IEEE Geoscience and Remote Sensing Magazine - December 2022 - 79

where (, )( ,)GxGxrr ~~ is the Green's function of the reverse,
or down-track, path and ))
rk
rr
kt
GxGxrr (, )( ,) is the
tr
00
rr
~~
complex conjugate of the forward path Green's function.
When rk
r ,0
coincides with r ,0 due to constructive interference,
the correlation becomes maximum, and when the arbitrary
point rk
moves away from the desired target location
the uncorrelated terms cancel out each other. Hence,
focusing will happen only when rk
coincides with r .0 The
final imaging functional is given by
(, )( ,)
M
N
= / / rk ~ji
IijR x
i =1 j =1
(37)
where M is the number of A-scans and N is the number of
frequency samples. The image reconstruction process using
TRI is depicted in Figure 9.
IMPACT OF SUBSURFACE WAVE VELOCITY
ON FOCUSING
The focusing obtained through TRI is primarily determined
by the velocity of the subsurface medium. If TRI
is performed with correct wave velocity, then focusing is
limited to the precise target position; otherwise, it diverges
from the true position. The PSNR can be used as a metric
to quantify the level of focusing for
different trial velocities.
PEAK SIGNAL-TO-NOISE RATIO
CALCULATION
The PSNR is a metric for evaluating
image quality. It depends on the
mean square error (MSE), which can
be calculated as the average squared
difference between the estimated
values and the actual value:
MSE(,)( )2
n
xy MN xy
1
==
m
M
/
/ mn mn
=
-
1
1
(38)
where xmn
is the image having pure
target information obtained by subtracting
the image without the target
from that with the target, ymn
is
a cluttered suppressed image, and M
and N represent the number of sampling
points and number of A-scans
involved, respectively:
PSNR = 10log10 MSE .
1
(39)
Z
IMPACT OF THE HEIGHT OF
OBSERVATION LINE ON
DIFFRACTION CURVE
Generally, on-ground (ground-coupled)
data acquisition is preferred
Z0
Target
(a)
Observation Line
X
X0
Observation Line
Air
Soil
X
θ1
X1
θ2
Z
h0
Air
X0
Z0
Target
(b)
FIGURE 10. The measurement view of (a) ground-coupled GPR and (b) air-coupled GPR.
DECEMBER 2022 IEEE GEOSCIENCE AND REMOTE SENSING MAGAZINE
79
Soil
Nx
Target
Reflection
(a)
Focusing at Target Position
(b)
FIGURE 9. The imaging process. (a) The imaging grid showing the imaging process at each
arbitrary location. (b) A focused image.
Arbitrary Point
during most of the GPR survey. However, this procedure
cannot be followed everywhere because of the intrinsic
danger of taking contact data. In such scenarios, the measurement
line can be kept at a distance from the air-soil
interface. A geometry view of different measurement scenarios
is presented in Figure 10. When the antenna is not in
contact with the soil, the estimation of media properties is
challenging because of the prominent ground clutters and
refraction phenomena at the air-soil interface. For an offground,
or air-coupled, GPR, the shape of the diffraction
curve depends on the distance of the observation line from
the air-soil interface. The diffraction curve can be obtained
by applying Snell's law at the interface to find the refraction
point/hitting point, which requires the solution of a fourthdegree
equation [105]. A mathematical description of the
diffraction curve is provided in closed form by taking into
account the refraction phenomena at the air-soil interface
to show the effect of the antenna height on the shape of the
diffraction curve.
A formal solution to the preceding problem is discussed,
and the sensitivity of the diffraction curve with respect
to the dielectric permittivity is analyzed. Increased difficulty
is experienced while retrieving the permittivity
Time Reversal of Target
Reflection
Focused Image
w
IEEE Geoscience and Remote Sensing Magazine - December 2022

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