Use of reciprocity theorem for obtaining Rayleigh wave radiation patterns

1966 ◽  
Vol 56 (4) ◽  
pp. 925-936 ◽  
Author(s):  
I. N. Gupta
Keyword(s):  
Rayleigh Wave ◽  
Rayleigh Waves ◽  
Three Dimensional ◽  
Reciprocity Theorem ◽  
Point Sources ◽  
Dimensional Case ◽  
Wave Radiation ◽  
Radiation Patterns ◽  
Homogeneous Half Space ◽  
Double Couple

abstract The reciprocity theorem is used to obtain Rayleigh wave radiation patterns from sources on the surface of or within an elastic semi-infinite medium. Nine elementary line sources first considered are: horizontal and vertical forces, horizontal and vertical double forces without moment, horizontal and vertical single couples, center of dilatation (two dimensional case), center of rotation, and double couple without moment. The results are extended to the three dimensional case of similar point sources in a homogeneous half space. Haskell's results for the radiation patterns of Rayleigh waves from a fault of arbitrary dip and direction of motion are reproduced in a much simpler manner. Numerical results on the effect of the depth of these sources on the Rayleigh wave amplitudes are shown for a solid having Poisson's ratio of 0.25.

Radiation pattern of mantle Rayleigh waves and the source mechanism of the Hindu Kush earthquake of July 6, 1962

1965 ◽  
Vol 55 (5) ◽  
pp. 805-819 ◽  
Author(s):  
Ramesh Chander ◽  
James N. Brune
Keyword(s):  
Rayleigh Waves ◽  
Focal Depth ◽  
Point Sources ◽  
The Body ◽  
P Wave ◽  
Source Mechanism ◽  
Radiation Patterns ◽  
Wave Data ◽  
Hindu Kush ◽  
Double Couple

abstract The source mechanism of the Hindu Kush earthquake of July 6, 1962 (magnitude 634-7, focal depth 218 km) was studied by comparing the observed amplitude and phase radiation patterns of mantle Rayleigh waves of 150 sec and 200 sec period with theoretical radiation patterns of Rayleigh waves from single- and double-couple point sources, and by considering evidence from Love waves and the shape of P and S pulses. The solution for the source mechanism, which is consistent with all the body wave and surface wave data available for this earthquake, is a double couple acting as a step function in time, with nodal planes oriented as determined from P wave data. Since for waves with periods greater than about 5 sec, the source appears to be an ideal point source, the radius of the equivalent source volume is estimated to be less than 10 km. For Rayleigh waves of 150 sec period, the agreement between observed and theoretical phases (for the above source model) is greatly improved by assuming regional phase velocities instead of a uniform phase velocity for all areas. It is concluded that with the accuracy currently attainable, a study of Rayleigh waves alone cannot determine the source mechanism of an earthquake uniquely.

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.

Radiation pattern of surface waves from point sources in a multi-layered medium

1964 ◽  
Vol 54 (1) ◽  
pp. 377-393 ◽  
Author(s):  
N. A. Haskell
Keyword(s):  
Radiation Pattern ◽  
Rayleigh Waves ◽  
Homogeneous Medium ◽  
Previous Treatment ◽  
Point Sources ◽  
Wave Radiation ◽  
Arbitrary Orientation ◽  
Double Couple ◽  
Single Force ◽  
Force Dipole

abstract A previous treatment of the radiation pattern of Rayleigh waves from single and double couple point sources of arbitrary orientation in a semi-infinite homogeneous medium is generalized to the case of a medium composed of an arbitrary number of parallel homogeneous layers by the use of layer matrices. Source coefficients are given for a single force, a force dipole without torque, a single couple, and a double couple without torque, all for arbitrary orientation vectors, and also for a spherically symmetrical compressional source. For the simplest case permitting the existence of Love waves, a single finite layer over a semi-infinite medium, expressions for the Love wave radiation pattern are given explicitly in a form comparable to that previously given for Rayleigh waves in the homogeneous case.

Body wave radiation patterns from elementary sources within a half space

1967 ◽  
Vol 57 (4) ◽  
pp. 657-674
Author(s):  
I. N. Gupta
Keyword(s):  
Free Surface ◽  
Half Space ◽  
Body Wave ◽  
Point Sources ◽  
The Body ◽  
Wave Radiation ◽  
Seismic Sources ◽  
Vertical Force ◽  
Radiation Patterns ◽  
Double Couple

abstract The known expressions for the polar radiation patterns due to a horizontal or a vertical force, applied at a point within a uniform half space, are used to obtain the body wave radiation patterns from several other elementary seismic sources. Polar radiation patterns from seven elementary line sources, i.e., horizontal and vertical double forces without moment, horizontal and vertical single couples, center of dilatation, center of ratation, and double couple without moment, are first derived. Similar point sources in the three-dimensional space are also considered and the corresponding polar as well as azimuthal radiation patterns are obtained for P, SV, and SH waves. The results obtained include the effect of finite depth of the source below the free surface. Some of the results of Burridge et al for double-couple type seismic sources near a free surface are reproduced in a simple manner. For the elementary point sources considered here, the azimuthal radiation patterns for a uniform half-space are found to be identical with those for an infinite homogeneous medium. However the polar radiation patterns appear to be profoundly affected by the proximity of the free surface.

Polar radiation patterns of P and SV waves in a multi-layered medium

1973 ◽  
Vol 63 (2) ◽  
pp. 529-547
Author(s):  
Tien-Chang Lee ◽  
Ta-Liang Teng
Keyword(s):  
Displacement Field ◽  
Layered Medium ◽  
Focal Depth ◽  
Focal Mechanisms ◽  
P Wave ◽  
Wave Radiation ◽  
Radiation Patterns ◽  
Double Couple ◽  
Spatial Coordinates ◽  
P And Sv Waves

abstract The displacement field in a multi-layered medium due to incident plane P or SV waves is formulated in terms of Haskell's layer matrices. Based on the reciprocity theorem, the far-field polar radiation patterns of single force, double force, single couple, double couple, and dilatation in a multi-layered medium can be obtained from the displacement field and its first derivatives with respect to the spatial coordinates. Numerical results for models of one layer overlying a half-space indicate that (1) the radiation patterns are sensitive to the variation of focal depth, (2) the layering has a more pronounced effect on SV-wave radiation patterns than on P-wave radiation patterns, (3) the radiation patterns become simpler as the wavelength increases, (4) polarity may reverse abruptly somewhere beyond the critical angle in SV-wave radiation patterns, (5) radiation may be discontinuous across interfaces for some assumed focal mechanisms applied slightly above and below the interfaces, and (6) no clearcut distinction among the various radiation patterns can be used to single out one type of the assumed focal mechanisms from the rest.

scholarly journals Hypothesis Tests on Rayleigh Wave Radiation Pattern Shapes: A Theoretical Assessment of Idealized Source Screening

2021 ◽  
Author(s):  
Joshua D Carmichael
Keyword(s):  
Rayleigh Wave ◽  
Ground Motion ◽  
Wave Radiation ◽  
Underground Explosion ◽  
Radiation Patterns ◽  
Hypothesis Tests ◽  
Unknown Parameters ◽  
Discrimination Power ◽  
Explosion Monitoring ◽  
Sensor Locations

Summary Shallow seismic sources excite Rayleigh wave ground motion with azimuthally dependent radiation patterns. We place binary hypothesis tests on theoretical models of such radiation patterns to screen cylindrically symmetric sources (like explosions) from non-symmetric sources (like non-vertical dip-slip, or non-VDS faults). These models for data include sources with several unknown parameters, contaminated by Gaussian noise and embedded in a layered half-space. The generalized maximum likelihood ratio tests that we derive from these data models produce screening statistics and decision rules that depend on measured, noisy ground motion at discrete sensor locations. We explicitly quantify how the screening power of these statistics increase with the size of any dip-slip and strike-slip components of the source, relative to noise (faulting signal strength), and how they vary with network geometry. As applications of our theory, we apply these tests to (1) find optimal sensor locations that maximize the probability of screening non-circular radiation patterns, and (2) invert for the largest non-VDS faulting signal that could be mistakenly attributed to an explosion with damage, at a particular attribution probability. Lastly, we quantify how certain errors that are sourced by opening cracks increase screening rate errors. While such theoretical solutions are ideal and require future validation, they remain important in underground explosion monitoring scenarios because they provide fundamental physical limits on the discrimination power of tests that screen explosive from non-VDS faulting sources.

A Robust Method for Assessing 3-D Topographic Site Effects: A Case Study at the LSBB Underground Laboratory, France

2012 ◽  
Vol 28 (3) ◽  
pp. 1097-1115 ◽  
Author(s):  
Emeline Maufroy ◽  
Víctor M. Cruz-Atienza ◽  
Stéphane Gaffet
Keyword(s):  
Site Effects ◽  
Amplification Factor ◽  
Three Dimensional ◽  
Reference Method ◽  
Spectral Ratio ◽  
Site Effect ◽  
Point Sources ◽  
Ratio Method ◽  
Double Couple

By means of three-dimensional (3-D) numerical simulations, including the Laboratoire Souterrain à Bas-Bruit (LSBB) topography, we carefully analyze site effects assessments yielded by two approaches: the classical site to reference spectral-ratio method (SRM) and the statistical median reference method (MRM). We show for both isotropic and double-couple point sources that a 94% reduction in the number of stations of a regularly spaced array yields MRM site-effect estimates within 5% of those obtained from the absolute regional median, and within 20% using a 98% station reduction with irregularly located sites. In contrast, the SRM yielded site-effect overestimates greater than 50% in some areas and up to 100% in specific sites, which makes the MRM much more robust than the SRM. We determined a 33% probability to exceed an amplification factor of 2, and an 8% probability to exceed a factor of 3 due to topography in the surroundings of the sharpest summit of the LSBB area.

Body wave radiation patterns from force applied within a half space

1966 ◽  
Vol 56 (1) ◽  
pp. 173-183 ◽  
Author(s):  
Indra N. Gupta
Keyword(s):  
Half Space ◽  
Body Wave ◽  
Elastic Half Space ◽  
Reciprocity Theorem ◽  
Numerical Results ◽  
Wave Radiation ◽  
Far Field ◽  
Radiation Patterns ◽  
Sh Waves ◽  
Vertical Displacements

abstract Expressions are derived for the horizontal and vertical displacements at an arbitrary depth within a homogeneous, isotropic, elastic half space when plane harmonic P, SV or SH waves are incident at any given angle. On the basis of the reciprocity theorem, these expressions represent also the far-field polar radiation patterns of P, SV and SH waves due to horizontal and vertical forces applied at a point within the half space. Numerical results for a few selected values of depth are shown for a solid half space.

Quantification of Effects of Ridge and Valley Topography on the Rayleigh Wave Characteristics

2018 ◽  
Vol 12 (03) ◽  
pp. 1850007 ◽  
Author(s):  
J. P. Narayan ◽  
A. Kumar
Keyword(s):  
Rayleigh Wave ◽  
Rayleigh Waves ◽  
Large Scale ◽  
Research Work ◽  
Vertical Component ◽  
Staggered Grid ◽  
Long Span ◽  
Shape Ratio ◽  
Day By Day ◽  
Ridge And Valley

The effects of ridge and valley on the characteristics of Rayleigh waves are presented in this paper. The research work carried out has been stimulated by the day by day increase of long-span structures in the hilly areas which are largely affected by the spatial variability in ground motion caused by the high-frequency Rayleigh waves. The Rayleigh wave responses of the considered triangular and elliptical ridge and valley models were computed using a fourth-order accurate staggered-grid viscoelastic P-SV wave finite-difference (FD) program. The simulated results revealed very large amplification of the horizontal component and de-amplification of the vertical component of Rayleigh wave at the top of a triangular ridge and de-amplification of both the components at the base of the triangular valley. The observed amplification of both the components of Rayleigh wave in front of elliptical valley was larger than triangular valley models. A splitting of the Rayleigh wave wavelet was inferred after interaction with ridge and valley. It is concluded that the large-scale topography acts as a natural insulator for the surface waves and the insulating capacity of the valley is more than that of a ridge. This insulation phenomenon is arising due to the reflection, diffraction and splitting of the surface wave while moving across the topography. It is concluded that insulating potential of the topography for the Rayleigh waves largely depends on their shape and shape-ratio.

Sign in / Sign up

Export Citation Format

Share Document