A novel integral equation for scattering by locally rough surfaces and application to the inverse problem: the Neumann case

# A novel integral equation for scattering by locally rough surfaces and application to the inverse problem: the Neumann case

Fenglong Qu School of Mathematics and Informational Science, Yantai University, Yantai, Shandong, 264005, China (fenglongqu@amss.ac.cn)    Bo Zhang NCMIS, LSEC and Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing, 100190, China and School of Mathematical Sciences, University of Chinese Academy of Sciences, Beijing 100049, China (b.zhang@amt.ac.cn)    Haiwen Zhang NCMIS and Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100190, China (zhanghaiwen@amss.ac.cn)
###### Abstract

This paper is concerned with direct and inverse scattering by a locally perturbed infinite plane (called a locally rough surface in this paper) on which a Neumann boundary condition is imposed. A novel integral equation formulation is proposed for the direct scattering problem which is defined on a bounded curve (consisting of a bounded part of the infinite plane containing the local perturbation and the lower part of a circle) with two corners and some closed smooth artificial curve. It is a nontrivial extension of our previous work on direct and inverse scattering by a locally rough surface from the Dirichlet boundary condition to the Neumann boundary condition [SIAM J. Appl. Math., 73 (2013), pp. 1811-1829]. For the Dirichlet boundary condition, the integral equation obtained is uniquely solvable in the space of bounded continuous functions on the bounded curve, and it can be solved efficiently by using the Nyström method with a graded mesh. However, the Neumann condition case leads to an integral equation which is solvable in the space of squarely integrable functions on the bounded curve rather than in the space of bounded continuous functions, making the integral equation very difficult to solve numerically. In this paper, we make us of the recursively compressed inverse preconditioning (RCIP) method developed by Helsing to solve the integral equation which is efficient and capable of dealing with large wave numbers. For the inverse problem, it is proved that the locally rough surface is uniquely determined from a knowledge of the far-field pattern corresponding to incident plane waves. Further, based on the novel integral equation formulation, a Newton iteration method is developed to reconstruct the locally rough surface from a knowledge of multiple frequency far-field data. Numerical examples are also provided to illustrate that the reconstruction algorithm is stable and accurate even for the case of multiple-scale profiles.

Key words. Integral equation, locally rough surface, inverse scattering problem, Neumann boundary condition, RCIP method, Newton iteration method.

AMS subject classifications. 35R30, 35Q60, 65R20, 65N21, 78A46

## 1 Introduction

This paper is concerned with the problem of scattering of plane acoustic or electromagnetic waves by a locally perturbed, perfectly reflecting, infinite plane (which is called a locally rough surface). Such problems arise in many applications such as geophysics, radar, medical imaging, remote sensing and nondestructive testing (see, e.g., [2, 6, 10, 13, 25]).

In this paper we are restricted to the two-dimensional case by assuming that the local perturbation is invariant in the direction. Precisely, let be the locally rough surface with a smooth function having a compact support in . Denote by the unbounded domain above the surface which is filled with a homogeneous medium. Denote by the wave number of the wave field in , where and are the wave frequency and speed, respectively. We assume throughout the paper that the incident field is the plane wave

 ui(x;d)=exp(ikd⋅x),

where is the incident direction, is the angle of incidence with and is the lower part of the unit circle . Notice that the incident wave is time-harmonic ( time dependence), so that the total field satisfies the Helmholtz equation

 Δu+k2u=0inD+. \hb@xt@.01(1.1)

Here, the total field satisfies the Neumann boundary condition on the surface :

 ∂u∂ν=∂ui∂ν+∂ur∂ν+∂us∂ν=0onΓ, \hb@xt@.01(1.2)

where is the unit normal vector on directed into , is the reflected wave of by the infinite plane :

 ur(x;d)=exp(ik[x1sinθ+x2cosθ])

and is the unknown scattered wave to be determined which is required to satisfy the Sommerfeld radiation condition

 limr→∞r12(∂us∂r−ikus)=0,r=|x|,x∈D+. \hb@xt@.01(1.3)

This problem models electromagnetic scattering by a locally perturbed, perfectly reflected, infinite plane in the TM polarization case; it also models acoustic scattering by a one-dimensional sound-hard, locally rough surface (see Figure LABEL:fig4_nr for the geometric configuration of the scattering problem). Further, it can be shown that has the following asymptotic behavior at infinity (see Remark LABEL:re3):

 us(x;d)=eik|x|√|x|(u∞(^x;d)+O(1|x|)),|x|→∞

uniformly for all observation directions with the upper part of the unit circle . Here, is called the far-field pattern of the scattered field , depending on the incident direction and the observation direction .

Direct scattering problems by locally rough surfaces have been studied both numerically and mathematically. For the Dirichlet case, the well-posedness of the scattering problem was first proved in [42] by the integral equation method. In [6], the scattering problem is studied by a variational method, based on a Dirichlet-to-Neumann (DtN) map, and then solved numerically with using the boundary element method. In [45], a novel integral equation defined on a bounded curve is proposed for the Dirichlet scattering problem, leading to a fast numerical algorithm for the scattering problem, even for large wave numbers. However, for the Neumann case, few results are available. For the special case when the local perturbation is below the infinite plane (which is called the cavity problem), the well-posedness was established in [1] via the variational method for the direct scattering problem with both Neumann and Dirichlet boundary conditions. A symmetric coupling method of finite element and boundary integral equations wasn developed in [2], which can be applied to arbitrarily shaped and filled cavities with Neumann or Dirichlet boundary conditions. Recently, in [3], the scattering problem by a locally perturbed interface is studied by a variational method coupled with a boundary integral equation method and numerically solved, based on the finite element method in a truncated bounded domain coupled with the boundary element method. It should be mentioned that some studies related to the scattering problem (LABEL:eq1)-(LABEL:rc) have also been conducted extensively. We refer to [36] for cavity scattering problems of Maxwell’s equations, and to [13, 14, 15, 16, 17, 18, 19, 43] for the non-local perturbation case which is called the rough surface scattering problem.

The inverse scattering problem of reconstructing the rough surfaces has also attracted many researchers’ attention. For example, many numerical algorithms have been developed for inverse scattering by locally rough surfaces with Dirichlet boundary conditions (see, e.g., [6, 21, 22, 27, 35, 45, 44] and the references quoted). A marching method based on the parabolic integral equation was proposed in [21] for the reconstruction of a locally rough surface with Dirichlet or Neumann boundary conditions from phaseless measurements of the single frequency scattering field at grazing angles, while a Kirsch-Kress decomposition method was given in [38] to recover a locally rough interface on which transmission boundary conditions are satisfied. For numerical recovery of non-locally rough surfaces, we refer to [4, 5, 7, 11, 10, 21, 20, 25, 26, 37, 39, 41]. For inverse cavity scattering, the reader is referred to [2, 28, 36].

In this paper, we extend our previous work in [45] on direct and inverse scattering by a locally rough surface from the Dirichlet condition to the Neumann condition. Precisely, we propose a novel integral equation formulation for the direct scattering problem (LABEL:eq1)-(LABEL:rc), which is defined on a bounded curve (consisting of a bounded part of the infinite plane containing the local perturbation and the lower part of a circle) with two corners and some closed smooth artificial curve. This extension is nontrivial because, for the Dirichlet condition case considered in [45], the integral equation obtained in [45] is uniquely solvable in the space of bounded continuous functions on the bounded curve, and it can be solved efficiently by using the Nyström method with a graded mesh. However, the Neumann boundary condition leads to an integral equation which is not solvable in the space of bounded continuous functions on the bounded curve (see, e.g., [8, 9]). In this paper, the integral equation formulation obtained is proved to be uniquely solvable in the space of squarely integrable functions on the bounded curve. But this result then leads to another difficulty in solving the integral equation numerically since the Nyström method with a graded mesh used previously in [45] does not work anymore. Instead, we make us of the recursively compressed inverse preconditioning (RCIP) method developed by Helsing to solve the integral equation (see [30, 31, 32, 33]) which is fast, accurate and capable of dealing with large wave numbers. In addition, a further difficulty in our analysis arises in the study of the property of the derived integral equation formulation in the neighborhood of the two corners of the bounded curve. To overcome this difficulty, we will use the operator boundedness from [12] in combination with some detailed energy estimates.

For the associated inverse scattering problem, based on the mixed reciprocity relation established in this paper, we will prove that the locally rough surface is uniquely determined from a knowledge of the far-field pattern associate with incident plane waves. Further, we develop a Newton iteration method to reconstruct the locally rough surface from a knowledge of multiple frequency far-field data. It should be mentioned that the proposed novel integral equation plays an essential role in solving the direct scattering problem in each iteration.

This paper is organized as follows. In Section LABEL:sec2, we first derive a novel integral equation formulation for the scattering problem (LABEL:eq1)-(LABEL:rc) and then prove its unique solvability. In section LABEL:se3, a fast numerical algorithm, based on the RCIP method, is proposed for solving the novel integral equation, and a numerical example is carried out to illustrate the performance of the algorithm. Section LABEL:sec3+ is devoted to the proof of uniqueness in the inverse scattering problem. In Section LABEL:se5, a Newton iteration algorithm with multi-frequency far-field data is developed to numerically solve the inverse problem, based on the proposed novel integral equation. Numerical experiments will also be provided to demonstrate that the reconstruction algorithm is stable and accurate even for the case of multiple-scale profiles. Finally, we will give some conclusion remarks in Section LABEL:se6.

## 2 Solvability of the direct problem via the integral equation method

Let on . It is easy to see that has a compact support on . Then the scattering problem (LABEL:eq1)-(LABEL:rc) can be reformulated as the Neumann problem (NP): Find which satisfies the Helmholtz equation (LABEL:eq1) in the Sommerfeld radiation condition (LABEL:rc) and the Neumann boundary condition:

 ∂us∂ν=fonΓ. \hb@xt@.01(2.1)

We will propose a novel boundary integral equation formulation for the Neumann problem (NP) which is defined on bounded curves and consequently can be solved with cheap computational cost. The boundary integral equation formulation will then be used to establish the well-posedness of the Neumann problem (NP). We first introduce some notations. Assume that . Let be a disk with large enough such that the local perturbation . Then represents the part of containing the local perturbation of the infinite plane. Let denote the endpoints of . For , let be the reflection of about the -axis. Write , and , where Further, let be an auxiliary domain in the interior of with a smooth boundary . See FIG. LABEL:fig4_nr for the geometry of the scattering problem and some notations introduced.

We now have the following uniqueness result for the Neumann problem (NP).

###### Theorem 2.1

The Neumann problem (NP) has at most one solution .

Proof. Let be the solution of the Neumann problem (NP) with the boundary data . Define the function in as follows: for , for . Then, by the Neumann boundary condition on and the reflection principle we know that satisfies the Helmholtz equation (LABEL:eq1) in and the Sommerfeld radiation condition (LABEL:rc) uniformly for all directions with .

Now, let . By the radiation condition (LABEL:rc) it follows that

 ∫∂B+R0{∣∣∣∂us∂ν∣∣∣2+k2|us|2+2kIm(us∂¯¯¯us∂ν)}ds=∫∂B+R0∣∣∣∂us∂ν−ikus∣∣∣2ds→0

as On the other hand, using Green’s theorem and the Neumann boundary condition on yields

 ∫∂B+R0us∂¯¯¯us∂νds=2∫BR0∩D+(|∇us|2−k2|us|2)dx. \hb@xt@.01(2.3)

Taking the imaginary part of (LABEL:201), we conclude from (LABEL:200) that

 limR0→+∞∫∂BR0(∣∣∣∂˜us∂ν∣∣∣2+k2|˜us|2)ds=limR0→+∞∫∂B+R0(∣∣∣∂us∂ν∣∣∣2+k2|us|2)ds=0.

Thus, by Lemma 2.12 in [23] we have in , and so, in . This, together with the unique continuation principle [23, Theorem 8.6], implies that in . The proof is thus complete.

We will derive a novel integral equation formulation for the Neumann problem (NP) to study the existence of solutions to the problem (NP) and to develop a fast algorithm to solve the problem numerically. To this end, we introduce the layer potentials on the curves , and . For the curve , define the single-layer potential by

 (Sk,IpφIp)(x):=∫IpΦk(x,y)φIp(y)ds(y),x∈R2∖Ip, \hb@xt@.01(2.4)

where , is the fundamental solution of the Helmholtz equation in . If the wave number , is the fundamental solution of the Laplace equation in . By the asymptotic behavior of the fundamental solution (see, e.g., [23]), the far-field pattern of the single-layer potential can be defined as

 (S∞k,IpφIp)(^x):=eiπ/4√8πk∫Ipe−ik^x⋅yφIp(y)ds(y),^x∈S1. \hb@xt@.01(2.5)

For , define the boundary integral operators :

 (Sk,Ip→IqφIp)(x) := ∫IpΦk(x,y)φIp(y)ds(y),x∈Iq, \hb@xt@.01(2.6) (K′k,Ip→IqφIp)(x) := ∫Ip∂Φk(x,y)∂ν(x)φIp(y)ds(y),x∈Iq, \hb@xt@.01(2.7)

where is the unit outward normal on and . Further, define the boundary operator :

 (K′,rek,Ip→∂B−RφIp)(x):=∫Ip∂Φk(xre,y)∂ν(xre)φIp(y)ds(y),x∈∂B−R,

where is the unit outward normal on directing into . Some properties of the boundary integral operators , and for are presented in Remark LABEL:re4 below. For a comprehensive discussion of mapping properties of the layer potentials, we refer to [23, 40] and the references therein.

###### Remark 2.2

(i) Let and . Since any function in can be extended by zero to a function in , it follows from the mapping properties of the single-layer potential over a closed piecewise-smooth boundary (see [40, Chapter 6]) that , satisfies the Helmholtz equation (LABEL:eq1) in and the jump relations on the curve :

 (Sk,IpφIp)±|Ip = Sk,Ip→IpφIp, \hb@xt@.01(2.8) (∂∂νSk,IpφIp)±∣∣Ip = ∓12φIp+K′k,Ip→IpφIp, \hb@xt@.01(2.9)

where

 (Sk,IpφIp)±(x) := limh→0+(Sk,IpφIp)(x±hν(x)),x∈Ip, limh→0+(∂∂νSk,IpφIp)±(x) := (ν⋅∇Sk,IpφIp)(x±hν(x)),x∈Ip.

For it is easy to see that , and are bounded operators.

(ii) Let . Then, from [40, Chapter 6] it follows that and satisfies the Helmholtz equation (LABEL:eq1) in as well as the jump relations (LABEL:eq43) and (LABEL:eq44) on the curve . Further, since is a -smooth curve, by [23] it is known that and are compact operators from to . Moreover, it can be seen that and are bounded operators since the kernels of those operators are smooth functions.

Let be the solution to the Neumann problem (NP). We now extend into by reflection, which we denote by again, such that for . Then, from the reflection principle and the Neumann boundary condition on it follows that and satisfies the Helmholtz equation (LABEL:eq1) in . Following the idea in [8], we seek the solution in the form

 us(x) = ∫ΓRΦk(x,y)φ1(y)ds(y)+∫∂B−RΦk(x,y)φ2(y)ds(y)+∫˜ΓΦk(x,y)φ3(y)ds(y), \hb@xt@.01(2.10) x∈R2∖(∂D−R∪˜Γ)

with with the norm . We will prove that, by choosing appropriate auxiliary curve the Neumann problem (NP) can be reduced to an equivalent boundary integral equation which is uniquely solvable (see Theorem LABEL:thm1 and Remark LABEL:re6 below).

Let be a continuous mapping from to such that

 ψre(x)={x,x∈ΓR,xre,x∈∂B−R∪{xA,xB}.

Then, by the boundary condition (LABEL:dbc) and the reflection principle, satisfies

 ∂us(x)∂ν(x)=−(∂ui(x)∂ν(x)+∂ur(x)∂ν(x)) forx∈ΓR, \hb@xt@.01(2.11) ∂us(x)∂ν(x)−∂us(ψre(x))∂ν(xre)=0 forx∈∂B−R. \hb@xt@.01(2.12)

We now impose the impedance boundary condition for on the auxiliary curve :

 ∂us(x)∂ν(x)+iρus(x)=0forx∈˜Γ, \hb@xt@.01(2.13)

where is a positive constant. Then the jump relations (see Remark LABEL:re4) and boundary conditions (LABEL:eq27)-(LABEL:eq28) lead to the boundary integral equation

 (I+A)Φ=G, \hb@xt@.01(2.14)

where is the identity operator and are given by

 A:=(Aij)3×3 \hb@xt@.01(2.15)

with , , , , , , , , and

 G:=⎛⎜ ⎜⎝2(∂ui/∂ν+∂ur/∂ν)∣∣ΓR00⎞⎟ ⎟⎠. \hb@xt@.01(2.16)

Obviously, . Further, it is seen from Remark LABEL:re4 that is a bounded linear operator on .

Conversely, we have the following result.

###### Lemma 2.3

Let be given by (LABEL:2.6) with which is the solution of the boundary integral equation (LABEL:2.7) with and defined in (LABEL:eq10) and (LABEL:eq11), respectively. Then and solves the Neumann problem (NP).

Proof. By the fact that and Rermark LABEL:re4, it is easy to see that defined by (LABEL:2.6) satisfies the Helmholtz equation (LABEL:eq1) in and the Sommerfeld radiation condition (LABEL:rc) and is in . Further, by the integral equation and the jump relations of the single-layer potentials (see Rermark LABEL:re4), it follows that for and for .

Let for . Then satisfies the Helmholtz equation (LABEL:eq1) in , the Sommerfeld radiation condition (LABEL:rc) and the condition for . Further, it follows from the uniqueness of the exterior Neumann problem (see, e.g., [23, Chapter 3]) that in , which implies that in . In particular, we obtain that on . Therefore, we have on . The proof is thus complete.

The following theorem gives the unique solvability of the integral equation .

###### Theorem 2.4

Assume that is not a Dirichlet eigenvalue of in . Let and be given by (LABEL:eq10) and (LABEL:eq11), respectively. Then the integral equation has a unique solution with the estimate

 \hb@xt@.01(2.17)

Proof. The proof is broken down into the following steps.

Step I. We show that is a Fredholm operator of index zero.

Step I.1. Define

 M0:=(M0ij)3×3 =⎛⎜ ⎜ ⎜ ⎜⎝−2K′0,ΓR→ΓR−2K′0,∂B−R→ΓR0−2(K′0,ΓR→∂B−R−K′,re0,ΓR→∂B−R)−2(K′0,∂B−R→∂B−R−K′,re0,∂B−R→∂B−R)0000⎞⎟ ⎟ ⎟ ⎟⎠.

Then it follows from Remark LABEL:re4 that is bounded in . In addition, the kernels of , , are bounded with the upper bound

 C[max(ln1|x−y|,1)+max(ln1|ψre(x)−y|,1)]

for with and thus weakly singular since for . Therefore, , , are compact. Further, the operators are compact since the kernels of those operators are smooth functions, and, by Remark LABEL:re4, the operator is also compact. Consequently, is compact in .

For and define . Obviously, we can choose a fixed such that for , and for . In the remaining part of the proof, we will choose . Now, choose a cut-off function satisfying that , in the region and in the region . Define by with

 (M0,r)ijφj:={ψr,xA(M0)ij(ψr,xAφj)+ψr,xB(M0)ij(ψr,xBφj),i,j=1,20,otherwise

for any . Since the kernel of vanishes in a neighborhood of and , is compact. Then the operator is compact from into .

Step I.2. We show that for a sufficiently small constant .

By the definition of , we have that for any ,

 ∥M0,rΦ∥X = ∥∥ ∥∥2∑j=1(ψr,xA(M0)1j(ψr,xAφj)+ψr,xB(M0)1j(ψr,xBφj))∥∥ ∥∥L2(ΓR) +∥∥ ∥∥2∑j=1(ψr,xA(M0)2j(ψr,xAφj)+ψr,xB(M0)2j(ψr,xBφj))∥∥ ∥∥L2(∂B−R) ≤ 2∑j=1∥(ψr,xA(M0)1j(ψr,xAφj)+ψr,xB(M0)1j(ψr,xBφj))∥L2(ΓR) +2∑j=1∥(ψr,xA(M0)2j(ψr,xAφj)+ψr,xB(M0)2j(ψr,xBφj))∥L2(∂B−R).

Recalling that and for , we find that

 ∥M0,rΦ∥X≤2∑j=1(∥ψr,xA(M0)1j(ψr,xAφj)∥2L2(ΓR)+∥ψr,xB(M0)1j(ψr,xBφj)∥2L2(ΓR))12 +2∑j=1(∥ψr,xA(M0)2j(ψr,xAφj)