Fractional White-Noise Limit and Paraxial Approximation for Waves in Random Media
- 105 Downloads
This work is devoted to the asymptotic analysis of high frequency wave propagation in random media with long-range dependence. We are interested in two asymptotic regimes, that we investigate simultaneously: the paraxial approximation, where the wave is collimated and propagates along a privileged direction of propagation, and the white-noise limit, where random fluctuations in the background are well approximated in a statistical sense by a fractional white noise. The fractional nature of the fluctuations is reminiscent of the long-range correlations in the underlying random medium. A typical physical setting is laser beam propagation in turbulent atmosphere. Starting from the high frequency wave equation with fast non-Gaussian random oscillations in the velocity field, we derive the fractional Itô–Schrödinger equation, that is, a Schrödinger equation with potential equal to a fractional white noise. The proof involves a fine analysis of the backscattering and of the coupling between the propagating and evanescent modes. Because of the long-range dependence, classical diffusion-approximation theorems for equations with random coefficients do not apply, and we therefore use moment techniques to study the convergence.
Unable to display preview. Download preview PDF.
- 4.Bal, G., Pinaud, O.: Imaging using transport models for wave-wave correlations. M3AS 21(5), 1071–1093, 2011Google Scholar
- 7.Borcea, L., Papanicolaou, G., Tsogka, C.: Interferometric array imaging in clutter. Inverse Probl. 21, 1419–1460, 2005Google Scholar
- 9.Çınlar, E.: Probability and Stochastics, Graduate Texts in Mathematics 261. Springer, New York (2011)Google Scholar
- 10.Claerbout, J.F.: Imaging the Earth's Interior. Blackwell Science, Palo Alto (1985)Google Scholar
- 16.Garcia, A., Rademich, E., Rumsey, H.: A real variable lemma and the continuity of paths of some Gaussian processes. Indiana Univ. Math. J. 20, 565–578, 1970/1971Google Scholar
- 21.C. Gomez.: Wave decoherence for the random Schrödinger equation with long-range correlations. Commun. Math. Phys. 320, 37–71, 2013Google Scholar
- 30.Reed, M., Simon, B.: Methods of Modern Mathematical Physics. IV. Analysis of Operators, 2nd edn. Academic Press, New York, 1980.Google Scholar
- 35.Tappert, F.D.: The parabolic approximation method in wave propagation and underwater acoustics. Lecture Notes in Physics 70, pp. 224–287. Springer, Berlin, 1977Google Scholar