To speed up multivariate geostatistical simulation it is common to transform the set of attributes into spatially uncorrelated factors that can be simulated independently. One decorrelation method is the method of minimum/maximum autocorrelation factors (MAF), where a two structure linear model of coregionalisation is assumed. The transformation decorrelates the theoretical model exactly, but the actual data are only approximately decorrelated. The MAF transformation can therefore be seen as a special case of a non-orthogonal approximate diagonaliser of a set of symmetric matrices. A more general approach for approximate joint diagonalisation (AJD) has been developed in the context of blind source separation. For these AJD algorithms no assumptions are made beyond symmetry of the individual matrices and so they can be applied to a family of experimental semivariogram matrices. In their application there are no restrictions on the number of structures in the linear model of coregionalisation (LMC) thus removing one of the conditions placed on the subsequent modelling of the spatial structure of the factors. In this paper the general background for the U-WEDGE AJD method is presented. It is shown that the U-WEDGE method achieves better spatial decorrelation than the MAF method. The factors derived from U-WEDGE are simulated using conditional turning bands simulation and then backtransformed to attribute space.