Geostatistics is intimately related to interpolation methods but extends far beyond simple interpolation problems. Geostatistical techniques rely on statistical models based on random function (orrandom variable) theory to model the uncertainty associated with spatial estimation and simulation.
A number of simpler interpolation methods/algorithms, such asinverse distance weighting,bilinear interpolation andnearest-neighbor interpolation, were already well known before geostatistics.[2] Geostatistics goes beyond the interpolation problem by considering the studied phenomenon at unknown locations as a set of correlated random variables.
LetZ(x) be the value of the variable of interest at a certain locationx. This value is unknown (e.g., temperature, rainfall,piezometric level, geological facies, etc.). Although there exists a value at locationx that could be measured, geostatistics considers this value as random since it was not measured or has not been measured yet. However, the randomness ofZ(x) is not complete. Still, it is defined by acumulative distribution function (CDF) that depends on certain information that is known about the valueZ(x):
Typically, if the value ofZ is known at locations close tox (or in theneighborhood ofx) one can constrain the CDF ofZ(x) by this neighborhood: if a high spatial continuity is assumed,Z(x) can only have values similar to the ones found in the neighborhood. Conversely, in the absence of spatial continuityZ(x) can take any value. The spatial continuity of the random variables is described by a model of spatial continuity that can be either a parametric function in the case ofvariogram-based geostatistics, or have a non-parametric form when using other methods such asmultiple-point simulation[3] orpseudo-genetic techniques.
By applying a single spatial model on an entire domain, one makes the assumption thatZ is astationary process. It means that the same statistical properties are applicable on the entire domain. Several geostatistical methods provide ways of relaxing this stationarity assumption.
In this framework, one can distinguish two modeling goals:
Estimating the value forZ(x), typically by theexpectation, themedian or themode of the CDFf(z,x). This is usually denoted as an estimation problem.
Sampling from the entire probability density functionf(z,x) by actually considering each possible outcome of it at each location. This is generally done by creating several alternative maps ofZ, called realizations. Consider a domain discretized inN grid nodes (or pixels). Each realization is a sample of the completeN-dimensional joint distribution function
In this approach, the presence of multiple solutions to the interpolation problem is acknowledged. Each realization is considered as a possible scenario of what the real variable could be. All associated workflows are then considering ensemble of realizations, and consequently ensemble of predictions that allow for probabilistic forecasting. Therefore, geostatistics is often used to generate or update spatial models when solvinginverse problems.[4][5]
A number of methods exist for both geostatistical estimation and multiple realizations approaches. Several reference books provide a comprehensive overview of the discipline.[2][6][7][8][9][10][11][12][13][14][15]
Kriging is a group of geostatistical techniques to interpolate the value of a random field (e.g., the elevation, z, of the landscape as a function of the geographic location) at an unobserved location from observations of its value at nearby locations.
Bayesian inference is a method of statistical inference in whichBayes' theorem is used to update a probability model as more evidence or information becomes available. Bayesian inference is playing an increasingly important role in geostatistics.[16] Bayesian estimation implements kriging through a spatial process, most commonly aGaussian process, and updates the process usingBayes' Theorem to calculate its posterior. High-dimensional Bayesian geostatistics.[17]
Considering the principle of conservation of probability, recurrent difference equations (finite difference equations) were used in conjunction with lattices to compute probabilities quantifying uncertainty about the geological structures. This procedure is a numerical alternative method to Markov chains and Bayesian models.[18]
^Krige, Danie G. (1951). "A statistical approach to some basic mine valuation problems on the Witwatersrand". J. of the Chem., Metal. and Mining Soc. of South Africa 52 (6): 119–139
^abIsaaks, E. H. and Srivastava, R. M. (1989),An Introduction to Applied Geostatistics, Oxford University Press, New York, USA.
^Mariethoz, Gregoire, Caers, Jef (2014). Multiple-point geostatistics: modeling with training images. Wiley-Blackwell, Chichester, UK, 364 p.
^Hansen, T.M., Journel, A.G., Tarantola, A. and Mosegaard, K. (2006). "Linear inverse Gaussian theory and geostatistics",Geophysics 71
^Kitanidis, P.K. and Vomvoris, E.G. (1983). "A geostatistical approach to the inverse problem in groundwater modeling (steady state) and one-dimensional simulations",Water Resources Research 19(3):677-690
^Remy, N., et al. (2009),Applied Geostatistics with SGeMS: A User's Guide, 284 pp., Cambridge University Press, Cambridge.
^Deutsch, C.V., Journel, A.G, (1997).GSLIB: Geostatistical Software Library and User's Guide (Applied Geostatistics Series), Second Edition, Oxford University Press, 369 pp.,http://www.gslib.com/
^Chilès, J.-P., and P. Delfiner (1999),Geostatistics - Modeling Spatial Uncertainty, John Wiley & Sons, Inc., New York, USA.
^Lantuéjoul, C. (2002),Geostatistical simulation: Models and algorithms, 232 pp., Springer, Berlin.
^Journel, A. G. and Huijbregts, C.J. (1978)Mining Geostatistics, Academic Press.ISBN0-12-391050-1
^Kitanidis, P.K. (1997)Introduction to Geostatistics: Applications in Hydrogeology, Cambridge University Press.
^Wackernagel, H. (2003).Multivariate geostatistics, Third edition, Springer-Verlag, Berlin, 387 pp.
^Pyrcz, M. J. and Deutsch, C.V., (2014).Geostatistical Reservoir Modeling, 2nd Edition, Oxford University Press, 448 pp.
^Tahmasebi, P., Hezarkhani, A., Sahimi, M., 2012, Multiple-point geostatistical modeling based on the cross-correlation functions, Computational Geosciences, 16(3):779-79742,
Hald, A, 1952, Statistical Theory with Engineering Applications, John Wiley & Sons, New York
Honarkhah, Mehrdad; Caers, Jef (2010). "Stochastic Simulation of Patterns Using Distance-Based Pattern Modeling".Mathematical Geosciences.42 (5):487–517.doi:10.1007/s11004-010-9276-7.S2CID73657847. (best paper award IAMG 09)
ISO/DIS 11648-1 Statistical aspects of sampling from bulk materials-Part1: General principles
Lipschutz, S, 1968, Theory and Problems of Probability, McCraw-Hill Book Company, New York.
Matheron, G. 1962. Traité de géostatistique appliquée. Tome 1, Editions Technip, Paris, 334 pp.
Matheron, G. 1989. Estimating and choosing, Springer-Verlag, Berlin.
McGrew, J. Chapman, & Monroe, Charles B., 2000. An introduction to statistical problem solving in geography, second edition, McGraw-Hill, New York.
