Analysis and elucidation of soil variation using wavelets

A wavelet is a compact analysing kernel that can be moved over a sequence of data to measure variation locally. There are several families of wavelet, and within any one family wavelets of different lengths and therefore smoothness and their corresponding scaling functions can be assembled into a collection of orthogonal functions. Such an assemblage can then be applied to filter spatial data into a series of independent components at varying scales in a single coherent analysis. The application requires no assumptions other than that of finite variance. The methods have been developed for processing signals and remote imagery in which data are abundant, and they need modification for data from field sampling. The paper describes the theory of wavelets. It introduces the pyramid algorithm for multiresolution analysis and shows how it can be adapted for fairly small sets of transect data such as one might obtain in soil survey. It then illustrates the application using Daubechies's wavelets to two soil transacts, one of gilgai on plain land in Australia and the other across a sedimentary sequence in England. In both examples the technique revealed strongly contrasting local features of the variation that had been lost by averaging in previous analyses and expressed them quantitatively in combinations of both scale and magnitude. Further, the results could be explained as the spatial effects of change in topography or geology underlying the variation in the soil.