Apparent attenuation of shear waves propagating through a randomly stratified anisotropic medium

2016 ◽  
Vol 16 (04) ◽  
pp. 1650009
Author(s):  
Josselin Garnier ◽  
Knut Sølna
Keyword(s):  
Elastic Waves ◽  
Seismic Waves ◽  
Stochastic Partial Differential Equation ◽  
Heterogeneous Media ◽  
Seismic Attenuation ◽  
The Earth ◽  
Separation Of Scales ◽  
Coherent Pulse ◽  
Apparent Attenuation ◽  
Media Experience

Waves propagating through heterogeneous media experience scattering that can convert a coherent pulse into small incoherent fluctuations. This may appear as attenuation for the transmitted front pulse. The classic O’Doherty–Anstey theory describes such a transformation for scalar waves in finely layered media. Recent observations for seismic waves in the earth suggest that this theory can explain a significant component of seismic attenuation. An important question to answer is then how the O’Doherty–Anstey theory generalizes to seismic waves when several wave modes, possibly with the same velocity, interact. An important aspect of the O’Doherty–Anstey theory is the statistical stability property, which means that the transmitted front pulse is actually deterministic and depends only on the statistics of the medium but not on the particular medium realization when the medium is modeled as a random process. It is shown in this paper that this property generalizes in the case of elastic waves in a nontrivial way: the energy of the transmitted front pulse, but not the pulse shape itself, is statistically stable. This result is based on a separation of scales technique and a diffusion-approximation theorem that characterize the transmitted front pulse as the solution of a stochastic partial differential equation driven by two Brownian motions.

scholarly journals Simulation of seismic waves at the Earth crust (brittle-ductile transition) based on the Burgers model

2014 ◽  
Vol 6 (1) ◽  
pp. 1371-1400 ◽  
Author(s):  
J. M. Carcione ◽  
F. Poletto ◽  
B. Farina ◽  
A. Craglietto
Keyword(s):  
Seismic Waves ◽  
Grid Method ◽  
In Situ Stress ◽  
Seismic Attenuation ◽  
S Wave ◽  
Fourier Pseudospectral Method ◽  
Stress Criterion ◽  
Brittle Ductile Transition ◽  
The Earth

Abstract. The Earth crust presents two dissimilar rheological behaviours depending on the in-situ stress-temperature conditions. The upper, cooler, part is brittle while deeper zones are ductile. Seismic waves may reveal the presence of the transition but a proper characterization is required. We first obtain a stress–strain relation including the effects of shear seismic attenuation and ductility due to shear deformations and plastic flow. The anelastic behaviour is based on the Burgers mechanical model to describe the effects of seismic attenuation and steady-state creep flow. The shear Lamé constant of the brittle and ductile media depends on the in-situ stress and temperature through the shear viscosity, which is obtained by the Arrhenius equation and the octahedral stress criterion. The P- and S-wave velocities decrease as depth and temperature increase due to the geothermal gradient, an effect which is more pronounced for shear waves. We then obtain the P-S and SH equations of motion recast in the velocity-stress formulation, including memory variables to avoid the computation of time convolutions. The equations correspond to isotropic anelastic and inhomogeneous media and are solved by a direct grid method based on the Runge–Kutta time stepping technique and the Fourier pseudospectral method. The algorithm is tested with success against known analytical solutions for different shear viscosities. A realistic example illustrates the computation of surface and reverse-VSP synthetic seismograms in the presence of an abrupt brittle-ductile transition.

scholarly journals Energy ratio of reflected and refracted seismic waves*

1944 ◽  
Vol 34 (2) ◽  
pp. 85-102
Author(s):  
B. Gutenberg
Keyword(s):  
Elastic Waves ◽  
Seismic Waves ◽  
Energy Ratio ◽  
The Earth

IV. Summary The investigation of amplitudes of elastic waves passing through layers of the earth, either in earthquakes or from artificial explosions, requires a knowledge of the distribution of energy at points of reflection or refraction, and also of the ratio between the amplitudes arriving at the surface of the earth and the displacements of the ground there. Some fundamental theoretical problems involved are discussed (section II) and typical curves for the quantities mentioned are given (section III).

scholarly journals Attenuation of seismic waves in the earth's mantle

1958 ◽  
Vol 48 (3) ◽  
pp. 269-282 ◽  
Author(s):  
B. Gutenberg
Keyword(s):  
Absorption Coefficient ◽  
Elastic Waves ◽  
Laboratory Experiments ◽  
Seismic Waves ◽  
Body Waves ◽  
The Core ◽  
Earth’S Mantle ◽  
The Earth ◽  
Theoretical Considerations

Abstract Contrasting with conclusions from laboratory experiments that the absorption coefficient k for amplitudes of elastic waves is proportional to 1/T, or, from theoretical considerations, that it should be proportional to 1/T or to 1/T2, observations of body waves through the mantle of the earth show little if any decrease in absorption with increasing period T. In teleseismic records S rarely shows periods of less than 4 seconds, while in P periods of 1 second are observed to the greatest distances. The value k = 0.06 per 1,000 km., found previously for P, P′P′ and P′P′P′ through the mantle and the core, is confirmed for P and PP and is found also for S in the mantle.

Separation of frequency-dependent scattering and inelastic absorption by waveform inversion of VSP data constrained by well logs

2019 ◽  
pp. 94-97
Author(s):  
A. S. Pirogova
Keyword(s):  
Seismic Waves ◽  
Waveform Inversion ◽  
Seismic Attenuation ◽  
Well Log ◽  
Frequency Dependent ◽  
Seismic Profiling ◽  
Vertical Seismic Profiling ◽  
The Earth ◽  
Two Factors ◽  
Vsp Data

The paper presents an approach to estimation of frequency-dependent attenuation of seismic waves propagating in the earth subsurface. The approach is based on the waveform inversion of vertical seismic profiling data acquired in a borehole. Incorporation of well log data (in particular, sonic and density logs) in the forward modelling routine allows for separation of two factors that cause frequency-dependent seismic attenuation. In particular, the inversion facilitates separation of 1D scattering versus inelastic absorption in the horizontally layered subsurface.

An analytic signal based accurate time-domain visco-acoustic wave equation from the Constant-Q theory

Geophysics ◽  
2021 ◽  
pp. 1-58
Author(s):  
Hongwei Liu ◽  
Yi Luo
Keyword(s):  
Wave Equation ◽  
Acoustic Wave ◽  
Time Domain ◽  
Seismic Waves ◽  
Heterogeneous Media ◽  
Relaxation Property ◽  
Analytic Signal ◽  
Acoustic Wave Equation ◽  
Real Component ◽  
Frequency Components

We present a concise time-domain wave equation to accurately simulate wave propagation in visco-acoustic media. The central idea behind this work is to dismiss the negative frequency components from a time-domain signal by converting the signal to its analytic format. The negative frequency components of any analytic signal are always zero, meaning we can construct the visco-acoustic wave equation to honor the relaxation property of the media for positive frequencies only. The newly proposed complex-valued wave equation (CWE) represents the wavefield with its analytic signal, whose real part is the desired physical wavefield, while the imaginary part is the Hilbert transform of the real component. Specifically, this CWE is accurate for both weak and strong attenuating media in terms of both dissipation and dispersion and the attenuation is precisely linear with respect to the frequencies. Besides, the CWE is easy and flexible to model dispersion-only, dissipation-only or dispersion-plus-dissipation seismic waves. We have verified these CWEs by comparing the results with analytical solutions, and achieved nearly perfect matching. Except for the homogeneous Q media, we have also extended the CWEs to heterogeneous media. The results of the CWEs for heterogeneous Q media are consistent with those computed from the nonstationary operator based Fourier Integral method and from the Standard Linear Solid (SLS) equations.

scholarly journals Simulation of seismic waves at the earth's crust (brittle–ductile transition) based on the Burgers model

Solid Earth ◽  
2014 ◽  
Vol 5 (2) ◽  
pp. 1001-1010 ◽  
Author(s):  
J. M. Carcione ◽  
F. Poletto ◽  
B. Farina ◽  
A. Craglietto
Keyword(s):  
Seismic Waves ◽  
In Situ Stress ◽  
Seismic Attenuation ◽  
Earth’S Crust ◽  
S Wave ◽  
Fourier Pseudospectral Method ◽  
Stress Criterion ◽  
Brittle Ductile Transition ◽  
Earth's Crust

Abstract. The earth's crust presents two dissimilar rheological behaviors depending on the in situ stress-temperature conditions. The upper, cooler part is brittle, while deeper zones are ductile. Seismic waves may reveal the presence of the transition but a proper characterization is required. We first obtain a stress–strain relation, including the effects of shear seismic attenuation and ductility due to shear deformations and plastic flow. The anelastic behavior is based on the Burgers mechanical model to describe the effects of seismic attenuation and steady-state creep flow. The shear Lamé constant of the brittle and ductile media depends on the in situ stress and temperature through the shear viscosity, which is obtained by the Arrhenius equation and the octahedral stress criterion. The P and S wave velocities decrease as depth and temperature increase due to the geothermal gradient, an effect which is more pronounced for shear waves. We then obtain the P−S and SH equations of motion recast in the velocity-stress formulation, including memory variables to avoid the computation of time convolutions. The equations correspond to isotropic anelastic and inhomogeneous media and are solved by a direct grid method based on the Runge–Kutta time stepping technique and the Fourier pseudospectral method. The algorithm is tested with success against known analytical solutions for different shear viscosities. A realistic example illustrates the computation of surface and reverse-VSP synthetic seismograms in the presence of an abrupt brittle–ductile transition.

JOINT 3D TRAVELTIME INVERSION OF P, S AND CONVERTED WAVES

2001 ◽  
Vol 09 (04) ◽  
pp. 1407-1416 ◽  
Author(s):  
GIULIANA ROSSI ◽  
ALDO VESNAVER
Keyword(s):  
Linear System ◽  
Seismic Waves ◽  
Velocity Fields ◽  
Traveltime Inversion ◽  
S Waves ◽  
The Earth ◽  
Converted Waves

Converted waves can play a basic role in the traveltime inversion of seismic waves. The sought velocity fields of P and S waves are almost decoupled, when considering pure P and S arrivals: their only connection are the possible common reflecting interfaces in the Earth. Converted waves provide new equations in the linear system to be inverted, which directly relates the two velocity fields. Since the new equations do not introduce additional unknowns, they increase the system rank or its redundancy, so making its solutions better constrained and robust.

Seismic Rays

2017 ◽  
Author(s):  
John A. Adam
Keyword(s):  
Inverse Problem ◽  
Seismic Tomography ◽  
Seismic Waves ◽  
Seismic Station ◽  
Three Dimensional ◽  
Arrival Times ◽  
The Earth ◽  
Straight Lines ◽  
Ray Path ◽  
Uniform Composition

This chapter focuses on the underlying mathematics of seismic rays. Seismic waves caused by earthquakes and explosions are used in seismic tomography to create computer-generated, three-dimensional images of Earth's interior. If the Earth had a uniform composition and density, seismic rays would travel in straight lines. However, it is broadly layered, causing seismic rays to be refracted and reflected across boundaries. In order to calculate the speed along the wave's ray path, the time it takes for a seismic wave to arrive at a seismic station from an earthquake needs to be determined. Arrival times of different seismic waves allow scientists to define slower or faster regions deep in the Earth. The chapter first presents the relevant equations for seismic rays before discussing how rays are propagated in a spherical Earth. The Wiechert-Herglotz inverse problem is considered, along with the properties of X in a horizontally stratified Earth.

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