Our analysis assumes that the
array has *N* transducers located at spatial positions
, for . (See Figure 1.)
When used in active mode, the array
probes the unknown acoustic medium containing *M*
small scatterers by emitting pulses and recording the time traces of
the back-scattered echos. We call the resulting data set the
multistatic array response (or transfer) matrix

(1) |

Figure 1

For the numerical examples considered here, we will treat ultrasonic
imaging problems.
Our simulations assume that ,where is the central wavelength, is a characteristic length
scale of the inhomogeneity (like a correlation
length), *a* is the array aperture, and *L*
is the approximate distance to the targets from the array.
This is the regime where multipathing, or multiple scattering, is
significant even when the standard deviation of sound speed fluctuations
is only a few percent.
Values used in the codes are mm, *a* = 2.5mm,
and a background wave speed of *c _{0}* = 1.5km/s. More details
concerning the simulations may be found in Borcea

Typical array processing methods assume that the targets are far away from the array and, therefore, they look like points. Similarly, the propagation medium is assumed homogeneous and so the observed wavefronts scattered by the targets look like plane waves at the array. Array noise has usually been treated as due either to diffuse sources of white noise coming simultaneously from all directions, or to isolated ``noise'' having the same types of source characteristics as the targets of interest. But in random media with significant multiple scattering, the resulting ``noise'' cannot be successfully treated in these traditional ways.

Real-space time-reversal processing of the array response data involves
an iterative procedure: sending a signal, recording and storing the scattered
return signal, time-reversing and then rebroadcasting the stored
signal, with subsequent repetitions. This procedure amounts to using
the power method for finding the singular vector of the data matrix
having the largest singular value. Alternatively,
when the full response/transfer matrix has
been measured for a multistatic active array, the resulting data
matrix can be analyzed directly by Singular
Value Decomposition (SVD) to determine not only the singular
vector having the largest singular value, but all singular vectors and
singular values -- simultaneously (Prada and Fink, 1994; Prada *et al.*, 1996; Mordant *et al.*, 1999).

Imaging is always done using a fictitious medium
for the simulated backpropagation that produces these images since
the real medium is not known.
Its large-scale features
could be estimated from other information, such as geological data
obtained by seismic methods. For example, migration methods
(Claerbout, 1976; Aki and Richards, 1980; Bleistein *et al.*, 2001)
can be used, where very large
arrays -- much larger than those we contemplate using here -- are required.
However, the small-scale random inhomogeneities are not known and
cannot be effectively estimated,
so the simplest thing to do is ignore them when imaging, and use
methods that are statistically stable and therefore
insensitive to the exact character of these small inhomogeneities.

6/7/2002