  Automated sensing instruments on satellites and aircraft have enabled the
collection of massive amounts of high-resolution observations of spatial fields
over large spatial regions. If these datasets can be efficiently exploited,
they can provide new insights on a wide variety of issues. However, traditional
spatial-statistical techniques such as kriging are not computationally feasible
for big datasets. We propose a multi-resolution approximation (M-RA) of
Gaussian processes observed at irregular locations in space. The M-RA process
is specified as a linear combination of basis functions at multiple levels of
spatial resolution, which can capture spatial structure from very fine to very
large scales. The basis functions are automatically chosen to approximate a
given covariance function, which can be nonstationary. All computations
involving the M-RA, including parameter inference and prediction, are highly
scalable for massive datasets. Crucially, the inference algorithms can also be
parallelized to take full advantage of large distributed-memory computing
environments. In comparisons using simulated data and a large satellite
dataset, the M-RA outperforms a related state-of-the-art method.
