Geostatistics is a branch of
statistics
Statistics (from German language, German: ''wikt:Statistik#German, Statistik'', "description of a State (polity), state, a country") is the discipline that concerns the collection, organization, analysis, interpretation, and presentation of ...
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 distribution
In probability theory and statistics, a probability distribution is the mathematical function that gives the probabilities of occurrence of different possible outcomes for an experiment. It is a mathematical description of a random phenomenon i ...
s of
ore grade
Ore is natural Rock (geology), rock or sediment that contains one or more valuable minerals, typically containing metals, that can be mined, treated and sold at a profit.Encyclopædia Britannica. "Ore". Encyclopædia Britannica Online. Ret ...
s 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 via ...
operations, it is currently applied in diverse disciplines including
petroleum geology
Petroleum geology is the study of origin, occurrence, movement, accumulation, and exploration of hydrocarbon fuels. It refers to the specific set of geological disciplines that are applied to the search for hydrocarbons (oil exploration).
Sedime ...
,
hydrogeology
Hydrogeology (''hydro-'' meaning water, and ''-geology'' meaning the study of the Earth) is the area of geology that deals with the distribution and movement of groundwater in the soil and rocks of the Earth's crust (commonly in aquif ...
,
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 calle ...
,
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 not ...
,
oceanography
Oceanography (), also known as oceanology and ocean science, is the scientific study of the oceans. It is an Earth science, which covers a wide range of topics, including ecosystem dynamics; ocean currents, waves, and geophysical fluid dynamic ...
,
geochemistry
Geochemistry is the science that uses the tools and principles of chemistry to explain the mechanisms behind major geological systems such as the Earth's crust and its oceans. The realm of geochemistry extends beyond the Earth, encompassing the e ...
,
geometallurgy Geometallurgy relates to the practice of combining geology or geostatistics with metallurgy, or, more specifically, extractive metallurgy, to create a spatially or geologically based predictive model for mineral processing plants. It is used in t ...
,
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, and ...
,
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. Th ...
,
environmental control
Heating, ventilation, and air conditioning (HVAC) is the use of various technologies to control the temperature, humidity, and purity of the air in an enclosed space. Its goal is to provide thermal comfort and acceptable indoor air quality. HV ...
,
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 organizati ...
,
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 to ...
(esp. in
precision farming
Precision agriculture (PA) is a farming management strategy based on observing, measuring and responding to temporal and spatial variability to improve agricultural production sustainability. It is used in both crop and livestock production. P ...
). 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, and ...
, 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 evidenc ...
), 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 network
A spatial network (sometimes also geometric graph) is a graph in which the vertices or edges are ''spatial elements'' associated with geometric objects, i.e., the nodes are located in a space equipped with a certain metric.M. Barthelemy, "M ...
s. Geostatistical algorithms are incorporated in many places, including
geographic information systems
A geographic information system (GIS) is a type of database containing geographic data (that is, descriptions of phenomena for which location is relevant), combined with software tools for managing, analyzing, and visualizing those data. In a br ...
(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
A random variable (also called random quantity, aleatory variable, or stochastic variable) is a mathematical formalization of a quantity or object which depends on random events. It is a mapping or a function from possible outcomes (e.g., the po ...
) 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
Nearest-neighbor interpolation (also known as proximal interpolation or, in some contexts, point sampling) is a simple method of multivariate interpolation in one or more dimensions.
Interpolation is the problem of approximating the value of ...
, 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
In probability theory and statistics, the cumulative distribution function (CDF) of a real-valued random variable X, or just distribution function of X, evaluated at x, is the probability that X will take a value less than or equal to x.
Ev ...
(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
A neighbourhood (British English, Irish English, Australian English and Canadian English) or neighborhood (American English; see spelling differences) is a geographically localised community within a larger city, town, suburb or rural area, ...
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
In spatial statistics the theoretical variogram 2\gamma(\mathbf_1,\mathbf_2) is a function describing the degree of spatial dependence of a spatial random field or stochastic process Z(\mathbf). The semivariogram \gamma(\mathbf_1,\mathbf_2) is ...
-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
In statistics and probability theory, the median is the value separating the higher half from the lower half of a data sample, a population, or a probability distribution. For a data set, it may be thought of as "the middle" value. The basic fe ...
or the
mode
Mode ( la, modus meaning "manner, tune, measure, due measure, rhythm, melody") may refer to:
Arts and entertainment
* '' MO''D''E (magazine)'', a defunct U.S. women's fashion magazine
* ''Mode'' magazine, a fictional fashion magazine which is ...
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
In probability theory and statistics, a Gaussian process is a stochastic process (a collection of random variables indexed by time or space), such that every finite collection of those random variables has a multivariate normal distribution, i.e. e ...
, 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
A Markov chain or Markov process is a stochastic model describing a sequence of possible events in which the probability of each event depends only on the state attained in the previous event. Informally, this may be thought of as, "What happe ...
*
Markov chain geostatistics Markov chain geostatistics uses Markov chain spatial models, simulation algorithms and associated spatial correlation measures (e.g., transiogram) based on the Markov chain random field theory, which extends a single Markov chain into a multi-d ...
Markov mesh models*
Support vector machine
In machine learning, support vector machines (SVMs, also support vector networks) are supervised learning models with associated learning algorithms that analyze data for classification and regression analysis. Developed at AT&T Bell Laboratorie ...
*
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
Regionalized variable theory (RVT) is a geostatistical method used for interpolation in space.
The concept of the theory is that interpolation from points in space should not be based on a smooth continuous object. It should be, however, based on ...
*
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 doma ...
*
Semi-variance
*
Variogram
In spatial statistics the theoretical variogram 2\gamma(\mathbf_1,\mathbf_2) is a function describing the degree of spatial dependence of a spatial random field or stochastic process Z(\mathbf). The semivariogram \gamma(\mathbf_1,\mathbf_2) is ...
*
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)
In spatial statistics the theoretical variogram 2\gamma(\mathbf_1,\mathbf_2) is a function describing the degree of spatial dependence of a spatial random field or stochastic process Z(\mathbf). The semivariogram \gamma(\mathbf_1,\mathbf_2) is hal ...
*
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
Stochastic (, ) refers to the property of being well described by a random probability distribution. Although stochasticity and randomness are distinct in that the former refers to a modeling approach and the latter refers to phenomena themselv ...
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
The International Association for Mathematical Geosciences (IAMG) is a nonprofit organization of geoscientists. It aims to promote international cooperation in the application and use of mathematics in geological research and technology. IAMG's act ...
See also
*
Arbia's law of geography
Arbia’s law of geography states, "Everything is related to everything else, but things observed at a coarse spatial resolution are more related than things observed at a finer resolution." Originally proposed as the 2nd law of geography, this is ...
*
Concepts and Techniques in Modern Geography
''Concepts and Techniques in Modern Geography'', abbreviated CATMOG, is a series of 59 short publications, each focused on an individual method or theory in geography.
Background and impact
''Concepts and Techniques in Modern Geography'' were p ...
*
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
In marketing, geodemographic segmentation is a multivariate statistical classification technique for discovering whether the individuals of a population fall into different groups by making quantitative comparisons of multiple characteristics with ...
*
Geodesy
Geodesy ( ) is the Earth science of accurately measuring and understanding Earth's figure (geometric shape and size), orientation in space, and gravity. The field also incorporates studies of how these properties change over time and equivale ...
*
Geographic Information Science
*
Geographic Information Systems
A geographic information system (GIS) is a type of database containing geographic data (that is, descriptions of phenomena for which location is relevant), combined with software tools for managing, analyzing, and visualizing those data. In a br ...
*
Geomatics
Geomatics is defined in the ISO/TC 211 series of standards as the "discipline concerned with the collection, distribution, storage, analysis, processing, presentation of geographic data or geographic information". Under another definition, it ...
*
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 Earth ...
*
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
''The Northern Miner'' is a weekly trade journal (formerly part of the Hollinger group) that reports on the mining industry. ''The Northern Miner'' began publication in Cobalt, Ontario, Canada, in 1915, and has since moved publication to Toront ...
, 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,