interpreted from a single time response is range information when depth is desired. The transfor-
mation into the x–z domain redistributes the energy in such a manner that these scattering
events are migrated to their true location in space where depth can now be interpreted.
The main goal of this paper is to provide a general
overview of
wave migration
for ground-penetrating radar using a matched-filter
approach. The motivation is to provide a theo- retical connection to traditional seismic tech-
niques so their use can be extended to radar applications. Using a generalized matched-filter
Ž .
definition Leuschen and Plumb, 1998 and spe- cific solutions and approximations to the wave
equation, this paper outlines four methods for transforming a monostatic radar survey from the
x–t domain to the x–z domain. Also, three of these methods are associated to traditional mi-
gration algorithms, not only showing the con- nection between a matched-filter and migration
but also a common link between existing tech- niques.
This paper will begin by examining the matched-filter technique and the general equa-
tion for an image. The basic steps of the formu- lation will be presented along with the neces-
sary approximations and assumptions. Next, four forward scattering techniques will be applied to
this image equation. These methods include ba- sic spherical spreading, propagation in the f–k
domain, a far-field approximation of the wave equation, and the finite-difference time-domain
Ž
. FDTD method. Finally, these methods will be
related to existing techniques and applied to measurements collected over four air-filled PVC
pipes buried in sand as a comparison.
2. The matched-filter technique
When a matched-filter is used to calculate the Ž
. value at a pixel, the signal-to-noise ratio SNR
is maximized when a specific target exists at Ž
. that location Levanon, 1988 . In the formula-
tion of the generalized matched-filter-based im- age, this target is a point scatterer; while dis-
tributed objects are approximated as a collection of independent point scatterers and multiple re-
flections are ignored. The match-filter proce- dure involves correlating the received data,
Ž .
u r,t , with an estimated response from a point
Ž
Y X
.
Y
scatterer, h r ,r ,r,t , where r is the transmit- ter location, r
X
is the pixel location of the point
scatterer, and r is the receiver location as shown by Fig. 1. The filter output is expressed as
S r
X
s h r
Y
,r
X
,r,t u r,y t d rdt. 1
Ž . Ž
. Ž .
Ž .
HH
This equation can be expressed as a convolu- tion.
X Y
X
S r s h r ,r ,r,t mu r,t d r 2
Ž . Ž
. Ž
. Ž .
H
ts0
We will begin by determining the point- Ž
Y X
.
scatter response, h r ,r ,r,t , in the frequency domain keeping in mind that the convolution
becomes multiplication.
For an
impulsive source, the general form of the estimated scat-
Ž
Y X
.
tered field due to a point target, H r ,r ,r, v , becomes
H r
Y
,r
X
,r, v
Ž .
X X
X Y
T
sa G r,r ,v O r G r ,r ,v a , 3
Ž . Ž .
Ž .
Ž . ˆ
ˆ
r r
i t
where G and G are the incident and reflected
i r
dyadic Green’s functions for the electric field,
O is the object scattering matrix, and a is the
ˆ
t
polarization of the transmitting antenna, and a
ˆ
r
is the polarization of the receiving antenna. The object scattering matrix relates how the incident
field induces current sources within the object, and if depolarization is ignored, the off-diagonal
Fig. 1. Configuration of wave scattering and collection. Ž
Y
. Ž
X
. Ž .
The transmitter is at r , object at r , and receiver at r .
components become zero resulting in the iden- tity matrix. For a monostatic radar configura-
tion,
r
Y
sr 4
Ž .
and a sa .
5
Ž . ˆ
ˆ
i r
Ž . Through reciprocity, Eq. 3 is simplified to
H r
X
,r, v s G
2
r
X
, r , v , 6
Ž .
Ž .
Ž .
Ý
a r a
where the subscript r corresponds to the direc- tion of the receiving antenna and a corresponds
to three orthogonal spatial directions. Finally, through the use of the exploding reflector model
Ž
. Loewenthal et al., 1976 , the squaring opera-
Ž . tion in Eq. 6 is expressed as
H r
X
,r, v s G
2r
X
,2r, v . 7
Ž .
Ž .
Ž .
Ý
a r a
Converting back to the time-domain, the general form of the image is expressed as
X X
S r s g
2r ,2r,t mu r,y t d r .
Ž . Ž
. Ž
.
Ý
H
ts0 a r
a
8
Ž .
At this point, we disregard the mathematics and interpret the physical processes implied by
this equation. The result is a forward scattering problem. First, the received signals become
time-reversed sources. Next, these sources are placed at the receiver locations in the exact
environment of the survey with all dimensions increased by a factor of two. The sources gener-
ate waves and these waves are propagated until time equals zero. At this time, the spatial distri-
bution of the scalar fields or the sum of the vector field components represents the image.
This process can be accomplished using any forward scattering technique as will be shown
later in this paper, but for now some properties
Ž
X
.
of the filter, h r ,r,t , are addressed with their implications on the image.
Ž
X
. Ž
X
First, h r ,r,t is not orthogonal with h r , r,
1 2
.
X X
t when r
is not equal to r . This can be
1 2
expressed mathematically as
h r
X
,r,t h r
X
, r ,t d rd t 0 9
Ž . Ž
. Ž .
HH
1 2
and implies that a non-zero output will occur at r
X
when a scatterer exists at r
X
. The result is
2 1
range sidelobes, causing ambiguities of the ob- ject’s actual location. However, we can assume
that this will only occur when r
X
is in close
2 X
X X
proximity to r and as the distance r yr
1 1
2
increases the output at r
X
will become insignifi-
2
cant. An example of this phenomenon is illus- trated by the finite bandwidth of the transmitted
signal. Reflections in the x–t domain due to a scatterer at r
X
do not exist as a single event but
1
are dispersed over time and tend to ring as they Ž
X
.
decay. Since h r ,r,t is not orthogonal to
1
Ž
X
.
X
h r ,r,t , values r can be chosen such that a
2 2
significant energy will occur at the output of Ž .
Eq. 9 , and this ringing will be translated into the x–z domain.
Ž
X
.
Second, the inner product of h r ,r,t is not normal or even constant for all values of r
X
. This is expressed as
h r
X
,r,t h r
X
,r,t d rd t sf r
X
constant
Ž . Ž
. Ž .
HH
10
Ž .
Ž
X
.
and implies that an amplitude variation, f r , will exist over the image. Usually, this function
is laterally independent and decreases with Ž
X
.
depth, and if f r can be approximated, its
effects can be eliminated by applying it as an inverse gain to the image.
3. Wave migration and the matched-filter
