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

2.5 Radar Imaging Methods Overview 28
�( xz , �0,
t)
�( xzt , , �0)
2D FFT
�( xz , �0, t) ��
( k, f)
vk
z
k � k
2 2
x z
2D Inverse FFT
0
0
� ( k , f) ��( x, z, t �0)
x
x
Interpolation / Mapping
f
k k
v
z �
2
� 2
2
x
Multiplication
vk / k � k
2 2
z x z
Fig. 2.5.6: Implementation of Stolt migration.
in blocks for practical implementation are shown in Fig. 2.5.6. The measured
data in time domain at receiver position ψ(x, z = 0, t) are transformed into the
frequency domain by 2D FFT. kx is interpolated and mapped into the kz. kz is
multiplied by v/ � k 2 x + k 2 z and transformed back into the time domain by inverse
FFT. The result ψ(x, z, t = 0) at time t = 0 is obtained.
The f-k migration algorithm, which has been just described, is limited to constant
velocity. Its use of Fourier transforms for all of the numerical integrations
means that it is computationally very efficient.
Stolt provided an approximate technique to adapt f-k migration to non constant
wave velocity v(z). This method used a pre-migration step called the Stolt stretch,
and after this step follows classical f-k migration. The idea was to perform a onedimensional
time-to-depth conversion with v(z) and then convert back to a pseudo
time with a constant reference velocity. The f-k migration is then performed with
this reference velocity. Stolt developed the time-to-depth conversion to be done
with a special velocity function derived from v(z) called the Stolt velocity. This
method is now known to progressively lose accuracy with increasing dip and has
lost favor. A technique that could handle this problem more precisely is the phaseshift
method of Gazdag and is described in [62]. Unlike the direct f-k migration,
phase shift is a recursive algorithm that processes v(z) as a system of constant
velocity layers. In the limit when layer thickness is going to be infinitely small, any
v(z) variation could be modeled. This method has quite complicated geometric
and long mathematically representation that is very precisely described by the
author of the idea in [62] and really user-friendly described in [96]. How does it
work for one layer will be described in this work.
The method can be derived starting from (2.5.39). Considering the first velocity
layer only, this result is valid for any depth within that layer provided that v is
replaced by the first layer velocity, v1. If the thickness of the first layer is ∆z1, then
the wavefield just above the interface between layers one and two can be written