عنوان مقاله [English]
Introduction: Drought is a very complex natural phenomenon which changes with time and space. Spatial and temporal variations of drought are analyzed separately. Geostatistical methods can be used for spatiotemporal analyses to find related spatial and temporal pattern changes. These methods, which use the spatio-temporal data, considering the spatial position of the data relative to each other, also take into account their temporal dependence. If needed, they can estimate values of their variable at any location and any time. Moreover, the drought spatial variations in the studied region can be drawn at every desired period. On the other hand, it is expected that intervening of the time dimension in the equations of these methods, as compared to the purely spatial methods, provide more precision in estimating the values of drought indices, which is studied in this research.
Materials and Methods: Monthly rainfall data of 48 stations in the northeast of Iran for the period of 1981-2012 were used in this study. The SPI drought index is calculated for the 12-month time scale. Data were divided into two groups of training data from 1981-2011 and experimental data of 2012. After analyzing the data regarding their stationarity and isotropic assumptions, the spatiotemporal data were formed and their spatiotemporal empirical variogram was drawn. Furthermore, the purely spatial and temporal variograms for the zero space and time steps were also drawn. Then, four models of the spatiotemporal variogram functions were applied on the training data. The performance of these models was tested and compared by estimating the parameters of the model based on the Square Error (MSE). Moreover, three-dimensional fitted variograms were drawn for different models. Mean The best spatiotemporal variogram model was selected by comparing the models prediction with experimental data using the Mean Square Prediction Error (MSPE). Using spatiotemporal kriging method, the predicted values of experimental data were interpolated and that of the observed values were interpolated by kriging method. Cross validation on experimental data was also performed using RMSE, MAE, ME and COR. Then spatiotemporal and purely spatial variogram models were investigated and compared.
Results and Discussion: The results showed that the 12-month SPI index had no spatial trend but had a decreasing trend against the time. Hence, a simple regression equation was used for fitting the trend of the data. After detrending the data, the SPI index values were considered as the dependent variable, while the time was taken as the independent variable. On the other hand, drawing the variogram in different directions (0°, 45°, 90°, and 135°) had no significant effect relative to each other, and the hypothesis of isotropic state was accepted. The plots of purely spatial and temporal variograms showed that the spherical variogram for space and the linear variogram for the time would have the best fitting. The empirical 3-D and 2-D spatiotemporal variograms of the training data were plotted. The empirical 3-D variogram showed that the data had reached to its temporal sill in a 1-year time lag, and had reached to its spatial sill, in about 25-kilometers, which are in conformity with the purely spatial and temporal variograms. The comparison of different variogram functions showed that the MSE values of the separable, metric, product-sum and sum-metric models were 0.00139, 0.00295, 0.00111, and 0.00112, respectively, the last two of which had fewer errors. Drawing the spatiotemporal variogram of these functions showed that the spatiotemporal variogram of product-sum and sum-metric models have more similarity to the sample one. Regarding the selection of the best model, the MSPE statistics of the product-sum and sum-metric models were 0.281 and 0.389, respectively. Therefore, the product-sum model could be selected as the best model. The least rate of errors was found in the exponential variogram model for space, and in the linear variogram for the time. The parameters of the nugget effect, partial sill and range for the spatial variogram would be 0.00, 0.063, and 5.78, and for the temporal variogram would be 0.00, 0.635, and 1.044, respectively. After predicting values of 12-month SPI in 2012 by the product-sum variogram model and adding the values of the trend, they were interpolated by using the spatiotemporal kriging, and the observed values were interpolated by the use of kriging. The obtained plot from the predicted values had great similarity with that of the observed values, which indicates the appropriate capability of the model in predicting the unobserved values. The cross-validation of different spatiotemporal and the spatial models with 25 and 47 neighborhoods showed that the performance of the models had no significant differences relative to each other, and they also had no better performance relative to the purely spatial model.
Conclusion: The results of this study showed that the product-sum model had a better performance among different spatiotemporal variogram models in predicting the 12-month SPI values of 2012. However, the performances of different spatiotemporal models were quite close to each other. There is no significant difference that could be observed between spatiotemporal and purely spatial models. Also, it is proposed to use the dynamic spatiotemporal models and the results to be compared with the classical models.