Geostatistical interpolation differs from the procedures described in Section 6.6, Deterministic Interpolation Methods, in that it assumes the data point values represent a sample from some underlying true population. By analyzing this sample it is often possible to derive a general model that describes how the sample values vary with distance (and optionally, direction). This model may be then used to interpolate, or predict, values at unsampled locations, in much the same way as with deterministic interpolation. If the samples meet various additional conditions it may also be possible to provide estimated confidence intervals for these predictions. Geostatistical interpolation methods attempt to address questions such as:
•how many points are needed to compute the local average?
•what size, orientation and shape of neighborhood should be chosen?
•what model and weights should be used to compute the local average?
•what errors (uncertainties) are associated with interpolated values?
A substantial number of specialist terms are used in the field of geostatistics and many of these are explained in the subsections below, before we proceed to describe specific geostatistical models in detail. These include:
•Sill, range and nugget
•Comments on geostatistical software packages
•Madograms and Rodograms, and
•Periodograms and Fourier analysis
We can define geostatistics as referring to “… models and methods for data observed at a discrete set of locations, such that the observed value, zi, is either a direct measurement of, or is statistically related to, the value of an underlying continuous spatial phenomenon, F(x,y), at the corresponding sampled location (xi,yi) within some spatial region A.” This definition is based on that drawn up by Professor Diggle and his colleagues at the University of Lancaster, Spatial Statistics Group. It emphasizes the fact that the term is particular to the analysis of continuous spatial phenomena, which is precisely the area that is of interest in 2D interpolation. It also highlights the statistical nature of the procedures involved, although rigid adherence to statistical requirements is not a pre-requisite for applying such techniques — rather it is a prerequisite for interpretation of some of the results, especially those relating to statistical measures such as the estimation of prediction errors and confidence bands. As a more specific resource on geospatial analysis, in particular geostatistics and associated software packages, the European Commission’s AI‑GEOSTATS website is highly recommended.
For the purposes of the present Section we focus on the application of geostatistical methods to interpolation problems. As noted in Section 6.7.1, geostatistical interpolation differs from deterministic interpolation in that it assumes the source data points are a specific statistical sample or realization from some true underlying surface function, and this sample must first be analyzed in order to create a suitable model that will provide the best possible estimate of this underlying surface.
The initial analysis and modeling stage involves examining the dataset for outliers, and then for spatial autocorrelation, essentially using similar techniques to those described in Section 5.5.2, Global spatial autocorrelation. Of particular relevance here is the Geary C statistics which (ignoring weights for the present) is given by:
and the correlogram form of the Moran I statistic:
where, as in Section 5.5.2, N(h) is the number of points in a distance band of width d or Δ and mean distance of h; zi is the (standardized) value at point i; and zj is the (standardized) value at a point j distance h from i (in practice within a distance band from h‑Δ/2 to h+Δ/2). The summations apply for all point pairs in the selected band.
There is an enormous volume of literature relating to geostatistics and geostatistical methods. Basic introductions in the GIS literature are provided in O’Sullivan and Unwin (2010) and Burrough and McDonnell (1998, 2015) amongst others. More advanced books include those by Goovaerts (1997, 1999), Deutsch and Journel (1992), Cressie (1993) and Isaaks and Srivastava (1989). The European Commission website devoted to Geostatistics (AI‑GEOSTATS website) is a useful starting point, as is the Center for Computational Geostatistics (CCG), a site that includes very useful course material by Prof Deutsch and colleagues on Monte Carlo simulation with applications in the petro-chemicals industry. This site lists articles, books, courses and conferences, software products and datasets available within the field of spatial statistics in general and geostatistics in particular. There is also some controversy about the usage of geostatistics, as there is with many attempts to apply statistical methods to spatial phenomena.