Reflection Kernel Research Articles

Transient high-frequency spherical wave propagation in porous medium using fractional calculus technique

AbstractTransient high-frequency spherical wave propagation in the porous medium is studied using the Biot-JKD theory. The porous media is considered to be a composed of deformable solid skeleton and viscous incompressible fluid inside the pores. In order to treat the fractional proportionality of the dynamic tortuosity to the frequency (or equivalently, to time) due to the viscous interaction between solid and fluid phases, the fractional calculus theory along with the Laplace and Fourier transforms are used to solve the coupled governing partial differential equations of the scaler and vector potential functions obtained from the Helmholtz’s decomposition in the Laplace domain. Both the longitudinal and transverse waves, additionally the reflection and transmission kernels are determined in detail at the Laplace domain. For the Laplace-to-time inversion transform, Durbin’s numerical formula is used and the independence of the results from the involved tuning and accuracy parameters is checked. The effects of the porosity, dynamic tortuosity, characteristics length, etc. on the reflected pressure and stress are investigated. The general pattern of the results is similar to our previous knowledge of wave propagation. Further works and experiments may be conducted in future works for various applications.

Frequency-domain reflection waveform inversion with generalized internal multiple imaging

Full-waveform inversion (FWI) has the potential to provide a high-resolution detailed model of the earth’s subsurface, but it often fails to do so if the starting model differs significantly from the true one. Reflection waveform inversion (RWI) is a popular method for building a sufficiently accurate initial model for FWI. In traditional RWI, the low-wavenumber updates are always computed and captured by smearing the data misfit along the reflection path with the help of migration/demigration. However, the success of RWI relies heavily on accurately reproducing the data in demigration. Thus, we have introduced a new generalized internal multiple imaging-based RWI (GIMI-RWI) implementation, in which we avoid the Born modeling and update the primary reflection kernel directly. In GIMI-RWI, we store one reflection kernel for each source-receiver pair, preserving the unique wavepath for every single source-receiver trace. Subsequently, the convolution between the data residuals and the corresponding reflection kernel can build the tomographic velocity updates. In this situation, the long-wavelength tomographic updates are free of migration footprints and will contribute a smoother background velocity to reduce the cycle-skipping risk and stabilize the followed FWI process. In addition, the GIMI-RWI method is source independent because it entirely relies on the data. Using a synthetic example extracted from the Sigsbee2A model, we find the reliable performance of the GIMI-RWI technique.

Scattering Kernel Kinetic Approach of Small Particles Transport in a Free Molecular Regime in Gaseous Media

We investigated through the scattering kernel kinetic concept the transport of small particles in gaseous media in a free molecular regime. The scattering kernel is generally used to provide gas-surface interaction boundary conditions for Boltzmann kinetic equation in flow problems (Maxwell, Phil. Trans. R. Soc. 170, 231, 1879; Sharipov, 2001). We developed a general expression for the transport force explicitly in terms of the scattering kernel and of gas molecule velocity distribution function. From the general expression, first, we could use the first-order Chapman-Enskog solution for the distribution function of gas molecules f(r, v, t) which shows three types of transportation mechanisms, or forces, such that F = FD+FT+FS. Where FD is the drag force on the particle due to its relative motion in the gas, FT is the thermophoretic force due to a gradient of temperature in the gas, and FS is due to shear forces on the particle. Second, it is possible to use any of the several scattering kernel models published in the literature. The traditional simple reflection kernels, specular, diffuse, and Maxwell, which comprises particular cases of the generalized force, are limited and do not fully describe the real reflection mode of molecule-particle interaction. So, we used the general (Struchtrup, Phys. Fluids. 25, 112001, 2013) reflection kernel model to study the drag force on small particles. The results were compared with available experimental data. We concluded that it is necessary to explicitly include impact velocity and particle surface rugosity effects which are predicted by the Struchtrup reflection kernel thermal activation model of molecule-particle interaction.

A survey of kernel-type estimators for copula and their applications

Copulas have been widely used to model nonlinear dependence structure. Main applications of copulas include areas such as finance, insurance, hydrology, rainfall to name but a few. The flexibility of copula allows researchers to model dependence structure beyond Gaussian distribution. Basically, a copula is a function that couples multivariate distribution functions to their one-dimensional marginal distribution functions. In general, there are three methods to estimate copula. These are parametric, nonparametric, and semiparametric method. In this article we survey kernel-type estimators for copula such as mirror reflection kernel, beta kernel, transformation method and local likelihood transformation method. Then, we apply these kernel methods to three stock indexes in Asia. The results of our analysis suggest that, albeit variation in information criterion values, the local likelihood transformation method performs better than the other kernel methods.

A moment model for phonon transport at room temperature

Heat transfer in solids is modeled by deriving the macroscopic equations for phonon transport from the phonon-Boltzmann equation. In these equations, the Callaway model with frequency-dependent relaxation time is considered to describe the Resistive and Normal processes in the phonon interactions. Also, the Brillouin zone is considered to be a sphere, and its diameter depends on the temperature of the system. A simple model to describe phonon interaction with crystal boundary is employed to obtain macroscopic boundary conditions, where the reflection kernel is the superposition of diffusive reflection, specular reflection and isotropic scattering. Macroscopic moments are defined using a polynomial of the frequency and wave vector of phonons. As an example, a system of moment equations, consisting of three directional and seven frequency moments, i.e., 63 moments in total, is used to study one-dimensional heat transfer, as well as Poiseuille flow of phonons. Our results show the importance of frequency dependency in relaxation times and macroscopic moments to predict rarefaction effects. Good agreement with data reported in the literature is obtained.

Fast Filtering of Reflection Probes

Game and movie studios are switching to physically based rendering en masse, but physically accurate filter convolution is difficult to do quickly enough to update reflection probes in real-time. Cubemap filtering has also become a bottleneck in the content processing pipeline. We have developed a two-pass filtering algorithm that is specialized for isotropic reflection kernels, is several times faster than existing algorithms, and produces superior results. The first pass uses a quadratic b-spline recurrence that is modified for cubemaps. The second pass uses lookup tables to determine optimal sampling in terms of placement, mipmap level, and coefficients. Filtering a full 1282 cubemap on an NVIDIA GeForce GTX 980 takes between 160 µs and 730 µs with out method, depending on the desired quality.

Reflection principles and kernels in $\mathbb R^n_+$ for the biharmonic and Stokes operators. Solutions in a large class of weighted Sobolev spaces

In this paper, we study the Stokes system in the half-space $\mathbb{R}^n_+$, with $n {\geqslant} 2$. We consider data and give solutions which live in weighted Sobolev spaces, for a whole scale of weights. We start to study the kernels of the biharmonic and Stokes operators. After the central case of the generalized solutions, we are interested in strong solutions and symmetrically in very weak solutions by means of a duality argument.

A transient boundary element method model of Schroeder diffuser scattering using well mouth impedance

Room acoustic diffusers can be used to treat critical listening environments to improve sound quality. One popular class is Schroeder diffusers, which comprise wells of varying depth separated by thin fins. This paper concerns a new approach to enable the modeling of these complex surfaces in the time domain. Mostly, diffuser scattering is predicted using steady-state single frequency methods. A popular approach is to use a frequency domain boundary element method (BEM) model of a box containing the diffuser, where the mouth of each well is replaced by a compliant surface with appropriate surface impedance. The best way of representing compliant surfaces in time domain prediction models, such as the transient BEM is, however, currently unresolved. A representation based on surface impedance yields convolution kernels which involve future sound, so is not compatible with the current generation of time-marching transient BEM solvers. Consequently, this paper proposes the use of a surface reflection kernel for modeling well behavior and this is tested in a time domain BEM implementation. The new algorithm is verified on two surfaces including a Schroeder diffuser model and accurate results are obtained. It is hoped that this representation may be extended to arbitrary compliant locally reacting materials.

A very shallow water ray propagation model

For applications such as minehunting, communications, high frequency time‐reversal, there has been a need for reliable and fast propagation models able to correctly treat scattering at the rough waveguide boundaries. Data and scattering models have shown that as signal frequency or surface roughness increases, the scattering pattern from the ocean bottom and ocean surface evolves from specular to quasi omni‐directional. Ray propagation models are fast but, a priori, they would treat surface bounces as perfectly specular which would not account for the angular spread observed on multipath structures. The idea followed by the authors over the past few years has been to introduce in the Bellhop ray tracing model a reflection kernel including deterministic properties of the sea surface and bottom. Simulations using this kernel display a significant effect of the multipath structure similar to the angular spread observed in in‐situ data. Simulation results are compared with a recent experimental study at 10‐20...

Time-Domain Techniques for Computation and Reconstruction of One-Dimensional Profiles

Abstract. This paper presents a time-domain technique to compute the electromagnetic fields and to reconstruct the permittivity profile within a one-dimensional medium of finite length. The medium is characterized by a permittivity as well as conductivity profile which vary only with depth. The discussed scattering problem is thus one-dimensional. The modeling tool is divided into two different schemes which are named as the forward solver and the inverse solver. The task of the forward solver is to compute the internal fields of the specimen which is performed by Green’s function approach. When a known electromagnetic wave is incident normally on the media, the resulting electromagnetic field within the media can be calculated by constructing a Green’s operator. This operator maps the incident field on either side of the medium to the field at an arbitrary observation point. It is nothing but a matrix of integral operators with kernels satisfying known partial differential equations. The reflection and transmission behavior of the medium is also determined from the boundary values of the Green's operator. The inverse solver is responsible for solving an inverse scattering problem by reconstructing the permittivity profile of the medium. Though it is possible to use several algorithms to solve this problem, the invariant embedding method, also known as the layer-stripping method, has been implemented here due to the advantage that it requires a finite time trace of reflection data. Here only one round trip of reflection data is used, where one round trip is defined by the time required by the pulse to propagate through the medium and back again. The inversion process begins by retrieving the reflection kernel from the reflected wave data by simply using a deconvolution technique. The rest of the task can easily be performed by applying a numerical approach to determine different profile parameters. Both the solvers have been found to have the ability to deal with different types of slabs and incident electromagnetic pulses. Slabs having continuous and discontinuous relative permittivity have already been tested successfully. The tested electromagnetic pulses are a Dirac, Gaussian and sinusoidal pulse. Due to sampling, the resolution of the system also plays a significant role in obtaining better outputs from this scheme.

Wave propagators for the Timoshenko beam

The propagation and scattering of waves on the Timoshenko beam are investigated by using the method of wave propagators. This method is more general than the scattering operators connected to the imbedding and Green function approaches; the wave propagators map the incoming field at an internal position onto the scattering fields at any other internal position of the scattering region. This formalism contains the imbedding method and Green function approach as special cases. Equations for the propagator kernels are derived, as are the conditions for their discontinuities. Symmetry requirements on certain coupling matrices originating from the wave splitting are considered. They are illustrated by two specific examples. The first being an unrestrained beam with a varying cross-section and the other a homogeneous, viscoelastically restrained beam. A numerical algorithm for solving the equations for the propagator kernels is described. The algorithm is tested for the case of a viscoelastically restrained, homogeneous beam. In a limit these results agree with the ones obtained for the reflection kernel by a previously developed algorithm for the imbedding reflection equation.

A time domain algorithm for the reflection of waves on a viscoelastically supported Timoshenko beam

This paper presents a time domain algorithm for solving the direct wave scattering problem for a Timoshenko beam. The beam is assumed to be made of a homogeneous material and to be restrained by a viscoelastic suspension of finite length. In the Timoshenko beam, the characteristic wave-fronts propagate with two distinct velocities. This implies that the equation satisfied by the reflection operator contains three separate families of characteristic curves. Furthermore, the equation is both temporally and spatially dependent due to the finite extent of the suspension. The mathematical problem this paper addresses is the construction of an algorithm to solve an associated partial integro-differential equation with three distinct wavefront velocities. The mathematical techniques that are developed are applicable to any set of matrix-valued, functional, first-order equations appropriate for reflection kernels. Some numerical examples are presented in order to validate the algorithm.

Second forerunners in reflection and transmission data

Second forerunner approximations to the transmission and reflection kernels for a temporally dispersive, nonmagnetic, isotropic slab are obtained. Numerical results are presented for two different dispersion models and are compared with the second forerunner approximations. Finally, transmission and reflection kernels are used to reconstruct the first susceptibility moments of the material of the slab.

Reconstruction of bi-isotropic material parameters using transient electromagnetic fields

A non-iterative time-domain method for solving the inverse scattering problem for the temporally dispersive, linear, homogeneous, and bi-isotropic slab at normal incidence is presented. The inverse scattering problem is to determine (finite time traces of) the susceptibility kernels of the medium given (finite time traces of) the reflection and transmission kernels, which are obtained by deconvolution of the scattered fields. Two numerical examples illustrate the method for this kind of noisy data. A series representation of the solution of the direct problem is obtained as well.

Permittivity Profile Reconstructions Using Transient Electromagnetic Reflection Data

This paper is concerned with the permittivity reconstruction of inhomogeneous dielectric media. The method applies to profiles that vary with depth only, i.e., it provides a one-dimensional profile reconstruction. The data are collected and analyzed in the time domain. In the first part of the paper the theory of the method is reviewed. It is shown that a finite time trace of reflection data suffices to uniquely reconstruct the permittivity profile of the medium. The latter part of the paper presents the experimental set-up and contains also a thorough discussion of the errors that affect the measurements. The inverse scattering algorithm that is used is either based upon an imbedding procedure or on a Green functions approach. The input to either of these algorithms is the reflection kernel or the impulse response of the medium, i.e., the delta function response of the medium. Therefore, a deconvolution of the the measured reflected field and the incident field must be performed. This deconvolution problem is also addressed briefly in this paper.

THE IMBEDDING EQUATIONS FOR THE TIMOSHENKO BEAM

Wave reflection in a Timoshenko beam is treated, using wave splitting and the imbedding technique. The beam is assumed to be inhomogeneous and restrained by a viscoelastic suspension. The viscoelasticity is characterized by constitutive relations that involve the past history of deflection and rotation of the beam through memory functions of the suspension. By applying wave splitting, the propagating fields are decomposed into left-and right-moving parts. An integral representation of the split fields in impulse responses is presented. This representation gives the reflected and transmitted fields as convolutions of the incident field with the reflection and transmission kernels, respectively. The kernels are independent of the incident field and depend only on the material properties. Invariant imbedding is used to obtain equations for these kernels. In general, the kernels contain discontinuities for which transport equations are derived and solved. Some numerical solutions are presented for the reflection by a homogeneous beam suspended on two separated, semi-infinite layers of continuously distributed, viscoelastically damped, local acting springs.

Reconstruction of the velocity and density in a stratified acoustic half-space using a short-pulse point source

The reconstruction of the velocity and density in a stratified half-space using the reflection data generated by a short-pulse point source is considered. Transverse spatial Hankel transform is applied to the transient reflected and incident pressures on the surface. The reflection kernel in the Hankel-transformed space is obtained through a deconvolution. The time-domain Green function approach based on the wave splitting is then used to reconstruct the parameters. This time-domain approach to the inverse problem is exact and noniterative. Numerical results for some synthetical examples are presented to illustrate the Hankel transform, deconvolution, and the reconstruction. The finite-difference time-domain method is used to generate the transient reflection data in the physical space in a numerical simulation. The reconstruction using obliquely incident plane waves is also considered in an Appendix.

Integrodifferential equations for the two-dimensional transition kernels of invariant imbedding

Invariant imbedding methods for two spatial dimensions are applied to derive a set of integrodifferential equations and related boundary conditions, in the kernels of transmission, left-turn, and right-turn transition operators for a two-dimensional particle transport theory problem. The operators describe the transition from incident to exiting physical states over rectangular regions. The set of equations is coupled only through the reflection kernel. This work is a continuation of and complements the earlier effort of the authors wherein an integrodifferential boundary-value problem for the reflection kernel is derived.

Design of reflectionless slabs for obliquely incident transient electromagnetic waves

A method for designing reflectionless plane-stratified conducting and dispersive slabs for obliquely incident transient plane waves is presented. The slabs are temporally dispersive with a spatially varying impedance and conductivity. It is shown that the effects from the variation of the impedance can be matched by the temporal dispersive effects and the conductivity for given incident directions. The problem of finding reflectionless slabs are formulated as an inverse problem for the TE and TM modes where the constitutive relation is to be determined as a function of depth given a reflection kernel that is zero for given incident directions. The inverse problem is solved by a time domain Green functions technique.

Permittivity Profile Reconstructions Using Transient Electromagnetic Reflection Data

This paper is concerned with the permittivity reconstruction of inhomogeneous dielectric media. The method applies to profiles that vary with depth only, i.e. it provides a one-dimensional profile reconstruction. The data are collected and analyzed in the time domain. In the first part of the paper the theory of the method is reviewed. It is showed that a finite time trace of reflection data suffices to uniquely reconstruct the permittivity profile of the medium. The latter part of the paper presents the experimental set-up and contains also a thorough discussion of the errors that affect the measurements. The inverse scattering algorithm that is used is either based upon an imbedding procedure or on a Green functions approach. The input to either of these algorithms is the reflection kernel or the impulse response of the medium, i.e. the delta function response of the medium. Therefore, a deconvolution of the the measured reflected field and the incident field must be performed. This deconvolution problem is also addressed briefly in this paper. (Less)