Through-Wall Imaging With UWB Radar System - KEMT FEI TUKE

2.5 Radar Imaging Methods Overview 22
where the δ (x0 − x) is a Dirac delta function. The Green’s function describes how
the wave changes during travel from point x0 to the point x.
Kirchhoff diffraction integral can be derived as follows. Let ψ be a solution to
the scalar wave equation:
∇ 2 ψ (x, t) = 1
v2 ∂2ψ (x, t)
∂2t (2.5.21)
where ψ (x, t) is a wave function, v is the wave velocity and do not has to be a
constant. When Green’s theorem from (2.5.18) is applied to the (2.5.21) with the
aid of Helmholtz equation, Hankel functions, and good mathematical skills (the
complete process can be find in [96]) the Kirchhoff’s diffraction integral can be
obtained:
�
ψ(x0, t) =
∂V
�
1
r
� �
∂ψ
−
∂n t+r/v0
1 ∂r
v0r ∂n
� �
∂ψ
+
∂t t+r/v0
1
r2 ∂r
∂n [ψ] �
dsurf
t+r/v0
(2.5.22)
where r = |x − x0|, x0 is the source position and v0 is a constant velocity. In many
cases this integral is derived for forward modeling with the result that all of the
terms are evaluated at the retarded time t − r/v0 instead of the advanced time.
This Kirchhoff diffraction integral expresses the wavefield at the observation point
xo at time t in terms of the wavefield on the boundary ∂V at the advanced time
t + r/v0. It is known from Fourier theory that knowledge of both ψ and ∂nψ is
necessary to reconstruct the wavefield at any internal point.
In order to obtain practical migration formula, two essential tasks are required.
First, the apparent need to know ∂nψ must be addressed. Second, the requirement
that the integration surface must extend all the way around the volume containing
the observation point must be dropped. There are various ways how to fulfill both
of these arguments. Schneider [128] solved the requirement of ∂nψ by using a dipole
Green’s function with an image source above the recording place, that vanished at
z = 0 and cancelled the ∂nψ term in (2.5.22). Wiggins in [143] adapted Schneider’s
technique to rough topography. Docherty in [51] showed that a monopole Green’s
can also leads to an acceptable result and once again challenged Schneider’s argument,
so the integral over the infinite hemisphere could be neglected. After all,
migration by summation along diffraction curves or by wavefront superposition
has been done for many years. Though Schneider’s derivation has been criticized,
his final expressions are considered correct.
As a first step in adapting (2.5.22) it is usually considered as appropriate to
discard the term 1 1
r2 ∂r
∂n [ψ(x)] . This is called the near-field term and decays
t+r/v0
more strongly with r than the other two terms. Then, the surface S = ∂V will be
taken as the z = 0 plane, S0, plus the surface infinitesimally below the reflector, Sz,
and finally these surfaces will be joined at infinity by vertical cylindrical walls, S∞,