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

TL;DR

This paper investigates the dimple problem in space-time covariance functions within the Lagrangian framework, emphasizing transport phenomena on Euclidean and spherical domains, and shows that dimples are natural and interpretable.

Contribution

It provides a comprehensive assessment of the dimple problem for transport-related covariance functions on Euclidean and spherical spaces, with general assumptions and illustrative examples.

Findings

Dimple phenomenon is natural in transport-based covariance functions.

Results apply to both Euclidean and spherical spatial domains.

Dimple properties have clear physical interpretations.

Abstract

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.