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.

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.

Source locations of microseisms in the North Atlantic from Matched Field Processing using full Green's Functions.

2021 ◽  
Author(s):  
Sven Schippkus ◽  
Céline Hadziioannou
Keyword(s):  
Wave Propagation ◽  
Travel Time ◽  
North Atlantic ◽  
Seismic Noise ◽  
Green's Functions ◽  
Green’S Functions ◽  
Regional Scale ◽  
Matched Field Processing ◽  
The North ◽  
The North Atlantic

<p>Precise knowledge of the sources of seismic noise is fundamental to our understanding of the ambient seismic field and its generation mechanisms. Two approaches to locating such sources exist currently. One is based on minimizing the misfit between estimated Green's functions from cross-correlation of seismic noise and synthetically computed correlation functions. This approach is computationally expensive and not yet widely adopted. The other, more common approach is Beamforming, where a beam is computed by shifting waveforms in time corresponding to the slowness of a potentially arriving wave front. Beamforming allows fast computations, but is limited to the plane-wave assumption and sources outside of the array.</p><p>Matched Field Processing (MFP) is Beamforming in the spatial domain. By probing potential source locations directly, it allows for arbitrary wave propagation in the medium as well as sources inside of arrays. MFP has been successfully applied at local scale using a constant velocity for travel-time estimation, sufficient at that scale. At regional scale, travel times can be estimated from phase velocity maps, which are not yet available globally at microseism frequencies.</p><p>To expand MFP’s applicability to new regions and larger scales, we replace the replica vectors that contain only travel-time information with full synthetic Green's functions. This allows to capture the full complexity of wave propagation by including relative amplitude information between receivers and multiple phases. We apply the method to continuous recordings of stations surrounding the North Atlantic and locate seismic sources in the primary and secondary microseism band, using pre-computed databases of Green's functions for computational efficiency. The framework we introduce here can easily be adapted to a laterally homogeneous Earth once such Green’s function databases become available, hopefully in the near future.</p>

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.

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.

WEIGHTING FACTORS IN THE CONSTRUCTION AND RECONSTRUCTION OF ACOUSTICAL WAVE FIELDS

Geophysics ◽  
1977 ◽  
Vol 42 (6) ◽  
pp. 1183-1198 ◽  
Author(s):  
Milos J. Kuhn ◽  
Khalid A. Alhilali
Keyword(s):  
Free Space ◽  
Green's Functions ◽  
Green’S Functions ◽  
Diffraction Theory ◽  
Weighting Factor ◽  
Total Response ◽  
Wave Fields ◽  
Seismic Migration ◽  
Weighting Factors ◽  
The Fourier Transform

The numerical solution of the Helmholtz equation is examined for a separated source and receiver over a model having a single planar interface. Expressions describing the construction and reconstruction of acoustical wave fields are derived in terms of Green’s functions. Their relation to the Fourier transform is briefly discussed. Three simple Green’s functions—free space, free surface, and rigid surface—are used to test the relative accuracy of the respective weighting factors by comparing the numerically calculated field for a simple model to a field obtained analytically by application of rigorous diffraction theory. The main purpose of this paper is to study the behavior of the total response (amplitude and phase) for models in which the aperture is not sufficiently sampled (e.g., close to half the wavelength). The degree of distortion in the response due to spatial undersampling is unacceptable for all three Green’s functions. A modified weighting factor relative to the free‐space Green’s function is introduced, which effectively reduces the degree of distortion in the total response under the same sampling condition. The importance of this finding to exploration geophysics in the construction of the synthetic seismograms by application of the Huygen’s principle and in seismic migration will be demonstrated.

Anisotropic finite-difference algorithm for modeling elastic wave propagation in fractured coalbeds

Geophysics ◽  
2012 ◽  
Vol 77 (1) ◽  
pp. C13-C26 ◽  
Author(s):  
Zhenglin Pei ◽  
Li-Yun Fu ◽  
Weijia Sun ◽  
Tao Jiang ◽  
Binzhong Zhou
Keyword(s):  
Wave Propagation ◽  
Finite Difference ◽  
Elastic Wave ◽  
Taylor Series Expansion ◽  
Real Data ◽  
Elastic Wave Propagation ◽  
Numerical Dispersion ◽  
Staggered Grid ◽  
S Wave ◽  
Data Set

The simulation of wave propagations in coalbeds is challenged by two major issues: (1) strong anisotropy resulting from high-density cracks/fractures in coalbeds and (2) numerical dispersion resulting from high-frequency content (the dominant frequency can be higher than 100 Hz). We present a staggered-grid high-order finite-difference (FD) method with arbitrary even-order ([Formula: see text]) accuracy to overcome the two difficulties stated above. First, we derive the formulae based on the standard Taylor series expansion but given in a neat and explicit form. We also provide an alternative way to calculate the FD coefficients. The detailed implementations are shown and the stability condition for anisotropic FD modeling is examined by the eigenvalue analysis method. Then, we apply the staggered-grid FD method to 2D and 3D coalbed models with dry and water-saturated fractures to study the characteristics of the 2D/3C elastic wave propagation in anisotropic media. Several factors, like density and direction of vertical cracks, are investigated. Several phenomena, like S-wave splitting and waveguides, are observed and are consistent with those observed in a real data set. Numerical results show that our formulae can correlate the amplitude and traveltime anisotropies with the coal seam fractures.

Frequency-domain Green’s functions for radar waves in heterogeneous 2.5D media

Geophysics ◽  
2009 ◽  
Vol 74 (3) ◽  
pp. J13-J22 ◽  
Author(s):  
Karl J. Ellefsen ◽  
Delphine Croizé ◽  
Aldo T. Mazzella ◽  
Jason R. McKenna
Keyword(s):  
Wave Propagation ◽  
Finite Difference ◽  
Frequency Domain ◽  
Hybrid Method ◽  
Green's Functions ◽  
Green’S Functions ◽  
Homogeneous Model ◽  
Electromagnetic Properties ◽  
Perfectly Matched Layers ◽  
Phase Errors

Green’s functions for radar waves propagating in heterogeneous 2.5D media might be calculated in the frequency domain using a hybrid method. The model is defined in the Cartesian coordinate system, and its electromagnetic properties might vary in the [Formula: see text]- and [Formula: see text]-directions, but not in the [Formula: see text]-direction. Wave propagation in the [Formula: see text]- and [Formula: see text]-directions is simulated with the finite-difference method, and wave propagation in the [Formula: see text]-direction is simulated with an analytic function. The absorbing boundaries on the finite-difference grid are perfectly matched layers that have been modified to make them compatible with the hybrid method. The accuracy of these numerical Green’s functions is assessed by comparing them with independently calculated Green’s functions. For a homogeneous model, the magnitude errors range from [Formula: see text] through 0.44%, and the phase errors range from [Formula: see text] through 4.86%. For a layered model, the magnitude errors range from [Formula: see text] through 2.06%, and the phase errors range from [Formula: see text] through 2.73%. These numerical Green’s functions might be used for forward modeling and full waveform inversion.

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