Geostatistics evolved in mineral exploration and mining of minerals, ores, and coals, and is currently applied in disciplines such as petroleum geology, hydrogeology, hydrology, meteorology, oceanography, geochemistry, geography, forestry, environmental control, landscape ecology, agriculture (esp. in precision farming) etc. Geostatistics, as defined on this page, explains not only its applications within Geographic Information Systems but also the numerous applications of mathematical analysis on varied spatial datasets, the most prominent being the Digital Elevation Model, from which any number of analysis may be derived. Geostistics is also applied in varied branches of human geography, particularly those involving the spread of disease (epidemiology), the practice of commerce and military planning (logistics), and the development of efficient spatial networks. Petroleum geology is a term used to refer to the specific set of geological disciplines that are applied to the search for hydrocarbons (oil exploration). ... Hydrogeology (hydro- meaning water, and -geology meaning the study of rocks) is the part of hydrology that deals with the distribution and movement of groundwater in the soil and rocks of the Earths crust (commonly in aquifers). ... Water covers 70% of the Earths surface. ... Satellite image of Hurricane Hugo with a polar low visible at the top of the image. ... Thermohaline circulation Oceanography (from Ocean + Greek γράφειν = write), also called oceanology or marine science is the branch of physical geography that studies the Earths oceans and seas. ... The field of geochemistry involves study of the chemical composition of the Earth and other planets, chemical processes and reactions that govern the composition of rocks and soils, and the cycles of matter and energy that transport the Earths chemical components in time and space, and their interaction with... A decidous beech forest in Slovenia. ... Landscape ecology is a subdiscipline of ecology and geography that is the study of spatial variation in interested in the of elements in the landscape (such as fields, hedgerows, woodlots, rivers or towns) and how their distribution affects the distribution and flow of energy and individuals in the environment (which... Precision farming or precision agriculture is an agricultural concept relying on the existence of in-field variability. ... A geographic information system (GIS) is a system for managing data that has a spatial specialized form of an information system. ... A digital elevation model (DEM) is a representation of the topography of the Earth in digital format, that is, by coordinates and numerical descriptions of altitude. ... Epidemiology is the scientific study of factors affecting the health and illness of populations, and serves as the foundation and logic of interventions made in the interest of public health and preventive medicine. ... Look up Logistics in Wiktionary, the free dictionary. ... A spatial network is abla bla bla network of spatial elements, typically in urban or building space. ...

Geographers study how and why things differ from place to place, as well as how spatial patterns change through time. All well trained geographers begin with the question 'Where?', exploring how features are distributed on a physical or cultural landscape, observing spatial patterns and the variation of phenomena. Contemporary geographical analysis has shifted to 'Why?', determining why a specific spatial pattern exists, what spatial or ecological processes may have affected a pattern, and why such processes operate. Only by approaching the 'why?' questions can social scientists begin to appreciate the mechanisms of change, which are infinite in their complexity.

When we measure any phenomena, our observation methodology will dictate the accuracy of subsequent analysis; in geography, this issue is complicated by unique variables and spatial patterns such as geospatial topology. An interesting feature in geostatistics, every location displays some form of spatial pattern, whether in the form of the environment, climate, pollution, ubanization, human health, etc.; this is not to state that all variables are spatially dependent, simply that variables are incapable of measurement separate from their surroundings, such that there can be no perfect control population. Whether our study is concerned with the nature of traffic patterns in an urban core, or with the analysis of weather patterns over the Pacific, there are always variables which escape our measurement; this is determined directly by the scale and distribution of our data collection, or survey, and its methodology. Limitations in data collection make impossible the direct measure of continuous spatial data without inferring probabilities, some of these probablility functions are applied to create an interpolation surface, predicting unmeasured variables at innumerable locations. Geospatial topology is the application of mathematical typology on geospatial problems. ... In the mathematical subfield of numerical analysis, interpolation is a method of constructing new data points from a discrete set of known data points. ...

Role of statistics in geography

Statistical techniques and procedures are applied in all fields of academic research; wherever data are collected and summarized or wherever any numerical information is analyzed or research is conducted, statistics are needed for sound analysis and interpretation of results.

Geographers use statistics in numerous ways:

• To describe and summarize spatial data.
• To make generalizations concerning complex spatial patterns.
• To estimate the probability of outcomes for an event at a given location.
• To use samples of geographic data to infer characteristics for a larger set of geographic data (population).
• To determine if the magnitude or frequency of some phenomenon differs from one location to another.
• To learn whether an actual spatial pattern matches some expected pattern.

This is not a comprehensive list, and should not be interpreted as such.

Spatial data and descriptive statistics

There are several potential difficulties associated with the analysis of spatial data, among these are boundary delineation, modifiable areal units, and the level of spatial aggregation or scale. In each of these cases, the absolute descriptive statistics of an area - the mean, median, mode, standard deviation, and variation - are changed through the manipulation of these spatial problems.

Boundary delineation

The location of a study area boundary and the positioning of internal boundaries affect various descriptive statistics. With respect to measures such as the mean or standard deviation, the study area size alone may have large implications; consider a study of per capita income within a city, if confined to the inner city, income levels are likely to be lower because of a less affluent population, if expanded to include the suburbs or surrounding communities, income levels will become greater with the influence of homeowner populations. Because of this problem, absolute descriptive statistics such as the mean, standard deviation, and variance should be evaluated comparatively only in relation to a particular study area. In the determination of internal boundaries this is also true, as these statistics may only have valid interpretations for the area and subarea configuration over which they are calculated.

Modifiable areal units

In many cases the subdivision of spatial data has already been determined, this is evident in demographic datasets, as the available information will be grouped into their respective counties or municipalities. For this type of data, analysts must use the same county or municipal boundaries delineated in the collected data for their subsequent analysis. When alternate boundaries are possible, an analyst must take into account that any new subdivision model may create different results.

Spatial aggregation/scale problem

Socio-economic data may be available at a variety of scales, for example: municipalities, regional districts, census tracts, enumeration districts, or at the provincial/state level. When this data is aggregated at different scales, the resulting descriptive statistics may exhibit variations, either in a systematic, predictable way, or in a more uncertain fashion. If we are observing economic data, we may notice a distinct reduction in manufacturing productivity for a country (the USA) over a certain period; since this is a general model, individual states may experience these effects differently. The result of this aggregation is that the standard deviation of the data in question is increased due to the variability among states.

Descriptive spatial statistics

For summarizing point pattern analysis, a set of descriptive spatial statistics has been developed that are areal equivalents to nonspatial measures. Since geographers are particularly concerned with the analysis of locational data, these descriptive spatial statistics (geostatistics) are often applied to summarize point patterns and to describe the degree of spatial variability of some phenomena.

Spatial measures of central tendency

Mean center

See also Mean center of U.S. population The mean center of U.S. population is determined by the United States Census Bureau after tabulating the results of each census. ...

The mean is an important measure of central tendency, which when extended to a set of points, located on a Cartesian coordinate system, the average location, or mean center, can be determined. Fig. ...

Weighted mean center

The weighted mean center is analogous to frequencies in the calculation of grouped statistics, such as the weighted mean. A point may represent a retail outlet, while its frequency will represent the volume of sales within the particular store.

Median center or Euclidean center

See also Manhattan distance Taxicab geometry, considered by Hermann Minkowski in the 19th century, is a form of geometry in which the usual metric of Euclidean geometry is replaced by a new metric in which the distance between two points is the sum of the (absolute) differences of their coordinates. ...

Spatial measures of dispersion

Standard distance

Relative distance

Topography

See main article Topography It has been suggested that Geomorphometry be merged into this article or section. ...

Topology

See main article Topology A Möbius strip, a surface with only one side and one edge; such shapes are an object of study in topology. ...

The Seven Bridges of Königsberg, one of the most famous problems in topology

The motivating insight behind topology is that some geometric problems depend not on the exact shape of the objects involved, but rather on the "way they are connected together". One of the first papers in topology was the demonstration, by Leonhard Euler, that it was impossible to find a route through the town of Königsberg (now Kaliningrad) that would cross each of its seven bridges exactly once. This result did not depend on the lengths of the bridges, nor on their distance from one another, but only on connectivity properties: which bridges are connected to which islands or riverbanks. This problem, the Seven Bridges of Königsberg, is now a famous problem in introductory mathematics, and led to the branch of mathematics known as graph theory. Image File history File links The seven bridges of Konigsberg - old map with bridges highlighted Public Domain Image, modified by me, released under GPL. File history Legend: (cur) = this is the current file, (del) = delete this old version, (rev) = revert to this old version. ... Image File history File links The seven bridges of Konigsberg - old map with bridges highlighted Public Domain Image, modified by me, released under GPL. File history Legend: (cur) = this is the current file, (del) = delete this old version, (rev) = revert to this old version. ... Euler redirects here. ... Government Russia District Subdivision Russia Northwestern Federal District Kaliningrad Oblast Mayor Yuri Savenko (2005) Geographical characteristics Area  - City 215. ... Map of Königsberg in Eulers time showing the actual layout of the seven bridges, highlighting the river Pregel and the bridges. ... A pictorial representation of a graph In mathematics and computer science, graph theory is the study of graphs, mathematical structures used to model pairwise relations between objects from a certain collection. ...

Topology rules

See main article Geospatial topology Geospatial topology is the application of mathematical typology on geospatial problems. ...

Topology rules are particularly important within GIS, and are used for a variety of correction and analytical procedures. The primary shapes in GIS are the point, line, and polygon, each of which implies different spatial characteristics; for instance, the only shape which has a distinguishable inside and outside is the polygon. Principles of connectivity associated with topology lead to applications in hydrology, urban planning, and logistics, as well as other fields; as such, topological analyses offer unique modelling capabilities, defining the vector nature of topological features and correcting spatial data errors from digitizing. A geographic information system (GIS) is a system for managing data that has a spatial specialized form of an information system. ... Point can refer to: Look up Point in Wiktionary, the free dictionary // Mathematics In mathematics: Point (geometry), an entity that has a location in space but no extent Fixed point (mathematics), a point that is mapped to itself by a mathematical function Point at infinity Point group Point charge, an... The word line derives from the Latin lingui, meaning flax plant from which linen is produced; at one time, a stretched linen thread was the most reliable way to determine a straight line. ... Look up Polygon in Wiktionary, the free dictionary. ... Water covers 70% of the Earths surface. ... Urban planning is concerned with the ordering and design of settlements, from the smallest towns to the worlds largest cities. ... Look up Logistics in Wiktionary, the free dictionary. ...

Sampling methodology

Statistical sampling

Geospatial sampling

Controversy

Professor D G Krige worked at the Witwatersrand gold reef complex in South Africa in the early 1950s when he discovered that two or more gold assays, determined in samples selected at positions with different coordinates in a finite sample space, define an infinite set of distance-weighted average gold grades. Professor Dr G Matheron was so taken with Krige’s aptitude in augmenting sparse sets of measured data with infinite sets of distance-weighted average gold grades that he conferred on Krige the ubiquitous krige eponym. Krige, Matheron and his following did not know that one-to-one correspondence between distance-weighted averages and variances is inviolable in mathematical statistics. As a result, pioneering geostatisticians replaced the genuine variance of the SINGLE distance-weighted average gold grade with the pseudo variance of a SET of degrees-of-freedom-and-variance-deprived functionally dependent distance-weighted averages gold grades. In time, kriging variances and kriging covariances of sets of kriged estimates became the cornerstones of geostatistics, an invalid variant of mathematical statistics that is applied to sparse data in small and large sample spaces alike.

One might suggest that spatial dependence should not be assumed to exist between widely spaced measured values such as boreholes in ore deposits but that it may well exist between closely spaced measured values. Spatial dependence between ore zones of ordered boreholes in a profile ought to be verified by applying analysis of variance to the variance of the set and the first variance term for the ordered set (see Ref 20). Each set of measured values of a stochastic variable in a sample space has a variance (see Ref 25). Variances of sets of functionally dependent kriged estimates are invalid measures for variability, precision and risk, particularly given that one-to-one correspondence between distance-weighted averages and variances is crucial in mathematical statistics (see Ref 25). A semivariance, of a subset of an infinite set does not replace the degrees of freedom which are removed from the functionally dependent distance weighted average, particularly for those points which are estimated using this function. Kriging assumes that ore concentrations are modelled by these autocorrelations; and inappropriate use makes the method susceptible to erroneous reading of results.^[1] In mathematical statistics, spatial dependence is a measure for the degree of associative dependence between independently measured values in a temporally or in situ ordered set, determined at different locations in a sample space or a sampling unit. ... In spatial statistics, semivariance can be described by where z is a data value at a particular location, h is the distance between data values, and n(h) counts the number of pairs of data values we are given, spaced a distance of h apart. ...

Practitioners who question the applicability of stochastic models to geological situations include Phillip and Watson.^[2]

Other practitioners advocate statistical tests to verify the spatial dependence of the data.^[3]

Related software

• gslib is a set of fortran 77 routines (open source) implementing most of the classical geostatistics estimation and simulation algorithms
• sgems is a cross-platform (windows, unix), open-source software that implements most of the classical geostatistics algorithms (kriging, Gaussian and indicator simulation, etc) as well as new developments (multiple-points geostatistics). It also provides an interactive 3D visualization and offers the scripting capabilities of python.
• gstat is an open source computer code for multivariable geostatistical modelling, prediction and simulation. The gstat functionaly is also available as an S extension, either as R package or S-Plus library.

• ISATIS,is a software today used by more than 250 companies worldwide (amongst them, we find numerous leading multinationals), is the fruit of 40 years of experience in the industrial application of geostatistics and applied geostatistical research.
  

ISATIS offers an exclusive range of proven methodologies providing you with the best expected results. While bridging the gap between industry and research, a close partnership with the Centre de Géostatistique in Fontainebleau (at the Ecole des Mines de Paris) - where, during the fifties, professor Georges Matheron gave birth to the concept of this new science - ensures that our different products remain at the cutting edge of technology.

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. Chilès, J.P., Delfiner, P. 1999. Geostatistics: modelling spatial uncertainty, Wiley Series in Probability and Mathematical Statistics, 695 pp.
8. 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/
9. Deutsch, C.V., 2002. Geostatistical Reservoir Modeling, Oxford University Press, 384 pp., http://www.statios.com/WinGslib/index.html
10. Isaaks, E.H., Srivastava R.M.: Applied Geostatistics. 1989.
11. ISO/DIS 11648-1 Statistical aspects of sampling from bulk materials-Part1: General principles
12. Journel, A G and Huijbregts, 1978, Mining Geostatistics, Academic Press
13. Kitanidis, P.K.: Introduction to Geostatistics: Applications in Hydrogeology, Cambridge University Press. 1997.
14. Lantuéjoul, C. 2002. Geostatistical simulation: models and algorithms. Springer, 256 pp.
15. Lipschutz, S, 1968, Theory and Problems of Probability, McCraw-Hill Book Company, New York.
16. Matheron, G. 1962. Traité de géostatistique appliquée. Tome 1, Editions Technip, Paris, 334 pp.
17. Matheron, G. 1989. Estimating and choosing, Springer-Verlag, Berlin.
18. McGrew, J. Chapman, & Monroe, Charles B., 2000. An introduction to statistical problem solving in geography, second edition, McGraw-Hill, New York.
19. Merks, J W, 1992, Geostatistics or voodoo science, The Northern Miner, May 18
20. Merks, J W, Abuse of statistics, CIM Bulletin, January 1993, Vol 86, No 966
21. Myers, Donald E.; "What Is Geostatics?
22. Philip, G M and Watson, D F, 1986, Matheronian Geostatistics; Quo Vadis?, Mathematical Geology, Vol 18, No 1
23. Sharov, A: Quantitative Population Ecology, 1996, http://www.ento.vt.edu/~sharov/PopEcol/popecol.html
24. 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
25. Volk, W, 1980, Applied Statistics for Engineers, Krieger Publishing Company, Huntington, New York.
26. Wackernagel, H. 2003. Multivariate geostatistics, Third edition, Springer-Verlag, Berlin, 387 pp.
27. Youden, W J, 1951, Statistical Methods for Chemists: John Wiley & Sons, New York.

See also

• statistics
• geology
• geographic information system (GIS)
• remote sensing
• kriging
• pedometrics

A graph of a bell curve in a normal distribution showing statistics used in educational assessment, comparing various grading methods. ... World geologic provinces Oceanic crust  0-20 Ma  20-65 Ma  >65 Ma Geologic provinces  Shield  Platform  Orogen  Basin  Large igneous province  Extended crust Geology (from Greek γη- (ge-, the earth) and λογος (logos, word, reason))[1] is the science and study of the solid matter of a celestial body, its composition... A geographic information system (GIS) is a system for creating, storing, analyzing and managing spatial data and associated attributes. ... Synthetic aperture radar image of Death Valley colored using polarimetry In the broadest sense, remote sensing is the measurement or acquisition of information of an object or phenomenon, by a recording device that is not in physical or intimate contact with the object. ... Kriging is a regression technique used in geostatistics to approximate or interpolate data. ... Pedometrics is defined as the application of mathematical and statistical methods for the study of the distribution and genesis of soils. ...

External links

• Kriging link, contains explanations of variance in geostats
• Arizona university geostats page
• A resource on the internet about geostatistics and spatial statistics
• A website critical of assuming spatial dependence rather than verifying it by analysis of variance.