Complex Anisotropic Media Research Articles

Polarization feature fusion and calculation of birefringence dynamics in complex anisotropic media.

As a complex anisotropic medium, variation in birefringence within biological tissues is closely associated with numerous physiological behaviors and phenomena. In this Letter, we propose a polarization feature fusion method and corresponding polarimetric parameters, which exhibit excellent performance of capturing the birefringence dynamic variation process in complex anisotropic media. By employing the feature fusion method, we combine and transform polarization basis parameters (PBPs) to derive fused polarization feature parameters (FPPs) with explicit expressions. Subsequently, we conduct Monte Carlo (MC) simulation to demonstrate the effectiveness of the proposed FPPs from two variation dimensions of birefringence direction θ and modulus Δn. Leveraging mathematical modeling and linear transformations, we investigate and abstract their response patterns concerning θ and Δn. Finally, the experiments confirm that the FPPs show superior adaptability and interpretability in characterizing the birefringence dynamic process of turbid media. The findings presented in this study provide new, to the best of our knowledge, methodological insights of information extraction for computational polarimetry in biomedical research.

Modeling 3-D Elastic Wave Propagation in TI Media Using Discontinuous Galerkin Method on Tetrahedral Meshes

Transversely isotropic (TI) medium is a widely studied anisotropic solid medium in seismology. The numerical simulation of seismic wave propagation in TI media is an effective tool to analyze the mechanism of seismic waves in complex anisotropic media. In this study, we introduce a double-weighted Runge-Kutta discontinuous Galerkin (RKDG) method for numerically solving wave propagation problems in 3D TI media with surface topography. This method incorporates the discontinuous Galerkin spatial discretization with an explicit double-weighted two-step iteration time discretization. The local Lax-Friedrichs flux is used as the numerical flux in the formulations. This method can solve the first-order velocity-stress seismic wave equations including TI media as well as more general anisotropic media. Due to the large scale of 3D problems, the parallel technology is adopted. Regions with irregular boundaries are discretized into unstructured tetrahedral meshes. Numerical experiments for various anisotropic media including the vertical and tilted TI media are presented. The results demonstrate the effectiveness of the double-weighted RKDG method in wavefield simulations in 3D complicated anisotropic media.

Solving elastic wave equations in 2D transversely isotropic media by a weighted Runge-Kutta discontinuous Galerkin method

Accurate wave propagation simulation in anisotropic media is important for forward modeling, migration and inversion. In this study, the weighted Runge-Kutta discontinuous Galerkin (RKDG) method is extended to solve the elastic wave equations in 2D transversely isotropic media. The spatial discretization is based on the numerical flux discontinuous Galerkin scheme. An explicit weighted two-step iterative Runge-Kutta method is used as time-stepping algorithm. The weighted RKDG method has good flexibility and applicability of dealing with undulating geometries and boundary conditions. To verify the correctness and effectiveness of this method, several numerical examples are presented for elastic wave propagations in vertical transversely isotropic and tilted transversely isotropic media. The results show that the weighted RKDG method is promising for solving wave propagation problems in complex anisotropic medium.

Orthotropic anisotropy analysis and parameter estimation from 3-D vertical seismic profile data

SUMMARY Orthotropic velocity models are commonly considered as to have the lowest complexity (or highest symmetry) that can realistically describe the combined effect of Earth’s velocity heterogeneity in the vertical and in the horizontal (radial) directions. Velocity heterogeneity in the vertical direction is commonly caused by horizontal stratification of subsurface rock layers with different elastic properties or caused intrinsically by microlayered rocks such as shales. In the horizontal direction, azimuthal variation of the Earth’s stress field can initially lead to preferential deformation of pore space in the rocks and eventual development of aligned (vertical) fractures that are commonly regarded as the main cause for seismic azimuthal anisotropy. Building an appropriate orthotropic velocity model that can explain seismic velocity variations in vertical planes and along any azimuthal direction can therefore enhance the quality of seismic data processing and imaging. This, however, requires knowledge of several anisotropy parameters defined in the three symmetry planes of the orthotropic model and also the orientation of these planes. A combination of P- and S-wave slowness and polarization data derived from vertical seismic profile (VSP) measurements has been routinely used to estimate parameters of less complex anisotropic media such as transverse isotropy with vertical axes of symmetry (VTI). The accuracy of anisotropy parameter estimation by these methods depends primarily on the availability of data that are used in the inversion and the size of anisotropy parameters. In this study, we develop a benchmark for the accuracy of orthotropic parameters that are estimated from 3-D VSP measurements using P-wave slowness only method. Through numerical analysis, we model a wide range of fracture-induced orthotropic symmetries by adding vertical fractures to a background VTI media and show that the accuracy of our orthotropic parameter estimation method is dependent on anisotropy in the host media, intensity of fracturing, noise level in slowness data and slowness data coverage in both the dip and azimuthal planes. We then expand our accuracy analysis work to multilayer media by examining the accuracy of P-wave slowness method for orthotropic anisotropy parameter estimation by inverting finite-difference data generated in a pseudo 3-D VSP experiment. We show that, for a typical 3-D VSP experiment, reasonable estimates of orthotropic parameters in the model can only be obtained with slowness data taken from Xmax/Z ≥ 1 and line azimuth interval ϕ ≤ 10° where Xmax is the maximum source to receiver offset and Z is the receiver depth. We finally describe how field VSP measurements can be inverted for orthotropic anisotropy parameters.

Characteristics of elastic wave dispersion and attenuation induced by microcracks in complex anisotropic media

AbstractThe mechanism of dispersion and attenuation induced by fluid flow among pores and microcracks in rocks is an important research topic in geophysical domain. A generalised frequency-dependent fourth-rank tensor is proposed and derived herein by combining Sayers's discontinuity tensor formula and Gurevich's squirt flow model. Furthermore, a proposed method for establishing a cracked model with cracks embedded in a transversely isotropic (TI) background medium is developed. Based on the new formulation, we investigate the characteristics of dispersion, attenuation and azimuthal anisotropy of three commonly encountered vertical crack distributions, including aligned cracks, monoclinic cracks and cracks with partial random orientations. We validate the developed model by comparing its predictions with those of the classic anisotropic squirt flow model for an aligned crack. The numerical analyses indicate that the azimuth is independent of frequency when the maximum attenuation is observed for all three crack distributions. In a low-frequency range in the case of an anisotropic background, the attenuation of the qP-wave is inversely proportional to velocity, whereas the attenuation of the qSV-wave is proportional to velocity. In addition, the inherent anisotropy of the rock does not significantly affect the dispersion and attenuation owing to squirt flow. Finally, to investigate the applicability of the theory, we model laboratory data of a synthetic porous sandstone with aligned cracks. Overall, the models agree well with laboratory data. The complex characteristics determined through this study may be useful for the seismic characterisation of fractured reservoirs.

Tensorial elastodynamics for anisotropic media

Elastic wavefield solutions computed by finite-difference (FD) methods in complex anisotropic media are essential elements of elastic reverse time migration and full-waveform inversion analyses. Cartesian formulations of such solution methods, though, face practical challenges when aiming to represent curved interfaces (including free-surface topography) with rectilinear elements. To forestall such issues, we have developed a general strategy for generating solutions of tensorial elastodynamics for anisotropic media (i.e., tilted transversely isotropic or even lower symmetry) in non-Cartesian computational domains. For the specific problem of handling free-surface topography, we define an unstretched coordinate mapping that transforms an irregular physical domain to a regular computational grid on which FD solutions of the modified equations of elastodynamics are straightforward to calculate. Our fully staggered grid (FSG) with a mimetic FD (MFD) (FSG + MFD) approach solves the velocity-stress formulation of the tensorial elastic wave equation in which we compute the stress-strain constitutive relationship in Cartesian coordinates and then transform the resulting stress tensor to generalized coordinates to solve the equations of motion. The resulting FSG + MFD numerical method has a computational complexity comparable with Cartesian scenarios using a similar FSG + MFD numerical approach. Numerical examples demonstrate that our solution can simulate anisotropic elastodynamic field solutions on irregular geometries; thus, it is a reliable tool for anisotropic elastic modeling, imaging, and inversion applications in generalized computational domains including handling free-surface topography.

Analysis of Variational Formulations and Low-regularity Solutions for Time-harmonic Electromagnetic Problems in Complex Anisotropic Media

We consider the time-harmonic Maxwell's equations with physical parameters, namely the electric permittivity and the magnetic permeability, that are complex, possibly non-Hermitian, tensor fields. Both tensor fields verify a general ellipticity condition. In this work, the well-posedness of formulations for the Dirichlet and Neumann problems (i.e. with a boundary condition on the electric field or its curl, respectively) is proven using well-suited functional spaces and Helmholtz decompositions. For both problems, the a priori regularity of the solution and the solution's curl is analysed. The regularity results are obtained by splitting the fields and using shift theorems for second-order divergence elliptic operators. Finally, the discretization of the formulations with a H(curl)-conforming approximation based on edge finite elements is considered. An a priori error estimate is derived and verified thanks to numerical results with an elementary benchmark.

An efficient vector elastic reverse time migration method in the hybrid time and frequency domain for anisotropic media

Vector elastic reverse time migration (ERTM) produces subsurface elastic images with correct polarities using multicomponent seismic data. However, the decomposition of elastic wavefields into vector P- and S-wavefields is computationally expensive, particularly in heterogeneous and complex anisotropic media. We have developed a computationally efficient vector ERTM method in the hybrid time and frequency domain by combining three existing techniques. Rather than decomposing elastic wavefields into vector qP- and qS-wavefields during time-domain wavefield propagation, we conduct the wavefield decomposition in the frequency domain for several selected frequencies. In general, the number of selected frequencies needed for migration imaging is much smaller than the number of time steps during forward and backward wavefield propagation, leading to greatly reduced computational costs associated with the qP-/qS-wavefield vector separation in complex heterogeneous anisotropic media. We further combine an implicit directional wavefield separation into the vector ERTM to enhance the image quality. The numerical results demonstrate that our method produces high-quality elastic-wave migration images with notably reduced computational costs compared to the conventional vector ERTM method.

Grid-Characteristic Numerical Method for Low-Velocity Impact Testing of Fiber-Metal Laminates

The grid-characteristic numerical method (GCM) for hyperbolic equations systems is applied in many science fields—gas dynamics, hydrodynamics, plasma dynamics, etc. Its application for problems of dynamics of deformable solids is less popular, especially in comparison with finite elements methods. GCM shows good results and high performance for elastic wave problems in the approximation of small deformations—seismic survey and ultrasound non-destructive testing in medicine, aviation and railway industry. Low-velocity impacts (hail, dropped tool, bird strike, etc.) are one of the most dangerous load types for polymer composites. They cause barely visible impact damage (BVID) that can only be detected by a thorough ultrasound testing, but severely reduces the residual strength of the material, especially for a compression load along the surface. This testing increases the operating cost, and its necessity can be easily missed, which greatly reduces the reliability of polymer composites. Hybrid fiber-metal composites (GLARE, ARALL, titanium composite laminates) were developed to unify the advantageous properties of polymer composites and metal. The addition of a thin metal layer (1–2 mm) helps to reduce the impact vulnerability of polymer composites in case of a penetration or significant deformations of the material. The application of GCM for low-velocity impact problems can help to explain the damage pattern in fiber-metal composites in case of low-velocity strike, including delamination effects, by modelling elastic wave processes in the complex anisotropicmedium. This article contains the brief description of the GCM and numerical results that were obtained for model problems of a low-velocity impact on titanium composite laminates.

An improved rotated staggered-grid finite-difference method with fourth-order temporal accuracy for elastic-wave modeling in anisotropic media

The rotated staggered-grid (RSG) finite-difference method is a powerful tool for elastic-wave modeling in 2D anisotropic media where the symmetry axes of anisotropy are not aligned with the coordinate axes. We develop an improved RSG scheme with fourth-order temporal accuracy to reduce the numerical dispersion associated with prolonged wave propagation or a large temporal step size. The high-order temporal accuracy is achieved by including high-order temporal derivatives, which can be converted to high-order spatial derivatives to reduce computational cost. Dispersion analysis and numerical tests show that our method exhibits very low temporal dispersion even with a large temporal step size for elastic-wave modeling in complex anisotropic media. Using the same temporal step size, our method is more accurate than the conventional RSG scheme. Our improved RSG scheme is therefore suitable for prolonged modeling of elastic-wave propagation in 2D anisotropic media.

Tracing Seismic Shortest Path Rays in Anisotropic Medium with Rolling Surface

AbstractIt has very important theoretical and practical values to take rolling surface and seismic anisotropy into consideration in seismic forward modeling and inversion. Based on previous researches, the shortest path calculation algorithm is introduced in this paper to calculate seismic travel times in anisotropic media with rolling surface. In the process of model gridding, the whole model is divided into square cells firstly and rolling boundary is regarded as the approximation of units with irregular shapes. Then shortest path calculation is achieved on irregular grids by setting irregular nodes. Sena's group velocity approximation is used to calculate seismic travel times in anisotropic media during ray tracing, and seismic shortest path ray tracing in a complex anisotropic medium with rolling surface is achieved in the end. Theoretical model calculations show that the method in this paper can be reliably applied to complex anisotropic models with higher calculation accuracy.

CDP mapping to obtain the fine structure of the crust and upper mantle from seismic sounding data: an example for the southeastern China

We present an efficient and general approach of common depth point (CDP) mapping to image the fine structure of the crust and upper mantle. We use velocity model and reflector geometries which are the results of conventional interpretation, as a reference model to construct our CDP mapping of crust and upper mantle. After three steps (the calculation of CDP points and reflecting traveltimes, the determination of CDP mapping trajectories and the implementation of CDP imaging), reflection images were created by CDP mapping technique in which each time element of a seismic trace, beginning with the direct arrival, is mapped to reflection point. Our test of the method presented in this paper shows that the algorithm can quickly, accurately image the subsurfaces of the given model. The result of CDP mapping in the southeastern China shows that the depth of Moho is about 34 km, there are numerous reflection interfaces within the upper crust; and the Moho is a transition zone with a thickness of 2–3 km. The CDP mapping approach presented here can easily be extended to three dimensions and complex anisotropic media. The method can also be used to develop seismic attribute images of the crust and upper mantle.

Nematic liquid crystals as a new challenge for radiative transfer

Radiative transfer has been studied for almost a century, but only recently have effects of broken symmetry in the diffusion of light been systematically studied. Familiar concepts such as the mean free path and the diffusion constant must be generalized. Nematic liquid crystals provide a realistic complex system in which the new concepts are relevant. Thermal fluctuations of the local optical axis generate a weak but very specific and anisotropic light scattering and can even be long range. In addition, two different modes of electromagnetic propagation exist, with different polarization, and with different speeds, that couple in multiple scattering. It becomes a challenge to describe optical phenomena such as birefringence, interference, polarization, and intensity fluctuations under such conditions. In this review, the authors first describe the interesting phenomena in radiative transfer in complex anisotropic media and nematic liquid crystals. They then develop the systematic theory of transport starting from the fundamental equations and going through a Green's-function formulation.

Numerical modelling of ultrasonic wave propagation using the efficient p-version finite element method

In this paper, the new p-version of the finite element method has been used for the numerical modelling of ultrasonic wave propagation in solid media. The p-method involves increasing the polynomial order of the interpolation functions to achieve convergence. This is unlike the conventional h-version in which the number of elements is increased to achieve the same goal. Illustrative examples of wave propagation through an isotropic aluminium block, including a case with a built-in defect in the form of a crack, have been discussed. Also, a convergence criterion has been developed for dynamic problems. The potential to extend this model to other applications such as medical ultrasonics or to a more complex anisotropic medium is significant.

Rigorous analysis of radiation properties of lossy patch resonators on complex anisotropic media and lossy ground metallization

An extended spectral-domain immittance approach for the rigorous analysis of resonant frequencies and radiation characteristics of microstrip resonators is presented. The dyadic Green's function in the spectral domain is modified to include a complex anisotropic substrate with both permeability and permittivity tensors, lossy ground metallization, and a lossy conducting patch of conventional or superconducting material. Closed-form expressions for the transverse propagation constants and related immittances of TE and TM waves in the spectral domain lead to a CPU-time efficient algorithm that is operational on standard workstations. Numerical results show how the radiation characteristics are affected by losses as well as uniaxial and biaxial anisotropies.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Determination of the Green's function in the spectral domain using a matrix method: Application to radiators or resonators immersed in a complex anisotropic layered medium

A planar structure having arbitrarily located conductor patches immersed in complex anisotropic layered media presents a very general field problem. This problem is solved here by a rigorous formulation technique characterizing each layer by a 6 \times 6 tensor and finding the appropriate Fourier transformed Green's function matrix G of 2n \times 2n size. The technique finds a set of field eigenvectors for each layer. Using G , a method of moments numerical solution for radiation characteristics of probe fed patch(es) can be had in the spectral domain employing, for example, a zero reaction method. Variation of real frequencies of the driving probe fed signal is allowed by that approach. Those workers desirous of radiator or resonator fields and frequency behavior at only selected resonant frequencies can use G to derive a matrix S_{X} given here. Setting the determinant of S_{X} equal to zero yields complex resonant frequency solutions, and the field solutions as a consequence to the nonprobe fed or free standing patch structure. The method is very versatile and can handle a large class of microwave or millimeter wave integrated circuit or monolithic circuit problems, no matter how simple or complex as long as they possess planar layers.

Electromagnetic theorems for complex anisotropic media

Complex anisotropic media can generally be described by a <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">6 \times 6</tex> macroscopic constitutive tensor <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">\hat{C}</tex> . Using <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">\hat{C}</tex> properties, a reaction integral formula is derived from which an anisotropic reaction theorem (modified reciprocity theorem) is developed. Reduction of the <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">\hat{C}</tex> medium into a reciprocal medium is discussed including tensor symmetry attributes and limiting cases. The anisotropic reaction theorem is utilized to derive a zero reaction theorem, and then treated in relation to the moment method. Mutual and self-impedance elements of a network are also derived in terms of reaction integrals, symmetry covered using the anisotropic reaction theorem, and impedance elements related to moment calculations. Use of spectral domain analysis is also covered.