This thesis has the De Wijs process as its primary focus. Theory on generalised functions as well as stochastic processes indexed on test functions, Borel sets, and measures is described. Estimation of the parameters in a De Wijs Plus White Noise (WWN) Process using restricted maximum likelihood is detailed. The covariance matrix of relevant contrasts is constructed, and a data-analysis is carried out. Parameter estimation is performed for the WWN Process on a soil dataset from Barro Colorado Island containing 19 different minerals, and the variogram is introduced and used as a summary statistic. Then kriging for set-indexed random fields is outlined and a 10-fold cross-validation is carried out on the WWN Process fitted to the data. Here it is found that for all of the minerals in the dataset the WWN Process performs better than predicting using the sample mean. Furthermore, on all but one mineral, the WWN process also performs better than an intrinsic random field of order $0$ with polynomial generalised covariance. When restricting only to the data on regularly spaced grid, there are a few minerals where the power or exponential model performs better than the WWN process. This suggests that Peter McCullagh's notion of a \textit{loi du terroir} does not extend to forest soil data.