Estimation of a point source wavelet in elastic media

Geophysics ◽  
1993 ◽  
Vol 58 (8) ◽  
pp. 1195-1199 ◽  
Author(s):  
Alexander Kagansky ◽  
Dan Loewenthal
Keyword(s):  
Point Source ◽  
Half Space ◽  
Simple Procedure ◽  
Estimation Algorithm ◽  
Vertical Displacement ◽  
Hankel Transform ◽  
P Wave ◽  
Wavelet Estimation ◽  
Acoustic Media ◽  
Correct Estimate

A new method of wavelet estimation in elastic or acoustic media is presented. The method is based on the simple procedure of weighted summation of seismic traces for all the distances of source‐receivers, with a horizontal offset r as a weight. The model treated consists of a homogeneous elastic (or acoustic) layer with the free surface above and a half‐space below. The Lamé parameters and the density of the half‐space can be any function of the depth. A P‐wave point source operates in the layer, and the vertical displacement field or the vertical particle velocity field (or the pressure in the acoustic case) is measured by two horizontal lines of receivers located at two depth levels in the same layer. To obtain the wavelet‐estimation algorithm, the Fourier‐Hankel transform of the field is used. It is shown that there are two possibilities of data measuring: (1) when both the lines of the receivers are below the source and (2) when one of the lines is above the source. Numerical examples show that the proposed method gives a correct estimate of the source wavelet.

A STUDY OF TWO‐DIMENSIONAL HEAD WAVES IN FLUID AND SOLID SYSTEMS

Geophysics ◽  
1963 ◽  
Vol 28 (4) ◽  
pp. 563-581 ◽  
Author(s):  
John W. Dunkin
Keyword(s):  
Half Space ◽  
Vertical Displacement ◽  
Point Of View ◽  
Head Wave ◽  
Apparent Velocity ◽  
Two Dimensional ◽  
Multiple Reflections ◽  
Transient Wave ◽  
Line Load ◽  
Head Waves

The problem of transient wave propagation in a three‐layered, fluid or solid half‐plane is investigated with the point of view of determining the effect of refracting bed thickness on the character of the two‐dimensional head wave. The “ray‐theory” technique is used to obtain exact expressions for the vertical displacement at the surface caused by an impulsive line load. The impulsive solutions are convolved with a time function having the shape of one cycle of a sinusoid. The multiple reflections in the refracting bed are found to affect the head wave significantly. For thin refracting beds in the fluid half‐space the character of the head wave can be completely altered by the strong multiple reflections. In the solid half‐space the weaker multiple reflections affect both the rate of decay of the amplitude of the head wave with distance and the apparent velocity of the head wave by changing its shape. A comparison is made of the results for the solid half‐space with previously published results of model experiments.

Diffraction of elastic waves by a fluid-filled crack in a fluid-saturated poroelastic half-space

2021 ◽  
Author(s):  
Zhongxian Liu ◽  
Jiaqiao Liu ◽  
Sibo Meng ◽  
Xiaojian Sun
Keyword(s):  
Half Space ◽  
Seismic Waves ◽  
Incident Angle ◽  
Excitation Frequency ◽  
Vertical Displacement ◽  
Crack Model ◽  
P Waves ◽  
Amplification Effect ◽  
Resonance Frequencies ◽  
Angle Crack

Summary An indirect boundary element method (IBEM) is developed to model the two-dimensional (2D) diffraction of seismic waves by a fluid-filled crack in a fluid-saturated poroelastic half-space, using Green's functions computed considering the distributed loads, flow, and fluid characteristics. The influence of the fluid-filled crack on the diffraction characteristics is investigated by analyzing key parameters, such as the excitation frequency, incident angle, crack width and depth, and medium porosity. The results for the fluid-filled crack model are compared to those for the fluid-free crack model under the same conditions. The numerical results demonstrate that the fluid-filled crack has a significant amplification effect on the surface displacements, and that the effect of the depth of the fluid-filled crack is more complex compared to the influence of other parameters. The resonance diffraction generates an amplification effect in the case of normally incident P waves. Furthermore, the horizontal and vertical displacement amplitudes reach 4.2 and 14.1, respectively. In the corresponding case of the fluid-free crack, the vertical displacement amplitude is only equal to 4.1, indicating the amplification effect of the fluid in the crack. Conversely, for normally incident SV waves at certain resonance frequencies, the displacement amplitudes above a fluid-filled crack may be lower than the displacement amplitudes observed in the corresponding case of a fluid-free crack.

Contact Stress Analysis of a Layered Transversely Isotropic Half-Space

1992 ◽  
Vol 114 (2) ◽  
pp. 253-261 ◽  
Author(s):  
C. H. Kuo ◽  
L. M. Keer
Keyword(s):  
Half Space ◽  
Mixed Boundary ◽  
Contact Region ◽  
Three Dimensional ◽  
Transversely Isotropic ◽  
Hankel Transform ◽  
Elastic Half Space ◽  
Contact Stresses ◽  
Stress Components ◽  
Contact Failure

The three-dimensional problem of contact between a spherical indenter and a multi-layered structure bonded to an elastic half-space is investigated. The layers and half-space are assumed to be composed of transversely isotropic materials. By the use of Hankel transforms, the mixed boundary value problem is reduced to an integral equation, which is solved numerically to determine the contact stresses and contact region. The interior displacement and stress fields in both the layer and half-space can be calculated from the inverse Hankel transform used with the solved contact stresses prescribed over the contact region. The stress components, which may be related to the contact failure of coatings, are discussed for various coating thicknesses.

ELECTRIC POTENTIAL NEAR A SPHERICAL BODY IN A CONDUCTING HALF‐SPACE

Geophysics ◽  
1971 ◽  
Vol 36 (4) ◽  
pp. 763-767 ◽  
Author(s):  
David B. Large
Keyword(s):  
Point Source ◽  
Electric Current ◽  
Half Space ◽  
Electric Potential ◽  
Spherical Body ◽  
Classical Potential

An extensive summary of classical potential solutions has been given recently by Van Nostrand and Cook (1966). This note presents a solution for the potential due to a point source of electric current placed on the earth’s surface in the vicinity of a buried spherical body of arbitrary resistivity. The analysis follows the procedure suggested by Van Nostrand and Cook and is similar to that used recently by Merkel (1969, 1971).

Computation of Green’s tensor integrals for three‐dimensional electromagnetic problems using fast Hankel transforms

Geophysics ◽  
1984 ◽  
Vol 49 (10) ◽  
pp. 1754-1759 ◽  
Author(s):  
Walter L. Anderson
Keyword(s):  
Half Space ◽  
Three Dimensional ◽  
Hankel Transform ◽  
Electromagnetic Modeling ◽  
Electromagnetic Problems ◽  
Hankel Transforms ◽  
Homogeneous Half Space ◽  
Green's Tensor ◽  
Green’S Tensor ◽  
The Many

A new method is presented that rapidly evaluates the many Green’s tensor integrals encountered in three‐dimensional electromagnetic modeling using an integral equation. Application of a fast Hankel transform (FHT) algorithm (Anderson, 1982) is the basis for the new solution, where efficient and accurate computation of Hankel transforms are obtained by related and lagged convolutions (linear digital filtering). The FHT algorithm is briefly reviewed and compared to earlier convolution algorithms written by the author. The homogeneous and layered half‐space cases for the Green’s tensor integrals are presented in a form so that the FHT can be easily applied in practice. Computer timing runs comparing the FHT to conventional direct convolution methods are discussed, where the FHT’s performance was about 6 times faster for a homogeneous half‐space, and about 108 times faster for a five‐layer half‐space. Subsequent interpolation after the FHT is called is required to compute specific values of the tensor integrals at selected transform arguments; however, due to the relatively small lagged convolution interval used (same as the digital filter’s), a simple and fast interpolation is sufficient (e.g., by cubic splines).

Closed-Form Solutions of the Elastic Half Space due to a Line Heat Source

2014 ◽  
Vol 638-640 ◽  
pp. 2082-2091
Author(s):  
John C.C. Lu ◽  
Feng Tsai Lin
Keyword(s):  
Heat Source ◽  
Closed Form ◽  
Half Space ◽  
Hankel Transform ◽  
Elastic Half Space ◽  
Line Heat Source ◽  
Closed Form Solutions ◽  
Line Sink ◽  
Vertical Displacements ◽  
Thermoelastic Response

Thermoelastic response due to a line heat source is analog to poroelastic reaction caused by a fluid line sink. In this study, the strata are modeled as a thermoelastic or poroelastic half space bounded by horizontal surface in the mathematical model. Thermomechanics and poromechanics are applied on the formulation of basic governing equations, and an analogy is drawn to show the similarity. Using Hankel transform technique and approaching symbolic integral through Mathematica, the closed-form solutions of the horizontal and vertical displacements due to a fluid line sink are obtained. The displacements produced by the line heat source are described through analog quantities between thermoelasticity and poroelasticity. The solutions can be applied to dewater operations and build waste repository.

Attenuation of seismograms obtained by the Cagniard‐Pekeris method

Geophysics ◽  
1983 ◽  
Vol 48 (9) ◽  
pp. 1204-1211 ◽  
Author(s):  
P. G. Kelamis ◽  
E. R. Kanasewich ◽  
F. Abramovici
Keyword(s):  
Reflection Coefficient ◽  
Point Source ◽  
Frequency Domain ◽  
Elastic Medium ◽  
Half Space ◽  
Ad Hoc ◽  
Synthetic Seismograms ◽  
Weathered Layer ◽  
Attenuation And Dispersion

Attenuation and dispersion are included in synthetic seismograms obtained by a Cagniard‐Pekeris formulation for the problem of a point source in a layer over a half‐space. The solution is decompose into generalized rays, and the effects of attenuation and dispersion are incorporated in an ad hoc manner in the frequency domain. The effects of the viscoelastic interfaces are taken into account by corrections to the reflection coefficient for an elastic medium. The results are illustrated with synthetics for a model simulating a weathered layer over a halt‐space. Both the SH and P‐SV cases are treated.

Reflection maxima for reflections from single interfaces

Geophysics ◽  
1988 ◽  
Vol 53 (2) ◽  
pp. 271-275 ◽  
Author(s):  
C. A. Rendleman ◽  
F. K. Levin
Keyword(s):  
Plane Wave ◽  
Point Source ◽  
Critical Angle ◽  
Maximum Amplitude ◽  
P Wave ◽  
Plane Interface ◽  
Wide Angle ◽  
Mathematical Expressions ◽  
Shed Light

At a workshop on refraction and wide‐angle reflections, Hilterman (1985) pointed out that, in contrast to the plane‐wave case, when there is a point source, a P-wave reflected from a plane interface attains its maximum amplitude at an offset greater than that corresponding to the critical angle (Figure 1). The same conclusion had been drawn earlier by Červený (1967). However, neither Červený’s results, which were based on very complicated mathematical expressions derived by Brekhovskikh (1960), nor Hilterman’s computer‐generated data shed light on the physics implied by the shifted maximum.

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