The NWP uncertainties of the rainfall forecast is based on non-stationary, nonlinear and dynamic effects as stated by Todini [18]. The solar radiation forecast, also a function of the cloud cover, is expected to be subjected to the same underlying effects. Time series models which using the DWT are capable to interpret non stationary effects as stated in [27] and time series models using the auto-regression (AR) models are capable to interpret dynamic effects [28]. Applying the AR model to the Haar and the bi-orthogonal DWT family, shows that the highest order bi-orthogonal wavelet transform of the Matlab™ tool set (bior6.8) :, has the best performance for the DWT-AR MOS of the solar radiation correction as shown in [29]. Thus, it can be considered, that the DWT, which uses the highest order bi-orthogonal wavelet corrects best the dynamic and non stationary characteristics of statistic MOS model. Improving the correction of the non linear effects, the AR model is substituted by an ANN in the present article.

3.1 Statistical correction of the NWP model output

The idea of the proposed statistical correction method is based on the time series correction MOS as presented in [15] for local correction. The Kalman Filter is substituted by the ANN-DWT (section 3.5). This method is utilized to estimate the correction of the solar radiation for the forecasted day, based on the recognized pattern of the NWP residual obtained from twelve previous days. The residual {sA} between the measured {H} and the forecasted daily mean solar radiation {HA} is estimated to correct the NWP output with this estimations {sA}. The forecasted {HA} and the measured {H} solar radiation time series are in a first step both decomposed in its sub-series {HAs} and {Hs} by the DWT. For mx time scales one obtains s = 1 … (mx + 1) sub­signals of residuals {sas} of the ARPS forecasts by the equation (2). For s = 1 … mx, the vectors {sas} are the sub-signals which represent the details and for s = (mx + 1), the {sas} is the sub­signal which represents the approximation of the ARPS residuals.

{sas} = {Has} — {Hs} (2)

The independent {sAS} sub-signal vectors can also be obtained by the direct decomposition of the residual vector {sA}. Each of these time series is utilized for the training of its corresponding ANN, minimizing the squared error (eqn. 3) in a supervised learning process [30] to estimate the predictand sAs i for the day i, based on the characteristic pattern of the predictors, the sAS — values of (i-1) to (i-k) previous days. The symbol E stands for the energy of the error [30].

E _ 2 ^^i-j (s As, i (sAs, (i-1) ■■■ sAs, (i-k)) — sAs, i) ^ тІП (3)

This function has a support length of thirteen sampled discrete values.

The forecast output of the NWP model is corrected with the sum of the (mx+1) estimated residual values ё а5,і.