#### 2.1. One-Dimensional Problem

In this subsection, we resume the simplified 1D-probing scheme proposed in [

30] to explain the basis of our approach.

Let us consider the 1D propagation of an electromagnetic pulse, with electric field

$E\left(ct,z\right)$, in a non-uniform half-space

$z>0$ characterized by a real-valued relative permittivity profile

$\epsilon \left(z\right)$ and a vacuum magnetic permeability

${\mu}_{0}$ (i.e., the half-space is assumed to be a lossless non-magnetic medium). Here and in the following,

t is the time,

z is the spatial coordinate and

c is the light velocity in vacuum. This phenomenon is governed by the wave equation

where

$s=ct$ is introduced for convenience, so that

${\partial}^{2}/\partial {s}^{2}={c}^{-2}{\partial}^{2}/\partial {t}^{2}$. The source is in

$z=0$. The trivial initial conditions

$E=0$ and

$\partial E/\partial t=0$ in

$t=0$,

$\forall z$, and a non-homogeneous boundary condition given by

define a transient field

$E\left(s,z\right)$ generated by the pulse

$f\left(s\right)$ entering the non-uniform half-space

$z>0$ with

${\epsilon}_{0}=\epsilon \left(z\to +0\right).$ The total wave field at

$z=0$, can be written as

$E\left(s,0\right)=f\left(s\right)+g\left(s\right)$, where

$g\left(s\right)$ is the cumulative backscattered signal born on the subsurface permittivity gradients.

To find a unique solution to the boundary-value problem, the radiation condition

has to be imposed, excluding the waves coming from

$z=\infty $. In Equation (3),

${\epsilon}_{\infty}=\epsilon \left(z\to \infty \right)$. The application of the Fourier integral transform

reduces Equation (1) to the 1D Helmholtz equation

or to an equivalent set of first-order ordinary differential equations (ODE) [

29]

with

${\epsilon}^{\prime}\left(z\right)=d\epsilon /dz.$Equations (6) govern the amplitudes

${A}^{+}\left(k,z\right)$ and

${A}^{-}\left(k,z\right)$ of the direct and backward waves in the total field representation

valid for

$z>0.$ The equation set (6) can be solved iteratively, starting from

$\partial {A}^{\pm}\left(k,z\right)/\partial z=0$. The first approximation gives

A backward Fourier transform yields an explicit formula relating the initial pulse

$f\left(s\right)$ with the total signal

$E\left(s,0\right)=f\left(s\right)+g\left(s\right)$, that can be measured in

$z=0$. In particular, the half-space response to the input electromagnetic pulse is

Equation (9), having the evident meaning of a sum of partial reflections due to the permittivity gradients, can be considered as an integral equation for the unknown function

$\epsilon \left(z\right)$. As shown in [

30], this equation, having a convolution form, can be solved by exploiting the Fourier–Laplace transform, yielding a parametric solution to the 1D inverse problem

where

and

$\tilde{f}\left(k\right),\tilde{g}\left(k\right)$ are the Fourier transforms of the initial pulse

$f\left(s\right)$ and received backscattered signal

$g\left(s\right)$, calculated accoding to Equation (4).

#### 2.2. 1.5-Dimensional Problem

In this subsection, we deal with a more realistic model, considering a GPR with separated antennas lying at the air-ground interface. We model the transmitting antenna as a line source, and develop an analytical method that allows to describe the electromagnetic field recorded by the receiving antenna, including the surface wave and all partial reflections by the subsurface permittivity discontinuities and gradients. As is well known, the line source is the two-dimensional counterpart of the hertzian dipole, for geometries with one invariant direction.

Although a line source is an idealized electromagnetic representation of an antenna, it allows a more realistic modeling of GPR problems than sources with an infinite extension, such as the plane wave. For example, in the presence of a two-dimensional variation of the subsurface physical properties, the line source allows to account for the position of the antenna in the scenario; furthermore, a suitable combination of line sources can be used to model the field distribution generated by a more complex antenna.

In our method, we exploit the Fourier–Laplace transform and reduce the time-domain boundary value problem to an ordinary differential equation, which is solved approximately by the Bremmer-Brekhovskikh method. A backward integral transform yields an approximate representation of the time-domain Green function, i.e., of the subsurface medium response to an elementary current jump in the GPR transmitting antenna. This result, in combination with the Duhamel principle [

34], gives an approximate solution to the forward electromagnetic scattering problem for an arbitrary electromagnetic pulse and permittivity profile.

Let us therefore consider the 1.5-dimensional scenario of short-pulsed radiation emitted by a line source stretched along the surface of a non-uniform dielectric half-space

$z>0.$ We assume that the half-space is horizontally layered, with a real-valued relative permittivity. We also assume a uniform current distribution along the thin wire, which is lying at

$x=z=0,-\infty y\infty $. The wave perturbation is excited by a current pulse

$I\left(t\right)$. The 2D wave equation governing the

y-component of the electric field

$E\left(t;x,z\right)$ is:

where

$\delta \left(\xb7\right)$ is the Dirac delta function. By using integral transforms and by imposing the initial conditions

E = 0 and

$\partial E/\partial t=0\epsilon \left(z\right)$ in

$t=0,\forall z$, Equation (12) can be reduced to an ordinary differential equation. In particular, we apply a Fourier transform with respect to the

x coordinate:

and we obtain the 2D counterpart of Equation (5):

Then, by using the Laplace transform with respect to the time variable:

we obtain the second-order ODE

where

$\widehat{I}\left(\gamma \right)$ is the Laplace transform of the antenna current

$I\left(t\right)$. Equation (16) can be reduced to a system of first-order ODE similar to Equation (6). Such a system, satisfying the boundary conditions at the air-ground interface, and the radiation condition for

$z\to \infty $, can be solved by iterations, starting from zero wave perturbation. The first approximation gives an integral representation of the initial probing wave and its subsurface reflections

as well as the “aerial” wave propagating in the upper half-space:

Here,

$\kappa \left(z\right)={\left[{q}^{2}\epsilon \left(z\right)+{p}^{2}\right]}^{1/2}$,

${\kappa}_{0}=\kappa \left(0\right)={\left[{q}^{2}{\epsilon}_{0}+{p}^{2}\right]}^{1/2}$,

${\kappa}_{A}={\left[{q}^{2}+{p}^{2}\right]}^{1/2}$, and

$q=\gamma /c$. The amplitude

${A}_{0}$ can be found from the excitation condition with a localized source

$2\gamma \delta \left(z\right)\widehat{I}\left(\gamma \right)/{c}^{2}$. The differentiation of Equations (17) and (18) yields:

and

where it can be noticed that the derivative

$\partial \widehat{E}/\partial z$ has a jump at the interface, which is approximately equal to

$-{A}_{0}({\kappa}_{0}+{\kappa}_{A})$. Taking this into account, we integrate Equation (16) over the small interval

$-0<z<+0$ and relate the wave amplitude

${A}_{0}$ to the Laplace image of the driving current

$\widehat{I}\left(\gamma \right)$:

The electromagnetic field amplitude at the interface

$z=0$, where by assumption the receiver antenna is placed, is given by the inverse Fourier–Laplace transform of the spectral distribution Equations (17) and (18):

where

In Equation (23), we simplified the expression by exploiting the formula of geometric series.

It is convenient to represent the electromagnetic field excited by an arbitrary current pulse as a convolution of the time-domain Green function with the current pulse

$I\left(t\right)$:

To find the Green function, it is necessary to calculate the radiation produced by a unit current step: $I\left(t\right)=1$ for $t>0$ and $I\left(t\right)=0$ for $t<0$, corresponding to $\widehat{I}\left(\gamma \right)=1/\gamma =1/cq$. Having no temporal scale, it is natural to use the uniform space-like variables $\left(s=ct;x,z\right)$.

From Equation (23), we find the boundary value of the spectral Green function:

This expression consists of two parts. The first term corresponds to direct pulse propagation along the ground surface (the so-called “direct” wave), the second term represents the cumulative reflection from the subsurface medium gradients.

The “direct” wave

${G}_{d}\left(s;x,z\right)$, with

$s=ct$, can be explicitly found by applying a backward Fourier–Laplace transform to the first term of Equation (25):

The inner integral in Equation (26) can be rewritten as two integrals over closed paths circumventing the corresponding branch points. After the substitution

$q=ip\eta $ and a change of integration order, the following formula arises, which describes the direct-wave propagation as the sum of two electromagnetic pulses (“aerial” and “ground” waves) moving along both sides of the

$z=0$ interface:

To find the cumulative signal reflected by the subsurface medium gradients,

${G}_{r}\left(s;x,0\right)$, we transform into the space-time domain the second part of the spectral function in Equation (25),

${G}_{r}\left(s;x,0\right)={{\displaystyle \int}}_{0}^{\infty}{\epsilon}^{\prime}\left(z\right)K\left(s;x,z\right)dz$, where:

In accordance with the problem geometry (absence of scaling parameters), the integrand in Equation (28) is homogeneous with respect to

$p$ and

$q$, which allows to simplify calculations by making the substitution

$q=\left|p\right|w$:

We consider the inner Laplace integral in Equation (29) under the two following conditions:

In the former case, the integration path can be closed in the right half-plane and the integral vanishes due to regularity of the integrand. In the latter case, the integration can be performed along the steepest-descent path Γ where the real part of the exponent is negative (red dashed line in

Figure 1). After such path deformation, we can change the integration order and calculate the inner integral:

Here, the following notations are introduced:

In the last integral of Equation (31), the integrand vanishes at infinity, so it can be reduced to residues:

where

${w}_{j}\left(s;x,z\right)$ are the roots of the transcendent equation

$\mathsf{\Phi}\left(s;w,z\right)=\pm ix$, lying in the right half-plane; the prime denotes differentiation with respect to

$w$, and

The poles of the integrand in Equation (31), lying at the level

$\mathrm{Re}\left[\mathsf{\Phi}\right]=0$, are schematically marked with crosses in

Figure 1. In

Figure 2, an example of exact solution to the functional equation

$\mathsf{\Phi}\left(s;w,z\right)=\pm ix$ is presented, for a linear transition layer with

$\mathsf{\epsilon}\left(\mathrm{z}\right)={\mathsf{\epsilon}}_{0}+\left({\epsilon}_{1}-{\epsilon}_{0}\right)\left(z-{z}_{0}\right)/\left({z}_{1}-{z}_{0}\right)$. Thus, for a given vertical permittivity distribution

$\mathsf{\epsilon}\left(\mathrm{z}\right),$the calculation of the essential Green function component, corresponding to the signal due to partial subsurface reflections, requires numerical localization of the poles, summation of the corresponding residues, and substitution of the kernel

$K\left(s;x,z\right)$ into the integral

${G}_{r}\left(s;x,0\right)$.

Let us finally comment that usually modification of the integration path is performed in order to achieve faster integration (and consequently accelerate forward-modelling calculations)—for example, [

23] is a proper reference on this topic. In our case, this procedure allows one to reduce the integral to a sum of residues (33), completely avoiding numerical quadrature.

#### 2.3. Geometrical-Optics Interpretation

Equations (31)–(33) provide an explicit approximate representation of the time-domain Green function for an arbitrary permittivity profile

$\mathsf{\epsilon}\left(\mathrm{z}\right)$, which, in combination with the Duhamel principle [

34], solves the electromagnetic forward problem for an arbitrary probing pulse. The key point in the numerical implementation resides in the evaluation of the following functional equation, to determine the poles

${w}_{j}\left(s;x,z\right)$.

By inspecting Equation (35), it can be noted that one of its solutions coincides with the geometro-optical (GO) one, rendering a minimum to the Fermat functional:

(optical path from an antenna element in

${x}_{0}={z}_{0}=0$,

${y}_{0}=x\mathrm{tan}\text{}\psi $, to the receiver point in

$\left(x,0,0\right),$ with intermediate specular reflection from

$\zeta =z$ plan).

By differentiating Equation (36) with respect to

$p,\psi $ and by equating the derivatives

$\partial S/\partial p$ and

$\partial S/\partial \psi $ to zero, we have:

Here, $p=P\left(x,z\right)$ is the solution of the second Equation (37), $s=S\left(x,z\right)$ being the result of its substitution into the last line of Equation (37), which, apparently, assures the fulfillment of the identity in Equation (35).

As follows from the laws of geometrical optics [

19], Equation (37) correspond to a ray trajectory in a horizontally-layered medium, which starts from

$\left(x=0,y=0,z=0\right)$ at an angle

${\theta}_{0}=\mathrm{arcsin}\left[P\left(x,z\right){\epsilon}_{0}^{-1/2}\right]$ with respect to the

z-axis and comes to the observation point

$\left(x=X,y=0,z=0\right)$ after specular reflection from a virtual mirror

$\zeta =z$ (see

Figure 3). This trajectory lies in the vertical plane

$y=0$ and, evidently, provides the shortest optical path from the line current source to the observation point, among ones touching the given level

$\zeta =z$.

From physical considerations, one may expect that the main contribution to the time-domain Green function

${G}_{r}$ is due to the values of

$w$ closest to GO. Ray interpretation suggests an efficient method to solve the functional Equation (36). Let us assume

$s=S\left(x,z\right)+\mu ,$$w=\pm i/\left(p+\nu \right),\left|\mu \right|\ll S,\left|\nu \right|\ll p$. Substitution of these quantities into Equation (36) gives an approximation, valid for small values of

$\nu $:

By taking into account the GO Equation (36) and defining

we get

$\mu \approx -T{\nu}^{2}/2$. As only the poles

$w=\pm i/\left(p+\nu \right)$ lying in the right half-plane give a contribution, we define

$\nu =\pm i{(2\mu /T)}^{1/2}=\pm i{\left\{2\left[s-S\left(x,z\right)\right]/T\left(x,z\right)\right\}}^{1/2}$ and obtain their approximate representation:

Now, it is easy to calculate the functions in Equations (32) and (34):

and the kernel of the time-domain Green function:

To conclude, in this quasi-optical approximation the search for the poles from Equation (35), which depend on the virtual reflection depth $z$ and normalized time $s$, is reduced to the calculation of the horizontal GO impulse$P\left(x,z\right)$, depending only on $z$, and to the computation of the integrals $S\left(x,z\right)$ and $T\left(x,z\right)$ via the explicit formulas given in Equations (37) and (39).