Geostatistics is an application of the theory of random functions for estimating natural phenomena. ‘Geostatistics offers a way of describing the spatial continuity of natural phenomena and provides adaptations of classical regression techniques to take advantage of this continuity’ (Isaaks and Srivastava 1989). The data that we have are never complete; we have either the wrong kind or insufficient or partial coverage. Naturally, we seek ways to predict the values between, or to extrapolate beyond, the limits of our data.
The basic concept of geostatistics is that of scales of spatial variation. Data which are spatially independent show the same variability regardless of the location of data points. However, spatial data in most cases are not spatially independent. Data values which are close spatially show less variability than data values which are farther away from each other. A fundamental concept in geography is that nearby entities often share more similarities than entities which are far apart (Miller 2004). This idea is often labelled ‘Tobler’s first law of geography’ and may be summarised as ‘everything is related to everything else, but near things are more related than distant things’ (Tobler 1970). The exact nature of this pattern varies from data set to data set; each set of data has its own unique function of variability and distance between data points. This variability is generally computed as a function called semivariance.
In one respect geostatistics might be viewed as simply a methodology for interpolating data on an irregular pattern but this is too simplistic. A number of interpolation methods/algorithms were already well known when geostatistics began to be known; for example, inverse distance weighting (IDW) and trend surface analysis as well as the much simpler nearest neighbor algorithm. Interpolation techniques use sample points to produce surfaces of the phenomena of interest. The interpolation techniques are divided into two main types: deterministic and geostatistical methods.
Deterministic interpolation techniques create surfaces from measured points, based on either the extent of similarity (e.g., IDW) or the degree of smoothing (e.g., radial basis functions). These techniques do not use a model of random spatial processes. A deterministic interpolation can either force the resulting surface to pass through the data values or not. An interpolation technique that predicts a value that is identical to the measured value at a sampled location is known as an exact interpolator (e.g., IDW and radial basis functions). An inexact interpolator (e.g., global and local polynomials) predicts a value that is different from the measured value. The latter can be used to avoid sharp peaks or troughs in the output surface.
Geostatistics assume that at least some of the spatial variation of natural phenomena can be modeled by random processes with spatial autocorrelation. Geostatistical techniques produce not only prediction surfaces but also error or uncertainty surfaces, giving us an indication of how good the predictions are. Geostatistical interpolators exhibit probabilistic behaviour, i.e., it can be considered that for one known condition there are many possible outcomes, some of which will be more likely than others.
Many methods are associated with geostatistics, but they generally fall in the kriging family, e.g., ordinary, simple, universal, probability, indicator, and disjunctive kriging, along with their counterparts in cokriging. Kriging is a method of estimation based on the trend and variability from the trend. Variability, in this context, refers to random errors about the trend or mean. In this context, ‘error’ does not imply a mistake but a fluctuation (error) about the trend is unknown and is not systematic; the fluctuation could be positive or negative. Kriging may be considered exact (or smoothed) or inexact. Kriging incorporates the principles of probability and prediction, and like the IDW, is a weighted average technique except that a surface produced by kriging may exceed the value range of the sample points while still not actually passing through them. Various statistical models can be chosen to produce map outputs (or surfaces) from the kriging process; such as, interpolated surface (the prediction), the standard prediction errors (variance), probability (that the prediction exceeds a threshold) and quantile (for any given probability) (Liu and Mason 2009). One may refer Isaaks and Srivastava (1989) for an introductory text on Geostatistics.
References
Isaaks, E.H. and R.M. Srivastava 1989, An Introduction to Applied Geostatistics, Oxford University Press, New York, 561 pp.
Miller, H.J. 2004, ‘Tobler’s first law and spatial analysis’, Annals of the Association of American Geographers 94: 284–289.
Tobler, W. 1970, ‘A computer movie simulating urban growth in the Detroit region’, Economic Geography 46: 234–240.
Liu, J.G. and P.J. Mason 2009, Essential Image Processing and GIS for Remote Sensing, Wiley-Blackwell, New York, 450 pp.
The basic concept of geostatistics is that of scales of spatial variation. Data which are spatially independent show the same variability regardless of the location of data points. However, spatial data in most cases are not spatially independent. Data values which are close spatially show less variability than data values which are farther away from each other. A fundamental concept in geography is that nearby entities often share more similarities than entities which are far apart (Miller 2004). This idea is often labelled ‘Tobler’s first law of geography’ and may be summarised as ‘everything is related to everything else, but near things are more related than distant things’ (Tobler 1970). The exact nature of this pattern varies from data set to data set; each set of data has its own unique function of variability and distance between data points. This variability is generally computed as a function called semivariance.
In one respect geostatistics might be viewed as simply a methodology for interpolating data on an irregular pattern but this is too simplistic. A number of interpolation methods/algorithms were already well known when geostatistics began to be known; for example, inverse distance weighting (IDW) and trend surface analysis as well as the much simpler nearest neighbor algorithm. Interpolation techniques use sample points to produce surfaces of the phenomena of interest. The interpolation techniques are divided into two main types: deterministic and geostatistical methods.
Deterministic interpolation techniques create surfaces from measured points, based on either the extent of similarity (e.g., IDW) or the degree of smoothing (e.g., radial basis functions). These techniques do not use a model of random spatial processes. A deterministic interpolation can either force the resulting surface to pass through the data values or not. An interpolation technique that predicts a value that is identical to the measured value at a sampled location is known as an exact interpolator (e.g., IDW and radial basis functions). An inexact interpolator (e.g., global and local polynomials) predicts a value that is different from the measured value. The latter can be used to avoid sharp peaks or troughs in the output surface.
Geostatistics assume that at least some of the spatial variation of natural phenomena can be modeled by random processes with spatial autocorrelation. Geostatistical techniques produce not only prediction surfaces but also error or uncertainty surfaces, giving us an indication of how good the predictions are. Geostatistical interpolators exhibit probabilistic behaviour, i.e., it can be considered that for one known condition there are many possible outcomes, some of which will be more likely than others.
Many methods are associated with geostatistics, but they generally fall in the kriging family, e.g., ordinary, simple, universal, probability, indicator, and disjunctive kriging, along with their counterparts in cokriging. Kriging is a method of estimation based on the trend and variability from the trend. Variability, in this context, refers to random errors about the trend or mean. In this context, ‘error’ does not imply a mistake but a fluctuation (error) about the trend is unknown and is not systematic; the fluctuation could be positive or negative. Kriging may be considered exact (or smoothed) or inexact. Kriging incorporates the principles of probability and prediction, and like the IDW, is a weighted average technique except that a surface produced by kriging may exceed the value range of the sample points while still not actually passing through them. Various statistical models can be chosen to produce map outputs (or surfaces) from the kriging process; such as, interpolated surface (the prediction), the standard prediction errors (variance), probability (that the prediction exceeds a threshold) and quantile (for any given probability) (Liu and Mason 2009). One may refer Isaaks and Srivastava (1989) for an introductory text on Geostatistics.
References
Isaaks, E.H. and R.M. Srivastava 1989, An Introduction to Applied Geostatistics, Oxford University Press, New York, 561 pp.
Miller, H.J. 2004, ‘Tobler’s first law and spatial analysis’, Annals of the Association of American Geographers 94: 284–289.
Tobler, W. 1970, ‘A computer movie simulating urban growth in the Detroit region’, Economic Geography 46: 234–240.
Liu, J.G. and P.J. Mason 2009, Essential Image Processing and GIS for Remote Sensing, Wiley-Blackwell, New York, 450 pp.
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