Anisotropic media with singular slowness surfaces

The wave quantities needed in constructing wave fields propagating in anisotropic elastic media are usually calculated as a function of the slowness vector, or of its direction called the wave normal. In some applications, however, it is desirable to calculate the wave quantities as a function of the ray direction. In this paper, a method of calculating the slowness vector for a specified ray direction is proposed. The method is applicable to general anisotropy of arbitrary strength with arbitrary complex wave surface. The slowness vector is determined by numerically solving a system of multivariate polynomial equations of the sixth order. By solving the equations, we obtain a complete set of slowness vectors corresponding to all wave types and to all branches of the wave surface including the slowness vectors along the acoustic axes. The wave surface can be folded to any degree. The system of equations is further specified for rays shot in the symmetry plane of an orthorhombic medium and for a transversely isotropic medium. The system is decoupled into two polynomial equations of the fourth order for the P –SV waves, and into equations for the SH wave, which yield an explicit closed-form solution. The presented approach is particularly advantageous in constructing ray fields, ray-theoretical Green functions, wavefronts and wave fields in strong anisotropy.

Three algebraic surfaces — the slowness surface, the phase-velocity surface, and the group-velocity surface — play fundamental roles in the theory of seismic wave propagation in anisotropic elastic media. While the slowness (sometimes called phase-slowness) and phase-velocity surfaces are fairly simple and their main algebraic properties are well understood, the group-velocity surfaces are extremely complex; they are complex to the extent that even the algebraic degree, [Formula: see text], of a system of polynomials describing the general group-velocity surface is currently unknown, and only the upper bound of the degree [Formula: see text] is available. This paper establishes the exact degree [Formula: see text] of the general group-velocity surface along with two closely related to [Formula: see text] quantities: the maximum number, [Formula: see text], of body waves that may propagate along a ray direction in a homogeneous anisotropic elastic solid [Formula: see text] and the maximum number, [Formula: see text], of isolated, singularity-unrelated cusps of a group-velocity surface [Formula: see text].

Summary Using ray perturbation theory, a linear relationship is derived that describes the first-order perturbation in slowness vectors due to elastic parameter and interface perturbations as well as to source and receiver position perturbations. For layered media. a propagator matrix, which has symplectic properties, is defined in Cartesian coordinates. The method is valid for heterogeneous isotropic or anisotropic media with interfaces, and allows the efficient calculation of Frechet derivatives of the polarization vectors. This makes it possible to use polarization data in tomographic inversion. Explicit integral expressions of the Frechet derivatives of the traveltime and the polarization vector are given for the qP wave in a transverse isotropic medium. The reference medium is a heterogeneous medium with elliptical anisotropy. A sensitivity study shows that the qP-wave polarization vectors provide information on the isotropic component of the slowness.

The dependence of the location of singular points of orthorhombic (ORT) media on the stiffness coefficients , and phase velocity at the singularity point is studied under the assumption that are larger than and . In this case, singular points appear only at the intersection of slowness surfaces of S1- and S2-waves. To simplify the presentation of the results, the values , are fixed and changed within the limits at which the stiffness matrix remains positive definite. We define the parameters, , , , which result in 0, 1, or 2 singular points in the symmetry planes of the ORT medium. The types of these singular points and their location on the unit circle are described. It is shown that, by selecting parameters , any singular point in the symmetry plane 13 can be combined with the limiting position of the singularity point in-between the symmetry planes, or this point can be included in the singular curve of the degenerate ORT medium. We derive equations for the semi-axes of an ellipse of conical refraction, which is the image of a singular point from plane 13 in the group domain. Conditions for degeneration of the ellipse of conical refraction into a segment or a point are defined. It is shown that there exists only one ORT medium with a fixed phase velocity of S1- and S2-waves in a given singular direction n. If we present the ORT media as projections of these vectors n onto the plane 12 and mark the value of the Poincaré index of the S1- or S2-wave at the point n, we get 2 regions with indices 1/2 and –1/2 separated by projection of the singular curve in the form of an ellipse or hyperbola. We compute equations for of a degenerate ORT medium in terms of the values, , and velocity of S1-, S2-waves on a singular curve. The singular curve is defined by the intersection of a unit sphere with an elliptic cone. It is proved that a degenerate ORT medium for or is, respectively, a transversally isotropic medium with a vertical or horizontal axis of symmetry. The results are illustrated in several examples.

We present an automated method to track and identify neurons in C. elegans, called "fast Deep Learning Correspondence" or fDLC, based on the transformer network architecture. The model is trained once on empirically derived synthetic data and then predicts neural correspondence across held-out real animals via transfer learning. The same pre-trained model both tracks neurons across time and identifies corresponding neurons across individuals. Performance is evaluated against hand-annotated datasets, including NeuroPAL [1]. Using only position information, the method achieves 80.0% accuracy at tracking neurons within an individual and 65.8% accuracy at identifying neurons across individuals. Accuracy is even higher on a published dataset [2]. Accuracy reaches 76.5% when using color information from NeuroPAL. Unlike previous methods, fDLC does not require straightening or transforming the animal into a canonical coordinate system. The method is fast and predicts correspondence in 10 ms making it suitable for future real-time applications.

Presents a new and more robust method of obtaining local projective and affine invariants. These shape descriptors are useful for object recognition because they eliminate the search for the unknown viewpoint. Being local, the invariants are much less sensitive to occlusion than the global ones used elsewhere. The basic ideas are: (i) employing an implicit curve representation without a curve parameter, thus increasing robustness; and (ii) using a canonical coordinate system which is defined by the intrinsic properties of the shape, regardless of any given coordinate system, and is thus invariant. Several configurations are treated including a general curve without any correspondence, and curves with known correspondence of feature points or lines. >

In this review we consider the propagation of nonlinear waves in magnetically structured plasmas. We start from studying nonlinear surface waves on magnetic interfaces and give a derivation of the equation governing propagation of these waves in incompressible plasmas. We present the results of a numerical solution of this equation. Then we discuss various generalizations of this equation.We proceed to study slow nonlinear waves in magnetic slabs. We present the Benjamin-Ono equation describing nonlinear slow sausage surface waves in magnetic slabs, and discuss its solutions in the form of algebraic solitons. Then we consider nonlinear slow body waves and write down the governing equation for these waves. We discuss its numerical solution which shows that the evolution of an initial perturbation results in a gradient catastrophe.After that we consider nonlinear slow sausage surface and slow body waves in magnetic tubes, which are similar to the corresponding counterparts in magnetic slabs. We present a system of two equations describing both types of waves. We briefly discuss the derivation of the Leibovich-Roberts equation for slow sausage surface waves from this system of equations, and present the results of its numerical study. Then we give a brief description of the derivation of the governing equation for slow body waves, which is very similar to its counterpart for slow body waves in magnetic slabs and has very similar properties. We complete this part of the review with presenting the governing equation for non-axisymmetric surface waves in a thin magnetic tube in an incompressible plasma.In the last part of the paper we consider the damping of nonlinear waves in weakly dissipative magnetically structured plasmas. We briefly discuss the application of the obtained results to the problem of heating of the solar corona.KeywordsSurface WaveSolitary WaveSolar PhysMagnetic TubeKink ModeThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Slowness surface for bulk wave propagation in anisotropic media can be divided into concave, saddle and convex regions by the parabolic lines. When a parabolic line crosses a symmetry plane, it leaves either an inflection point or a parabolic point. Surface normal at these points is associated with cuspidal point and swallowtail point, respectively, on the wave surface and in phonon focusing patterns. By examining the degeneracies in the Stroh eigenvalue equation, we have calculated the cuspidal points in cubic crystals analytically. In this work, the parabolic point and its surface normal are discussed. The main idea is to establish a connection between the parabolic point and the extraordinary transonic state that is related to a degeneracy with a multiplicity of four in the Stroh eigenvalue equation. Such a connection yields a series of simple expressions, which determine the locations of parabolic points and the corresponding swallowtail points. The result is demonstrated using phonon focusing patterns of cubic crystals, and the method also provides a tool for general discussion of the slowness surface geometry.

The singularity points are very important for elastic waves propagation in low-symmetry anisotropic media (Stovas et al., 2021a). Being converted into the group velocity domain, they result in internal refraction cone with anomalous amplitudes and very complicated polarization fields. In elastic orthorhombic (ORT) media, there is always one singularity point in the essential symmetry plane, (0,1,2) singularity points in remaining non-essential symmetry planes and (0,1) points in-between the symmetry planes (Stovas et al., 2021b).I analyze the conditions for existence of a singularity point in-between the symmetry planes.In order to do that I fix the diagonal elements of the stiffness coefficient matrix, cjj , j=1,6, and introduce new variables d12 = c12 +c66, d13 = c13 +c55 and d23 = c23 +c44. I also assume that the symmetry plane 2-3 is the essential one by introducing the inequality c55 < c44 < c66 . If c66 < c44 < c55, the 2-3 plane is still essential one but the properties of non-essential planes will interchange. In case of other inequalities for “S wave” stiffness coefficients, the corresponding properties of singularity points can be obtained by a cyclic rotation of stiffness coefficients and symmetry planes (Stovas et al., 2023).For selected essential symmetry plane (2-3), I propose to fix two variables d12 and d13, and set the variable d23 as a free variable. By changing d23 only, I can define the trajectory of a singularity point in-between the symmetry planes. This trajectory is given by a continuous line connecting the symmetry planes. Then I define the traces of this trajectory on symmetry planes (maximum two points for each plane) by a specific value of the variable d23. These 6 values can be used for intervals of d23 where the singularity point in-between the symmetry planes exists. Analysis shows that there are 7 zones in (d12 , d13) plane with different intervals of d23, which guarantee the existence of singularity point in-between the symmetry planes. There are 3 intervals of d23 in one zone, two intervals in two zones and one interval in three zones. There is no singularity point in-between the symmetry planes for any d23 in remaining zone. These zones are separated by three straight lines that defined by d12 = d12(critical) , d13 = d13(critical)  and d13 = α d12, where α guarantees that trajectory of singularity point meets the essential symmetry plane. ReferencesStovas, A., Roganov, Yu., and V. Roganov, 2021a, Geometrical characteristics of P and S wave phase and group velocity surfaces in anisotropic media, Geophysical Prospecting, 68(1), 53-69.Stovas, A., Roganov, Yu., and V. Roganov, 2021b, Wave characteristics in elliptical orthorhombic medium, Geophysics, 86(3), C89-C99.Stovas, A., Roganov, Yu., and V. Roganov, 2023, On singularity points in elastic orthorhombic media, Geophysics, 88(1), C11-C32.

Subject. This article focuses on the problems of solving systems of homogeneous linear algebraic equations in the pricing problems of an alternative financial model. Objectives. The article aims to develop a non-trivial approach to solving the problem of pricing from the scratch in the construction of a national alternative financial model, which boils down to solving large homogeneous systems of linear algebraic equations, complicated by the fuzzy setting of initial parameters (initial data) within the framework of creating a State model of a socially oriented economy, guaranteed to ensure economic, military and other types of integrated security of the Russian Federation. Methods. For the study, we used tensorial notations involving the use of elements of linear algebra, including operations with matrices and vectors, the apparatus of the fuzzy set theory (interval arithmetic), as well as methods and principles of scientific research and complex logical analysis and modeling of pricing and technological processes. Results. The article shows that an effective non-zero (different from zero, i.e. classical, trivial) solution of systems of homogeneous algebraic equations in pricing problems of an alternative financial model from the scratch is possible if the initial parameters are specified by fuzzy sets, which significantly simplifies the procedure of the computational process and the volume of arithmetic operations. Conclusions. The results obtained can be used in structures involved in the creation and testing of financial and economic models both at the federal and the Russian Federation constituent entity levels, as well as in governmental and non-governmental organizations dealing with the issues of pricing, planning and logistics support for business entities not only in the Russian Federation, but also in other countries experiencing problems in the field of financing.

Wavefield extrapolation in pseudoacoustic orthorhombic anisotropic media suffers from wave-mode coupling and stability limitations in the parameter range. We use the dispersion relation for scalar wave propagation in pseudoacoustic orthorhombic media to model acoustic wavefields. The wavenumber-domain application of the Laplacian operator allows us to propagate the P-waves exclusively, without imposing any conditions on the parameter range of stability. It also allows us to avoid dispersion artifacts commonly associated with evaluating the Laplacian operator in space domain using practical finite-difference stencils. To handle the corresponding space-wavenumber mixed-domain operator, we apply the low-rank approximation approach. Considering the number of parameters necessary to describe orthorhombic anisotropy, the low-rank approach yields space-wavenumber decomposition of the extrapolator operator that is dependent on space location regardless of the parameters, a feature necessary for orthorhombic anisotropy. Numerical experiments that the proposed wavefield extrapolator is accurate and practically free of dispersion. Furthermore, there is no coupling of qSv and qP waves because we use the analytical dispersion solution corresponding to the P-wave.

Formulae for the zeroth-order principal term plus the first-order additional term of the qP- and qS-wave Green's functions in the so-called quasi-isotropic (QI) approximation are derived for an unbounded inhomogeneous weakly anisotropic medium. The basic idea of this approximation is to seek the asymptotic solution of the elastodynamic equation as an expansion with respect to two small parameters of the same order: the small parameter used in the standard ray method and a parameter characterizing differences of tensors of elastic parameters of a weakly anisotropic medium and of a nearby 'background' isotropic medium. As a result, the procedure of constructing the Green's functions is split into two steps: (1) calculation of rays, traveltimes, the geometrical spreading and polarization vectors in the background isotropic medium; (2) calculation of corrections of traveltimes, amplitudes and polarization vectors due to the deviation of the weakly anisotropic medium from the isotropic background.

The purpose of this work is to draw attention to several differences between wave propagation in dissipative anisotropic media and purely elastic anisotropic media. In an elastic medium, the wavefront is defined as the envelope of the family of planes that makes the phase of the plane waves zero. It turns out that this definition coincides with the wavefronts obtained from the group and energy velocities, i.e., the three concepts are equivalent. However, for plane waves traveling in dissipative anisotropic media these concepts are different. Despite these differences, the velocity of the envelope of plane waves closely approximates the energy velocity, and therefore can represent the wavefront from a practical point of view. On the other hand, the group velocity describes the wavefront only when the attenuation is relatively low, i.e., for Q values higher than 100. The values of the different velocities and the shape of the wavefront are considerably influenced by the relative values of the attenuation along the principal axes of the anisotropic medium. This means that the anisotropic coefficients in attenuating anisotropic media may differ substantially from the corresponding elastic coefficients. Moreover, it is shown that the usual orthogonality properties between the slowness surface and energy velocity vector and the wavefront and wavenumber vector does not hold for dissipative anisotropic media.

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.

nd qP waves in the middle of snapshots, since we use the exact dispersion relation and no constrains on Thomsen’s parameters for stability.