Geostatistics

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Not to be confused with statistical geography.
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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, 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

[edit] Background

Geostatistics is intimately related to interpolation methods, but extends far beyond simple interpolation problems. It consists of a collection of numerical and mathematical techniques dealing with the characterization of spatial phenomena. Geostatistical techniques rely on statistical model that is 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.[1] 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 pdf 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:

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.[2][3]

A number of methods exist for both geostatistical estimation and multiple realizations approaches. Several reference books provide a comprehensive overview of the discipline.[4][5][6][7][8][9][10][11][12]

[edit] Methods

[edit] Exploratory data analysis

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[edit] Estimation

Kriging
Indicator kriging

[edit] Simulation

Aggregation
Dissagregation
Turning bands
Spectral simulation
SGS
Transition probabilities
Markov chain geostatistics
Support vector machine
Boolean simulation
Genetic models
Pseudo-genetic models
Cellular automata
Multiple-Point Geostatistics (MPS)

[edit] Definitions and tools

[edit] Main scientific journals related to geostatistics

[edit] Related software

[edit] See also

[edit] Notes

  1. ^ Isaaks, E. H. and Srivastava, R. M. (1989), An Introduction to Applied Geostatistics, Oxford University Press, New York, USA.
  2. ^ Hansen, T.M., Journel, A.G., Tarantola, A. and Mosegaard, K. (2006). "Linear inverse Gaussian theory and geostatistics", Geophysics 71
  3. ^ 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
  4. ^ Remy, N., et al. (2009), Applied Geostatistics with SGeMS: A User's Guide, 284 pp., Cambridge University Press, Cambridge.
  5. ^ 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/
  6. ^ Chilès, J.-P., and P. Delfiner (1999), Geostatistics - Modeling Spatial Uncertainty, John Wiley & Sons, Inc., New York, USA.
  7. ^ Lantuéjoul, C. (2002), Geostatistical simulation: Models and algorithms, 232 pp., Springer, Berlin.
  8. ^ Journel, A. G. and Huijbregts, C.J. (1978) Mining Geostatistics, Academic Press. ISBN 0-12-391050-1
  9. ^ Kitanidis, P.K. (1997) Introduction to Geostatistics: Applications in Hydrogeology, Cambridge University Press.
  10. ^ Wackernagel, H. (2003). Multivariate geostatistics, Third edition, Springer-Verlag, Berlin, 387 pp.
  11. ^ Deutsch, C.V., (2002). Geostatistical Reservoir Modeling, Oxford University Press, 384 pp., http://www.statios.com/WinGslib/index.html
  12. ^ Isaaks, E.H., Srivastava R.M. (1989) Applied Geostatistics.

[edit] References

  1. Armstrong, M and Champigny, N, 1988, A Study on Kriging Small Blocks, CIM Bulletin, Vol 82, No 923
  2. Armstrong, M, 1992, Freedom of Speech? De Geeostatisticis, July, No 14
  3. Champigny, N, 1992, Geostatistics: A tool that works, The Northern Miner, May 18
  4. Clark I, 1979, Practical Geostatistics, Applied Science Publishers, London
  5. David, M, 1977, Geostatistical Ore Reserve Estimation, Elsevier Scientific Publishing Company, Amsterdam
  6. Hald, A, 1952, Statistical Theory with Engineering Applications, John Wiley & Sons, New York
  7. 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)
  8. ISO/DIS 11648-1 Statistical aspects of sampling from bulk materials-Part1: General principles
  9. Lipschutz, S, 1968, Theory and Problems of Probability, McCraw-Hill Book Company, New York.
  10. Matheron, G. 1962. Traité de géostatistique appliquée. Tome 1, Editions Technip, Paris, 334 pp.
  11. Matheron, G. 1989. Estimating and choosing, Springer-Verlag, Berlin.
  12. McGrew, J. Chapman, & Monroe, Charles B., 2000. An introduction to statistical problem solving in geography, second edition, McGraw-Hill, New York.
  13. Merks, J W, 1992, Geostatistics or voodoo science, The Northern Miner, May 18
  14. Merks, J W, Abuse of statistics, CIM Bulletin, January 1993, Vol 86, No 966
  15. Myers, Donald E.; "What Is Geostatistics?
  16. Philip, G M and Watson, D F, 1986, Matheronian Geostatistics; Quo Vadis?, Mathematical Geology, Vol 18, No 1
  17. Sharov, A: Quantitative Population Ecology, 1996, http://www.ento.vt.edu/~sharov/PopEcol/popecol.html
  18. 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
  19. Strahler, A. H., and Strahler A., 2006, Introducing Physical Geography, 4th Ed., Wiley.
  20. Volk, W, 1980, Applied Statistics for Engineers, Krieger Publishing Company, Huntington, New York.
  21. Yang, X. S., 2009, Introductory Mathematics for Earth Scientists, Dunedin Academic Press, 240pp.

[edit] External links

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