  Earth-observing satellite instruments obtain a massive number of observations
every day. For example, tens of millions of sea surface temperature (SST)
observations on a global scale are collected daily by the Moderate Resolution
Imaging Spectroradiometer (MODIS) instrument. Despite their size, such datasets
are incomplete and noisy, necessitating spatial statistical inference to obtain
complete, high-resolution fields with quantified uncertainties. Such inference
is challenging due to the high computational cost, the nonstationary behavior
of environmental processes on a global scale, and land barriers affecting the
dependence of SST. In this work, we develop a multi-resolution approximation
(M-RA) of a Gaussian process (GP) whose nonstationary, global covariance
function is obtained using local fits. The M-RA requires domain partitioning,
which can be set up application-specifically. In the SST case, we partition the
domain purposefully to account for and weaken dependence across land barriers.
Our M-RA implementation is tailored to distributed-memory computation in
high-performance-computing environments. We analyze a MODIS SST dataset
consisting of more than 43 million observations, to our knowledge the largest
dataset ever analyzed using a probabilistic GP model. We show that our
nonstationary model based on local fits provides substantially improved
predictive performance relative to a stationary approach.
