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## Tensor Analysis (tensor + analysis)
## Selected Abstracts## An ellipticity criterion in magnetotelluric tensor analysis GEOPHYSICAL JOURNAL INTERNATIONAL, Issue 1 2004M. BeckenSUMMARY We examine the magnetotelluric (MT) impedance tensor from the viewpoint of polarization states of the electric and magnetic field. In the presence of a regional 2-D conductivity anomaly, a linearly polarized homogeneous external magnetic field will generally produce secondary electromagnetic fields, which are elliptically polarized. If and only if the primary magnetic field vector oscillates parallel or perpendicular to the 2-D structure, will the horizontal components of the secondary fields at any point of the surface also be linearly polarized. When small-scale inhomogeneities galvanically distort the electric field at the surface, only field rotations and amplifications are observed, while the ellipticity remains unchanged. Thus, the regional strike direction can be identified from vanishing ellipticities of electric and magnetic fields even in presence of distortion. In practice, the MT impedance tensor is analysed rather than the fields themselves. It turns out, that a pair of linearly polarized magnetic and electric fields produces linearly polarized columns of the impedance tensor. As the linearly polarized electric field components generally do not constitute an orthogonal basis, the telluric vectors, i.e. the columns of the impedance tensor, will be non-orthogonal. Their linear polarization, however, is manifested in a common phase for the elements of each column of the tensor and is a well-known indication of galvanic distortion. In order to solve the distortion problem, the telluric vectors are fully parametrized in terms of ellipses and subsequently rotated to the coordinate system in which their ellipticities are minimized. If the minimal ellipticities are close to zero, the existence of a (locally distorted) regional 2-D conductivity anomaly may be assumed. Otherwise, the tensor suggests the presence of a strong 3-D conductivity distribution. In the latter case, a coordinate system is often found, in which three elements have a strong amplitude, while the amplitude of the forth, which is one of the main-diagonal elements, is small. In terms of our ellipse parametrization, this means, that one of the ellipticities of the two telluric vectors approximately vanishes, while the other one may not be neglected as a result of the 3-D response. The reason for this particular characteristic is found in an approximate relation between the polarization state of the telluric vector with vanishing ellipticity and the corresponding horizontal electric field vector in the presence of a shallow conductive structure, across which the perpendicular and tangential components of the electric field obey different boundary conditions. [source] ## Retrospective measurement of the diffusion tensor eigenvalues from diffusion anisotropy and mean diffusivity in DTI MAGNETIC RESONANCE IN MEDICINE, Issue 1 2006Khader M. HasanAbstract A simple theoretical framework to compute the eigenvalues of a cylindrically symmetric prolate diffusion tensor (D) from one of the rotationally-invariant diffusion anisotropy indices and average diffusivity is presented and validated. Cylindrical or axial symmetry assumes a prolate ellipsoid shape (,, = ,1 > ,, = (,2 + ,3)/2; ,2 = ,3). A prolate ellipsoid with such symmetry is largely satisfied in a number of white matter (WM) structures, such as the spinal cord, corpus callosum, internal capsule, and corticospinal tract. The theoretical model presented is validated using in vivo DTI measurements of rat spinal cord and human brain, where eigenvalues were calculated from both the set of diffusion coefficients and a tensor analysis. This method was used to retrospectively analyze literature data that reported tensor-derived average diffusivity, anisotropy, and eigenvalues, and similar eigenvalue measurements were obtained. The method provides a means to retrospectively reanalyze literature data that do not report eigenvalues. Other potential applications of this method are also discussed. Magn Reson Med, 2006. © 2006 Wiley-Liss, Inc. [source] ## Absolute eigenvalue diffusion tensor analysis for human brain maturation NMR IN BIOMEDICINE, Issue 5 2003Yuji SuzukiAbstract The absolute eigenvalues of the diffusion tensor of white matter in sixteen normal subjects in two groups representing the early developmental stage (ages 1,10 years, n,=,8) and young adult stage (ages 18,34 years, n,=,8) were assessed using a high-field (3.0,T) magnetic resonance (MR) system. All three eigenvalues, including the largest eigenvalue, decreased significantly with brain maturation. The rate of the decline in the two small eigenvalues was, however, much higher than that of the largest eigenvalue, resulting in an actual increase in fractional anisotropy, a commonly measured relative index. The data demonstrate that an increase in anisotropy associated with brain maturation represents a significant decline in the small eigenvalue components, rather than an increase in the largest eigenvalue. The observed pattern of eigenvalue changes is best explained by the simultaneous occurrence of two of several independent phenomena within the axonal microenvironment during the myelination process, namely, (1) decline in unrestricted water content in extra-axonal space, and (2) increase in apparent diffusivity within the axon. Copyright © 2003 John Wiley & Sons, Ltd. [source] ## Raman tensor analysis of baddeleyite single-crystal and its application to define crystallographic domains in polycrystalline zirconia PHYSICA STATUS SOLIDI (B) BASIC SOLID STATE PHYSICS, Issue 2 2010Kyoju FukatsuAbstract The angular dependence of polarized Raman intensity for the Ag and Bg modes was investigated and the full set of Raman tensor elements defined for a baddeleyite single-crystal, namely the monoclinic polymorph of zirconia (ZrO2). Based on the quantitative knowledge of the tensor elements, a method has been proposed for the determination of unknown crystallographic textures in monoclinic zirconia. An application of this method is also shown, which consists of a Raman analysis of crystal orientation on the microscopic scale in polycrystalline ZrO2 after its tetragonal-to-monoclinic (t,m) polymorphic transformation (i.e., occurred under an externally applied stress field). This working example not only confirms the well-known phenomenon of stress-induced phase transformation in polycrystalline zirconia, but also proves the existence of textured domain patterns in the monoclinic phase on a scale larger than that of individual grains. This finding might suggest that the structural and functional properties of polycrystalline zirconia after partial phase transformation should be reinterpreted with taking into account a crystallographic reorientation effect. [source] |