 
Summary: STATISTICAL INVERSION FROM REFLECTIONS OF SPHERICAL
WAVES BY A RANDOMLY LAYERED MEDIUM
M. ASCH , W. KOHLER y, G. PAPANICOLAOU z, M. POSTELx AND B. WHITE
1. Introduction. Waves re ected by randomly layered media are very noisy be
cause of the intense multiple scattering they undergo before returning to the surface.
As a result, it is di cult to extract useful information about the medium from ob
servations of re ected waves. In 1 and the references cited there we showed that if
the large scale variations of medium properties, such as density and sound speed, can
be distinguished well from their small scale uctuations due to the layering, then it is
possible to use re ection data to solve some statistical inverse problems. More speci 
cally, the statistical properties of re ected signals arising from plane or spherical wave
sound pulses incident on such a random medium can be calculated accurately when
the pulse width is intermediate between the two scales characterizing the medium
properties, with the ratio between the two scales acting as a small parameter in an
asymptotic analysis. Within the framework of this threescale theory where the scale
of macroscopic variations is much larger than the pulse width which in turn is much
larger than the scale of the random layering we formulated and solved in 1 and 2
statistical inverse problems for plane wave pulses. That is, from the re ected acous
tic pressure or velocity measured at the surface of the randomly layered medium we
recovered the large scale variations in the medium properties.
