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

2.5 Radar Imaging Methods Overview 24
where θ is the vertical angle between the receiver location and the ray to the
scatterpoint. With this, the final formula for the scattered wavefield just above
the reflector is:
� � �
2 cos θ ∂ψ
ψ(x0, t) =
dsurf. (2.5.27)
vr ∂t t+r/v
S0
The fact that (2.5.27) is a form of direct wavefield extrapolation, but not recursive
is the reason why it is not yet a migration equation. A migration equation has to
estimate reflectivity, not just the scattered wavefield, and for this purpose a model
relating the wavefield to the reflectivity is required. The most simple model is
the exploding reflector model [93] which asserts that the reflectivity is identical to
the downward continued scattered wavefield at t = 0 provided that the downward
continuation is done with v = v/2. Thus, a wave migration equation follows
immediately from equation (2.5.27) as:
� � �
�
2 cos θ ∂ψ
ψ(x0, 0) =
dsurf =
vr ∂t
S0
r/v
S0
4 cos θ
vr
� �
∂ψ
dsurf. (2.5.28)
∂t 2r/v
Finally, (2.5.28) is the Kirchhoff migration equation. This result was derived by
many authors including Schneider [128] and Scales [124]. It expresses migration
by summation along hyperbolic travel paths through the input data space. The
hyperbolic summation does not have to be seen at the first view, but it can be
indicated by [∂tψ] 2r/v , notation means that partial derivation is to be evaluated
at the time 2r/v. ∂tψ (x, t) is integrated over the z = 0 plane, only those specific
traveltimes values are selected by:
t = 2r
�
2 (x − x0)
= 2 + (y − y0) 2 + (z − z0) 2
(2.5.29)
v
which is the equation of a zero-offset diffraction hyperbola.
In addition to diffraction summation, (2.5.28) requires that the data be scaled
by 4cosθ/(vr) and that the time derivative be taken before summation. These additional
details were not indicated by the simple geometric theory in Section 2.5.2
and are major benefits of Kirchhoff theory. The same correction procedures are
contained implicitly in f-k migration that will be described in Section 2.5.6. Kirchhoff
migration is one of the most adaptable migration schemes available. It can be
easily modified for account of such difficulties as topography, irregular recording
geometry, pre-stack migration, converted wave imaging as well as through-wall
imaging. Computational cost is one of the biggest disadvantages of this method.
2.5.6 f-k Migration
Wave equation based migration could be done also in the frequency domain. Stolt
in 1978 showed that migration problem could be solved by Fourier transform [133].
v