Arikah Map

Geostatistics

This article or section is in need of attention from an expert on the subject.
Please help recruit one, or improve this page yourself if you can. See discussion page for details.

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.

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:

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 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
Geostatistics:The Seven Bridges of Königsberg, one of the most famous problems in topology
Enlarge
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.

Topology rules

See main article Geospatial topology

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.

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]

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]

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

Categories


Pages needing expert attention | Geostatistics

Find

Find

Find