Consider a spatially distributed variable that has been observed (possibly with error) at \(n\) distinct points in a bounded study region of interest, \(D\), where \(D \subset \mathbb R^2\). Our goal is to make inferences about the process that governs how this variable is distributed spatially and about values of the variable at locations where it was not observed.

The geostatistical approach bases on the assumption that the observed data is a realization of a continuously indexed spatial stochastic process, also refereed to as random field

\[Z(\cdot) \equiv \{Z(s):s \in D\}\]

The classical geostatistical model decomposes a spatial stochastic process as

\[Z(s) = \mu(s) + \epsilon(s)\]

where \(\mu(s) = E(Z(s))\), the mean function, corresponds to the first-order structure and \(\epsilon(s)\), a zero-mean random *error* process, corresponds to the second-order structure.

In the geostatistical model the mean function \(\mu(s)\) accounts for large-scale spatial variation, often referred to as *global trend*, and \(\epsilon(s)\) accounts for the small-scale spatial variation, also denoted as *spatial dependence*. In addition to capturing the small-scale spatial variation, the error process \(\epsilon(s)\) accounts for measurement error that may occur in the data collection process. As this measurement error component typically has no spatial structure we explicitly separate it from the spatially dependent component. Thus we may write the geostatistical model as

\[Z(s) = \mu(s) + \eta(s) + \epsilon(s)\]

where \(\eta(s)\) is the spatially dependent component and the spatially uncorrelated mean zero errors, \(\epsilon(s)\), is the measurement error. Note that the processes \(\eta(s)\) and \(\epsilon(s)\) are independent.

The error process \(\epsilon(s)\) is often referred to as a *nugget effect*. The terminology stems from mining applications, where the occurrence of gold nuggets shows substantial variability (see Gelfand at al. 2010).