4. SMMAR(1) IMPLEMENTATION It was developed a Pascal routine for generating synthetic total monthly precipitation series. The model was calibrated with the existing monthly rainfall data for the period between jan/44 and dec/05 in nine stations inside Paraná and Uruguay Rivers basins (La Plata Basin Brazilian portion). However, in order to reduce the covariance matrices sizes, principal component analysis was applied. By reducing the matrices size, inconsistencies occurrences in the SMMAR (1) implementation are avoided. So, 26 groups were determined, with dimensions ranging between 2 and 12; 24 of them are smaller than 10. Depending on the analysis performed, four SMMAR (1) versions were prepared. All can be used to generate synthetic total monthly precipitation series for these stations. The proposed versions for the model SMMAR (1) can be classified according to the Box-Cox transformation parameter and covariance matrices between total monthly rainfall series (between stations for a given month and the previous month). In summary, the model SMMAR (1) was implemented for each month of the year for precipitation transformed by the logarithm and by Box-Cox transformation. Thus, for each model it was necessary a 18 dimension covariance matrix for each month of the year (9 rainfall stations with the precipitation of the month and previous month). In the SMMAR (1) calibration phase for the rainfall transformed by Box-Cox transformation, the covariance matrices inverses did not show inconsistencies. 1. MONTHLY SEASONAL MULTIVARIATE AUTOREGRESSIVE MODEL – SMMAR(1) This model deals with seasonality by standardizing rainfall and considers non-stationarity in the correlation structure. Time-varying parameters are required to include seasonal variability in the correlation structure. The first order multivariate seasonal autoregressive model preserves all seasonal means of all variables in the state vector, all seasonal variances, all correlations among all elements of the state vector, in addition to lag-one correlations between adjacent seasons and between all variables. As the proposed model is valid for stationary processes with marginal normal distribution, the monthly average rainfall original series should be submitted to transformations, in order to obtain normally distributed data. 2. GENERAL STRUCTURE Mainmodel equation follows: SYNTHETIC RAINFALL SERIES GENERATION – SMMAR(1) Eloy Kaviski, Miriam Rita Moro Mine Contacts: BRAS, R.; RODRÍGUEZ-ITURBE, I. (1985) Random Functions and Hydrology. Addison-Wesley Publishing Company, California. CLARKE, ROBIN T. (1994) Statistical modeling in hydrology. John Wiley & Sons Ltd. Chichester, England. LEE, K.Y., EL-SHARKAWI, M.A. Modern Heuristic Optimization Techniques, IEEE Press, J. Wiley, New Jersey, MATALAS, N. (1967) Mathematical assessment of synthetic hydrology. Water Resources Res. 3(40:937-46). METROPOLIS, N., ROSENBLUTH, A.W., ROSENBLUTH, M.N., TELLER, A.H. Equation of state calculations by fast computing machines, J. Chem.Phys. 21 (6), 1087 (1953). NEWMAN, M.E.J., BARKEMA, G.T. Monte Carlo Methods in Statistical Physics, Clarendon Press,1999. ACKNOWLEDGEMENTS: The research leading to these results has received funding from the European Community's Seventh Framework Programme (FP7/ ) under Grant Agreement N° Second author also would like to thank “Conselho Nacional de Desenvolvimento Científico e Tecnológico-CNPq” for the financial support. Where, known as the zero-mean vector. In turn, is the random variables state vector, during the year i and season j at location l with mean. The standard normal deviates vector is. The similarity between the model above and a multivariate stationary autoregressive model is obvious; the only difference lies on the seasonal parameters and, represented by matrices. SMMAR (1) get its final equation by redefining vectors and as and, respectively: All bars above the letters represent vectors or matrices Parameters Estimation Bras and Rodríguez-Iturbe (1985) derived parameters estimation to stationary AR(1) models as function of covariances. The process used in the present work is analogous: The interest covariances are given by: With the matrices and calculated in variances, standard deviations and correlations terms Marginal Distributions Transformation to normality by taking logarithms is a log-normal distribution property and it is a common technique among researchers. In working with this marginal distribution, one must use the proper formulation in order to apply the parameters successfully. Matalas (1967) brings equations that allows to perform log-normal transformation preserving the original variables statistics, without bias. However, in practice, hydrological data follows a probability distribution that is known to be skewed but not necessarily log-normal. A more general transformation, of which the log-normal transformation is a particular case, is the Box-Cox transformations family. This technique is used in this work and is detailed in the following. 3. BOX COX TRANSFORMATIONS Box Cox transformations family is defined by: Where is estimated from the data. Clarke (1994), applied the Maximum Likelihood Method to estimate this parameter, in order to fit Box Cox transformation in a data sequence from Ibirama city, Brazil, locate inside CLARIS-LPB study area. The value found was , clearly close to zero; if used, it would neutralize the transformation intended. As solution, in the present work, Box Cox parameter is estimated through the Moments Method application Moments Method Application Box Cox parameter can be estimated by imposing the condition that the skew coefficient of z is zero (corresponding to the normal distribution skew coefficient). From a different point of view, this condition is treated as an optimization problem objective function. To solve this query, the simulated annealing method (Lee and El-Sharkawi, 2008) is employed. This method was devised by analogy with the Metropolis algorithm (Metropolis et al., 1953), which was proposed to simulate statistical physics problems by Monte Carlo method (Newman and Barkema, 1999). Once is calculated, the parameters m and s, transformed variable average and standard deviation respectively, can be estimated using the equations: And K is determined by: The equations system above must be solved numerically. With the obtained solution, the variable y (precipitation) mean and variance are preserved.