A general deformation matrix for three-dimensions

A deformation that is obtained by any simultaneous combination of two steady-state progressive deformations: simple shearing and a coaxial progressive deformation, involving or not a volume change, can be expressed by a single transformation, or deformation matrix. In the general situation of simple shearing in a direction non-orthogonal with the principal strains of the coaxial progressive deformation, this deformation matrix is a function of the strain components and the orientation of shearing. In this example, two coordinate systems are defined: one for the coaxial progressive deformation (x_i system), where the principal and intermediate strains are two horizontal coordinate axes, and another for the simple shear (x _i ^t’ system), with any orientation in space. For steady-state progressive deformations, from the direction cosines matrix that defines the orientation of shear strains in the x_i coordinate system, an asymmetric finite-deformation matrix is derived. From this deformation matrix, the orientation and ellipticity of the strain ellipse, or the strain ellipsoid for three-dimensional deformations, can be determined. This deformation matrix also can be described as a combination of a rigid-body rotation and a stretching represented by a general coaxial progressive deformation. The kinematic vorticity number (W _k is derived for the general deformation matrix to characterize the non-coaxiality of the three-dimensional deformation. An application of the deformation matrix concept is given as an example, analyzing the changes in orientation and stretching that variously-oriented passive linear markers undergo after a general two-dimensional deformation. The influence of the kinematic vorticity number, the simple and pure shear strains, and the obliquity between the two deformation components, on the linear marker distribution after deformation is discussed.     

Variogram Fitting by Generalized Least Squares Using an Explicit Formula for the Covariance Structure

In the context of spatial statistics, the classical variogram estimator proposed by Matheron can be written as a quadratic form of the observations. If data are Gaussian with constant mean, then the correlation between the classical variogram estimator at two different lags is a function of the spatial design matrix and the variance matrix. When data are independent with unidimensional and regular support, an explicit formula for this correlation is available. The same is true for a multidimensional and regular support as can be shown by using Kronecker products of matrices. As variogram fitting is a crucial stage for correct spatial prediction, it is proposed to use a generalized least squares method with an explicit formula for the covariance structure (GLSE). A good approximation of the covariance structure is achieved by taking account of the explicit formula for the correlation in the independent situation. Simulations are carried out with several types of underlying variograms, as well as with outliers in the data. Results show that this technique (GLSE), combined with a robust estimator of the variogram, improves the fit significantly.     

Growth of plagioclase rims around metastable kyanite during decompression of high‐pressure felsic granulites (Bohemian Massif)

Samples of high‐pressure felsic granulites from the Bohemian Massif (Variscan belt of Central Europe) characterized by a peak metamorphic (high‐pressure) mineral assemblage of garnet – kyanite – plagioclase – K‐feldspar – quartz ± biotite show well‐developed plagioclase reaction rims around kyanite grains in two microstructural settings. In one setting, kyanite is randomly distributed in the polyphase matrix, whereas in the other setting, it is enclosed within large perthitic K‐feldspar. Kyanite is regarded as a relict of the high‐pressure metamorphic assemblage that became metastable during transition to a low‐pressure overprint. Plagioclase rims from both microstructural settings show continuous outwards decrease of the anorthite content from An_32–25 at the contact with kyanite to An_20–19 at the contact with the matrix or to the perthitic K‐feldspar respectively. Based on mass balance considerations, it is shown that in some cases, a small amount of kyanite was consumed in the rim‐forming reaction to provide the Al₂O₃ component for the growth of plagioclase, whereas in other cases no Al₂O₃ from kyanite was necessary. In a majority of examples, the necessary Al₂O₃ was supplied with CaO and Na₂O from the surrounding matrix material. For kyanite in perthite, a thermodynamic analysis reveals that the kyanite became metastable at the interface with the host perthite at the peak metamorphic pressure, and therefore the plagioclase rim started to grow at ～ 18 kbar. In contrast, kyanite in the polyphase matrix remained stable down to pressures of ～ 16 kbar, and the plagioclase rim only started to grow at a later stage during the decompression. Plagioclase rims around kyanite inclusions within large perthite have a radial thickness of up to 50 μm. In contrast, the radial thickness of plagioclase rims around kyanite in the polycrystalline matrix is significantly larger, up to 200 μm. Another peculiarity is that the plagioclase rims around kyanite in the matrix are polycrystalline, whereas the plagioclase rims around kyanite inclusions in perthitic hosts are single crystals with the same crystallographic orientation as the host perthite. The difference in rim thickness for the two microstructural settings is ascribed to the differences in the efficiency of chemical mass transfer next to the reaction site. The comparatively large thickness of the plagioclase rims grown around kyanite in the matrix is probably due to efficient material transport along the grain and phase boundaries in the matrix. In contrast, chemical mass transfer was comparatively slow in the large perthitic K‐feldspar grains.     

The influence of matrix rheology and vorticity on fabric development of populations of rigid objects during plane strain deformation

The influence of vorticity and rheology of matrix material on the development of shape-preferred orientation (SPO) of populations of rigid objects was experimentally studied. Experiments in plane strain monoclinic flow were performed to model the fabric development of two populations of rectangular rigid objects with object aspect ratios (R_ob) 2 and 3. The density of the rigid object populations was 14% of the total area. Objects were dispersed in a Newtonian and a non-Newtonian, power law matrix material with a power law exponent n of 1.2. The kinematic vorticity number (W_n) of the plane strain monoclinic flow was 1, 0.8 and 0.6 with finite simple shear strain of 4.6, 3.0 and 0.9, respectively. In experiments with R_ob=3, the SPO is strongly influenced by W_n and the material properties of the matrix. Deformation of a power law matrix material and low W_n resulted in a stronger SPO than deformation of a linear viscous matrix and high W_n. Strain localization coupled with particle interaction plays a significant role in the development of a shape-preferred orientation. High strain simple shear zones separate trains of rigid objects that are surrounded by low strain zones with W_n lower than the bulk W_n. In fabrics involving populations of objects with R_ob=2, rheology of the matrix materials does not systematically influence the intensity of the SPO.     

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.     

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).     

Using calculated chemical potential relationships to account for coronas around kyanite: an example from the Bohemian Massif

Corona textures around kyanite, involving for example zoned plagioclase separating kyanite from the matrix, reflect the instability of kyanite with the matrix on changing P–T conditions, commonly related to decompression. The chemical potential gradients set up between the kyanite and the matrix as a consequence of slow Al diffusion drive corona development, with the zoning of the plagioclase reflecting the gradients. Calculated mineral equilibria are used to account for corona textures involving plagioclase ± garnet around kyanite, and replacement of kyanite by plagioclase + spinel symplectite, in quartz + plagioclase + K‐feldspar + garnet + kyanite granulite facies gneiss from the Blanský les massif in the Bohemian massif, Czech Republic. In the garnet‐bearing coronas, a commonly discontinuous garnet layer lies between the kyanite and the continuous plagioclase layer in the corona, with both the garnet and the plagioclase appearing mainly to replace matrix rather than kyanite. The garnet layer commonly extends around kyanite from original matrix garnet adjacent to the kyanite. Where garnet is missing in the corona, the kyanite itself may be replaced by a spinel–plagioclase corona. In a local equilibrium model, the mineral and mineral compositional spatial relationships are shown to correspond to paths in μ(Na₂O)–μ(CaO)–μ(K₂O)–μ(FeO)–μ(MgO)–μ(SiO₂) in the model chemical system, Na₂O–CaO–K₂O–FeO–MgO–Al₂O₃–SiO₂ (NCKFMAS). The discontinuous nature of the garnet layer in coronas is accounted for by the effect of the adjacent original garnet on the chemical potential relationships. The replacement of kyanite by spinel + plagioclase appears to be metastable with respect to replacement by corundum + plagioclase, possibly reflecting the difficulty of nucleating corundum.     

On matrix diffusion: formulations, solution methods and qualitative effects

 Matrix diffusion has become widely recognized as an important transport mechanism. Unfortunately, accounting for matrix diffusion complicates solute-transport simulations. This problem has led to simplified formulations, partly motivated by the solution method. As a result, some confusion has been generated about how to properly pose the problem. One of the objectives of this work is to find some unity among existing formulations and solution methods. In doing so, some asymptotic properties of matrix diffusion are derived. Specifically, early-time behavior (short tests) depends only on φ _m ² R _m D _m  / L _m ², whereas late-time behavior (long tracer tests) depends only on φ _m R _m , and not on matrix diffusion coefficient or block size and shape. The latter is always true for mean arrival time. These properties help in: (a) analyzing the qualitative behavior of matrix diffusion; (b) explaining one paradox of solute transport through fractured rocks (the apparent dependence of porosity on travel time); (c) discriminating between matrix diffusion and other problems (such as kinetic sorption or heterogeneity); and (d) describing identifiability problems and ways to overcome them. Received, October 1997 · Revised, November 1997 · Accepted, December 1997     

Examination of the Chayes-Kruskal procedure for testing correlations between proportions

The Chayes-Kruskal procedure for testing correlations between proportions uses a linear approximation to the actual closure transformation to provide a null value,p _ij , against which an observed closed correlation coefficient,r _ij , can be tested. It has been suggested that a significant difference betweenp _ij andr _ij would indicate a nonzero covariance relationship between theith andjth open variables. In this paper, the linear approximation to the closure transformation is described in terms of a matrix equation. Examination of the solution set of this equation shows that estimation of, or even the identification of, significant nonzero open correlations is essentially impossible even if the number of variables and the sample size are large. The method of solving the matrix equation is described in the appendix.     

Cold subduction and the formation of lawsonite eclogite – constraints from prograde evolution of eclogitized pillow lava from Corsica

A new discovery of lawsonite eclogite is presented from the Lancône glaucophanites within the Schistes Lustrés nappe at Défilé du Lancône in Alpine Corsica. The fine‐grained eclogitized pillow lava and inter‐pillow matrix are extremely fresh, showing very little evidence of retrograde alteration. Peak assemblages in both the massive pillows and weakly foliated inter‐pillow matrix consist of zoned idiomorphic Mg‐poor (<0.8 wt% MgO) garnet + omphacite + lawsonite + chlorite + titanite. A local overprint by the lower grade assemblage glaucophane + albite with partial resorption of omphacite and garnet is locally observed. Garnet porphyroblasts in the massive pillows are Mn rich, and show a regular prograde growth‐type zoning with a Mn‐rich core. In the inter‐pillow matrix garnet is less manganiferous, and shows a mutual variation in Ca and Fe with Fe enrichment toward the rim. Some garnet from this rock type shows complex zoning patterns indicating a coalescence of several smaller crystallites. Matrix omphacite in both rock types is zoned with a rimward increase in X_Jd, locally with cores of relict augite. Numerous inclusions of clinopyroxene, lawsonite, chlorite and titanite are encapsulated within garnet in both rock types, and albite, quartz and hornblende are also found included in garnet from the inter‐pillow matrix. Inclusions of clinopyroxene commonly have augitic cores and omphacitic rims. The inter‐pillow matrix contains cross‐cutting omphacite‐rich veinlets with zoned omphacite, Si‐rich phengite (Si = 3.54 apfu), ferroglaucophane, actinolite and hematite. These veinlets are seen fracturing idiomorphic garnet, apparently without any secondary effects. Pseudosections of matrix compositions for the massive pillows, the inter‐pillow matrix and the cross‐cutting veinlets indicate similar P–T conditions with maximum pressures of 1.9–2.6 GPa at temperatures of 335–420 °C. The inclusion suite found in garnet from the inter‐pillow matrix apparently formed at pressures below 0.6–0.7 GPa. Retrogression during initial decompression of the studied rocks is only very local. Late veinlets of albite + glaucophane, without breakdown of lawsonite, indicate that the rocks remained in a cold environment during exhumation, resulting in a hairpin‐shaped P–T path.     

Effects of matrix grain size on the kinetics of intergranular diffusion

A linear relationship exists between the mean volume of garnet porphyroblasts and the squared inverse of mean matrix grain diameter for six samples of garnetiferous mica quartzite with identical thermal histories and similar mineralogy and modes. This relationship accords with theoretical predictions of the dependence of intergranular diffusive fluxes on the volume fraction of grain edges that function as diffusional pathways during porphyroblast growth. The impact of matrix grain size is large: compared to a rock with a 1‐mm matrix, a rock with a 10‐μm matrix would experience rates of diffusion‐controlled porphyroblast growth that are 10 000 times faster, and characteristic length scales for chemical equilibration that are 100 times larger. Precursor grain sizes may therefore exert a major influence on crystallization kinetics. If matrix coarsening occurs during prograde reaction, a decrease in the volume fraction of diffusional pathways will tend to counteract the exponential thermal increase in diffusive fluxes. The impact of such matrix grain growth, although difficult to assess without firm knowledge of coarsening rates in polymineralic aggregates, might be significant for matrices finer than c. 100 μm at temperatures above c. 500–600 °C, but is likely negligible for coarser grain sizes and lower temperatures.     

Undrained torsional shear tests on gravelly soils

Slope instability and landslides are frequently triggered during heavy rainfalls in mountainous areas. Geomaterials that are subject to this type of failure normally include coarse grains, which are made by weathering of mother rocks. These materials are called sandy gravel or gravelly sand in soil mechanics, depending upon the amount of gravelly components. This situation suggests a need for laboratory investigations that aim to understand the shear behavior of sand with gravel toward failure. Another feature of this type of failure is a quick rate of slope failure that is reasonably considered as an undrained process of shear distortion. Hence, the present study investigated by means of experiments the undrained shear behavior of sand with gravel. The torsion shear tests on hollow cylindrical specimens concerned the effects of gravel content on the undrained shear behavior. The tests revealed that the effect of gravel content is twofold. When the gravel content is relatively small, the effects of the sandy component are more important. Hence, the relative density of the sandy matrix among gravels has a predominant influence. In this situation, therefore, the overall relative density is a less important index to account for the shear behavior. In contrast, when the gravel content exceeds a threshold value, the amount of gravel comes to have a more predominant influence than the sandy matrix. This is probably because gravel particles start to have contact with each other to form a structural matrix of gravel grains that governs the overall stress–strain behavior. These results were summarized in three-dimensional diagrams that related the strength properties of gravelly sand varying with the density of sand matrix or the density of sand–gravel mixtures as well as the gravel content.     

Performance of a flux splitting scheme when solving the single-phase pressure equation discretized by MPFA

In this article we present a series of tests to study how well suited the TPFA coefficient matrix is as a preconditioner for the MPFA discrete system of equations in an iterative solver, using a flux splitting method. These tests have been conducted for single-phase flow for a wide range of anisotropy, heterogeneity, and grid skewness (mainly parallelogram grids). We use the K-orthogonal part of the MPFA transmissibilities for a parallelogram grid to govern the TPFA transmissibilities. The convergence of the flux splitting method is for each test case measured by the spectral radius of the iteration matrix.     

Numerical Method for Conditional Simulation of Levy Random Fields

Stochastic simulations of subsurface heterogeneity require accurate statistical models for spatial fluctuations. Incremental values in subsurface properties were shown previously to be approximated accurately by Levy distributions in the center and in the start of the tails of the distribution. New simulation methods utilizing these observations have been developed. Multivariate Levy distributions are used to model the multipoint joint probability density. Explicit bounds on the simulated variables prevent nonphysical extreme values and introduce a cutoff in the tails of the distribution of increments. Long-range spatial dependence is introduced through off-diagonal terms in the Levy association matrix, which is decomposed to yield a maximum likelihood type estimate at unobserved locations. This procedure reduces to a known interpolation formula developed for Gaussian fractal fields in the situation of two control points. The conditional density is not univariate Levy and is not available in closed form, but can be constructed numerically. Sequential simulation algorithms utilizing the numerically constructed conditional density successfully reproduce the desired statistical properties in simulations.     

Some properties of principal component scores

Commonly used methods for calculating component scores are reviewed. Means, variances, and the covariance structures of the resulting sets of scores are examined both by calculations based on a large set of electron microprobe analyses of melilite (supplied by D. Velde)and by a survey of recent geological applications of principal component analysis. Most of the procedures used to project raw data into the new vector space yield uncorrelated scores. In exceptions so far encountered, correlations between scores seem to have been occasioned by the use of unstandardized variables with components calculated from a correlation matrix. In a number of cases substantive interpretations of such correlations have been proposed. A different set of correlations results for the same data if scores are computed from standardized variables and components based on the covariance matrix. If unscaled components are rotated by the varimax procedure, the result is a return to the original space. In the work reported here, nevertheless, scores calculated from varimax-rotated scaled vectors are uncorrelated.     

A new form of the cokriging equations

Myers developed a matrix form of the cokriging equations, but one that entails the solution of a large system of linear equations. Large systems are troublesome because of memory requirements and a general increase in the matrix condition number. We transform Myers’s system into a set of smaller systems, whose solution gives the classical kriging results, and provides simultaneously a nested set of lower dimensional cokriging results. In the course of developing the new formulation we make an interesting link to the Cauchy-Schwarz condition for the invertibility of a system, and another to a simple situation of coregionalization. In addition, we proceed from these new equations to a linear approximation to the cokriging results in the event that the crossvariograms are small, allowing one to take advantage of a recent results of Xie and others which proceeds by diagonalizing the variogram matrix function over the lag classes.     

Strontium isotope zoning in garnet: implications for metamorphic matrix equilibration,geochronology and phase equilibrium modelling

In principle, garnet growth rates may be calculated from ⁸⁷Rb/⁸⁶Sr and ⁸⁷Sr/⁸⁶Sr measurements in garnet subsamples and the surrounding rock matrix. Because of low Rb/Sr, garnet should passively record the matrix decay of ⁸⁷Rb to ⁸⁷Sr as a progressive increase in ⁸⁷Sr/⁸⁶Sr from core to rim. This concept was tested by collecting Rb‐Sr data for five garnet grains from four major orogenic belts: eastern Vermont (c. 380 Ma), western New Hampshire (c. 320 Ma), southern Chile (c. 75 Ma) and northwestern Italy (c. 35 Ma). Both normal Sr isotope zoning (increasing ⁸⁷Sr/⁸⁶Sr from core to rim) and inverse Sr zoning (decreasing ⁸⁷Sr/⁸⁶Sr from core to rim) were observed. Garnet and matrix isotope data commonly yielded grossly inaccurate model ages. Incomplete Rb and Sr equilibration among matrix minerals is invoked to explain the deviations between theoretical v. measured zoning patterns and the age disparities. Initially, the reactive matrix is dominated by rapidly equilibrating Rb‐rich mica, which imparts high ⁸⁷Sr/⁸⁶Sr values in garnet cores. Progressive participation of slower equilibrating Sr‐rich plagioclase buffers or even reduces ⁸⁷Sr/⁸⁶Sr, possibly leading to flat or decreasing ⁸⁷Sr/⁸⁶Sr from garnet cores to rims. Unusually high ⁸⁷Sr/⁸⁶Sr in garnet in combination with bulk matrix compositions causes erroneously young apparent ages, so metamorphic ages, growth rates, and associated heating and loading rates are likely suspect. Although Rb‐Sr may be the most susceptible because of the profound disparities between mica and feldspar, zircon reactivity might influence the Lu‐Hf system by up to a few per cent. The Sm‐Nd system seems generally immune to these effects. Pseudosection analysis and conventional garnet geochronology, which presume complete matrix equilibration during metamorphism, may require modification to account for differences between whole‐rock v. reactive matrix compositions.     

Blueschist metamorphism during accretion in the Lachlan Orogen, south-eastern Australia

Low‐T, intermediate to high‐P assemblages indicative of the prehnite–pumpellyite, greenschist and blueschist facies are preserved in mélange zones and slivers of oceanic crust within two major fault zones of the turbidite‐dominated Lachlan Orogen. In one of these fault zones (Governor Fault Zone), blueschists occur as Franciscan‐like blocks in a serpentinite/talc matrix that is interleaved with phyllites and slates, and structurally overlain by a fault slice or duplex of predominantly pillow basalt, chert, and turbidite. The blueschist metavolcanics are interpreted to have formed at < 450 °C and at a depth of approximately 21–27 km. The presence of blue amphibole in the blocks, rinds and matrix indicate that the metavolcanics were emplaced in the matrix prior to blueschist metamorphism. Blocks and matrix were partially exhumed, interleaved with tectonic slices of phyllite and slate, and subsequently folded at about 10–12 km depth, inferred from b_o values of the dominant mica fabric in the phyllites and slates. Metamorphic P–T is highest in the structurally lowest slice (mélange zone) and lowest in the overlying ophiolitic fault slice, suggestive of an accretionary burial metamorphic pattern formed by underplating of the mélange. In the other fault zone (Heathcote Fault Zone), blueschists transitional to greenschist facies are interpreted to have formed at < 450 °C and at a depth of approximately 15–21 km. They occur as blocks in serpentinite/talc‐matrix mélange and are also associated with fault slices of oceanic crust. Textural and mineralogical evidence suggests that the protoliths for the blueschists in both fault zones were boninitic pillow lavas. The metamorphic facies and patterns, and the structural and lithological associations, can be interpreted in terms of disruption of oceanic crust and overlying sediments during subduction, and formation of serpentinite‐matrix mélange overprinted by blueschist metamorphism either prior to or during underplating of the mélange and duplex formation. The presence of blueschist metavolcanics indicate that these processes occurred at considerable depth. These interpretations have implications for the evolution of large‐scale fault zones in noncollisional, convergent oceanic settings.     

Calculation of the Inverse of the Covariance

In reservoir characterization, the covariance is often used to describe the spatial correlation and variation in rock properties or the uncertainty in rock properties. The inverse of the covariance, on the other hand, is seldom discussed in geostatistics. In this paper, I show that the inverse is required for simulation and estimation of Gaussian random fields, and that it can be identified with the differential operator in regularized inverse theory. Unfortunately, because the covariance matrix for parameters in reservoir models can be extremely large, calculation of the inverse can be a problem. In this paper, I discuss four methods of calculating the inverse of the covariance, two of which are analytical, and two of which are purely numerical. By taking advantage of the assumed stationarity of the covariance, none of the methods require inversion of the full covariance matrix.