A Derivation of the Relation between Spatial Correlation and Green’s Function Based on the Slowness Method: Isotropic Incidence of Random Scalar Waves

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  • NAKAHARA Hisashi
    Solid Earth Physics Laboratory, Department of Geophysics, Graduate School of Science, Tohoku University

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  • 波動場の空間相関とグリーン関数との関係のスローネス法に基づく導出: 等方入射するランダムスカラー波の場合
  • ハドウバ ノ クウカン ソウカン ト グリーン カンスウ トノ カンケイ ノ スローネスホウ ニ モトズク ドウシュツ トウホウ ニュウシャスル ランダム スカラーハ ノ バアイ
  • A Derivation of the Relation between Spatial Correlation and Green^|^rsquo;s Function Based on the Slowness Method: Isotropic Incidence of Random Scalar Waves

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Abstract

It has been demonstrated theoretically and experimentally that Green’s function for a source and a receiver can be retrieved from the spatial correlation of a wave field between the two points. Recently, it was pointed out that the normalized cross spectrum of a random wave field can be connected to the time-domain Green’s function via the inverse Fourier transform: Hilbert transform and differentiation have to be operated on the normalized spatial correlation to retrieve Green’s function for 2D and 3D, respectively. In these derivations, a multiple integration with respect to frequency and slowness appears in the estimation of the normalized spatial correlation of the wave field. So far, the spectral method is usually used, which performs the frequency integration after the slowness integration. However, adopting the slowness method, which reverses the order of the integration, in this study, I have succeeded in deriving the same results as were derived by the spectral method. The derivation using the slowness method seems more advantageous because the stationary points, which make principal contribution to the integration, can be graphically understood.

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