# Combination of Models to Generate the First PAR Maps for Spain
## 2 Materials and Methods
\[\text{PAR}=\text{a}\cdot\text{PAR}_{\text{MODIS Model}}+\text{b}\cdot\text{PAR}_{ \text{CM-SAF Model}}+\text{c} \tag{2}\]  
where the \(\text{PAR}_{\text{MODIS Model}}\) is the estimates from the MODIS model, the \(\text{PAR}_{\text{CM-SAF Model}}\) is the estimates from the CM-SAF model and a, b, and c are the fitting coefficients. To obtain estimates from the MODIS and CM-SAF models, the coefficients of these models were linearly interpolated to the coordinates of the location of each station and then both models were fed with the GHI data measured at the stations to calculate the PAR estimates. In this way, it is possible to obtain the coefficients a, b, and c that correspond to each station. Therefore, the fitting coefficients obtained for the location of the stations were linearly extrapolated on a least-squares approximation of the gradient at the boundary to each location of the original data grid, in order to obtain a PAR model that covers the entire territory of mainland Spain. To validate the new model, its estimates were compared with PAR measurements in the seven stations. Statistics such as the slope and intercept of the scatterplot between the PAR estimates and the PAR measurements, the coefficient of determination (R\({}^{2}\)), the mean bias error (MBE) and the root mean square error (RMSE) were used to address the goodness of the models. \[\text{slope}=\frac{\sum_{i=1}^{n}(\text{PAR}_{\text{Model}}(\text{i})-\overline {\text{PAR}}_{\text{Model}})\big{(}\text{PAR}_{\text{Measured}}(\text{i})- \overline{\text{PAR}}_{\text{Measured}}\big{)}}{\sum_{i=1}^{n}\big{(}\text{ PAR}_{\text{Measured}}(\text{i})-\overline{\text{PAR}}_{\text{Measured}}\big{)}^{2}} \tag{3}\]  
\[\text{intercept}=\overline{\text{PAR}}_{\text{Model}}-\text{slope}\times \overline{\text{PAR}}_{\text{Measured}} \tag{4}\]  
\[\text{MBE}=\frac{1}{n}\sum_{i=1}^{n}(\text{PAR}_{\text{Model}}(\text{i})- \text{PAR}_{\text{Measured}}(\text{i})) \tag{5}\]  
\[\text{RMSE}=\sqrt{\frac{1}{n}\sum_{i=1}^{n}\big{(}\text{PAR}_{\text{Model}}( \text{i})-\text{PAR}_{\text{Measured}}(\text{i})\big{)}^{2}} \tag{6}\]  
\[\text{R}^{2}=\frac{\sigma_{\text{ModelPAR MeasuredPAR}}^{2}}{\sigma_{\text{ ModelPAR}}^{2}\text{* }\sigma_{\text{MeasurePAR}}^{2}} \tag{7}\]  
where, \(\text{PAR}_{\text{Model}}\) are the estimates of the model, \(\text{PAR}_{\text{Measured}}\) are the PAR measurements, n is the number of recordings, \(\sigma_{\text{ModelPAR MeasuredPAR}}^{2}\) is the covariance of the measured and modeled PAR, \(\sigma_{\text{ModelPAR}}^{2}\) is the variance of the modeled PAR, and \(\sigma_{\text{MeasurePAR}}^{2}\) is the variance of the measured PAR. The daily PAR average of each month was then calculated using the estimates of the new model for every point of the grid corresponding to mainland Spain. Similarly, annual averages of PAR estimates were calculated. In order to make these calculations, monthly and annual averages of the CM-SAF and MODIS PAR estimates were used to feed the PAR model. These monthly and annual PAR averages, estimated using the PAR model, were subsequently used to develop PAR maps over mainland Spain. To calculate these estimates from the PAR model estimates, it was necessary to also use estimates from the CM-SAF and MODIS datasets. The daily data collected from the seven stations were filtered, eliminating any data that did not meet any of the following conditions. The PAR/GHI ratio (both variables in W/m\({}^{2}\) so that the ratio was dimensionless) was between 0.3 and 0.6, relative humidity between 0 and 100%, air temperatures between \(-40\) and \(60\)\({}^{\circ}\)C, and clearness indexes (k\({}_{\text{t}}\)) between 0 and 1. These criteria were used to discard any data recorded under environmental conditions outside the tolerance range of the instruments and to discard any data whose values do not make physical sense (for example, PAR values in umol m\({}^{-2}\) s\({}^{-1}\) should be higher than GHI values in W m\({}^{-2}\)).