The representation of complex soil variation on wavelet packet bases

The discrete wavelet packet transform (DWPT) is an advanced wavelet technique that has distinct advantages over the discrete wavelet transform (DWT) previously used in soil science. Because the DWT divides the spatial frequencies of the data into non-uniform intervals, it may fail to resolve features of the spatial variation that a simpler spectral analysis might identify under stationary conditions. However, the DWPT allows us to retain the advantages of the DWT (particularly the lack of stationarity assumptions) while achieving much better resolution in the spatial frequency domain. This is at the cost of poorer spatial resolution at the higher frequencies. However, by selecting a best wavelet packet basis from among the many possible, according to some appropriate criterion, we can find a compromise between frequency and spatial resolution that is best suited to the representation of our particular data set. In this paper, I describe a number of analyses based on the DWPT and apply them to two data sets on the soil. The advantages of the DWPT with best basis selection are clearly illustrated. In particular, nonstationary behaviour of a significant periodic component of variation was identified that could not be effectively resolved by the DWT. Improved resolution in the frequency domain allowed aspects of the non-stationary variation of soil to be resolved in a multiresolution analysis (MRA) adapted to the variation of the data.