Three-dimensional Green’s function for an anisotropic multi-layered half-space

A new numerical approach is presented to calculate the Green’s function for an anisotropic multi-layered half space. The formulation is explicit and unconditionally stable. It imposes no limit to the thickness of the layered medium and the magnitude of the frequency. In the analysis, the Fourier transform and the precise integration method (PIM) are employed. Here, the Fourier transform is employed to transform the wave motion equation from the spatial domain to the wavenumber domain. A second order ordinary differential equation (ODE) is observed. Then, the dual vector representation of the wave motion equation is used to reduce the second order ODE to first order. It is solved by the PIM. Finally, the Green’s function in the wavenumber domain is obtained. For the evaluation of the Green’s function in the spatial domain, the double inverse Fourier transform over the wavenumber is employed to derive the solutions. Especially, for the transversely isotropic medium, the double inverse Fourier transform can be further reduced to a single integral by the cylindrical polar coordinate transform. Numerical examples are provided. Comparisons with other methods are done. Very promising results are obtained.