Scattering equations and the Born approximation
In this auxiliary material section, we just introduce the equations governing the electromagnetic
inverse scattering phenomena. Furthermore, we also give a description of the simplified model of
the electromagnetic scattering under the Born approximation (Chew, 1995; Soldovieri and
Solimene, 2010)., which is at the basis of the inversion model used in the work.
An inverse scattering problem is defined as a problem where one attempts to infer the
electromagnetic properties of a target (scatterer) starting from the scattered field measured outside
of the target.
The statement of the problem is then the following: an incident field is given, Einc , that is the
electromagnetic field generated in the background medium by known sources, in the absence of
scattering objects. By the interaction of Einc with the object(s), the scattered field ES arises; from
the knowledge of the scattered field ES some properties about the scatterer(s), either geometrical
and/or structural, have to be retrieved.
Hence, it is mandatory to introduce the mathematical equations describing the scattering
phenomena to solve the above stated problem. To this end, we refer to a two-dimensional and
scalar geometry. We consider a dielectric object having cylindrical symmetry enclosed within the
domain D , illuminated by an incident field linearly polarized along the axis of invariance. The
scattered field is observed over the domain  (not necessarily rectilinear). Moreover, we denote
by  eq (r ) and  0 the equivalent dielectric permittivity function of the unknown object and of the
known background free space medium, respectively. The magnetic permeability is assumed equal
the one of the free space  0 everywhere.
The problem, thus, amounts to retrieving the dielectric permittivity profile  eq (r ) of the unknown
object by the knowledge of the scattered field ES . It can be proven that the frequency domain
mathematical relationship between the data and the unknown is given by the following scalar
Helmoltz equation (Chew, 1995)
 2 E  k02 E  k02  (r ) E
with E  Einc  ES is the total field, k 0 is the
(A1)
subsurface (background) wave-number and
 (r )   eq (r ) /  0  1 is the dimensionless contrast function. The equivalent dielectric permittivity
 (r )
 eq (r ) of the target, is defined as  eq (r )   (r ) 
where  (r ) and  (r ) are the dielectric
j
permittivity and conductivity of the target, respectively.
By adopting the Green’s function method, eq. (.A1) leads to the following pair of scalar integral
equations (Colton and Kress, 1992):
E (r , r S ; k0 )  Einc (r , r S ; k0 )  k02  G (r , r '; k0 ) E (r ', r S ; k0 )  (r ')d r'
r D
D
ES ( r O , r S ; k 0 )  k
2
b
 G ( r O , r ; k0 ) E (r , r S ; k 0 )  ( r ) d r
rO  
(A2)
D
where G is the pertinent Green’s function, r O is the observation point and r S is the position of the
source.
In accordance to the theorem of the volumetric equivalence (Harrington, 1961), the above integral
formulation permits to interpret the scattered field as being radiated by a secondary sources (the
”polarization currents”) which are just located within the spatial region occupied by the scatterers.
The reconstruction problem thus consists of inverting the system of equations (A2) for the contrast
function. However, since in (A1) the field inside the scatterers depends upon the unknown
contrast function, the relationship between the contrast function and the scattered field is
nonlinear.
The problem can be cast as a linear one if the first line equation in (A2) is arrested at the first term
of its Neumann expansion. This way, E  Einc is assumed within the scatterer region and the socalled Born linear model is obtained (Chew, 1995). Thus, the scattering model now becomes
ES (r O , r S ; k0 )  kb2  G(r O , r; k0 ) Einc (r , r S ; k0 )  (r )d r
r O  (A3)
D
We just remark that, within the linear approximation, the internal field does not depend on the
dielectric profile which is the same to say that mutual interactions between different parts of the
object and between different objects are neglected. In other words, each part of the object is
considered as an elementary scatterer independent on the presence of other scatterers.
Now, by considering a homogeneous background medium, a monostatic data collection ( r O  r S ),
and by resorting to asymptotic arguments for the Green’s function, eq. (A3) can be rewritten as:
exp( 2 jkb | r O  r |)
 ( r )d r
| rO  r |
D
ES (r O ; kb )  kb 
rO  
(A4)
where a two-dimensional dipole with current I ( )  1 /  is assumed as source of the incident
field and a scalar factor has been omitted.
Therefore, within the framework of the linear Born approximation we have established a clear
connection between the migrated field and scatterers in terms of their contrast functions. Similar
results can be obtained for different kind of scatterers, where other linear approximations can be
adopted (Pierri et al., 2006; Solimene et al., 2008).
References
Chew, W. C. (2005). Waves and fields in inhomogeneous media. IEEE Press, New York.
Colton, D. and Kress, R. (1992). Inverse Acoustic and Electromagnetic Scattering Theory. SpringerVerlag, Berlin.
Harrington, R. F. (1961). Time-Harmonic Electromagnetic Fields. McGraw-Hill, New York.
Pierri, R.; Liseno, A.; Solimene, R. and Soldovieri, F. (2006). Beyond physical optics SVD shape
reconstruction of metallic cylinders. IEEE Trans. Anten. Propag. Vol. 54, pp. 655-665.
Soldovieri F., Solimene R., (2010): Ground Penetrating Radar Subsurface Imaging of Buried
Objects, in Radar Technology, IN-TECH, Vienna Austria, ISBN 978-953-307-029-2, Edited by: Guy
Kouemou.
Solimene, R.; Brancaccio, A.; Romano, J. and Pierri, R. (2008). Localizing thin metallic cylinders by
a 2.5 D linear distributional approach: experimental results. IEEE Trans. Anten. Propag., Vol. 56, pp.
2630-2637.