Consider a spatially distributed variable that has been observed (possibly with error) at \(n\) distinct points within a bounded study region of interest \(D\), where \(D \subset \mathbb R^2\). Our goal is to make inferences about the process that governs the spatial distribution and locations without observed values of the variable.
The geostatistical approach is based 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. Because measurement errors usually lack spatial structure, we explicitly separate them from the spatially dependent component. Thus, we may rewrite the geostatistical model as:
\[Z(s) = \mu(s) + \eta(s) + \epsilon(s)\]
where \(\eta(s)\) represents the spatially dependent component and \(\epsilon(s)\) represents the measurement error, which is spatially uncorrelated and has a mean of zero. Note that the processes \(\eta(s)\) and \(\epsilon(s)\) are independent.
The error process \(\epsilon(s)\) is often referred to as the nugget effect. This term originates from mining applications, where the occurrence of gold nuggets shows substantial variability (see Gelfand at al. 2010).
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The E-Learning project SOGA-R was developed at the Department of Earth Sciences by Kai Hartmann, Joachim Krois and Annette Rudolph. You can reach us via mail by soga[at]zedat.fu-berlin.de.
You may use this project freely under the Creative Commons Attribution-ShareAlike 4.0 International License.
Please cite as follow: Hartmann, K., Krois, J., Rudolph, A. (2023): Statistics and Geodata Analysis using R (SOGA-R). Department of Earth Sciences, Freie Universitaet Berlin.