The dimple problem related to space-time modeling under the Lagrangian framework

Space-time covariance modeling under the Lagrangian framework has been especially popular to study atmospheric phenomena in the presence of transport effects, such as prevailing winds or ocean currents, which are incompatible with the assumption of full symmetry. In this work, we assess the dimple problem (Kent et al., 2011) for covariance functions coming from transport phenomena. We work under two important cases: the spatial domain can be either the $d$-dimensional Euclidean space $\mathbb{R}^d$ or the spherical shell of $\mathbb{R}^d$. The choice is relevant for the type of metric chosen to describe spatial dependence. In particular, in Euclidean spaces, we work under very general assumptions with the case of radial symmetry being deduced as a corollary of a more general result. We illustrate through examples that, under this framework, the dimple is a natural and physically interpretable property.