  This work is motivated by constructing a weather simulator for precipitation.
Temperature and humidity are two of the most important driving forces of
precipitation, and the strategy is to have a stochastic model for temperature
and humidity, and use a deterministic model to go from these variables to
precipitation. Temperature and humidity are empirically positively correlated.
Generally speaking, if variables are empirically dependent, then multivariate
models should be considered. In this work we model humidity and temperature in
southern Norway. We want to construct bivariate Gaussian random fields (GRFs)
based on this dataset. The aim of our work is to use the bivariate GRFs to
capture both the dependence structure between humidity and temperature as well
as their spatial dependencies. One important feature for the dataset is that
the humidity and temperature are not necessarily observed at the same
locations. Both univariate and bivariate spatial models are fitted and
compared. For modeling and inference the SPDE approach for univariate models
and the systems of SPDEs approach for multivariate models have been used.
  To evaluate the performance of the difference between the univariate and
bivariate models, we compare predictive performance using some commonly used
scoring rules: mean absolute error, mean-square error and continuous ranked
probability score. The results illustrate that we can capture strong positive
correlation between the temperature and the humidity. Furthermore, the results
also agree with the physical or empirical knowledge. At the end, we conclude
that using the bivariate GRFs to model this dataset is superior to the approach
with independent univariate GRFs both when evaluating point predictions and for
quantifying prediction uncertainty.
