Numerical evaluation of Green’s functions for axisymmetric boreholes using leaking modes

Geophysics ◽  
1987 ◽  
Vol 52 (8) ◽  
pp. 1099-1105 ◽  
Author(s):  
R. A. W. Haddon
Keyword(s):  
Wave Propagation ◽  
Fourier Transform ◽  
Elastic Wave ◽  
Numerical Evaluation ◽  
Green's Functions ◽  
Green’S Functions ◽  
Complete Solution ◽  
Body Waves ◽  
Complex Frequency ◽  
Residue Theory

By choosing appropriate paths of integration in both the complex frequency (ω) and complex wavenumber (k) planes, exact Green’s functions for elastic wave propagation in axisymmetric boreholes are expressed completely as sums of modes. The integrations with respect to k are performed exactly using Cauchy residue theory. The remaining integrations with respect to ω are then carried out using the fast Fourier transform (FFT). The complete solution, including all possible body waves, is expressed simply as a superposition of modes without any contributions from branch line integrals. There are no spurious arrivals and, provided that the number of points in the FFT can be taken sufficiently large, no restrictions on distance. The method is fast, accurate, and easy to apply.

Exact Green’s functions using leaking modes for axisymmetric boreholes in solid elastic media

Geophysics ◽  
1989 ◽  
Vol 54 (5) ◽  
pp. 609-620 ◽  
Author(s):  
R. A. W. Haddon
Keyword(s):  
Wave Propagation ◽  
Fourier Transform ◽  
Elastic Wave ◽  
Fast Fourier Transform ◽  
Numerical Method ◽  
Green's Functions ◽  
Green’S Functions ◽  
Elastic Media ◽  
Complex Frequency ◽  
Residue Theory

By choosing appropriate paths of integration in both the complex frequency ω and complex wavenumber k planes, exact Green’s functions for elastic wave propagation in axisymmetric fluid‐filled boreholes in solid elastic media are expressed completely as sums of modes. There are no contributions from branch line integrals. The integrations with respect to k are performed exactly using Cauchy residue theory. The remaining integrations with respect to ω are then carried out partly by using the fast Fourier transform (FFT) and partly by using another numerical method. Provided that the number of points in the FFT can be taken sufficiently large, there are no restrictions on distance. The method is fast, accurate, and easy to apply.

scholarly journals Matched Field Processing for complex Earth structure

2021 ◽  
Author(s):  
Sven Schippkus ◽  
Celine Hadziioannou
Keyword(s):  
Wave Propagation ◽  
Elastic Wave ◽  
Green's Functions ◽  
Green’S Functions ◽  
Real Data ◽  
Earth Structure ◽  
Wave Fields ◽  
Matched Field Processing ◽  
Source Localisation ◽  
Northeastern Atlantic

Matched Field Processing (MFP) is a technique to locate the source of a recorded wave field. It is the generalization of beamforming, allowing for curved wavefronts. In the standard approach to MFP, simple analytical Green's functions are used as synthetic wave fields that the recorded wave fields are matched against. We introduce an advancement of MFP by utilizing Green's functions computed numerically for real Earth structure as synthetic wave fields. This allows in principle to incorporate the full complexity of elastic wave propagation, and through that provide more precise estimates of the recorded wave field's origin. This approach also further emphasizes the deep connection between MFP and the recently introduced interferometry-based source localisation strategy for the ambient seismic field. We explore this connection further by demonstrating that both approaches are based on the same idea: both are measuring the (mis-)match of correlation wave fields. To demonstrate the applicability and potential of our approach, we present two real data examples, one for an earthquake in Southern California, and one for secondary microseism activity in the Northeastern Atlantic and Mediterranean Sea. Tutorial code is provided to make MFP more approachable for the broader seismological community.

scholarly journals Asymptotic analysis for dispersion relations and travel times in noise cross-correlations: spherically symmetric case

2018 ◽  
Vol 474 (2218) ◽  
pp. 20180382
Author(s):  
Tianshi Liu ◽  
Haiming Zhang
Keyword(s):  
Asymptotic Analysis ◽  
Surface Waves ◽  
Green's Functions ◽  
Green’S Functions ◽  
Dispersion Relations ◽  
Body Waves ◽  
Spherically Symmetric ◽  
Travel Times ◽  
Cross Correlations ◽  
The Cross

The cross-correlations of ambient noise or earthquake codas are massively used in seismic tomography to measure the dispersion curves of surface waves and the travel times of body waves. Such measurements are based on the assumption that these kinematic parameters in the cross-correlations of noise coincide with those in Green's functions. However, the relation between the cross-correlations of noise and Green's functions deserves to be studied more precisely. In this paper, we use the asymptotic analysis to study the dispersion relations of surface waves and the travel times of body waves, and come to the conclusion that for the spherically symmetric Earth model, when the distribution of noise sources is laterally uniform, the dispersion relations of surface waves and the travel times of SH body-wave phases in noise correlations should be exactly the same as those in Green's functions.

2.5D AND 3D GREEN'S FUNCTIONS FOR ACOUSTIC WEDGES: IMAGE SOURCE TECHNIQUE VERSUS A NORMAL MODE APPROACH

2013 ◽  
Vol 21 (01) ◽  
pp. 1250025 ◽  
Author(s):  
A. TADEU ◽  
E. G. A. COSTA ◽  
J. ANTÓNIO ◽  
P. AMADO-MENDES
Keyword(s):  
Wave Propagation ◽  
Normal Mode ◽  
Green's Functions ◽  
Green’S Functions ◽  
Three Dimensional ◽  
Element Formulation ◽  
Two Dimensional ◽  
Image Source ◽  
Point Pressure ◽  
Fluid Channel

2.5D and 3D Green's functions are implemented to simulate wave propagation in the vicinity of two-dimensional wedges. All Green's functions are defined by the image-source technique, which does not account directly for the acoustic penetration of the wedge surfaces. The performance of these Green's functions is compared with solutions based on a normal mode model, which are found not to converge easily for receivers whose distance to the apex is similar to the distance from the source to the apex. The applicability of the image source Green's functions is then demonstrated by means of computational examples for three-dimensional wave propagation. For this purpose, a boundary element formulation in the frequency domain is developed to simulate the wave field produced by a 3D point pressure source inside a two-dimensional fluid channel. The propagating domain may couple different dipping wedges and flat horizontal layers. The full discretization of the boundary surfaces of the channel is avoided since 2.5D Green's functions are used. The BEM is used to couple the different subdomains, discretizing only the vertical interfaces between them.

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