locations within the ground. In seismic tech- niques, migration is used to approximate the
reflectivity at a point in space by estimating what the response would be given that both the
transmitter and receiver were collocated at this point during the survey. Using this model, the
reflectivity would be the received waveform at time equal to zero. Physically, this task is im-
possible without digging into the ground, and, therefore, the fields that were collected at the
surface are extrapolated back into the ground. These extrapolation processes that have been
based on many ideas can usually be reduced to
Ž . a forward scattering problem similar to Eq. 8
with the main differences lying in the physical interpretation of wave propagation. In the next
sections, four different solutions of the wave equation are applied to the matched-filter image
Ž . shown by Eq. 8 and resulting equations are
compared to current migration techniques.
4. Diffraction summation
The simplest form of wave migration is diffraction summation or wavefront interference
Ž migration
Schneider, 1978;
Gazdag and
. Sguazzero, 1984 . This method uses a geometric
interpretation of wave propagation rather than a solution of the wave equation. In this method,
waves are assumed to propagate spherically with no dispersion, distortion, or attenuation. The
function describing this process is expressed as
X
2 r yr
X
g 2r,2r ,t sd ty .
11
Ž .
Ž .
ž
Õ Ž .
Substituting the delta function into Eq. 8 and integrating over time is equivalent to sampling
the scattered-field response at times correspond- ing to the two-way travel distance divided by
the velocity of the medium, Õ.
X
2 r yr
X
S r s u r,ts d r
12
Ž . Ž .
H
ž
Õ Ž
. The output in Eq.
12 is equivalent to the
conventional process of summing over a hyper- bola described by the two-way travel times from
the receiver array to a pixel in space.
5. f–k migration
The fast f–k migration described by Stolt Ž
. 1978 is an efficient means of focusing the
scattered-field response collected by a monos- tatic radar configuration back to the original
location in the x–z domain via a two-dimen- sional Fourier transform. This method involves
converting to the f–k domain, a translation of
x
the energy into the k –k domain, and finally
x z
converting to x–z. In this section, migration from the f–k
to the k –k will be developed
x x
z
using a matched-filter definition based on elec- tromagnetic principles. In addition, the method
will be expanded for the inclusion of a ground. The general form for the scalar two-dimen-
sional matched-filter image in the frequency domain is expressed as
S x
X
, z
X
;v
Ž .
s U x ;v G
U
2 x
X
yx ,2 z
X
;v d xd v ,
Ž .
Ž .
Ž .
HH
13
Ž .
Ž .
where U x;v is the measured radar response Ž
. at the location x on the surface
z s0 and Ž
. Ž
. S x, z;v
is the image at location x, z . The
integral on the right-hand side is in the form of a convolution with respect to x
X
, and when transformed to the spatial frequency domain,
k , can be represented as multiplication.
x
1 k
X x
X X
X X
U
S k , z ;v s U k ;v G
,2 z ;v d v
Ž .
Ž .
H
x x
ž
2 2
14
Ž .
Assuming the medium is homogeneous in x, the Green’s function for the electric field is
obtained analytically by solving the one-dimen- sional problem
2
E
2 2
q k z yk
G k , z ;v
Ž . Ž
.
Ž .
x x
2
E z syjvmd z .
15
Ž . Ž .
For a medium that is also homogeneous in z, the conjugate of the Green’s function, shown in
Ž .
Eq. 14 , takes the form k
X x
X U
G ,2 z ;v
ž
2 vm
s
1r2 X
2 2
v k
2
2 y
ž
ž
Õ 2
=
1r2 X
2 2
° ¶
v k
2
~ •
exp j y
2 z 16
Ž .
ž
ž
¢ ß
Õ 2
were Õ is the wave velocity in the medium. Ž
. Ž
. Substituting Eq. 16 into Eq. 14 and convert-
ing to the spatial frequency domain, k , results
z
in a delta function on the right-hand side. S k
X
, k
X
;v
Ž .
x z
vm
X
s U k ;v
Ž .
H
x 1r2
X 2
2
v k
x
4 y
ž
ž
Õ 2
=
1r2 X
2 2
v k
x X
d 2 y
yk d v 17
Ž .
z
ž
ž
Õ 2
This integration is performed by sampling v when the argument of the delta function is zero
and scaling by the inverse of the derivative of this argument with respect to v. By doing so, a
Fig. 2. One-dimensional interpretation of a source at the interface of a dielectric ground.
representation of the image in the spatial fre- quency domains, k
X
and k
X
, is expressed as
x z
2
Õ m
Õ
1r2 X
X X
X 2 X 2
S k , k s
U k ;v s k qk
.
Ž .
x z
x z
x
ž
8 2
18
Ž .
Ž .
The image shown by Eq. 18 is based on the one-dimensional free-space Green’s function,
Ž .
Eq. 16 , and is equivalent, excluding a scaling function, to the Stolt f–k migration. If the
one-dimensional Green’s function is represented by only the exponential, the original form of the
Stolt f–k migration can be reproduced exactly.
k
X x
X U
G ,2 z ;v
ž
2
1r2 X
2 2
° ¶
v k
2
~ •
sexp j y
2 z 19
Ž .
ž
ž
¢ ß
Õ 2
S k
X
, k
X
Ž .
x z
Õ k
X z
s
1r2 X 2
X 2
4 k qk
x z
= Õ
1r2 X
X 2 X 2
U k ;v s k qk
20
Ž .
x z
x
ž
2 The one-dimensional Green’s function above
can also be modified to include the presence of a homogeneous ground. This function is ana-
lyzed using two outward propagating waves that represent the fields in both the ground and the
air. Each wave is characterized by a propagation constant and scaling factor shown in Fig. 2,
where
k sv ´ m 21
Ž .
z
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
