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
The final type of propagation algorithm de- Ž
scribed in this paper is the FDTD Taflove,
. 1995 . The FDTD method is an excellent tool
for the simulation of wave propagation in the presence of complex environments, and, there-
Fig. 6. Resulting image from the diffraction summation approach.
Fig. 7. Resulting image from f – k propagation.
fore, is used for wave migration in the presence of a ground and any other known objects. The
method is a numerical technique in which Maxwell’s curl equations are solved in both
space and time using a Taylor approximation.
D t
E t qDt sE t q = = H t
37
Ž .
Ž . Ž .
Ž .
´ D t
H t qDt sH t y = = E t
38
Ž .
Ž . Ž .
Ž .
m Ž .
A closed-form solution to Eq. 8 is not used. Rather, this equation is interpreted as a forward
scattering problem and FDTD is used to propa-
Fig. 8. Resulting image from Stolt’s migration. Fig. 9. Resulting image from the far-field approach.
gate the fields numerically. The basic method is outlined as reversing the data in time, increasing
all dimensions by a factor of two, introducing these
time-reversed waveforms
as current
sources into the FDTD lattice, and finally evalu- ating the fields at time equal to zero.
8. Experimental set up and results
An experiment consisting of four 10-cm di- ameter PVC pipes buried in dry sand was con-
ducted in the sand-filled GPR test facility at The University of Kansas; it is 5 = 4.25 = 1.8 m
Fig. 10. Resulting image from Kirchhoff migration.
Fig. 11. Resulting image from the FDTD-based reverse- time migration.
deep. All the pipes were oriented along the y-axis simplifying the experiment into a two-di-
mensional space of x and z. The configuration shown by Fig. 4 indicates the relative locations
of the PVC pipes, the concrete walls of the sandbox, and 30 monostatic antenna positions.
The three pipes at the surface were purposely separated by both 20 and 30 cm center-to-center
to investigate the lateral resolution of the algo- rithms, while the fourth pipe near the bottom is
a permanent fixture of the box. The fourth pipe near the bottom is inline with the middle pipe
near the surface and centered with respect to the synthetic radar aperture on the surface.
Data were recorded in a monostatic configu- ration. All measurements were collected using a
Sensors and Software Pulse Ekko 1000 impul- sive radar system with a bandwidth and center
frequency of about 900 MHz. The sampling interval of 20 ps was chosen to meet the stabil-
ity criteria of the FDTD method, while 1500 samples were collected for each trace corre-
sponding to a time interval of 30 ns. The traces shown in Fig. 5 were collected with y-polarized
antennas at each of the 30 locations. A linear gain was applied to the data only to bring out
some of the deeper events in the figure, and it should be noted that this gain was not applied to
the data prior to the matched-filter processing.
Next, the data were migrated using each of the methods described above. Fig. 6 shows the
results of applying a spherical-spreading-based matched-filter to the data. The implementation
of both the f–k domain methods, matched- filter-based migration and the corresponding
Stolt’s migration, are shown in Figs. 7 and 8, respectively. Kirchhoff migration is shown in
Fig. 9, while the far-field matched-filter method is shown in Fig. 10. Finally, FDTD was used to
implement the filter and the resulting image is shown in Fig. 11. Table 1 shows a brief sum-
mary of all the techniques discussed in this paper including the domain transfers, equations
governing wave propagation, and figure number of the resulting image.
9. Conclusions
