Inference about soil variability from the structure of the best wavelet packer basis

The plausibility of the assumption that soil variation can be treated as a realization of a random spatial process that is stationary in the variance can break down in various ways. It is possible to test the assumption using methods based on the wavelet transform. To date these approaches have been applied using the discrete wavelet transform. A drawback of this approach is that it uses a partition of the spatial frequencies represented in the data into intervals (scales) that are arbitrarily defined in advance and are not necessarily suitable for the representation of the variation of the data in question. A solution to this problem is to identify the best basis for the data from a wavelet packet library. An interesting question is whether the structure of this best basis is in itself informative about the plausibility of the stationarity assumption. In this paper, I show that this is indeed the case. The best basis for a stationary random variable from some packet library is the basis on the maximum dilation of the mother wavelet, which gives the finest resolution in the frequency domain. I propose the ratio of the entropy cost functional for this basis to that of the empirical best basis as a measure of evidence against the null hypothesis of stationarity in the variance. Critical values of this statistic may be obtained by Monte Carlo methods. I demonstrate the method using data on the clay content of soil on a transect in central England. The null hypothesis of stationarity in the variance may be rejected. Tests for the uniformity of variance can then be applied to wavelet packets in the best basis. The dominant local feature that is reflected in this behaviour is the unique pattern of variation in alluvium around a drainage channel that crosses the transect. This variation contrasts with that seen at most positions on the transect, variation that arises from a more or less regular pattern of boundaries between contrasting Jurassic strata.