New classes of covariance and spectral density functions for spatio-temporal modelling

In the nonseparable spatio-temporal context, several efforts have been made in order to obtain general classes of spatio-temporal covariances. Our aim in this paper is to join several approaches coming from different authors and provide some ideas for the construction of new models of spatio-temporal covariance and spectral density functions. On one hand, we build new covariance families while removing some undesirable features of the previously proposed models, particularly following Stein's (in J Am Stat Assoc 100:310-321, 2005) remark about Gneiting's (in J Am Stat Assoc 97:590-600, 2002) approach and about some tensorial product covariance models. We show some of the theoretical results and examples obtained with the product or the sum of spatio-temporal covariance functions or even better with the mixed forms. On the other hand, we define new models for spectral densities through the product of two other spectral densities. We give some characterizations and properties as well as several examples. Finally, we present a practical modelling of Irish wind speed data based on some of the space-time covariance models presented in this paper.