Space-time data fusion under error in computer model output: an application to modeling air quality

Abstract

We provide methods that can be used to obtain more accurate environmental exposure assessment. In particular, we propose two modeling approaches to combine monitoring data at point level with numerical model output at grid cell level, yielding improved prediction of ambient exposure at point level. Extending our earlier downscaler model (Berrocal, V. J., Gelfand, A. E., and Holland, D. M. (2010b). A spatio-temporal downscaler for outputs from numerical models. Journal of Agricultural, Biological and Environmental Statistics 15, 176-197), these new models are intended to address two potential concerns with the model output. One recognizes that there may be useful information in the outputs for grid cells that are neighbors of the one in which the location lies. The second acknowledges potential spatial misalignment between a station and its putatively associated grid cell. The first model is a Gaussian Markov random field smoothed downscaler that relates monitoring station data and computer model output via the introduction of a latent Gaussian Markov random field linked to both sources of data. The second model is a smoothed downscaler with spatially varying random weights defined through a latent Gaussian process and an exponential kernel function, that yields, at each site, a new variable on which the monitoring station data is regressed with a spatial linear model. We applied both methods to daily ozone concentration data for the Eastern US during the summer months of June, July and August 2001, obtaining, respectively, a 5% and a 15% predictive gain in overall predictive mean square error over our earlier downscaler model (Berrocal et al., 2010b). Perhaps more importantly, the predictive gain is greater at hold-out sites that are far from monitoring sites.

Figure 1

Training and validation sites used to fit and assess the out-of-sample predictive performance of the ordinary kriging model, the downscaler, the GMRF smoothed downscaler and the smoothed downscaler using spatially varying random weights.

Figure 2

(a) Daily mean (filled circles) and standard deviation (empty circles) of square root of observed ozone concentration at all the 800 monitoring sites. (b) Daily correlation betweeen square root of observed ozone concentration and square root of CMAQ output of ozone concentration. In both plots, the three days for which we will present results in Section 5 are surrounded by a box. They are, respectively, July 4, July 20, and August 9, 2001.

Figure 3

Spatial maps of: (a) the square root of the CMAQ output, x(B, t), (b) the posterior mean of Ṽ(B, t), and (c) the posterior mean of x̃(B, t), for July 4, 2001 for a subregion in the Northeast.

Figure 4

(a)–(c) Observed ozone concentration (ppb) on August 9, 2001 in two subregions of the Eastern US. (b)–(d) Posterior predictive mean of ozone concentration on August 9, 2001 as yielded by the smoothed downscaler with spatially varying random weights.

Figure 5

(a) Location of the four sites for which we are displaying the posterior predictive mean of the spatially varying random weights w_k (s, t). (b)–(e) Posterior predictive mean of the spatially varying random weights w_k (s, t) for sites: (b) s₁; (c) s₂; (d) s₃; and (e) s₄ on July 4, 2001.