A hierarchical modelling approach to combining environmental data at different scales

Summary. Long-transported air pollution in Europe is monitored by a combination of a highly complex mathematical model and a limited number of measurement stations. The model predicts deposition on a 150 km × 150 km square grid covering the whole of the continent. These predictions can be regarded as spatial averages, with some spatially correlated model error. The measurement stations give a limited number of point estimates, regarded as error free. We combine these two sources of data by assuming that both are observations of an underlying true process. This true deposition is made up of a smooth deterministic trend, due to gradual changes in emissions over space and time, and two stochastic components. One is non- stationary and correlated over long distances; the other describes variation within a grid square. Our approach is through hierarchical modelling with predictions and measurements being independent conditioned on the underlying non-stationary true deposition. We assume Gaussian processes and calculate maximum likelihood estimates through numerical optimization. We find that the variation within a grid square is by far the largest component of the variation in the true deposition. We assume that the mathematical model produces estimates of the mean over an area that is approximately equal to a grid square, and we find that it has an error that is similar to the long-range stochastic component of the true deposition, in addition to a large bias.