is the propagation constant in free space and k sv ´ ´ m
22
Ž .
zg g
is the propagation constant in the ground. The scaling factors, A
q
and A
y
, are determined by Ž
. evaluating Eq. 15 at the interface z s0.
vm
q y
A sA s 23
Ž .
k qk
Ž .
z zg
Ž .
Using this solution of Eq. 15 , the conjugate of Ž
. the Green’s function, shown in Eq. 14 , takes
the form k
X x
X U
G , z ;v
ž
2 vm
s
1r2 1r2
X X
2 2
2 2
v k
v k
x 2
y q
y
ž ž
ž ž
Õ 2
c 2
=
1r2 X
2 2
° ¶
v k
2
~ •
exp j y
2 z 24
Ž .
ž
ž
¢ ß
Õ 2
while the image is expressed as S k
X
, k
X
Ž .
x z
Õ
2
mk
X z
s
1r2 2
2
Õ Õ
X X 2
X 2 X 2
4 k q k q
k yk
z z
x x
ž ž
½ 5
c c
= Õ
1r2 X
X 2 X 2
U k ;vs k qk
. 25
Ž .
x z
x
ž
2
6. Far-field approximation and the Kirchhoff method
The general form for the matched-filter defi- nition of the image is expressed in the fre-
quency domain as
X X
U T
S r , v s U
r ;v a G r , r , v O
Ž .
Ž .
Ž .
ˆ
HH
r r
X X
Y
r G r , r , v a d rd v . 26
Ž . Ž
. Ž .
ˆ
i t
Fig. 3. Far-field approximation to monostatic scattering and collection. The configuration includes an x-directed
X
antenna, and a scatterer at a distance r yr from the antenna with only the u
component of the scattering
x
matrix.
In the monostatic radar configuration, both the transmitting and receiving antennas have the
same location and like polarization.
r sr
Y
27
Ž .
a sa sx 28
Ž . ˆ
ˆ ˆ
t r
Furthermore, using a far-field approximation, the induced currents within the scattering object
can be limited only to the u direction relative
x
to the transmitting antenna as shown by Fig. 3. This is expressed mathematically in the object
scattering matrix.
X X
O r sO r 29
Ž . Ž .
Ž .
uu
When the monostatic configuration and far-field Ž
. approximation are applied to Eq. 26 , the im-
age can be expressed with scalar values as
S r
X
;v s U
U
r;v G 2r
X
,2r;v d r , 30
Ž .
Ž .
Ž .
Ž .
H
u x
where G is the u -directed field due to an
u x x
x-directed source and all dimensions have been increased by a factor of two due to the explod-
ing reflector model. An expression for G
is obtained from the
u x
Ž free-space dyadic Green’s function
Chew, .
1990 ,
X 2
= = = = G yk Gsyjvmd ryr , 31
Ž .
Ž .
where 1
X
4
G r,r syjvm
exp yjkR sin u ,
Ž .
Ž .
u x x
4p R 32
Ž .
X
R s r yr 33
Ž .
and u is the angle between the x-axis and the
x
vector r
X
yr. An expression for the image is Ž
. Ž
. realized by substituting Eq. 32 into Eq. 30 .
sin u
Ž .
x X
S r ;v s jvm
Ž .
HH
8p R =
4
U
exp yjk2 R U r,v d rd v 34
Ž .
Ž .
The integral over v can be divided into three operations on the measured response, U. The
exponential represents sampling at y2 RrÕ while the complex conjugate translates into
a time reversal. The combination of these two operations is equivalent to sampling the
response at 2 RrÕ, which corresponds to the two-way travel time. Finally, the jv term is
interpreted as a derivative operation in the time-domain.
m sin u E
2 R
Ž .
x X
S r s u r ;t s
d r . 35
Ž . Ž .
H
ž
8p R Et
Õ Ž
. The form of Eq. 35 is similar to Kirchhoff
migration, cos u
E 2 R
Ž .
z X
S r s u r ;t s
d r . 36
Ž . Ž .
H
ž
2p RÕ Et Õ
Ž .
Ž . In Eq. 35 , the sin u
term is a result of the
Fig. 4. Two-dimensional configuration of four 4-in. diame- ter air-filled PVC pipes buried in dry sand.
Fig. 5. Raw data with a linear gain applied.
radiation pattern of a infinitesimal dipole, while Ž
. Ž .
in Eq. 36 the cos u is a result of the solution to Kirchhoff integral.
7. FDTD reverse-time migration
