Elsevier

Journal of Computational Physics

Volume 386, 1 June 2019, Pages 350-364
Journal of Computational Physics

Data recovery in inverse scattering: From limited-aperture to full-aperture

https://doi.org/10.1016/j.jcp.2018.10.036Get rights and content

Highlights

  • Recovery of the full-aperture data from limited-aperture data in inverse scattering.

  • A novel approach to seek an analytic function in a suitable space governing the scattering problem.

  • Two schemes to recover the full-aperture data using the Green's formula and the single layer potential.

Abstract

Inverse scattering has been an active research area for the past thirty years. While very successful in many cases, progress has lagged when only limited-aperture measurement is available. In this paper, we perform some elementary study to recover data that can not be measured directly. In particular, we aim at recovering the full-aperture far field data from limited-aperture measurement. Due to the reciprocity relation, the multi-static response matrix (MSR) has a symmetric structure. Using the Green's formula and single layer potential, we propose two schemes to recover full-aperture MSR. The recovered data is tested by a recently proposed direct sampling method and the factorization method. Numerical results show that it is possible to, at least, partially recover the missing data and consequently improve the reconstruction of the scatterer.

Introduction

Inverse scattering has been a fast-developing area for the past thirty years. The aim is to detect and identify the unknown objects using acoustic, electromagnetic, elastic waves, etc. Many methods using the full-aperture data have been proposed, e.g., iterative methods, decomposition methods, the linear sampling method, the factorization method and direct sampling methods [4], [5], [6], [9], [11], [15], [16], [17], [21], [22], [28], [29], [18]. Such methods usually provide satisfactory reconstructions.

However, in practical cases such as underground mineral prospection, it is not possible to measure the full-aperture data. The above methods might not produce satisfactory results [12]. To process limited-aperture data, various reconstruction algorithms have been developed [1], [3], [13], [27], [16], [20], [14], [24], [25], [31], [7], [23]. Although uniqueness of the inverse problems can be proved in some cases [10], the quality of the reconstructions are not satisfactory as well. A typical feature is that the “shadow region” is elongated in down range [20]. Physically, the information from the “shadow region” is very weak, especially for high frequency waves [24]. For two-dimensional problems, the numerical experiments of the decomposition methods in [27], [31] indicate that satisfactory reconstructions need an aperture not smaller than 180 degrees.

In this paper, other than directly processing the limited-aperture data, we first recover the data that can not be measured directly. Then methods using full-aperture data can be employed. We take the acoustic scattering by time-harmonic plane waves as the model problem. The measurement data are only available for limited-aperture observation angles but for all incident directions. The goal is to recover data for all observation angles. The case to recover full-aperture data from limited-aperture observation angles due to limited-aperture incident directions will be considered in future.

For scattering problems, it is well-known that the full-aperture data can be uniquely determined by the limited-aperture data by analyticity. However, because of the severely ill-posed nature of analytic continuation, it is difficult to recover full-aperture data using techniques such as extrapolation [2]. We also refer to [8], [7], [23] for a conditional stability estimation on a line or an analytic curve. We take a different way by seeking the solution based on the PDE theory governing the scattering problem. More precisely, we look for density functions of layer potentials that generate the measured data approximately by regularization. Then these densities are used to obtain the full-aperture data. The method can be viewed as a regularization method to reconstruct full-aperture data in a specific functional space, i.e., the space of radiating solutions to Helmholtz equations. Note that similar problems have been considered for near field measurements for acoustic imaging, see, e.g., [30], [19].

The rest of the paper is organized as follows. In Section 2, we briefly introduce the scattering problem of interests and the MSR matrix. Due to the reciprocity relation of the far field pattern, the MSR has a symmetry property, which can be used to recover partial missing data. In Section 3.1, we propose a technique using the Green's formula to recover the full MSR. Another recovery technique based on the single layer potential is proposed in Section 3.2. Combining these techniques and the symmetry property, a novel algorithm is proposed to recover the full-aperture MSR. In Section 4, numerical examples are presented to demonstrate the performance of the data recover techniques. The recovered data are tested using a direct sampling method and the factorization method. We draw some conclusions and discuss future works in Section 5.

Section snippets

The multi-static response matrix

Let k be the wave number of a time harmonic wave and ΩRn,n=2,3, be a bounded domain with Lipschitz-boundary ∂Ω such that Rn\Ω is connected. Let the incident field ui be given byui(x)=ui(x;d)=eikxd,xRn, where dSn1,Sn1:={xRn:|x|=1}, denotes the direction of the plane wave.

The scattering problem for an inhomogeneous medium is to find the total field u=ui+us such thatΔu+k2(1+q)u=0in Rn,limr:=|x|rn12(usrikus)=0, where qL(Rn) such that its imaginary part (q)0 and q=0 in Rn\Ω. The

Data recovery schemes

In this section, we propose two methods to recover Ffull from Flimit(l). The first one is based on the Green's formula. The second one is based on the single layer potential.

Numerical examples and discussions

In this section, we present some results for sound-soft scatterers. The performance of the method for other scatterers is similar. The numerical examples are divided into two groups. We first present some numerical examples to demonstrate how to use the proposed methods to recover data. The second group of numerical examples are to test the recovered data by some non-iterative methods to reconstruct the support of the scatterer. The results show that the reconstruction improves significantly,

Concluding remarks

The limited-aperture problems arise in various practical applications such as nondestructive testings. It is well-known that the illuminated part can be reconstructed well, while the shadowed part is difficult to reconstructed. In this paper, based on the PDE theory for scattering problems, we introduce two techniques to recover the missing data. Using the recovered full-aperture data, a direct sampling method proposed in a recent paper [21] and the factorization method yield satisfactory

Acknowledgements

The research is partially supported by the Youth Innovation Promotion Association CAS, the NNSF of China under grant 11571355 and 11771068.

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