Estimation of covariance matrix from geochemical data with observations below detection limits

Multivariate statistical analyses have been extensively applied to geochemical measurements to analyze and aid interpretation of the data. Estimation of the covariance matrix of multivariate observations is the first task in multivariate analysis. However, geochemical data for the rare elements, especially Ag, Au, and platinum-group elements, usually contain observations the below detection limits. In particular, Instrumental Neutron Activation Analysis (INAA) for the rare elements produces multilevel and possibly extremely high detection limits depending on the sample weight. Traditionally, in applying multivariate analysis to such incomplete data, the observations below detection limits are first substituted, for example, each observation below the detection limit is replaced by a certain percentage of that limit, and then the standard statistical computer packages or techniques are used to obtain the analysis of the data. If a number of samples with observations below detection limits is small, or the detection limits are relatively near zero, the results may be reasonable and most geological interpretations or conclusions are probably valid. In this paper, a new method is proposed to estimate the covariance matrix from a dataset containing observations below multilevel detection limits by using the marginal maximum likelihood estimation (MMLE) method. For each pair of variables, sayY andZ whose observations containing below detection limits, the proposed method consists of three steps: (i) for each variable separately obtaining the marginal MLE for the means and the variances, , , , and forY andZ: (ii) defining new variables by and and lettingA=C+D andB=C–D, and obtaining MLE for variances, and forA andB; (iii) estimating the correlation coefficient _YZ by and the covariance _YZ by . The procedure is illustrated by using a precious metal geochemical data set from the Fox River Sill, Manitoba, Canada.     

Estimation for partially observed Markov processes

Many stochastic process models for environmental data sets assume a process of relatively simple structure which is in some sense partially observed. That is, there is an underlying process (X_n, n 0) or (X_t, t 0) for which the parameters are of interest and physically meaningful, and an observable process (Y_n, n 0) or (Y_t, t 0) which depends on the X process but not otherwise on those parameters. Examples are wide ranging: the Y process may be the X process with missing observations; the Y process may be the X process observed with a noise component; the X process might constitute a random environment for the Y process, as with hidden Markov models; the Y process might be a lower dimensional function or reduction of the X process. In principle, maximum likelihood estimation for the X process parameters can be carried out by some form of the EM algorithm applied to the Y process data. In the paper we review some current methods for exact and approximate maximum likelihood estimation. We illustrate some of the issues by considering how to estimate the parameters of a stochastic Nash cascade model for runoff. In the case of k reservoirs, the outputs of these reservoirs form a k dimensional vector Markov process, of which only the kth coordinate process is observed, usually at a discrete sample of time points.     

Application of X-ray computed tomography for analyzing cleat spacing and cleat aperture in coal samples

New techniques to determine distributions of cleat aperture, cleat orientation and cleat spacing from CT scans have been developed. For cleat orientation and spacing distributions, two different coal blocks were scanned. The CT scans have been analyzed for the three orthogonal directions. Histograms of the cleat orientations are bimodal, expressing the typical cleat texture of face and butt cleats and bedding perpendicular relaxation fractures. Deviations up to 20° from the peak values in the cleat orientation distributions were used as input for automated image analysis of cleat spacing. Distributions of the cleat spacing measurements are related to the face and butt cleat directions. The term “relevant cleat length” is introduced as a measure to extract the amount of cleat length involved with the cleat spacing measurements. The ratio ranges from 0.03 to 0.38 and expresses the difference in cleat texture in both samples. Cleat spacing versus relevant cleat length shows sample specific patterns for face cleat, butt cleat and bedding. To describe cleat aperture quantitatively, peak height and missing attenuation have been used. The image of a cleat was seen as a convolution of a rectangular fracture profile with a Gaussian point spread function.     

Correlations in aftershock and seismicity patterns

Correlations in space and time play a fundamental role in earthquake processes. One direct manifestation of the effects of correlations is the occurrence of aftershocks due to the stress transfer in the vicinity of a main shock. Less obvious and more speculative changes in correlations may occur in the background seismicity before large earthquakes. Using statistical physics it is possible to introduce a measure of spatial correlations through a correlation length. This quantity characterizes how local fluctuations can influence the occurrence of earthquakes over distances comparable with the correlation length. In this work, the physical basis of spatial correlations of earthquakes is discussed in the context of critical phenomena and the percolation problem. The method of two-point correlation function is applied to the seismicity of California. Well defined variations in time of the correlation length are found for aftershock sequences and background seismicity. The scaling properties of our obtained distributions are analyzed with respect to changes in several scaling parameters such as lower magnitude cutoff of earthquakes, the maximum time interval between earthquakes, and the spatial size of the area considered. This scaling behavior can be described in a unified manner by utilizing the multifractal fit. Utilizing the percolation approach the time evolution of clusters of earthquakes is studied with the correlation length defined in terms of the radius of gyration of clusters. This method is applied to the seismicity of California.     

Modèles géostatistiques de concentrations ou de débits le long des cours d'eau

Estimating concentrations or flow rates along a stream network requires specific models. Two classes of models, recently proposed in the literature, are generalized, to the intrinsic case in particular. We present a global construction by ‘streams’, i.e. on the whole set of paths between sources and outlet. Combining stationary or intrinsic one-dimensional random functions leads to stationary or intrinsic models on segments, with discontinuities at the forks. A construction from outlet to sources, leads to stationary or intrinsic models on each stream, without any discontinuity at the forks. The linear variogram is found as a particular case. The extension to the linear model of coregionalization is immediate, allowing a multivariate modelling of concentrations. To cite this article: C. de Fouquet, C. Bernard-Michel, C. R. Geoscience 338 (2006).     

Mineralogical and chemical transformation of Oldoinyo Lengai natrocarbonatites, Tanzania

The minerals of Oldoinyo Lengai natrocarbonatite lavas are unstable under atmospheric conditions. Subsolidus mineral assemblages in natrocarbonatites were studied in 105 samples from contemporary eruptions ranging from present day to about 100 years old. The subsolidus minerals in natrocarbonatites were formed (i) along cracks on the lava surface from hot gases escaping during cooling, (ii) as atmospheric alteration by solution of water-soluble minerals, in particular halides and gregoryite, and by hydration of nyerereite under the influence of meteoric water and (iii) by reaction with fumarole gases. After solidification, the lavas were cut by a network of thin cracks, the edges of which are covered by polymineralic encrustations. Samples collected 2–24 h after eruption contain nahcolite, trona, sylvite, and halite with accessory kalicinite and villiaumite. Atmospheric humidity results immediately (≥ 2 h after eruption) in alteration of black lavas that is marked by the appearance of white powdery thermonatrite with nahcolite on the lava surface. Subsequent reaction (weeks, months, years) of natrocarbonatite with meteoric water and the atmosphere results in the formation of pirssonite, gaylussite, shortite, trona, thermonatrite, nahcolite and calcite. Generally, the first important step is the formation of pirssonite and the end-members are calcite carbonate rocks or loose aggregates. Fumarolic activity is common for the active northern crater of the volcano. Reaction of hot (54–141 °C) fumarolic gases with natrocarbonatite leads to the formation of sulphur, gypsum, calcite, anhydrite, monohydrocalcite, barite and celestine. Changes in mineralogy of the natrocarbonatite lead to substantial chemical transformation. The most obvious chemical changes in this process are the loss of Na, K, Cl and S, combined with an increase in H₂O, Ca, Sr, Ba, F and Mn. The oxygen and carbon isotopic composition of altered natrocarbonatites shows a significant shift from the primary “Lengai Box” to high values of δ¹⁸O and δ¹³C. Calcite exhibits δ¹³C values between − 2‰ and − 4‰ PDB and δ¹⁸O values of + 23‰ to + 26‰ SMOW. The observed assemblages of secondary minerals formed by reaction with atmosphere and meteoric water, the changes in chemical composition of the natrocarbonatite and field observations suggest that alteration of natrocarbonatite is an open-system low-temperature process. It takes place at temperatures between 8 and 43 °C with the addition of H₂O to the system and the removal of Na, K, Cl and S from the carbonatites. Low-temperature thermodynamic models developed for alkali carbonate systems can be used for the interpretation of Oldoinyo Lengai subsolidus mineralization.     

A comparison of chironomid biostratigraphy from Lake Vuolep Njakajaure with vegetation, lake-level, and climate changes in Abisko National Park, Sweden

Chironomid remains from the sediment of Lake Vuolep Njakajaure reflect limnological conditions resulting from changing climate and vegetation throughout the Holocene, but do not strictly follow accepted climate trends or the vegetation history based on regional pollen and macrofossil analyses. Chironomid community changes appear to be influenced by organic nutrient input from the surrounding catchment vegetation and lake hydrology, both of which are indirectly responding to some combination of climate change, hypolimnetic oxygen concentration, and changes in basin morphology. The chironomid-based quantitative mean July air-temperature reconstruction differs from other regional quantitative records; this discrepancy is likely related to limnological conditions particular to Lake Vuolep Njakajaure. Comparison of a northern Swedish temperature transfer function and one from western Canada reveals differences in the mean July air-temperature optima of several common taxa, suggesting that the existing conservative estimates of Holocene climate change in northern Sweden may be underestimated due to the limited temperature gradient captured by the Swedish training set.