Computing spatiotemporal variogram and covariance model

A large number of environmental phenomena may be regarded as the realizations of spatiotemporal random fields. In practice, these environmental phenomena are sparsely sampled generally. In order to research deeply, it is necessary to construct a continuous spatiotemporal data surface, so the prediction or interpolation must be done. The spatiotemporal variogram and covariance model is useful means of describing the spatiotemporal correlation structure. For the straightforward extension of variogram and covariance from pure spatial to spatiotemporal fields, there are a number of statistical studies about theoretical spatiotemporal model but very less research on model computing. After making some theoretical spatiotemporal statistical analysis, this paper focused mainly on the computation of spatiotemporal variogram and covariance model and implement effective variogram and covariance model. Firstly, the spatiotemporal product-sum model is deduced into the form of calculable in theory. Secondly, the most likely variogram model and its parameters of sill, nugget, and range are derived through computing the spatial and temporal variogram respectively. Thirdly, the policy of how to determine the parameters k1,k2 and k3 in the product-sum model are put forward. The objective to introduce k1,k2 and k3 is to ensure the effectiveness of variogram and covariance model. Lastly, the spatiotemporal variogram and covariance model are implemented. The results have shown the positive definite characteristics of the spatiotemporal variogram and covariance varying with the parameter k1 and reverse variation characteristics between variogram and covariance, which proves that the theoretical model chosen is effective and the computing approach about spatiotemporal variogram and covariance model is feasible. The research of this paper has laid the foundation for spatiotemporal prediction or interpolation, because prediction or interpolation can do only basing on suitable variogram or covariance model.