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Sharma, M. D. "Rayleigh wave at the surface of a general anisotropic poroelastic medium: derivation of real secular equation." Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 474, no. 2211 (March 2018): 20170589. http://dx.doi.org/10.1098/rspa.2017.0589.
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A secular equation governs the propagation of Rayleigh wave at the surface of an anisotropic poroelastic medium. In the case of anisotropy with symmetry, this equation is obtained as a real irrational equation. But, in the absence of anisotropic symmetries, this secular equation is obtained as a complex irrational equation. True surface waves in non-dissipative materials decay only with depth. That means, propagation of Rayleigh wave in anisotropic poroelastic solid should be represented by a real phase velocity. In this study, the determinantal system leading to the complex secular equation is manipulated to obtain a transformed equation. Even for arbitrary (triclinic) anisotropy, this transformed equation remains real for the propagation of true surface waves. Such a real secular equation is obtained with the option of boundary pores being opened or sealed. A numerical example is solved to study the existence and propagation of Rayleigh waves in porous media for the top three (i.e. triclinic, monoclinic and orthorhombic) anisotropies.
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Tleukenov, S. K., M. K. Zhukenov, and N. A. Ispulov. "Propagation of electromagnetic waves in anisotropic magnetoelectric medium." Bulletin of the Karaganda University. "Physics" Series 94, no. 2 (June 28, 2019): 29–34. http://dx.doi.org/10.31489/2019ph2/29-34.
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Liu, Weiqiang, Pinrong Lin, Qingtian Lü, Rujun Chen, Hongzhu Cai, and Jianhua Li. "Time Domain and Frequency Domain Induced Polarization Modeling for Three-dimensional Anisotropic Medium." Journal of Environmental and Engineering Geophysics 22, no. 4 (December 2017): 435–39. http://dx.doi.org/10.2113/jeeg22.4.435.
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Time domain induced polarization (TDIP) and frequency domain induced polarization (FDIP) synthetic models, incorporating three-dimensional (3D) anisotropic medium, were tested. In TDIP modeling, both resistivity and chargeability of the medium were anisotropic, and the apparent chargeability values were calculated by carrying out two resistivity forward calculations using resistivity with and without an IP effect. We analyzed the TDIP response of a 3D isotropic cube model embedded in the anisotropic subsurface half-space. In FDIP modeling, the complex resistivity of the medium at various frequencies was anisotropic. The complex resistivity was determined by a Cole-Cole model with anisotropic model parameters. We then analyzed the FDIP response of a 3D anisotropic cube model embedded in an isotropic subsurface half-space. Both of the TDIP and FDIP simulation results suggest that IP responses acquired in two orthogonal directions on the surface are different when the same arrays are used and acquisition in orthogonal directions helps resolve the presence of anisotropy. The anisotropy should be taken into account in practice for TDIP and FDIP exploration.
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Zhu, Tieyuan. "Numerical simulation of seismic wave propagation in viscoelastic-anisotropic media using frequency-independent Q wave equation." GEOPHYSICS 82, no. 4 (July 1, 2017): WA1—WA10. http://dx.doi.org/10.1190/geo2016-0635.1.
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Seismic anisotropy is the fundamental phenomenon of wave propagation in the earth’s interior. Numerical modeling of wave behavior is critical for exploration and global seismology studies. The full elastic (anisotropy) wave equation is often used to model the complexity of velocity anisotropy, but it ignores attenuation anisotropy. I have presented a time-domain displacement-stress formulation of the anisotropic-viscoelastic wave equation, which holds for arbitrarily anisotropic velocity and attenuation [Formula: see text]. The frequency-independent [Formula: see text] model is considered in the seismic frequency band; thus, anisotropic attenuation is mathematically expressed by way of fractional time derivatives, which are solved using the truncated Grünwald-Letnikov approximation. I evaluate the accuracy of numerical solutions in a homogeneous transversely isotropic (TI) medium by comparing with theoretical [Formula: see text] and [Formula: see text] values calculated from the Christoffel equation. Numerical modeling results show that the anisotropic attenuation is angle dependent and significantly different from the isotropic attenuation. In synthetic examples, I have proved its generality and feasibility by modeling wave propagation in a 2D TI inhomogeneous medium and a 3D orthorhombic inhomogeneous medium.
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Berthier, Serge. "Anisotropic effective medium theories." Journal de Physique I 4, no. 2 (February 1994): 303–18. http://dx.doi.org/10.1051/jp1:1994139.
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Qin, Xilin, Zhixian Gui, Fei Yang, and Yuanyuan Liu. "Anisotropic frequency-dependent characteristics of PP- and PS-waves in partially saturated double-porosity rocks." Journal of Geophysics and Engineering 18, no. 3 (June 2021): 355–68. http://dx.doi.org/10.1093/jge/gxab019.
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Abstract Most frequency-dependent AVO inversions are currently based on an approximate equation derived using an isotropic medium. However, actual reservoirs usually show anisotropy, such as shale reservoirs, tight sandstone reservoirs and fractured reservoirs. We propose a joint frequency-dependent AVO (JFAVO) inversion in an anisotropic medium based on a periodic layered double-porosity medium. This JFAVO will allow us to quantitatively study the influence of fluids on the dispersion of PP- and PS-wave velocities and anisotropic parameters. First, we used a double-porosity medium to analyse the frequency-dependent characteristics of velocities and anisotropy parameters. We found that the anisotropic parameters show obvious dispersions, similar to those of velocities. Then, we derived the JFAVO inversion based on Rüger's equation to extract the dispersion of velocities and anisotropic parameters. Finally, we analysed the stability and applicability of the inversion algorithm, and used three sets of models to analyse the sensitivity of dispersion properties to fluids. The numerical analysis results show that PP-wave velocity dispersion and anisotropic parameter δ dispersion are sensitive to fluids, whereas, the velocity dispersion of the PS-wave is not. When saturation exceeds 80%, the velocity dispersion and anisotropic parameter dispersion properties are not sensitive to fluids.
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Аshcheulov, А. А., D. А. Lavreniuk, and M. Ya Derevianchuk. "Electric field transformation effect in anisotropic dielectric medium." Технология и конструирование в электронной аппаратуре, no. 3-4 (2020): 24–27. http://dx.doi.org/10.15222/tkea2020.3-4.24.
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The authors consider the aspects of the electric field distribution in an anisotropic medium and establish how its longitudinal and transverse components depend on the geometric factors. A rectangular plate of dimensions a×b×c is studied, its selected crystallographic axes located in the plane of the side face (a×b), while one of the axes is oriented at a certain angle α to the edge a. It is shown that applying a certain potential difference to the upper and lower faces electrically polarizes the volume of the plate and causes the appearance of the longitudinal and transverse components of the internal electric field. The authors investigate the possibility of transforming the magnitude of the electric field and methods for its optimization. The transformation coefficient of such a device is determined by the anisotropy of the dielectric permeability of the plate material and its shape coefficient k = a/b. The paper considers one of the design options for an anisotropic dielectric transformer and proposes its equivalent electrical circuit. Structural elements based on anisotropic dielectric transformers may be widely used both in power supplies of various electronic devices and for coordination of radar transceiver systems with antenna arrays of centimeter, millimeter and submillimeter wavelength ranges. The possibility of simultaneous transformation of constant and alternating electric fields allows them to be used in devices of simultaneous comparison, enabling to determine the current values of voltage, as well as the power of electromagnetic radiation in a wide range of wavelengths. The vortex nature of the electric field in the plate’s volume caused by the coefficient anisotropy of the dielectric permeability also creates the preconditions for the emergence of new principles for generating high-power electromagnetic radiation in a wide spectral range. The generation frequency of such devices is determined by the geometric dimensions of the anisotropic plate. The use of the described transformation effect will significantly expand the possibilities of practical application of the considered electrostatic phenomena, which will lead to the emergence of a new generation of devices for microwave technology, electronics and electric power.
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Sams, Mark, Annushia Annamalai, and Jeremy Gallop. "Seismic amplitude variation with offset inversion in a vertical transverse isotropy medium." Interpretation 7, no. 3 (August 1, 2019): T581—T593. http://dx.doi.org/10.1190/int-2018-0137.1.
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Vertical transverse isotropy (VTI) will affect seismic inversion, but it is not possible to solve for the full set of anisotropic elastic parameters from amplitude variation with offset inversion because there exists an isotropic solution to every VTI problem. We can easily approximate the pseudoisotropic properties that result from the isotropic solution to the anisotropic problem for well-log data. We can then use these well-log properties to provide a low-frequency model for inversion and/or a framework for interpreting either absolute or relative inversion results. This, however, requires prior knowledge of the anisotropic properties, which are often unavailable or poorly constrained. If we ignore anisotropy and assume that the amplitude variations caused by VTI are going to be accounted for by effective wavelets, the inversion results would be in error: The impact of anisotropy is not merely a case of linear scaling of seismic amplitudes for any particular angle range. Ignoring VTI does not affect the prediction of acoustic impedance, but it does affect predictions of [Formula: see text] and density. For realistic values of anisotropy, these errors can be significant, such as predicting oil instead of brine. If the anisotropy of the rocks is known, then we can invert for the true vertical elastic properties using the known anisotropy coefficients through a facies-based inversion. This can produce a more accurate result than solving for pseudoelastic properties, and it can take advantage of the sometimes increased separation of isotropic and anisotropic rocks in the pseudoisotropic elastic domain. Because the effect of anisotropy will vary depending on the strength of the anisotropy and the distribution of the rocks, we strongly recommend forward modeling for each case prior to inversion to understand the potential impact on the study objectives.
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Mavko, Gary, and Kaushik Bandyopadhyay. "Approximate fluid substitution for vertical velocities in weakly anisotropic VTI rocks." GEOPHYSICS 74, no. 1 (January 2009): D1—D6. http://dx.doi.org/10.1190/1.3026552.
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Despite the prevalence of elastic anisotropy in rocks, most models used routinely for wave propagation, imaging, and rock physics assume an isotropic Earth. The main reason for using isotropic approximations is that we seldom measure enough parameters to characterize the stiffness tensor of a rock completely. Fluid substitution is an important example: because of an incomplete knowledge of the stiffness tensor, often we choose isotropic equations over their anisotropic form. Assuming weak anisotropy, we derive an approximate form of the anisotropic fluid-substitution equation for seismic waves propagating along the symmetry axis of a transversely isotropic medium. The approximation takes the form of the usual isotropic calculation, with a simple first-order correction proportional to the Thomsen anisotropic parameter [Formula: see text], thus requiring only three elastic constants. Because [Formula: see text] can be either greater than or less than zero in a VTI medium, we can explain why isotropic fluid substitution sometimes overpredicts or underpredicts the full anisotropic result. Numerical simulations show that the approximate equation is valid for anisotropic medium with absolute value of [Formula: see text] as high as 0.3.
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Аshcheulov, А. А., M. Ya Derevianchuk, D. А. Lavreniuk, and I. S. Romaniuk. "Electric current transformation by anisotropic electrically conductive medium." Технология и конструирование в электронной аппаратуре, no. 5-6 (2020): 28–32. http://dx.doi.org/10.15222/tkea2020.5-6.28.
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The authors consider the aspects of electric current distribution in electrically conductive anisotropic medium and establish how geometrical factors affect its longitudinal and transverse components. In the case of an a×b×с rectangular plate, its selected crystallographic axes are located on the plane of the side face a×b, whereas one of these axes is oriented at an angle α to the edge a. Applying a certain potential difference to the upper and lower end faces of the plate causes the appearance of longitudinal and transverse components of the internal electric current. The paper demonstrates the possibility of transforming the magnitude of the electric current and a way to optimize this magnitude. The transformation coefficient of such a device is determined by the anisotropy of the electrical conductivity of the plate and the coefficient of its shape k = a/b. The authors consider a few versions of anisotropic dielectric transformer design and offer their equivalent electric circuits. Another suggested transformer design is spiral in shape, compact and is characterized by high transformation coefficient value n. For example, at external radius r1 = 12,5 mm, internal radius r2 = 2 mm, height b = 2 mm and plate thickness c = 2,0 mm, its transformation coefficient n = 103. The information is given on existing monocrystalline and artificial anisotropic materials that can be used for the proposed device. High-temperature superconducting materials characterized by a high value of residual resistance anisotropy hold special promise in this case. Using the described transformation effect will significantly expand the possibilities of practical application of the considered electroohmic phenomenon. This will lead to the emergence of a new generation of devices for microwave technology, electronics and power engineering.
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Bouden, M., and S. K. Datta. "Rayleigh and Love Waves in Cladded Anistropic Medium." Journal of Applied Mechanics 57, no. 2 (June 1, 1990): 398–403. http://dx.doi.org/10.1115/1.2892003.
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Guided waves in coated anisotropic medium are of interest in electronics. Also, in recent years, cladded fiber-reinforced composites are being developed for use as aerospace structures. This paper deals with guided wave propagation in a cladded or coated anisotropic medium. The cladding (coating) is assumed to be a thin isotropic layer, which is bonded to a transversely isotropic substrate with the axis of symmetry parallel to the layer. It is shown that the anisotropy of the substrate affects the dispersion behavior in a manner that is substantially different than in the case of isotropic substrate.
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Collet, Olivia, and Boris Gurevich. "Fluid dependence of anisotropy parameters in weakly anisotropic porous media." GEOPHYSICS 78, no. 5 (September 1, 2013): WC137—WC145. http://dx.doi.org/10.1190/geo2012-0499.1.
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Predicting seismic velocities in isotropic fluid-saturated rocks is commonly done using the isotropic Gassmann theory. For anisotropic media, the solution is expressed in terms of stiffness or compliance, which does not provide an intuitive understanding on how the fluid affects wave propagation in anisotropic media. Assuming weak anisotropy, we expressed the anisotropy parameters of transversely isotropic saturated media as a function of the anisotropy parameters in the dry medium, the bulk and shear moduli of the saturated and dry media, the grain and fluid bulk moduli, and the porosity. By deriving an approximation of the anellipticity parameter [Formula: see text], we discovered that if the dry medium was elliptical, the saturated medium was also elliptical but only if the porosity exceeded a certain threshold value. This result can provide a way of differentiating between stress- and fracture-induced anisotropy.
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SILI, ALI. "HOMOGENIZATION OF A NONLINEAR MONOTONE PROBLEM IN AN ANISOTROPIC MEDIUM." Mathematical Models and Methods in Applied Sciences 14, no. 03 (March 2004): 329–53. http://dx.doi.org/10.1142/s0218202504003258.
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This paper deals with an analysis of a nonlinear monotone conduction problem posed on a medium which we assume to be anisotropic and periodically reinforced by thin fibers also assumed to be anisotropic. We consider the case where the conductivity in the fibers are more important than in the material which surrounds it so that the operator we have to consider loses his uniform ellipticity with respect to the size ε of the period; we then investigate the effect of the anisotropy in the homogenization process.
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Xie, Kunkun, Haopeng Song, and Cunfa Gao. "Electric and heat conduction across an elliptic cavity in an anisotropic medium." Mathematics and Mechanics of Solids 24, no. 10 (April 17, 2019): 3279–94. http://dx.doi.org/10.1177/1081286519840739.
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It is well known that the anisotropy of materials will significantly affect heat conduction, and the corresponding results have been applied to the thermal analysis of materials. An elliptic cavity in a nonlinearly coupled anisotropic medium, on the other hand, is much more difficult to analyze. Based on the complex variable method, the problem of a two-dimensional elliptical cavity in an anisotropic material is analyzed in this paper, and the field distributions have been obtained in closed-form. The field intensity factors are discussed in detail. The results show that both the temperature and electric potential gradients at a crack tip are always perpendicular to the crack surface, regardless of the anisotropy and the nonlinearity in the constitutive equations and the arbitrariness of loading direction. These results provide a powerful tool to analyze the effective behavior and reliability of anisotropic materials with cavities.
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Zhang, Xiangxiang, J. G. Wang, Xiaolin Wang, and Feng Gao. "Numerical Simulations on the Front Motion of Water Permeation into Anisotropic Porous Media." Geofluids 2019 (March 4, 2019): 1–13. http://dx.doi.org/10.1155/2019/7692490.
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Water permeation into a porous medium is a common but important phenomenon in many engineering fields such as hydraulic fracturing. The water permeation front moves with time and may significantly impact the field variable evolution near the water front. Many algorithms have been developed to calculate this water front motion, but few numerical algorithms have been available to calculate the water front motion in anisotropic fluid-solid couplings with high computational efficiency. In this study, a numerical model is proposed to investigate the front motion of water permeation into an anisotropic porous medium. This model fully couples the mechanical deformation, fluid flow, and water front motion. The water front motion is calculated based on a directional Darcy’s flow in the anisotropic porous medium, and a revised formula with a correction coefficient is developed for the estimation of permeation depth. After verification with three sets of experimental data, this model is used to numerically investigate the impacts of permeability, viscosity, permeability anisotropy, and mechanical anisotropy on water front motion. Numerical results show that the proposed model can well describe the anisotropic water permeation process with reasonable accuracy. The permeation depth increases with permeability, mobility, and mechanical anisotropy but decreases with viscosity and permeability anisotropy. The correction coefficient mainly depends on porosity evolution, flow pattern, mobility, permeability anisotropy, and mechanical anisotropy.
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Castro, Caio Leandro Perdigão, José Jadsom Sampaio de Figueiredo, and Isadora Augusta Soares de Macedo. "COMPARING TWO APPROACHES ON THE ESTIMATIVE ANISOTROPIC PARAMETERSFROM WELL LOGS: AN APPLICATION ON THE NORNE FIELD DATASET." Revista Brasileira de Geofísica 36, no. 4 (December 21, 2018): 1. http://dx.doi.org/10.22564/rbgf.v36i4.1971.
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ABSTRACT. Estimating the elastic properties of the rocks in the subsurface is a task with many challenges. The main goal of this work is to estimate the Thomsen anisotropic parameters from the inversion of elastic stiffness coefficients using data from five wells of the Norne Field, located at Norway. We compare the results of these parameters with the Backus average, using Li’s empirical method. Further, aspect ratio and crack density are calculated from the results of the elastic stiffness coefficients. It is considered a transversely isotropic medium. The results from the two methods showed similarities in estimating anisotropic parameters, aspect ratio and fracture density. The anisotropy of the study area is weak with some regions with moderate anisotropy. Some patterns suggest the possibility of calculating the anisotropic parameters for the adjacent wells and interpolate values for use in seismic processing.Keywords: Transversally isotropic medium, well logs, Thomsen parameters, Backus AverageRESUMO. Estimar as propriedades elásticas das rochas em subsurperfície é uma tarefa com muitos desafios. O principal objetivo deste trabalho é estimar os parâmetros de anisotropia de Thomsen a partir da inversão dos coeficientes de rigidez elástica, utilizando dados de cinco diferentes poços do campo de Norne, localizado na Noruega. Comparamos os resultados obtidos para esses parâmetros com a média de Backus, usando o método empírico de Li. Em seguida, a razão de aspecto e a densidade de fratura foram calculadas a partir dos resultados dos coeficientes de rigidez elástica. O meio transversalmente isotrópico é considerado neste trabalho. Os resultados obtidos a partir dos dois métodos mostraram similaridades na estimativa dos parâmetros de anisotropia, razão de aspecto e densidade de fratura. A anisotropia da área de estudo é fraca com algumas regiões de anisotropia moderada. Alguns padrões encontrados sugerem a possibilidade de calcular os parâmetros de anisotropia para os poços vizinhos e interpolá-los para uso futuro no processamento sísmico.Palavras-chave: Meios transversalmente isotrópicos, perfis de poços, parâmetros de Thomsen, média de Backus 1UFPA,
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Shao, Siyi, and Laura J. Pyrak-Nolte. "Interface waves along fractures in anisotropic media." GEOPHYSICS 78, no. 4 (July 1, 2013): T99—T112. http://dx.doi.org/10.1190/geo2012-0464.1.
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The detection of fractures in an anisotropic medium is complicated by discrete modes that are guided or confined by fractures such as fracture interface waves. Fracture interface waves are generalized coupled Rayleigh waves whose existence and velocity in isotropic media depend on the stiffness of the fracture, frequency of the source, and shear-wave polarization. We derived the analytic solution for fracture interface waves in an orthorhombic medium and found that the existence and velocity of interface waves in anisotropic media are also affected by the orientation of a fracture relative to the layering. Laboratory measurements of fracture interface waves using ultrasonic transducers (central frequency [Formula: see text] MHz) on garolite specimens confirmed that the presence of fracture interface waves can mask the textural shear-wave anisotropy of waves propagating parallel to the layering. At low stresses, a layered medium appears almost isotropic when a fracture is oriented perpendicular to the layering, and conversely, a layered medium exhibits stronger anisotropy than the matrix for a fracture oriented parallel to the layering. The matrix shear-wave anisotropy is recovered when sufficient stress is applied to close a fracture. The theory and experimental results demonstrated that the interpretation of the presence of fractures in anisotropic material can be unambiguously interpreted if measurements are made as a function of stress, which eliminates many fractured-generated discrete modes such as fracture interface waves.
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Kamra, Veenu, and Achim Dreher. "Multilayered Transmission Lines on Quasi-planar Substrates With Anisotropic Medium." Advances in Radio Science 17 (September 19, 2019): 77–82. http://dx.doi.org/10.5194/ars-17-77-2019.
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Abstract. This paper exhibits the extension of the discrete mode matching (DMM) method to analyze conformal structures with anisotropy. It represents a simple formalism as a basis to analyze multilayered structures with quasi-planar anisotropic dielectric layers. The dyadic Green's function is then calculated using a full-wave equivalent circuit (FWEC) of the structure, where each layer is represented with the hybrid block consisting of the tangential field components. The application is demonstrated by computing propagation constants for partially filled quasi-planar waveguides and microstrip lines with isotropic, uniaxial and biaxial anisotropic dielectrics.
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Okoye, Patrick N., and Norm F. Uren. "Fresnel zones and spatial resolution for P- and SH-waves in transversely isotropic media." GEOPHYSICS 65, no. 4 (July 2000): 1168–78. http://dx.doi.org/10.1190/1.1444810.
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In an elastically anisotropic medium where the seismic wave velocity is a function of direction, the wavefront shape is nonspherical and, in most cases, assumes a nonelliptical shape. Numerical modelling techniques have been used to calculate the Fresnel‐zone diameter for compressional (P) and shear (SH) waves in transversely isotropic and isotropic media, respectively. The size of the Fresnel zone is found to be predominantly dependent on the curvatures and wavelength of the wavefront as well as the dip angle of the reflector. In addition, the anisotropic elastic parameters δ* (critical near‐vertical anisotropy), ε (the P-wave anisotropy), and γ (the SH-wave anisotropy) are found to significantly affect the size of the Fresnel zone. Numerical modeling results show considerable differences between the Fresnel zones for anisotropic and isotropic velocity functions at various reflector dips. In addition, the Fresnel‐zone dimensions for anisotropic media exhibit asymmetry and considerable change with dip. By way of contrast, those of the corresponding isotropic velocity field exhibit symmetry and negligible variation with dip. The spatial resolution of unmigrated seismic data in an anisotropic medium would consequently be significantly different from that determined for the same medium if it is assumed to be isotropic. Physical modeling results demonstrate that anisotropy can significantly affect the spatial resolving power of seismic waves. The degree of these effects depends on the wavefront curvature, which changes with dip and orientation of the symmetry axis. This observation indicates that the spatial imaging of unmigrated reflection events from the base of thick shale sediments will be affected by anisotropy.
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Abu-shady, M., and H. M. Fath-Allah. "Melting of quarkonium in an anisotropic hot QCD medium in the presence of a generalized Debye screening mass and Nikiforov–Uvarov’s method." International Journal of Modern Physics A 35, no. 21 (July 20, 2020): 2050110. http://dx.doi.org/10.1142/s0217751x20501109.
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Generalized temperature and anisotropy dependent Debye screening mass is introduced into the real part of a potential in an anisotropic plasma. The N-radial Schrödinger equation (SE) is approximately solved by using the Nikiforov–Uvarov (NU) method which based on the expansion of power series. Binding energies and dissociation temperatures of charmonium and bottomonium are calculated. In addition, we have calculated the screening mass values for different parameters. Comparing to their values in an isotropic medium, the charmonium and bottomonium binding energies within an anisotropic medium are found to be increased. Also, the dissociation temperatures of both the charmonium and the bottomonium within anisotropic environments appear larger relative to those found within an isotropic medium. Finally, one observes that in any medium the bottomonium dissociation temperature is higher than the charmonium one.
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Helbig, Klaus, and Patrick N. Rasolofosaon. "Kelvin’s eigensystems in anisotropic poroelasticity." GEOPHYSICS 74, no. 5 (September 2009): WB97—WB105. http://dx.doi.org/10.1190/1.3173805.
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Correct interpretation and processing of seismic data must integrate a correct description of the mechanical behavior of rocks, taking into account facts such as the presence of anisotropy and porosity with or without a saturating fluid. This work discusses elasticity of porous media of arbitrary anisotropy type, with emphasis on the study of deformation states and the associated elastic constants. The stress-strain law is represented in seven dimensions. Dynamic parameters (i.e., the six stress components and fluid pressure) are linked with kinematical parameters (i.e., the six strain components and the local increase of fluid content) by a 7D poroelastic tensor. The model is based on the following mechanical interpretation: each eigenvector (eigenstrain) of the poroelastic tensor defines a fundamental deformation state of the medium and the seven eigenvalues (eigenstiffnesses) representthe genuine poroelastic parameters. The set of seven eigenstrains and corresponding eigenstiffnesses constitute the eigensystem of the poroelastic medium. Complete characterization of the eigensystems corresponding to different types of anisotropies encountered in geologic media is achieved. The first six eigenstrains do not differ substantially from the six eigenstrains in elastic nonporous media, which are well documented in the literature. In contrast, the next result is the existence of a seventh eigenstrain characterized by reduction of the total volume of the porous medium associated with an increase in fluid content. Finally, the analysis is applied to experimental data on a rock sample of Pfalz sandstone, considered as an arbitrarily anisotropic porous medium. Thus, more complex mechanical behaviors of rocks can be introduced naturally, including viscoelastic rheology (already published), frequency dependence, nonlinearity, and even hysteresis, as has been done recently in nonporous media.
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Shen, Linfang, Jin-Jei Wu, and Tzong-Jer Yang. "Anisotropic medium with parabolic dispersion." Applied Physics Letters 92, no. 26 (June 30, 2008): 261905. http://dx.doi.org/10.1063/1.2953546.
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Martins, Jorge L. "Elastic impedance in weakly anisotropic media." GEOPHYSICS 71, no. 3 (May 2006): D73—D83. http://dx.doi.org/10.1190/1.2195448.
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The original formulation for the P-wave elastic impedance (EI) equation ignores seismic anisotropy. Incorporation of anisotropy effects into the EI formula requires a suitable approximation for reflection coefficients. In order to derive an anisotropic EI equation, this paper uses an approximation for PP-wave reflection [Formula: see text] coefficients which holds for weak-contrast interfaces separating weakly anisotropic media of arbitrary symmetry. Inserting the chosen [Formula: see text] coefficient approximation into the original formalism provides an anisotropic EI formula, which is written as a product of two terms: a modified version for the isotropic EI equation and a correction because of weak anisotropy. The latter term shows dependence of the anisotropic EI formula on the so-called weak anisotropy (WA) parameters, on a reference isotropic medium, and on the azimuthal and incident phase angles. Numerical tests show the performance of the EI formula in calculating anisotropic [Formula: see text] coefficients and in constructing azimuthal far-offset EI logs. Since EI allows applying poststack algorithms without modification, an inversion methodology can be designed for investigating anisotropy in sedimentary formations.
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Sarkar, Debashish, Andrey Bakulin, and Robert L. Kranz. "Anisotropic inversion of seismic data for stressed media: Theory and a physical modeling study on Berea Sandstone." GEOPHYSICS 68, no. 2 (March 2003): 690–704. http://dx.doi.org/10.1190/1.1567240.
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Nonhydrostatic stress, an often‐ignored source of seismic anisotropy, is universally present in the subsurface and may be as common as intrinsic or fracture‐induced anisotropy. Nonhydrostatic stress, applied to an initially transversely isotropic solid with vertical symmetry axis (VTI), results in an effective medium having almost orthorhombic symmetry (provided that one of the principal stresses is aligned with the symmetry axis). The symmetry planes observed in this orthorhombic medium are aligned with the orientations of the principal stresses, and anisotropic parameters (ε(1,2), δ(1,2,3), and γ(1,2)) can reveal information about the stress magnitudes. Thus, time‐lapse monitoring of changes in anisotropy potentially can provide information on temporal variations in the stress field. We use nonlinear elasticity theory to relate the anisotropic parameters to the magnitudes of the principal stresses and verify these relationships in a physical modeling study. Under the assumption of weak background and stress‐induced anisotropy, each effective anisotropic parameter reduces to the sum of the corresponding Thomsen parameter for the unstressed VTI background and the corresponding parameter associated with the nonhydrostatic stress. The stress‐related anisotropic parameters depend only on the differences between the magnitudes of principal stresses; therefore, principal stresses can influence anisotropic parameters only if their magnitudes differ in the symmetry plane in which the anisotropic parameters are defined. We test these predictions on a physical modeling data set acquired on a block of Berea Sandstone exhibiting intrinsic VTI anisotropy. Uniaxial stress, applied normal to the VTI symmetry axis, i.e., horizontally, produces an effective medium that is close to orthorhombic. We use two different methods to estimate the anisotropic parameters and study their variation as a function of stress. The first method utilizes conventional measurements of transmission velocities along the principal axes of the sample. The second method uses PP and PS reflection data acquired along seven different azimuths on the surface of the block. In accordance with theoretical predictions, the anisotropic parameters in the vertical plane normal to the stress are almost insensitive to the magnitude of the stress. In contrast, anisotropic parameters in the vertical plane of the applied stress increase approximately in a linear fashion with increasing stress. Except for the parameter δ(1), comparison of the measured values of anisotropic parameters with theoretical predictions shows satisfactory agreement. Despite some documented discrepancies, we believe that nonlinear elasticity may provide a suitable framework for estimating pore pressure and 3D stresses from seismic data.
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Mohd Rusdi, Nadia Diana, Nor Fadzillah Mohd Mokhtar, Norazak Senu, and Siti Suzilliana Putri Mohamed Isa. "Stability Convection in a Couple Stress Fluid Saturated in an Anisotropic Porous Medium with Internal Heating Effect." Journal of Advanced Research in Fluid Mechanics and Thermal Sciences 76, no. 2 (October 23, 2020): 75–84. http://dx.doi.org/10.37934/arfmts.76.2.7584.
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Internal heating effect with stability convection in a couple stress fluid saturated in an anisotropic porous medium has been studied numerically using linear stability analysis. The presence of internal heating on couple stress fluid in an anisotropic porous medium heated from below has been verified. The momentum equation and Boussinesq approximation is used for the density variation in the porous medium. By using Chebyshev Tau method numerically, the eigenvalue problems of the perturbed state were obtained from a normal mode analysis. The effect of the Rayleigh number, internal heat source and anisotropy parameter has been shown graphically. The critical Rayleigh number also has been obtained and plotted on the system. From the result, it is found that the mechanical anisotropy parameter and internal heating effect destabilized the system while couple stress fluid and thermal anisotropy parameter help in stabilizing the system.
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Alkhalifah, Tariq. "Scanning anisotropy parameters in complex media." GEOPHYSICS 76, no. 2 (March 2011): U13—U22. http://dx.doi.org/10.1190/1.3553015.
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Parameter estimation in an inhomogeneous anisotropic medium offers many challenges; chief among them is the trade-off between inhomogeneity and anisotropy. It is especially hard to estimate the anisotropy anellipticity parameter η in complex media. Using perturbation theory and Taylor’s series, I have expanded the solutions of the anisotropic eikonal equation for transversely isotropic (TI) media with a vertical symmetry axis (VTI) in terms of the independent parameter η from a generally inhomogeneous elliptically anisotropic medium background. This new VTI traveltime solution is based on a set of precomputed perturbations extracted from solving linear partial differential equations. The traveltimes obtained from these equations serve as the coefficients of a Taylor-type expansion of the total traveltime in terms of η. Shanks transform is used to predict the transient behavior of the expansion and improve its accuracy using fewer terms. A homogeneous medium simplification of the expansion provides classical nonhyperbolic moveout descriptions of the traveltime that are more accurate than other recently derived approximations. In addition, this formulation provides a tool to scan for anisotropic parameters in a generally inhomogeneous medium background. A Marmousi test demonstrates the accuracy of this approximation. For a tilted axis of symmetry, the equations are still applicable with a slightly more complicated framework because the vertical velocity and δ are not readily available from the data.
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Elhag, S. H., and F. S. Bayones. "Effect of Rotation, Gravity, Primary Stress and Magnetic Field on Shear Waves Spreading in an Anisotropic Incompressible Sandy Elastic Medium." Journal of Computational and Theoretical Nanoscience 16, no. 11 (November 1, 2019): 4443–54. http://dx.doi.org/10.1166/jctn.2019.8610.
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In this paper, we investigated the spreading of shear wave in an anisotropic non-homogeneous elastic medium under effect of dry sand, rotation, gravity, primary stress, and magnetic field. We have reached equation of variation of shear wave velocity c1 in an anisotropic incompressible medium according to sand, rotation, gravity, primary stress, and magnetic field, then we used graphs to illustrate to show the effect of direction of spreading of shear wave. The results indicate that the effect of dry sand, rotation, gravity, primary stress, and magnetic field on the spreading of shear wave in an anisotropic inhomogeneous elastic medium are very pronounced. The results have been obtained are discussed and presented visually, the results demonstrate that the effect of sand, gravity field, primary stress, magnetic field, anisotropy and rotation are noticeable.
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Schleicher, Jörg, and Rafael Aleixo. "Time and depth remigration in elliptically anisotropic media using image-wave propagation." GEOPHYSICS 72, no. 1 (January 2007): S1—S9. http://dx.doi.org/10.1190/1.2374857.
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The image-wave equations for the problems of depth and time remigration in elliptically anisotropic media are second-order partial differential equations similar to the acoustic-wave equation. The propagation variable is the vertical velocity or the medium ellipticity rather than time. These differential equations are derived from the kinematic properties of anisotropic remigration. The objective is to construct subsurface images that correspond to different vertical velocity and/or different degrees of medium anisotropy directly from a single migrated image. In this way, anisotropy panels can be obtained in a way completely analogous to velocity panels for migration velocity analysis. A simple numerical example demonstrates the validity of the theory.
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Alkhalifah, Tariq. "Transformation to zero offset in transversely isotropic media." GEOPHYSICS 61, no. 4 (July 1996): 947–63. http://dx.doi.org/10.1190/1.1444044.
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Nearly all dip‐moveout correction (DMO) implementations to date assume isotropic homogeneous media. Usually, this has been acceptable considering the tremendous cost savings of homogeneous isotropic DMO and considering the difficulty of obtaining the anisotropy parameters required for effective implementation. In the presence of typical anisotropy, however, ignoring the anisotropy can yield inadequate results. Since anisotropy may introduce large deviations from hyperbolic moveout, accurate transformation to zero‐offset in anisotropic media should address such nonhyperbolic moveout behavior of reflections. Artley and Hale’s v(z) ray‐tracing‐based DMO, developed for isotropic media, provides an attractive approach to treating such problems. By using a ray‐tracing procedure crafted for anisotropic media, I modify some aspects of their DMO so that it can work for v(z) anisotropic media. DMO impulse responses in typical transversely isotropic (TI) models (such as those associated with shales) deviate substantially from the familiar elliptical shape associated with responses in homogeneous isotropic media (to the extent that triplications arise even where the medium is homogeneous). Such deviations can exceed those caused by vertical inhomogeneity, thus emphasizing the importance of taking anisotropy into account in DMO processing. For isotropic or elliptically anisotropic media, the impulse response is an ellipse; but as the key anisotropy parameter η varies, the shape of the response differs substantially from elliptical. For typical η > 0, the impulse response in TI media tends to broaden compared to the response in an isotropic homogeneous medium, a behavior opposite to that encountered in typical v(z) isotropic media, where the response tends to be squeezed. Furthermore, the amplitude distribution along the DMO operator differs significantly from that for isotropic media. Application of this anisotropic DMO to data from offshore Africa resulted in a considerably better alignment of reflections from horizontal and dipping reflectors in common‐midpoint gather than that obtained using an isotropic DMO. Even the presence of vertical inhomogeneity in this medium could not eliminate the importance of considering the shale‐induced anisotropy.
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Nefedkina, T. V., P. A. Lykhin, and G. A. Dugarov. "DETERMINATION OF AZIMUTHAL ANISOTROPIC MEDIA ELASTIC PARAMETERS FROM MULTIWAVE AVOA DATA BY NONLINEAR OPTIMIZATION METHOD." Russian Journal of geophysical technologies, no. 2 (January 29, 2019): 14–26. http://dx.doi.org/10.18303/2619-1563-2018-2-2.
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In this paper, we investigate optimization algorithm of joint nonlinear AVOA inversion of PP+PS reflections in anisotropic media. Algorithm is based on the exact solution for PP and PS waves reflection coefficients in anisotropic HTI medium. The PP and PS wave’s reflections from the top of the anisotropic layer are examined. We use synthetic seismograms generated by ray method for the algorithm testing. We show that joint compressional and converted wave’s inversion allows increasing the robustness of the method and the accuracy of medium-parameter estimates. Coefficients of anisotropy are determined with better accuracy if signal-to-noise ratio is bigger than 5 for PP wave and bigger than 2 for PS wave.
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Araújo, Iury, Murillo Nascimento, Jessé Costa, Alan Souza, and Jörg Schleicher. "Anisotropic Born scattering for the qP scalar wavefield using a low-rank symbol approximation." GEOPHYSICS 86, no. 5 (August 18, 2021): T337—T348. http://dx.doi.org/10.1190/geo2020-0764.1.
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We have developed a procedure to derive low-rank evolution operators in the mixed space-wavenumber domain for modeling the qP Born-scattered wavefield at perturbations of an anisotropic medium under the pseudoacoustic approximation. To approximate the full wavefield, this scattered field is then added to the reference wavefield obtained with the corresponding low-rank evolution operator in the background medium. Being built upon a Hamiltonian formulation using the dispersion relation for qP-waves, this procedure avoids pseudo-S-wave artifacts and provides a unified approach for linearizing anisotropic pseudoacoustic evolution operators. Therefore, it is immediately applicable to any arbitrary class of anisotropy. As an additional asset, the scattering operators explicitly contain the sensitivity kernels of the Born-scattered wavefield with respect to the anisotropic medium parameters. This enables direct access to important information such as its offset dependence or directional characteristics as a function of the individual parameter perturbations. For our numerical tests, we specify the operators for a mildly anisotropic tilted transversely isotropic (TTI) medium. We validate our implementation in a simple model with weak contrasts and simulate reflection data in the BP TTI model to indicate that the procedure works in a more realistic scenario. The Born-scattering results indicate that our procedure is applicable to strongly heterogeneous anisotropic media. Moreover, we use the analytical capabilities of the kernels by means of sensitivity tests to demonstrate that using two different medium parameterizations leads to different results. The mathematical formulation of the method is such that it allows for an immediate application to least-squares migration in pseudoacoustic anisotropic media.
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Apresyan, L. A., and D. V. Vlasov. "On depolarization factors of anisotropic ellipsoids in an anisotropic medium." Technical Physics 59, no. 12 (December 2014): 1760–65. http://dx.doi.org/10.1134/s1063784214120020.
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Ivanov, Yuriy, and Alexey Stovas. "Traveltime parameters in tilted orthorhombic medium." GEOPHYSICS 82, no. 6 (November 1, 2017): C187—C200. http://dx.doi.org/10.1190/geo2016-0486.1.
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Traveltime parameters, defined through the series coefficients of the traveltime squared as a function of the horizontal offset projections, play an important role in moveout approximations and corrections, and in model parameter inversion. We evaluate an approach to derive the traveltime parameters in a single homogeneous anisotropic layer of tilted orthorhombic symmetry for one- and two-way traveling waves. The approach allows us to obtain the traveltime parameters of pure and converted modes. We use numerical models to illustrate the dependence of the high-order traveltime parameters on the Euler angles and the anisotropy parameters. The traveltime parameter inversion is a strongly ill-posed problem in anisotropic media, and improvements due to inclusion of the high-order traveltime parameters can sufficiently reduce the space of equivalent kinematic models. We perform a numerical model parameter inversion using the concept of artificial neural networks to demonstrate the accuracy improvements due to inclusion of the high-order traveltime parameters over the inversion of the second-order coefficient, conventionally known as normal moveout velocity, only. We demonstrate algebraically and numerically that the presented approach to calculate the traveltime parameters is easily extended to multilayered media. It can be used for Dix-type inversion to obtain the interval medium parameters.
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Chang, Ching S., and Yang Chang. "Green’s Function for Elastic Medium With General Anisotropy." Journal of Applied Mechanics 62, no. 3 (September 1, 1995): 573–78. http://dx.doi.org/10.1115/1.2895983.
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It is difficult to obtain explicit expressions of Green’s function for elastic medium with general anisotropy. The difficulty is associated with an integration of functions with high degrees of singularity. In this paper, we propose a method employing extend functions. This method avoids the difficulty of singularities and renders an explicit series expression of Green’s function for general anisotropic conditions. Analytical expression of the coefficients in the series are provided. Numerical examples are given to evaluate the applicability of this method.
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Ivanov, Yuriy, and Alexey Stovas. "Upscaling in orthorhombic media: Behavior of elastic parameters in heterogeneous fractured earth." GEOPHYSICS 81, no. 3 (May 2016): C113—C126. http://dx.doi.org/10.1190/geo2015-0392.1.
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A stack of horizontal homogeneous elastic arbitrary anisotropic layers in welded contact in the long-wavelength limit is equivalent to an elastic anisotropic homogeneous medium. Such a medium is characterized by an effective average description adhering to previously derived closed-form formalism. We have used this formalism to study three different inhomogeneous orthorhombic (ORT) models that could represent real geologic scenarios. We have determined that a stack of thin orthorhombic layers with arbitrary azimuths of vertical symmetry planes can be approximated by an effective orthorhombic medium. The most suitable approach for this is to minimize the misfit between the effective anisotropic medium, monoclinic in that case, and the desirable orthorhombic medium. The second model is an interbedding of VTI (transversely isotropic with a vertical symmetry axis) layers with the same layers containing vertical fractures (shales are intrinsically anisotropic and often fractured). We have derived a weak-anisotropy approximation for important P-wave processing parameters as a function of the relative amount of the fractured lithology. To accurately characterize fractures, inversion for the fracture parameters should use a priori information on the relative amount of a fractured medium. However, we have determined that the cracks’ fluid saturation can be estimated without prior knowledge of the relative amount of the fractured layer. We have used field well-log data to demonstrate how fractures can be included in the interval of interest during upscaling. Finally, the third model that we have considered is a useful representation of tilted orthorhombic medium in the case of two-way propagation of seismic waves through it. We have derived a weak anisotropy approximation for traveltime parameters of the reflected P-wave that propagates through a stack of thin beds of tilted orthorhombic symmetry. The tilt of symmetry planes in an orthorhombic medium significantly affects the kinematics of the reflected P-wave and should be properly accounted for to avoid mispositioning of geologic structures in seismic imaging.
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Pšenčík, Ivan, and Véronique Farra. "First-order ray tracing for qP waves in inhomogeneous, weakly anisotropic media." GEOPHYSICS 70, no. 6 (November 1, 2005): D65—D75. http://dx.doi.org/10.1190/1.2122411.
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We propose approximate ray-tracing equations for qP-waves propagating in smooth, inhomogeneous, weakly anisotropic media. For their derivation, we use perturbation theory, in which deviations of anisotropy from isotropy are considered to be the first-order quantities. The proposed ray-tracing equations and corresponding traveltimes are of the first order. Accuracy of the traveltimes can be increased by calculating a secondorder correction along first-order rays. The first-order ray-tracing equations for qP-waves propagating in a general weakly anisotropic medium depend on only 15 weak-anisotropy parameters (generalization of Thomsen’s parameters). The equations are thus considerably simpler than the exact ray-tracing equations. For higher-symmetry anisotropic media the equations differ only slightly from equations for isotropic media. They can thus substitute for the traditional isotropic ray tracers used in seismic processing. For vanishing anisotropy, the first-order ray-tracing equations reduce to standard, exact ray-tracing equations for isotropic media. Numerical tests for configuration and models used in seismic prospecting indicate negligible dependence of accuracy of calculated traveltimes on inhomogeneity of the medium. For anisotropy of about 8%, considered in the examples presented, the relative errors of the traveltimes, including the second-order correction, are well under 0.05%; for anisotropy of about 20%, they do not exceed 0.3%.
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Maisheev, V. A. "-beam propagation in an anisotropic medium." Nuclear Instruments and Methods in Physics Research Section B: Beam Interactions with Materials and Atoms 168, no. 1 (May 2000): 11–20. http://dx.doi.org/10.1016/s0168-583x(99)00828-9.
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Gjerde, Jon. "The pressure equation in anisotropic medium." Stochastics and Stochastic Reports 59, no. 1-2 (October 1996): 47–69. http://dx.doi.org/10.1080/17442509608834084.
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Shaikh, Dastgeer, and Gary P. Zank. "Anisotropic Cascades in Interstellar Medium Turbulence." Astrophysical Journal 656, no. 1 (January 18, 2007): L17—L20. http://dx.doi.org/10.1086/512051.
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Křepelka, Jaromír. "Reversibility Theorem for Anisotropic Stratified Medium." Journal of Modern Optics 40, no. 8 (August 1993): 1581–86. http://dx.doi.org/10.1080/09500349314551611.
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Alessandrini, Giovanni, Maarten V. de Hoop, Romina Gaburro, and Eva Sincich. "EIT in a layered anisotropic medium." Inverse Problems & Imaging 12, no. 3 (2018): 667–76. http://dx.doi.org/10.3934/ipi.2018028.
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SAFONOV, VLADIMIR L. "CONDUCTION ELECTRON IN THE ANISOTROPIC MEDIUM." International Journal of Modern Physics B 07, no. 22 (October 10, 1993): 3899–905. http://dx.doi.org/10.1142/s0217979293003528.
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A new model for describing free electrons and holes in crystals in the long-wavelength approximation is proposed. The crystalline anisotropy in the framework of this model is introduced by means of corresponding space-time geometry. The generalized Dirac’s equation is constructed and non-relativistic Hamiltonian containing energy terms of the order of c–2 is calculated. It is shown that the spin magnetic components depend on corresponding effective cyclotron masses. Applicability of the results of the proposed model to different experiments is discussed. For the one-dimensional case, a hypothesis of para-Fermi statistics is suggested which may appear to explain one more mechanism of high-T c superconductivity.
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Sekar, R., G. Vaidyanathan, and A. Ramanathan. "Ferroconvection in an anisotropic porous medium." International Journal of Engineering Science 34, no. 4 (March 1996): 399–405. http://dx.doi.org/10.1016/0020-7225(95)00113-1.
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Das, Debojyoti. "Turing pattern formation in anisotropic medium." Journal of Mathematical Chemistry 55, no. 3 (November 10, 2016): 818–31. http://dx.doi.org/10.1007/s10910-016-0709-5.
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Pšenčk, Ivan, and Dirk Gajewski. "Polarization, phase velocity, and NMO velocity of qP-waves in arbitrary weakly anisotropic media." GEOPHYSICS 63, no. 5 (September 1998): 1754–66. http://dx.doi.org/10.1190/1.1444470.
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We present approximate formulas for the qP-wave phase velocity, polarization vector, and normal moveout velocity in an arbitrary weakly anisotropic medium obtained with first‐order perturbation theory. All these quantities are expressed in terms of weak anisotropy (WA) parameters, which represent a natural generalization of parameters introduced by Thomsen. The formulas presented and the WA parameters have properties of Thomsen’s formulas and parameters: (1) the approximate equations are considerably simpler than exact equations for qP-waves, (2) the WA parameters are nondimensional quantities, and (3) in isotropic media, the WA parameters are zero and the corresponding equations reduce to equations for isotropic media. In contrast to Thomsen’s parameters, the WA parameters are related linearly to the density normalized elastic parameters. For the transversely isotropic media with vertical axis of symmetry, the equations presented and the WA parameters reduce to the equations and linearized parameters of Thomsen. The accuracy of the formulas presented is tested on two examples of anisotropic media with relatively strong anisotropy: on a transversely isotropic medium with the horizontal axis of symmetry and on a medium with triclinic anisotropy. Although anisotropy is rather strong, the approximate formulas presented yield satisfactory results.
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Boitz, Nepomuk, Anton Reshetnikov, and Serge A. Shapiro. "Visualizing effects of anisotropy on seismic moments and their potency-tensor isotropic equivalent." GEOPHYSICS 83, no. 3 (May 1, 2018): C85—C97. http://dx.doi.org/10.1190/geo2017-0442.1.
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Radiation patterns of earthquakes contain important information on tectonic strain responsible for seismic events. However, elastic anisotropy may significantly impact these patterns. We systematically investigate and visualize the effect of anisotropy on the radiation patterns of microseismic events. For visualization, we use a vertical-transverse-isotropic (VTI) medium. We distinguish between two different effects: the anisotropy in the source and the anisotropy on the propagation path. Source anisotropy mathematically comes from the matrix multiplication of the anisotropic stiffness tensor with the source strain expressed by the potency tensor. We analyze this effect using the corresponding radiation pattern and the moment tensor decomposition. Propagation anisotropy mathematically comes from the deviation between the polarization and the propagation direction of a quasi P-wave in an anisotropic medium. We investigate both effects separately by either assuming the source to be anisotropic and the propagation to be isotropic or vice versa. We find that both effects have a significant impact on the radiation pattern of a pure-slip source. Finally, we develop an alternative visualization of source mechanisms by plotting beach balls proportional to their potency tensors. For this, we multiply the potency tensor with an isotropic elasticity tensor having the equivalent shear modulus [Formula: see text] and [Formula: see text]. In this way, we visualize the tectonic deformation in the source, independently of the rock anisotropy.
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Carcione, José M., Stefano Picotti, Fabio Cavallini, and Juan E. Santos. "Numerical test of the Schoenberg-Muir theory." GEOPHYSICS 77, no. 2 (March 2012): C27—C35. http://dx.doi.org/10.1190/geo2011-0228.1.
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The Schoenberg-Muir theory states that an equivalent, homogeneous and anisotropic medium can be constructed from a layered medium composed of several thin layers, each anisotropic, under the assumption of stationarity. To test the theory we considered single transversely isotropic layers with different orientations of the symmetry axis and performed numerical simulations of wave propagation with a full-wave solver. The equivalent media have orthorhombic and monoclinic symmetries, respectively. The theory performed very well from the kinematical and dynamical points of view, even for strong anisotropy and layers described by media whose symmetry axes have different orientations.
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Barret, C., and S. Baste. "Effective Elastic Stiffnesses of an Anisotropic Medium Permeated by Tilted Cracks." Journal of Applied Mechanics 66, no. 3 (September 1, 1999): 680–86. http://dx.doi.org/10.1115/1.2791562.
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This paper is concerned with the relationship between the effective stiffness tensor and the intensity of damage in individual modes for an anisotropic material with tilted cracks. The predictions are compared favorably with the experimentally measured load-induced changes of the 13 stiffnesses of a two-dimensional C/C-SiC ceramic matrix composite subjected to an off-axis solicitation. By taking into account the thickness of the cracks, it is possible to understand the change of the elastic anisotropy of the material and of its inelastic strain.
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Jin, Song, and Alexey Stovas. "Reflection and transmission coefficient approximations for P, S1 and S2 waves in triclinic media." Geophysical Journal International 224, no. 1 (October 22, 2020): 558–80. http://dx.doi.org/10.1093/gji/ggaa493.
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SUMMARY The conventional assumptions, in most published approximations of the reflection and transmission (R/T) coefficients of plane waves at a plane interface between two anisotropic half-spaces, confine their applications to weakly anisotropic and/or weak contrast models. We consider the horizontal interface enclosed by two triclinic half-spaces to approximate the R/T coefficients normalized by the vertical energy flux. The homogeneous background medium can be anisotropic with arbitrary symmetry to better simulate the strongly anisotropic media. The second-order approximations are proposed to accommodate the strong contrast interface. We also consider an isotropic background medium under the weak anisotropy assumption. The obtained approximations can be applied to P, S1 and S2 waves, except for the transmission coefficients between the S1 and S2 waves. The S-wave transmission coefficients are insensitive to the model parameter contrasts and predominately rely on the S-wave polarization directions in the half-spaces above and below the interface. The proposed approximations are tested numerically.
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Yuan, S. P., and R. M. C. So. "Turbulent rotating flow calculations: An assessment of two-equation anisotropic and Reynolds stress models." Proceedings of the Institution of Mechanical Engineers, Part G: Journal of Aerospace Engineering 212, no. 3 (March 1, 1998): 193–212. http://dx.doi.org/10.1243/0954410981532270.
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The stress field in a rotating turbulent internal flow is highly anisotropic. This is true irrespective of whether the axis of rotation is aligned with or normal to the mean flow plane. Consequently, turbulent rotating flow is very difficult to model. This paper attempts to assess the relative merits of three different ways to account for stress anisotropies in a rotating flow. One is to assume an anisotropic stress tensor, another is to model the anisotropy of the dissipation rate tensor, while a third is to solve the stress transport equations directly. Two different near-wall two-equation models and one Reynolds stress closure are considered. All the models tested are asymptotically consistent near the wall. The predictions are compared with measurements and direct numerical simulation data. Calculations of turbulent flows with inlet swirl numbers up to 1.3, with and without a central recirculation, reveal that none of the anisotropic two-equation models tested is capable of replicating the mean velocity field at these swirl numbers. This investigation, therefore, indicates that neither the assumption of anisotropic stress tensor nor that of an anisotropic dissipation rate tensor is sufficient to model flows with medium to high rotation correctly. It is further found that, at very high rotation rates, even the Reynolds stress closure fails to predict accurately the extent of the central recirculation zone.