Introduction

Three common characteristics are often observed with spatial data (that is, data indexed by their spatial locations):

1. slowly varying, large-scale variations in the measured values

2. irregular, small-scale variations

3. similarity of measurements at locations close together

As an illustration, consider a hypothetical example in which an organic solvent leaks from an industrial site and spreads over a large area. Assume the solvent is absorbed and immobilized into the subsoil above any groundwater level, so you can ignore any time dependence.

To find the areal extent and the concentration values of the solvent, you need measurements. Although the problem is inherently three-dimensional, if you measure total concentration in a column of soil or take a depth-averaged concentration, it can be handled reasonably well with two-dimensional techniques.

You usually assume that measured concentrations are higher closer to the source and decrease at larger distances from the source. On top of this smooth variation, measured concentrations typically have small-scale variations, due perhaps to the inherent variability of soil properties.

You also tend to suspect that measurements made close together yield similar concentration values, while measurements made far apart can have very different values.

These physically reasonable qualitative statements have no explicit probabilistic content. A number of numerical smoothing techniques, such as inverse distance weighting and splines, make use of large-scale variations and "close distance-close value" characteristics of spatial data to interpolate the measured concentrations for contouring purposes.

While characteristics (i) and (iii) are handled by such smoothing methods, characteristic (ii), the small-scale residual variation in the concentration field, is not accounted for.

There can be situations, due to the use of the prediction map or the relative magnitude of the irregular fluctuations, where you cannot ignore these small-scale irregular fluctuations. In other words, the smoothed or predicted values of the concentration field alone are not a sufficient characterization; you also need the possible spread around these contoured values.