Geostatistics

Geostatistics is a branch of statistics focusing on spatial or spatiotemporal datasets. Developed originally to predict probability distributions of ore grades for mining operations,^[1] it is currently applied in diverse disciplines including petroleum geology, hydrogeology, hydrology, meteorology, oceanography, geochemistry, geometallurgy, geography, forestry, environmental control, landscape ecology, soil science, and agriculture (esp. in precision farming). Geostatistics is applied in varied branches of geography, particularly those involving the spread of diseases (epidemiology), the practice of commerce and military planning (logistics), and the development of efficient spatial networks. Geostatistical algorithms are incorporated in many places, including geographic information systems (GIS).

Background

Geostatistics is intimately related to interpolation methods, but extends far beyond simple interpolation problems. Geostatistical techniques rely on statistical models that are based on random function (or random variable) theory to model the uncertainty associated with spatial estimation and simulation.

A number of simpler interpolation methods/algorithms, such as inverse distance weighting, bilinear interpolation and nearest-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.

Let $Z (x)$ be the value of the variable of interest at a certain location $x$ . This value is unknown (e.g. temperature, rainfall, piezometric level, geological facies, etc.). Although there exists a value at location $x$ that could be measured, geostatistics considers this value as random since it was not measured, or has not been measured yet. However, the randomness of $Z (x)$ is not complete, but defined by a cumulative distribution function (CDF) that depends on certain information that is known about the value $Z (x)$ :

F({\mathit {z}},\mathbf {x} )=\operatorname {Prob} \lbrace Z(\mathbf {x} )\leqslant {\mathit {z}}\mid {\text{information}}\rbrace .

Typically, if the value of $Z$ is known at locations close to $x$ (or in the neighborhood of $x$ ) one can constrain the CDF of $Z (x)$ 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 continuity $Z (x)$ 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 of variogram-based geostatistics, or have a non-parametric form when using other methods such as multiple-point simulation^[3] or pseudo-genetic techniques.

By applying a single spatial model on an entire domain, one makes the assumption that $Z$ is a stationary 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 for $Z (x)$ , typically by the expectation, the median or the mode of the CDF $f (z, x)$ . This is usually denoted as an estimation problem.
Sampling from the entire probability density function $f (z, x)$ by actually considering each possible outcome of it at each location. This is generally done by creating several alternative maps of $Z$ , called realizations. Consider a domain discretized in $N$ grid nodes (or pixels). Each realization is a sample of the complete $N$ -dimensional joint distribution function

F(\mathbf {z} ,\mathbf {x} )=\operatorname {Prob} \lbrace Z(\mathbf {x} _{1})\leqslant z_{1},Z(\mathbf {x} _{2})\leqslant z_{2},...,Z(\mathbf {x} _{N})\leqslant z_{N}\rbrace .

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 solving inverse 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.^[6]^[2]^[7]^[8]^[9]^[10]^[11]^[12]^[13]^[14]^[15]

Methods

Estimation

Kriging

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 estimation

Bayesian inference is a method of statistical inference in which Bayes' 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 a Gaussian process, and updates the process using Bayes' Theorem to calculate its posterior. High-dimensional Bayesian Geostatistics ^[17]

Finite difference method

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]

Simulation

Aggregation
Dissagregation
Turning bands
Cholesky decomposition
Truncated Gaussian
Plurigaussian
Annealing
Spectral simulation
Sequential Indicator
Sequential Gaussian
Dead Leave
Transition probabilities
Markov chain geostatistics
Support vector machine
Boolean simulation
Genetic models
Pseudo-genetic models
Cellular automata
Multiple-Point Geostatistics

Definitions and tools

Notes

^ 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
^ ^a ^b Isaaks, 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. ISBN 0-12-391050-1
^ 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,
^ Schnetzler, Manu. "Statios - WinGslib". Archived from the original on 2015-05-11. Retrieved 2005-10-10.
^ Banerjee S., Carlin B.P., and Gelfand A.E. (2014). Hierarchical Modeling and Analysis for Spatial Data, Second Edition. Chapman & Hall/CRC Monographs on Statistics & Applied Probability. ISBN 9781439819173
^ Banerjee, Sudipto. High-Dimensional Bayesian Geostatistics. Bayesian Anal. 12 (2017), no. 2, 583--614. doi:10.1214/17-BA1056R. https://projecteuclid.org/euclid.ba/1494921642
^ Cardenas, IC (2023). "A two-dimensional approach to quantify stratigraphic uncertainty from borehole data using non-homogeneous random fields". Engineering Geology. doi:10.1016/j.enggeo.2023.107001.

References

Armstrong, M and Champigny, N, 1988, A Study on Kriging Small Blocks, CIM Bulletin, Vol 82, No 923
Armstrong, M, 1992, Freedom of Speech? De Geeostatisticis, July, No 14
Champigny, N, 1992, Geostatistics: A tool that works, The Northern Miner, May 18
Clark I, 1979, Practical Geostatistics, Applied Science Publishers, London
David, M, 1977, Geostatistical Ore Reserve Estimation, Elsevier Scientific Publishing Company, Amsterdam
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. S2CID 73657847. (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.
Merks, J W, 1992, Geostatistics or voodoo science, The Northern Miner, May 18
Merks, J W, Abuse of statistics, CIM Bulletin, January 1993, Vol 86, No 966
Myers, Donald E.; "What Is Geostatistics?
Philip, G M and Watson, D F, 1986, Matheronian Geostatistics; Quo Vadis?, Mathematical Geology, Vol 18, No 1
Pyrcz, M.J. and Deutsch, C.V., 2014, Geostatistical Reservoir Modeling, 2nd Edition, Oxford University Press, New York, p. 448
Sharov, A: Quantitative Population Ecology, 1996, https://web.archive.org/web/20020605050231/http://www.ento.vt.edu/~sharov/PopEcol/popecol.html
Shine, J.A., Wakefield, G.I.: A comparison of supervised imagery classification using analyst-chosen and geostatistically-chosen training sets, 1999, https://web.archive.org/web/20020424165227/http://www.geovista.psu.edu/sites/geocomp99/Gc99/044/gc_044.htm
Strahler, A. H., and Strahler A., 2006, Introducing Physical Geography, 4th Ed., Wiley.
Tahmasebi, P., Hezarkhani, A., Sahimi, M., 2012, Multiple-point geostatistical modeling based on the cross-correlation functions, Computational Geosciences, 16(3):779-79742.
Volk, W, 1980, Applied Statistics for Engineers, Krieger Publishing Company, Huntington, New York.

External links

[1] 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

[IandS1989-2] Isaaks, E. H. and Srivastava, R. M. (1989), An Introduction to Applied Geostatistics, Oxford University Press, New York, USA.

[3] Mariethoz, Gregoire, Caers, Jef (2014). Multiple-point geostatistics: modeling with training images. Wiley-Blackwell, Chichester, UK, 364 p.

[4] Hansen, T.M., Journel, A.G., Tarantola, A. and Mosegaard, K. (2006). "Linear inverse Gaussian theory and geostatistics", Geophysics 71

[5] 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

[6] Remy, N., et al. (2009), Applied Geostatistics with SGeMS: A User's Guide, 284 pp., Cambridge University Press, Cambridge.

[7] 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/

[8] Chilès, J.-P., and P. Delfiner (1999), Geostatistics - Modeling Spatial Uncertainty, John Wiley & Sons, Inc., New York, USA.

[9] Lantuéjoul, C. (2002), Geostatistical simulation: Models and algorithms, 232 pp., Springer, Berlin.

[10] Journel, A. G. and Huijbregts, C.J. (1978) Mining Geostatistics, Academic Press. ISBN 0-12-391050-1

[11] Kitanidis, P.K. (1997) Introduction to Geostatistics: Applications in Hydrogeology, Cambridge University Press.

[12] Wackernagel, H. (2003). Multivariate geostatistics, Third edition, Springer-Verlag, Berlin, 387 pp.

[13] Pyrcz, M. J. and Deutsch, C.V., (2014). Geostatistical Reservoir Modeling, 2nd Edition, Oxford University Press, 448 pp.

[14] Tahmasebi, P., Hezarkhani, A., Sahimi, M., 2012, Multiple-point geostatistical modeling based on the cross-correlation functions, Computational Geosciences, 16(3):779-79742,

[15] Schnetzler, Manu. "Statios - WinGslib". Archived from the original on 2015-05-11. Retrieved 2005-10-10.

[16] Banerjee S., Carlin B.P., and Gelfand A.E. (2014). Hierarchical Modeling and Analysis for Spatial Data, Second Edition. Chapman & Hall/CRC Monographs on Statistics & Applied Probability. ISBN 9781439819173

[17] Banerjee, Sudipto. High-Dimensional Bayesian Geostatistics. Bayesian Anal. 12 (2017), no. 2, 583--614. doi:10.1214/17-BA1056R. https://projecteuclid.org/euclid.ba/1494921642

[18] Cardenas, IC (2023). "A two-dimensional approach to quantify stratigraphic uncertainty from borehole data using non-homogeneous random fields". Engineering Geology. doi:10.1016/j.enggeo.2023.107001.

[1]

[2]

[3]

[4]

[5]

[6]

[7]

[8]

[9]

[10]

[11]

[12]

[13]

[14]

[15]

[16]

[17]

[18]