Planar Layered Media Research Articles

An image treatment of elastostatic transmission from an interface layer

A basic theorem for representing the Airy stress function for two perfectly bonded semi-infinite planes in terms of the corresponding Airy function for the unbounded homogeneous plane is applied in a systematic stepwise fashion to generate the corresponding Airy stress function for a three-phase composite comprising two semi-infinite planes separated by a thick layer. The loading of the three-phase composite is arbitrary, and may be in or near the interface layer. The basic theorem is first illustrated by applying it to an elastic medium which is bounded by two unloaded straight edges which intersect at an angle π/ n, where n is a positive integer. This example illustrates a case of a finite system of images, while the plane-layered medium problem leads to an infinite series of images.

On the fast approximation of Green's functions in MPIE formulations for planar layered media

The numerical implementation of the complex image approach for the Green's function of a mixed-potential integral-equation formulation is examined and is found to be limited to low values of /spl kappa//sub o//spl rho/ (in this context /spl kappa//sub 0//spl rho/ = 2/spl pi//spl rho///spl lambda//sub 0/, where /spl rho/ is the distance between the source and the field points of the Green's function and /spl lambda//sub 0/ is the free space wavelength). This is a clear limitation for problems of large dimension or high frequency where this limit is easily exceeded. This paper examines the various strategies and proposes a hybrid method whereby most of the above problems can be avoided. An efficient integral method that is valid for large /spl kappa//sub 0//spl rho/ is combined with the complex image method in order to take advantage of the relative merits of both schemes. It is found that a wide overlapping region exists between the two techniques allowing a very efficient and consistent approach for accurately calculating the Green's functions. In this paper, the method developed for the computation of the Green's function is used for planar structures containing both lossless and lossy media.

A new technique for the derivation of closed-form electromagnetic Green's functions for unbounded planar layered media

A closed-form electromagnetic Green's function for unbounded, planar, layered media is derived in terms of a finite sum of Hankel functions. The derivation is based on the direct inverse Hankel transform of a pole-residue representation of the spectral-domain form of the Green's function. Such a pole-residue form is obtained through the solution of the spectral-domain form of the governing Green's function equation numerically, through a finite-difference approximation, rather than analytically. The proposed methodology can handle any number of layers, including the general case where the planar media exhibit arbitrary variation in their electrical properties in the vertical direction. The numerical implementation of the proposed methodology is straightforward and robust, and does not require any preprocessing of the spectrum of the Green's function for the extraction of surface-wave poles or its quasi-static part. The number of terms in the derived closed-form expression is chosen adaptively with the distance between source and observation point as parameter. The development of the closed-form Green's function is presented for both vertical and horizontal dipoles. Its accuracy is verified through a series of numerical examples and comparisons with results from other established methods.

A Complete Set of Spatial-Domain Dyadic Green's Function Components for Cylindrically Stratified Media in Fast Computational Form

Numerical characterization of electromagnetic waves in cylindrically multilayered structures can be efficiently and rigorously performed by employing the method of moments (MoM) in spatial-domain. Thus the idea of obtaining spatial-domain Greens' functions in closed form or fast computational form was proposed [1] and since then many useful results were obtained for planar layered media. For cylindrically layered media, the closed form Green's functions in spatial-domain were reported in [2], where the spatial-domain Green's function components due to z- and ø-oriented electrical and magnetic sources have been obtained using the generalized pencil of function (GPOF) method. This paper, as a further extension, presents spatial-domain Green's function components in fast computational form due to ρ-oriented electrical source embedded in cylindrically layered media. The Green's function components in approximate but fast computational form are compared with the exact Green's function components obtained from direct numerical evaluations of the corresponding integrals, and a good agreement between the fast computational form and the exact form. Furthermore, the spatial-domain Green's function components due to z- and ø-oriented electrical and magnetic sources have also been obtained but for all the possible azimuth angles.

Wave diffraction by a rough boundary of an arbitrary plane-layered medium

The problem of electromagnetic (EM) wave scattering by a slightly rough boundary of an arbitrary layered medium is solved by a small perturbation method. The bistatic amplitude of scattering as well as scattering cross sections for a statistically rough surface are calculated for linear and circular polarized waves. Along with the scattering into the upgoing waves in the homogeneous medium, the scattering cross sections in the downgoing waves into a layered medium are obtained. Analytical results are applied to the modeling of natural layered media (ice and sand layers) remote sensing problems employing global positioning system (GPS) technics.

Reflection and Transmission of X-Waves in the Presence of Planarly Layered Media: The Pulsed Plane Wave Representation

Reflection and Transmission of X-Waves in the Presence of Planarly Layered Media: The Pulsed Plane Wave Representation

Beam pattern formation and on-/off-axis pressure in the near field through elastic structures

Detailed measurements have been done and compared to analytical models of the beam pattern and on-/off-axis pressure in the near-field region of plane circular transducers in order to model the reflection and transmission of ultrasonic waves from and through steel pipes (casing) and plates immersed in oil. It is shown that for broadband and pure tone pulses in the near field, for free field conditions and through a curved or planar layered media, the sidelobes are smaller and the half-power beamwidth is narrower than in the far field. The beamwidth is slightly dispersed (<1 deg) through the casing and plates of similar thickness, being larger through the planar geometry. Off-axis pressure through these elastic structures reveals condensing and dispersing effects (<2 mm) in the curved and planar geometry, respectively. On-axis pressure, however, for λ < structure’s thickness and plane ultrasonic transducers with size a/λ>5, where a is the transducer radius, the reflected and transmitted sound field is the same from and through both structures and behaves as a well-collimated beam in the near-field region. [Work supported by the University of Texas at Austin and the Mexican Petroleum Institute.]

Modelling teleseismic waves in dipping anisotropic structures

SUMMARY The existence of seismic discontinuities within the continental upper mantle has long been recognized, with more recent studies often indicating an association with elastic anisotropy. Their near-vertical sampling renders teleseismic P and S waves suitable for characterization of mantle discontinuities, but computationally e⁄cient methods of calculating synthetic seismograms are required for structures that exhibit lateral variability. We consider lithospheric models consisting of planar, homogeneous anisotropic layers with arbitrary dip.We adopt the traveltime equation of Diebold for dipping, plane-layered media as the basis for a high-frequency asymptotic method that does not require ray tracing. Traveltimes of plane waves in anisotropic media are calculated from simple analytic formulae involving the depths of layers beneath a station and the vertical components of phase slowness within the layers. We compute amplitudes using the re£ection and transmission matrices for planar interfaces separating homogeneous anisotropic media. Modelling indicates that upper-mantle seismic responses depend in a complex fashion on both layer dip and anisotropy, particularly in the case of converted phases. Azimuthal anisotropy generally displays a distinctive 180 0 backazimuthal periodicity in Ps conversion amplitude, as opposed to the 360 0 symmetry produced by dip. In contrast, anisotropy with a steeply plunging axis may under certain conditionsbe di⁄cultto distinguish from interface dip, asboth exhibit a 360 0 symmetry. We demonstrate the application of the method on Ps and Sp conversion data from the Yellowknife Array, which show evidence for both dipping and anisotropic layering, attributed to layers of anisotropic fabric in the upper mantle associated with ancient subducted slabs.

Wave diffraction by rough interfaces in an arbitrary plane-layered medium

The problem of electromagnetic wave scattering by a slightly rough interface in an arbitrarily layered medium is solved by a small-perturbation method. The bistatic amplitude of scattering as well as the scattering cross sections for statistically rough surfaces are calculated for linear polarized waves. Along with scattering into up-going waves in a homogeneous medium and scattering cross sections in down-going waves into a layered medium, scattering amplitudes from a rough interface in the arbitrarily layered medium are obtained.

Novel closed-form Green's function in shielded planar layered media

A new method is proposed for the construction of closed-form Green's function in planar, stratified media between two conducting planes. The new approach does not require the a priori extraction of the guided-wave poles and the quasi-static part from the Green function spectrum. The proposed methodology can be easily applied to arbitrary planar media without any restriction on the number of layers and their thickness. Based on the discrete solution of one-dimensional ordinary differential equations for the spectral-domain expressions of the appropriate vector potential components, the proposed method leads to the simultaneous extraction of all Green's function values associated with a given set of source and observation points. Krylov subspace model order reduction is used to express the generated closed-form Green's function representation in terms of a finite sum involving a small number of Hankel functions. The validity of the proposed methodology and the accuracy of the generated closed-form Green's functions are demonstrated through a series of numerical experiments involving both vertical and horizontal dipoles.

Reflection and Transmission of X-Waves in the Presence of Planarly Layered Media: the Pulsed Plane Wave Representation - Abstract

In this paper, a general ray-theoretical representation of the X-wave pulse is introduced. This representation is based on pulsed plane waves having wave vectors lying on a conic surface. The pulsed plane wave representation is used to study the behavior of X-waves in the presence of planarly layered media. In the case of an interface separating two semi-infinite media, the pulsed plane wave representation provides a clear explanation for the disintegration of the transmitted field. It shows that unlike the incident and reflected fields, the wave vectors associated with the transmitted field do not lie on a cone. Furthermore, this approach allows one to estimate several characteristic properties of the layered media (e.g., dielectric constant and thickness of each layer) from the time of arrival of the reflected X-waves.

3-D FEM/BEM-hybrid approach based on a general formulation of Huygens' principle for planar layered media

Huygens' principle is used to exclude inhomogeneous regions and ideal conducting regions from planar layered structures. The fields in the inhomogeneous regions are modeled by the finite-element method (FEM) with tetrahedral edge elements in terms of the electric-field strength E/spl I.oarr/. The fields in the layered structure are described by an integral representation of the electric-field strength E/spl I.oarr/ in terms of equivalent electric and magnetic Huygens' surface current densities for the inhomogeneous regions, and in terms of electric Huygens' surface current densities for ideal conducting regions. It is formulated with the help of electromagnetic (EM) potentials resulting in low-order singular integral kernels to facilitate the numerical handling of the integral representation. A general formulation of the integral representation is given for observation points lying in the Huygens' surface. As compared to the homogeneous-space case, additional terms in the integral representation have to be considered if parts of the Huygens' surface lie in an interface of layers with different material properties. An integral equation is formulated and discretized by a Galerkin testing procedure (boundary-element method), together with the finite-element (FE) system resulting in an unequivocal discretized description of the entire field problem. The method is validated with the help of a canonical test problem. Further numerical results are presented for dielectric resonators coupled to microstrip circuits.

Insulated antenna in a layered medium

The article presents a generalization of the integral equation of an insulated linear antenna immersed in a cylindrically layered lossy dielectric medium. The insulation is provided by a lossless dielectric layer. The kernel of the integral equation is represented as a superposition of the fundamental solutions of the wave equation with equivalent propagation constants for the given media. A generalization to a plane-layered medium is proposed. The problem of a vertical radiator above a layered half-space is considered.

Dynamic stress variations due to shear faults in a plane-layered medium

SUMMARY A complete set of expressions is presented for the computation of elastic dynamic stress in plane-layered media. We use a discrete-wavenumber reflectivity method to compute the stress field radiated by arbitrary moment-tensor sources. The expressions derived here represent an interesting tool for both-the observational and theoretical analysis of dynamic stress changes associated with earthquake phenomena. Dynamic stress changes associated with a strike-slip fault having unilateral rupture are shown. This modelling, which is similar to the 1992 Landers California earthquake, illustrates the effects of distance, directivity and depth on transient stress changes.

Spectral diffusion of elastic wave energy

SUMMARY Elastic-wave propagation in strongly scattering media is described by an energy diffusion equation derived directly from the coupled mode equations. The physical model we assume consists of a stack of layers over a homogeneous, non-scattering half-space. The overlying stack contains the heterogeneities that cause the diffusion of energy by scattering and may have an arbitrary mixture of fluid and solid layers. The coupled mode equations provide a formally exact representation of the displacementstress field in 2-D heterogeneous media, so the resulting diffusion equation is physically based rather than phenomenological. By allowing the mode spacing to become very small, the diffusion equation is derived from a differential-difference coupled energy equation. All assumptions and approximations must be explicitly stated and implemented in the derivation of the energy diffusion equation. Strong forward scattering is assumed to dominate. The mean free path for multiple scattering must be large compared with the size of the medium fluctuations responsible for the scattering, and the spatial scale of the heterogeneities must be large compared with the wavelength. The energy diffusivity is a range- and frequency-dependent functional of the displacementstress field components and the horizontal gradients of the medium properties including anisotropy. The diffusivity functional is derived directly from the continuum limit of the mode-coupling matrix; it is essentially the spatial autospectrum of the coupling matrix weighted by a function describing the density of modes in spectral space. The approach presented in this paper is in contrast to energy diffusion equations derived from radiative transfer theory in which the diffusivity must be specified separately in an ad hoc manner. Although our energy diffusion equation is range-dependent, the computation of the diffusivity depends on local medium properties and field values for a plane-layered medium. An additional difference between the diffusion equation of this paper and previously published treatments of elastic-energy diffusion is that this paper describes energy diffusion in spectral space, for example slowness, as a function of range rather than diffusion in physical space as a function of time. The dimensions of the diffusivity in slowness space are [slowness2/length].

3D FEM/BEM-Hybrid Approach for Planar Layered Media

ABSTRACT A finite element method (FEM)/boundary element method (BEM)-hybrid approach is presented which allows electromagnetic field calculations in planar layered media including three-dimensional inhomogeneous regions. The fields within the inhomogeneities are modeled by the FEM in terms of the electric field strength Ē with edge elements on tetrahedral meshes. Based on Huygens’ principle the BEM is formulated with equivalent electric and magnetic surface currents on a surface completely enclosing the inhomogeneous regions. Additionally, ideal conducting objects can be included into the BEM-model. The electric field integral equation (EFIE) for the BEM is formulated as a mixed potential integral equation (MPIE). Applications of the method to field calculations in a microstrip resonator with finite metallization thickness and to the coupling of a cylindrical dielectric resonator to a microstrip line are presented.

Circularly polarized absolute‐value rays in stratified absorbing media

AbstractTo provide an example of circularly polarized absolute‐value rays in a nondispersive absorbing medium, the theory of ray tracing is presented in the case in which a plane wave is incident on a planar layered medium. Specific tracing methods are presented for the loci of the real phase plane normal, the loci of the projection ray related to the flow line of the normalized Poynting flux, the loci of the circularly polarized absolute‐value rays, and the intensities of these rays. The relationships to Poynting flux and their tracing method are described. Further, in the planar layered medium, an anisotropic projection ray index of refraction is introduced to the projection ray. It is shown that Fermat's principle holds under this index of refraction. On the other hand, Fermat's principle also holds in the tracing of the trajectory of the real‐phase plane normal. In Fermat's principle of the projection ray, its relationship to the energy flux density is new.

High-frequency wave packets in inhomogeneous plasma with striction nonlinearity

The study of the evolution of 1 —D intense high-frequency (HF) packets in a smoothly inhomogeneous plane-layer medium with local and nonlocal striction nonlinearities is reviewed. The role of low-frequency (LF) radiation from an HF packet in an inhomogeneous medium with nonlocal nonlinearity is discussed.

Preliminary analysis of long‐period basin response in the Los Angeles Region from the 1994 Northridge Earthquake

Long‐period (1–10 sec) ground motions recorded in the Los Angeles basin region during the Northridge earthquake show complex waveforms, extended durations and multiple sets of arrivals which cannot be attributed solely to source processes or wave propagation within a plane‐layered medium. These features suggest a strong interaction between the propagating seismic wave field and the laterally varying subsurface geologic structure of this region. Recorded motions at a hard rock site in the Santa Monica Mountains (scrs) are smaller by nearly a factor of three in peak velocity compared to recordings at more distant sites in the adjacent, northwest portion of the Los Angeles basin. In addition, although the rock site recording has a relatively simple wave shape, the basin site recordings are dominated by late arriving, large amplitude pulses of energy. We interpret these arrivals to be surface waves, which are generated by body waves interacting with the thickening margin of the basin. A preliminary modeling analysis of these data indicates that a combination of both large‐scale (deep basin) and small‐scale (shallow micro‐basin) structures are needed to explain the observed responses.

DSM synthetic seismograms using analytic trial functions: planelayered, isotropic, case

SUMMARY The Direct Solution Method (DSM) (Geller et al. 1990; Geller & Ohminato 1994) is a Galerkin weak-form method for solving the elastic equation of motion. In previous applications of the DSM to both laterally homogeneous and laterally heterogeneous media the vertically dependent part of the trial functions has been either linear splines (e.g. Cummins et al. 1994a, b) or the vertically dependent part of modal eigenfunctions (e.g. Hara, Tsuboi & Geller 1993). In this paper we formulate the DSM using analytic trial functions which are solutions of the homogeneous (source-free) equation of motion in locally homogeneous portions of the medium. We present explicit formulations for SH and P-SV wave propagation in an isotropic, laterally homogeneous, plane-layered model. The trial functions so that it is easy to satisfy continuity of displacement at internal interfaces. For the laterally homogeneous SH problem we first find a set of R+ 1 analytic trial functions that satisfy continuity of displacement at the R - 1 internal interfaces between the R layers. Each of the trial functions is non-zero in at most two layers. We then solve the weak form of the equation of motion, which in effect enforces the upper and lower boundary conditions, and continuity of traction at the internal interfaces. The trial functions are chosen so that the equation of motion becomes a tridiagonal (R+ 1) X (R+ 1) system of linear equations. For the P-SV problem we define a set of 2R+ 2 analytic trial functions that satisfy continuity of displacement, but not continuity of traction, at internal interfaces; the trial functions are chosen so that the equation of motion then becomes a (2R+ 2) X (2R+ 2) system with a bandwidth of 7. In contrast, previous global solution methods (e.g. Chin, Hedstrom & Thigpen 1984; Schmidt & Tango 1986), which solve simultaneously for both internal continuity of displacement and traction as well as the external boundary conditions, solve a 2R X 2R system of linear equations for the SH problem or a 4R X 4R system for the P-SV problem, each having approximately twice the bandwidth of our systems of equations. We show that through an appropriate choice of the form of the homogeneous solutions in each layer our approach can also be readily incorporated into strong-form global solution methods, thereby leading to exactly the same system of equations obtained by our weak form derivation. We also present the DSM equation of motion for a plane-layered medium composed of a combination of fluid and solid layers. The dependent variable in the fluid layers is a scalar quantity proportional to the pressure change, while the dependent variable in the solid is the displacement. Continuity of displacement and traction at fluid-solid boundaries is enforced by augmenting the weak-form operator by appropriate surface integrals.