scholarly journals Dynamic Green's functions for layered half space by using 3 dimensional Cartesian coordinates / twofold Fourier transform and their verification

2012 ◽  
Vol 68 (2) ◽  
pp. I_805-I_812
Author(s):  
Takanori Harada ◽  
Ryosuke Matsuda ◽  
Kazuhiro Yoshida ◽  
Yushi Awada
Keyword(s):  
Fourier Transform ◽  
Half Space ◽  
Green's Functions ◽  
Green’S Functions ◽  
Cartesian Coordinates ◽  
3 Dimensional ◽  
Layered Half Space

An efficient method for computing Green's functions for a layered half-space with sources and receivers at close depths

1994 ◽  
Vol 84 (5) ◽  
pp. 1456-1472 ◽  
Author(s):  
Yoshiaki Hisada
Keyword(s):  
Half Space ◽  
Green's Functions ◽  
Green’S Functions ◽  
Point Sources ◽  
Asymptotic Solutions ◽  
Reflection And Transmission ◽  
Reflection And Transmission Coefficient ◽  
Dipole Sources ◽  
Coefficient Method ◽  
Layered Half Space

Abstract We propose an analytical method to compute efficiently the displacement and stress of static and dynamic Green's functions for viscoelastic layered half-spaces. When source and receiver depths are close, it is difficult to compute Green's functions of the layered half-space, because their integrands, whose variable of integration is the horizontal wavenumber, oscillate with only slowly decreasing amplitude. In particular, when the depths are equal, it is extremely difficult to compute the stress Green's functions, because their integrands oscillate with increasing amplitude. To remedy this problem, we first derive the asymptotic solutions, which converge to the integrands of Green's functions with increasing wavenumber. For this purpose, we modify the generalized R/T (reflection and transmission) coefficient method (Luco and Apsel; 1983) to be completely free from growing exponential terms, which are the obstacles to finding the asymptotic solutions. By subtracting the asymptotic solutions from the integrands of the corresponding Green's functions, we obtain integrands that converge rapidly to zero. We can, therefore, significantly reduce the range of wavenumber integration. Since the asymptotic solutions are expressed by the products of Bessel functions and simple exponential functions, they are analytically integrable. Finally, we obtain accurate Green's functions by adding together numerical and analytical integrations. We first show this asymptotic technique for Green's functions due to point sources, and extend it to Green's functions due to dipole sources. Finally, we demonstrate the validity and efficiency of our method for various cases.

An efficient method for computing Green's functions for a layered half-space with sources and receivers at close depths (part 2)

1995 ◽  
Vol 85 (4) ◽  
pp. 1080-1093
Author(s):  
Yoshiaki Hisada
Keyword(s):  
Free Surface ◽  
Half Space ◽  
Efficient Method ◽  
Green's Functions ◽  
Green’S Functions ◽  
New Method ◽  
Direct Wave ◽  
Transmitted Waves ◽  
Dipole Sources ◽  
Layered Half Space

Abstract In this study, we improve Hisada's (1994) method to efficiently compute Green's functions for viscoelastic layered half-spaces with sources and receivers located at equal or nearly equal depths. Compared with Hisada (1994), we can significantly reduce the range of wavenumber integration especially for the case that sources and receivers are close to the free surface or to boundaries of the source layer. This can be done by deriving analytical asymptotic solutions for both the direct wave and the reflected/transmitted waves from the boundaries, which are neglected in Hisada (1994). We demonstrate the validity and efficiency of our new method for several cases. The FORTRAN codes for this method for both point and dipole sources are open to academic use through anonymous FTP.

On the Green's functions for a layered half-space. Part II

1983 ◽  
Vol 73 (4) ◽  
pp. 931-951
Author(s):  
Randy J. Apsel ◽  
J. Enrique Luco
Keyword(s):  
Integral Representation ◽  
Half Space ◽  
Numerical Evaluation ◽  
Green's Functions ◽  
Green’S Functions ◽  
Numerical Procedure ◽  
Number Of Solutions ◽  
Viscoelastic Media ◽  
Layered Half Space

abstract A numerical procedure to obtain the dynamic Green's functions for layered viscoelastic media is presented. The procedure is based on numerical evaluation of certain Hankel-type integrals which appear in an integral representation derived previously by the authors. Comparisons illustrating the accuracy and flexibility of the approach are made with a number of solutions obtained by other methods.

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