An analytical approach to GPR probing of a horizontally layered subsurface medium is developed, based on the coupled-wave WKB approximation. An empirical model of current in dipole transmitter antenna is used.

1. GPR Probing of Smoothly Layered Subsurface
Medium: 3D Analytical Model
A. Popov1
, I. Prokopovich1
, P. Morozov1
, M Marciniak2
1
Pushkov Institute of Terrestrial Magnetism, Ionosphere and Radio Wave Propagation,
Troitsk, Moscow, Russia, popov@izmiran.ru
2
National Institute of Telecommunications, Warsaw, Poland, m.marciniak@itl.waw.pl
Abstract—An analytical approach to GPR probing of a
horizontally layered subsurface medium is developed, based on
the coupled-wave WKB approximation. An empirical model of
current in dipole transmitter antenna is used.
Index Terms—ground penetrating radar (GPR),
horizontally layered medium, time-domain coupled WKB
approximation.
I. INTRODUCTION
Despite the growing possibilities of computers and
modern numerical techniques, the problem of subsurface
sensing requires the development of analytical approaches
revealing fundamental laws of electromagnetic pulse
propagation in non-uniform natural environments and
simplifying the solution of direct and inverse problems.
Without denying the usefulness of rigorous numerical
calculations – see, e.g. [1-3], it is sometimes useful to
consider a simplified model to obtain a closed-form
analytical solution. A good example present GPR return
signals (not always recognized) from smooth ground
permittivity gradients. Such protracted signals, often
encountered in deep GPR sounding, were usually attributed
to low-frequency noise in receiving equipment or alien
subsurface echoes and eliminated by the use of “dewowing”
filters. However, as was understood from many years of low-
frequency GPR practice and confirmed by model numerical
calculations [4], they may reflect gradual vertical variations
in the material properties of the subsurface medium and yield
useful information for karst detection and mineral
prospecting.
While wave propagation through material environments
with gradually changing parameters is successfully described
by geometric optics or its analogue – WKB method of
quantum mechanics, backward reflected waves in a smoothly
non-uniform medium can be exponentially small and lie
beyond the accuracy of these methods. A promising
analytical technique, based on the coupled-wave WKB
approximation [5-6] transformed to time domain, has been
developed in [7-8] for one-dimensional and “1.5-D” models
of GPR probing. In this paper, we consider a more realistic
scenario, taking into account finite length and non-uniform
current distribution in the transmitter GPR antenna. Coupled-
wave WKB method yields an analytical representation of the
electromagnetic pulse emitted from the earth surface,
propagating into horizontally layered subsurface medium and
being partly reflected by smooth permittivity gradients. To
complete the obtained solution one just has to introduce a
realistic current distribution in the resistively loaded antenna
lying on the ground surface, which implies solving an
integro-differential equation, similar to Hallen-Pocklington
or Leontovich-Levin ones [9-10]. In order to avoid excessive
computational difficulties we propose a physically
meaningful analytic model following from the results of
GPR experiments [11].
II. ANALYSIS
Consider a resistively loaded GPR dipole antenna
stretched along y axis on the ground-air interface 0z = –
see Fig.1. Let the horizontally layered non-magnetic
subsurface medium 0z > be characterized by relative
dielectric permittivity ( )zε . The electromagnetic field
1
,
A
E H A
c t
∂
= −∇Φ − = ∇×
∂

 
generated by the transient
current ( , )I y t in the transmitter antenna can be expressed in
terms of two-component vector and scalar potentials
(0, , )y zA A A=

and Φ . The potentials components satisfy
scalar wave equations
Fig. 1. Schematic geometry of GPR probing.

2. 2
2 2
2
2 2
2
2 2
( ) 4
( ) ( ) ( , ),
( ) ( )
div ,
( )
( )
0.
y
y
z
z
Az
A x z I y t
cc t
Az z
A A
zc t
z
c t
ε π
δ δ
ε ε
ε
ε
 ∂
Δ − = −
 ∂

 ′∂
Δ − = −
∂

∂ Φ
ΔΦ − =
∂

(1)
The longitudinal component ( , , , )yA A x y z t= can be found
independently from the first equation of the set (1). The
matching conditions at the ground-air interface 0x =
, 0, 0.
A A
A A x
z z
+ −+ − ∂ ∂   
= − = ≠   
∂ ∂   
(2)
follow from the continuity of the horizontal components of
the electric and magnetic field. Generalizing the procedure
used in our paper [8] we apply 2D Fourier transform
( , , , ) ( , , , )A x y z t A p q z t  where
( )
2
1
( , , , ) ( , , , )
4
i px qy
A p q z t e A x y z t dxdy
π
∞ ∞
− +
−∞ −∞
=   (3)
satisfies the “telegraph” equation
2 2
2 2
2 2 2
( ) 2
( ) ( ) ( , )
A z A
p q A z I q t
cz c t
ε
δ
∂ ∂
− − + = −
∂ ∂
 
  (4)
with
1
( , ) ( , )
2
l
iqy
l
I q t e I y t dy
π
−
−
=  , l is the half-length of
the antenna vibrator. Equation (4), in turn, with the help of
Laplace transform can be reduced to an ODE:
0
2
2
2
ˆ( , , , ) ( , , , ) ,
ˆ 2ˆ ˆ( , , , ) ( ) ( , )
tA p q z e A p q z t dt
A
p q z A z I q
cz
γγ
κ γ δ γ
∞
−=
∂
− = −
∂
 
(5)
where 2 2 2 2( , , , ) ( ) / ,p q z z c p qκ γ γ ε= + +
( )( )
0 0
1ˆ( , ) ( , ) ( , )
2
6
l
t iqy t
l
I q e I q t dt dy e I y t dtγ γγ
π
∞ ∞
− − +
−
= =  
Equation (5) is formally equivalent to the 1.5D case
considered in [8], with substitution 2 2p p q + ; we
solve it using Bremmer-Brekhovskikh approximation [5-6]:
0 0 0 0
0
0 0
0
( ) ( ) 2 ( )
2 ( )
0
ˆ( , , , )
1 ( )
( , , ) ,
( ) 2 ( )
0; (7)
1 ( )
( , , ) 1 , 0.
( ) 2 ( )
z z
A
d d d
z
d
z
A p q z
A p q e e e d
z
z
A p q e d e z
z
ς
ς
κ ς ς κ ς ς κ η η
κ η η
κ
γ
κ κ ς
γ ς
κ κ ς
κ κ ς
γ ς
κ κ ς
∞− −
∞ −
=
  
  ′  
−  
   
<

  ′  − > 
   


By integrating Eq. (5) over the interface 0z = we relate the
vector potential jump with the antenna current spectrum:
ˆ ˆ 2 ˆ( , )
A A
I q
z z c
γ
+ −
   ∂ ∂
− = −      ∂ ∂   
(8)
and determine spectral amplitude 0
( , , )A p q γ :
( )
0
0
2 ( )
0 0
2 ˆ( , )
( , , )
1 ( )
2 ( )
d
A A
z
I q
cA p q
e d
ς
κ η η
γ
γ
κ ς
κ κ κ κ ς
κ ς
∞ −
=
′
+ + − 
(9)
Here,
2 2
2 2 2 2
0 2 2
( , ,0, ) (0) , .Ap q p q p q
c c
γ γ
κ κ γ ε κ= = + + = + +
The backward Fourier-Laplace transform of the
approximate solution (7-9) yields a closed-form
representation of the wave field measured by a GPR
receiver on the earth surface:
( )1 ˆ( , ,0, ) ( , ,0, ) (10)
2
i
t i px qy
i
A x y t e d e A p q dpdq
i
α
γ
α
γ γ
π
+ ∞ ∞ ∞
+
− ∞ −∞−∞
=   
For 0z = , both lines of Eq. (7) coincided and the solution
can be written in the form
( )
0 0
00
2 ( )
0
ˆ2 ( , ) ( )ˆ( , ,0, ) 1 (11)
( )
d
AA
I q
A p q e d
c
ς
κ η ηκγ κ ς
γ ς
κ κ κ ςκ κ
∞ − 
′ 
≈ − ++   

Physical meaning of its components was discussed in [8].
The first term in the brackets yields a contribution

3. proportional to ( ) 0
0
1
0
~
1
A
A
κ κ
κ κ
ε
− −
+
−
corresponding to the
surface waves propagating along both sides of the ground-air
interface with different phase velocities. The second term,
whose integrand is proportional to ( )ε ς′ , describes partial
reflection of the probing pulse by smooth gradients of the
ground dielectric permittivity. This effect, resulting in
protracting subsurface echoes, is typical for deep low-
frequency GPR probing [4].
In order to complete the analysis we have to give an
adequate model of current pulse in the transmitter antenna (it
is not easy as the antenna current dynamics depends on the
soil parameters and, at this stage of technology development,
can’t be measured in the process of GPR survey. Accurate
measurements done by S. Arcone [11] and practical
experience with deep penetration GPR [4, 8] allow us to put
forward an empirical model of the current waveform in a
resistively loaded GPR transmitter antenna:
( ) ( )
0 cos sin 2 ,
( , ) 2
0,
a y vt a y vty
I e e y vt
I y t l
y vt
π
π
− −  − <   = 
 >
(12)
Here, l is antenna half-length, 0 log2 /t a= is a reference
pulse duration, and is the measured velocity of its
propagation along the antenna wire. Both parameters a and
v depend on the soil permittivity (e.g. ~ 0.9v c for a wet soil
[11]); they may vary along the GPR B-scan [12] but can be
estimated using the initial part of the radargram.
III. CONCLUSION
We suggest an analytical approach to the calculation of
the radar return pulse by GPR probing of the horizontally
layered subsurface medium. The method is based on the
time-domain version of the coupled-wave WKB
approximation allowing for accounting exponentially small
backward reflections from smooth vertical gradients of the
dielectric permittivity.
The obtained analytical solution and a realistic model of
the transient current distribution in the transmitter dipole
antenna can be applied to a wide variety scenarios of GPR
probing.
ACKNOWLEDGMENTS
This work was supported in part by the Russian Foundation
for Basic Research, grant No. 18-02-00185.
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(a)
(b)
Fig. 2. Measured current pulse in a resistively loaded antenna [11]
(a); model GPR pulse [4] (b).