Modeling and emulation of nonstationary Gaussian fields

Abstract

Geophysical and other natural processes often exhibit nonstationary covariances and this feature is important for statistical models that attempt to emulate the physical process. A convolution-based model is used to represent nonstationary Gaussian processes that allows for variation in the correlation range and the variance of the process across space. We apply this model in two steps: windowed estimates of the covariance function under the assumption of local stationarity and encoded local estimates into a single spatial process model that allows for efficient simulation. We show that nonstationary covariance functions based on the Matérn family can be reproduced by the LatticeKrig (LK) model, a flexible, multi-resolution representation of Gaussian processes. Stationary models based on the Matérn covariance are fit in local windows and these estimates are assembled into a single, global LK model. The LK model is efficient for simulating nonstationary fields even at 105 locations. This work is motivated by the interest in emulating spatial fields derived from numerical model simulations such as Earth system models. We successfully apply these ideas to emulate fields that describe the uncertainty in the pattern scaling of mean summer (JJA) surface temperature from a series of climate model experiments. The spatial covariance structure developed in this paper is not limited to emulation, and could also be used for spatial prediction and conditional simulation for observational data and leverages embarrassingly parallel strategies for computational efficiency.