International Journal of Antennas and Propagation

Volume 2019, Article ID 6956809, 13 pages

https://doi.org/10.1155/2019/6956809

## Imaging Internal Defects with Synthetic and Experimental Data

Correspondence should be addressed to Ling Ma; moc.361@36gnilam

Received 29 March 2019; Accepted 19 June 2019; Published 14 July 2019

Guest Editor: Lulu Wang

Copyright © 2019 Hongwei Zhou et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

This work concerns an inverse time-dependent electromagnetic scattering problem of imaging internal defects in a homogeneous isotropic medium. The position and cross section of the defects are detected by transient electromagnetic pulses in the case of TE polarization. We apply the Kirchhoff migration scheme to locate the position of small objects from both synthetic and experimental data. The multiple-input-multiple-out scheme is used to recover extended scatterers from the data generated by the software GprMax. Numerical experiments show that the Kirchhoff migration method is not only efficient but also robust with respect to polluted data at high noise levels. Experimental results show good quantitative agreement with numerical simulations.

#### 1. Introduction

Inverse scattering is to recover physical and geometrical information of inaccessible objects from scattered fields measured outside. It has been one of the most challenging problems with considerable practical applications in many areas of technology such as nondestructive evaluation, subsurface and ground-penetrating radar, geophysical remote sensing, medical imaging, seismology, and target identification; see [1–5]. One of the most prominent features of inverse scattering problem is its noninvasiveness, along with the affordability due to cheap nonionizing sensors. However, solving inverse scattering problems is difficult due to the inherent ill-posedness and nonlinearity. Small variation in the measured data can lead to large errors in the reconstruction of the scatterer, unless regularization methods are used. Extensive investigation has been carried out and a variety of inversion algorithms have been proposed.

In the time-harmonic regime, Kirsch and Kress [6, 7], Colton and Monk [8, 9], and Angrll, Kleiman, and Roach [10] theoretically separate the nonlinearity and the ill-posedness of inverse obstacle problems, giving rise to the decomposition method. Muhammad [11], Liang Yang [12], and Hui H* et al*. [13] employed iterative methods to solve inverse obstacle problems. In these optimization-based iterative approaches, an efficient forward solver is needed for each iteration, and good a priori information might be required in order to choose an initial guess that ensures numerical convergence. Noniterative sampling methods were intensively studied over the last twenty years, for instance, linear sampling method [14], factorization method [15], enclosure method [16], and singular point source method [17]. The key ingredient of the sampling approach is to design an appropriate indicator function from measured data for characterizing the region occupied by the scatterer. Forward solvers are not needed in the process of inversion. The above-mentioned approaches are mostly applicable to measured data irradiated at a fixed energy with many incoming directions. A recursive linearization method with multifrequency data was investigated in [18–22].

There exist a number of reconstruction algorithms in the time domain with dynamic measurement data. Here we mention the time reversal techniques [23, 24], the reversed time migration [25], and the boundary control method [26]. For time-dependent sampling type methods, we refer to the point source method [27], the enclosure method [28], and the total focusing method (TFM) arising from nondestructive evaluation [29, 30]. In one of the author's previous works [31], the TFM (which is also known as the Kirchhoff migration approach used in geophysics; see, e.g., [32]) was examined for imaging acoustically extended scatterers in two dimensions and the indicator behavior was mathematically analyzed. This work concerns imaging internal defects in a homogeneous isotropic medium. It is shown that the TFM is not only efficient but also robust to measurement noise. The aim of this paper is to test the TFM for inverse electromagnetic scattering problem of imaging perfectly conducting cylinders buried in a half-space homogeneous isotropic background medium. Such kind of inverse problems has many applications in searching for internal defects with radar techniques like ground-penetrating radar (GPR); see, e.g., [5, 33, 34]. We note that the light speed is much faster than the sound speed, leading to additional difficulties in computational simulation [35]. In this paper, we make use of the open software GprMax [35, 36] to generate the forward scattering data. The GprMax is an electromagnetic simulation tool based on the Finite-Difference Time-Domain (FDTD) approach coupled with the perfectly matched layer (PML) technique. In our experiments, the real data are gained by GPR.

In this paper, we apply the TFM (or Kirchhoff migration scheme) to inverse electromagnetic scattering problems. In comparison to other algorithms, the implementation of TFM is quite simple, since the image is formed through a superposition of the scattered signals irradiated by each transducer. In fact, our imaging functions are explicit and involve only integral calculations on the measurement surface. Hence, our inversion algorithm is totally “direct”. This also explains why this method is very robust with respect to measurement noise at high levels. Although the inverse scattering problem is difficult due to the inherent ill-posedness and nonlinearity, we do not need to solve the ill-posed and nonlinear problems, even without approximation and iteration. In this work, experiments with synthetic and experimental data are conducted to demonstrate the feasibility and applicability of this approach in the TE polarization case of inverse electromagnetic scattering problems.

The remaining part of this paper is organized as follows. In Section 2 we rigorously formulate the forward and inverse electromagnetic scattering problems. Section 3 is devoted to the description and implementation of the TFM for imaging small scatterers from both synthetic and real data. Extended scatterers will be reconstructed in Section 4 by using GprMax simulated data. Conclusion follows in Section 5.

#### 2. Problem Setting

Consider the electromagnetic wave propagation in a homogeneous isotropic medium. This background medium can be characterized by the electric permittivity , magnetic permeability , and electric conductivity , all of which are assumed to be constant. The propagation of the electric and magnetic fields is governed by the Maxwell system.Assume that is an infinitely long perfectly conducting cylinder buried in the lower half-space , which remains invariant in -direction; see Figure 1(a). The medium in the exterior of* D* is assumed to be dielectric, which means that . The cross section of in the -plane is denoted by . Incident cylindrical source waves emitted from the surface will be utilized for the purpose of detecting the position and shape of . Throughout the paper, we assume that the incident wave with , , is a filamentary current generated at the source location with a temporal pulse signal . The temporal function is supposed to have a compact support for some . Moreover, we consider the Transverse Electric (TE) case of the monopole with the polarization direction . Then, the incident electric and magnetic waves are governed by the systemfor all , , and , where is the Dirac distribution. Denote by the scattered fields and write the total field as , . Then, we havetogether with the zero initial condition at in . Here is the unit outward normal to the boundary . Eliminating the magnetic field, we arrive atIn the TM polarization case, the electric field takes the form . Hence, the perfectly conducting boundary condition can be written asimplying that on . Usingwe deduce the reduced wave equation from the Maxwell system (5) as follows:In the particular case that is a -periodic function, where denotes the frequency, the incident field can be explicitly represented asHere is the Hankel function of first kind of order zero; see [37, Chapter 3.4].