EDGE DIFFRACTION IN AN ARBITRARY ANISOTROPIC MEDIUM. I

1967 ◽  
Vol 45 (11) ◽  
pp. 3479-3502 ◽  
Author(s):  
Shalom Rosenbaum
Keyword(s):  
Anisotropic Medium ◽  
Short Wavelength ◽  
Dielectric Tensor ◽  
Asymptotic Results ◽  
Asymptotic Field ◽  
Lateral Field ◽  
Reflected Waves ◽  
Edge Diffraction ◽  
Diffracted Waves ◽  
Formal Solutions

Diffraction by a perfectly conducting half-plane embedded in a transversely unbounded region filled with a uniform, lossless but arbitrarily anisotropic medium characterized by a Hermitian dielectric tensor ε is studied. Formal solutions in terms of a plane-wave modal superposition (with the modal amplitudes determined via the Wiener–Hopf technique) are obtained. Asymptotic (short-wavelength) contributions (saddle-point contributions as well as contributions due to intercepted singular points) are considered. The asymptotic results are then cast into invariant, ray optical forms, amenable to a distinct physical interpretation.Mode coupling at the boundary gives rise to diffracted (lateral) waves as well as to geometric-optical (incident and reflected) waves. In addition to the "conventional" radially diffracted waves, one observes "secondary" lateral waves generated by the edge. In the short-wavelength limit the edge behaves as a virtual line source whose magnitude is proportional to the total (direct and lateral) field incident upon it. The asymptotic field solutions are valid, subject to the exclusion of some suitably defined transition regions, which are distinctly determined by the geometric-optical expressions. The various (asymptotic) wave constituents are shown to correspond to the anticipated results of geometrical optics.

EDGE DIFFRACTION IN AN ARBITRARY ANISOTROPIC MEDIUM. II

1967 ◽  
Vol 45 (11) ◽  
pp. 3503-3519
Author(s):  
Shalom Rosenbaum
Keyword(s):  
Anisotropic Medium ◽  
Line Source ◽  
Short Wavelength ◽  
Entire Range ◽  
Dielectric Tensor ◽  
Total Field ◽  
Reflected Waves ◽  
Edge Diffraction ◽  
Wave Numbers ◽  
Formal Solutions

Diffraction by a perfectly conducting half-plane embedded in a transversely unbounded region filled with a uniform, lossless but arbitrarily anisotropic medium characterized by a Hermitian dielectric tensor ε is studied. Excitation is by a linearly phased line source of arbitrarily polarized electric and magnetic currents. Formal solutions are obtained in terms of a two-dimensional plane-wave modal superposition (over the entire range of the transverse wave numbers representing the incident (ηn) and scattered (η) waves). The original contours of integration in both the complex ηn and η planes are deformed simultaneously into their respective paths of steepest descent, and contributions to intercepted singular points are considered. Asymptotic (short-wavelength) analysis yields results which, despite their relative complexity, are cast into invariant, ray-optical forms, amenable to distinct physical (geometric-optical) interpretation.Mode coupling at the boundary gives rise to source-excited lateral waves as well as geometric-optical (incident and reflected) waves. In the short-wavelength limit, the edge behaves as a virtual line source whose magnitude is proportional to the total field incident upon it (which is composed of direct rays emerging from the source, as well as a lateral wave propagating along the boundary, towards the edge). Both the direct and the lateral waves are diffracted by the edge (i.e., coupled to edge-excited "radial" and "secondary" lateral waves) in a manner identical with that discussed in Part I.

Radiation Field of a Uniformly Moving Charge in an Anisotropic Medium

1974 ◽  
Vol 29 (5) ◽  
pp. 687-692
Author(s):  
G. P. Sastry ◽  
S. Datta Majumdar
Keyword(s):  
Radiation Field ◽  
Anisotropic Medium ◽  
Point Charge ◽  
Potential Distribution ◽  
Asymptotic Form ◽  
Dielectric Tensor ◽  
Arbitrary Direction ◽  
Hankel Functions ◽  
Anisotropic Fields ◽  
Set Up

Abstract Fourier integrals are set up for the field of a point charge moving uniformly in an arbitrary direction in a uniaxial medium anisotropic in ε only. The integrals break up into several parts two of which yield the ordinary and extraordinary cones with uniform azimuthal potential distribution. The remaining integrals neither contribute to the energy radiated nor affect the size and the shape of the cones, but merely distort the field within the cones. The integrals are evaluated exactly in the non-dispersive case and closed expressions for the potential are obtained. In the dispersive case, the radiation field is determined by using the asymptotic form of the Hankel functions occurring in the integrand. The resulting expressions exhibit the high azimuthal asymmetry characteristic of anisotropic fields. From the expressions derived for a pure dielectric the potential in a doubly anisotropic medium is obtained, without a fresh calculation, by making appropriate substitutions for the coordinates of the field point and the components of the dielectric tensor.

scholarly journals Separation of diffracted waves via SVD filter

2020 ◽  
Vol 17 (5) ◽  
pp. 1259-1271
Author(s):  
Hong-Yan Shen ◽  
Qin Li ◽  
Yue-Ying Yan ◽  
Xin-Xin Li ◽  
Jing Zhao
Keyword(s):  
Seismic Waves ◽  
Original Data ◽  
Identification Accuracy ◽  
Reflected Waves ◽  
Wave Separation ◽  
Imaging Results ◽  
Diffracted Waves ◽  
Fast Processing ◽  
The Difference ◽  
Value Decomposition

Abstract Diffracted seismic waves may be used to help identify and track geologically heterogeneous bodies or zones. However, the energy of diffracted waves is weaker than that of reflections. Therefore, the extraction of diffracted waves is the basis for the effective utilization of diffracted waves. Based on the difference in travel times between diffracted and reflected waves, we developed a method for separating the diffracted waves via singular value decomposition filters and presented an effective processing flowchart for diffracted wave separation and imaging. The research results show that the horizontally coherent difference between the reflected and diffracted waves can be further improved using normal move-out (NMO) correction. Then, a band-rank or high-rank approximation is used to suppress the reflected waves with better transverse coherence. Following, separation of reflected and diffracted waves is achieved after the filtered data are transformed into the original data domain by inverse NMO. Synthetic and field examples show that our proposed method has the advantages of fewer constraints, fast processing speed and complete extraction of diffracted waves. And the diffracted wave imaging results can effectively improve the identification accuracy of geological heterogeneous bodies or zones.

Inversion of seismic refraction and reflection data for building long-wavelength velocity models

Geophysics ◽  
2015 ◽  
Vol 80 (2) ◽  
pp. R81-R93 ◽  
Author(s):  
Haiyang Wang ◽  
Satish C. Singh ◽  
Francois Audebert ◽  
Henri Calandra
Keyword(s):  
Wave Equation ◽  
Short Wavelength ◽  
Velocity Model ◽  
Density Model ◽  
Data Set ◽  
Reflected Waves ◽  
Long Wavelength ◽  
Velocity Models ◽  
Computation Efficiency ◽  
Refracted Waves

Long-wavelength velocity model building is a nonlinear process. It has traditionally been achieved without appealing to wave-equation-based approaches for combined refracted and reflected waves. We developed a cascaded wave-equation tomography method in the data domain, taking advantage of the information contained in the reflected and refracted waves. The objective function was the traveltime residual that maximized the crosscorrelation function between real and synthetic data. To alleviate the nonlinearity of the inversion problem, refracted waves were initially used to provide vertical constraints on the velocity model, and reflected waves were then included to provide lateral constraints. The use of reflected waves required scale separation. We separated the long- and short-wavelength subsurface structures into velocity and density models, respectively. The velocity model update was restricted to long wavelengths during the wave-equation tomography, whereas the density model was used to absorb all the short-wavelength impedance contrasts. To improve the computation efficiency, the density model was converted into the zero-offset traveltime domain, where it was invariant to changes of the long-wavelength velocity model. After the wave-equation tomography has derived an optimized long-wavelength velocity model, full-waveform inversion was used to invert all the data to retrieve the short-wavelength velocity structures. We developed our method in two synthetic tests and then applied it to a marine field data set. We evaluated the results of the use of refracted and reflected waves, which was critical for accurately building the long-wavelength velocity model. We showed that our wave-equation tomography strategy was robust for the real data application.

Pre-stack diffraction separation by parameterizing the reflection local slope

Geophysics ◽  
2021 ◽  
pp. 1-49
Author(s):  
Chuangjian Li ◽  
Suping Peng ◽  
Xiaoqin Cui ◽  
Qiannan Liu ◽  
Peng Lin
Keyword(s):  
Plane Wave ◽  
Separation Method ◽  
Small Scale ◽  
Reflected Waves ◽  
Local Slope ◽  
Illumination Direction ◽  
Novel Approach ◽  
Diffracted Waves ◽  
Slope Estimation ◽  
Ray Parameter

Diffracted waves provide the opportunity to detect small-scale subsurface structures because they give wide illumination direction of geological discontinuities such as faults, pinch-outs, and collapsed columns. However, separating diffracted waves is challenging because diffracted waves have greater geometrical amplitude losses and are generally weaker than reflections. To retain more diffracted waves, a pre-stack diffraction separation method is proposed based on the local slope pattern and plane-wave destruction method. Generally, it is difficult to distinguish between the hyperbolic reflections and hyperbolic diffractions using the data-driven local slope estimation in the shot domain. Therefore, we transfer the slope estimation in the shot domain to the velocity analysis in the common midpoint domain and the ray parameter calculation in the stack domain. The connection between the local slope and the normal move-out velocity and the surface-ray parameter is known, which provides a novel approach for estimating the local slope of the hyperbolic reflected waves in the shot domain. The estimated slope can provide an exact slope-based operator for the plane-wave destruction (PWD) method, thus allowing the PWD to separate diffracted waves from reflected waves in the shot domain. Synthetic and field data tests demonstrate the feasibility and effectiveness of the proposed pre-stack diffraction separation method.

Propagation of a P-pulse in a solid sphere

1965 ◽  
Vol 55 (5) ◽  
pp. 821-861
Author(s):  
Z. Alterman ◽  
F. Abramovici
Keyword(s):  
Rayleigh Waves ◽  
Group Velocity Dispersion ◽  
Surface Displacement ◽  
Elastic Solid ◽  
Solid Sphere ◽  
Reflected Waves ◽  
Arrival Times ◽  
Angular Component ◽  
Diffracted Waves ◽  
Deep Focus

abstract An exact solution is obtained for the displacement of the surface of a uniform elastic solid sphere of radius a due to an impulsive compressional pulse from a point-source situated at a distance b from the center. The duration of the source is δa/c where c denotes the shear-wave velocity, and its time-variation is such that the surface-displacement stays finite when the time tends to infinity. The solution is applied to a source at a distance of one-eighth of the radius below the surface, approximating a deep-focus earthquake. Theoretical seismograms, radial and angular component, are given at distances 0 < ϑ < π for a source of duration 0.03a/c. Rayleigh waves are clearly seen at ϑ ≧ 45 °. Groups of reflected waves, especially predominant in the angular component, have the velocity of the lowest Airy phase in the group-velocity dispersion-curves. Diffracted waves, discussed in a previous paper, are found here again and in certain cases have an amplitude seven times larger than the amplitude of the direct pulse and also larger than any of the reflected pulses at the same distance. The transformed phases PSn, P2Sn have in general larger amplitude than the reflected Pn. Arrival times, initial amplitudes, reflection and convergence coefficients of pulses are obtained by steepest descents analysis and compared with the complete results.

Internal waves in a wedge-shaped region

1970 ◽  
Vol 43 (1) ◽  
pp. 97-120 ◽  
Author(s):  
D. G. Hurley
Keyword(s):  
Internal Waves ◽  
Diffraction Problem ◽  
Acute Angle ◽  
Vertex Angle ◽  
Ray Theory ◽  
Reflected Waves ◽  
Quarter Wavelength ◽  
Diffracted Waves ◽  
Ill Posed ◽  
Characteristic Angle

The Green's functions are found for a line source of internal waves in a wedge of stratified fluid of constant Brunt–Väisälä frequency, and are used to discuss the diffraction of internal waves by a wedge in all cases when the vertex angle of the wedge of fluid exceeds the acute angle between a characteristic and the horizontal. Robinson's (1970) results are confirmed and extended.It is found that the diffracted waves are as important as the incident and reflected ones at all points that lie within a quarter-wavelength or so of either characteristic that passes through the apex. Also, in cases when all the reflected waves are inclined forwards, the diffracted waves lead to a positive backscatter of energy. When the vertex angle of the fluid wedge is less than the characteristic angle, the diffraction problem appears to be ill-posed, and, instead, the motion due to a vibrating body in the wedge of fluid is considered.A general conclusion is that the so-called ray theory for internal waves, in which the incident and reflected waves alone are considered, has similar limitations to the geometrical theory of optics. Both theories involve the assumption that the typical dimensions in the problem are large compared to the wavelength.

Scattering of plane elastic waves at rough surfaces. I

1962 ◽  
Vol 58 (1) ◽  
pp. 136-157 ◽  
Author(s):  
Iya Abubakar
Keyword(s):  
Elastic Waves ◽  
Angle Of Incidence ◽  
Dimensional Problem ◽  
Irregular Boundary ◽  
Periodic Surface ◽  
Reflected Waves ◽  
Diffracted Waves ◽  
Incident Waves ◽  
Phase Angles ◽  
The Waves

ABSTRACTAn approximate solution of the two-dimensional problem of reflexion of plane harmonic P and S V waves by an irregular boundary is obtained using a modification of Rice's perturbation method of approximation on the assumption that the curvature of the surface is everywhere small. The case of a periodic surface is treated in more detail. It is found that the reflected waves are composed of specularly reflected waves and various diffracted waves, propagating in both horizontal directions if the wavelength of the incident waves is long compared with that of the surface. If the wavelength of incident distortional waves is long compared with that of the surface, the amplitudes of some of the scattered waves decrease exponentially with depth. In general the phases of the waves change on reflexion and the phase angles of the reflected waves are functions of the wavelength of the corrugation and the angle of incidence. It is verified, in the case of zero angle of incidence, that the energy going into the scattered radiation is obtained at the expense of the energy of the specularly reflected waves.

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