Abstract
A novel computational method to evaluate the Sommerfeld integral (SI) efficiently and accurately is presented. The method rewrites the SI into two parts, applying discrete complex image method (DCIM) to evaluate the infinite integral while using double exponential quadrature rules (DE rules) for the computation of the finite part. Estimation of signal parameters via rotational invariance techniques (ESPRIT) is used to improve the accuracy and efficiency of extracting DCIM compared to the generalized pencil of function (GPOF). Due to the symmetry of the horizontal layered media, the Green function, representing the seismic fields due to a point source, can be written in the form of Sommerfeld integral in cylindrical coordinate system and be calculated by the proposed method. The performance of the method is then compared to the DE rules with weighted average partition extrapolation (WA), which shows a good agreement, with computational time reduced by about 40%.
Highlights
Green function for the horizontal layered seismic field is usually derived by reflectivity method, which was proposed by Fuchs and Müller [1] and extended to many other kinds [2], like reflection and transmission coefficient matrix method [3], discrete wavenumber method [4], discrete wavenumber finite element method [5], and generalized reflection transmission coefficient matrix method [6]
Given the advantages and disadvantages of these two methods, we propose a method by combining DE rules and discrete complex image method (DCIM) to calculate Sommerfeld integral
This paper first presents the Green function of a point source in a multilayer half-space in Section 3, explaining the mathematical manipulation required to obtain the solution as a Sommerfeld integral form in the frequency domain; it describes the principle of the proposed method with DE quadrature rules and DCIM in Section 4; it corroborates the correctness of the algorithm by the frequency responses obtained from the proposed approach with those where DE rules and weighted average partition extrapolation (WA) partition-extrapolation are used for half-space model, and the finite element method is used for three layers model
Summary
Green function for the horizontal layered seismic field is usually derived by reflectivity method, which was proposed by Fuchs and Müller [1] and extended to many other kinds [2], like reflection and transmission coefficient matrix method [3], discrete wavenumber method [4], discrete wavenumber finite element method [5], and generalized reflection transmission coefficient matrix method [6]. Discrete complex image method (DCIM), which approximates the integrand of Sommerfeld integral by a series of complex exponential functions, is commonly used for the advantages of high computational efficiency, but it needs to handle the surface wave poles contributions, which makes the computation complicated and brings singularity to the near region [8], and the calculation accuracy and effective range are difficult to be accurately estimated. This paper first presents the Green function of a point source in a multilayer half-space in Section 3, explaining the mathematical manipulation required to obtain the solution as a Sommerfeld integral form in the frequency domain; it describes the principle of the proposed method with DE quadrature rules and DCIM in Section 4; it corroborates the correctness of the algorithm by the frequency responses obtained from the proposed approach with those where DE rules and WA partition-extrapolation are used for half-space model, and the finite element method is used for three layers model
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