Fields In Layered Media Research Articles

Multi‐mode inversion of dispersive surface‐wave free fields in layered media

AbstractInversion of surface‐wave free fields is of great importance in providing seismic excitations for the finite element (FE) analysis of soil–structure interaction (SSI) systems. In the case of layered media, the propagating surface waves are characterized by their unique dispersion property, resulting in dispersive multi‐mode surface wavefields. Available studies primarily focus on only the fundamental mode and neglect the influence of higher modes, which fail to identify the real structural response characteristics. To enhance the precision of seismic excitations for SSI systems situated in layered media, a multi‐mode inversion method for dispersive surface‐wave free fields is proposed in this study. The proposed method employs the energy flow density to characterize the participation volume of dispersive modes, and further calculates corresponding modal participation factors by combining with the dispersion curve and dynamic‐stiffness matrix (DSM) method. The modal participation factors are then utilized to assign the ground surface components of each mode and invert respective single‐mode surface‐wave free fields. Based on the mode superposition theory, all the single‐mode surface‐wave free fields are finally superimposed to construct the multi‐mode wavefields. The accuracy of the proposed method is thoroughly verified and validated across various types of layered media. Furthermore, the engineering application of the proposed method is illustrated by performing the FE analysis on an SSI example. The analysis results highlight the significant participation of surface waves' higher modes in structural seismic responses, particularly in the weak‐interbed type media.

Active control of electromagnetic fields in layered media

This article presents a numerical strategy for actively manipulating electromagnetic (EM) fields in layered media. In particular, we develop a scheme to characterize an EM source that will generate some predetermined field patterns in prescribed disjoint exterior regions in layered media. The proposed question of specifying such an EM source is not an inverse source problem (ISP) since the existence of a solution is not guaranteed. Moreover, our problem allows for the possibility of prescribing different EM fields in mutually disjoint exterior regions. This question involves a linear inverse problem that requires solving a severely ill-posed optimization problem (i.e. suffering from possible non-existence or non-uniqueness of a solution). The forward operator is defined by expressing the EM fields as a function of the current at the source using the layered media Green's function (LMGF), accounting for the physical parameters of the layered media. This results to integral equations that are then discretized using the method of moments (MoM), yielding an ill-posed system of linear equations. Unlike in ISPs, stability with respect to data is not an issue here since no data is measured. Rather, stability with respect to input current approximation is important. To get such stable solutions, we applied two regularization methods, namely, the truncated singular value decomposition (TSVD) method and the Tikhonov regularization method with the Morozov Discrepancy Principle. We performed several numerical simulations to support the theoretical framework and analyzes, and to demonstrate the accuracy and feasibility of the proposed numerical algorithms.

Three-dimensional tensor controlled-source electromagnetic modeling based on the vector finite-element method

Scalar CSAMT is only suitable for measurements in one and two dimensions perpendicular to geological structures. For complex 3D geoelectric structure, tensor CSAMT is more suitable. In this paper, we discuss 3D tensor CSAMT forward modeling using the vector finite-element method. To verify the feasibility of the algorithm, we calculate the electric field, magnetic field, and tensor impedance of the 3D CSAMT far-zone field in layered media and compare them with theoretical solutions. In addition, a three-dimensional anomaly in half-space is also simulated, and the response characteristics of the impedance tensor and the apparent resistivity and impedance phase are analyzed. The results suggest that the vector finite-element method produces high-precision electromagnetic field and impedance tensor data, satisfies the electric field discontinuity, and does not require divergence correction using the vector finite-element method.

Effects of source and cavity depths on wave fields in layered media

Abstract The multiple modes of Rayleigh waves (R-waves) or leaky Lamb waves (L-waves) may be present in the activated and/or the back-scattered wave fields. The presence of these waves in the wave fields is related to the source and the cavity depths. Two types of layered media are used to investigate the influences of the source and the cavity depths. One is the layered half space where the half space is stiffer than all the overlying layers; the other is the pavement system where there is significant velocity contrast between layers and the shear velocity of layers decreases with the layer depth. The discrete displacement expressions developed for spherical point sources are used to study the effects of the source depth on the excitability and the propagation behavior of R-waves or leaky L-waves. In layered half spaces, the multiple modes of R-waves can be activated by the sources within a certain depth. The effective phase velocity of these activated modes is derived. In pavement systems, the leaky L-waves can be activated by the sources at any depth. The discrete displacement expressions of the scattered waves are derived and used to evaluate the effects of the cavity depth on the presence of R-waves or the leaky L-waves in the back-scattered wave field.

Complex plane integration in the modelling of electromagnetic fields in layered media: part 2. Pole location

This part 2 companion paper derives several related methods to determine pole location for guided electromagnetic waves in a planarly layered medium in the quasi-static approximation. Such analysis can be useful for the numerical evaluation of field representation integrals of layered earth models. Results compare favourably with a numerical nonlinear optimizer in cases where the poles are suitably distinct. A transverse resonance method is also shown to yield reasonable results. Bounds for surface wave and guided wave pole locations are also included.

Complex plane integration in the modelling of electromagnetic fields in layered media: part 1. Application to a very large loop

This paper analyses the details of a procedure for the numerical integration of Hankel transforms in the calculation of the electromagnetic fields generated by a large horizontal loop over a 1D earth. The method performs the integration by deforming the integration path into the complex plane and applying Cauchy's theorem on a modified version of the integrand. The modification is the replacement of the Bessel functions J0 and J1 by the Hankel functions and respectively. The integration in the complex plane takes advantage of the exponentially decaying behaviour of the Hankel functions, allowing calculation on very small segments, instead of the infinite line of the original improper integrals. A crucial point in this problem is the location of the poles. The companion paper shows two methods to estimate the pole locations. We have used this method to calculate the fields of very large loops. Our results show that this method allows the estimation of the integrals with fewer evaluations of the integrand functions than other methods.

Mathematical modeling of wave fields in layered media with additional stresses

We consider the model of a vertically inhomogeneous medium, when one or more layers are under more stress. We analyzed the results of research of influence of hydrostatic pressure act-ing in one of the layers of horizontally-layered structure of the wave field excited by pulse point seismic source, which is represented by seismic moment tensor localized in the isotropic layer. We suggested to use the matrix method of Thomson - Haskell for solving the problem to build field movements in order to use relevant analytical relations for solving the inverse problem in respect of stress tensor and/or parameters of the source. The results of mathematical modelling are simple in utilization and can be used for interpretation of seismic records.

Equilibrium ensembles of electromagnetic thermal radiation in layered media

AbstractThis paper describes equilibrium ensembles of thermally excited electromagnetic fields in layered media. The obtained results complement Planck's law of thermal radiation that determines the spectrum of the radiation but supplies little information about the ensemble of eigenfields (normal modes) excited in the medium. The developments regarding these ensembles presented in this paper make it possible to apply perturbation techniques for the analysis of the ensembles of radiated fields in layered media with a steady heat flux.

Reconstruction of the Electromagnetic Field in Layered Media Using the Concept of Approximate Transmission Conditions

We propose a fast and stable method to reconstruct the electromagnetic field in layered media from overdetermined data on the outer boundary. This procedure can be used for instance as a data pre-processing in inversion algorithms to accurately image buried objects under known stratified structures. Our method is based on successive use of so-called approximate transmission conditions that provide accurate reconstructions for layers with sufficiently small width. The feasibility and efficiency of the proposed method are demonstrated by numerical experiments.

Beam summation algorithm for wave radiation and propagation in stratified media

An efficient Gaussian beams summation (GBS) algorithm for tracking source excited wavefields in plane stratified media is introduced. It has two important features: (a) it involves an efficient calculation of the GB propagators, and (b) it involves rather sparse lattice of beams. For (a) we approximate the medium using layers with constant wavespeed gradient, and derive an efficient recursive algorithm for tracking the GB through such medium. This model not only reduces the number of layers, and thereby the algorithm complexity, in comparison with the conventional uniform layers model, but it also eliminates the reflection artifacts at layers interfaces. Property (b) is achieved by determining the beam expansion parameters for an efficient discretization of the source-excited beam spectra using a sparse lattice of beams. The algorithm has been validated and calibrated via thorough numerical comparisons with closed form ray solutions for source-excited fields in layered media. Perfect agreement between these independent solutions has been obtained in regions where the ray solution is valid, but the beam formulation also provided smooth and physically meaningful solutions in caustic regions where the ray solution fails.

Propagation of Seismic Wave Fields in Layered Media

The general theory of propagation of seismic wave fields of signal type is presented in a mathematically precise and physically descriptive form. Basic relations and notions of the theory are given for general cases of the anisotropy of a medium. Consequences of the theory are discussed by comparing the phenomena in anisotropic and isotropic seismic media. Bibliography: 6 titles.

Problem of modeling of the interaction of nonstationary electric, thermal, and diffusion fields in layered media

The interaction of nonstationary electric and thermal fields in a layered medium with allowance for mass transfer is considered. The heating of an electrochemical cell is modeled numerically; the influence of an electric double layer at the metal–electrolyte interface is investigated.

Method of asymptotic perturbation of fundamental solution and electromagnetic waves in layered media

The method of asymptotic perturbation of the fundamental solution of the wave equation is applied to compute the rapidly oscillating integrals that arise in wave fields in layered media. The principal part is separated for one of the integrals determining the electromagnetic waves in a layer medium.

Three-dimensional waves in layered media with nonparallel and curved interfaces: A theoretical approach

A calculation method is presented for transducer-generated three-dimensional, transient wave fields in layered media with curved interfaces by using a separation approach in combination with integral transform methods and point source synthesis. The transducer-specimen module is decomposed into different layers and the wave propagation is calculated separately in each layer. To simplify the boundary conditions, the existence of a thin liquid layer between two solid media is used. Using generalized ray theory it can be shown that the normal stress on the interface caused by an incident wave provides sufficient information to calculate the transmitted waves in the adjacent media. Two different functions for the point sources of the exciting element and the source function for normal stress are derived. By means of a source function for normal stress, the transmitted transient wave can be determined from the normal-stress distribution on the interface. The method is demonstrated by two examples.

The mean acoustic field in layered media with rough interfaces

An algorithm is presented for determining the mean acoustic field in a layered medium containing rough interfaces. It is assumed that scattering by the rough interfaces when considered separately and in the absence of sound speed and density variation can be well-approximated. It is also assumed that propagation in layered media with flat interfaces can be well approximated. The present work shows how these results can be combined to yield the mean field in a stack of layers with variable sound speeds and densities which are separated by rough interfaces.

Marching and spatial-spectral filtering of localized phase-space representations of wave fields in layered media

Spatial and spectral (directional) information of the wave data are of equal importance in the analysis and synthesis of wave phenomena. Their simultaneous collaboration in a phase-space format may be achieved by incorporating windowing techniques into (conventional) spectral integrals. The result is a local-spectral representation of the field that is concentrated inherently around physically meaningful regions in phase space. This representation is most naturally suited to perform simultaneous spatial and spectral processing of wave data and therefore may serve as a powerful tool in a variety of applications such as remote sensing, imaging, field reconstruction, and more. In this work a method to propagate the local spectra of wave fields through plane stratified media is developed and discussed. An adaptive marching algorithm that permits control of the localization process will be presented. The ability of the new scheme to perform simultaneous spatial and spectral data filtering of wave phenomena, and its resolving power in phase space, will be discussed and demonstrated by means of numerical examples.

A note on the mixed potential representation of electric fields in layered media

A mixed potential formulation is given for electrical fields in layered environments. Contributions to the field from charges are identified explicitly through a scalar Green's function for layered media. The outcome is a computationally expedient Sommerfeld integral representation. >

Weight functions and fundamental fields in layered media

A variational principle introduced by Sham to determine elastic fields with unbounded strain energy in finite homogeneous bodies is extended to piecewise homogeneous bodies. It is shown that the modification involves only an additional known body force like term in the functional. The resulting displacement-based finite element method determines the weight functions for all three fracture modes and for both traction and displacement boundaries in a unified manner. Numerical examples for two-dimensional cracks in layered isotropic bodies are presented.