Geostatistics

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Geostatistics applies the theories of stochastic processes and statistical inference to geographic phenomena. It was traditionally used in geo-sciences. Methods of geostatistics are used in petroleum geology, hydrogeology, hydrology, meteorology, oceanography, geochemistry, geography, forestry, environmental control, landscape ecology, agriculture (esp. in precision farming) etc. Widely practiced within Geographic Information Systems, geostatistics are 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. Applications also exist 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.

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.

Contents

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

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.

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

Spatial Measures of Dispersion

Standard Distance

Relative Distance

Topography

See main article Topography

Topology

See main article 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.

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

Topology Rules

Sampling Methodology

Statistical Sampling

Geospatial Sampling

History

The discipline of geostatistics emerged from the application of a questionable variant of mathematical statistics to sampling methods in geology, hydrology, and other earth sciences.Myers. Dr Herbert Sichel, Daniel G. Krige, and Georges Matheron have pioneered geostatistics since the 1950s. Professor D G Krige discovered during his work at the Witwatersrand complex in South Africa in the early 1950s that two or more gold grades, 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 G Matheron in the 1960s prefixed “geo” to “statistics” and created the term geostatistics because, in his own words, geologists stress structure and statisticians stress randomness. When Matheron found out about Krige’s discovery, he himself conferred on Krige the ubiquitous krige eponym. In time, the distance-weighted average metamorphosed into a kriged estimate (see Ref 2, 3, 9).

In those early days, Krige, Matheron and his following were unaware that each distance-weighted average has its own variance in mathematical statistics. In geostatistics, however, the true variance of the single distance-weighted average was replaced with the pseudo kriging variance of a set of kriged estimates formerly known as distance-weighted averages. Geostatistics is an invalid variant of mathematical statistics because the variance of a set of functionally dependent values violates the requirement of functional independence and ignores the concept of degrees of freedom.

Controversy

It is claimed that spatial dependence should not be assumed to exist between measured values, instead this should be verified by applying analysis of variance to the variance of the set and the first variance term for the ordered set.^{[[Citing sources citation needed]]} Each measured value of a stochastic variable in a sample space has its own variance.^{[[Citing sources citation needed]]} Variances of sets of functionally dependent kriged estimates are inaccurate, particularly given that one-to-one correspondence between distance-weighted averages and variances is crucial in mathematical statistics.^{[[Citing sources citation needed]]} 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.Cressie.

Practitioners who question the applicability of stochastic models to geological situations include Phillip and Watson.Philip.

Other practitioners advocate statistical tests to verify the spatial dependence of the data.Fortin, Ullah and Schabenberger.

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.

Notes

References

Armstrong, M and Champigny, N, 1988, A Study on Kriging Small Blocks, CIM Bulletin, Vol 82, No 923
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
Chilès, J.P., Delfiner, P. 1999. Geostatistics: modelling spatial uncertainty, Wiley Series in Probability and Mathematical Statistics, 695 pp.
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/
Deutsch, C.V., 2002. Geostatistical Reservoir Modeling, Oxford University Press, 384 pp., http://www.statios.com/WinGslib/index.html
Isaaks, E.H., Srivastava R.M.: Applied Geostatistics. 1989.
Journel, A G and Huijbregts, 1978, Mining Geostatistics, Academic Press
Kitanidis, P.K.: Introduction to Geostatistics: Applications in Hydrogeology, Cambridge University Press. 1997.
Lantuéjoul, C. 2002. Geostatistical simulation: models and algorithms. Springer, 256 pp.
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], Abuse of statistics, CIM Bulletin, Jan 1993.
Myers, Donald E.; ["What Is Geostatics?]
Philip, G M and Watson, D F, 1986, Matheronian Geostatistics; Quo Vadis?, Mathematical Geology, Vol 18, No 1
Sharov, A: Quantitative Population Ecology, 1996, 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, http://www.geovista.psu.edu/sites/geocomp99/Gc99/044/gc_044.htm
Volk, W, 1980, Applied Statistics for Engineers, Krieger Publishing Company, Huntington, New York.
Wackernagel, H. 2003. Multivariate geostatistics, Third edition, Springer-Verlag, Berlin, 387 pp.
Youden, W J, 1951, Statistical Methods for Chemists: John Wiley & Sons, New York.

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