STUDIES ON SEISMIC WAVES: I. REFLECTION AND REFRACTION OF PLANE WAVES

Geophysics ◽  
1946 ◽  
Vol 11 (1) ◽  
pp. 1-9 ◽  
Author(s):  
C. Y. Fu
Keyword(s):  
Rayleigh Waves ◽  
Seismic Waves ◽  
Plane Waves ◽  
The Body ◽  
Body Waves ◽  
Angle Of Incidence ◽  
Apparent Velocity ◽  
Square Root ◽  
Absolute Value ◽  
Reflection And Refraction

By taking the apparent velocity along the boundary as the parameter instead of the angle of incidence, the equations for the different wave amplitudes may be put in more symmetrical forms. In this way, it is more convenient to discuss both the body waves and the Rayleigh waves at the same time. A difficulty in the plotting of the square root of the wave intensity against the angles is also discussed. When the reflection or refraction coefficient is not real, the meaning of the intensity, as obtained by squaring the absolute value of the latter quantity, needs clarification.

Simultaneous body and surface wave retrieval from the seismic ambient field and discrimination from unavoidably arising spurious artifacts

2020 ◽  
Author(s):  
Ali Riahi ◽  
Zaher-Hossein Shomali ◽  
Anne Obermann ◽  
Ahmad Kamayestani
Keyword(s):  
Surface Waves ◽  
Rayleigh Waves ◽  
Ambient Noise ◽  
Low Frequency ◽  
The Body ◽  
Body Waves ◽  
Apparent Velocity ◽  
Noise Sources ◽  
P Waves ◽  
Seismic Ambient Noise

<p>We simultaneously extract both, direct P-waves and Rayleigh waves, from the seismic ambient noise field recorded by a dense seismic network in Iran. With synthetics, we show that the simultaneous retrieval of body and surface waves from seismic ambient noise leads to the unavoidable appearance of spurious arrivals that could lead to misinterpretations.</p><p>We work with 2 months of seismic ambient noise records from a dense deployment of 119 sensors with interstation distances of 2 km in Iran. To retrieve body and surface waves, we calculate the cross-coherency in low-frequency ranges, i.e. frequencies up to 1.2 Hz, to provide the empirical Green’s functions between each pair of stations. To separate the P and Rayleigh waves, we use the polarization method that also enhances the small amplitude body waves.</p><p>We observe both P and Rayleigh waves with an apparent velocity of 4.9±0.3 and 1.8±0.1 km/s in the studied area, respectively, as well as S or higher mode of Rayleigh waves, with an apparent velocity of 4.1±0.1 km/s. Besides these physical arrivals, we also observe two spurious arrivals with similar amplitudes before/after the P and/or Rayleigh waves that render the discrimination challenging.</p><p>To better understanding these arrivals, we perform synthetic tests. We show that simultaneously retrieving the body and surface waves from seismic ambient noise sources will unavoidably lead to the appearance of superior arrivals in the calculation of empirical Green’s functions.</p>

Two-dimensional scattering of a plane compressional wave by surface imperfections

1969 ◽  
Vol 59 (3) ◽  
pp. 1349-1364
Author(s):  
Ivor K. McIvor
Keyword(s):  
Order Approximation ◽  
Strong Dependence ◽  
Plane Waves ◽  
Compressional Wave ◽  
Elastic Half Space ◽  
Kernel Functions ◽  
The Body ◽  
Angle Of Incidence ◽  
Small Surface ◽  
Surface Imperfections

abstract A perturbation method for treating the scattering of plane waves by small surface imperfections on an elastic half space is presented. The solution to the first order approximation is given as convolution integrals of the surface imperfection with kernel functions defined by Fourier inversion integrals. The evaluation of these integrals is discussed and their asymptotic representations determined. The far field scattered displacements are explicitly obtained for arbitrary imperfections. The scattered field consists of a Rayleigh surface wave and four body phases which at the free surface travel with the speed of dilational or distortional waves. Numerical examples are given. In particular the error in the apparent angle of emergence due to the scattered waves is obtained. The body phases exhibit the familiar 3/2 geometric attenuation, but still may make a significant contribution at moderately long distances. A strong dependence of the magnitude of the error on the angle of incidence is demonstrated.

The azimuthal and polar radiation patterns obtained from a horizontal stress applied at the surface of an elastic half space

1962 ◽  
Vol 52 (1) ◽  
pp. 27-36
Author(s):  
J. T. Cherry
Keyword(s):  
Surface Wave ◽  
Half Space ◽  
Rayleigh Waves ◽  
Poisson’S Ratio ◽  
Horizontal Stress ◽  
Elastic Half Space ◽  
The Body ◽  
Body Waves ◽  
Poisson's Ratio ◽  
A Value

Abstract The body waves and surface waves radiating from a horizontal stress applied at the free surface of an elastic half space are obtained. The SV wave suffers a phase shift of π at 45 degrees from the vertical. Also, a surface wave that is SH in character but travels with the Rayleigh velocity is shown to exist. This surface wave attenuates as r−3/2. For a value of Poisson's ratio of 0.25 or 0.33, the amplitude of the Rayleigh waves from a horizontal source should be smaller than the amplitude of the Rayleigh waves from a vertical source. The ratio of vertical to horizontal amplitude for the Rayleigh waves from the horizontal source is the same as the corresponding ratio for the vertical source for all values of Poisson's ratio.

Body waves as normal and leaking modes. Part I: Introduction

1967 ◽  
Vol 57 (2) ◽  
pp. 191-198
Author(s):  
J. Cl. De Bremaecker
Keyword(s):  
Rayleigh Waves ◽  
Fourier Transformation ◽  
Normal Modes ◽  
Synthetic Seismograms ◽  
Body Waves ◽  
Apparent Velocity ◽  
S Wave ◽  
Sh Waves ◽  
P And Sv Waves ◽  
Method Of Residues

abstract Realistic artificial seismograms may be computed by considering body waves as sums of normal or leaking modes of surface waves: the S wave and those arriving after S may be considered as sums of higher normal modes of Rayleigh waves (RiN) and Love waves (LiN); in this case the apparent velocity c < βn. Earlier arrivals are generally due to the first kind of leaking modes of Rayleigh waves (RiL1) for which βn < c < αn. Deep reflections in seismic prospecting are RiL2 for which c > αn. Synthetic seismograms can be computed by double Fourier transformation in those two last cases. Alternately the method of residues followed by a single Fourier (or Laplace) transformation may be used in all cases. Earth-stretching approximations should give excellent results for SH waves and may give satisfactory results for P and SV waves.

scholarly journals Observation of the Long-Period Monotonic Seismic Waves of the November 11, 2018, Mayotte event by the Iranian Broadband Seismic Stations

2020 ◽  
Author(s):  
Hossein Sadeghi ◽  
Sadaomi Suzuki
Keyword(s):  
Surface Waves ◽  
Rayleigh Waves ◽  
Seismic Waves ◽  
Maximum Amplitude ◽  
Body Waves ◽  
Unusual Event ◽  
Long Period ◽  
Phase Velocities ◽  
Clear Peak ◽  
Rayleigh And Love Waves

Abstract On November 11, 2018, an event generating long-lasting, monotonic long-period surface waves was observed by seismographs around the world. This event occurred at around 09:30 (UTC) east of the Mayotte Island, east Africa. This event is unusual due to the absence of body waves in the seismograms and people’s lack of sense. The purpose of this study is to investigate this unusual event using the waveforms recorded by the Iranian National Broadband Seismic Network. The network consisted of 26 stations in operation on November 11, 2018. The stations are located from 4542 km to 5772 km north-northeast of the event’s epicentre. The arrival of monochromatic long-period signals is visible around 10 UTC in the recordings of all the stations and lasts for more than 30 minutes. Frequency analysis of the seismograms shows a clear peak at 0.064 Hz (15.6 sec/cycle). The maximum amplitude of the transverse components is less than a half of the radial components. This is in agreement with the theoretical radiation pattern of Rayleigh and Love waves at a frequency of 0.06 Hz from a vertical Compensated Linear Vector Dipole (CLVD) source mechanism. The average apparent phase velocities are calculated as 3.31 km/s and 2.97 km/s, in the transverse and radial directions, corresponding respectively to the Love and Rayleigh waves in the range of 0.05 to 0.07 Hz. The surface wave magnitude of Ms 5.07 ± 0.22 was estimated. Just before the monochromatic signal, there is some dispersion in the surface waves. This observation may suggest a regular earthquake that triggered the strange Mayotte event.

Composite absorbing boundaries for the numerical simulation of seismic waves

1994 ◽  
Vol 84 (1) ◽  
pp. 185-191
Author(s):  
Nanxun Dai ◽  
Antonios Vafidis ◽  
Ernest Kanasewich
Keyword(s):  
Numerical Simulation ◽  
Transition Zone ◽  
Seismic Waves ◽  
Absorbing Boundary Conditions ◽  
The Body ◽  
Body Waves ◽  
Filter Method ◽  
Characteristic Analysis ◽  
Absorbing Boundary ◽  
The One

Abstract Composite absorbing boundary methods are developed for the numerical simulation of seismic waves. These methods combine low-angle absorbing boundary conditions, based on the characteristic analysis of one-dimensional wave equations, with either of two novel wave field modification approaches, namely the anisotropic filters and the one-way sponge filters. The anisotropic filter method adjusts the propagation direction of the waves, so that they reach the boundary at normal angles. The one-way sponge filter method endows the transition zone with a dissipation mechanism that selectively damps the incoming waves. These methods absorb not only the body waves but also the surface waves. A narrow transition zone adjacent to a computational boundary is introduced whose width is smaller than in the sponge filter approach. Numerical examples illustrate the effectiveness of these methods in absorbing the artificial reflections.

Propagation of SH-Waves Through Non Planer Interface between Visco-Elastic and Fibre-Reinforced Solid Half-Spaces

2017 ◽  
Vol 33 (4) ◽  
pp. 545-557
Author(s):  
B. Prasad ◽  
P. C. Pal ◽  
S. Kundu
Keyword(s):  
Closed Form ◽  
Half Space ◽  
Seismic Waves ◽  
Order Approximation ◽  
Elastic Half Space ◽  
Angle Of Incidence ◽  
Sh Waves ◽  
Transmission Coefficients ◽  
Reflection And Refraction ◽  
Special Cases

AbstractIn the propagation of seismic waves through layered media, the boundaries play crucial role. The boundaries separating the different layers of the earth are irregular in nature and not perfectly plane. It is, therefore, necessary to take into account the corrugation of the boundaries while dealing with the problem of reflection and refraction of seismic waves. The present study explores the reflection and refraction phenomena of SH-waves at a corrugated interface between visco-elastic half-space and fibre-reinforced half-space. Method of approximation given by Rayleigh is adopted and the expressions for reflection and transmission coefficients are obtained in closed form for the first and second order approximation of the corrugation. The closed form formulae of these coefficients are presented for a corrugated interface of periodic shape (cosine law interface). It is found that these coefficients depend upon the amplitude of corrugation of the boundary, angle of incidence and frequency of the incident wave. Numerical computations for a particular type of corrugated interface are performed and a number of graphs are plotted. Some special cases are derived.

Two-dimensional simulation of Rayleigh waves with staggered sine/cosine transforms and variable grid spacing

Geophysics ◽  
2010 ◽  
Vol 75 (4) ◽  
pp. T133-T140 ◽  
Author(s):  
Dan Kosloff ◽  
José M. Carcione
Keyword(s):  
Boundary Conditions ◽  
Rayleigh Waves ◽  
Time Integration ◽  
Fourier Method ◽  
The Body ◽  
Body Waves ◽  
Spatial Sampling ◽  
Grid Points ◽  
Dimensional Simulation ◽  
Stress Free Boundary

Simulation of Rayleigh waves requires high accuracy and an adequate spatial sampling at the surface. Discrete cosine and sine transforms are used to compute spatial derivatives along the direction perpendicular to the surface of the earth. Unlike the standard Fourier method, these transforms allow nonperiodic boundary conditions to be satisfied, in particular, the stress-free conditions at the surface. Because simulation of surface waves requires more points per minimum wavelength at the surface than simulation of body waves, the equispaced grid is not efficient. To overcome this problem, a grid compression is performed at the surface to obtain a denser spatial sampling. Grid size is minimal at the surface and increases with depth until reaching, at a relatively shallow depth, the grid points per wavelength required by the body waves. The stress-free boundary conditions are naturally handled by expanding the appropriate stress components in terms of the discrete sine transform. The wave equation is solved in the particle-velocity and stress formulation using a Runge-Kutta time integration and the convolutional PML (CPML) method to prevent reflections from the mesh boundaries. The simulations are very accurate for shallow sources and receivers and large offsets.

Reflection of spherical seismic waves in elastic layered media

Geophysics ◽  
1983 ◽  
Vol 48 (6) ◽  
pp. 655-664 ◽  
Author(s):  
Paul M. Krail ◽  
Henry Brysk
Keyword(s):  
Plane Wave ◽  
Seismic Waves ◽  
Plane Waves ◽  
Spherical Wave ◽  
Bright Spot ◽  
Angle Of Incidence ◽  
Complicated Manner ◽  
Plane Wave Incident ◽  
Wave Limit ◽  
The Impact

The solution of the elastic wave equation for a plane wave incident on a plane interface has been known since the turn of the century. For reflections from reasonably shallow beds, however, it is necessary to treat the incident wave as spherical rather than plane. The formalism for expressing spherical wavefronts as contour integrals over plane waves goes back to Sommerfeld (1909) and Weyl (1919). Brekhovskikh (1960) performed a steepest descent evaluation of the integrals to attain analytic results in the acoustic case. We have extended his approach to elastic waves to obtain spherical‐wave Zoeppritz coefficients. We illustrate the impact of the curvature correction parametrically (as the velocity and density contrasts and Poisson’s ratios are varied). In particular, we examine conditions appropriate to “bright spot” analysis; expectedly, the situation becomes less simple than in the plane‐wave limit. The curvature‐corrected Zoeppritz coefficients vary more strongly (and in a more complicated manner) with the angle of incidence than do the original ones. The determination of material properties (velocities and densities) from the reflection coefficients is feasible in principle, with exacting prestack processing and interpretation. For orientation, we outline the procedure for the simple case of a separated single source and detector pair over a multilayered horizontal earth.

Radiation pattern of Rayleigh waves from a fault of arbitrary dip and direction of motion in a homogeneous medium

1963 ◽  
Vol 53 (3) ◽  
pp. 619-642
Author(s):  
N. A. Haskell
Keyword(s):  
Rayleigh Wave ◽  
Rayleigh Waves ◽  
Fault Plane ◽  
Cylindrical Wave ◽  
Point Sources ◽  
The Body ◽  
Wave Functions ◽  
Body Waves ◽  
Surface Boundary ◽  
Double Couple

Abstract Expressions for the displacements in the body waves radiated in an unbounded, homogeneous elastic medium by dipolar point sources of arbitrary orientation may be readily derived in Cartesian coordinates from formulae given by Love. The free-surface boundary conditions are, however, most conveniently expressed in terms of Sezawa's cylindrical wave functions. The necessary transformation between the two representations is provided by the Sommerfeld integral and others that may be derived from it by differentiations with respect to the radial and axial (vertical) coordinates. By this means the total radiation field (direct plus surface reflected) is expressed in terms of integrals of cylindrical wave functions. The Rayleigh wave component may then be separated out by calculating the residue at the Rayleigh pole of the integrand. The azimuthal dependence of the Rayleigh wave displacements appears as the sum of three terms, one independent of the azimuth angle, φ, another depending upon sin φ and cos φ, and a third depending upon sin 2φ and cos 2φ. The coefficients of these terms are functions of the direction cosines of the normal to the fault plane and the direction of the relative displacement vector in the fault plane. Equations are presented for sources of both single and double couple types. The effect of fault propagation with finite velocity over a finite distance may be included by multiplying these expressions by the finite source factor previously derived by Ben-Menahem. Polar plots of the amplitude and initial phase are presented for single and double-couple representations of a number of different types of faults. It is noted that for one certain orientation a shallow double-couple source generates no Rayleigh waves.

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