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The spectrum of backscatter for pulse reflection from a randomly stratified medium. (English) Zbl 0622.73010

Ramdom media, IMA Vol. Math. Appl. 7, 29-43 (1987).
[For the entire collection see Zbl 0619.00026.]
We consider the problem of a plane wave pulse normally incident on a plane stratified acoustic or elastic half space with material parameters which are stationary random functions of position. For the elastic case we study SH waves, so that the problem is completely one dimensional. It is assumed that the wave length is large compared to the typical size of an inhomogeneity, and that sufficient time has elapsed for the pulse to have travelled through many inhomogeneities. More specifically, let \(\epsilon\) be the ratio of the size of a typical inhomogeneity to a wavelength characteristic of the incident pulse. It is assumed that \(\epsilon <<1\). We consider a subsection, or ”time window”, of the backscattered signal of width \(O(1/\epsilon)\), the order of magnitude of the pulse duration, but centered at a large time \(r/\epsilon^ 2\), where r is O(1). Then this section of the backscattered process is approximately stationary and Gaussian, with power spectral density \(S_ r(\omega)\) given by \(S_ r(\omega)=| \hat f(\omega)|^ 2(1/r)\mu \sqrt{\alpha r}\omega)\), where \(\hat f\) is the Fourier transform of the pulse shape, \(\alpha\) is a constant which may be computed from the statistics of the random medium, and \(\mu\) is a universal function, which we characterize.

MSC:

74A40 Random materials and composite materials
74J10 Bulk waves in solid mechanics
60G35 Signal detection and filtering (aspects of stochastic processes)
60H15 Stochastic partial differential equations (aspects of stochastic analysis)