Analysis and elucidation of soil variation using wavelets

A wavelet is a compact analysing kernel that can be moved over a sequence o f data to measure variation locally. There are several families of wavelet, and within any one family wavelets of different lengths and therefore smoo thness and their corresponding scaling functions can be assembled into a co llection of orthogonal functions. Such an assemblage can then be applied to filter spatial data into a series of independent components at varying sca les in a single coherent analysis. The application requires no assumptions other than that of finite variance. The methods have been developed for pro cessing signals and remote imagery in which data are abundant, and they nee d modification for data from field sampling. The paper describes the theory of wavelets. It introduces the pyramid algor ithm for multiresolution analysis and shows how it can be adapted for fairl y small sets of transect data such as one might obtain in soil survey. It t hen illustrates the application using Daubechies's wavelets to two soil tra nsacts, one of gilgai on plain land in Australia and the other across a sed imentary sequence in England. In both examples the technique revealed stron gly contrasting local features of the variation that had been lost by avera ging in previous analyses and expressed them quantitatively in combinations of both scale and magnitude. Further, the results could be explained as th e spatial effects of change in topography or geology underlying the variati on in the soil.