Ground detection by a single electromagnetic far-field measurement

Abstract

We consider detecting objects on a flat ground by using the electromagnetic (EM) measurement made from a height. Our study is conducted in a very general and practical setting. The number of the target scatterers is not required to be known in advance, and each scatterer could be either an inhomogeneous medium or an impenetrable perfectly conducting (PEC) obstacle. Moreover, there might be multiscale components of small-size and extended-size (compared to the detecting wavelength) presented simultaneously. Some a priori information is required on scatterers of extended-size. The inverse problem is nonlinear and ill-conditioned. We propose a “direct” locating method by using a single EM far-field measurement. The results extend those obtained in [17], [18] for locating multiscale EM scatterers located in a homogeneous space.

Introduction

This paper concerns locating objects on a ground by using the electromagnetic (EM) scattering measurement made from a height. In Fig. 1, we give a schematic illustration of our study, where one wants to detect the multiple objects on the ground $\mathcal{G}$. To that end, one sends certain detecting wave fields and then measures the scattered wave fields from a height, from which to infer knowledge about the target objects. A practical scenario for our study is the scoutplane detection in the battlefield.

In what follows, we present the mathematical formulation for the current study. The detecting waves are chosen to be the time-harmonic electromagnetic plane waves of the following form

$$E^i(x)=p{e}^{\mathrm{i}\omega x\cdot d},H^i(x)=\frac{1}{\mathrm{i}\omega }\mathrm{\nabla }\wedge E^i(x),x\in {\mathbb{R}}^3$$

where $\mathrm{i}=\sqrt{-1}$, $\omega \in {\mathbb{R}}_{+}$ denotes the frequency, $d\in {\mathbb{S}}^2:=\{x\in {\mathbb{R}}^3;|x|=1\}$ denotes the impinging direction, and $p\in {\mathbb{R}}^3$ denotes the polarization with $p\cdot d=0$. $E^i$ and $H^i$ are entire solutions to the Maxwell equations in the free space

$$\mathrm{\nabla }\wedge E^i-\mathrm{i}\omega H^i=0,\mathrm{\nabla }\wedge H^i+\mathrm{i}\omega E^i=0.$$

The ground $\mathcal{G}$ is assumed to be perfectly electric conducting (PEC). The EM waves cannot penetrate inside the ground and propagate only in the space above the ground. If there is no object presented on the ground, one would have a reflected wave field $E_{\mathcal{G}}^i$ such that the total wave field $E=E^i-E_{\mathcal{G}}^i$ satisfies the following PEC boundary condition

$$\nu \wedge E=\nu \wedge (E^i-E_{\mathcal{G}}^i)=0\text{on}\mathcal{G},$$

where ν is the unit upward normal vector to $\mathcal{G}$. If $\mathcal{G}$ is flat, the reflected wave field $E_{\mathcal{G}}^i$ is well-understood through the work [22], [23], and if $\mathcal{G}$ is non-flat/rough, the reflection would be much more complex. Throughout the present work, we assume that $\mathcal{G}$ is flat and shall leave the rough case for a future study. Furthermore, without loss of generality, we assume that $\mathcal{G}:=\{x:=({\mathrm{x}}_1,{\mathrm{x}}_2,{\mathrm{x}}_3)\in {\mathbb{R}}^3;x^{\prime }:=({\mathrm{x}}_1,{\mathrm{x}}_2)\in {\mathbb{R}}^2,{\mathrm{x}}_3=0\}$. Denote ${\mathbb{R}}_{\pm }^3:=\{x:=({\mathrm{x}}_1,{\mathrm{x}}_2,{\mathrm{x}}_3)\in {\mathbb{R}}^3;x^{\prime }:=({\mathrm{x}}_1,{\mathrm{x}}_2)\in {\mathbb{R}}^2,{\mathrm{x}}_3\lessgtr 0\}$ and ${\mathbb{S}}_{\pm }^2:={\mathbb{R}}_{\pm }^3\cap {\mathbb{S}}^2$. Moreover, we let Π denote the usual reflection with respect to $\mathcal{G}$, i.e., $\Pi v=(v_1,v_2,-v_3)$ for a generic 3-vector $v=(v_1,v_2,v_3)$. Then, we have that (cf. [22], [23])

$$E_{\mathcal{G}}^i=\Pi \circ E^i\circ \Pi .$$

Next, we consider that there are EM objects presented on the ground. Let $\psi (x^{\prime })$, $x^{\prime }\in {\mathbb{R}}^2$, be a non-negative Lipschitz continuous function such that $\psi (x^{\prime })=0$ for $|x^{\prime }|>R$, where R is a large enough positive constant. Let ${\Sigma }^{\prime }:={\{x^{\prime }\in {\mathbb{R}}^2;\psi (x^{\prime })>0\}}^{\prime }:={\bigcup }_{j=1}^l{\Sigma }_j^{\prime }$, where ${\Sigma }_j^{\prime }$, $j=1,2,\dots ,l$ denote the simply connected components of ${\Sigma }^{\prime }$. Define

$${\Sigma }_j^{+}:=\{(x^{\prime },{\mathrm{x}}_3)\in {\mathbb{R}}_{+}^3;x^{\prime }\in {\Sigma }_j^{\prime },0<{\mathrm{x}}_3<\psi \left(x^{\prime }\right)\},1\le j\le l;{\Sigma }^{+}:=\bigcup _{j=1}^l{\Sigma }_j^{+}.$$

Each ${\Sigma }_j^{+}$, $1\le j\le l$, represents an EM object on the ground, and will be referred to as a scatterer in the sequel. Let ${\epsilon }_j,{\mu }_j$ and ${\Sigma }_j$ be the EM parameters for the object supported in ${\Sigma }_j$, respectively, representing the electric permittivity, magnetic permeability and electric conductivity. It is assumed that ${\epsilon }_j$, ${\mu }_j$ and ${\Sigma }_j$ are all constants, satisfying $0<{\epsilon }_j<+\infty$, $0<{\mu }_j<+\infty$ and $0\le {\Sigma }_j\le +\infty$. Furthermore, it is assumed that $|{\epsilon }_j-1|+|{\mu }_j-1|+|{\Sigma }_j|>0$ for $j=1,2,\dots ,l$. If ${\Sigma }_j=+\infty$, then ${\Sigma }_j^{+}$ is taken to be a PEC obstacle, disregarding ${\epsilon }_j$ and ${\mu }_j$. In the free space, $\epsilon =\mu =1$ and $\Sigma =0$. We set

$$(\epsilon (x),\mu (x),\Sigma (x)):=\{\begin{array}{cc}({\epsilon }_j,{\mu }_j,{\Sigma }_j)\hfill & \text{when}x\in {\Sigma }_j^{+},j=1,2,\dots ,l;\hfill \\ \hfill \\ (1,1,0)\hfill & \text{when}x\in {\mathbb{R}}_{+}^3\backslash {\Sigma }^{+}.\hfill \end{array}$$

The presence of the scatterer $(\Sigma ;\epsilon ,\mu ,\Sigma )$ on the ground would further perturb the propagation of the EM field $E^i-E_{\mathcal{G}}^i$, inducing the so-called scattered wave field $E^s$ in ${\mathbb{R}}_{+}^3$. The scattered wave field is radiating in nature, characterized by the Silver–Müller radiation condition

$$\underset{|x|\to +\infty }{\lim }|x||(\mathrm{\nabla }\wedge E^s)(x)\wedge \frac{x}{|x|}-\mathrm{i}\omega E^s(x)|=0,$$

which holds uniformly for all directions $\hat{x}:=x/|x|\in {\mathbb{S}}_{+}^2$. The total electric wave field $E:=E^i-E_{\mathcal{G}}^i+E^s$, together with the corresponding magnetic wave field H, is governed by the following Maxwell system

$$\mathrm{\nabla }\wedge E-\mathrm{i}\omega \mu H=0,\mathrm{\nabla }\wedge H+\mathrm{i}\omega (\epsilon +\mathrm{i}\frac{\Sigma }{\omega })E=0\text{in}{\mathbb{R}}_{+}^3,$$

where $\epsilon ,\mu$ and Σ are given in (1.5). Similar to (1.2), we have that

$$\nu \wedge E=\nu \wedge (E^i-E_{\mathcal{G}}^i+E^s)=0\text{on}\mathcal{G}.$$

We seek a pair of solutions $(E,H)\in H_{\mathit{loc}}(\text{curl},{\mathbb{R}}_{+}^3)\wedge H_{\mathit{loc}}(\text{curl},{\mathbb{R}}_{+}^3)$ to the scattering system (1.6), (1.7), (1.8). Particularly, the radiating wave field $E^s(x)$ has the following asymptotic expansion as $|x|\to +\infty$,

$$E^s(x)=\frac{e^{\mathrm{i}\omega |x|}}{|x|}A(\frac{x}{|x|};d,p,\omega )+\mathcal{O}\left(\frac{1}{{|x|}^2}\right).$$

$A(\hat{x};d,p,\omega )$ with $\hat{x}:=x/|x|\in {\mathbb{S}}_{+}^2$ is known as the electric far-field pattern, which encodes the scattering measurement illustrated in Fig. 1.

The ground detection problem can be abstractly formulated as

$$\mathcal{F}\left(({\Sigma }^{+};\epsilon ,\mu ,\Sigma )\right)=A(\hat{x};d,p,\omega ),$$

where $\mathcal{F}$ is the operator sending the scatterer to the corresponding far-field pattern, defined by the Maxwell system (1.6), (1.7), (1.8). It is easily verified that $\mathcal{F}$ is nonlinear and moreover it is ill-conditioned since $\mathcal{F}$ is completely continuous (cf. [12]). In what follows, $A(\hat{x};d,p,\omega )$ is always assumed to be given with all $\hat{x}\in {\mathbb{S}}_{+}^2$. Furthermore, if $d\in {\mathbb{S}}_{+}^2,p\in {\mathbb{R}}^3$ and $\omega \in {\mathbb{R}}_{+}$ are all fixed, then $A(\hat{x};d,p,\omega )$ is called a single EM measurement; otherwise, it is called multiple EM measurements. In practice, a single EM measurement can be obtained by sending a single incident plane wave, and then collecting the scattered electric wave in all the observation angles. Throughout the present study, we shall take a single EM measurement for the ground detection. Moreover, our study shall be conducted in a very general and practical setting. The number of the target scatterers is not required to be known in advance, and each scatterer could be either an inhomogeneous medium or an impenetrable perfectly conducting (PEC) obstacle. Furthermore, there might be multiscale components of small-size and extended-size (compared to the detecting wavelength) presented simultaneously. Some realistic a priori information is required on scatterers of extended-size. We propose a “direct” locating method without any inversion involved. To our best knowledge, both the direct scattering model and the inverse scattering schemes are new to the literature. The results extend those obtained in [17], [18] for locating multiscale EM scatterers located in a homogeneous space. The present study is closely related to the inverse electromagnetic scattering problems from rough surfaces; see, e.g., [6], [7], [8], [9], [10], [11], [16]. We also refer to [3], [4], [5], [6], [12], [13], [14], [19], [20], [21], [25], [26], [27] for the recent progress on the inverse scattering theory and numerical study. It remarked that for highly conductive objects with large Σ (1.7) reduces to the eddy current model, which is justified in [1]. A similar asymptotic formalism for detecting and characterizing small conductive inclusions from electromagnetic induction data is developed in a recent work [2]. The mechanisms proposed in this work could be migrated to the eddy current model to detect complicated settings, e.g., multiple regular-size or even multiscale conductive inclusions.

The rest of the paper is organized as follows. In Section 2, we present some results concerning the direct scattering problem for our subsequent use. Section 3 is devoted to the inverse scattering scheme. Numerical results and discussion are presented in Section 4.