The reverse dimple in potentially negative-value space-time covariance models

Recently several efforts have been made to model space-time covariance functions. The majority of covariance model structures are monotonic decreasing, typically remains non-negative. One class of spatially isotropic models proposed by Gneiting (J Am Stat Assoc 97(458): 590-600, 2002) has been used as a building block to model various complicated non-separable models. Kent et al. (Biometrika 98(2): 489-494, 2011) draw out attention on the counterintuitive presence of possible dimple property associated with these covariance models. In this paper, we first attempt to propose a simple approach to model potentially negative-value stationary space-time models. Second, we show that in certain circumstances such space-time models possess a reverse (pointing upward) dimple. To illustrate analytical findings, results of numerical calculations and numerous plots are presented.