Generalized Reflection Coefficients Research Articles

A GPR layered media parameters inversion method under rough surface

Abstract Ground Penetrating Radar (GPR) plays an important role in pavement detection and road exploration due to its efficiency. Traditional media parameter inversion algorithms only consider ideal smooth surfaces and neglect the roughness of interfaces, leading to significant errors in practical applications. Therefore, a GPR rough surface-layered medium parameter inversion method based on genetic algorithm is proposed in this paper. The method uses the generalized reflection coefficient and considers the influence of the surface roughness of the medium. Then, a radar echo signal time-domain model is established. Finally, the cost function is optimized by genetic algorithm. The effectiveness and accuracy are validated and analysed by utilizing simulated data through random rough surfaces models.

Stationary time correlations for fermions after a quench in the presence of an impurity

We consider the quench dynamics of non-interacting fermions in one dimension in the presence of a finite-size impurity at the origin. This impurity is characterized by general momentum-dependent reflection and transmission coefficients which are changed from to at time t = 0. The initial state is at equilibrium with such that the system is cut in two independent halves with , , respectively, to the right and to the left of the impurity. We obtain the exact large time limit of the multi-time correlations. These correlations become time translationally invariant, and are non-zero in two different regimes: i) for where the system reaches a non-equilibrium steady state (NESS), ii) for , i.e., the ray regime. For a repulsive impurity these correlations are independent of , , while in the presence of bound states they oscillate and memory effects persist. We show that these nontrivial relaxational properties can be retrieved in a simple manner from the large time behaviour of the single particle wave functions.

Tapered Graded Index Lens Antenna With Enhanced Penetration for Near-Field Torso Imaging

A tapered graded index (GRIN) lens antenna that aims at enhancing electromagnetic penetration into human torso by generating focused planar near-field radiation is presented. The antenna is composed of a four-layer lens excited by a modified slot antenna. Each layer of the lens case is filled with fabricated mixtures of engineered materials that have specific relative permittivity values. The exciting antenna is made compact by utilizing meandered feed lines and substrate folded technique. The number of layers of the lens and the specification of each layer are determined by minimizing the generalized reflection coefficient of the antenna. A focused plan-wave radiation is achieved by tapering the lens based on the theory of total internal reflection. The study of <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$E$ </tex-math></inline-formula> -field quality inside the torso demonstrates that the proposed lens successfully converts spherical wave radiation into a local plane-wave radiation, resulting in 11 dB improvement in wave penetration inside the torso compared to the body-matched antenna (without lens). The measured results show that the antenna operates over a wide band from 0.43 to 1.85 GHz (125% fractional bandwidth), which is a merit for the torso imaging application. Furthermore, the antenna has the physical size of 110 <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\times 110\times48$ </tex-math></inline-formula> mm3, corresponding to <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$0.15\times 0.15\times 0.06\lambda _{0}^{3}$ </tex-math></inline-formula> , where <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\lambda _{0}$ </tex-math></inline-formula> is the wavelength at the lowest working frequency. The results indicate that the antenna accomplishes 4.9 dB (209% improvement) stronger wave penetration inside the torso compared to other existing GRIN lens antennas.

Exploiting evanescent bending waves for the identification of near-boundary cracks in slender beams

Bending evanescent waves are among the promising approaches for the local damage identification of slender beams due to the high sensitivity and spatial concentrating nature. However, such local damage usually appears near the boundary of the beam in practices. It is therefore difficult to distinguish the evanescent wave components resulting from the boundary and the crack, leading to confusion in damage identification. In order to tackle this issue, the generalized reflection coefficient of the evanescent wave with respect to the incident propagating wave (GRC-EP) is exploited in the present work, which allows to quantitatively identify the severity of the near-boundary crack. Taking the semi-infinite Timoshenko beam having a crack damage at the fixed end, the generalized reflections of the evanescent and propagating waves at the crack, with respect to a unit incident wave, are derived analytically, showing that the amplitude of the evanescent wave is negatively related to the damage severity. Based on this, the amplitude ratio between the reflected evanescent wave and the incident propagating wave, i.e., the GRC as mentioned, is proposed to be the damage indicator. Referring to the theoretical GRC values, the damage severity can then be identified from the estimated GRCs that are obtained from the inverse algorithm, as long as the acceleration responses are given. Finally, the feasibility of the proposed indicator is verified by taking a slender beam having a local damage at the fixed end as an example. The results show that the proposed GRC-EP is capable to indicate the severity of the near-boundary damage despite the consideration of noise, confirming its potential value in the health monitoring of slender beam structures.

Finite-Region Approximation of EM Fields in Layered Biaxial Anisotropic Media

A new algorithm is developed to accurately compute the electromagnetic (EM) fields in the layered biaxial anisotropic media. We enclose the computational region in an infinitely long rectangular region by four vertical truncation planes and establish the corresponding algorithm to approximate the EM fields in the entire space. The EM fields in this region are expanded as a two-dimensional (2-D) Fourier series of the transverse variables. By using the spectral state variable method, the generalized reflection coefficient matrices and transmission matrices are then derived to determine the Fourier coefficients per layer. Therefore, we can obtain the spatial-domain EM fields by summing the 2-D Fourier series. To enhance the accuracy and efficiency of this algorithm, we apply the method of images to estimate the influence of the artificial boundaries on the EM fields at the observer. We then further develop a quantitative principle to choose the proper size of the region according to the desired error tolerance. With the proper choice, the summation of the series can achieve satisfactory accuracy. This algorithm is finally applied to simulate the responses of the triaxial logging tool in transversely isotropic and biaxial anisotropic media and is verified through comparisons to the other method.

Parameter Inversion by a Modified Reflected Signal Reconstruction Method for Thin-Layered Media

Parameter estimation of layered media is an important application of ground penetrating radar (GPR). However, in the case of thin-layer detection, the limitation of GPR vertical resolution can lead to overlapping reflected signals, which seriously deteriorates the detection results. To solve this problem, a modified reflected signal reconstruction method based on reflected signal reconstruction is proposed in this letter. First, a forward model of the GPR reflected signal is constructed using the generalized reflection coefficient definition. Then, the parameter inversion is transformed into an optimization problem. The residue cost function is defined as the error square between the model and the actual reflected signal, which is minimized by using genetic algorithm. Furthermore, the generalized reflection coefficient spectrum is feasible to estimate the parameters of each layer as the initial values of the parameters required by the algorithm, thus improving the accuracy and convergence speed of the algorithm. Numerical, experimental, and field tests demonstrate that the method has a high time resolution and antinoise capability in the inversion of layered media parameters.

Substrate effect on scattering by a chiral sphere

The effect of stratified substrates on scattering from a chiral sphere is analytically and numerically examined. A combination of vector Mie solution and the field transformations between vector spherical functions (VSFs) and plane waves (PWs) is used to determine the scattered fields of different orders. The generalized reflection coefficients of the stratified half-space and vector Mie solution for the chiral sphere are used to calculate successive interactions of the sphere and the interface and a series solution is obtained. Commercial software FEKO is used to numerically validate the derived expressions for the multilayered substrate. Various numerical results are presented and discussed. Specifically, a slab with exponential profile is considered as an example of an inhomogeneous substrate and its scattering coefficients are compared with the homogeneous case.

Однопозиционный метод определения координат источника радиоизлучения на земных трассах по совокупности отражений его сигнала от местных предметов

Present work is devoted to the problem of validity increasing of an experiment to detect the motion of a continuous medium by the waveguide method. The study attempts to optimize the experiment to find the optimal scheme for its installation. The target effect, the magnitude of which should be maximum, is the generalized reflection coefficient, which is influenced by both the Fresnel electromagnetic drag coefficient and the polarization plane rotation due to motion of the medium. The main optimization task is to determine the output parameters for the experiment for stable and reliable motion detection of a continuous medium in a waveguide. In this work, a heuristic analysis algorithm is used. Optimization factors: frequency of the signal at microwave, physical dimensions of objects, shape of the waveguide cross-section, linear velocity of the medium. ECAD PathWave EM Design (EMPro) 2021 was used to the experiment designing stages. Distilled water is the moving medium, which is not subject to optimization. The motion of a continuous medium is considered through its refined refractive index. The maximum magnitude of the target effect is observed for the topology with a rectangular waveguide and with a tube oriented along a wide wall at f = [9.3 ÷ 9.5] GHz. These values coincide with a good degree of accuracy with the already conducted experiments.

Investigation of borehole effects on azimuthal resistivity measurements using novel pseudo‐analytic formulas

ABSTRACTDownhole resistivity measurements provide valuable information for geosteering and formation evaluation. It is important to understand and correct the environmental effects, such as the borehole, the tool eccentricity and the resistivity anisotropy effects, of the measurements based on fast and accurate modelling methods. A new pseudo‐analytic solution for eccentric coaxial and tilted‐coil antennas in the cylindrically multilayered medium with transverse isotropic conductivity was developed. This method can take the eccentric tool, the borehole, the mud invasion and the resistivity anisotropy into account. These formulas enable us to simulate the responses of the downhole induction logging tools, including induction wireline and azimuthal propagation logging‐while‐drilling measurements. Instead of using the prior fictitious boundary, the generalized reflection coefficients are proposed to construct the linear matrix functions to deal with the tool eccentricity. The cylindrical functions are reorganized and presented in forms of ratios. Thus, the proposed formulas obviate the overflow issue in the computation and are more stable and efficient. The proposed approach is compared and validated with other well‐established methods. Simulations and case studies show that: (1) the responses of azimuthal resistivity logging‐while‐drilling tool are affected by the tool's eccentricity in both isotropic and anisotropic medium; (2) sine curve behaviours exist for the phase difference and amplitude ratio of azimuthal resistivity measurements along with different eccentricity azimuth.

Design of Multi-Resonant Cavities Based on Metal-Coated Dielectric Nanocylinders

Abstract In this paper, the light scattering properties for multiple silver-coated dielectric nanocylinders with the symmetrical distribution were investigated. Based on the transfer matrix method, we derive the general transmission and reflection coefficient matrices for multiple dielectric nanocylinders. When the incident light frequencies are less than the plasma frequencies, the surface plasmons (SPs) appear in the interface between the silver and dielectrics. Numerical simulations show that there are three peaks of absorption cross-section (ACS) in the relationship between the ACS and the frequencies of the incident light, when the distance between the silver-coated dielectric nanocylinders is chosen properly. These SPs resonance peaks are characterised as resonances intrinsic to the cylindrically periodic system corresponding to different inner cavity structures. These multi-resonant cavities may have potential applications in integrated devices, optical sensors and optical storage devices.

Application of Taguchi Method for Parametric Studies of a Funnel Shaped Structure Used for Noise Reflection with Source on the Center

In this paper, attempt has been made to minimize sound reflection from the wall by using Taguchi’s method and to find optimal structure for the suggested test-section inside the cavitation tunnel. The suggested structure which was added to the test-section is funnel-shaped with a performance like a check valve. In order to obtain approximate values of five independent parameters, three levels were taken into account for each parameter. By combining parameters of different levels, 27 tests were designed using Taguchi’s method and Minitab Software. Different acoustic analyses were conducted in COMSOL Multiphysics software, and defined parameter of general reflection coefficient was obtained for 21 observer points. Applying the general reflection coefficients to Minitab Software and drawing the SNR graph, approximate values of the parameters were obtained. However, these values did not produce enough accuracy to design the optimal structure. For this reason, five levels around optimal values, obtained from the previous analysis, were considered for each parameter. Same steps were repeated again for the parameters at five levels and optimal values were obtained. Optimal structure was modelled and analyzed. Consequently, appropriate defined parameters of general and local reflection coefficients were extracted which represented an optimal structure for the intended test section.

Generalized Reflection Coefficients

I consider general reflection coefficients for arbitrary one-dimensional whole line differential or difference operators of order 2. These reflection coefficients are semicontinuous functions of the operator: their absolute value can only go down when limits are taken. This implies a corresponding semicontinuity result for the absolutely continuous spectrum, which applies to a very large class of maps. In particular, we can consider shift maps (thus recovering and generalizing a result of Last–Simon) and flows of the Toda and KdV hierarchies (this is new). Finally, I evaluate an attempt at finding a similar general setup that gives the much stronger conclusion of reflectionless limit operators in more specialized situations.

Derivation of a General Reflection Coefficient of Blast-Loaded Buried Structures

AbstractA general reflection coefficient (GRC) is defined to determine the equivalent uniform stress on an underground arched structure subjected to a subsurface explosion. The single-degree-of-freedom model approach is adopted and the structure–medium interaction and curvature effect are considered. The derivation lies on a virtual work method, which provides the GRC of the buried arch-shaped underground structure. The GRC is also suitable for box-shaped underground structures when the curvature of the arch tends to be zero. Case studies are discussed to investigate the effect of dimensionless parameters. The developed method is applied to a practical example. It is shown that the present approach is effective and applicable for practical use designing buried underground structures subjected to subsurface explosions. It also complements the current design codes, which lack details for designing arch-shaped buried protective structures.

Determination of Parameters of Subsurface Layers Using GPR Spectral Inversion Method

A step-by-step spectral inversion algorithm is proposed in this paper to estimate the thickness, depth, permittivity, and conductivity of layered media from ground-penetrating radar (GPR) signals. Due to the distinct GPR spectral responses to media parameters, different attributes of the spectra are used to estimate, separately, one or several parameters within each step. Such estimated parameters are then used as starting values in a nonlinear optimization problem for the inversion of all parameters simultaneously from the whole GPR spectral data. The cost function of the optimization problem is defined as the difference between the modeled generalized reflection coefficient spectrum of subsurface layers and the actual one. The optimization problem is solved by using a modified stochastic hill-climbing (SHC) algorithm. The method is applied on synthetic data of a wedge model and real data from two highway GPR detections. The results show that the step-by-step spectral inversion method can reduce the ambiguity of the inversion and improve its accuracy and efficiency.

Analytical perturbation methods for studying a transversely isotropic medium in multipole acoustic logging

A new analytical perturbation method is developed in this study to investigate the general reflection coefficients in the frequency-wavenumber domain of the acoustic field in a fluid-filled borehole surrounded by a transversely isotropic medium (TIM). The transversely isotropic medium with a symmetric axis parallel to the borehole axis, which is usually called a VTI medium, was adopted because its exact solutions exists, and a corresponding isotropic medium was adopted as a reference state of perturbation solution. The general reflection coefficients were originally calculated by using the perturbation method and were compared with the analytical solutions. The zero-, first- and second-order perturbation solutions for the general reflection coefficients excited by monopole, dipole and quadrupole sources were investigated for a transversely isotropic elastic solid. The results showed that the general reflection coefficients obtained by using the perturbation solutions and the analytical solutions were similar for all three sources. In summary, our study demonstrated that the perturbation method is valid and effective in acoustical logging. This work provided a theoretical foundation for extending perturbation analyses to complicated anisotropic acoustical logging applications.

Characteristics of the Sum of Cross-Components of Triaxial Induction Logging Tool in Layered Anisotropic Formation

There are two nonzero cross-components of a triaxial induction logging tool in the medium coordinate system in a transversely isotropic (TI) formation. The sum of the cross-components holds dependence on only the horizontal conductivities with little nonlinearity and small shoulder bed effect in a horizontally layered formation. Based on the generalized reflection coefficient method, simplified formulas of the sum are derived to analyze its response characteristics. Numerical results show that the apparent conductivity obtained from the sum is a square-looking log with high vertical resolution. Moreover, the readings almost equal to the true horizontal conductivity of the formation beds. These characteristics can be applied to reconstruct the horizontal conductivity and the formation boundaries with triaxial induction logging data.

Reflection coefficient due to a discontinuity in the nth order derivative of the refractive index

Surfaces within a dielectric material, where the derivatives of a continuous and real refractive index profile are discontinuous, are shown to enhance reflection. To this end, the amplitude and phase representation of electromagnetic waves is used to model light propagating normally through a transparent medium with a continuous refractive index profile that varies only in one spatial direction. The amplitude equation is solved under the slowly varying refractive index approximation (SVRI). To isolate the effect of a single surface where the refractive index derivatives are discontinuous, an n(z) profile is proposed that is analytical, smooth and slowly varying except for a single piecewise junction. At this junction, n(z) is continuous but some of the mth order derivatives are not. Two different SVRI approximated solutions are joined at the discontinuity plane and, by demanding that boundary conditions are satisfied, a general complex reflection coefficient is obtained. By categorizing profiles according to the lowest order discontinuous derivative at the junction, a simple expression for the reflection coefficient can be written. Results are compared favorably with previous numerical solutions. Furthermore, a conjecture by the authors in a previous paper, ‘for a Cm−1 refractive index profile type, where m stands as the order of the lowest order discontinuous derivative, phase change upon reflection at the discontinuity plane is for an increasing lowest order discontinuous derivative and for the decreasing case’, is proved here.

Forward modelling of multi-component induction logging tools in layered anisotropic dipping formations

Multi-component induction logging provides great assistance in the exploration of thinly laminated reservoirs. The 1D parametric inversion following an adaptive borehole correction is the key step in the data processing of multi-component induction logging responses. To make the inversion process reasonably fast, an efficient forward modelling method is necessary. In this paper, a modelling method has been developed to simulate the multi-component induction tools in deviated wells drilled in layered anisotropic formations. With the introduction of generalized reflection coefficients, the analytic expressions of magnetic field in the form of a Sommerfeld integral were derived. The fast numerical computation of the integral has been completed by using the fast Fourier–Hankel transform and fast Hankel transform methods. The latter is so time efficient that it is competent enough for real-time multi-parameter inversion. In this paper, some simulated results have been presented and they are in excellent agreement with the finite difference method code's solution.

An Inverse Electromagnetic Scattering Method for One-Dimensional Inhomogeneous Media

A frequency-domain inversion scheme for reconstructing the permittivity profile of one-dimensional inhomogeneous media is proposed. The generalized reflection coefficients and the generalized transmission coefficients of the inhomogeneous media are used as the input data of the inverse model. A Newton-like iterative algorithm known as the generalized pulse-spectrum technique with the Tikhonov regularization is applied to solve the inverse problem. Novel boundary conditions are proposed for the inverse problem and therefore the permittivity at the boundary of the inhomogeneous media is not required as prior knowledge. The choice of frequency points of the frequency-domain method is also investigated. Numerical examples are carried out to validate the inversion technique. Good agreements between the reconstructed profiles and the true profiles are shown.

Scattering From Layered Structures With One Rough Interface: A Unified Formulation of Perturbative Solutions

In this paper, we investigate analytically the connection between the existing first-order small perturbation method solutions for the scattering from a layered structure with one rough interface. First of all, by using effectively the concept of generalized reflection coefficients, we cast the existing models in a unified more compact formulation and point out the connection between the different analytical solutions. The obtained reformulations of the available analytical solutions allow us to subsequently prove the consistency of the considered models. Finally, a suitable expansion is performed that leads us to understand the physical meaning of the analytical expressions. The obtained unified formulation also opens the way toward a general closed-form solution for the problem of scattering by a layered structure with an arbitrary number of corrugated interfaces.