A generalization of extendedQ-mode factor analysis to data matrices with variable row sums

The extended method of Q-mode factor analysis developed by Miesch for data matrices with constant row sums is generalized to data matrices with variable row sums. With the algorithm provided it is possible to compute factor scores in the metric of the original data and compute goodness-of-fit statistics and model geological systems unconstrained by constancy of row sums of data points.     

The use of correspondence analysis in geochemistry

When factor analysis is used in geochemistry, it may be useful for factors to be transformed by rotations in order to be identified either to the end members of a mixing model (Miesch, 1976a),or to known chemical equilibriums. It requires that the formula for recomputing data from the factors may be written in a factored manner, which is generally not the case in correspondence analysis. The present paper shows that this becomes possible with data having constant row sums. As an example, the method is tested on the lavas of Paricutin Volcano, already examined by using an extended Q-mode factor analysis (Miesch, 1979).Recomputation of the data after projection gives simular results for both methods. Otherwise, the fact that correspondence analysis provides centered factors makes it well suited to the study of chemical reactions leading to constant mass transformations.     

Multigroup discrimination of river, beach, and dune sands using axial-ratio sample statistics of quartz grains

Optimal discrimination among several groups can be achieved by simultaneous diagonalization of pooled within-group, W, and among-group, A, sums of squares and cross-product matrices formed by utilizing axial-ratio sample statistics of quartz grains belonging to different sieve grades. This method maximizes the ratio of among-group cross products to within-group cross product quadratic forms (V'AV/V'WV)and simultaneously yields discriminant scores whose correlation coefficients are zero for group means as well as for within each group. This procedure enables a simple Euclidean distance measure for partitioning the discriminant space for assignment. Although W^–1 and Amatrices are symmetric, the W^–1 Amatrix needed for multigroup discrimination is asymmetric and hence the eigenstructure of W^–1 Ais obtained by simultaneous diagonalization of Wand Amatrices. The first four sample statistics (mean, standard deviation, skewness, kurtosis) of normalized axial-ratios are required for discrimination, although the mean and standard deviation are the most important discriminators.     

Efficient Computation of Linearized Cross-Covariance and Auto-Covariance Matrices of Interdependent Quantities

In many geostatistical applications, spatially discretized unknowns are conditioned on observations that depend on the unknowns in a form that can be linearized. Conditioning takes several matrix–matrix multiplications to compute the cross-covariance matrix of the unknowns and the observations and the auto-covariance matrix of the observations. For large numbers n of discrete values of the unknown, the storage and computational costs for evaluating these matrices, proportional to n ², become strictly inhibiting. In this paper, we summarize and extend a collection of highly efficient spectral methods to compute these matrices, based on circulant embedding and the fast Fourier transform (FFT). These methods are applicable whenever the unknowns are a stationary random variable discretized on a regular equispaced grid, imposing an exploitable structure onto the auto-covariance matrix of the unknowns. Computational costs are reduced from O(n ²) to O(nlog₂ n) and storage requirements are reduced from O(n ²) to O(n).     

Improved Bell's method for the stability analysis of slopes

A new procedure based on the approximation to the total normal pressure along the slip surface is developed to compute the factor of safety of slopes for slip surfaces of all shapes. By taking the whole sliding body, instead of an individual slice, as the loaded object, all the equilibrium equations are formulated according to the three‐moment equilibrium conditions rather than the two force equilibrium conditions and one‐moment equilibrium condition. The system of nonlinear equations deduced in this way is well‐scaled and enjoys excellent numerical properties such as the existence of solution with a positive factor of safety, a nearly unlimited scope of convergence and a rapid convergence rate associated with the Newton method. In the case of ?_u =0—the situation where no drainage and no consolidation are involved, furthermore, the system has a unique solution and the factor of safety has an explicit expression. Some typical examples are analyzed to demonstrate the numerical properties of the proposed procedure. Copyright © 2009 John Wiley & Sons, Ltd.     

松科二井深地震反射数据Q值分析及其对含气层位的确定

地震波在地下介质中传播时,地震波能量会出现一定程度的衰减,品质因子Q作为衡量地下岩石吸收衰减属性的重要参数,对描述岩性特征以及预测油气分布具有重要意义。本文针对深地震反射数据,利用基于S变换谱比法的Q值分析技术,获得了更加准确的Q值。以松辽盆地沙河子组为主要目标层,对穿过松科二井的叠后深地震反射剖面进行Q值计算,生成Q值剖面,总结出沙河子组Q值分布特征,同时结合松科二井测井、分层和气测异常资料进行分析,推断沙河子组为含气储层,为下一步深部储层预测提供思路,为东北地区深地勘查工程提供服务。     

A method for determining the reversibility of a Markov sequence

This paper describes, given a tally matrix with strictly positive entries, a method to determine whether the associated Markov process is reversible, and (for reversible Markov processes) methods to compute the reversibility matrix from the tally matrix. If the tally matrixN is symmetric, then it is shown that the Markov process must be reversible and the reversibility matrixC equalss (R ^–1NR^–1), whereR is the diagonal matrix whosei ^th diagonal entry is the sum of the entries of thei ^th row ofN (for everyi) ands denotes the sum of all the entries ofN. Because a symmetric tally matrix is of special importance in applications, a ² test is proposed for determining, in the presence of experimental errors, whether such a matrix is symmetric.     

Estimation of Probabilities of Three Kinds of Petrologic Hypotheses with Bayes Theorem

Physical-chemical explanations of the causes of variations in rock suites are evaluated by comparing predicted to measured compositions. Consistent data turn an explanation into a viable hypothesis. Predicted and measured values seldom are equal, creating problems of defining consistency and quantifying confidence in the hypthesis. Bayes theorem leads to methods for testing alternative hypotheses. Information available prior to data collection provides estimates of prior probabilities for competing hypotheses. After consideration of new data, Bayes theorem updates the probabilities for the hypotheses being correct, returning posterior probabilities. Bayes factors, B, are a means of expressing Bayes theorem if there are two hypotheses, H ₀ and H ₁. For fixed values of the prior probabilities, B > 1 implies an increased posterior probability for H ₀ over its prior probability, whereas B < 1 implies an increased posterior probability for H ₁ over its prior probability. Three common problems are: (1) comparing variances in sets of data with known analytical uncertainties, (2) comparing mean values of two datasets with known analytical uncertainties, and (3) determining whether a data point falls on a predicted trend. The probability is better than 0.9934 that lava flows of the 1968 eruption of Kilauea Volcano, Hawaii, are from a single magma batch. The probability is 0.99 that lava flows from two outcrops near Mount Edziza, British Columbia, are from different magma batches, suggesting that the two outcrops can be the same age only by an unlikely coincidence. Bayes factors for hypotheses relating lava flows from Volcano Mountain, Yukon Territory, by crystal fractionation support the hypothesis for one flow but the factor for another flow is so small it practically guarantees the fractionation hypothesis is wrong. Probabilities for petrologic hypotheses cannot become large with a single line of evidence; several data points or datasets are required for high probabilities.     

Minres factor analysis of hypersthene data, Strathcona Mine (Ontario, Canada)

This is the first application of minimum residuals (minres),a type of factor analysis, in the study of hypersthene minerals from a mafic norite formation at the Strathcona Mine near Sudbury, Ontario. Minres, because it yields highest communalities for some variables, is preferred to other types of factoring solutions including a common factor model with Chayes' null correlations as factor input. Oblique rotation of factors is rejected as a model for statistical and geochemical reasons. A five oxide-variable model that reasonably well determines hypersthene is reduced by minres to a two-factor model which is statistically significant. Because of the small number of variables in the analysis, it is difficult to interpret the isolated factors in terms of specific geologic processes. The factors, however, even if surrogate, are linked with substitution phenomena in the hypersthene.     

In Situ Determination of First‐Row Transition Metal,Ga and Ge Abundances in Geological Materials via Medium‐Resolution LA‐ICP‐MS

An in situ, medium‐resolution LA‐ICP‐MS method was developed to measure the abundances of the first‐row transition metals, Ga and Ge in a suite of geological materials, namely the MPI‐DING reference glasses. The analytical protocol established here hinged on maximising the ablation rate of the ultraviolet (UV) laser system and the sensitivity of the ICP‐MS, as well minimising the production of diatomic oxides and argides, which serve as the dominant sources of isobaric interferences. Non‐spectral matrix effects were accounted for by using multiple external calibrators, including NIST SRM 610 and the USGS basaltic glasses BHVO‐2G, BIR‐1G and BCR‐2G, and utilising ⁴³Ca as an internal standard. Analyses of the MPI‐DING reference glasses, which represent geological matrices ranging from basaltic to rhyolitic in composition, included measurements of concentrations as low as < 10⁰ μg g^?1 and as high as > 10⁴ μg g^?1. The new data reported here were found to statistically correlate with the ‘preferred’ reference values for these materials at the 95% confidence level, though with significantly better precision, typically on the order of ≤ 3% (2s_m). This analytical method may be extended to any matrix‐matched geological sample, particularly oceanic basalts, silicate minerals and meteoritic materials.     

A type curve approach to estimatingp for ap-normal transformation

The p-normal transformation plays an important role in reservoir characterization for data sets that are neither normally nor log-normally distributed. The key step in the transformation is to estimate the value of pfor a given data set. Even though there are several ways to determine p,these are more inconvenient than the quicker and easier type curve approach to estimate pwe present in this paper. In addition, the method provides the p-normal transformation with a visual interpretation. We demonstrate the technique by analyzing reservoir permeability and porosity data from the East Velma West Block Sims Sand Unit, Oklahoma.     

A geomicrobiological study of soils collected from auriferous areas of Argentina

Geomicrobial and geochemical studies were carried out in Argentina (Patagonia, Chubut Province) on four Au and polymetallic sulfide vein-type deposits. A horizon soils were analyzed for Bacillus reacting to lecithin [Bacillus L.(+)], Au and 12 additional elements. In two of the four sampling sites, exhibiting known and relatively simple mineralized structures, Bacillus L.(+) populations are clearly related to Au, As, Pb, Zn, Cu-sulfide mineralization. In areas containing more complex mineralized structures, the spatial relationship between Bacillus L.(+) and metals in the A horizon is more difficult to interpret. Results of a factor analysis performed on all analytical data (n = 130) suggest a partial relationship between Bacillus L.(+) and Au-As-Y pedochemical associations located above known Au mineralization. Bacillus L.(+) was first analyzed in Argentina in December 1994 and re-analyzed in Belgium five to seven months later. Most of the Bacillus contents (85%) of the Belgian tests are higher than those determined in Argentina. The present results and data of a previous study in Mexico (Melchior et al., 1994a; Melchior et al., 1994b) suggest that this may be the result of temperature variations during sample storage between periods of microbial analysis. From a strictly analytical point of view, the geomicrobial method is not an accurate, reproducible technique. However, Bacillus L.(+) can be used as a microbiological indicator of Au and polymetallic mineralization at a reconnaissance-level regional survey. At a local scale, this microbiological tool should be combined with classical exploration techniques such as soil geochemistry. It is recommended that the collection of all A horizon samples (for microbial study) should be accompanied by B or C horizon soils (for potential geochemical study, after prioritizing targets) so that a second field sampling program does not have to be undertaken.     

Scaling variables and interpretation of eigenvalues in principal component analysis of geologic data

The dominant feature distinguishing one method of principal components analysis from another is the manner in which the original data are transformed prior to the other computations. The only other distinguishing feature of any importance is whether the eigenvectors of the inner product-moment of the transformed data matrix are taken directly as the Q-mode scores or scaled by the square roots of their associated eigenvalues and called the R-mode loadings. If the eigenvectors are extracted from the product-moment correlation matrix, the variables, in effect, were transformed by column standardization (zero means and unit variances), and the sum of the p-largest eigenvalues divided by the sum of all the eigenvalues indicates the degree to which a model containing pcomponents will account for the total variance in the original data. However, if the data were transformed in any manner other than column standardization, the eigenvalues cannot be used in this manner, but can only be used to determine the degree to which the model will account for the transformed data. Regardless of the type of principal components analysis that is performed—even whether it is Ror Q-mode—the goodness-of-fit of the model to the original data is given better by the eigenvalues of the correlation matrix than by those of the matrix that was actually factored.     

Electron density distribution and bond critical point properties for forsterite, Mg2 SiO4, determined with synchrotron single crystal X-ray diffraction data

A generalized X-ray scattering factor model experimental electron density distribution has been generated for the orthosilicate forsterite, using an essentially extinction and absorption free set of single crystal diffraction data recorded with intense, high energy synchrotron X-ray radiation (E=100.6 keV). A refinement of the model converged with an R(F)=0.0061. An evaluation of the bond critical point, bcp, properties of the distribution at the (3, –1) stationary points for the SiO and MgO bonded interactions, yielded values that agree typically within ~5%, on average, with theoretical values generated with quantum chemical computational strategies, using relatively robust basis sets. On the basis of this result, the modeling of the experimental distribution is considered to be adequate. As the bcp properties increase in magnitude, the MgO and SiO bonds decrease in length as calculated for a number of rock forming silicates. As asserted by Coppens (X-ray charge densities and chemical bonding. Oxford University Press, Oxford, 1997), large negative ²(r_c) values, characteristic of shared interactions involving first row atoms, may not be characteristic of closed shell covalent bonded interactions involving second row Si, P and S atoms bonded to O. This study adds new evidence to the overall relatively good agreement between theoretical bcp properties generated with computational quantum strategies, on the one hand, and experimental properties generated with single crystal high energy synchrotron diffraction data on the other. The similarity of results not only provides a basis for using computational strategies for studying and modeling structures, defects and the reactivity of representative structures, but it also provides a basis for improving our understanding of the crystal chemistry of earth materials and the character of the SiO bonded interaction.     

Upper bound solutions of bearing capacity of strip footing by pseudo-dynamic approach

This paper presents the pseudo-dynamic analysis of seismic bearing capacity of a strip footing using upper bound limit analysis. However, in the literature, the pseudo-static approach was frequently used by several researchers to compute the seismic bearing capacity factor theoretically, where the real dynamic nature of the earthquake accelerations cannot be considered. Under the seismic conditions, the values of the unit weight component of bearing capacity factor N _γE are determined for different magnitudes of soil friction angle, soil amplification and seismic acceleration coefficients both in the horizontal and vertical directions. The results obtained from the present study are shown both graphically as well as in the tabular form. It is observed that the bearing capacity factor N _γE decreases significantly with the increase in seismic accelerations and amplification. The results are thoroughly compared with the existing values in the literature and the significance of the present methodology for designing the shallow footing is discussed.     

Metamorphic evolution of the Maud Belt: P–T–t path for high-grade gneisses in Gjelsvikfjella, Dronning Maud Land, East Antarctica

A metamorphic petrological study, in conjunction with recent precise geochronometric data, revealed a complex P–T–t path for high-grade gneisses in a hitherto poorly understood sector of the Mesoproterozoic Maud Belt in East Antarctica. The Maud Belt is an extensive high-grade, polydeformed, metamorphic belt, which records two significant tectono-thermal episodes, once towards the end of the Mesoproterozoic and again towards the late Neoproterozoic/Cambrian. In contrast to previous models, most of the metamorphic mineral assemblages are related to a Pan-African tectono-thermal overprint, with only very few relics of late Mesoproterozoic granulite-facies mineral assemblages (M₁) left in strain-protected domains. Petrological and mineral chemical evidence indicates a clockwise P–T–t path for the Pan-African orogeny. Peak metamorphic (M_2b) conditions recorded by most rocks in the area (T = 709–785 °C and P = 7.0–9.5 kbar) during the Pan-African orogeny were attained subsequent to decompression from probably eclogite-facies metamorphic conditions (M_2a).The new data acquired in this study, together with recent geochronological and geochemical data, permit the development of a geodynamic model for the Maud Belt that involves volcanic arc formation during the late Mesoproterozoic followed by extension at 1100 Ma and subsequent high-grade tectono-thermal reworking once during continent–continent collision at the end of the Mesoproterozoic (M₁; 1090–1030 Ma) and again during the Pan-African orogeny (M_2a, M_2b) between 565 and 530 Ma. Post-peak metamorphic K-metasomatism under amphibolite-facies conditions (M_2c) followed and is ascribed to post-orogenic bimodal magmatism between 500 and 480 Ma.     

On the swelling potential of compacted high plasticity clays

We conducted a study of the spatial distributions of seismicity and earthquake hazard parameters for Turkey and the adjacent areas, applying the maximum likelihood method. The procedure allows for the use of either historical or instrumental data, or even a combination of the two. By using this method, we can estimate the earthquake hazard parameters, which include the maximum regional magnitude Mˆ_max, the activity rate of seismic events and the well-known bˆ value, which is the slope of the frequency-magnitude Gutenberg-Richter relationship. These three parameters are determined simultaneously using an iterative scheme. The uncertainty in the determination of the magnitudes was also taken into consideration. The return periods (RP) of earthquakes with a magnitude M ≥ m are also evaluated. The whole examined area is divided into 24 seismic regions based on their seismotectonic regime. The homogeneity of the magnitudes is an essential factor in such studies. In order to achieve homogeneity of the magnitudes, formulas that convert any magnitude to an M_S-surface scale are developed. New completeness cutoffs and their corresponding time intervals are also assessed for each of the 24 seismic regions. Each of the obtained parameters is distributed into its respective seismic region, allowing for an analysis of the localized seismicity parameters and a representation of their regional variation on a map. The earthquake hazard level is also calculated as a function of the form Θ = (Mˆ_max,RP_6.0), and a relative hazard scale (defined as the index K) is defined for each seismic region. The investigated regions are then classified into five groups using these parameters. This classification is useful for theoretical and practical reasons and provides a picture of quantitative seismicity. An attempt is then made to relate these values to the local tectonics.     

An algorithm andFortran-iv program for large-scaleQ-mode factor analysis and calculation of factor scores

An algorithm andFortran-iv computer program,cabfac, forQ-mode factor analysis is described. The program will accept up to 1500 items and 50 variables on a moderate-size computer.     

Bidirectional Reflectance of Gaussian Random Surfaces and Its Scaling Properties

Using rigorous probabilistic techniques we define and compute the shadowing factor, the conditional mean slope and the conditional standard deviation of the slope for a random rough surface described as an isotropic Gaussian stochastic process with Gaussian autocorrelation function. We use these quantities in order to obtain the bidirectional reflectance distribution function for incoherent light scattering by rough surfaces. The calculated quantities depend on reduced, affine invariant parameters, such that changing roughness is equivalent to rescaling the slopes of incident and emergent direction. We discuss some possible applications of these scaling properties to remote sensing.