3D bivariate spatial modelling of Argo ocean temperature and salinity

Variables within the global oceans can reveal the impacts of a warming climate, as the oceans absorb huge amounts of solar energy. Understanding the joint spatial distribution of key ocean variables is, therefore, essential. In this paper we investigate the spatial dependence structure between ocean temperature and salinity using Argo observations and construct a bivariate spatial model covering from the surface through the ocean interior. We develop a flexible class of multivariate nonstationary covariance models defined in three-dimensional (3D) space (longitude × latitude × depth) that allow the variances and correlations to vary with depth, capturing the ocean’s vertical structure. These models describe the joint spatial distribution of the two variables while incorporating the underlying vertical structure of the ocean. We apply this framework to Argo temperature and salinity data and address the computational challenges of large data volumes through the Vecchia approximation. Our results show that the proposed bivariate covariance model effectively represents the complex vertical cross-covariance structure of the processes and their first- and second-order differences, whereas classical bivariate models, including the bivariate Matérn, poorly fit the empirical cross-covariance structure.