# Geostatistics

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**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) and the R statistical environment.

## Contents

## 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**):

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

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 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

- <math>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 .</math>

- 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.
^{[3]}^{[4]}

A number of methods exist for both geostatistical estimation and multiple realizations approaches. Several reference books provide a comprehensive overview of the discipline.^{[5]}^{[6]}^{[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.

#### Indicator kriging

Multiple-indicator kriging (MIK) is a recent advance on other techniques for mineral deposit modeling and resource block model estimation, such as ordinary kriging. Initially, MIK showed considerable promise as a new method that could more accurately estimate overall global mineral deposit concentrations or grades.

### Simulation

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

## Definitions and tools

- Regionalized variable theory
- Covariance function
- Semi-variance
- Variogram
- Kriging
- Range (geostatistics)
- Sill (geostatistics)
- Nugget effect
- Training image

- Water Resources Research
- Advances in Water Resources
- Ground Water
- Mathematical Geosciences
- Computers & Geosciences
- Computational Geosciences
- J. Soil Science Society of America
- Environmetrics
- Remote Sensing of the Environment
- Stochastic Environmental Research and Risk Assessment

- European Forum for GeoStatistics (see note below, under External Links)
- GeoEnvia promotes the use of geostatistical methods in environmental applications

## Related software

- The R programming language has around 20 other packages dedicated to geostatistics, and around 30 dedicated to other areas of spatial statistics.
- D-STEM is a software based on the MATLAB language able to handle spatiotemporal univariate and multivariate datasets. The software allows to produce dynamic maps of the observed variables over geographic regions.

## See also

- Multivariate interpolation
- Spline interpolation
- Geodemographic segmentation
- Remote sensing
- Pedometrics

## 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**^**Isaaks, E. H. and Srivastava, R. M. (1989),*An Introduction to Applied Geostatistics,*Oxford University Press, New York, USA.**^**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.**^**Deutsch, C.V., (2002).*Geostatistical Reservoir Modeling*, Oxford University Press, 384 pp.,**^**Tahmasebi, P., Hezarkhani, A., Sahimi, M., 2012, Multiple-point geostatistical modeling based on the cross-correlation functions, Computational Geosciences, 16(3):779-79742,**^**http://www.statios.com/WinGslib/index.html**^**Isaaks, E.H., Srivastava R.M. (1989)*Applied Geostatistics*.

## 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, M and Caers, J, 2010, Stochastic Simulation of Patterns Using Distance-Based Pattern Modeling, Mathematical Geosciences, 42: 487 - 517 (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
- Sharov, A: Quantitative Population Ecology, 1996, 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, 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

- GeoENVia promotes the use of geostatistical methods in environmental applications, and organizes bi-annual conferences.
- European Forum for GeoStatistics is a forum that uses the word geostatistics in another way as used here: they take it as the plural of "geostatistic". In a project called "GEOSTAT", the
*... goals are to develop the guidelines for datasets and methods to link 2010/11 Population and Housing Census results to a common harmonised grid.*See also the difference between Statistics and Statistic. - Kriging link, contains explanations of variance in geostats
- Arizona university geostats page
- AI-Geostats, a resource on the internet about geostatistics and spatial statistics
- On-Line Library that chronicles Matheron's journey from classical statistics to the new science of geostatistics
- http://www.geostatscam.com Is the site of Jan W. Merks, who claims that geostatistics is "voodoo science" and a "scientific fraud"
- [1] It is a group for exchanging of ideas and discussion on multiple point geostatistics (MPS).

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