Space-time data are ubiquitous in the environmental sciences. Often, a
s is the case with atmospheric and oceanographic processes, these data
contain many different scales of spatial and temporal variability. Su
ch data are often non-stationary in space and time and may involve man
y observation/prediction locations. These factors can limit the effect
iveness of traditional spacetime statistical models and methods. In th
is article, we propose the use of hierarchical space-time models to ac
hieve more flexible models and methods for the analysis of environment
al data distributed in space and time. The first stage of the hierarch
ical model specifies a measurement-error process for the observational
data in terms of some 'state' process. The second stage allows for si
te-specific time series models for this state variable. This stage inc
ludes large-scale (e.g. seasonal) variability plus a space-time dynami
c process for the 'anomalies'. Much of our interest is with this anoma
ly process. In the third stage, the parameters of these time series mo
dels, which are distributed in space, are themselves given a joint dis
tribution with spatial dependence (Markov random fields). The Bayesian
formulation is completed in the last two stages by specifying priors
on parameters. We implement the model in a Markov chain Monte Carlo fr
amework and apply it to an atmospheric data set of monthly maximum tem
perature.