Spatial Random Fields

One method of incorporating characteristic (ii) into the construction of a contour map is to model the concentration field as a spatial random field (SRF). The mathematical details of SRF models are given in a number of texts, such as Cressie (1993) and Christakos (1992). The mathematics of SRFs is formidable. However, under certain simplifying assumptions, it produces classical linear predictors with very simple properties, enabling easy implementation for prediction purposes. These predictors, primarily ordinary kriging (OK), give both a prediction and a standard error of prediction at unsampled locations. This allows the construction of a map of both predicted values and level of uncertainty about the predicted values.

The key assumption in applying the SRF formalism is that the measurements come from a single realization of the SRF. However, in most geostatistical applications, the focus is on a single, unique realization. This is unlike most other situations in stochastic modeling in which there will be future experiments or observational activities (at least conceptually) under similar circumstances. This renders many traditional ideas of statistical inference ambiguous and somewhat counterintuitive.

Additional logical and methodological problems could stand in the way of applying a stochastic model to a unique but partly unknown natural process; see the introduction in Matheron (1971) and Cressie (1993, section 2.3). These difficulties have resulted in attempts to frame the prediction problem in a completely deterministic way (Isaaks and Srivastava 1988; Journel 1985). Also, some issues with kriging, and with spatial prediction methods in general, are related to the necessary assumption of ergodicity of the spatial process. This assumption is required to estimate the covariance or semivariogram from sample data. Details are provided in Cressie (1993, pp. 52–58).

Despite these difficulties, ordinary kriging remains a popular and widely used tool in modeling spatial data, especially in generating surface plots and contour maps. An abbreviated derivation of the OK predictor for point prediction and the associated standard error is discussed in the following section. Full details are given in Journel and Huijbregts (1978), Christakos (1992), and Cressie (1993).