Spatially Varying Anisotropy for Gaussian Random Fields in Three-Dimensional Space

TL;DR

This paper develops a non-stationary anisotropic Gaussian random field model in three dimensions using SPDEs, effectively capturing spatial variability in ocean phenomena and outperforming stationary models in real-world predictions.

Contribution

It introduces a novel SPDE-based approach for modeling non-stationary anisotropic GRFs in 3D, tailored for oceanographic applications.

Findings

The method accurately captures spatially varying anisotropy in ocean data.

Simulation studies demonstrate the model's data efficiency.

Application to real ocean data shows improved prediction over stationary models.

Abstract

Isotropic covariance structures can be unreasonable for phenomena in three-dimensional spaces such as the ocean. In the ocean, the variability of the response may vary with depth, and ocean currents may lead to spatially varying anisotropy. We construct a class of non-stationary anisotropic Gaussian random fields (GRFs) in three dimensions through stochastic partial differential equations (SPDEs) where computations are done using Gaussian Markov random field approximations. The approach is proven in a simulation study where the amount of data required to estimate these models is explored. Then, the method is applied to construct a GRF prior on an ocean mass outside Trondheim, Norway, based on simulations from the complex numerical ocean model SINMOD. This GRF prior is compared to a stationary anisotropic GRF using in-situ measurements collected with an autonomous underwater vehicle where our approach outperforms the stationary anisotropic GRF for real-time prediction of unobserved locations.