Geostatistics is a branch of
statistics focusing on spatial or
spatiotemporal dataset A data set (or dataset) is a collection of data. In the case of tabular data, a data set corresponds to one or more database tables, where every column of a table represents a particular variable, and each row corresponds to a given record of the ...
s. Developed originally to predict
probability distributions of
ore grades for
mining
Mining is the extraction of valuable minerals or other geological materials from the Earth, usually from an ore body, lode, vein, seam, reef, or placer deposit. The exploitation of these deposits for raw material is based on the economic ...
operations, it is currently applied in diverse disciplines including
petroleum geology,
hydrogeology,
hydrology
Hydrology () is the scientific study of the movement, distribution, and management of water on Earth and other planets, including the water cycle, water resources, and environmental watershed sustainability. A practitioner of hydrology is call ...
,
meteorology
Meteorology is a branch of the atmospheric sciences (which include atmospheric chemistry and physics) with a major focus on weather forecasting. The study of meteorology dates back millennia, though significant progress in meteorology did no ...
,
oceanography,
geochemistry,
geometallurgy,
geography
Geography (from Greek: , ''geographia''. Combination of Greek words ‘Geo’ (The Earth) and ‘Graphien’ (to describe), literally "earth description") is a field of science devoted to the study of the lands, features, inhabitants, an ...
,
forestry
Forestry is the science and craft of creating, managing, planting, using, conserving and repairing forests, woodlands, and associated resources for human and environmental benefits. Forestry is practiced in plantations and natural stands. ...
,
environmental control,
landscape ecology
Landscape ecology is the science of studying and improving relationships between ecological processes in the environment and particular ecosystems. This is done within a variety of landscape scales, development spatial patterns, and organizatio ...
,
soil science
Soil science is the study of soil as a natural resource on the surface of the Earth including soil formation, classification and mapping; physical, chemical, biological, and fertility properties of soils; and these properties in relation to th ...
, and
agriculture
Agriculture or farming is the practice of cultivating plants and livestock. Agriculture was the key development in the rise of sedentary human civilization, whereby farming of domesticated species created food surpluses that enabled people t ...
(esp. in
precision farming). Geostatistics is applied in varied branches of
geography
Geography (from Greek: , ''geographia''. Combination of Greek words ‘Geo’ (The Earth) and ‘Graphien’ (to describe), literally "earth description") is a field of science devoted to the study of the lands, features, inhabitants, an ...
, particularly those involving the spread of diseases (
epidemiology
Epidemiology is the study and analysis of the distribution (who, when, and where), patterns and determinants of health and disease conditions in a defined population.
It is a cornerstone of public health, and shapes policy decisions and evide ...
), the practice of commerce and military planning (
logistics
Logistics is generally the detailed organization and implementation of a complex operation. In a general business sense, logistics manages the flow of goods between the point of origin and the point of consumption to meet the requirements of ...
), 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
Inverse distance weighting (IDW) is a type of deterministic method for multivariate interpolation with a known scattered set of points. The assigned values to unknown points are calculated with a weighted average of the values available at the kn ...
,
bilinear interpolation
In mathematics, bilinear interpolation is a method for interpolating functions of two variables (e.g., ''x'' and ''y'') using repeated linear interpolation. It is usually applied to functions sampled on a 2D rectilinear grid, though it can be ...
and
nearest-neighbor interpolation, were already well known before geostatistics.
[Isaaks, E. H. and Srivastava, R. M. (1989), ''An Introduction to Applied Geostatistics,'' Oxford University Press, New York, USA.] Geostatistics goes beyond the interpolation problem by considering the studied phenomenon at unknown locations as a set of correlated random variables.
Let be the value of the variable of interest at a certain location . This value is unknown (e.g. temperature, rainfall,
piezometric level, geological facies, etc.). Although there exists a value at location 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 is not complete, but defined by a
cumulative distribution function (CDF) that depends on certain information that is known about the value :
:
Typically, if the value of is known at locations close to (or in the
neighborhood of ) one can constrain the CDF of by this neighborhood: if a high spatial continuity is assumed, can only have values similar to the ones found in the neighborhood. Conversely, in the absence of spatial continuity 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 is a
stationary process
In mathematics and statistics, a stationary process (or a strict/strictly stationary process or strong/strongly stationary process) is a stochastic process whose unconditional joint probability distribution does not change when shifted in time. Con ...
. 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
Estimation (or estimating) is the process of finding an estimate or approximation, which is a value that is usable for some purpose even if input data may be incomplete, uncertain, or unstable. The value is nonetheless usable because it is der ...
the value for , typically by the
expectation, the
median or the
mode of the CDF . This is usually denoted as an estimation problem.
#
Sampling from the entire probability density function by actually considering each possible outcome of it at each location. This is generally done by creating several alternative maps of , called realizations. Consider a domain discretized in grid nodes (or pixels). Each realization is a sample of the complete -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 solving
inverse problem
An inverse problem in science is the process of calculating from a set of observations the causal factors that produced them: for example, calculating an image in X-ray computed tomography, source reconstruction in acoustics, or calculating the ...
s.
A number of methods exist for both geostatistical estimation and multiple realizations approaches. Several reference books provide a comprehensive overview of the discipline.
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. 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
[Banerjee, Sudipto. High-Dimensional Bayesian Geostatistics. Bayesian Anal. 12 (2017), no. 2, 583--614. . https://projecteuclid.org/euclid.ba/1494921642]
Simulation
* Aggregation
* Dissagregation
*
Turning bands
*
Cholesky decomposition
In linear algebra, the Cholesky decomposition or Cholesky factorization (pronounced ) is a decomposition of a Hermitian, positive-definite matrix into the product of a lower triangular matrix and its conjugate transpose, which is useful for effici ...
* 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
A cellular automaton (pl. cellular automata, abbrev. CA) is a discrete model of computation studied in automata theory. Cellular automata are also called cellular spaces, tessellation automata, homogeneous structures, cellular structures, tessel ...
* Multiple-Point Geostatistics
Definitions and tools
*
Regionalized variable theory
*
Covariance function In probability theory and statistics, the covariance function describes how much two random variables change together (their ''covariance'') with varying spatial or temporal separation. For a random field or stochastic process ''Z''(''x'') on a ...
*
Semi-variance
*
Variogram
*
Kriging
In statistics, originally in geostatistics, kriging or Kriging, also known as Gaussian process regression, is a method of interpolation based on Gaussian process governed by prior covariances. Under suitable assumptions of the prior, kriging giv ...
*
Range (geostatistics)
*
Sill (geostatistics)
*
Nugget effect
*
Training image
Related academic journals
Water Resources ResearchAdvances in Water ResourcesGround Water
*
Mathematical Geosciences
Computers & GeosciencesComputational GeosciencesJ. Soil Science Society of America
EnvironmetricsRemote Sensing of the Environment*
Stochastic Environmental Research and Risk Assessment
Scientific organisations related to geostatistics
European Forum for Geography and Statistics(EFGS; formerly the ''European Forum for Geostatistics'')
GeoEnviapromotes the use of geostatistical methods in environmental applications
*
International Association for Mathematical Geosciences
See also
*
Arbia's law of geography
*
Concepts and Techniques in Modern Geography
*
Multivariate interpolation
In numerical analysis, multivariate interpolation is interpolation on functions of more than one variable; when the variates are spatial coordinates, it is also known as spatial interpolation.
The function to be interpolated is known at given poi ...
*
Spline interpolation
In the mathematical field of numerical analysis, spline interpolation is a form of interpolation where the interpolant is a special type of piecewise polynomial called a spline. That is, instead of fitting a single, high-degree polynomial to all ...
*
Geodemographic segmentation
*
Geodesy
*
Geographic Information Science
*
Geographic Information Systems
*
Geomatics
*
Remote sensing
Remote sensing is the acquisition of information about an object or phenomenon without making physical contact with the object, in contrast to in situ or on-site observation. The term is applied especially to acquiring information about Eart ...
*
Pedometrics
*
Time geography Time geography or time-space geography is an evolving transdisciplinary perspective on spatial and temporal processes and events such as social interaction, ecological interaction, social and environmental change, and biographies of individuals. T ...
*
Tobler's first law of geography The First Law of Geography, according to Waldo Tobler, is "everything is related to everything else, but near things are more related than distant things." This first law is the foundation of the fundamental concepts of spatial dependence and spati ...
*
Tobler's second law of geography The second law of geography, according to Waldo Tobler, is "the phenomenon external to a geographic area of interest affects what goes on inside."
Background
Tobler's second law of geography, "the phenomenon external to a geographic area of inte ...
Notes
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
# (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.
#
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
GeoENViapromotes the use of geostatistical methods in environmental applications, and organizes bi-annual conferences.
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 geostatisticshttps://web.archive.org/web/20040326205028/http://geostatscam.com/Is the site of Jan W. Merks, who claims that geostatistics is "voodoo science" and a "scientific fraud"
It is a group for exchanging of ideas and discussion on multiple point geostatistics (MPS).
{{Authority control
Geostatistics,