Pressure fields generated by acoustical pulses propagating in randomly layered media

This paper investigates the pressure field generated at the bottom of a high-contrast randomly layered slab by an acoustical pulse emitted at the surface of the slab. This analysis takes place in the framework introduced by Asch et al. [SIAM Rev., 33 (1991), pp. 519-625], where the incident pulse wave length is long compared to the correlation length of the random inhomogeneities but short compared to the size of the slab. This problem has been studied in the one-dimensional case simultaneously by Clouet and Fouque [Ann. Appl. Probab., 4 (1994), pp. 1083-1097] and Lewicki, Burridge, and Papanicolaou [Wave Motion, 20 (1994), pp. 177-195] and also for multimode plane wave pulses by Lewicki, Burridge, and De Hoop [SIAM J. Appl. Math., 56 (1996), pp. 256-276]. These situations require only the use of classical diffusion-approximation results whereas the point-source problem studied in this paper requires a nontrivial combination of diffusion-approximation results with stationary phase methods. The stationary phase method has been used by De Hoop, Chang, and Burridge [Geophys. J. Int., 104 (1991), pp. 489-506] for weakly fluctuating media and by Asch et al. for the study of the reflected pressure. The main statement of this paper is that, in order to simultaneously apply diffusion-approximation and stationary phase results, it is correct to apply them consecutively. We believe that this situation will be encountered again and again in this field and the goal of this paper is to present a clear statement in the most simple case of an acoustical pulse propagating in a randomly layered medium. In that case, the main result gives a formula describing the spreading of the pulse around its arrival time at the bottom of the slab which is obviously not contained in the classical geometrical acoustic approximation.