. That which we try to approach.

3 It is necessary to make the distinction etween reality itself (which is more complex than the representation of the state vector) and the est possile representation of reality y a state vector than we will call : true state at the time of the analysis. Another important value of the state vector is, who is the a priori or a ackground estimate (draft) of the true state. The analysis is indicated y. Therefore, the prolem of the analysis reduces to find a correction or called analysis increment such as :, eing as close as possile of. For a given analysis we use several values of oservation (or data and other information related), which are contained in an oservation vector. To use its values in the procedure of analysis requires eing ale to compare them with the state vector. The key of the data analysis is the use of contradictions etween the oservations and the state vector. Those are given y the vector of departures at the oservation points : analysis can e reduced to a simple scalar expression, so that : X X k(x X ) a Let us notice that the oservations here y. o are expressed The variance of the error of the estimate (of the analysis) is given y : ( 1 k) k a Therefore, the optimal analysis is : x xo. o x a (4) o o eing as close as possile of the true state ; the oservations are expressed here y ; and is the a priori or a ackground (the first estimate) of the true state. The «gain», or the weight of the analysis is given y : o When they are calculated from the rough outline (first estimation or ackground), they are called innovations, and when they are calculated from the analysis, they are called analysis residuals. Studying these departure vectors provides important information aout the quality of the procedure of assimilation. Where, the oservation operator, is a linear operator - the dimension of the model state is the dimension of the vector of oservations is. ) Optimal least-squares analysis equations. The optimal estimators of least-squares, or analysis BLUE, are defined y the following interpolation equations : - Optimal analysis : we should seek a weighted linear average whose form is : X X K(y H[X ]) a - Gain or weight of the analysis : K BH T (HBH T R) Where the linear operator is called «gain», or weight matrix of the analysis. The weights are noted (minuscule)., and are respectively: the operator of oservations, the matrix of covariance of the outline (ackground) errors and the matrix of covariance of the oservation errors. The optimal least-squares 1 k (5) Where and are respectively the error variances of the ackground (or first estimate) and of the oservation. The error variance analysis is : o (6) In the limiting case of a very low quality of oservation, we will have, so, and then ( ) : and the analysis remains equal to the ackground. Because as we saw : ) ). However, if the oservation is of very good quality (contrary case)., then, and the analysis remains equal to the oservation ( ) If oth have the same precision :, then, and analysis is simply the arithmetic average of and, which reflects the fact that we trust as much the oservation as the ackground. We have then :, so we make a compromise. 3

4 But in all cases, which means that the analysis is a weighted average of the ackground and the oservation. existing differences etween methods AA and POLY6. The analysis variance for the optimal is : 6. Choice of the oservation and the first estimate (ackground) for the calculation of optimal values of dates of eginning and end of rainy season We take as oservations the dates of eginning and end oserved on the seasonal variations of precipitations from 1 st August until 31 st July, which are provided y the polynomial of the sixth degree. Indeed, these dates are comparale to direct oservations, ecause they are readale on the graph of rain variations. We will take as first estimate (or outline, ackground), the dates of eginning and end of season estimated y the method Anomalous Accumulation. We will take the Root Mean Squared Error (noted RMSE), as value of (standard deviation of oservation errors provided y the polynomial of the sixth degree). The RMSE is regarded as an estimate of the standard deviation of calculated answers. Because we do not know really, which of the two methods (polynomial of the sixth degree and anomalous accumulation) is closest to the true state, we will give them the same importance while imposing as and consequently that That is equivalent supposing that the values resulting from the two methods have the same proaility ( equiproale ). That suits us, ecause the average Root Mean Squared Error of the anomalous accumulation is much larger (188,8), compared to that of the polynomial of the sixth degree, and in this case the analysis will e almost equal to the oservation (polynomial of the sixth degree). Results of the average duration for the whole of the Northern part of Madagascar, (average of 33 seasons: from 1979 to 01) are given in the following tale. Figure 4 : Average dates of eginning and end of rainy season, otained y the methods of the polynomial of sixth degree (POLY6) and anomalous accumulation (AA), compared with those calculated y the optimal least squares analysis (ANALYSIS). In the previous sections, we worked with gloal daily averages. Now, to determine year per year the dates of eginning and end for ten (10) rain seasons, etween 1979 and 1989, we will reduced and sudivided the Northern part of Madagascar in four parts. 7. Reduction and division of the study area The reduced area is located etween 1 and 15 of South latitude and 47 and 51 of East longitude. We will divide the area into four parts (following figure). Tale 1 : Duration of the rain season (average of 33 seasons : from 1979 to 01) in the whole of the Northern part of Madagascar, and parameters of the analysis. DURATIONS OF THE RAIN SEASON POLY6 AA optimal ANALYSIS 88 Days : 9 months et 18 days 137 Days : 4 months et 17 days 13 Days : 7 months et 3 days = 1,47 Compromise : = 0,8660 Note: We have named : - POLY6: Polynomial of sixth degree - AA: anomalous accumulation - ANALYSIS: optimal estimator of least-squares We see the tendency of anomalous accumulation method to reduce the duration of the season of rains, and the capacity of least-squares optimal estimator to optimize the duration of the season y reducing the Figure 5 : Area sudivided into four parts 4

5 Index of dates (in days) Duration of season (in days) Duration of season (in days) 8. Determination of indices of start and end of the rainy season for ten seasons going to to , and their duration The results we considered were those provided y the optimal least squares analysis. They are supposed to e etter ecause they are optimal according to the optimal least squares analysis. We have summarized in Figures 6, the average dates of eginning and end of seasons, and their average earliest and latest dates for each of the four areas. - Duration of seasons: In zone 3 there was a very significant polynomial trend of the sixth degree for the durations of the seasons. y = -0,1053x 6 + 3,6457x 5-48,946x ,08x ,6x ,9x - 633,37 DURATION (in Day) R² = 0,8808 Linéaire (DURATION (in Day)) Poly. (DURATION (in Day)) y = -5,84x + 03,73 R² = 0, Rank of the seasons: from 1979 to 1989 Figure 8: Behavior of the durations of the seasons of rains of 1979 to 1989 in Zone 3 (Northern Part: South-west) Figure 6 : Average dates of eginning and end of season (average of ten seasons studied) for each of the four zones. 9. Behavior of the dates of eginning and end, and of the durations of the rain seasons etween 1979 and 1989 (10 seasons) We made this oservations: these variales did not present a significant linear trend for the ten studied seasons; a significant polynomial regression trend of the sixth degree was oserved, as shown in the following examples. In the graphs which will follow, Linéaire means linear regression and Poly means polynomial regression. - Dates of eginning: In zone 4, there was a significant polynomial trend of the sixth degree. y = -0,0144x 6 + 0,3397x 5 -,4778x 4 + 3,8001x ,107x - 49,743x Index of BEGINNING (in Day) R² = 0,775 Linéaire (Index of BEGINNING (in Day)) y = -1,3636x + 116,8 Poly. (Index of BEGINNING (in Day)) R² = 0, Rank of the seasons: from 1979 to 1989 Remark: We oserved in zone 1, a weak linear falling trend with a coefficient of determination of 0.5 (figure 9 elow). We shall see what will give the linear regression over thirty-three (33) years (i.e. up to 01). If this tendency is confirmed, it would mean a reduction in the duration of the seasons of rains. This will e the oject of the continuation of our work Figure 9: Behavior of the durations of the seasons of rains of 1979 to 1989 in Zone 1 (Northern Part: North-West) Conclusion y = -0,075x 6 + 0,9138x 5-11,655x ,188x 3-11,61x + 73,68x + 40,667 R² = 0,5878 DURATION (in Day) Linéaire (DURATION (in Day)) y = -,5394x + 159,87 Poly. (DURATION (in Day)) R² = 0, Rank of the seasons: from 1979 to 1989 In this study, we oserved that on average, the rainy season egin in the East and advances towards the West. As for the end of the season, it propagate from West to East in the Northern zone of the Northern Part of Madagascar, and from East in West in the Southern zone of this Northern part. In most cases, the dates of eginning and end, and durations of seasons of rains, showed a strong polynomial trend of the sixth degree.. Figure 7: Behavior of the dates of eginning of seasons of rains from 1979 to 1989 in Zone 4 (Northern Part : South-east) 5

6 References : [1] RAMIHARIJAFY Rodolphe : «Inter corrélation entre la pluviométrie et le déit sauvage en amont de la centrale hydroélectrique du site d Andekaleka a Madagascar». article&id_article=3144 Thèse Ecole Doctorale Physique et Application Université d Antananarivo. Page [] Apport des données TRMM 3B4 à l étude des précipitations au MATO GROSSO. Université Paris 7 - Universidade do Estado do Rio de Janeiro - UERJ /746/40_arvor.pdf Page [3] ECMWF, Bouttier and Courtier March 1999 : Data assimilation : concepts and methods milation%0concepts%0and%0methods.pdf 6