Spatial Inference of Nitrate Concentrations in Groundwater
We develop a method for multiscale estimation of pollutant concentrations, based on a nonparametric spatial statistical model. We apply this method to estimate nitrate concentrations in groundwater over the mid-Atlantic states, using measurements gathered during a period of 10 years. A map of the fine-scale estimated nitrate concentration is obtained, as well as maps of the estimated county-level average nitrate concentration and similar maps at the level of watersheds and other geographic regions. The fine-scale and coarse-scale estimates arise naturally from a single model, without refitting or ad hoc aggregation. As a result, the uncertainty associated with each estimate is available, without approximations relying on high spatial density of measurements or parametric distributional assumptions.
Several risk measures are also obtained, including the probability of the pollutant concentration exceeding a particular threshold. These risk measures can be obtained at the fine scale, or at the level of counties or other regions.
The nonparametric Bayesian statistical model allows for this flexibility in estimation while avoiding strong assumptions. This method can be applied directly to estimate ozone concentrations in air, pesticide concentrations in groundwater, or any other quantity that varies over a geographic region, based on approximate measurements at some locations and perhaps of associated covariates. An S-PLUS package with this capability is provided as supplemental material.
Key WordsBayesian Geostatistics Kriging Lévy processes Nonparametrics Response surface Spatial moving average
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- Ator, S. W. (1998), “Nitrate and Pesticide Data for Waters of the Mid-Atlantic Region,” USGS Open File Report 98-158, U.S. Geological Survey, Reston, VA. Google Scholar
- Ator, S. W., and Denis, J. M. (1997), “Relation of Nitrogen and Phosphorus in Ground Water to Land Use in Four Subunits of the Potomac River Basin,” USGS Water-Resources Investigations Report 97-4268, U.S. Geological Survey, Reston, VA. Google Scholar
- Ator, S. W., and Ferrari, M. J. (1997), “Nitrate and Selected Pesticides in Ground Water of the Mid-Atlantic Region,” USGS Water-Resources Investigations Report 97-4139, U.S. Geological Survey, Reston, VA. Google Scholar
- Clyde, M. A., House, L. L., and Wolpert, R. L. (2006), “Nonparametric Models for Proteomic Peak Identification and Quantification,” in Bayesian Inference for Gene Expression and Proteomics, eds. K. A. Do, P. Muller, and M. Vannucci, Cambridge, U.K.: Cambridge University Press, pp. 293–308. CrossRefGoogle Scholar
- Cressie, N. (1993), Statistics for Spatial Data, New York: Wiley. Google Scholar
- Faulkner, B. R. (2003), “Confronting the Modifiable Areal Unit Problem for Inference on Nitrate in Regional Shallow Ground Water,” in Groundwater Quality Modeling and Management Under Uncertainty, ed. S. Mishra, Reston, VA: American Society of Civil Engineers, pp. 248–259. Google Scholar
- Fields Development Team (2004), Fields: Tools for Spatial Data, Boulder, CO: National Center for Atmospheric Research. Available at http://www.cgd.ucar.edu/stats/Software/Fields/.
- Hamilton, P. A., Denver, J. M., Phillips, P. J., and Shedlock, R. J. (1993), “Water-Quality Assessment of the Delmarva Peninsula, Delaware, Maryland, and Virginia—Effects of Agricultural Activities on, and Distribution of, Nitrate and Other Inorganic Constituents in the Surficial Aquifer,” USGS Open File Report 93-40, U.S. Geological Survey, Reston, VA. Google Scholar
- House, L. L., Clyde, M. A., and Wolpert, R. L. (2006), “Nonparametric Models for Peak Identification and Quantification in Mass Spectroscopy, With Application to MALDI-TOF,” Discussion Paper 2006-24, Duke University, Dept. of Statistical Science, available at ftp://ftp.isds.duke.edu/pub/WorkingPapers/06-24.html.
- Ickstadt, K., and Wolpert, R. L. (1997), “Multiresolution Assessment of Forest Inhomogeneity,” in Case Studies in Bayesian Statistics, Vol. III, eds. C. Gatsonis, J. S. Hodges, R. E. Kass, R. E. McCulloch, P. Rossi, and N. D. Singpurwalla, New York: Springer-Verlag, pp. 371–386. Google Scholar
- — (1999), “Spatial Regression for Marked Point Processes” (with comments), in Bayesian Statistics, Vol. 6, eds. J. M. Bernardo, J. O. Berger, A. P. Dawid, and A. F. M. Smith, Oxford: Oxford University Press, pp. 323–341. Google Scholar
- Lophaven, S. N., Nielsen, H. B., and Søndergaard, J. (2002), “DACE: A Matlab Kriging Toolbox, Version 2.0,” Technical Report IMM-TR-2002-12, Technical University of Denmark, available at http://www.imm.dtu.dk/~hbn/dace/dace.pdf.
- S-Plus (2007), S-PLUS 8 Programmer’s Guide, Seattle, WA: Insightful Corporation. Google Scholar
- Tu, C. (2006), “Bayesian Nonparametric Modeling Using Lévy Process Priors With Applications for Function Estimation, Time Series Modeling, and Spatio-Temporal Modeling,” Ph.D. thesis, Duke University, Dept. of Statistical Science. Google Scholar
- U. S. Environmental Protection Agency (1991), “Fact Sheet: National Primary Drinking Water Standards,” U.S. Government Printing Office, Washington, DC. Google Scholar
- — (1998b), “Simulation of Lévy Random Fields,” New York: Springer-Verlag. Google Scholar
- Wolpert, R. L., Clyde, M. A., and Tu, C. (2006), “Lévy Adaptive Regression Kernels,” Discussion Paper 2006-08, Duke University, Dept. of Statistical Science, available at http://ftp.stat.duke.edu/WorkingPapers/06-08.html.