  Spatio-temporal processes in environmental applications are often assumed to
follow a Gaussian model, possibly after some transformation. However,
heterogeneity in space and time might have a pattern that will not be
accommodated by transforming the data. In this scenario, modelling the variance
laws is an appealing alternative. This work adds flexibility to the usual
Multivariate Dynamic Gaussian model by defining the process as a scale mixture
between a Gaussian and log-Gaussian processes. The scale is represented by a
process varying smoothly over space and time which is allowed to depend on
covariates. State-space equations define the dynamics over time for both mean
and variance processes resulting infeasible inference and prediction. Analysis
of artificial datasets show that the parameters are identifiable and simpler
models are well recovered by the general proposed model. The analyses of two
important environmental processes, maximum temperature and maximum ozone,
illustrate the effectiveness of our proposal in improving the uncertainty
quantification in the prediction of spatio-temporal processes.

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