Spatial prediction by linear kriging

Abstract

Kriging denotes a body of techniques to predict data in Euclidean space. By prediction 1 we mean that the target quantity is estimated at an arbitrary location, given its coordinates and some observations recorded at a set of known locations. Developed mainly by G. Matheron and co-workers in the 1960s and 1970s at the Ecole des Mines in Paris [256, 257], kriging is embedded in the framework of stochastic mean square prediction, and it is therefore related closely to earlier work by N. Wiener and A. N. Kolmogorov on the optimal linear prediction of times series. Typically, kriging is used to answer questions of the kind 'where and how much?'. A surveyor, as an example, might have to delineate the part of a polluted region where the concentration of some soil pollutant exceeds a threshold value, or a remote sensing specialist might need to know what kind of land use is likely for a patch of land that was obscured by clouds when the reflected radiation of the ground was recorded by some airborne sensor. In some applications, however, the part 'where?' of the question is of lesser importance, and the quantity of interest might be the mean of an attribute over the whole study domain or the means over some sub-domains. Although kriging can deal with such tasks as well, the methods of classical sampling theory are generally better suited to tackle the so-called global estimation problems, which focus on the question 'how much?'. In this chapter, we shall restrict ourselves to the problem of local estimation. For global estimation the interested reader should consult any of the numerous textbooks about sampling theory [69, 361, 362]. Although not immediately obvious because remotely sensed imagery normally provides full coverage data, geostatistical interpolation techniques have proven useful to tackle various problems in remote sensing. First, kriging was used to restore remotely sensed images that contained missing or faulty data [22]. A particular example is the aforementioned estimation of some characteristic of the ground surface that was hidden by clouds during the scan [311, 5]. Second, multi-variable linear 1. In accordance with [83] we denote both smoothing and non-smoothing, i.e., interpolating methods as spatial prediction techniques. 83 A. Stein et al. (eds.), Spatial Statistics for Remote Sensing, 83-113.