Time-reversal acoustics can be understood in a straightforward way from
the scattering theory presented so far. First, define a complex
vector relating scattering at point with all *M* of the
acoustic sensors in a sensor array located at positions
. Then,

**H**_n^T = G_0(**y**_n,**x**_1), & ..., & G_0(**y**_n,**x**_M).
By analogy to (Hn), we define a general vector of the same form
, indexed by a general location in the model space
, such that

**H**_r^T = G_0(**r**,**x**_1), & ..., & G_0(**r**,**x**_M).
We will term a ``trial vector'' at , and
one of the set of *N* ``solution vectors.''

[Note that, for inhomogeneous media, if we use *G* instead of *G _{0}* in
(Hn) and (Hr), then we have the method known as
``matched field processing.'' Similarly, if we replace

With these definitions, the fundamental solution in the Born approximation can be rewritten for as

G(**x**_m,**x**_m') G_0(**x**_m,**x**_m') +
K(**x**_m,**x**_m'),
where the ``response matrix'' (or transfer matrix)

**K** = _n=1^N q_n**H**_n**H**_n^T.
Elements of the matrix are given by

K_m,m' = K(**x**_m,**x**_m') = _n=1^N q_n G_0(**x**_m,**y**_n)G_0(**y**_n,**x**_m')
Clearly, the response matrix is complex and symmetric.

9/18/2001