  This paper investigates the cross-correlations across multiple climate model
errors. We build a Bayesian hierarchical model that accounts for the spatial
dependence of individual models as well as cross-covariances across different
climate models. Our method allows for a nonseparable and nonstationary
cross-covariance structure. We also present a covariance approximation approach
to facilitate the computation in the modeling and analysis of very large
multivariate spatial data sets. The covariance approximation consists of two
parts: a reduced-rank part to capture the large-scale spatial dependence, and a
sparse covariance matrix to correct the small-scale dependence error induced by
the reduced rank approximation. We pay special attention to the case that the
second part of the approximation has a block-diagonal structure. Simulation
results of model fitting and prediction show substantial improvement of the
proposed approximation over the predictive process approximation and the
independent blocks analysis. We then apply our computational approach to the
joint statistical modeling of multiple climate model errors.
