R. Tolosana-Delgado

Simplifying compositional multiple regression: Application to grain size controls on sediment geochemistry

Modern geochemical data sets have typically around 20-30 compositional variables measured on some tens or hundreds of samples. A statistical analysis of data sets with so many variables should take as a priority the reduction of dimensionality of the model, in order to increase its reliability and enhance its interpretation. In the framework of compositional data analysis with multiple regression, such simplification can be achieved taking some geometric concepts into account. First, the sample space of compositions, the simplex, is given an Euclidean space structure by the compositional operations of perturbation, powering and Aitchison inner product. Then, given some qualitative information on which subcompositions might depend on each explanatory variable, one can decompose the simplex in a set of orthogonal subspaces, in such a way that the composition projected onto each subspace is independent of a subset of the explanatory variables. This is achieved with a series of singular value decomposition computations. The method is applied to a data set of 88 observations of six major oxides in molar proportions, from modern glacial and fluvio-glacial sediments, with grain size ranging from coarse sand to clay. The goal is to assess the influence of chemical weathering processes (expected to impose a linear relation of composition and grain size) against purely physical processes (expected to show step-wise functions following the largest characteristic crystal sizes of specific minerals in the source rock). We exhaustively explore all patterns of uncorrelation of the composition with three explanatory variables: grain size in @f scale, and two step functions for the silt and clay domains. The best pattern, chosen with a likelihood ratio test, has only a smooth trend of (Mg,Fe) vs. (Al,K,Ca+Na) enrichment towards finer grain sizes-explained as differential mechanical behaviour of phyllosilicates vs. feldspar-and coefficients for the two step functions related to the sharp decrease of quartz in silt fractions, and the sudden enrichment of mafic accessory minerals, alteration products and mechanically unstable phyllosilicates in the clay fraction. We could thus be confident that weathering is almost absent in this data set.

Joint Consistent Mapping of High-Dimensional Geochemical Surveys

Geochemical surveys often contain several tens of components, obtained from different horizons and with different analytical techniques. These are used either to obtain elemental concentration maps or to explore links between the variables. The first task involves interpolation, the second task principal component analysis (PCA) or a related technique. Interpolation of all geochemical variables (in wt% or ppm) should guarantee consistent results: At any location, all variables must be positive and sum up to 100xa0%. This is not ensured by any conventional geostatistical technique. Moreover, the maps should ideally preserve any link present in the data. PCA also presents some problems, derived from the spatial dependence between the observations, and the compositional nature of the data. Log-ratio geostatistical techniques offer a consistent solution to all these problems. Variation-variograms are introduced to capture the spatial dependence structure: These are direct variograms of all possible log ratios of two components. They can be modeled with a function analogous to the linear model of coregionalization (LMC), where for each spatial structure there is an associated variation matrix describing the links between the components. Eigenvalue decompositions of these matrices provide a PCA of that particular spatial scale. The whole data set can then be interpolated by cokriging. Factorial cokriging can also be used to map a certain spatial structure, eventually projected onto those principal components (PCs) of that structure with relevant contribution to the spatial variability. If only one PC is used for a certain structure, the maps obtained represent the spatial variability of a geochemical link between the variables. These procedures and their advantages are illustrated with the horizon C Kola data set, with 25 components and 605 samples covering most of the Kola peninsula (Finland, Norway, Russia).

Constructing modal mineralogy from geochemical composition: A geometric-Bayesian approach

Modal mineralogical composition is known to carry more information than major element geochemistry, though the latter is far easier to determine in the lab. Constructing mineral compositions from geochemistry can be seen as a typical end-member problem, where one assumes that some multivariate observations are generated by a convex linear mixture of a few pure end-members: these end-member characteristics as well as the coefficients of the linear mixture for the observations can be then estimated from geochemical data. We propose a mixed geometric-probabilistic solution to this problem. First, we assume known end-members, in number and properties, and study the set of solutions from a purely geometric perspective. Second, we discuss how to select representative solutions from this space, in particular under some distributional assumptions. Third, we allow the end-member properties to randomly vary in a controlled, interpretable fashion. Finally we build a Bayesian model, with a parsimonious parametrization characterizing each of these three steps, that can be treated by conventional Markov-Chain Monte Carlo techniques. In the illustration case study, we apply the method to reconstruct the mineralogy of a set of fluvio-glacial monomictic sediments from an Alpine granitoid massif. Results suggest a trend of enrichment in chlorite, muscovite and Ti-bearing minerals, along with depletion in quartz and feldspar. This is tentatively interpreted as an effect of comminution combined with differential mechanical properties. Moreover, mineral chemistry is estimated to exhibit very low Na in muscovite, Fe-rich garnet, Na-rich plagioclase, K-feldspar with up to 10% Na-component (albite), and biotite with Mg>Fe. The reconstructed modal mineralogy stays in a reasonable agreement with quantitative XRD phase analyses.

On the Joint Multi Point Simulation of Discrete and Continuous Geometallurgical Parameters

Geometallurgical parameters are descriptions of the mineralogy and microstructure of the ore determining its mineralogical and microstructural characteristics. From a conditional geostatistical simulation of such properties, a processing model can compute recovery, equipment usage, processing costs, and thus the monetary value for mining and processing a block with certain processing parameters. The output can be used for optimising mining sequences or finding optimal processing parameters by solving the corresponding stochastic optimisation problem. The approach requires two properties of the simulation not provided by established geostatistical techniques: n n(1) n nMany relevant geometallurgical parameters are from non-Euclidean statistical scales such as (mineral) compositions, (grain size) distribution, (grain) geometry, and (stratigraphic type) categorical which might produce nonsensical values (for example, negative proportions, negative facies probabilities, planar grains) when simulated with standard geostatistical techniques. n n n n n(2) n nDue to the nonlinearity of processing, the entire conditional distribution of the geometallurgical parameters is relevant, not only its mean and variance. The geostatistical simulation needs to reproduce the joint conditional distributions of all the geometallurgical parameters. n n n n n n nThe multi-point conditional geostatistical simulation technique discussed here allows for jointly simulating dependent spatial variables from various sample spaces. The technique combines an infill simulation, similar to the one used in multi-point geostatistics (MPS), with a new form of distributional regression to estimate conditional distributions of arbitrary scales from different information sources, including training images, training models and observed data. The distributional regression is based on a generalisation of logistic regression and is related to both Bayesian Maximum Entropy (BME) geostatistics and high order cumulants. The method ensures that simulated data reside in the set of possible values and honour the characteristics of the joint distribution to be reproduced. The computational effort is substantial, but affordable for a useful application with standard problems: from processing-aware block value prediction and block processing optimisation as shown in the test application to a mathematically completely defined simulated model situation with a complex processing model.

Compositionally Compliant Contact Analysis

Contact analysis assesses the evolution of the average value along boreholes of a given variable at increasing distances from the contact between two facies or domains. The concept is long established in the geostatistical literature and software, albeit for studying the behavior of a single variable. This contribution explores practical ways for studying this transient behavior of a set of variables forming a composition, in such a way that spurious correlation effects are avoided. This is obtained with contact diagrams for each possible pairwise logratio of two components, as well as with a contact analysis of the centered-logratio transformed components. This approach is particularly promising when the set of components considered account for a considerable amount of the total mass, or dilution effects are suspected to have affected only a subset of the components. These concepts are illustrated with data from Murrin Murrin, WA, a Ni-Co laterite deposit where intensive remobilization of both value and deleterious components is known to have occurred.

Predictive Geometallurgy: An Interdisciplinary Key Challenge for Mathematical Geosciences

Predictive geometallurgy tries to optimize the mineral value chain based on a precise and quantitative understanding of: the geology and mineralogy of the ores, the minerals processing, and the economics of mineral commodities. This chapter describes the state of the art and the mathematical building blocks of a possible solution to this problem. This solution heavily relies on all classical fields of mathematical geosciences and geoinformatics, but requires new mathematical and computational developments. Geometallurgy can thus become a new defining challenge for mathematical geosciences, in the same fashion as geostatistics has been in the first 50 years of the IAMG.

The inherent link between ore formation and geometallurgy as documented by complex tin mineralization at the Hämmerlein deposit (Erzgebirge, Germany)

A comprehensive quantitative mineralogical study on the Hämmerlein tin deposit in the Erzgebirge, Germany, not only yields insights into the genesis of Sn mineralization but also provides also important clues for beneficiation. The lithological units of the skarn and greisen deposit show significant differences in modal mineralogy and Sn deportment. These systematic differences are attributed to several stages of ore formation. Of greatest significance is a paragenetically late cassiterite-chlorite-fluorite-sulfide assemblage. This assemblage replaces pre-existing skarn lithologies and also forms stockwork mineralization in greisen-type ores developed at the expense of mica schist that surrounds the skarn. The co-genetic formation of the cassiterite-chlorite-fluorite-sulfide assemblage is captured by the mineral association parameter—a parameter that can be easily quantified from data acquired during automated mineralogy studies. To document the preferred mineral association, a ratio is introduced that illustrates how closely cassiterite—the only Sn mineral of economic relevance—is associated with chlorite, fluorite, and sulfides. This so-called MAMA ratio illustrates the strongly preferred association between cassiterite and chlorite. The results also illustrate that the abundance of rock-forming chlorite may be used as a proxy for the abundance of the much less common cassiterite. This proxy is well-suited to sort ore from poorly mineralized/unmineralized rock fragments early during the beneficiation process. Such separation may well be achieved by using a short wave infrared detector that is already deployed in commercially available sorting equipment. The case study illustrates the inherent link between the processes responsible for ore genesis, the definition of geometallurgical domains, and the selection of suitable beneficiation strategies.

Geostatistical Simulation of Geochemical Compositions in the Presence of Multiple Geological Units: Application to Mineral Resource Evaluation

An accurate prediction of benefit in ore deposits with heterogeneous spatial variations requires the definition of geological domains that differentiate the types of mineralogy, alteration, and lithology, as well as the prediction of full mineral and geochemical compositions within each modeled domain and across boundaries between different domains. This paper proposes and compares various approaches (different combinations of log-ratio transformation, Gaussian and flow anamorphosis, and deterministic or probabilistic geological models) for geostatistical simulation of geochemical compositions in the presence of several geological domains. Different approaches are illustrated through an application to a nickel–cobalt laterite deposit located in Western Australia. Four rock types (ferruginous, smectite, saprolite, and ultramafic) are considered to define compositionally homogeneous domains. Geochemical compositions are comprised of six different components of interest (Fe, Al, Mg, Ni, Co, and Filler). The results suggest that the flow anamorphosis is a vital element for geostatistical modeling of geochemical composition due to its invariance properties and capability for reproducing complex patterns in input data, including: presence of outliers, presence of several populations (due to the presence of several geological domains), nonlinearity, and heteroscedasticity.

Joint Compositional Calibration: An Example for U–Pb Geochronology

This contribution explores several issues arising in the measurement of a (geo)chemical composition with Laser Ablation Inductively Coupled Plasma Mass Spectrometry (LA-ICP-MS), specially in the case that the quantities of interest are linear functions of (log)-ratios. These quantities are scale invariant, but in general cannot be estimated without taking into account possible additive noise effects of the instrumentation, incompatible with a purely compositional approach. The proposed ways to a solution heavily build upon the multi-Poisson distribution, highlighting the counting nature of the readings delivered by these instruments. The model can be fitted using a generalized linear model formalism, and it allows for a joint calibration of all components at once. Relevance of these considerations is shown with some simulation studies and in a real case of multi-isotopic geochronological analyses. Results suggest that the most critical aspect of this analytical technique is the assumption that the amount of ablated mass per second between samples of unknown and known compositions is similar (matrix matching): if this cannot be ensured, absolute estimations of the abundance of each of these isotopes fails, while their (log)ratios are perfectly estimable. This opens the door to using the model for a joint calibration by loosening the condition of matrix matching and using several standards of different composition.