  Estimation of stationary dependence structure parameters using only a single
realisation of the spatial process, typically leads to inaccurate estimates and
poorly identified parameters. A common way to handle this is to fix some of the
parameters, or within the Bayesian framework, impose prior knowledge. In many
applied settings, stationary models are not flexible enough to model the
process of interest, thus non-stationary spatial models are used. However, more
flexible models usually means more parameters, and the identifiability problem
becomes even more challenging. We investigate aspects of estimation of a
Bayesian non-stationary spatial model for annual precipitation using
observations from multiple years. The model contains replicates of the spatial
field, which increases precision of the estimates and makes them less prior
sensitive. Using R-INLA, we analyse precipitation data from southern Norway,
and investigate statistical properties of the replicate model in a simulation
study. The non-stationary spatial model we explore belongs to a recently
introduced class of stochastic partial differential equation (SPDE) based
spatial models. This model class allows for non-stationary models with
explanatory variables in the dependence structure. We derive conditions to
facilitate prior specification for these types of non-stationary spatial
models.
