Abstract
A fundamental problem facing the physical sciences today is analysis of natural variations and mapping of spatiotemporal processes. Detailed maps describing the space/time distribution of groundwater contaminants, atmospheric pollutant deposition processes, rainfall intensity variables, external intermittency functions, etc. are tools whose importance in practical applications cannot be overestimated. Such maps are valuable inputs for numerous applications including, for example, solute transport, storm modeling, turbulent-nonturbulent flow characterization, weather prediction, and human exposure to hazardous substances. The approach considered here uses the spatiotemporal random field theory to study natural space/time variations and derive dynamic stochastic estimates of physical variables. The random field model is constructed in a space/time continuum that explicitly involves both spatial and temporal aspects and provides a rigorous representation of spatiotemporal variabilities and uncertainties. This has considerable advantages as regards analytical investigations of natural processes. The model is used to study natural space/time variations of springwater calcium ion data from the Dyle River catchment area, Belgium. This dataset is characterized by a spatially nonhomogeneous and temporally nonstationary variability that is quantified by random field parameters, such as orders of space/time continuity and random field increments. A rich class of covariance models is determined from the properties of the random field increments. The analysis leads to maps of continuity orders and covariances reflecting space/time calcium ion correlations and trends. Calcium ion estimates and the associated statistical errors are calculated at unmeasured locations/instants over the Dyle region using a space/time kriging algorithm. In practice, the interpretation of the results of the dynamic stochastic analysis should take into consideration the scale effects.
Similar content being viewed by others
References
Barton, G., 1991, Elements of Green's functions and propagation: Oxford Sci. Publ., Oxford, 465 P.
Berkowitz, B., Ben-Zvi, M., and Berkowitz, J., 1992, A spatial, time-dependent approach to estimation of hydrologic data: Jour. Hydrology, v. 135, no. 1–4, p. 133–142.
Bilonick, R. A., 1985, The space-time distribution of sulfate deposition in the northeastern U.S: Atmospheric Environment, v. 19, no. 11, p. 2513–2524.
Bras, R. L., and Rodriguez-Iturbe, I., 1985, Random functions and hydrology, Addison-Wesley, New York, 559 p.
Christakos, G., 1991, A theory of spatiotemporal random fields and its application to space-time data processing: IEEE Trans. Systems, Man and Cybernetics, v. 21, no. 4, p. 861–875.
Christakos, G., 1992, Random field models in earth sciences: Academic Press, San Diego, California, 474 p.
Clifton, P. M., and Newman, S. P., 1982, Effects of kriging and inverse modeling on conditional simulation of the Awa Valley aquifer in southern Arizona: Water Res. Research, v. 18, no. 4, p. 1215–1234.
Cox, D. R., and Isham, V., 1988, A simple spatial-temporal model of rainfall: Proc. Roy. Soc. London, Ser. A, v. 415, p. 317–328.
Delfiner, P., 1976, Linear estimation of nonstationary spatial phenomena:in Adv. Geostatistics in the Mining Industry, (ed.) M. Guarascio, Reidel, Dordrecht, p. 49–68.
Dimitrakopoulos, R., and Luo, X., 1993, Spatiotemporal modelling: covariances and ordinary kriging systems,in Geostatistics for the next century: Kluwer Acad. Publ., Dortrecht, The Netherlands, p. 88–93.
Egbert, G. D., and Lettenmaier, D. P., 1986, Stochastic modeling of space-time structure of atmospheric chemical deposition: Water Res. Research, v. 22, no. 2, p. 165–179.
Einstein, A., and Infeld, L., 1942, The evolution of physics: Simon and Schuster, New York, 319 p.
Goovaerts, P., Sonnet, P., and Navarre, A., 1993, Factorial kriging analysis of springwater contents in the Dyle River basin, Belgium: Water Res. Research, v. 29, no. 7, p. 2115–2125.
Gupta, V. K., and Waymire, E., 1987, On Taylor's hypothesis and dissipation in rainfall: Jour. Geophy. Res., v. 92, no. D8, p. 9657–9660.
Guttorp, P., Sampson, P. D., and Neuman, K., 1992, Nonparametric estimation of spatial covariance with application to monitoring network evaluation:in Statistics in the Environmental and Earth Sciences, (eds.) A. T. Whalden and P. Guttorp, London, Edward Arnold, p. 39–51.
Hasselmann, K., 1988, PIPs and POPs—the reduction of sampling errors of precipitation from space-borne and ground sensors: Jour. Geophys. Res., v. 93, no. D9, p. 11015–11021.
Kashyap, R. L., and Rao, A., 1976, Dynamic stochastic models from empirical data: Academic Press, New York, 334 p.
Journel, A. G., 1989, Fundamentals of geostatistics in five lessons: Am. Geophys. Union, Washington, D.C., v. 8, 40 p.
Matheron, G., 1973, The intrinsic random functions and their applications: Adv. Appl. Prob., v. 5, no. 3, p. 439–468.
Navarre, A., Lecomte, P., and Martin, H., 1976, Analyse des tendances de donnees hydrogeochimiques du bassin de la Dyle en amont d'Archennes: Ann. Soc. Geol. Belgium, v. 99, p. 299–313.
Newman, S. P., 1982, Statistical characterization of aquifer heterogeneities: an overview,in Recent trends in hydrogeology: Geol. Soc. America Spec. Paper, v. 189, p. 81–102.
Petersen, D. P., and Middleton, D., 1964, Reconstruction of multidimensional stochastic fields from discrete measurements of amplitude and gradient: Info. and Control, v. 7, no. 4, p. 445–476.
Polyak, I., North, G. R., and Valdes, J. B., 1994, Multivariate space-time analysis of PRE-STORM precipitation: Jour. Appl. Meteorology, v. 33, no. 9, p. 1079–1087.
Raghu, V., 1995, Numerical implementation of the spatiotemporal random model in the study of water quality data: unpubl. masters thesis, Dept. of Environmental Sci. and Engin., Univ. North Carolina, Chapel Hill, 51 p.
Rodriguez-Iturbe, I., and Eagleson, P. S. 1987, Mathematical models of rainstorm events in space and time: Water Res. Research, v. 23, no. 1, p. 181–190.
Rouhani, S., and Wackernagel, H., 1990, Multivariate geostatistical approach to space-time data analysis: Water Res. Research, v. 26, no. 4, p. 585–591.
SANLIB, 1995, “Stochastic Analysis Software Library and User's Guide,”Stochastic Research Group, Research Rept. No. SM/1.95, Dept. of Environmental Sci. and Engin., Univ. of North Carolina, Chapel Hill, NC. 63 p.
Schlick, M., 1920, Space and time in contemporary physics: Oxford Univ. Press, Oxford, 88 p.
Shapiro, D. E., and Switzer, P., 1989, Extracting time series trends from multiple monitoring sites: Dept. Statistics, Stanford Univ., Tech. Rept. No. 132, 34 p.
Storch, H. von, Weese, U., and Xu, J. S., 1989, Simultaneous analysis of space-time variability: POPs and PIPs with applications to the southern oscillation: Max-Planck Institut fur Meteorologie, Hamburg, Germany, Rept. No. 34, 12 p.
Tatarskii, V. I., 1961, Wave propagation in a turbulent medium: McGraw-Hill Book Co., New York, 285 p.
Yaglom, A. M., 1955, Correlation theory of processes with stationary random increments of order n: Mat. USSR Sb., p. 37–141 (English translation in Am. Math. Soc. Trans. Ser. 2, p. 8–87, 1958).
Yaglom, A. M., 1957, Some classes of random fields in n-dimensional space, related to stationary random processes: Theory Probability and its Appl., English Transl. No. 3, p. 273–320.
Yaglom, A. M., 1991, Private Communication.
Vyas, V., 1994, Development of non-separable generalized spatiotemporal covariance models: Dept. of Environmental Sci. and Engin., Univ. North Carolina, Chapel Hill, Research Rept. no. SM/5.94, 10 p.
Weyl, H., 1987, The continuum. a critical examination of the foundation of analysis: Dover Publ., Inc., New York, 130 p.
Zauderer, E., 1989, Partial differential equations of applied mathematics: Wiley, New York, 891 p.
Zwart, P. J., 1976, About time: North Holland, Amsterdam, 266 p.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Christakos, G., Raghu, V.R. Dynamic stochastic estimation of physical variables. Math Geol 28, 341–365 (1996). https://doi.org/10.1007/BF02083205
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF02083205