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.

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.

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.

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