Gaussian beams, complex rays, and the analytic extension of the Green's function in smoothly inhomogeneous media |
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| Authors: | Ru-Shan Wu |
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| Institution: | Earth Resources Laboratory, Department of Earth, Atmospheric and Planetary Sciences, Massachusetts Institute of Technology, Cambridge, MA 02139, USA |
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| Abstract: | Summary. Some relations between Gaussian beams, complex rays and the analytic extension of the Green's function in smoothly inhomogeneous media are shown in this paper. It is found that: (1) a single Gaussian beam is a paraxial approximation of the analytical extension of the ray-approximated Green's function in smoothly inhomogeneous media by putting the source point into a complex space. The Gaussian beam approximation of the Green's function has an advantage in computational efficiency and stability and can avoid the singularity problems at caustics, but also introduces a parabolic approximation to the wavefront and an angle-dependent amplitude damping. Therefore the validity of the Gaussian beam approximation should be checked using other methods. (2) Complex-ray tracing, which does not involve the paraxial approximation, can also avoid the singularity problemsm though without the computational efficiency. Therefore, it should be used to verify the Gaussian beam approximation, whenever possible. (3) The decomposition of a plane wave into an ensemble of Gaussian beams is equivalent to approximating the Green's function (the kernel of the ray-Kirchhoff method) with a single Gaussian beam. This introduces a parabolic approximation to the wavefront and a Gaussian windowing for arrival angles which may cause some problems in modelling wave propagation and scattering and has no advantages over other methods. (4) The representation of a point source field by a superposition of Gaussian beams, on the other hand, is equivalent to approximating the Green's function with a bundle of overlapped Gaussian beams. This representation is similar to a Maslov uniform asymptotic representation. It has no caustics and has improved accuracies at the caustics for quasi-plane waves compared to the extended WKBJ method. |
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