×

Reflection of waves generated by a point source over a randomly layered medium. (English) Zbl 0735.73021

Summary: We consider a randomly layered half space adjoined to a homogeneous half space at the plane interface \(z=0\). An acoustic source in the homogeneous medium generates a time-limited pulse which is then multiply reflected and backscattered from the random medium. We compute here the time dependent statistics of the signals recorded at receivers located on the interface \(z=0\).
We do not assume that the random fluctuations in material parameters are of small magnitude, and we do not assume that the medium is statistically stationary in space. Our fundamental assumption is that of a separation of spatial scales. The random fluctuations occur on a small microscale, while their statistics, e.g., mean properties, are allowed to vary on a much larger macroscale. The pulse wavelength is assumed intermediate between the two scales, so that it can be an effective probe of the macroscopic variation only. Using asymptotic methods for stochastic differential equations, we compute power spectra and cross power spectra for the receivers. Since the backscattered signals are not stationary random functions of time, these power spectra can only be defined for segments, or time windows, during which the signals are approximately statistically stationary. We show how these statistics change with time, i.e. the location of the time window. In this paper we present an extension of previous, one-dimensional results [e.g.: R. Burridge, G. Papanicolaou and B. White, SIAM J. Appl. Math. 47, 146-168 (1987; Zbl 0625.73038), and with P. Sheng, ibid. 49, No. 2, 582-607 (1989; Zbl 0686.73027)]to the more realistic three dimensional, but stratified, case.

MSC:

74J20 Wave scattering in solid mechanics
74A40 Random materials and composite materials
35R60 PDEs with randomness, stochastic partial differential equations
60H30 Applications of stochastic analysis (to PDEs, etc.)
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Burridge, R.; Papanicolaou, G.; White, B., Statistics for pulse reflection from a randomly layered medium, Siam. J. Appl. Math., 47, 146-168 (1987) · Zbl 0625.73038
[2] (Knopps, J. R.; Lacey, A. A., Non-Classical Continuum Mechanics (1987), Cambridge Univ. Press), 3-21, An abridged version appeared in: · Zbl 0686.73027
[3] Sheng, P.; Zhang, Z.-Q.; White, B.; Papanicolaou, G., Multiple scattering noise in one dimension: universality through localization length scales, Phys. Rev. Letters, 57, 8, 1000-1003 (1986)
[4] White, B.; Sheng, P.; Zhang, Z.-Q.; Papanicolaou, G., Wave localization characteristics in the time domain, Phys. Rev. Letters, 59, 1918-1921 (1987)
[5] Asch, M.; Papanicolaou, G.; Postel, M.; Sheng, P.; White, B., Frequency content of randomly scattered signals I, Wave Motion, 12, 5, 429-450 (1990) · Zbl 0727.73019
[6] Papanicolaou, G.; Postel, M.; Sheng, P.; White, B., Frequency content of randomly scattered signals II: Inversion, Wave Motion, 12, 6, 527-549 (1990) · Zbl 0727.73020
[7] Kohler, W.; Papanicolaou, G., Power statistics for wave propagation in one dimension and comparison with transport theory, J. Math. Phys., 15, 2186-2197 (1974)
[8] Brekhovskikh, L. M., Waves in Layered Media (1960), Academic Press: Academic Press New York · Zbl 0558.73018
[9] Marcuvitz, N., Waveguide Handbook (1951), McGraw-Hill: McGraw-Hill New York
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.