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.     

Spatial relationships of multivariate data

Spatial factor analysis (SFA) is a multivariate method that determines linear combinations of variables with maximum autocorrelation at a given lag. This is achieved by deriving estimates of auto-/cross-correlations of the variables and calculating the corresponding eigenvectors of the covariance quotient matrix. A two-point spatial factor analysis model derives factors by the formation of transition matrixU comparing auto-/cross-correlations at lag 0,R ₀, with those at a specified lag d,R _d, expressed asU _d=R ₀ ^–1 R_d. The matrixU _d can be decomposed into its spectral components which represent the spatial factors. The technique has been extended to include three points of reference. Spatial factors can be derived from the relationship:

Multi-Scale Texture Modeling

Patterns of crystallographic preferred orientation are referred to as texture. The specific subject of texture analysis is the experimental determination and interpretation of the statistical distribution of orientations of crystals within a specimen of polycrystalline material, which could be metals or rocks. The objective is to relate an observed pattern of preferred orientation to its generating processes and vice versa. In geosciences, texture of minerals in rocks is used to infer constraints on their tectono-metamorphic history. Since most physical properties of crystals, such as elastic moduli, the coefficients of thermal expansion, or chemical resistance to etching depends on crystal symmetry and orientation, the presence of texture imparts directional properties to the polycrystalline material. A major issue of mathematical texture analysis is the resolution of the inverse problem to determine a reasonable orientation density function on SO(3) from measured pole intensities on , which relates to the inverse of the totally geodesic Radon transform. This communication introduces a wavelet approach into mathematical texture analysis. Wavelets on the two-dimensional sphere and on the rotational group SO(3) are discussed, and an algorithms for a wavelet decomposition on both domains following the ideas of Ta-Hsin Li is given. The relationship of these wavelets on both domains with respect to the totally geodesic Radon transform is investigated. In particular, it is shown that the Radon transform of these wavelets on SO(3) are again wavelets on . A novel algorithm for the inversion of experimental pole intensities to an orientation density function based on this relationship is developed.     

Maximum entropy and shear strain of shear zone

The principle of maximum entropy can be used to determine the shear strain in natural shear zones. When the margin of a shear zone is assumed, the principle leads to the truncated exponential distribution of the shear strain. Ifx is the distance remote from the shear zone center, which possesses the maximum shear strain, the shear strain (x) is given by

Single crystal raman studies of pyrite-type RuS2, RuSe2, OsS2, OsSe2, PtP2, and PtAs2

Single crystal Raman spectra of pyrite-type RuS₂, RuSe₂, OsS₂, OsSe₂, PtP₂, and PtAs₂ are presented and discussed with reference to the energies of the X-X stretching modes _x-x (A _g, F _g) and the X₂ librations (E, 2F_g). The main results obtained are (i) strong Raman resonance effects, (ii) different sequences for _x-x (A _g) and (E _g), i.e., R_{x_2 } $$ " align="middle" border="0"> for PtP₂ and PtAs₂ and R_{x_2 } $$ " align="middle" border="0"> for OsS₂, owing to the interplay of intraionic and interionic lattice forces, (iii) greater strengths for the intraionic P-P and As-As bonds compared to the S-S and Se-Se bonds, respectively, and (iv) a strong influe_gnce of the metal ions on the strength of the X-X bonds.This is contribution LXI of a series of papers on lattice vibration spectra     

Empirical estimation of isostatic uplift using the lake-tilting method at lake Fegen and lake S?ven, southwestern Sweden

A nonuniform glacio-isostatic uplift results in differential uplift for different parts of a lake. If the lake outlet is situated in the area with the greatest rate of uplift, then the lake will be continuously transgressed. Ancient lake levels can be estimated by dating transgressed peat at different depths in such a lake. Two lakes in southwestern Sweden have been investigated by this method and the course of glacio-isostatic uplift has been determined empirically. The uplift can be expressed by an exponential function through the following formula

Estimating relative bed-load contributions by lithology counts

In gravel-bedded streams where bed material of a tributary differs distinctly in lithology from that of the main stream, rock-type percentages can be used to estimate bed-load contributions of the two streams. The rock type that shows the greatest difference in abundance between the two streams is selected as the indicator lithology. Percentages of this lithology are estimated in both the main stream and tributary stream above their junction, and also in the main stream at a distance sufficiently downstream from the junction to allow complete mixing. The fraction of bed load contributed by the main stream, p,is estimated by ,where is an estimate of the proportion of indicator rock fragments in the bed of the main stream above the junction, is an estimate of the proportion in the bed of the tributary above the junction, and is an estimate of the proportion in the bed of the main stream below the junction. The variance of is obtained as var ( )= [p₁q₁(p_r – p₂)²/n(p₁ – p₂)⁴] + [p₂q₂(p_r – p₁)²/n(p₁ – p₂)⁴] + [p_rq_r/n(p₁ – p₂)²].Although no estimate of actual quantity of bed load is provided, the indicator rock technique supplies data that can serve as a check on data obtained by means of empirical formulas or actual transport measurements.     

A simple method for estimating excess pressure over horsts from seismic sections

Observations in the North Sea Basin max indicate significant overpressure in sediments over horst blocks but not over grabens at the same submudline depth. The purpose is to show that over a horst, of top width W, with grabens on either side of top widths G₁ and G₂, respectively, the equivalent mud density. _r can be estimated approximately from the simple equation.

A Coupled Riverine-Marine Fractionation Model for Dissolved Rare Earths and Yttrium

Fractionation of yttrium (Y) and the rare earth elements (REEs) begins in riverine systems and continues in estuaries and the ocean. Models of yttrium and rare earth (YREE) distributions in seawater must therefore consider the fractionation of these elements in both marine and riverine systems. In this work we develop a coupled riverine/marine fractionation model for dissolved rare earths and yttrium, and apply this model to calculations of marine YREE fractionation for a simple two-box (riverine/marine) geochemical system. Shale-normalized YREE concentrations in seawater can be expressed in terms of fractionation factors ( _ij ) appropriate to riverine environments ( ) and seawater ( ):

Topological properties of random crack networks

The junctions of cracks in mudcrack, patterned ground, and columnar joint patterns can be categorized into Y, T,and Xtypes. The mean number of sides, ,to the polygonal areas in such nets is = 2(2J_T + 3J_Y + 4J_X)/(J_T + J_Y + 2J_X)where J_T, J_Y,and J_X are the proportions of T, Y,and Xjunctions, respectively.     

Aluminum partitioning between olivine and ultrabasic silicate liquid to 6 GPa

Trace element analyses of 1-atm and high-pressure experiments show that in komatiite and peridotite, the olivine (OL)/liquid (L) distribution coefficient for Al₂O₃ ( ) increases with pressure and temperature. Olivine in equilibrium with liquid accepts as much as 0.2 wt% Al₂O₃ in solution at 6 GPa. Convergence to equilibrium compositions at this high level is shown by cation diffusion of Al into synthetic forsterite crystals of low-Al contents in the presence of melt. Convergence to low-Al equilibrium compositions at lower P and T is shown by diffusion of Al out of synthetic forsterite with high initial Al content. Isobaric and isothermal experimental data subsets reveal that temperature and pressure variations both have real effects on . Variation in silicate melt composition has no detectable effect on within the limited range of experimentally investigated mixtures. Least-squares regression for 24 experiments, using komatiite and peridotite, performed at 1 atm to 6 GPa and 1300 to 1960°C, gives the best fit equation: Increase in with increasingly higher-pressure melting is consistent with incorporation of a spinel-like component of low molar volume into olivine, although other substitutions possibly involving more complex coupling cannot be ruled out. High P-T ultrabasic melting residues, if pristine, may be recognized by the high calculated from microprobe analyses of Al₂O₃ concentrations in residual olivines and estimated Al₂O₃ concentration in the last liquid removed. In general the low levels of Al in natural olivine from mantle xenoliths suggest that pristine residues are rarely recovered.     

Reversed experimental calibration of the garnet-clinopyroxene Fe — Mg exchange thermometer

The temperature-sensitive Fe,Mg exchange equilibrium,

Fitting matrix-valued variogram models by simultaneous diagonalization (Part I: Theory)

Suppose that ¯(x₁),...,¯Z(x_n). are observations of vector-valued random function ¯(x). In the isotropic situation, the sample variogram γ^*(h) for a given lag h is $$\bar \gamma ^ * (h) = \frac{1}{{2N(h)}}\mathop \sum \limits_{s(h)} (\overline Z (x_1 ) - \overline Z (x_1 )) \overline {(Z} (x_1 ) - \overline Z (x_1 ))^T $$ where s(h) is a set of paired points with distance h and N(h) is the number of pairs in s(h).. For a selection of lags h₁, h₂, .... h_k such that N (h₁) > O. we obtain a ktuple of (semi) positive definite matrices $\bar \gamma ^ * (h_{ 1} ),. . . ., \bar \gamma ^ * (h_{ k} )$ . We want to determine an orthonormal matrix B which simultaneously diagonalizes the $\bar \gamma ^ * (h_{ 1} ),. . . ., \bar \gamma ^ * (h_{ k} )$ or nearly diagonalizes them in the sense that the sum of squares of offdiagonal elements is small compared to the sum of squares of diagonal elements. If such a B exists, we linearly transform $\overline Z (x)$ by $\overline Y (x) = B\overline Z (x)$ . Then, the resulting vector function $\overline Y (x)$ has less spatial correlation among its components than $\overline Z (x)$ does. The components of $\overline Y (x)$ with little contribution to the variogram structure may be dropped, and small crossvariograms fitted by straightlines. Variogram models obtained by this scheme preserve the negative definiteness property of variograms (in the matrix-valued function sense). A simplified analysis and computation in cokriging can be carried out. The principles of this scheme arc presented in this paper.     

Some exact sampling distributions for variogram estimators

For equally spaced observations from a one-dimensional, stationary, Gaussian random function, the characteristic function of the usual variogram estimator for a fixed lag k is derived. Because the characteristic function and the probability density function form a Fourier integral pair, it is possible to tabulate the sampling distribution of a function of a using either analytic or numerical methods. An example of one such tabulation is given for an underlying model that is simple transitive.     

Use of Pitzer Equations to Examine the Formation of Mercury(II) Hydroxide and Chloride Complexes in NaClO<Subscript>4</Subscript> Media

The thermodynamic stability constants for the hydrolysis and formation of mercury (Hg²⁺) chloride complexes

Influence of hydrogen on Fe–Mg interdiffusion in (Mg,Fe)O and implications for Earth’s lower mantle

Interdiffusion of Fe and Mg in (Mg,Fe)O has been investigated experimentally under hydrous conditions. Single crystals of MgO in contact with (Mg_0.73Fe_0.27)O were annealed hydrothermally at 300 MPa between 1,000 and 1,250°C and using a Ni–NiO buffer. After electron microprobe analyses, the dependence of the interdiffusivity on Fe concentration was determined using a Boltzmann–Matano analysis. For a water fugacity of ∼300 MPa, the Fe–Mg interdiffusion coefficient in Fe _x Mg_1−x O with 0.01 ≤ x ≤ 0.25 can be described by with and C = −80 ± 10 kJ mol⁻¹. For x = 0.1 and at 1,000°C, Fe–Mg interdiffusion is a factor of ∼4 faster under hydrous than under anhydrous conditions. This enhanced rate of interdiffusion is attributed to an increased concentration of metal vacancies resulting from the incorporation of hydrogen. Such water-induced enhancement of kinetics may have important implications for the rheological properties of the lower mantle.

Thermodynamic modelling of non-convergent ordering in orthopyroxenes: a comparison of classical and Landau approaches

The excess Gibbs free energy due to non-convergent ordering is described by a Landau expansion in which configurational and non-configurational entropy contributions are separated:

Microphenocrystic pyrrhotite from dacite rocks of Satsuma-Iwojima,southwest Kyushu,Japan and the solubility of sulfur in dacite magma

Microphenocrystic pyrrhotites were observed in the glassy groundmass of two dacite rocks from Satsuma-Iwojima, southwest Kyushu, Japan. It suggests that the dacite magma was saturated with respect to pyrrhotite at the time of eruption, and thus the sulfur contents in the groundmass can be taken as the solubility of sulfur in the dacite magma. The solubility of sulfur in the dacite rocks thus calculated is 65 to 72 ppm sulfur at the estimated conditions of T=900±50°C, and atm.     

Kinetic rate laws as derived from order parameter theory II: Interpretation of experimental data by laplace-transformation,the relaxation spectrum,and kinetic gradient coupling between two order parameters

A unifying theory of kinetic rate laws, based on order parameter theory, is presented. The time evolution of the average order parameter is described by $$\langle Q\rangle \propto \smallint P(x)e^{^{^{^{^{^{^{ - xt} } } } } } } dx = L(P)$$ where t is the time, x is the effective inverse susceptibility, and L indicates the Laplace transformation. The probability function P(x) can be determined from experimental data by inverse Laplace transformation. Five models are presented:

1. Polynomial distributions of P(x) lead to Taylor expansions of 〈Q〉 as $$\langle Q\rangle = \frac{{\rho _1 }}{t} + \frac{{\rho _2 }}{{t^2 }} + ...$$
2. Gaussian distributions (e.g. due to defects) lead to a rate law $$\langle Q\rangle = e^{ - x_0 t} e^{^{^{^{^{\frac{1}{2}\Gamma t^2 } } } } } erfc\left( {\sqrt {\frac{\Gamma }{2}} t} \right)$$ where x ₀ is the most probable inverse time constant, Γ is the Gaussian line width and erfc is the complement error integral.
3. Maxwell distributions of P are equivalent to the rate law 〈Q〉∝e^?k√t .
4. Pseudo spin glasses possess a logarithmic rate law 〈Q〉∝lnt.
5. Power laws with P(x)=x ^a lead to a rate law: ln〈Q〉=-(α + 1) ln t.

The power spectra of Q are shown for Gaussian distributions and pseudo spin glasses. The mechanism of kinetic gradient coupling between two order parameters is evaluated.