Editing Geostatistics (section)

==Background==
Geostatistics is intimately related to interpolation methods but extends far beyond simple interpolation problems. Geostatistical techniques rely on statistical models 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.<ref name="IandS1989">Isaaks, E. H. and Srivastava, R. M. (1989), ''An Introduction to Applied Geostatistics,'' Oxford University Press, New York, USA.</ref> Geostatistics goes beyond the interpolation problem by considering the studied phenomenon at unknown locations as a set of correlated random variables.

Let {{math|''Z''('''x''')}} be the value of the variable of interest at a certain location {{math|'''x'''}}. This value is unknown (e.g., temperature, rainfall, [[Potentiometric surface|piezometric level]], geological facies, etc.). Although there exists a value at location {{math|'''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 {{math|''Z''('''x''')}} is not complete. Still, it is defined by a [[cumulative distribution function]] (CDF) that depends on certain information that is known about the value {{math|''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 {{math|''Z''}} is known at locations close to {{math|'''x'''}} (or in the [[Neighbourhood (mathematics)|neighborhood]] of {{math|'''x'''}}) one can constrain the CDF of {{math|''Z''('''x''')}} by this neighborhood: if a high spatial continuity is assumed, {{math|''Z''('''x''')}} can only have values similar to the ones found in the neighborhood. Conversely, in the absence of spatial continuity {{math|''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]]<ref>Mariethoz, Gregoire, Caers, Jef (2014). Multiple-point geostatistics: modeling with training images.  Wiley-Blackwell, Chichester, UK, 364 p.</ref> or [[pseudo-genetic]] techniques.

By applying a single spatial model on an entire domain, one makes the assumption that {{math|''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:

# [[Estimation theory|Estimating]] the value for {{math|''Z''('''x''')}}, typically by the [[Expected value|expectation]], the [[median]] or the [[Mode (statistics)|mode]] of the CDF {{math|''f''(''z'','''x''')}}. This is usually denoted as an estimation problem.
# [[Sampling (statistics)|Sampling]] from the entire probability density function {{math|''f''(''z'','''x''')}} by actually considering each possible outcome of it at each location. This is generally done by creating several alternative maps of {{math|''Z''}}, called realizations. Consider a domain discretized in {{math|''N''}} grid nodes (or pixels). Each realization is a sample of the complete {{math|''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 problem]]s.<ref>Hansen, T.M., Journel, A.G., Tarantola, A. and Mosegaard, K. (2006). "Linear inverse Gaussian theory and geostatistics", ''Geophysics'' 71</ref><ref>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</ref>

A number of methods exist for both geostatistical estimation and multiple realizations approaches. Several reference books provide a comprehensive overview of the discipline.<ref name="IandS1989" /><ref>Remy, N., et al. (2009), ''Applied Geostatistics with SGeMS: A User's Guide,'' 284 pp., Cambridge University Press, Cambridge.
</ref><ref>
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/
</ref><ref>Chilès, J.-P., and P. Delfiner (1999), ''Geostatistics - Modeling Spatial Uncertainty,'' John Wiley & Sons, Inc., New York, USA.
</ref><ref>Lantuéjoul, C. (2002), ''Geostatistical simulation: Models and algorithms,'' 232 pp., Springer, Berlin.
</ref><ref>Journel, A. G. and Huijbregts, C.J. (1978) ''Mining Geostatistics,'' Academic Press. {{ISBN|0-12-391050-1}}
</ref><ref>Kitanidis, P.K.  (1997) ''Introduction to Geostatistics: Applications in Hydrogeology,'' Cambridge University Press.
</ref><ref>Wackernagel, H. (2003). ''Multivariate geostatistics,'' Third edition, Springer-Verlag, Berlin, 387 pp.
</ref><ref>Pyrcz, M. J. and Deutsch, C.V., (2014). ''Geostatistical Reservoir Modeling, 2nd Edition'', Oxford University Press, 448 pp.
</ref><ref>Tahmasebi, P., Hezarkhani, A., Sahimi, M., 2012, Multiple-point geostatistical modeling based on the cross-correlation functions, Computational Geosciences, 16(3):779-79742,  
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