Analysing spatially intermittent variation of nitrous oxide emissions from soil with wavelets and the implications for sampling

Emissions of gases from the soil are known to vary spatially in a complex way. In this paper we show how such data can be analysed with the wavelet transform. We analysed data on rates of N2O emission from soil cores collected at 4-m intervals on a 1024-m transect across arable land at Silsoe in England. We used a thresholding procedure to represent intermittent variation in N2O emission from the soil as a sparse wavelet process, i.e. one in which most of the wavelet coefficients are not significantly different from zero. This analysis made clear that the rate of N2O emission varied more intermittently on this transect than did soil pH, for which many more of the wavelet coefficients had to be retained. This account of intermittent variation motivated us to consider a class of random functions, which we call wavelet random functions, for the simulation of spatially intermittent variation. A wavelet random function (WRF) is an inverse wavelet transform of a set of random wavelet coefficients with specified variance at each scale. We generated intermittent variation at a particular scale in the WRF by specifying a binormal process for the wavelet coefficients at this scale. We showed by simulation that adaptive sampling schemes are more efficient than ordinary stratified random sampling to estimate the mean of a spatial variable that is intermittent at a particular scale. This is because the sampling can be concentrated in the more variable regions. When we simulated values that emulate the intermittency of our data on N2O we found that the gains in efficiency from simple adaptive sampling schemes were small. This was because the emission of N2O is intermittent over several disparate scales. More sophisticated adaptive sampling is needed for these conditions, and it should embody knowledge of the relevant soil processes.