Pseudo-elastic PP and PSSP modeling in stratified media

Geophysics ◽  
2021 ◽  
pp. 1-56
Author(s):  
Lasse Amundsen ◽  
Bjørn Ursin
Keyword(s):  
Green’S Function ◽  
Green's Function ◽  
Fourier Transforms ◽  
P Wave ◽  
P Waves ◽  
Simple Modification ◽  
Time Space ◽  
P Domain ◽  
Band Limited ◽  
Time Band

Many modeling techniques have been developed to find the acoustic and elastic responses of a stack of plane layers to a plane spectral wave. For an elastic medium bounded above by an acoustic half-space, the acoustic wave propagator matrix modeling method can be modified to model pseudoelastic PP arrivals and PSSP arrivals. PP arrivals propagate as pure longitudinal (P) waves in the layers, whereas PSSP arrivals propagate as shear (S) waves in the elastic part of the model. A simple modification of the pseudoelastic PP response modeling scheme allows modeling of primary P reflections. A primary reflection event involves just one reflection in the plane stratified model and thus excludes internal multiples. The propagator modeling scheme is formulated in the frequency-horizontal slowness domain. By applying inverse Fourier transforms over the frequency and horizontal wavenumbers, where the wavenumber is the horizontal slowness divided by the frequency, modeled seismograms are computed and displayed in the time-space domain. By applying an inverse Fourier transform over the frequency for selected horizontal slowness components, the computed seismograms can be shown in the intercept time-horizontal slowness ( τ- p) domain. When the source wavelet is unity for frequencies of interest, the τ- p domain seismograms become plane-wave Green’s function seismograms. The p-traces of the Green’s function primary P-wave seismograms accumulate with increasing time band-limited step functions weighted by reflection strengths.

On the Green function of the Lord–Shulman thermoelasticity equations

2019 ◽  
Vol 220 (1) ◽  
pp. 393-403 ◽  
Author(s):  
Zhi-Wei Wang ◽  
Li-Yun Fu ◽  
Jia Wei ◽  
Wanting Hou ◽  
Jing Ba ◽  
...  
Keyword(s):  
Green’S Function ◽  
Green's Function ◽  
Parabolic Type ◽  
Second Order ◽  
P Wave ◽  
Thermal Mode ◽  
S Wave ◽  
Order Tensor ◽  
P Waves ◽  
Thermal Conductivities

SUMMARY Thermoelasticity extends the classical elastic theory by coupling the fields of particle displacement and temperature. The classical theory of thermoelasticity, based on a parabolic-type heat-conduction equation, is characteristic of an unphysical behaviour of thermoelastic waves with discontinuities and infinite velocities as a function of frequency. A better physical system of equations incorporates a relaxation term into the heat equation; the equations predict three propagation modes, namely, a fast P wave (E wave), a slow thermal P wave (T wave), and a shear wave (S wave). We formulate a second-order tensor Green's function based on the Fourier transform of the thermodynamic equations. It is the displacement–temperature solution to a point (elastic or heat) source. The snapshots, obtained with the derived second-order tensor Green's function, show that the elastic and thermal P modes are dispersive and lossy, which is confirmed by a plane-wave analysis. These modes have similar characteristics of the fast and slow P waves of poroelasticity. Particularly, the thermal mode is diffusive at low thermal conductivities and becomes wave-like for high thermal conductivities.

Green's function of the Lord–Shulman thermo-poroelasticity theory

2020 ◽  
Vol 221 (3) ◽  
pp. 1765-1776 ◽  
Author(s):  
Jia Wei ◽  
Li-Yun Fu ◽  
Zhi-Wei Wang ◽  
Jing Ba ◽  
José M Carcione
Keyword(s):  
Plane Wave ◽  
Green’S Function ◽  
Green's Function ◽  
Heterogeneous Media ◽  
Numerical Algorithms ◽  
Heat Diffusion ◽  
P Wave ◽  
Wave Analysis ◽  
S Wave ◽  
S Waves

SUMMARY The Lord–Shulman thermoelasticity theory combined with Biot equations of poroelasticity, describes wave dissipation due to fluid and heat flow. This theory avoids an unphysical behaviour of the thermoelastic waves present in the classical theory based on a parabolic heat equation, that is infinite velocity. A plane-wave analysis predicts four propagation modes: the classical P and S waves and two slow waves, namely, the Biot and thermal modes. We obtain the frequency-domain Green's function in homogeneous media as the displacements-temperature solution of the thermo-poroelasticity equations. The numerical examples validate the presence of the wave modes predicted by the plane-wave analysis. The S wave is not affected by heat diffusion, whereas the P wave shows an anelastic behaviour, and the slow modes present a diffusive behaviour depending on the viscosity, frequency and thermoelasticity properties. In heterogeneous media, the P wave undergoes mesoscopic attenuation through energy conversion to the slow modes. The Green's function is useful to study the physics in thermoelastic media and test numerical algorithms.

On estimating the impulse response between receivers in a controlled ultrasonic experiment

Geophysics ◽  
2006 ◽  
Vol 71 (4) ◽  
pp. SI79-SI84 ◽  
Author(s):  
K. van Wijk
Keyword(s):  
Data Processing ◽  
Laboratory Experiment ◽  
Green’S Function ◽  
Detailed Analysis ◽  
Elastic Medium ◽  
Impulse Response ◽  
Green's Function ◽  
Geophysical Data ◽  
Seismic Interferometry ◽  
Band Limited

A controlled ultrasonic laboratory experiment provides a detailed analysis of retrieving a band-limited estimate of the Green's function between receivers in an elastic medium. Instead of producing a formal derivation, this paper appeals to a series of intuitive operations, common to geophysical data processing, to understand the practicality of seismic interferometry. Whereas the retrieval of the full Green's function is based on the crosscorrelation of receivers in the presence of equipartitioned signal, an estimate of the impulse response is recovered successfully with 40 sources in a line covering six wavelengths at the surface.

Numerical Evaluation of a Band-Limited Green’s Function Using the Finite Element Method for Acoustic Emission Analysis

Volume 1 ◽  
2004 ◽  
Author(s):  
Ramez-Robert Naber ◽  
Hamid Bahai ◽  
Barry E. Jones
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Acoustic Emission ◽  
Green’S Function ◽  
Green's Function ◽  
The Finite Element Method ◽  
Transient Wave ◽  
Emission Analysis ◽  
Band Limited ◽  
Element Method

The ability to model transient wave propagation in solids and determine the Green’s function plays a major role in improving the reliability of quantitative source characterization of acoustic emission. In this work, the finite element method is employed to determine a numerical solution of the Green’s function of an isotropic plate due to a point source applied normally to the surface. The advantage of using the finite element method is that it can be extended to model realistic geometries that cannot be treated analytically. The numerical results presented here are based on a two-dimensional axisymmetric transient finite element analysis. A limited bandwidth approximation of a delta function is used (Hanning function) for modeling the source. Hence the solution is called the band-limited Green’s function. The exact analytical solutions of the Green’s function of an isotropic infinite plate are used to validate the numerical solutions. Further analysis is carried out to investigate the effects of varying the spatial resolution of the finite element model on the accuracy of the solutions. Finally, it is demonstrated how the results of the band-limited Green’s function can be used to accurately convolve the response of an arbitrary source function.

A modal solution for reflection and transmission at a Chiral–Chiral interface

2005 ◽  
Vol 83 (12) ◽  
pp. 1267-1290 ◽  
Author(s):  
P E Crittenden ◽  
E Bahar
Keyword(s):  
Green’S Function ◽  
Electromagnetic Fields ◽  
Green's Function ◽  
Fourier Transforms ◽  
Modal Expansion ◽  
Function Solution ◽  
Flat Interface ◽  
Polarized Waves ◽  
Chiral Materials ◽  
Modal Solution

A harmonic Green's function solution for a magnetic line source above and below a flat interface between two chiral materials is derived. The solution is expressed in terms of the characteristic right and left circularly polarized waves. The harmonic Green's function formulation is converted into a modal representation. The modal representation is suitable for the complete expansion of the electromagnetic fields above and below a rough interface between two chiral materials with laterally varying material properties. The modal expansion is written in terms of orthogonal-basis and reciprocal-basis functions, which have been used to formulate generalized Fourier transforms and derive the generalized telegraphists' equations for electromagnetic fields in irregular chiral media.PACS Nos.: 33.55.Ad, 78.20.Ek

Monte Carlo Simulation Compared With Classic Deterministic Solutions for Neutron Transport and Diffusion

2010 ◽  
Author(s):  
Zafar Ullah Koreshi ◽  
Sadaf Siddiq
Keyword(s):  
Monte Carlo ◽  
Green’S Function ◽  
Green's Function ◽  
Transport Theory ◽  
Fourier Transforms ◽  
Neutron Transport ◽  
Isotropic Scattering ◽  
Finite Slab ◽  
Deterministic Methods ◽  
Mc Simulation

The Monte Carlo (MC) simulation method, known to handle complex problems which may be formidable for deterministic methods, will always require validation with classic problems that have evolved historically from deterministic methods [1–5] based on Chandrasekhar’s method in radiative transfer, Fourier transforms, Green’s functions, Weiner-Hopf method etc which are restricted to simple geometries, such as infinite or semiinfinite media, and simple scattering laws too for practical application. This work compares deterministic results with MC simulation results for neutron flux in a slab. We consider mono-energetic transport problem in an infinite medium and in a 1-D finite slab with isotropic scattering. The transport theory solutions used in infinite geometry are the Green’s function solution and the spherical harmonics (P1, P3) solutions, while for the 1-D finite slab, we refer to a transport benchmark for which an exact solution is available. For diffusion theory, we consider the Green’s function infinite geometry solution, and the exact and eigen-function numerical solution for finite geometry (1-D slab). The objective of this work is to illustrate the results from all the methods considered especially near the source and boundaries, and as a function of the scattering probability. The results are plotted for six elements that include a strong absorber, such as gadolinium, and a strong “scaterrer” such as aluminium. The present work is didactic and focuses on problems which are simple enough, yet important, to illustrate the conceptual difference and computational complexity of the deterministic and stochastic approaches.

scholarly journals Modeling P waves in seismic noise correlations: Application to fault monitoring using train traffic sources

2021 ◽  
Author(s):  
Korbinian Sager ◽  
Victor Tsai ◽  
Yixiao Sheng ◽  
Florent Brenguier ◽  
Pierre Boué ◽  
...  
Keyword(s):  
Green’S Function ◽  
Surface Waves ◽  
Green's Function ◽  
Seismic Waves ◽  
Low Frequency ◽  
Body Waves ◽  
Noise Sources ◽  
P Waves ◽  
Promising Alternative ◽  
Fault Monitoring

The theory of Green's function retrieval essentially requires homogeneously distributed noise sources. Even though these conditions are not fulfilled in nature, low-frequency (<1 Hz) surface waves generated by ocean-crust interactions have been used successfully to image the crust with unprecedented spatial resolution. In contrast to low-frequency surface waves, high-frequency (>1 Hz) body waves have a sharper, more localized sensitivity to velocity contrasts and temporal changes at depth. In general, their retrieval using seismic interferometry is challenging, and recent studies focus on powerful, localized noise sources. They have proven to be a promising alternative but break the assumptions of Green's function retrieval. In this study, we present an approach to model correlations between P waves for these scenarios and analyze their sensitivity to 3D Earth structure. We perform a series of numerical experiments to advance our understanding of these signals and prepare for an application to fault monitoring. In the considered cases, the character of the signals strongly diverges from Green's function retrieval, and the sensitivity to structure has significant contributions in the source direction. An accurate description of the underlying physics allows us to reproduce observations made in the context of monitoring the San Jacinto Fault in California using train-generated seismic waves. This approach provides new perspectives for detecting and localizing temporal velocity changes previously unnoticed by commonly exploited surface-wave reconstructions.

scholarly journals Seismic Interferometry from Correlated Noise Sources

2021 ◽  
Vol 13 (14) ◽  
pp. 2703
Author(s):  
Daniella Ayala-Garcia ◽  
Andrew Curtis ◽  
Michal Branicki
Keyword(s):  
Green’S Function ◽  
Green's Function ◽  
Ambient Noise ◽  
Ocean Waves ◽  
Seismic Interferometry ◽  
Correlated Noise ◽  
Highway Traffic ◽  
Noise Sources ◽  
Estimation Errors ◽  
Band Limited

It is a well-established principle that cross-correlating seismic observations at different receiver locations can yield estimates of band-limited inter-receiver Green’s functions. This principle, known as Green’s function retrieval or seismic interferometry, is a powerful technique that can transform noise into signals which enable remote interrogation and imaging of the Earth’s subsurface. In practice it is often necessary and even desirable to rely on noise already present in the environment. Theory that underpins many applications of ambient noise interferometry assumes that the sources of noise are uncorrelated in time. However, many real-world noise sources such as trains, highway traffic and ocean waves are inherently correlated in space and time, in direct contradiction to the these theoretical foundations. Applying standard interferometric techniques to recordings from correlated energy sources makes the Green’s function liable to estimation errors that so far have not been fully accounted for theoretically nor in practice. We show that these errors are significant for common noise sources, always perturbing or entirely obscuring the phase one wishes to retrieve. Our analysis explains why stacking may reduce the phase errors, but also shows that in commonly encountered circumstances stacking will not remediate the problem. This analytical insight allowed us to develop a novel workflow that significantly mitigates effects arising from the use of correlated noise sources. Our methodology can be used in conjunction with already existing approaches, and improves results from both correlated and uncorrelated ambient noise. Hence, we expect it to be widely applicable in ambient noise studies.

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