Inverse Scattering Fundamentals - Introduction to Imaging from Scattered Fields - page 43

Four
Inverse Scattering Fundamentals
4.1 CAtegoRIzAtIon oF InveRSe SCAtteRIng pRobleMS
In general, inverse scattering problems can be placed in one of the two
categories:
1. Weak scattering
2. Nonweak or strong scattering
Weak scattering occurs when the incident wave is only “scattered” once,
and this incident wave basically undergoes very little perturbation as it trav-
els through or interacts with the target. This is significant in that in most of
these cases the wave inside the target can generally be approximated as the
incident wave, which allows the problem to be linearized in order to find a
solution. This approximation is what is exploited in the approach to the solu-
tion used in the Born and Rytov approximation methods (Avish and Slaney,
1988; Lin and Fiddy, 1990). In these approaches, by linearizing the problem,
one is able to establish a Fourier relationship between the measured scattered
field data and the target or scattering function. In principle, these methods are
only supposed to work well with weak scatterers due to the dependence on
this approximation. More precisely, the Rytov approximation requires targets
whose permittivity or index varies only very slowly on the scale of the wave-
length. This concept of weak scattering will be examined, and in some sense
challenged, in this research to understand the extent to which these methods
actually work. In the case of strong scatterers, the field inside the target is scat-
tered multiple times and can incur significant perturbation that introduces
nonlinearities to the integral equation of scattering (Chew, 1995). This negates
the linear approach to this problem, making it quite difficult to solve if and
when the above-mentioned methods do not perform well. A solution for the
problem involving strong scattering is highly desired, as most targets of inter-
est in real life would fall into this category.
4.2 InveRSe SCAtteRIng In two dIMenSIonS
A common setup for a 2-D inverse scattering experiment is shown in Figure
4.1. Typically, there is some fixed number of receivers or receiver locations
set up in some configuration around the center of the target, usually equally
spaced. An incident plane wave then illuminates the target at some known
angle ϕ inc with respect to the x -axis and the scattered field is measured at an
angle of ϕ s in the far field Now, a target or scattering object represented by V ( r )
is placed in a homogeneous background, which has a permittivity of ε 0 , where
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