Asymptotics for the Space-Time Wigner Transform with Applications to Imaging

We consider the space-time Wigner transform of the solution of the random Schrödinger equation in the white noise limit and for high frequencies. We analyze in particular the strong lateral diversity limit in which the space-time Wigner transform becomes weakly deterministic. We also show how to use these asymptotic results in broadband array imaging in random media.

[1]  R. Khas'minskii A Limit Theorem for the Solutions of Differential Equations with Random Right-Hand Sides , 1966 .

[2]  Gregory Beylkin,et al.  Linearized inverse scattering problems in acoustics and elasticity , 1990 .

[3]  J. Vesecky,et al.  Wave propagation and scattering. , 1989 .

[4]  Donald A. Dawson,et al.  Measure-valued Markov processes , 1993 .

[5]  Fred D. Tappert,et al.  The parabolic approximation method , 1977 .

[6]  W. Grassman Approximation and Weak Convergence Methods for Random Processes with Applications to Stochastic Systems Theory (Harold J. Kushner) , 1986 .

[7]  F. Bailly,et al.  Parabolic and Gaussian White Noise Approximation for Wave Propagation in Random Media , 1996, SIAM J. Appl. Math..

[8]  G. Papanicolaou,et al.  Theory and applications of time reversal and interferometric imaging , 2003 .

[9]  Albert Fannjiang,et al.  White-Noise and Geometrical OpticsLimits of Wigner–Moyal Equation for Beam Waves in Turbulent Media II: Two-Frequency Formulation , 2005 .

[10]  Hongkai Zhao,et al.  Super-resolution in time-reversal acoustics. , 2002, The Journal of the Acoustical Society of America.

[11]  Mtw,et al.  Stochastic flows and stochastic differential equations , 1990 .

[12]  Guillaume Bal,et al.  SELF-AVERAGING IN TIME REVERSAL FOR THE PARABOLIC WAVE EQUATION , 2002, nlin/0205025.

[13]  Akira Ishimaru,et al.  Wave Propagation in Random Media (Scintillation) , 1993 .

[14]  G. Papanicolaou,et al.  Stability and Control of Stochastic Systems with Wide-band Noise Disturbances. I , 1978 .

[15]  G. Papanicolaou,et al.  Stability and control of stochastic systems with wide-band noise disturbances , 1977 .

[16]  Liliana Borcea,et al.  Adaptive interferometric imaging in clutter and optimal illumination , 2006 .

[17]  Jean-Pierre Fouque,et al.  La convergence en loi pour les processus à valeurs dans un espace nucléaire , 1984 .

[18]  G. Papanicolaou,et al.  Interferometric array imaging in clutter , 2005 .

[19]  George Papanicolaou,et al.  Self-Averaging from Lateral Diversity in the Itô-Schrödinger Equation , 2007, Multiscale Model. Simul..

[20]  Albert C. Fannjiang White-Noise and Geometrical Optics Limits of Wigner-Moyal Equation for Wave Beams in Turbulent Media , 2003 .

[21]  George Papanicolaou,et al.  Forward and Markov approximation: the strong-intensity-fluctuations regime revisited , 1998 .

[22]  Donald A. Dawson,et al.  A random wave process , 1984 .

[23]  G. Papanicolaou,et al.  Imaging and time reversal in random media , 2001 .

[24]  Guillaume Bal,et al.  Time Reversal and Refocusing in Random Media , 2003, SIAM J. Appl. Math..

[25]  George Papanicolaou,et al.  Statistical Stability in Time Reversal , 2004, SIAM J. Appl. Math..

[26]  Guillaume Bal On the Self-Averaging of Wave Energy in Random Media , 2004, Multiscale Model. Simul..

[27]  Uriel Frisch,et al.  WAVE PROPAGATION IN RANDOM MEDIA. , 1970 .

[28]  Etienne Pardoux,et al.  Asymptotic analysis of P.D.E.s with wide–band noise disturbances, and expansion of the moments , 1984 .

[29]  Jon F. Claerbout,et al.  DOWNWARD CONTINUATION OF MOVEOUT‐CORRECTED SEISMOGRAMS , 1972 .