Planar Stratified Media Research Articles

A Hybrid Loop-Tree FEBI Method for Low-Frequency Well Logging of 3-D Structures in Layered Media

Electromagnetic simulation plays an essential role in formation conductivity determination by electromagnetic well-logging tools, especially when the nonvertical borehole and invasion zones are present in a layered earth. A hybrid finite-element boundary integral (FEBI) method is suitable for such well-logging simulations. The usage of layered medium Green's functions allows the simulation background to be a planar stratified medium to characterize the earth formation. However, conventional FEBI using the Rao-Wilton-Glisson (RWG) basis function produces inaccurate results due to the low-frequency breakdown of the boundary integral equation solution. The new contribution of this work is to apply the loop-tree (LT) basis functions in the FEBI method to reduce the numerical error at low frequencies. Such an LT-FEBI method can handle all specific requirements of well-logging simulations at low frequencies. A direct solver is applied to make it more efficient to obtain logging curves with multiple source locations. The LT-FEBI simulation results are validated in several numerical examples, including a challenging well-logging model with a deviated borehole and invasion zones in the Oklahoma formation having 28 layers.

Influence of the illumination direction on the optical extinction by a scatterer close to a dielectric interface

The intrinsic optical response of a scattering object is known to be modified when it is placed on a substrate. Moreover, the total extinction through ohmic losses and Rayleigh scattering is basically dependent on the reversal of the direction of the excitation wave. Considering, as stated by the generalized optical theorem, that the total extinction of the incident wave is shared among the extinction of the waves transmitted and reflected by the non-absorbing substrate, we show that only the contribution of the transmitted wave is insensitive to the direction of illumination, by analogy with the textbook case of transmission through a planar stratified medium. This property was recently confirmed experimentally [J. Phys. Chem. C 123, 15217 (2019)] and is established here on theoretical grounds in the frameworks of the Green's function integral equation method and the Lorentz reciprocity theorem. Some of its limitations with regard to the incidence and the polarization of the excitation wave are also discussed in detail.

A Procedure for Primary-Field Compensation in the Problems of Radiowave Diagnostics of Media Using Linear Antennas

A possibility of radio-wave diagnostics of plane-stratified media using a single linear antenna and primaryfield compensation in the microwave range (300 MHz) is analyzed. The antenna-coupled impedance of the monitored medium is investigated numerically taking into account primary-field compensation. The investigations are performed for three-layer dielectric media as a function of their dielectric properties, level of field compensation, and antenna position with respect to the air-medium boundary.

Modal Analysis of Gratings With Conducting Strip-Loaded Bars and Sandwiched Between Multiple Dielectric Layers

This communication describes a full-wave modal analysis of plane-wave scattering by periodic arrays of penetrable conducting bars with strip covers on both the lower and upper surfaces of each ridge to create iris-type slits across groove apertures, which altogether is sandwiched by multiple dielectric layers on both sides of the gratings. The approach entails the moment method using parallel-plate waveguide cavity Green’s functions and a numerical spectral-domain Green’s function for planar stratified media accompanied by the Floquet theorem. Computed results obtained from codes written based on this formulation are validated with those simulated by a commercial full-wave software solver as well as results from the literature. Validation is performed in terms of reflection and transmission spectra as well as modal dispersion. The new parameters arising from the special complex geometry herein studied provide the means to enhance the performances of various applications such as beam deflectors, mode converters, and grating couplers, as well as the resolution of diffraction gratings in the context of spectroscopy. Measurement results acquired from experimentations carried out on a manufactured grating that corroborate well with theoretical predictions are also presented.

A Hybrid 3-D Electromagnetic Method for Induction Detection of Hydraulic Fractures Through a Tilted Cased Borehole in Planar Stratified Media

As one of the most important nondestructive characterization techniques, electromagnetic (EM) methods can be used in the subsurface fracture detection, especially for hydraulic fracture evaluation in unconventional petroleum exploration and development. The multiscale nature of long but extremely thin 3-D fractures is difficult for conventional EM modeling methods such as the finite element method (FEM) in numerical simulation. The problem becomes even more challenging when the effects of tilted borehole, casing, and planar stratified media need to be considered. So far, modeling a tilted borehole in layered media is still a major challenge for conventional methods. In this paper, we present the hybrid numerical mode-matching method with the stabilized biconjugate gradient fast Fourier transform method as the forward modeling algorithm that can efficiently model 3-D fractures in planar stratified media with a cased borehole environment. Numerical results validate the accuracy of the hybrid forward method and show orders of magnitude higher efficiency of this forward solver than the FEM.

Analytic growth rate of gravitational instability in self-gravitating planar polytropes

Gravitational instability is a key process that may lead to fragmentation of gaseous structures (sheets, filaments, haloes) in astrophysics and cosmology. We introduce here a method to derive analytic expressions for the growth rate of gravitational instability in a plane stratified medium. First, the main strength of our approach is to reduce this intrinsically fourth-order eigenvalue problem to a sequence of second-order problems. Second, an interesting by-product is that the unstable part of the spectrum is computed by making use of its stable part. Third, as an example, we consider a pressure-confined, static, self-gravitating slab of a fluid with an arbitrary polytropic exponent, with either free or rigid boundary conditions. The method can naturally be generalised to analyse the stability of richer, more complex systems. Finally, our analytical results are in excellent agreement with numerical solutions. Their second-order expansions provide a valuable insight into how the rate and wavenumber of maximal instability behave as functions of the polytropic exponent and the external pressure (or, equivalently, the column density of the slab).

Second-Order Perturbative Solutions for 3-D Electromagnetic Radiation and Propagation in a Layered Structure With Multilayer Rough Interfaces

We present a fast method for deriving a general explicit closed-form expression of second-order perturbative solutions of the problem of radiation and propagation electromagnetic (EM) fields in a hybrid layered structure with an arbitrary number of rough interfaces and planar stratified medium based on the classical small perturbation method (SPM). The second-order SPM expression of the electric and magnetic fields in each region of the rough-layered structure is provided by theoretical analysis and fully formula derivation instead of considering any other approximation and equivalent process. In addition, we give the complete understanding of physical radiation and propagation mechanism of double-bounce second-order SPM solutions. Finally, the second-order SPM solutions derived by employing the proposed method are verified with existing methods and numerical results. What is more, we investigate completely the influence of various parameters of layered structure and configuration of radar system on EM scattering fields.

Bistatic Scattering From Three-Dimensional Layered Structures With Multilayer Rough Interfaces

We develop a fast method for deriving the general closed-form perturbative solutions of bistatic scattering from layered structure with multilayer rough interfaces based on the small perturbation method (SPM). The proposed method is developed by theoretical analysis and fully formula derivation instead of considering any other approximation and equivalent process. In addition, our proposed method for completeness of SPM can give the general closed-form solution for the problem of radiation and propagation of electromagnetic fields in a hybrid layered structure with an arbitrary number of rough interfaces and planar stratified medium, which is not available in the previous work yet, and it is highly desirable. The demonstration of the consistency of the presented method is analytically provided.

On the Limitations of the Time-Domain Impedance Boundary Condition

Limitations of the time-domain Leontovich impedance boundary condition are discussed for a canonical problem consisting of a planar interface with a conducting half-space. The exact analytical solution obtained using an extension of the Cagniard-DeHoop technique is presented and compared with the approximate one found via the impedance boundary concept. Ranges of applicability of the latter are introduced and numerically tested on illustrative examples. A proper understanding of these limitations may prove to be crucial in the emerging field of time-domain integral equation formulations in planar stratified media.

A Unified Microwave Radiative Transfer Model for General Planar Stratified Media: Slab Formulation

A unified microwave radiative transfer (UMRT) model is presented for computing the thermal radiation emitted from any geophysical medium composed of planar layers of either densely or tenuously distributed moderately sized spherical scatterers. UMRT employs the discrete-ordinate eigenanalysis (DOE) method with layer adding to solve the differential radiative transfer equation for such multilayer structures. UMRT inherits the symmetrization and analytical diagonalization and factorization techniques of symmetric and positive definite matrices from the discrete-ordinate tangent linear radiative transfer (DOTLRT) model presented by Voronovich These techniques ensure accuracy, numerical stability, and rapid computation for all matrix operations required for DOE along with a fast Jacobian calculation for radiance assimilation purposes. UMRT extends the applicability of DOTLRT by including both the Mie theory and the dense media radiative transfer (DMRT) theory. Other nontrivial extensions within UMRT are the following: 1) The vertical and horizontal radiation intensities are coupled within each layer by applying the reduced Mie or DMRT phase matrices, and 2) the physical temperature profile of a layer is allowed to be linear in height. The symmetry properties of both the reduced Mie and DMRT phase matrices are proved, and the associated scattering and absorption coefficients are compared and discussed. The UMRT slab formulation is validated by imposing energy conservation, and the numerical results for some nominal cases are produced and discussed.

Object-oriented electrodynamic S-matrix code with modern applications

The S-matrix algorithm for the propagation of an electromagnetic wave through planar stratified media has been implemented in a modern object-oriented programing language. This implementation is suitable for the study of such applications as the Anderson localization of light and super-resolution (perfect lensing). For our open-source code to be as useful as possible to the scientific community, we paid particular attention to the pathological cases that arise in the limit of vanishing absorption.

Characteristics of exact retrieval of the structure of plane-stratified media using reflected and transmitted pulse signals

We consider the exact iterative algorithm for solution of the direct and inverse scattering problems in 1D inhomogeneous half-space using piecewise-constant approximation of the constitutive parameter profiles (e.g., refractive index). A novel modification of the Gelfand–Levitan equation is derived for solution of the inverse scattering problem. Numerical examples are given and the effects related to a finite frequency band of the sounding pulse are discussed.

A Vector Transform Solution Procedure for Solving Electromagnetic Problems in Cartesian Coordinates

Some vector transforms of importance to solving electromagnetic problems were introduced by Chew and Habashy in 1986. The primary advantage gained in the use of these transforms is that the analytical development of field solutions becomes streamlined by what amounts to a very powerful bookkeeping method. The aforementioned paper presents transforms for Cartesian, cylindrical, and elliptical geometries. A later paper by Weiss and Kahn demonstrated that these transforms emerge from careful consideration of the longitudinal components of the electric and magnetic fields. Applying the techniques to the Cartesian geometry, one finds a variation to the vector transform postulated by Chew and Habashy. This letter will discuss the analytical formulation of the variation and demonstrate its usefulness in the development of Green's functions for a planar stratified medium. The foundation of this variation to the well-established development in Chew and Habashy's work lies in obtaining it solely from consideration of the longitudinal electric and magnetic field components. This method then simplifies the derivations and necessary bookkeeping. Although demonstrated here for a simple geometry, the approach applies to multiple layers as effectively.

Numerical simulation of the incoherent electro-optical imaging process in plane-stratified media

A high-fidelity numerical model is developed for the incoherent electro-optical imaging process in media where the inherent optical properties vary with range. Four contributions to the image data are considered: laser light reflected from the scene, volume backscatter of laser light, reflected solar light, and solar backscatter. The model implements rigorous mathematical formulations, incorporates adaptive error control, uses computationally efficient algorithms, and has been rigorously validated. The model is flexible enough to handle many different sensor configurations and sampling techniques, continuous-wave and pulsed laser sources, and range-dependent properties of the optical environment. The derivation of relevant mathematical models is reviewed and unified into a comprehensive and flexible system that can be tailored to fit the various incoherent electro-optical imaging systems currently in use or in development. The numerical implementations are described for each key component of the overall model, and a sampling of new verification and validation results is provided.

On the efficient computation of closed-form Green's functions in planar stratified media

The spatial-domain Green's functions for the vector and scalar potentials in planar stratified media are cast into closed forms via a two-level approximation of their spectral-domain counterparts. The proposed methodology begins with the approximation of the spectral-domain Green's functions over large values of the spectral variable by complex exponentials, and continues with the approximation of the remainder by rational functions. Finally, the closed-form Green's functions in terms of spherical and cylindrical waves are derived, making use of some well-known integral identities. A key-feature of the proposed approach is that although it does not call for an analytical extraction of the quasistatic terms and the surface wave poles, it provides the means for the accurate description of both the near-field and far-field physics. Moreover, the rational function spectrum fitting proposed here overcomes the problem of the spurious singular behavior of the spatial-domain Green's functions because of the use of Hankel functions. © 2008 Wiley Periodicals, Inc. Int J RF and Microwave CAE, 2008.

A beam summation algorithm for wave radiation and guidance in stratified media

A Gaussian beams summation (GBS) algorithm for tracking source excited wave fields in plane stratified media is presented. In the present application the medium is described by layers with constant gradient of the wave speed, and the GB propagators are calculated recursively in a closed form. The algorithm is calibrated for numerical efficacy and accuracy by defining simple physical criteria for choosing the expansion parameters (the beam collimation and the spectral discretization and truncation) that allow for sparse representation of the source-excited angular spectrum of beams. It is validated for a source-excited example in layered media, where it provides a smooth and physically meaningful solution under multipath and caustic conditions and remains accurate for long propagation ranges where phase error tends to accumulate.

Accurate Approximation of Green's Functions in Planar Stratified Media in Terms of a Finite Sum of Spherical and Cylindrical Waves

A robust and computationally-expedient methodology is presented for accurate, closed-form approximation of the Green's functions used in the mixed-potential integral equation statement of the electromagnetic boundary value problem in planar stratified media. The proposed methodology is based on the fitting of the spectrum of the Green's function, after the extraction of the quasistatic part, making use of rational functions. The effectiveness and robustness of the proposed methodology rely upon the proper sampling of the spectrum in order to improve the accuracy of the rational function fit. The resulting closed-form approximation is in terms of both spherical and cylindrical waves. Thus, high accuracy is obtained in the approximation of the Green's function irrespective of the distance of the observation point from the source. The methodology is validated through its application to the approximation of the Green's function for a multi-layered, planar dielectric stack

An inverse problem for the wave equation in plane-stratified media

Assume that two media are laying in a half-space and the interface wall is parallel to the boundary of the half-space. We can directly observe the data near the boundary of the half-space, but we cannot directly observe inside the half-space. In this situation, we try to identify these unknown things by creating an artificial explosion and observing on the boundary the waves generated by the explosion. In the previous works related to this problem, only the speeds of the waves were treated, but we also take into account the impedances of the media in our setting.

An inverse problem for the elastic equation in plane-stratified media

Assume that two media are laying in a half-space and the interface wall is parallel to the boundary of the half-space. We can directly observe the data near the boundary of the half-space, but we cannot directly observe inside the half-space. In this situation, we try to identify these unknown things by creating artificial explosions and observing on the boundary the waves generated by the explosions. Here, the waves are described by the elastic equation.

Study of Stratified Dielectric Slab Medium Structures Using Pseudo-spectral Time Domain (PSTD) Algorithm

A planar stratified dielectric slab medium, which is an interesting problem in optics and geophysics, is studied using a pseudo-spectral time-domain (PSTD) algorithm. Time domain electric fields and frequency domain propagation characteristics of both single and periodic dielectric slab-layer structures are analyzed. Results show that this method matches satisfactorily the Nyquist sampling theorem in terms of spatial discretization. By comparing the given results, it is found that the PSTD method outperforms the finite-difference time-domain (FDTD) method in general, especially in terms of the locations of critical frequencies and coarseness in spatial discretization.