Geostatistical modelling of compositional variability across granitoid complexes: Its relevance to petrogenetic interpretations and specification of parent-rock properties in sediment-generation models

Sediment-generation models need an accurate specification of the fundamental properties of parent rocks and their variability at the scale of first-order (mono-lithologic) drainage basins. Georeferenced point-count data of five extensively surveyed plutons were extracted from the literature to examine the spatial variability of modal composition in granitoids. Point-count data must be considered inherently noisy from the point of view of geospatial analysis because sampled areas (thin sections) are small compared to the average crystal size of granitoids. Geostatistical modelling of such data is further complicated by the fact that crucial information on short-range variability is unavailable because sampling was carried out according to more or less regular patterns to achieve an equal density of data coverage across plutons. Geostatistical modelling of compositional data was carried out by transforming the data to centred log-ratios and calculating their Principal Components (PCs). The PC scores were used as input for a geostatistical workflow based on Ordinary Kriging, coupled with cross-validation and stochastic simulation to assess the predictive capabilities of the models. Sets of omnidirectional exponential variogram models with fixed range and sill, and variable nugget were used for each pluton. The local neighbourhood (search radius) for geostatistical interpolation was set equal to the range. Variogram modelling was formulated as an optimisation problem aimed at estimating the nugget for which the mean squared cross-validation (prediction) error reaches its minimum. The best model of each pluton was selected from all possible combinations of models obtained from the PCs. Results of a permutation test indicate that the residuals of observed and predicted compositions do not exhibit significant cross-covariance within the search radius adopted and may be interpreted as stationary random errors, which suggests that little may be gained by application of co-kriging. Random sampling from the geostatistical models indicates that up to several hundreds of specimens must be analysed to successfully predict the area-weighted mean modal composition of plutons and its spatial covariance structure as depicted in single-component and QAPF (Streckeisen) maps. An illustration of the internal consistency of the log-ratio approach and petrogenetic models is provided by the analysis of the compositional pattern in one of the granite complexes, which can be explained by the combined effects of mixing and fractionation. The volumes sampled by bulk geochemical analysis are equivalent to the areas of thin sections, and geochemical data of coarse-crystalline rocks are thus subject to the same limitations as modal analyses. The combined data-reduction and geostatistical modelling strategy outlined in this study is expected to be particularly efficient for the modelling of such high-dimensional data. The advent of quick non-destructive measurement techniques such as XRF and NIR is expected to play a crucial role in future attempts at rigorous quantitative mapping of lithosomes. If sediment generation can be simulated, the uncertainties associated with the initial conditions (area-weighted means of parent-rock properties) can be propagated all the way through to uncertainties of predicted sediment properties.