We propose a method of analyzing spatiotemporal data by decomposition into deterministic nonparametric functions of time and space, linear functions of other covariates, and a random component that is spatially, though not temporally, correlated. The resulting model is used for spatial interpolation and especially for estimation of a spatially dependent temporal average. The results are applied to part of the PM2.5 network established by the U.S. Environmental Protection Agency, covering three southeastern U.S. states. A novel feature of the analysis is a variant of the expectation-maximization algorithm to account for missing data. The results show, among other things, that a substantial part of the region is in violation of the proposed long-term average standard for PM2.5.