A Spatio-Temporal Downscaler for Output From Numerical Models

  • Veronica J. BerrocalEmail author
  • Alan E. Gelfand
  • David M. Holland


Often, in environmental data collection, data arise from two sources: numerical models and monitoring networks. The first source provides predictions at the level of grid cells, while the second source gives measurements at points. The first is characterized by full spatial coverage of the region of interest, high temporal resolution, no missing data but consequential calibration concerns. The second tends to be sparsely collected in space with coarser temporal resolution, often with missing data but, where recorded, provides, essentially, the true value. Accommodating the spatial misalignment between the two types of data is of fundamental importance for both improved predictions of exposure as well as for evaluation and calibration of the numerical model. In this article we propose a simple, fully model-based strategy to downscale the output from numerical models to point level. The static spatial model, specified within a Bayesian framework, regresses the observed data on the numerical model output using spatially-varying coefficients which are specified through a correlated spatial Gaussian process.

As an example, we apply our method to ozone concentration data for the eastern U.S. and compare it to Bayesian melding (Fuentes and Raftery 2005) and ordinary kriging (Cressie 1993; Chilès and Delfiner 1999). Our results show that our method outperforms Bayesian melding in terms of computing speed and it is superior to both Bayesian melding and ordinary kriging in terms of predictive performance; predictions obtained with our method are better calibrated and predictive intervals have empirical coverage closer to the nominal values. Moreover, our model can be easily extended to accommodate for the temporal dimension. In this regard, we consider several spatio-temporal versions of the static model. We compare them using out-of-sample predictions of ozone concentration for the eastern U.S. for the period May 1–October 15, 2001. For the best choice, we present a summary of the analysis. Supplemental material, including color versions of Figures 4, 5, 6, 7, and 8, and MCMC diagnostic plots, are available online.

Key Words

Bayesian melding Calibration Markov chain Monte Carlo Ordinary kriging Spatial misalignment Spatially varying coefficient model 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Supplementary material

13253_2009_4_MOESM1_ESM.pdf (37 kb)
Below is the link to the electronic supplementary material. (PDF 38 kB)
13253_2009_4_MOESM2_ESM.pdf (60 kb)
Below is the link to the electronic supplementary material. (PDF 61 kB)
13253_2009_4_MOESM3_ESM.pdf (72 kb)
Below is the link to the electronic supplementary material. (PDF 73 kB)
13253_2009_4_MOESM4_ESM.pdf (160 kb)
Below is the link to the electronic supplementary material. (PDF 161 kB)
13253_2009_4_MOESM5_ESM.pdf (138 kb)
Below is the link to the electronic supplementary material. (PDF 139 kB)
13253_2009_4_MOESM6_ESM.pdf (216 kb)
Below is the link to the electronic supplementary material. (PDF 217 kB)
13253_2009_4_MOESM7_ESM.pdf (81 kb)
Below is the link to the electronic supplementary material. (PDF 81 kB)
13253_2009_4_MOESM8_ESM.pdf (1.4 mb)
Below is the link to the electronic supplementary material. (PDF 1.445 kB)


  1. Banerjee, S., Carlin, B. P., and Gelfand, A. E. (2004), Hierarchical Modeling and Analysis for Spatial Data, Boca Raton, FL: Chapman & Hall/CRC. zbMATHGoogle Scholar
  2. Carroll, S. S., Day, G., Cressie, N., and Carroll, T. R. (1995), “Spatial Modeling of Snow Water Equivalent Using Airborne and Ground-Based Snow Data,” Environmetrics, 6, 127–139. CrossRefGoogle Scholar
  3. Carter, C., and Kohn, R. (1994), “On Gibbs Sampling for State Space Models,” Biometrika, 81, 541–553. zbMATHCrossRefMathSciNetGoogle Scholar
  4. Chilès, J.-P., and Delfiner, P. (1999), Geostatistics: Modeling Spatial Uncertainty, New York: Wiley. zbMATHGoogle Scholar
  5. Cowles, M. K., and Zimmerman, D. L. (2003), “A Bayesian Space-Time Analysis of Acid Deposition Data Combined From Two Monitoring Networks,” Journal of Geophysical Research, 108 (D24), 9006, doi:  10.1029/2003JD004001. CrossRefGoogle Scholar
  6. Cowles, M. K., Zimmerman, D. L., Christ, A., and McGinnis, D. L. (2002), “Combining Snow Water Equivalent Data From Multiple Sources to Estimate Spatio-Temporal Trends and Compare Measurement Systems,” Journal of Agricultural, Biological and Environmental Statistics, 7, 536–557. CrossRefGoogle Scholar
  7. Cressie, N. A. C. (1993), Statistics for Spatial Data, New York: Wiley. Google Scholar
  8. Denison, D. G. T., Mallick, B. K., and Smith, A. G. M. (1998), “Automatic Bayesian Curve Fitting,” Journal of the Royal Statistical Society, Ser. B, 60, 333–350. zbMATHCrossRefMathSciNetGoogle Scholar
  9. Diggle, P. J., Moyeed, R. A., and Tawn, J. A. (1998), “Model-Based Geostatistics (with discussion),” Applied Statistics, 47, 299–350. zbMATHMathSciNetGoogle Scholar
  10. Dominici, F., Daniels, M., Zeger, S. L., and Samet, J. M. (2002), “Air Pollution and Mortality: Estimating Regional and National Dose-Response Relationships,” Journal of the American Statistical Association, 97, 100–111. zbMATHCrossRefMathSciNetGoogle Scholar
  11. Foley, K. M., and Fuentes, M. (2008), “A Statistical Framework to Combine Multivariate Spatial Data and Physical Models for Hurricane Wind Prediction,” Journal of Agricultural, Biological and Environmental Statistics, 13, 37–59. CrossRefMathSciNetGoogle Scholar
  12. Fuentes, M., and Raftery, A. E. (2005), “Model Evaluation and Spatial Interpolation by Bayesian Combination of Observations With Outputs From Numerical Models,” Biometrics, 61, 36–45. zbMATHCrossRefMathSciNetGoogle Scholar
  13. Fuentes, M., Song, H.-R., Ghosh, S. K., Holland, D. M., and Davis, J. M. (2006), “Spatial Association Between Speciated Fine Particles and Mortality,” Biometrics, 62, 855–863. zbMATHCrossRefMathSciNetGoogle Scholar
  14. Gelfand, A. E., Banerjee, S., and Gamerman, D. (2005), “Spatial Process Modelling for Univariate and Multivariate Dynamic Spatial Data,” Environmetrics, 16, 465–479. CrossRefMathSciNetGoogle Scholar
  15. Gelfand, A. E., Kim, H.-J., Sirmans, C. F., and Banerjee, S. (2003), “Spatial Modeling With Spatially Varying Coefficient Processes,” Journal of the American Statistical Association, 98, 387–396. zbMATHCrossRefMathSciNetGoogle Scholar
  16. Gelfand, A. E., Schmidt, A. M., Banerjee, S., and Sirmans, C. F. (2004), “Nonstationary Multivariate Process Modeling Through Spatially Varying Coregionalization,” TEST, 13, 263–312. zbMATHCrossRefMathSciNetGoogle Scholar
  17. Gotway, C. A., and Young, L. J. (2002), “Combining Incompatible Spatial Data,” Journal of the American Statistical Association, 97, 632–648. zbMATHCrossRefMathSciNetGoogle Scholar
  18. — (2007), “A Geostatistical Approach to Linking Geographically Aggregated Data From Different Sources,” Journal of Computational and Graphical Statistics, 16, 115–135. CrossRefMathSciNetGoogle Scholar
  19. Green, P. J. (1995), “Reversible Jump Markov Chain Monte Carlo Computation and Bayesian Model Determination,” Biometrika, 82, 711–732. zbMATHCrossRefMathSciNetGoogle Scholar
  20. McMillan, N., Holland, D. M., Morara, M., and Feng, J. (2009), “Combining Numerical Model Output and Particulate Data Using Bayesian Space-Time Modeling,” Environmetrics, to appear. Google Scholar
  21. Poole, D., and Raftery, A. E. (2000), “Inference for Deterministic Simulation Models: The Bayesian Melding Approach,” Journal of the American Statistical Association, 95, 1244–1255. zbMATHCrossRefMathSciNetGoogle Scholar
  22. Sahu, S. K., Gelfand, A. E., and Holland, D. M. (2006), “Spatio-Temporal Modeling of Fine Particulate Matter,” Journal of Agricultural, Biological and Environmental Statistics, 11, 61–86. CrossRefGoogle Scholar
  23. Schmidt, A. M., and Gelfand, A. E. (2003), “A Bayesian Coregionalization Approach for Multivariate Pollutant Data,” Journal of the Geophysical Research, 108 (D24), 8783, doi:  10.1029/2002JD002905. CrossRefGoogle Scholar
  24. Smith, B. J., and Cowles, M. K. (2007), “Correlating Point-Referenced Radon and Areal Uranium Data Arising From a Common Spatial Process,” Applied Statistics, 56, 313–326. MathSciNetGoogle Scholar
  25. Tolbert, P., Mulholland, J., MacIntosh, D., Xu, F., Daniels, D., Devine, O., Carlin, B. P., Klein, M., Dorley, J., Butler, A., Nordenberg, D., Frumkin, H., Ryan, P. B., and White, M. (2000), “Air Pollution and Pediatric Emergency Room Visits for Asthma in Atlanta,” American Journal of Epidemiology, 151, 798–810. Google Scholar
  26. Wackernagel, H. (1998), Multivariate Geostatistics (2nd ed.), Berlin: Springer. zbMATHGoogle Scholar
  27. West, M., and Harrison, J. (1999), Bayesian Forecasting and Dynamic Models (2nd ed.), New York: Springer-Verlag. Google Scholar
  28. Wikle, C. K., and Berliner, L. M. (2005), “Combining Information Across Spatial Scales,” Technometrics, 47, 80–91. CrossRefMathSciNetGoogle Scholar
  29. Zhang, H. (2004), “Inconsistent Estimation and Asymptotically Equal Interpolations in Model-Based Geostatistics,” Journal of the American Statistical Association, 99, 250–261. zbMATHCrossRefMathSciNetGoogle Scholar
  30. Zhu, L., Carlin, B. P., and Gelfand, A. E. (2003), “Hierarchical Regression With Misaligned Spatial Data: Relating Ambient Ozone and Pediatric Asthma ER Visits in Atlanta,” Environmetrics, 14, 537–557. CrossRefGoogle Scholar

Copyright information

© International Biometric Society 2009

Authors and Affiliations

  • Veronica J. Berrocal
    • 1
    Email author
  • Alan E. Gelfand
    • 1
  • David M. Holland
    • 2
  1. 1.Department of Statistical ScienceDuke UniversityDurhamUSA
  2. 2.27711U.S. Environmental Protection Agency, National Exposure Research LaboratoryResearch Triangle ParkUSA

Personalised recommendations