We assume that the problems of interest are well-approximated by the inhomogeneous Helmholtz equation

[^2 + k_0^2n^2(**x**)]u(**x**) = s(**x**),
where is the wave amplitude,
is a localized source function,
is the wavenumber of the
homogeneous background, with being angular frequency,
*f* frequency,
*c _{0}* the assumed homogeneous background wave speed, and
wavelength.
The acoustic index of refraction is

n(**x**) = c_0c(**x**),
where is the wave speed at spatial location .Thus, in the background and measures the change in the wave speed at the scatterers.

Pertinent fundamental solutions for this problem satisfy:

[^2 + k_0^2]G_0(**x**,**x**') =
-(**x** - **x**')
and

[^2 + k_0^2n^2(**x**)]G(**x**,**x**') =
-(**x** - **x**'),
for the homogeneous and inhomogeneous media, respectively.
In both cases, we assume the radiation (out-going) boundary condition
at infinity.

The well-known solution of (g0) for the homogeneous medium in 3D is

G_0(**x**,**x**') = e^ik_0**x** - **x**'
4**x** - **x**'.
The fundamental solution of (g) for the inhomogeneous medium can be written
in terms of that for the homogeneous one (in the usual way) as

G(**x**,**x**') = G_0(**x**,**x**') +
k_0^2a(**y**)G_0(**x**,**y**')G(**y**,**x**')d^3y.
Note that the right hand side depends on the values of the *G*,
which is to be determined by the same equation. So this is an implicit
integral equation that must be solved for .The regions of nonzero are assumed to be finite in number
(*N*), in compact domains , all of which are small compared
to the wavelength . Then, there will be some position
(certainly for convex domains) inside each domain ,characterizing the location of each of the *N* scatterers.
We also call these scatterers ``targets,'' since it is their
locations that we seek.

With these assumptions, it is a good approximation to set the
arguments of *G _{0}* and

q_n k_0^2__n a(**y**)d^3y,
which then characterizes the strength of the scattering from the *n*th
target domain.

With these definitions, we finally have

G(**x**,**x**') G_0(**x**,**x**') +
_n=1^N q_n G_0(**x**,**y**_n)G(**y**_n,**x**').
Furthermore, if the scatterers are sufficiently far apart and the
scattering strengths *q*_{n} are not too large, then on the far right can be replaced by , giving
the explicit formula

G(**x**,**x**') G_0(**x**,**x**') +
_n=1^N q_n G_0(**x**,**y**_n)G_0(**y**_n,**x**').
Equation (Born) is the Born approximation to
for small scatterers.

9/18/2001