High-order Statistics of Spatial Random Fields: Exploring Spatial Cumulants for Modeling Complex Non-Gaussian and Non-linear Phenomena

The spatial distributions of earth science and engineering phenomena under study are currently predicted from finite measurements and second-order geostatistical models. The latter models can be limiting, as geological systems are highly complex, non-Gaussian, and exhibit non-linear patterns of spatial connectivity. Non-linear and non-Gaussian high-order geostatistics based on spatial connectivity measures, namely spatial cumulants, are proposed as a new alternative modeling framework for spatial data. This framework has two parts. The first part is the definition, properties, and inference of spatial cumulants—including understanding the interrelation of cumulant characteristics with the in-situ behavior of geological entities or processes, as examined in this paper. The second part is the research on a random field model for simulation based on its high-order spatial cumulants. Mathematical definitions of non-Gaussian spatial random functions and their high-order spatial statistics are presented herein, stressing the notion of spatial cumulants. The calculation of spatial cumulants with spatial templates follows, including anisotropic experimental cumulants. Several examples of two- and three-dimensional images, including a diamond bearing kimberlite pipe from the Ekati Mine in Canada, are analyzed to assess the relations between cumulants and the spatial behavior of geological processes. Spatial cumulants of orders three to five are shown to capture directional multiple-point periodicity, connectivity including connectivity of extreme values, and spatial architecture. In addition, they provide substantial information on geometric characteristics and anisotropy of geological patterns. It is further shown that effects of complex spatial patterns are seen even if only subsets of all cumulant templates are computed. Compared to second-order statistics, cumulant maps are found to include a wealth of additional information from underlying geological patterns. Further work seeks to integrate this information in the predictive capabilities of a random field model.     

Two-stage reverse flotation process for removal of carbonates and silicates from phosphate ore using anionic and cationic collectors

Phosphate rock contains various gangue minerals including silicates and carbonates which need to be reduced in content in order to meet the requirements of the phosphate industry. Froth flotation has become an integral part of phosphate concentration process. In this study, double reverse flotation was applied to recover apatite from phosphate ore. H₃PO₄ and CaO were used as phosphate depressants, in acidic and alkaline conditions. Fatty acids and amines were added as carbonate and silicate collectors respectively. An experimental protocol devised to optimize the grade and recovery of phosphate using anionic–cationic method was found effective. Consequently, a required high quality of phosphate concentrate containing 30.1% P₂O₅ was obtained, with a recovery of 94%. X-ray diffraction and optical microscopy studies were performed to define the main minerals.