This dissertation is concerned with data exhibiting spatial and spatio-temporal dependence. It is based on three separate research works.

One chapter is concerned with the biogeochemical Argo data in the Southern Ocean, which aims to collect measurements of oxygen, temperature and salinity as well as other variables at varying depths in the ocean. The biogeochemical Argo data is important to improve our understanding of vital biogeochemical processes such as the biological carbon pump and air-sea CO2 exchanges, monitor changes such as ocean deoxygenation and acidification, and improve estimates of the carbon budget. We introduce and estimate a functional regression model for oxygen, temperature, and salinity data. Our model elucidates important aspects of the joint distribution of temperature, salinity, and oxygen across the entire ocean depth covered by the Argo data and improves location estimates of so-called oceanographic fronts, which are of significant scientific interest in their own right. In addition, it enables us to use the more pervasively available temperature and salinity data to recover biogeochemical data at locations where it is not observed.

Another chapter, motivated by the work on the Argo data, contributes to the solution of an open problem in the spatial statistics literature. Concretely, we study the smoothness estimation of so-called Whittle-Matérn kernels on closed Riemannian manifolds. The smoothness of Matérn kernels controls, for example, optimal error bounds for kriging and posterior contraction rates in Gaussian process regression. However, it has been an open problem whether their smoothness can always be consistently estimated. On closed Riemannian manifolds, we show that their smoothness can be consistently estimated from the maximizer(s) of the Gaussian likelihood when the underlying data stem from point evaluations of a Gaussian process and, perhaps surprisingly, even when the data comprise evaluations of a non-Gaussian process. Moreover, we generalize a well-known equivalence of measures phenomenon related to Matérn kernels to the non-Gaussian case by using Kakutani's theorem.

The remaining chapter extends this work to processes observed on the vertices of graphs. Due to increased tractability of the problem is this setting, we are able to provide more complete results. In addition, we establish connections to processes observed on smooth domains such as Riemannian manifolds. In this way, we believe that our results for processes on graphs provide additional insights for such cases as well.