  Geophysical and other natural processes often exhibit non-stationary
covariances and this feature is important to take into account for statistical
models that attempt to emulate the physical process. A convolution-based model
is used to represent non-stationary Gaussian processes that allows for
variation in the correlation range and vari- ance of the process across space.
Application of this model has two steps: windowed estimates of the covariance
function under the as- sumption of local stationary and encoding the local
estimates into a single spatial process model that allows for efficient
simulation. Specifically we give evidence to show that non-stationary
covariance functions based on the Mat`ern family can be reproduced by the Lat-
ticeKrig model, a flexible, multi-resolution representation of Gaussian
processes. We propose to fit locally stationary models based on the Mat`ern
covariance and then assemble these estimates into a single, global LatticeKrig
model. One advantage of the LatticeKrig model is that it is efficient for
simulating non-stationary 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 de- scribe the uncertainty in the pattern scaling of mean
summer (JJA) surface temperature from a series of climate model experiments.
This example is significant because it emulates tens of thousands of loca-
tions, typical in geophysical model fields, and leverages embarrassing parallel
computation to speed up the local covariance fitting
