  Spatio-temporal data sets are rapidly growing in size. For example,
environmental variables are measured with ever-higher resolution by increasing
numbers of automated sensors mounted on satellites and aircraft. Using such
data, which are typically noisy and incomplete, the goal is to obtain complete
maps of the spatio-temporal process, together with proper uncertainty
quantification. We focus here on real-time filtering inference in linear
Gaussian state-space models. At each time point, the state is a spatial field
evaluated on a very large spatial grid, making exact inference using the Kalman
filter computationally infeasible. Instead, we propose a multi-resolution
filter (MRF), a highly scalable and fully probabilistic filtering method that
resolves spatial features at all scales. We prove that the MRF matrices exhibit
a particular block-sparse multi-resolution structure that is preserved under
filtering operations through time. We also discuss inference on time-varying
parameters using an approximate Rao-Blackwellized particle filter, in which the
integrated likelihood is computed using the MRF. We compare the MRF to existing
approaches in a simulation study and a real satellite-data application.
