IEEE Geoscience and Remote Sensing Magazine - December 2021 - 177

IMAGE FORMATION
Once the scattering data have been collected or generated
by solving the forward problem, a postprocessing procedure
is needed to produce an image of the illuminated
scene [12], [40]. In most cases, when the detection of concealed
targets is of interest, the image is 2D and formed
over a horizontal plane within an area ahead of the moving
FL system [see Figures 2(a) and (b)]. Although the height of
the 2D image can be arbitrarily chosen, the capability of
penetrating lossy soils at microwave frequencies is on the
order of 2-10 cm, which is comparable to the achievable
resolution. Therefore, varying the height by a few centimeters
will not significantly affect the image quality and
target-detection capability.
Several image formation approaches have been proposed
in the literature, with a majority being simple adaptations of
conventional algorithms used for focusing SAR data. More
involved strategies based on electromagnetic formulations
of the problem have been presented to account for the presence
of the dielectric interface and near-field conditions
arising due to shorter distances between the antennas and
imaging region of interest. In the following sections, we give
an overview of these methods.
MIGRATION
Among the most well-known image formation algorithms,
migration has been broadly used to focus GPR data. Migration
is a family of imaging techniques that originate from
the seismic literature [41], [42]. Over the years, this class of
algorithms has been adapted within radar imaging frameworks,
including SAR and GPR (see, e.g., [43] and [44]).
From a practical viewpoint, the algorithm essentially operates
on the scattered field at the receiver to compensate for
the different delays encountered by the signal generated by
point-like scatterers, which are illuminated within a certain
time interval during the movement of the system (the FLGPR
platform in this case).
In radar imaging, the migration algorithm is sometimes
assimilated to beamforming approaches since both essentially
compensate for the hyperbolic patterns representing
raw data in a time-range scattering diagram. Image reconstruction
in terms of migrated data can be achieved by numerically
implementing a double integral function of time
and range, which includes the scattered field and migration
operator (see, e.g., [45] and [46]). A number of contributions
on the application of migration and beamforming
algorithms to GPR data have been proposed. A comprehensive
review of these approaches can be found in [17].
MICROWAVE TOMOGRAPHY
GPR imaging methods based on an electromagnetic formulation
constitute a so-called inverse problem [12], [47].
Mathematically, a solution to the direct problem exists that
is unique and has a continuous dependence on the data
(see, e.g., [40] and [48]). The problem becomes ill posed
when the uniqueness of the solution and/or continuity of
DECEMBER 2021 IEEE GEOSCIENCE AND REMOTE SENSING MAGAZINE
its data dependence do not hold. The latter implies that
even a small error in the scattered field (e.g., the presence
of additive thermal noise) can cause a considerable error
in the reconstruction of the background dielectric characteristics.
In practice, a regularization is applied to solve
the inverse problem [48]. The
main objective of the numerous
methods that exist to regularize
the inverse problem is
renouncing an ideal solution
and looking for suitable robustness
in the results.
Imaging procedures based
on a linear solution of the
scattering equation have been
shown to be simple and particularly
suitable for the processing
of GPR data [4], [49]-[51],
including FL configurations
[23], [33]. These procedures are
mainly based on the Born approximation
(BA) (see e.g., [40] and [49] and the references
therein), which essentially approximates the internal field
of a dielectric object with the incident field; the latter being
a known term.
By suitably defining the Green's function of the problem
[12], the electromagnetic formulation of the scattering field
based on a linear solution allows near-field consideration
consistent with the nature of the illumination. We can also
describe in the formulation a flat air-soil interface by defining
a Green's function for multilayered media.
Methods based on the inversion of the linear scattering
equation are often referred to as microwave tomography approaches
[49], which essentially consist of retrieving the unknown
profile of a dielectric object, i.e., the contrast function,
from the knowledge of the scattered field collected at
the receiving antenna. The contrast function is defined as
the relative difference between the (complex) permittivity
of the target and that of the reference propagation scenario
(free space, in the case at hand).
By modeling the transmitting antennas as vertically oriented
Hertzian dipoles and measuring only the VV-polarization
scattered field from the investigation domain D, the
linear relationship under BA for shallow-buried targets can
be expressed as [12], [23]
Ejk2
sr
^
rz Gr r
Gr rz rr
,,,
,,
~~n~
~|
h =- b
6
$$ 0@ ()d ,
^
00 $ ##D
h
t
^
r
h
(1)
where Es is the VV-polarized scattered field corresponding
to angular frequency ~ collected at point r ,r | is the unknown
scene reflectivity, G is the free-space dyadic Green's
function, kk
= f
=~nf
br 0 is the wavenumber in the medium,
and k00 0 is the free-space wavenumber. The vectors
rr and rt represent the positions of the receive and
transmit antennas, respectively; r denotes a generic point
177
ONCE THE SCATTERING
DATA HAVE BEEN
COLLECTED OR GENERATED
BY SOLVING THE FORWARD
PROBLEM, A
POSTPROCESSING
PROCEDURE IS NEEDED TO
PRODUCE AN IMAGE OF THE
ILLUMINATED SCENE.
IEEE Geoscience and Remote Sensing Magazine - December 2021

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