# Optimized impedance mismatch through an inverse algorithm applied to a dielectric layer for subsurface imaging applications

Simon Marcellin, Slimane Arhab, Quentin Didier, Gaëlle Lefeuve-Mesgouez

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Simon Marcellin, Slimane Arhab, Quentin Didier, Gaëlle Lefeuve-Mesgouez. Optimized impedance mismatch through an inverse algorithm applied to a dielectric layer for subsurface imaging applications. 4th URSI Atlantic RadioScience Conference, May 2024, Gran Canaria, France. 10.46620/UR-SIATRASC24/UHXK4580. hal-04650063

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# Optimized impedance mismatch through an inverse algorithm applied to a dielectric layer for subsurface imaging applications

Simon Marcellin(1), Slimane Arhab(1)(2), Quentin Didier(1) and Gaëlle Lefeuve-Mesgouez(1)

(1) UMR 1114 EMMAH, Avignon Université — INRAE, Avignon 84000, France. https://www6.paca.inrae.fr/emmah

(2) UAR LSBB 3538 CNRS - Avignon Université, La Grande Combe, F-84400 Rustrel, France.

## Abstract

In this paper, we present our numerical results of the enhanced electromagnetic propagation when a dielectric layer (DL) is placed at the air-ground interface for non-destructive testing and evaluation (NDTE) in ground penetrating radar (GPR) applications. The DL permittivity is controlled by a Constrained Gauss-Newton optimization algorithm in order to maximize its transmission at a single frequency. We easily show a 10% gain compared to the case where no such DL is in place, with only one optimized parameter, the dielectric permittivity.

<figure><img src="image_2.png" /><figcaption>**Figure 1.** Principle of the optimized metasurface applied to a bistatic radar, aiming at maximizing the transmitted and the scattered signals. We first consider this metasurface as a dielectric layer with no periodic conditions.</figcaption></figure>

## 1 Introduction

Electromagnetic characterization of the near subsoil (up to 10 meters deep) by imaging system suffers from a weak coupling between the antennas and the air-ground interface [1]. For a bistatic mode, most of the wave radiated by the transmitting antenna is immediately reflected by the interface and therefore provides no information on the heterogeneities of the subsurface. This interface is also at the origin of a strong decrease of the wave scattered by these heterogeneities, which is recorded by the receiving antenna. This is a problem when the imaging system is used on agricultural or karstic soils for which the Fresnel reflection coefficient reaches very high values, especially when there is a high water content. To reduce the reflection at the interface and achieve a significant signal-to-noise ratio at the receiving antenna, some works have emerged in the last decade dealing with the conception of two-dimensional metamaterials. Commonly called metasurfaces (MTS), they are positioned between both transmitting and receiving antennas and the air-ground interface (as shown in green in Figure 1) in order to achieve a higher transmission coefficient at a given frequency of the electromagnetic signal [2, 3].

MTS have the advantages of light weight, ease of fabrication for microwave applications, and good adaptability to a large frequency range [4]. However, an easy-to-design structure will not perfectly fit the initial requirements, and vice versa: some MTS designs perfectly fits their objectives during modeling process but have too much details and requires too much resources to be properly manufactured,

even in a laboratory but especially on a larger scale. Moreover, there is no automatic design for a given frequency range. MTS unit-cells always look different from a publication to another, even if the goal is the same, because different patterns could lead to the same results. It is impossible to know if the initial design (before the optimization process) is the best to solve a particular problem. Actual development of MTS suffer from this time-consuming process. Both numerical analysis and experiments show that High-transmission MTS for GPR applications are already working well for impedance matching [5, 6, 7, 8]. We want to improve this existing work through an inverse algorithm, in order to make MTS versatile enough in any frequency range with complex radiated field shape. The preliminary work in this paper is oriented to test an inverse algorithm in a very simple condition, on a dielectric layer (DL) before applying it on a MTS for a frequency range. Thus our DL has no periodic condition and is only characterized by its permittivity $\epsilon_r$.

We will first introduce our 2D model, and then develop our forward and inverse algorithm process, to finally show the numerical results, which we will briefly discuss.

## 2 FEM model of the dielectric layer

We use the commercial software COMSOL Multiphysics for the numerical simulations in the frequency domain, at 1 GHz. We work on a 2D model as shown in Figure 2 with a punctual emitting source 4 cm above the ground and a 2 cm thick DL with a dielectric permittivity $\epsilon_r \in [1; 8]$. The
medium under test has a permittivity of $ε_r = 16$. We apply a perfectly matched layers (PML) around it to avoid any reflection problem. In order to generate the best permittivity map $ε_r$ in the DL we use a Constrained Gauss-Newton optimization algorithm which is described in section 4.

<figure><img src="image_3.png" /><figcaption>**Figure 2.** FEM 2D model of the dielectric layer at the air-ground interface. One point source emitter at the top ($\mathbf{r}_s$ position), and 12 points detectors below the surface ($\mathbf{r}_d$ positions).</figcaption></figure>

### 3 Forward Model

A first step consists in accurately and efficiently performing synthetic simulations to generate reference data, and also to modelize the field involved in the inverse model. To do so Maxwell's equations have to be solved in the harmonic regime with pulsation $ω$. The presence of an invariance axis in the z direction leads to a 2D scalar problem studied through its electric component ($\mathbf{E}(\mathbf{r}) = E(\mathbf{r})\mathbf{z}$) which satisfies the following Helmholtz equation for a purely dielectric medium [9]:

$$ \Delta E(\mathbf{r}) + \frac{\omega^2}{c_0^2} \varepsilon_r(\mathbf{r}) E(\mathbf{r}) = j \omega \mu_0 A \delta(\mathbf{r} - \mathbf{r}_s) \quad (1) $$

with $c_0 = \frac{1}{\sqrt{\varepsilon_0 \mu_0}}$ the speed of light in vacuum, $\Delta$ the Laplacian operator, A the amplitude of the monopole source set to one, $j^2 = -1$ and $\delta(\mathbf{r} - \mathbf{r}_s)$ the Dirac delta function centered on the source location $\mathbf{r}_s = x_s \mathbf{e}_x + y_s \mathbf{e}_y$. The relative dielectric permittivity $\varepsilon_r$ depends on the spatial position. Equation (1) is solved thanks to the finite element method with a triangular mesh involving second-order Lagrange polynomial basis functions. To simulate the free space and avoid any reflection on the boundaries of the domain, stretching complex coordinate perfectly matched layers are used [10].

The reference data that have to be fitted are the transmitted electric field without the DL in a fictitious medium with lower permittivity ($\varepsilon_r = 4$) than the actual studied one ($\varepsilon_r = 16$). These reference data are calculated at a finite number of detection points (12 detectors) below the surface at $z = 6$ cm depth.

### 4 Inverse Model

To optimize the permittivity distribution inside the DL, the Constrained Gauss-Newton method minimizes a cost function depending on reference data and simulated ones. At k

iteration this cost function writes:

$$ \mathcal{F}_k(\varepsilon_r^k) = \frac{\sum_{d=1}^{N_d} |E^{\text{ref}}(\mathbf{r}_d) - E^k(\mathbf{r}_d)|^2}{||E^{\text{ref}}||_2^2} \quad (2) $$

where $E^{\text{ref}}(\mathbf{r}_d)$ the electric field calculated in the fictitious medium of permittivity $\varepsilon_r = 4$, $E^k(\mathbf{r}_d)$ the electric field simulated at the current iteration k in the actual medium of permittivity $\varepsilon_r = 16$, $N_d = 12$ is the number of detector positions involved. Minimizing the cost function $\mathcal{F}_k(\varepsilon_r^k)$ yields updated parameter $\varepsilon_r^{k+1} = \varepsilon_r^k + \delta\varepsilon_r^k$. The Fréchet derivative of $\mathcal{F}_k(\varepsilon_r^k)$ is involved in the following linear system [11]:

$$ \left\{ \begin{array}{l} E^{\text{ref}}(\mathbf{r}_d) - E^k(\mathbf{r}_d) = j\omega\varepsilon_0 \int_{\mathcal{S}} E^k(\mathbf{r})\tilde{E}^k(\mathbf{r}) \delta\varepsilon_r^k(\mathbf{r}) \, d\mathbf{r} \\ \text{written in the compact form for all detector positions:} \\ \\ \delta E^k = \mathcal{D}_k \delta\varepsilon_r^k \end{array} \right. \quad (3) $$

with each element $\delta E_d^k = E^{\text{ref}}(\mathbf{r}_d) - E^k(\mathbf{r}_d)$ is the misfit between reference and simulated data at the current iteration. Here $\mathcal{D}_k$ corresponds to the Fréchet derivative of the cost function with respect to $\varepsilon_r$ [12]. Equation (3) is derived using the adjoint method, involving two electric fields calculated in the $\mathcal{S}$ domain of the DL: The electric field of the forward model $E^k$ solution of equation 1, and the adjoint electric field $\tilde{E}^k$ solution of the same equation but by replacing $\mathbf{r}_s$ with $\mathbf{r}_d$, which means that the incident field is in this case radiated from the detector position $\mathbf{r}_d$.

The matrix system appearing in equation (3) is ill-conditioned and therefore cannot be solved as it stands. This aspect is handled with the standard Tikhonov regularization [13, 14], which leads to the following normal equation:

$$ \mathcal{D}_k^\dagger \delta E^k = (\mathcal{D}_k^\dagger \mathcal{D}_k + \nu I) \delta \varepsilon_r^k \quad (4) $$

Here I is the identity matrix and $\nu$ is the positive Tikhonov parameter which balances two contributions, a minimum norm on misfit with respect to reference data, and a minimum norm of the correction on the parameter. The symbol $\dagger$ stands for the conjugated transpose. In addition to standard Tikhonov regularization, we bounded the permittivity values in the DL between two extrema (1 and $\varepsilon_r^{\text{max}} = 8$). We introduced this a priori information through the following bi-parameter variable change:

$$ \varepsilon_r = (1 - \varepsilon_r^{\text{max}})e^{-\gamma^2} + \varepsilon_r^{\text{max}} \quad (5) $$

Consequently, the parameter to retrieve is not anymore $\varepsilon_r$ but $\gamma$ which is unbounded. The conjugate gradient (CG) algorithm [15] is then applied to solve the linear system (4) after introducing the variable change (5).

### 5 Results and discussions

Firstly, we present in the Figure 3 the effect of the optimized DL on the electric field inside the ground, at the last step of
the iteration process. We clearly see an amplification of
the signal compared to the absence of the DL in the same
situation.

<figure><img src="image_4.png" /><figcaption>**Figure 3.** Transmitted electric field modulus in the ground enhanced with an optimized dielectric layer (bottom) through our inverse algorithm.</figcaption></figure>

From a quantitative point of view, the optimization is not
perfectly balance between the different reference positions.
The Figure 4 show the electric field modulus along the de-
tection points. If the amplitude is close enough to the desire
reference data (top), not every position has been improved
in the same way (bottom). There is a slight variability in
the process, but symmetric, which can be explained by the
punctual nature of the source. Even so, we obtained at best
a 10% gain compared to the absence of DL in the first place.

These numerical results confirm the possibility of apply-
ing an inverse algorithm to generate a permittivity map
(ε_r) capable of amplifying the electromagnetic signal into
the ground, in a simple configuration and without any pe-
riodic pattern in the DL. The next challenges will be to
find a better reference data, to add other adjustment crite-
ria than permittivity alone, and gradually evolve towards a
metasurface-type structure. Our goal is to improve this in-
verse algorithm towards shape optimization process, with
more physical properties and less intuitive patterns.

<figure><img src="image_5.png" /><figcaption>**Figure 4.** Electric field amplitude according to the detector positions in the numerical simulation. Top: with and without DL, according to our reference data. Bottom: relative amplitude variation.</figcaption></figure>

6 Acknowledgements

We thank all PHYSIQUE team members at Avignon Uni-
versité for their contribution during various stages of the
paper preparation.

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