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# Spatial Prediction 2: Geostatistics

DOI link for Spatial Prediction 2: Geostatistics

Spatial Prediction 2: Geostatistics book

# Spatial Prediction 2: Geostatistics

DOI link for Spatial Prediction 2: Geostatistics

Spatial Prediction 2: Geostatistics book

## ABSTRACT

Geostatistics is discussed separately to the methods outlined in Chapter 6 on the grounds that geostatistics entails different operational principles, as well as a different underlying philosophy (use of a random function model, as defined below). In practice, as detailed in this chapter, there are similarities between methods outlined in this chapter and some of those discussed in Chapter 6. Geostatistics is based upon the recognition that in the Earth sciences there

is usually a lack of sufficient knowledge concerning how properties vary in space. Therefore, a deterministic model may not be appropriate. If we wish to make predictions at locations for which we have no observations, we must allow for uncertainty in our description as a result of our lack in knowledge. So, the uncertainty inherent in predictions of any property we cannot describe deterministically is accounted for through the use of probabilistic models. With this approach, the data are considered as the outcome of a random process. Isaaks and Srivastava (200) caution that we should remember that use of a probabilistic model is an admission of our ignorance, but it does not mean that any spatially-referenced property varies randomly in reality. For a criticism of geostatistics, see Philip and Watson (317), with responses to this paper given by Journel (209) and Srivastava (340). In geostatistics, spatial variation is modelled as comprising two distinct

parts, a deterministic (trend) component (m(s)) and a stochastic (‘random’) component (R(s)):

Z(s) = m(s) +R(s) (7.1)

This is termed a random function (RF) model. In geostatistics, a spatiallyreferenced variable, z(s), is treated as the outcome of a RF, Z(s), defined as a set of random variables (RV) that vary as a function of spatial location s. A realisation of a RF is called a regionalised variable (ReV). The Theory of Regionalised Variables (265), (266) is the fundamental framework on which geostatistics is based. Note the common division into two components with the thin plate spline model of Equation 6.6. Our observations (that is, measurements of the property of interest) are

modelled as ReVs. So, if the property of interest was precipitation, the ReV may be viewed as a realisation of the set of RVs ‘precipitation’ (see Journel and Huijbregts (212) for a clear and detailed overview). In this chapter, the initial focus is on the theoretical basis of linear

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geostatistics based on a stationary model (Sections 7.3 through 7.12), and a short review of other approaches is given in Section 7.13. This is necessary to provide a basis upon which the following account of nonstationary (that is, locally-adaptive) geostatistics builds (Sections 7.14 through 7.18). Finally, the chapter summarises some key issues. Illustrated examples of the applications of some key methods are presented in the body of the text. Many introductions to geostatistics are available (for example, (18), (85),

(153), (384)). An introduction to geostatistics is provided by Burrough and McDonnell (70) and others by Atkinson and Lloyd (28) and Lloyd (245). Introductions have been written for users of GISystems (297), physical geographers (298), (299), the remote sensing community (98), and archaeologists (249).