Mathematical Geology

, Volume 31, Issue 6, pp 651–684 | Cite as

Geostatistical Space–Time Models: A Review

  • Phaedon C. Kyriakidis
  • André G. Journel
Article

Abstract

Geostatistical space–time models are used increasingly for addressing environmental problems, such as monitoring acid deposition or global warming, and forecasting precipitation or stream flow. Each discipline approaches the problem of joint space–time modeling from its own perspective, a fact leading to a significant amount of overlapping models and, possibly, confusion. This paper attempts an annotated survey of models proposed in the literature, stating contributions and pinpointing shortcomings. Stochastic models that extend spatial statistics (geostatistics) to include the additional time dimension are presented with a common notation to facilitate comparison. Two conceptual viewpoints are distinguished: (1) approaches involving a single spatiotemporal random function model, and (2) approaches involving vectors of space random functions or vectors of time series. Links between these two viewpoints are then revealed; advantages and shortcomings are highlighted. Inference from space–time data is revisited, and assessment of joint space–time uncertainty via stochastic imaging is suggested.

space–time models geostatistics time series trend models stochastic simulation 

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REFERENCES

  1. Almeida, A., and Journel, A. G., 1994, Joint simulation of multiple variables with a Markov-type coregionalization model: Math. Geology, v. 26, no. 5, p. 565–588.Google Scholar
  2. Armstrong, M., Chetboun, G., and Hubert, P., 1993, Kriging the rainfall in Lesotho, in A. Soares, ed., Geostatistics Tróia' 92, Vol. 2: Kluwer Academic Publ., Dordrecht, p. 661–672.Google Scholar
  3. Bacchi, B., and Kottegoda, N. T., 1995, Identification and calibration of spatial correlation patterns of rainfall: Jour. Hydrology, v. 165, p. 311–348.Google Scholar
  4. Bennett, R. J., 1979, Spatial time series: Analysis-forecasting-control: Pion, London, 674 p.Google Scholar
  5. Bilonick, R. A., 1985, The space-time distribution of sulfate deposition in the northeastern United States: Atmosph. Environment, v. 19, no. 11, p. 1829–1845.Google Scholar
  6. Bilonick, R. A., 1988, Monthly hydrogen ion deposition maps for the northeastern U.S. from July 1982 to September 1984: Atmosph. Environment, v. 22, no. 9, p. 1909–1924.Google Scholar
  7. Bogaert, P., 1996, Comparison of kriging techniques in a space-time context: Math. Geology, v. 28, no. 1, p. 73–86.Google Scholar
  8. Bogaert, P., and Christakos, G., 1997a, Spatiotemporal analysis and processing of thermometric data over Belgium: Jour. Geoph. Res., v. 102, no. D22, p. 25831–25846.Google Scholar
  9. Bogaert, P., and Christakos, G., 1997b, Stochastic analysis of spatiotemporal solute content measurements using a regression model: Stoch. Hydrol. Hydraulics, v. 11, no. 4, p. 267–295.Google Scholar
  10. Boulanger, F., 1989, Géostatistique et methods autorégressifs: Une nouvelle méthode de modélisation, in M. Armstrong, ed., Geostatistics, Vol. 1: Kluwer Academic Publ., Dordrecht, p. 259–271.Google Scholar
  11. Box, G. E. P., Jenkins, G. M., and Reinsel, G. C., 1994, Time series analysis, forecasting and control: Prentice-Hall, Englewood Cliffs, NJ, 3rd ed., 598 p.Google Scholar
  12. Bras, R. L., and Colón, R., 1978, Time-averaged areal mean of precipitation: Estimation and network design: Water Resources Res., v. 14, no. 5, p. 878–888.Google Scholar
  13. Bras, R. L., and Rodrígues-Iturbe, I., 1984, Random functions and hydrology: Addison-Wesley, Reading, MA, 559 p.Google Scholar
  14. Brown, P. J., Le, N. D., and Zidek, J. V., 1994, Multivariate spatial interpolation and exposure to air pollutants: Can. Jour. Statistics, v. 22, no. 4, p. 489–509.Google Scholar
  15. Buxton, B. E., and Plate, A. D., 1994, Joint temporal-spatial modeling of concentrations of hazardous pollutants in urban air, in R. Dimitrakopoulos, ed., Geostatistics for the next century: Kluwer Academic Publ., Dordrecht, p. 75–87.Google Scholar
  16. Carroll, R. J., Chen, R., George, E. I., Li, T. H., Newton, H. J., Schmiediche, H., and Wang, N., 1997, Ozone exposure and population density in Harris county, Texas (with discussion): Jour. Amer. Statistical Assoc., v. 92, no. 438, p. 392–415.Google Scholar
  17. Carroll, S. S., and Cressie, N., 1997, Spatial modeling of snow water equivalent using covariances estimated from spatial and geomorphic attributes: Jour. Hydrology, v. 190, p. 42–59.Google Scholar
  18. Chirlin, G. R., and Wood, E. F., 1982, On the relationship between kriging and state estimation: Water Resources Res., v. 18, no. 2, p. 432–438.Google Scholar
  19. Christakos, G., 1992, Random field models in earth sciences: Academic Press, San Diego, CA, 474 p.Google Scholar
  20. Christakos, G., and Bogaert, P., 1996, Spatiotemporal analysis of spring water ion processes derived from measurements at the Dyle basin in Belgium: IEEE Transactions on Geoscience and Remote Sensing, v. 34, no. 3, p. 626–642.Google Scholar
  21. Christakos, G., and Hristopulos, D. T., 1998, Spatiotemporal environmental health modelling: A tractatus stochasticus: Kluwer Academic Publ., Boston, 424 p.Google Scholar
  22. Christakos, G., and Lai, J., 1997, A study of breast cancer dynamics in North Carolina: Social Sci. Medicine, v. 45, no. 10, p. 1503–1517.Google Scholar
  23. Christakos, G., and Raghu, R., 1996, Dynamic stochastic estimation of physical variables: Math. Geology, v. 28, no. 3, p. 341–365.Google Scholar
  24. Cleveland, W., and Devlin, S., 1988, Locally weighted regression: An approach to regression analysis by local fitting: Jour. Amer. Statistical Assoc., v. 83, no. 403, p. 596–610.Google Scholar
  25. Cliff, A. D., Haggett, P., Ord, J. K., and Bassett, K. A., 1975, Elements of spatial structure: A quantitative approach: Cambridge University Press, New York, 258 p.Google Scholar
  26. Cressie, N. A. C., 1993, Statistics for spatial data: John Wiley & Sons, New York, rev. ed., 900 p.Google Scholar
  27. Creutin, J. D., and Obled, C., 1982, Objective analyses and mapping techniques for rainfall fields: An objective comparison: Water Resources Res., v. 18, no. 2, p. 413–431.Google Scholar
  28. De Cesare, L., Myers, D. E., and Posa, D., 1997, Spatio-temporal modeling of SO 2 in Milan district, in Baaffi, E., and Schofield, N., eds., Geostatistics Wollongong'96, Vol. 2: Kluwer Academic Publ., Dordrecht, p. 1310–1042.Google Scholar
  29. Delfiner, P., 1976, Linear estimation of non-stationary spatial phenomena, in Guarascio, M., David M., and Huijbregts Ch. J., eds., Advanced Geostatistics in the Mining Industry: Reidel, Dordrecht, The Netherlands, p. 49–68.Google Scholar
  30. Deutsch, C. V., and Journel, A. G., 1998, GSLIB: Geostatistical software library and user's guide: Oxford University Press, New York, 2nd ed., 368 p.Google Scholar
  31. Dimitrakopoulos, R., and Luo, X., 1994, Spatiotemporal modeling: Covariances and ordinary kriging systems, in R. Dimitrakopoulos, ed., Geostatistics for the next century: Kluwer Academic Publ., Dordrecht, p. 88–93.Google Scholar
  32. Dimitrakopoulos, R., and Luo, X., 1997, Joint space-time modeling in the presence of trends, in Baaffi, E., and Schofield, N., eds., Geostatistics Wollongong'96, Vol. 1: Kluwer Academic Publ., Dordrecht, p. 138–149.Google Scholar
  33. Egbert, G. D., and Lettenmaier, D. P., 1986, Stochastic modeling of the space-time structure of atmospheric chemical deposition: Water Resources Res., v. 22, no. 2, p. 165–179.Google Scholar
  34. Eynon, B. P., 1988, Statistical analysis of precipitation chemistry measurements over the eastern United States. Part II: Kriging analysis of regional patterns and trends: Jour. Appl. Meteorology, v. 27, p. 1334–1343.Google Scholar
  35. Eynon, B. P., and Switzer, P., 1983, The variability of rainfall acidity: Can. Jour. Statistics, v. 11, no. 1, p. 11–24.Google Scholar
  36. Gandin, L., 1963, Objective Analysis of Meteorological Fields: Gidrometeorologicheskoe Izdatel'stvo (GIMEZ), Leningrad. Reprinted by Israel Program for Scientific Translations, Jerusalem, 1965.Google Scholar
  37. Gelb, A., 1974, Applied optimal estimation: MIT Press, Cambridge, MA, 374 p.Google Scholar
  38. Goovaerts, P., 1997, Geostatistics for natural resources evaluation: Oxford University Press, New York, 483 p.Google Scholar
  39. Goovaerts, P., and Chiang, C., 1993, Temporal persistence of spatial patterns for mineralizable nitrogen and selected soil properties: Soil Sci. Soc. America Jour., v. 57, no. 2, p. 372–381.Google Scholar
  40. Goovaerts, P., and Sonnet, P., 1993, Study of spatial and temporal variations of hydrogeochemical variables using factorial kriging analysis, in A. Soares, ed., Geostatistics Tróia '92, Vol. 2: Kluwer Academic Publ., Dordrecht, p. 745–756.Google Scholar
  41. Goulard, M., and Voltz, M., 1992, Linear coregionalization model: Tools for estimation and choice of cross-variogram matrix: Math. Geology, v. 24, no. 3, p. 269–286.Google Scholar
  42. Guttorp, P., and Sampson, P., 1994, Methods for estimating heterogeneous spatial covariance functions with environmental applications, in Patil, G. P., and Rao, C. R., eds., Handbook of statistics, Vol. 12: Elsevier Science, North-Holland, Amsterdam, p. 661–689.Google Scholar
  43. Guttorp, P., Sampson, P. D., and Newman, K., 1992, Non-parametric estimation of spatial covariance with application to monitoring network evaluation, in Walden, A. T., and Guttorp, P., eds., Statistics in environmental and earth sciences: Edward Arnold, London, p. 39–51.Google Scholar
  44. Haas, T. C., 1995, Local prediction of a spatio-temporal process with an application to wet sulfate deposition: Jour. Am. Statistical Assoc., v. 90, no. 432, p. 1189–1199.Google Scholar
  45. Haas, T. C., 1998, Statistical assessment of spatio-temporal pollutant trends and meteorological transport models: Atmosph. Environment, v. 32, no. 11, p. 1865–1879.Google Scholar
  46. Handcock, M. S., and Wallis, J. R., 1994, An approach to statistical spatial-temporal modeling of meteorological fields (with discussion): Jour. Am. Statistical Assoc., v. 89, no. 426, p. 368–390.Google Scholar
  47. Haslett, J., 1989, Space time modeling in meteorology: A review, Proceedings of the 47th Session: International Statistical Institute, Paris, p. 229–246.Google Scholar
  48. Haslett, J., and Raftery, A. E., 1989, Space-time modeling with long-memory dependence: Assessing Ireland's wind power resource (with discussion): Appl. Statistics, v. 38, no. 1, p. 1–50.Google Scholar
  49. Heuvelink, G. B. M., Musters, P., and Pebesma, E. J., 1997, Spatio-temporal kriging of soil water content, in Baaffi, E., and Schofield, N., eds., Geostatistics Wollongong '96, Vol. 2: Kluwer Academic Publ., Dordrecht, p. 1020–1030.Google Scholar
  50. Hohn, M. E., Liebhold, A. M., and Gribko, L. S., 1993, Geostatistical model for forecasting spatial dynamics of defoliation caused by the gypsy moth (Lepidoptera: Lymantriidae): Environmental Entomology, v. 22, no. 5, p. 1066–1075.Google Scholar
  51. Høst, G., Omre, H., and Switzer, P., 1995, Spatial interpolation errors for monitoring data: Jour. Am. Statistical Assoc., v. 90, no. 431, p. 853–861.Google Scholar
  52. Hudson, G., and Wackernagel, H., 1994, Mapping temperature using kriging with external drift: Theory and an example from Scotland: Int. Jour. Climatology, v. 14, no. 1, p. 77–91.Google Scholar
  53. Jones, R. H., and Vecchia, A. V., 1993, Fitting continuous ARMA models to unequally spaced spatial data: Jour. Am. Statistical Assoc., v. 88, no. 423, p. 947–954.Google Scholar
  54. Jones, R. H., and Zhang, Y., 1997, Models for continuous stationary space-time processes, in Gregoire, T. G., Brillinger, D. R., Diggle, P. J., Russek-Cohen, E., Warren, W. G., and Wolfinger, R. D., eds., Modelling longitudinal and spatially correlated data, Lecture Notes in Statistics, Vol. 122: Springer-Verlag, New York, p. 289–298.Google Scholar
  55. Journel, A. G., 1986, Geostatistics: Models and tools for the earth sciences: Math. Geology, v. 18, no. 1, p. 119–140.Google Scholar
  56. Journel, A. G., 1989, Fundamentals of geostatistics in five lessons, Volume 8 Short Course in Geology: American Geophysical Union, Washington, DC, 40 p.Google Scholar
  57. Journel, A. G., 1993, Modeling uncertainty: Some conceptual thoughts, in R. Dimitrakopoulos, ed., Geostatistics for the Next Century: Kluwer Academic Publ., Dordrecht, p. 30–43.Google Scholar
  58. Journel, A. G., and Huijbregts, Ch. J., 1972, Estimation of lateritic-type orebodies, Proceedings of the 10th International APCOM Symposium: Society of Mining Engineers, Johannesburg, p. 202–212.Google Scholar
  59. Journel, A. G., and Huijbregts, Ch. J., 1978, Mining geostatistics: Academic Press, New York, 600 p.Google Scholar
  60. Journel, A. G., and Rossi, M., 1989, When do we need a trend model in kriging?: Math. Geology, v. 21, no. 7, p. 715–739.Google Scholar
  61. Kyriakidis, P. C., 1998, Stochastic models for spatiotemporal distributions: Application to sulphate deposition over Europe, Rep. 11, Vol. 2: Stanford Center for Reservoir Forecasting, Stanford University, Stanford, CA.Google Scholar
  62. Kyriakidis, P. C., 1999, Stochastic Simulation of Spatiotemporal Phenomena: Unpublished doctorate dissertation: Stanford University, Stanford, CA, 221 p.Google Scholar
  63. Loader, C., and Switzer, P., 1992, Spatial covariance estimation for monitoring data, in Walden, A. T., and Guttorp, P., eds., Statistics in Environmental and Earth Sciences: Edward Arnold, London, p. 52–70.Google Scholar
  64. Mardia, K. V., and Goodall, C. R., 1993, Spatial-temporal analysis of multivariate environmental data, in Patil, G. P., and Rao, C. R., eds., Multivariate environmental statistics: Elsevier, Amsterdam, p. 347–386.Google Scholar
  65. Matérn, B. 1980, Spatial Variation, Lecture Notes in Statistics, Vol. 36: Springer Verlag, New York, 2nd ed., 151 p. First edition published in Meddelanden fran Statens Skogsforskningsinstitut, Band 49, No. 5, 1960.Google Scholar
  66. Matheron, G., 1962, Traitéde géostatistique appliquée, Tome I: Mémoires du Bureau de Recherches Géologiques et Miniéres, Editions Technip, Paris, 333 p.Google Scholar
  67. Matheron, G., 1973, The intrinsic random functions and their applications: Adv. Appl. Probab., v. 5, p. 439–468.Google Scholar
  68. Meiring, W., Guttorp, P., and Sampson, P. D., 1998, Space-time estimation of grid-cell hourly ozone levels for assessment of a deterministic model: Environmental and Ecological Statistics, v. 5, in press.Google Scholar
  69. Meiring, W., Monestiez, P., Sampson, P. D., and Guttorp, P., 1997, Developments in the modeling of nonstationary spatial covariance structure from space-time monitoring data, in Baaffi, E., and Schofield, N., eds., Geostatistics Wollongong '96, Vol. 1: Kluwer Academic Publ., Dordrecht, p. 162–173.Google Scholar
  70. Myers, D. E., and Journel, A. G., 1990, Variograms with zonal anisotropies and noninvertible kriging systems: Math. Geology, v. 22, no. 7, p. 779–785.Google Scholar
  71. Obled, C., and Creutin, J. D., 1986, Some developments in the use of Empirical Orthogonal Functions for mapping meteorological fields: Jour. Climate Appl. Meteorology, v. 25, no. 9, p. 1189–1204.Google Scholar
  72. Oehlert, G. W., 1993, Regional trends in sulfate wet deposition: Jour. Am. Statistical Assoc., v. 88, no. 422, p. 390–399.Google Scholar
  73. Papritz, A., and Flühler, H., 1994, Temporal change of spatially autocorrelated soil properties: Optimal estimation by cokriging: Geoderma, v. 62, p. 29–43.Google Scholar
  74. Pereira, M. J., Soares, A., and Branquinho, C., 1997, Stochastic simulation of fugitive dust emissions, in Baaffi, E., and Schofield, N., eds., Geostatistics Wollongong '96, Vol. 2: Kluwer Academic Publ., Dordrecht, p. 1055–1065.Google Scholar
  75. Rao, A. R., and Hsieh, C. H., 1991, Estimation of variables at ungaged locations by Empirical Orthogonal Functions: Jour. Hydrology, v. 123, p. 51–67.Google Scholar
  76. Rehman, S. U., 1995, Semiparametric Modeling of Cross-Semivariograms: Unpublished doctorate dissertation, Georgia Institute of Technology, Athens, GA, 143 p.Google Scholar
  77. Reichenbach, H., 1958, The philosophy of space & time: Dover, New York, 295 p.Google Scholar
  78. Rodríguez-Iturbe, I., and Mejía, J. M., 1974, The design of rainfall networks in time and space: Water Resources Res., v. 10, no. 4, p. 713–728.Google Scholar
  79. Rouhani, S., Ebrahimpour, R. M., Yaqub, I., and Gianella, E., 1992, Multivariate geostatistical trend detection and network evaluation of space-time acid deposition data–I. Methodology: Atmosph. Environment, v. 26, no. 14, p. 2603–2614.Google Scholar
  80. Rouhani, S., and Hall, T. J., 1989, Space-time kriging of groundwater data, in M. Armstrong, ed., Geostatistics, Vol. 2: Kluwer Academic Publ., Dordrecht, p. 639–650.Google Scholar
  81. Rouhani, S., and Myers, D. E., 1990, Problems in space-time kriging of geohydrological data: Math. Geology, v. 22, no. 5, p. 611–623.Google Scholar
  82. Rouhani, S., and Wackernagel, H., 1990, Multivariate geostatistical approach to space-time data analysis: Water Resources Res., v. 26, no. 4, p. 585–591.Google Scholar
  83. Salas, J. D., 1993, Analysis and modeling of hydrologic time series, in D. R. Maidment, ed., Handbook of hydrology: McGraw-Hill, New York, p. 19.1–19.71.Google Scholar
  84. Sampson, P., and Guttorp, P., 1992, Non parametric estimation of nonstationary spatial structure: Jour. Am. Statistical Assoc., v. 87, no. 417, p. 108–119.Google Scholar
  85. Séguret, S. A., 1989, Filtering periodic noise by using trigonometric kriging, in M. Armstrong, ed., Geostatistics, Vol. 1: Kluwer Academic Publ., Dordrecht, p. 481–491.Google Scholar
  86. Séguret, S., and Huchon, P., 1990, Trigonometric kriging: A new method for removing the diurnal variation fom geomagnetic data, Jour. Geophys. Res., v. 95, no. B13, p. 21383–21397.Google Scholar
  87. Simard, Y., and Marcotte, D., 1993, Assessing similarities and differences among maps: A study of temporal changes in distribution of northern shrimp (pandalus borealis) in the gulf of St. Lawrence, in A. Soares, ed., Geostatistics Tróia '92, Vol. 2: Kluwer Academic Publ., Dordrecht, p. 865–874.Google Scholar
  88. Soares, A., Patinha, P. J., and Pereira, M. J., 1996, Stochastic simulation of space-time series: Application to a river water quality modeling, in Srivastava, R. M., Rouhani, S., Cromer, M. V., Johnson, A. I., and Desbarats, A. J., eds., Geostatistics for environmental and geotechnical applications: American Society for Testing and Materials, West Conshohocken, PA, p. 146–161.Google Scholar
  89. Sølna, K., and Switzer, P., 1996, Time trend estimation for a geographic region: Jour. Am. Statistical Assoc., v. 91, no. 434, p. 577–589.Google Scholar
  90. Solow, A. R., 1984, The analysis of second-order stationary processes: Time series analysis, spectral analysis, harmonic analysis, and geostatistics, in Verly, G., David, M., Journel, A. G., and Marechal, A., eds., Geostatistics for Natural Resources Characterization, Vol. 1: Reidel, Dordrecht, The Netherlands, p. 573–585.Google Scholar
  91. Solow, A. R., and Gorelick, S. M., 1986, Estimating monthly streamflow values by cokriging: Math. Geology, v. 18, no. 8, p. 785–809.Google Scholar
  92. Srivastava, R., 1992, Reservoir characterization with probability field simulation, SPE Annual Conference and Exhibition, Washington, DC: Society of Petroleum Engineers, Washington, DC, p. 927–938. SPE paper number 24753, preprint.Google Scholar
  93. Stein, M., 1986, A simple model for spatial-temporal processes: Water Resources Res., v. 22, no. 13, p. 2107–2110.Google Scholar
  94. Switzer, P., 1979, Statistical considerations in network design: Water Resources Res., v. 15, no. 6, p. 1712–1716.Google Scholar
  95. Switzer, P., 1989, Non-stationary spatial covariances estimated from monitoring data, in M. Armstrong, ed., Geostatistics, Vol. 1: Kluwer Academic Publ., Dordrecht, p. 127–138.Google Scholar
  96. Tabios, G., and Salas, J., 1985, A comparative analysis of techniques for spatial interpolation of precipitation: Water Resources Bull., v. 21, no. 3, p. 365–380.Google Scholar
  97. Venkatram, A., 1988, On the use of kriging in the spatial analysis of acid precipitation data: Atmosph. Environment, v. 22, no. 9, p. 1963–1975.Google Scholar
  98. Voltz, M., and Goulard, M., 1994, Spatial interpolation of soil moisture retention curves: Geoderma, v. 62, p. 109–123.Google Scholar
  99. Vyas, V., and Christakos, G., 1997, Spatiotemporal analysis and mapping of sulfate deposition data over eastern U.S.A.: Atmosph. Environment, v. 31, no. 21, p. 3623–3633.Google Scholar
  100. Wackernagel, H., 1995, Multivariate Geostatistic: Springer, Berlin, 256 p.Google Scholar
  101. Weyl, H., 1952, Space-time-matter: Dover, New York, 330 p.Google Scholar
  102. Wikle, C., Berliner, L. M., and Cressie, N., 1998, Hierarchical Bayesian space-time models: Environmental and Ecological Statistics, v. 5, in press.Google Scholar
  103. Wikle, C. K., and Cressie, N., 1997, A dimension reduction approach to space-time Kalman filtering, Technical Report Preprint No. 97–24: Statistical Laboratory, Department of Statistics, Iowa State University, Ames, IA.Google Scholar
  104. Xu, W., and Journel, A. G., 1994, Dssim: A general sequential simulation algorithm, Report. 7: Stanford Center for Reservoir Forecasting, Stanford University, Stanford, CA.Google Scholar
  105. Yao, T., 1996, Hole effect or not?, Rep. 9: Stanford Center for Reservoir Forecasting, Stanford University, Stanford, CA.Google Scholar
  106. Yao, T., 1998. Conditional spectral simulation with phase identification: Math. Geology, v. 30, no. 3, p. 285–308.Google Scholar
  107. Yao, T., and Journel, A. G., 1998, Automatic modeling of covariance tables with Fast Fourier Transform: Math. Geology, v. 30, no. 6, p. 589–615.Google Scholar

Copyright information

© International Association for Mathematical Geology 1999

Authors and Affiliations

  • Phaedon C. Kyriakidis
    • 1
  • André G. Journel
    • 2
  1. 1.Department of Geological and Environmental SciencesStanford UniversityStanford
  2. 2.Department of Petroleum EngineeringStanford UniversityStanford

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