lvlodel Validation and Spatial Interpolation by Combining Observations \ vith Outputs from Numerical lvlodels via Bayesian lvlelding 1

Constructing maps of pollution levels is vital for air quality management, and presents statistical problems typical of many environmental and spatial applications. Ideally, such maps would be based on a dense network of monitoring stations, but this does not exist. Instead, there are two main sources of information in the U.S.: one is pollution measurements at a sparse set of about 50 monitoring stations called CASTNet, and the other is pollution emissions data. The pollution emissions data do not give direct information about pollution levels, but instead are combined with numerical models of weather and the emissions process and information about land use and cover (collectively called Models-3), to produce maps. Here we develop a formal method for combining these two sources of information. vVe specify a simple model for both the Models-3 output and the CASTNet observations in terms of the unobserved ground truth, and estimate the model in a Bayesian way. This yields solutions to the spatial prediction, model validation and bias removal problems simultaneously. It provides improved spatial prediction via the posterior distribution of the ground truth, allows us to validate Models-3 via the posterior predictive distribution of the CASTNet observations, and enables us to remove the bias in the Models-3 output by estimating additive and multiplicative bias parameters in the model. vVe apply our methods to data on 802 concentrations. Key tlJords: air pollution, Ba~yesian inference, change of support, likelihood approaches, Matern covariance, nonstationary process, spatial-temporal statistics.