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

2.5 Radar Imaging Methods Overview 29
as
where
��
ψ(x, z = ∆z1, t) =
φ0(kx, f)e 2πi(kxx−kz1∆z1−ft) dkxdf (2.5.44)
kz1 =
�
f 2
v 2 1
− k 2 x. (2.5.45)
(2.5.44) is an expression for downward continuation or extrapolation of the wavefield
to the depth ∆z1. The wavefield extrapolation expression in (2.5.44) is more
simply written in the Fourier domain to suppress the integration that performs
the inverse Fourier transform:
��
φ(kx, z = ∆z1, f) = φ0(kx, f)e 2πikz1∆z1dkxdf. (2.5.46)
Now consider a further extrapolation to estimate the wavefield at the bottom of
the layer two (z = ∆z1 + ∆z2). This can be written as:
φ(kx, z = ∆z1 + ∆z2, f) = φ(kx, z = ∆z1, f)T (kx, f)e 2πikz2∆z2 (2.5.47)
where T (kx, f) is a correction factor for the transmission loss endured by the wave
as it crossed from layer two into layer one.
If this method want to be applied in praxis, phase shift algorithm [96] would
have to be studied into more depth in order to prevent many of unwanted effects
and incorrect implementations.
The Fourier method discussed in this section is based upon the solution of
the scalar wave equation using Fourier transforms. Though Kirchhoff methods
seem superficially quite different from Fourier methods, the uniqueness theorems
from partial differential equation theory guarantee that they are equivalent [96].
However, this equivalence only applies to the problem for which both approaches
are exact solutions to the wave equation and that is the case of a constant velocity
medium, with regular wavefield sampling. In all other cases, both methods
are implemented with differing approximations and can give very distinct results.
Furthermore, even in the exact case, the methods have different computational
artifacts [96].
2.5.7 Improvements in f-k Migration
f-k migration obtained title of the most often used migration method in praxis
due to its fast computation and very precise image results. A lot of improvements
and modifications have been done in f-k migration. They are mostly described in
airborne, or GPR scenarios. The most important of them will be shortly described
in this section.