Inverse Elastic Scattering for a Random Source

TL;DR

This paper addresses the inverse problem of recovering the covariance structure of a random elastic source in 2D, demonstrating unique reconstruction using frequency-averaged displacement data and advanced microlocal analysis techniques.

Contribution

It introduces a novel method to uniquely determine the covariance operator's principal symbol from displacement measurements in a random elastic scattering setting.

Findings

Unique solvability of the direct problem via Lippmann--Schwinger equation

Almost sure unique determination of the covariance's principal symbol

Effective use of Born approximation and microlocal analysis in inverse scattering

Abstract

Consider the inverse random source scattering problem for the two-dimensional time-harmonic elastic wave equation with an inhomogeneous, anisotropic mass density. The source is modeled as a microlocally isotropic generalized Gaussian random function whose covariance operator is a classical pseudo-differential operator. The goal is to recover the principle symbol of the covariance operator from the displacement measured in a domain away from the source. For such a distributional source, we show that the direct problem has a unique solution by introducing an equivalent Lippmann--Schwinger integral equation. For the inverse problem, we demonstrate that, with probability one, the principle symbol of the covariance operator can be uniquely determined by the amplitude of the displacement averaged over the frequency band, generated by a single realization of the random source. The analysis employs the Born approximation, asymptotic expansions of the Green tensor, and microlocal analysis of the Fourier integral operators.

Equations

$$\mu \Delta \mathbf{u}+(\lambda +\mu )\nabla \nabla \cdot \mathbf{u}+{\omega }^2\mathbf{u}-\mathbf{M}\mathbf{u}=\mathbf{f}\mathrm{in}{\mathbb{R}}^2,$$

$$\mathbf{u}=-\frac{1}{{\kappa }_{\mathrm{p}}^2}\nabla \nabla \cdot \mathbf{u}+\frac{1}{{\kappa }_{\mathrm{s}}^2}\mathbf{curl}\mathrm{curl}\mathbf{u}:={\mathbf{u}}_{\mathrm{p}}+{\mathbf{u}}_{\mathrm{s}}\mathrm{in}{\mathbb{R}}^2\setminus \overline{D}.$$

$$\mathbf{curl}u=({\partial }_{x_2}u,-{\partial }_{x_1}u{)}^{\top },\mathrm{curl}\mathbf{u}={\partial }_{x_1}{u}_2-{\partial }_{x_2}{u}_1.$$

$$\underset{r\to \infty }{\lim }{r}^{\frac{1}{2}}\left({\partial }_r{\mathbf{u}}_{\mathrm{p}}-\mathrm{i}{\kappa }_{\mathrm{p}}{\mathbf{u}}_{\mathrm{p}}\right)=0,\underset{r\to \infty }{\lim }{r}^{\frac{1}{2}}\left({\partial }_r{\mathbf{u}}_{\mathrm{s}}-\mathrm{i}{\kappa }_{\mathrm{s}}{\mathbf{u}}_{\mathrm{s}}\right)=0,r=|x|,$$

$${\kappa }_{\mathrm{p}}=\frac{\omega }{(\lambda +2\mu {)}^{1/2}}=c_{\mathrm{p}}\omega ,{\kappa }_{\mathrm{s}}=\frac{\omega }{{\mu }^{1/2}}=c_{\mathrm{s}}\omega .$$

$$c_{\mathrm{p}}=(\lambda +2\mu {)}^{-1/2},c_{\mathrm{s}}={\mu }^{-1/2}.$$

$$\underset{Q\to \infty }{\lim }\frac{1}{Q-1}{\int }_1^Q{\omega }^{m+1}|\mathbf{u}(x,\omega ){|}^{2}d\omega =a{\int }_{{\mathbb{R}}^2}\frac{1}{|x-y|}\varphi (y)dy,$$

$$H^{s,p}({\mathbb{R}}^2)=\{h=(I-\Delta {)}^{-\frac{s}{2}}g:g\in L^p({\mathbb{R}}^2)\},$$

$$\Vert h{\Vert }_{H^{s,p}({\mathbb{R}}^2)}=\Vert (I-\Delta {)}^{\frac{s}{2}}h{\Vert }_{L^p({\mathbb{R}}^2)}.$$

$$\Vert h{\Vert }_{H^{s,p}(V)}=\inf \{\Vert g{\Vert }_{H^{s,p}({\mathbb{R}}^2)}:g{|}_V=h\}.$$

$$\Vert h{\Vert }_{H_0^{s,p}(V)}=\Vert h{\Vert }_{H^{s,p}({\mathbb{R}}^2)}.$$

$$H_0^{-s,q}(V)=(H^{s,p}(V){)}^{\prime }\mathrm{and}{H}^{-s,q}(V)=(H_0^{s,p}(V){)}^{\prime },$$

$$\Vert gh{\Vert }_{H_0^{-ϵ,\tilde{r}}({\mathbb{R}}^2)}\lesssim \Vert g{\Vert }_{H^{ϵ,2r}({\mathbb{R}}^2)}||h|{|}_{H_0^{-ϵ,r^{\prime }}({\mathbb{R}}^2)},$$

$$\Vert gh{\Vert }_{H^{s,\tilde{p}}({\mathbb{R}}^2)}\lesssim \Vert g{\Vert }_{L^{q_1}({\mathbb{R}}^2)}||h|{|}_{H^{s,q_2}({\mathbb{R}}^2)}+\Vert h{\Vert }_{L^{r_1}({\mathbb{R}}^2)}\Vert g{\Vert }_{H^{s,r_2}({\mathbb{R}}^2)}.$$

$$\hat{\omega }\in \Omega \to ⟨h(\hat{\omega }),\psi ⟩\in \mathbb{R}$$

$$\mathbb{E}h:\psi \in C_0^{\infty }({\mathbb{R}}^d)$$

$$\mathrm{Cov}h:({\psi }_1,{\psi }_2)\in C_0^{\infty }({\mathbb{R}}^d{)}^2$$

$$\mathrm{Cov}(⟨h,{\psi }_1⟩,⟨h,{\psi }_2⟩)=\mathbb{E}((⟨h,{\psi }_1⟩-\mathbb{E}⟨h,{\psi }_1⟩)(⟨h,{\psi }_2⟩-\mathbb{E}⟨h,{\psi }_2⟩))$$

$$⟨{\mathrm{Cov}}_h{\psi }_1,{\psi }_2⟩=\mathrm{Cov}(⟨h,{\psi }_1⟩,⟨h,{\psi }_2⟩)=\mathbb{E}(⟨h-\mathbb{E}h,{\psi }_1⟩⟨h-\mathbb{E}h,{\psi }_2⟩).$$

$$⟨C_h,{\psi }_1\otimes {\psi }_2⟩=⟨{\mathrm{Cov}}_h{\psi }_1,{\psi }_2⟩,\forall {\psi }_1,{\psi }_2\in C_0^{\infty }({\mathbb{R}}^2).$$

$$C_h(x,y)=\mathbb{E}((h(x)-\mathbb{E}h(x))(h(y)-\mathbb{E}h(y))).$$

$$C_h(x,y)=\{\begin{array}{ll}{c}_0(x,y)\log |x-y|+r_1(x,y)& \mathrm{for}m=2,\\ c_0(x,y)|x-y{|}^{m-2}+r_1(x,y)& \mathrm{for}m\in (2,5/2),\end{array}$$

$$H_n^{(1)}(t)=J_n(t)+\mathrm{i}{Y}_n(t),$$

$$J_n(t)$$

$$Y_n(t)$$

$$-\frac{1}{\pi }\sum _{p=0}^{\infty }\frac{(-1{)}^p}{p!(n+p)!}{\left(\frac{t}{2}\right)}^{n+2p}\{\psi (p+n)+\psi (p)\},$$

$$H_0^{(1)}(t)$$

$$H_1^{(1)}(t)$$

$$H_2^{(1)}(t)$$

$$H_3^{(1)}(t)$$

$${\Gamma }_n(z,\omega ):={\kappa }_{\mathrm{s}}^n{H}_n^{(1)}({\kappa }_{\mathrm{s}}|z|)-{\kappa }_{\mathrm{p}}^n{H}_n^{(1)}({\kappa }_{\mathrm{p}}|z|).$$

$${\Gamma }_1(z,\omega )=\frac{\mathrm{i}}{\pi }|z|\left({\kappa }_{\mathrm{s}}^2\ln \frac{{\kappa }_{\mathrm{s}}|z|}{2}-{\kappa }_{\mathrm{p}}^2\ln \frac{{\kappa }_{\mathrm{p}}|z|}{2}\right)+b_1({\kappa }_{\mathrm{s}}^2-{\kappa }_{\mathrm{p}}^2)|z|+O(|z{|}^3\ln \frac{|z|}{2}),$$

$${\Gamma }_2(z,\omega )=\frac{\mathrm{i}}{4\pi }\left({\kappa }_{\mathrm{s}}^4\ln \frac{{\kappa }_{\mathrm{s}}|z|}{2}-{\kappa }_{\mathrm{p}}^4\ln \frac{{\kappa }_{\mathrm{p}}|z|}{2}\right)|z{|}^2-\frac{\mathrm{i}}{\pi }({\kappa }_{\mathrm{s}}^2-{\kappa }_{\mathrm{p}}^2)+O(|z{|}^2),$$

$${\Gamma }_3(z,\omega )=\frac{2\mathrm{i}}{\pi }({\kappa }_{\mathrm{p}}^2-{\kappa }_{\mathrm{s}}^2)\frac{1}{|z|}+\frac{\mathrm{i}}{4\pi }({\kappa }_{\mathrm{p}}^4-{\kappa }_{\mathrm{s}}^4)|z|+O(|z{|}^3\ln \frac{|z|}{2}).$$

Full text

1 Introduction

The inverse source scattering problems are to recover the unknown sources from the radiated wave field which is generated by the unknown sources. These problems are motivated by significant applications in diverse scientific areas such as medical imaging [3, 23, 34], and antenna design and synthesis [20]. Driven by these applications, the inverse source scattering problems have been extensively studied by many researchers in both mathematical and engineering communities. Consequently, a great deal of mathematical and numerical results are available, especially for deterministic sources [1, 6, 13, 20, 22]. It is known that the inverse source problem, in general, does not have a unique solution at a single frequency due to the existence of non-radiating sources [8, 17, 21, 24]. There are two approaches to overcome the issue non-uniqueness: one is to seek the minimum energy solution [33], which represents the pseudo-inverse solution for the inverse source problem; the other is the use of multi-frequency data to achieve uniqueness and gain increasing stability [12, 14, 15, 19, 30].

In many situations, the source, hence the wave field, may not be deterministic but are rather modeled by random processes [7]. Due to the extra challenge of randomness and uncertainties, little is known for the inverse random source scattering problems. In [9, 11, 28, 27, 10, 16], the random source was assumed to be driven by an additive white noise. Mathematical modeling and numerical computation were proposed for a class of inverse source problems for acoustic and elastic waves. The method requires to know the expectation of the scattering data, which needs to be measured corresponding to a fairly large number of realizations of the source.

Recently, a different model is proposed in [18, 32] to describe random functions. The random function is considered to be a generalized Gaussian random function whose covariance is represented by a classical pseudo-differential operator. The authors studied an inverse problem for the two-dimensional random Schrödinger equation where the potential function was random. It is shown that the principle symbol of the covariance operator can be uniquely determined by the backscattered far field [18] or backscattered field [32], generated by a single realization of the random potential and plane waves [18] or a point source [32] as the incident field. A related work can be found in [25] where the authors considered an inverse scattering problem in a half-space with an impedance boundary condition where the impedance function was random. In [29], the inverse random source scattering problems were considered for the time-harmonic acoustic and elastic waves in a homogeneous and isotropic medium. The source is assumed to be a microlocally isotropic generalized Gaussian random function. It is shown that the amplitude of the scattering field averaged over the frequency band, obtained from a single realization of the random source, determines uniquely the principle symbol of the covariance operator. In this paper, we study an inverse random source scattering problem for the two-dimensional elastic wave equation with an inhomogeneous, anisotropic mass density. This paper significantly extends our previous work on the inverse random source problem for elastic waves. The techniques also differ greatly because a more complicated model equation is considered.

The wave propagation is governed by the stochastic elastic wave equation

[TABLE]

where $\bm{u}\in\mathbb{C}^{2}$ is the complex-valued displacement vector, $\omega>0$ is the angular frequency, $\lambda$ and $\mu$ are the Lamé constants satisfying $\mu>0,\lambda+\mu>0$, and $\bm{M}\in\mathbb{R}^{2\times 2}$ is a deterministic real-valued symmetric matrix with a compact support contained in $D\subset\mathbb{R}^{2}$ and represents either a linear load acting on the elastic medium or an inhomogeneous, anisotropic mass density of the elastic medium inside $D$. The randomness of (1) comes from the external source $\bm{f}=(f_{1},f_{2})^{\top}$. Throughout, we make the following assumption.

Assumption 1**.**

The domain $D$ is bounded, simply connected, and Lipschitz. The source $\bm{f}=(f_{1},f_{2})^{\top}$ is compactly supported in $D$ and $f_{j},j=1,2$ are microlocally isotropic Gaussian random fields of the same order $m\in[2,\frac{5}{2})$ in $D$. Each covariance operator $C_{f_{j}}$ is a classical pseudo-differential operator having the same principle symbol $\phi(x)|\xi|^{-m}$ with $\phi\in C_{0}^{\infty}(D),\phi\geq 0$. Moreover, the source $\bm{f}$ is assumed to be bounded almost surely with $\mathbb{E}(f_{j})=0$ and $\mathbb{E}(f_{1}f_{2})=0$.

Since (1) is imposed in the whole space $\mathbb{R}^{2}$, an appropriate radiation condition is needed to complete the problem formulation. By the Helmholtz decomposition, the displacement $\bm{u}$ can be decomposed into the compressional part $\bm{u}_{\rm p}$ and the shear part $\bm{u}_{\rm s}$ away from the source:

[TABLE]

For a scalar function $u$ and a vector function ${\bm{u}}=(u_{1},u_{2})^{\top}$, the vector and scalar cur operators are defined by

[TABLE]

The Kupradze–Sommerfeld radiation condition requires that $\bm{u}_{\rm p}$ and $\bm{u}_{\rm s}$ satisfy the Sommerfeld radiation condition:

[TABLE]

where $\kappa_{\rm p}$ and $\kappa_{\rm s}$ are known as the compressional wavenumber and the shear wavenumber, respectively, and are defined by

[TABLE]

Here

[TABLE]

Note that $c_{\rm p}$ and $c_{\rm s}$ are independent of $\omega$ and $c_{\rm p}<c_{\rm s}$.

Given $\omega,\lambda,\mu,\bm{M}$, and $\bm{f}$, the direct scattering problem is to determine $\bm{u}$ which satisfies (1)–(2). For $m\in[2,5/2)$, the random source is a rough field and belongs to the Sobolev space with a negative smoothness index almost surely. A careful study is needed to show the well-posedness of the direct scattering problem for such a distributional source. Using Green’s theorem and the Kupradze–Sommerfeld radiation condition, we show that the direct scattering problem is equivalent to a Lippmann–Schwinger equation. By the Fredholm alternative along with the unique continuation principle, we prove that the Lippmann–Schwinger equation has a unique solution which belongs to the Sobolev space with a negative smoothness index almost surely. Thus the well-posedness is established for the direct scattering problem.

Given $\omega,\lambda,\mu,\bm{M}$, the inverse scattering problem is to determine $\phi(x)$, the micro-correlation strength of the source, from the displacement measured in a bounded domain $U\subset\mathbb{R}^{2}\setminus\overline{D}$ standing for the measurement domain, which is required to satisfy the following assumption.

Assumption 2**.**

The measurement domain $U$ is bounded, simply connected, Lipschitz, convex, and has a positive distance to $D$.

In addition, the following assumption is imposed on $\bm{M}$.

Assumption 3**.**

The matrix $\bm{M}=(M_{ij})_{2\times 2}$ is a deterministic and real-valued symmetric matrix with $M_{ij}\in C_{0}^{1}(\overline{D})$ for $i,j=1,2$.

The following result concerns the uniqueness of the inverse scattering problem and is the main result of this paper.

Theorem 4**.**

Let $\bm{f},U$, and $\bm{M}$ satisfy Assumptions 1, 2, and 3, respectively. Then for all $x\in U$, it holds almost surely that

[TABLE]

where $a=\frac{1}{32\pi}\left({c_{\rm s}^{3-m}}+{c_{\rm p}^{3-m}}\right)$ is a constant. Moreover, the function $\phi$ can be uniquely determined from the integral equation (3) for all $x\in U$.

For any finite $Q$, the scattering data given in the left-hand side of (3) is random in the sense of that it depends on the realization of the source, while (3) shows that in the limit $Q\to\infty$, the scattering data becomes statistically stable, i.e., it is independent of realization of the source. Hence, Theorem 4 shows that the amplitude of the displacement averaged over the frequency band, measured from a single realization of the random source, can uniquely determine the micro-correlation strength function $\phi$. The proof of Theorem 4 combines the Born approximation, asymptotic expansions of the Green tensor, and microlocal analysis of integral operators

The paper is organized as follows. In Section 2, we briefly introduce some necessary notations including the Sobolev spaces, the generalized Gaussian random function, and some properties of the Hankel function of the first kind. Section 3 addresses the direct scattering problem; Sections 4 and 5 study the inverse scattering problem. In Section 3, the well-posedness of the direct scattering problem is established for a distributional source. Using the Riesz–Fredholm theory and the Sobolev embedding theorem, we show that the direct scattering problem is equivalent to a uniquely solvable Lippmann–Schwinger equation. Section 4 presents the Born approximation of the solution to the Lippmann–Schwinger integral equation. Section 5 examines the second term in the Born approximation via the microlocal analysis. The paper is concluded with some general remarks in Section 6.

2 Preliminaries

In this section, we introduce some notations and properties of the Sobolev spaces, the generalized Gaussian random functions, and the Hankel function of the first kind.

2.1 Sobolev spaces

Let $C_{0}^{\infty}(\mathbb{R}^{2})$ be the set of smooth functions with compact support, and $\mathcal{D^{\prime}}(\mathbb{R}^{2})$ be the set of generalized (distributional) functions. Given $1<p<\infty,s\in\mathbb{R}$, define the Sobolev space

[TABLE]

which has the norm

[TABLE]

With the definition of Sobolev spaces in the whole space, the Sobolev space $H^{s,p}(V)$ for any Lipschitz domain $V\subset\mathbb{R}^{2}$ can be defined as the restriction to $V$ of the elements in $H^{s,p}(\mathbb{R}^{2})$ with the norm

[TABLE]

By [26], for $s\in\mathbb{R}$ and $1<p<\infty$, $H^{s,p}_{0}(V)$ can be defined as the space of all distributions $h\in H^{s,p}(\mathbb{R}^{2})$ satisfying ${\rm supp}h\subset\overline{V}$ with the norm

[TABLE]

It is known that $C_{0}^{\infty}(V)$ is dense in $H_{0}^{s,p}(V)$ for any $1<p<\infty,s\in\mathbb{R}$; $C_{0}^{\infty}(V)$ is dense in $H^{s,p}(V)$ for any $1<p<\infty,s\leq 0$; $C^{\infty}(\overline{V})$ is dense in $H^{s,p}(V)$ for any $1<p<\infty,s\in\mathbb{R}$. In addition, by [26, Propositions 2.4 and 2.9], for any $s\in\mathbb{R}$ and $p,q\in(1,\infty)$ satisfying $\frac{1}{p}+\frac{1}{q}=1$, we have

[TABLE]

where the prime denotes the dual space.

The following two lemmas will be used in the subsequent analysis. The proofs of Lemma 5 and Lemma 6 can be found in [32, Lemma 2] and [35, Proposition 1], respectively.

Lemma 5**.**

Assume that $\epsilon>0$, $1<r<\infty$, $\frac{1}{r}+\frac{1}{r^{\prime}}=1$, $g\in H^{\epsilon,2r}_{\rm loc}(\mathbb{R}^{2})$, $h\in H_{0}^{-\epsilon,r^{\prime}}(\mathbb{R}^{2})$. Then $gh\in H_{0}^{-\epsilon,\tilde{r}}(\mathbb{R}^{2})$ and satisfies

[TABLE]

where $\tilde{r}=\frac{2r}{2r-1}$.

Lemma 6**.**

Assume that $s>0$, $1<\tilde{p}<\infty$ and $\frac{1}{\tilde{p}}=\frac{1}{q_{1}}+\frac{1}{q_{2}}=\frac{1}{r_{1}}+\frac{1}{r_{2}},q_{1},r_{1}\in(1,\infty],q_{2},r_{2}\in(1,\infty)$. Then the following estimate holds

[TABLE]

Throughout the paper, $a\lesssim b$ stands for $a\leq Cb$, where $C$ is a positive constant and its specific value is not required but should be clear from the context.

2.2 Generalized Gaussian random functions

Let $(\Omega,\mathcal{F},\mathcal{P})$ be a complete probability space. The function $h$ is said to be a generalized Gaussian random function if $h:\Omega\rightarrow\mathcal{D}^{\prime}(\mathbb{R}^{2})$ is a mapping such that, for each $\hat{\omega}\in\Omega$, the realization $h(\hat{\omega})$ is a real-valued linear functional on $C_{0}^{\infty}(\mathbb{R}^{2})$ and the function

[TABLE]

is a Gaussian random variable for all $\psi\in C_{0}^{\infty}(\mathbb{R}^{2})$. The distribution of $h$ is determined by its expectation $\mathbb{E}h$ and the covariance ${\rm Cov}h$ defined as follows

[TABLE]

where $\mathbb{E}\langle h,\psi\rangle$ denotes the expectation of $\langle h,\psi\rangle$ and

[TABLE]

denotes the covariance of $\langle h,\psi_{1}\rangle$ and $\langle h,\psi_{2}\rangle$. The covariance operator ${\rm Cov}_{h}:C_{0}^{\infty}(\mathbb{R}^{2})\rightarrow\mathcal{D^{\prime}}(\mathbb{R}^{2})$ is defined by

[TABLE]

Since the covariance operator ${\rm Cov}_{h}$ is continuous, the Schwartz kernel theorem shows that there exists a unique $C_{h}\in\mathcal{D}^{\prime}(\mathbb{R}^{2}\times\mathbb{R}^{2})$, usually called the covariance function, such that

[TABLE]

By (4) and (5), it is easy to see that

[TABLE]

In this paper, we are interested in the generalized, microlocally isotropic Gaussian random function which is defined as follows.

Definition 7**.**

A generalized Gaussian random function $h$ on $\mathbb{R}^{2}$ is called microlocally isotropic of order $m$ in D, if the realizations of $h$ are almost surely supported in the domain $D$ and its covariance operator ${\rm Cov}_{h}$ is a classical pseudo-differential operator having the principal symbol $\phi(x)|\xi|^{-m}$, where $\phi\in C_{0}^{\infty}(\mathbb{R}^{2})$ satisfies ${\rm supp}\phi\subset D$ and $\phi(x)\geq 0$ for all $x\in\mathbb{R}^{2}$.

In particular, we pay attention to the case $m\in[2,5/2)$, which corresponds to rough fields. The following results will also be used in the subsequent analysis. The proofs of Lemmas 8 and 9 can be found in [32, Theorem 2 and Proposition 1].

Lemma 8**.**

Let $h$ be a generalized, microlocally isotropic Gaussian random function of order $m$ in $D$. If $m=2$, then $h\in H^{-\varepsilon,p}(D)$ almost surely for all $\varepsilon>0,1<p<\infty$. If $m\in(2,5/2)$, then $h\in C^{\alpha}(D)$ almost surely for all $\alpha\in(0,\frac{m-2}{2})$.

Lemma 9**.**

Let $h$ be a microlocally isotropic Gaussian random field of order $m\in[2,5/2)$. Then the Schwartz kernel of the covariance operator ${\rm Cov}_{h}$ has the form

[TABLE]

where $c_{0}\in C_{0}^{\infty}(D\times D)$ and $r_{1}\in C_{0}^{\alpha}(D\times D)$ for any $\alpha<1$.

2.3 The properties of the Hankel function of the first kind

In this subsection, we present some asymptotic expansions of the Hankel function of the first kind for small and large arguments. Let $H_{n}^{(1)}$ be the Hankel function of the first kind. Recall the definition

[TABLE]

where $J_{n}$ and $Y_{n}$ are the Bessel functions of the first and second kind with order $n$, respectively. They admit the following expansions

[TABLE]

where $\gamma:=\lim_{p\rightarrow\infty}\left\{\sum_{j=1}^{p}j^{-1}-\ln p\right\}$ denotes the Euler constant, $\psi(0)=0$, $\psi(p)=\sum_{j=1}^{p}j^{-1}$, and the finite sum in (7) is set to be zero for $n=0$.

Using the expansions (6) and (7), we may verify as $t\to 0$ that

[TABLE]

where $b_{0}=1+\frac{2{\rm i}}{\pi}\gamma$, $b_{1}=\frac{1}{2}+\frac{{\rm i}}{\pi}\gamma-\frac{\rm i}{2\pi}$, $b_{2}=\frac{\gamma{\rm i}}{4\pi}-\frac{3{\rm i}}{16\pi}+\frac{1}{8},b_{3}=\frac{{\gamma\rm i}}{24\pi}+\frac{1}{48}-\frac{11{\rm i}}{288\pi}$. Denote

[TABLE]

Noting (8)–(11), we have from a direct calculation as $|z|\to 0$ that

[TABLE]

For a large argument, i.e., as $|z|\to\infty$, it follows from [5, (9.2.7)–(9.2.10)] and [31, (5.11.4)] that the Hankel function of the first kind $H_{n}^{(1)}$ has the following asymptotics

[TABLE]

where $\delta$ is a small positive number and the coefficients $a_{j}^{(n)}=(-2{\rm i})^{j}\sqrt{\frac{2}{\pi}}(n,j)$ with

[TABLE]

Using the first $N$ terms in the asymptotic of $H_{n}^{(1)}(\kappa|z|)$, we define

[TABLE]

Denote $\Gamma_{n,N}(\kappa|z|):=H_{n}^{(1)}(\kappa|z|)-H_{n,N}^{(1)}(\kappa|z|)$, it is easy to show from (2.3) that

[TABLE]

3 The direct scattering problem

This section aims to establish the well-posedness of the direct scattering problem for a distributional source. Based on Green’s theorem and the Kupradze–Sommerfeld radiation, the direct problem is equivalently formulated as a Lippmann–Schwinger equation, which is shown to have a unique solution by using the Riesz–Fredholm theory and the Sobolev embedding theorem.

By Lemma 8, we have that $\bm{f}\in H^{-\varepsilon,p}(D)^{2}$ almost surely for all $\varepsilon>0$, $1<p<\infty$ if $m=2$; $\bm{f}\in C^{0,\alpha}(D)^{2}$ almost surely for all $\alpha\in(0,\frac{m-2}{2})$ if $m\in(2,5/2)$. Therefore, it suffices to show that the scattering problem (1)–(2) has a unique solution for such a deterministic source $\bm{f}\in H^{-\varepsilon,p}(D)^{2}$.

Introduce the Green tensor $\bm{G}(x,y,\omega)\in\mathbb{C}^{2\times 2}$ for the Navier equation

[TABLE]

where $\bm{I}$ is the $2\times 2$ identity matrix, $\Phi(x,y,\kappa)=\frac{\rm i}{4}H_{0}^{(1)}(\kappa|x-y|)$ is the fundamental solution for the two-dimensional Helmholtz equation, and $\nabla_{x}\nabla_{x}^{\top}$ is defined by

[TABLE]

for some scalar function $\varphi$ defined in $\mathbb{R}^{2}$. It is easy to note that the Green tensor $\bm{G}(x,y,\omega)$ is symmetric with respect to the variables $x$ and $y$.

In order to obtain the well-posedness of the scattering problem (1)–(2), we first derive a Lippmann–Schwinger equation which is equivalent to the direct scattering problem, then we show that the Lippmann–Schwinger equation has a unique solution.

Theorem 10**.**

For some $p\geq 2$, $\frac{1}{p}+\frac{1}{p^{\prime}}=1$, $0<\varepsilon<\frac{2}{p}$, $\bm{f}\in H_{0}^{-\varepsilon,p^{\prime}}(D)^{2}$, if $\bm{M}$ satisfies Assumption 3, then the scattering problem (1)–(2) is equivalent to the Lippmann–Schwinger equation

[TABLE]

Proof.

Let $\bm{u}\in H^{\varepsilon,p}_{\rm loc}(\mathbb{R}^{2})^{2}$ be a solution to (19), then we have

[TABLE]

Since the Green tensor $\bm{G}(x,y,\omega)$ and its derivatives satisfy the Kupradze–Sommerfeld radiation condition, we conclude that $\bm{u}$ also satisfies the Kupradze–Sommerfeld radiation condition. By (18), the Green tensor $\bm{G}(x,y,\omega)$ satisfies

[TABLE]

Letting $y=0$ and taking the Fourier transform with respect to $x$ on both side of (20) yields

[TABLE]

Note that the integral in (19) is a convolution since $\bm{G}(x,y,\omega)$ is a function of $x-y$. Taking the Fourier transform on both sides of (19) and using (21) lead to

[TABLE]

which gives

[TABLE]

Taking the inverse Fourier transform yields

[TABLE]

Hence, $\bm{u}$ is the solution of the direct scattering problem (1)–(2).

Conversely, if $\bm{u}$ is a solution of the direct scattering problem (1)–(2), we show that $\bm{u}$ satisfies the Lippmann–Schwinger equation (19). Since

[TABLE]

where $\bm{u}\in H_{\rm loc}^{\varepsilon,p}(\mathbb{R}^{2})^{2}$ and $M_{ij}\in C_{0}^{1}(\overline{D})$. Note that $\bm{f}\in H_{0}^{-\varepsilon,p^{\prime}}(\mathbb{R}^{2})^{2}$, we have that $\bm{M}\bm{u}+\bm{f}\in H_{0}^{-\varepsilon,p^{\prime}}(\mathbb{R}^{2})^{2}$. An application of Lemma 4.1 in [29] shows that for some fixed $x\in\mathbb{R}^{2}$, $\bm{G}(x,\cdot,\omega)\in[L^{2}_{\rm loc}(\mathbb{R}^{2})\cap H_{\rm loc}^{1,\hat{p}}(\mathbb{R}^{2})]^{2\times 2}$ for $\hat{p}\in(1,2)$. Since $0<\varepsilon<\frac{2}{p}$, a simple calculation gives that $\frac{1}{p}-\frac{\varepsilon}{2}>0$. Let $\widetilde{\delta}=\frac{1}{p}-\frac{\varepsilon}{2}$ and define $\widetilde{p}:=\frac{2}{1+\widetilde{\delta}}<2$, then $\frac{1}{\widetilde{p}}-\frac{1}{2}<\frac{1}{p}-\frac{\varepsilon}{2}$. It follows from the Sobolev embedding theorem that $H_{\rm loc}^{1,\widetilde{p}}(\mathbb{R}^{2})$ is embedded into $H_{\rm loc}^{\varepsilon,p}(\mathbb{R}^{2})$, which implies that $\bm{G}(x,\cdot,\omega)\in[H_{\rm loc}^{\varepsilon,p}(\mathbb{R}^{2})]^{2\times 2}$. Choose a large enough ball $B_{r}$ such that $D\subset B_{r}$, then we have in the sense of distributions that

[TABLE]

Denote by $T$ the operator that maps $\bm{u}$ to the left-hand side of the above equation. For $\bm{\psi}\in C^{\infty}(\mathbb{R}^{2})^{2}$, by the similar arguments as those in the proof of Lemma 4.3 in [29], we obtain

[TABLE]

where $P\bm{\psi}:=\mu\frac{\partial\bm{\psi}}{\partial\bm{\nu}}+(\lambda+\mu)(\nabla\cdot\bm{\psi})\bm{\nu}$ and $\bm{\nu}$ is the unit normal vector on the boundary $\partial B_{r}$.

Approximating $\bm{u}$ with smooth functions, we get

[TABLE]

Using the radiation condition yields

[TABLE]

Therefore,

[TABLE]

which shows that $\bm{u}$ satisfies the Lippmann–Schwinger equation (19) and completes the proof. ∎

The Lippmann–Schwinger equation (19) can be written in the operator form

[TABLE]

where the operators $H_{\omega}$ and $K_{\omega}$ are defined by

[TABLE]

Lemma 11**.**

Assume that $p\geq 2$, $\frac{1}{p}+\frac{1}{p^{\prime}}=1$, $0<\varepsilon<\frac{2}{p}$, and $\bm{M}$ satisfies Assumption 3. Then the operators $H_{\omega}:H_{0}^{-s}(D)^{2}\rightarrow H^{s}(D)^{2}$ and $K_{\omega}:H^{\varepsilon,p}(D)^{2}\rightarrow H^{\varepsilon,p}(D)^{2}$ are bounded for $s\in(0,1)$. Moreover, $K_{\omega}:H^{\varepsilon,p}(D)^{2}\rightarrow H^{\varepsilon,p}(D)^{2}$ is compact.

Proof.

We study the asymptotic expansion of the Green tensor $\bm{G}(x,y,\omega)$ when $|x-y|\to 0$. Recall the Green tensor:

[TABLE]

and the recurrence relation for the Hankel function of the first kind [31, (5.6.3)]:

[TABLE]

A direct calculation shows for $i,j=1,2$ that

[TABLE]

where $\delta_{ij}$ is the Kronecker delta function. Substituting (12)–(13) into (25) gives

[TABLE]

Comparing (3) with (8), we conclude that the singularity of $\nabla_{x}\nabla_{x}^{\top}(\Phi(x,y,\kappa_{\rm s})-\Phi(x,y,\kappa_{\rm p}))$ is not exceeding the singularity of $\Phi(x,y,\kappa_{\rm s})\bm{I}$ when $|x-y|\to 0$. It follows from Lemma 5 that $H_{\omega}:H_{0}^{-s}(D)^{2}\rightarrow H^{s}(D)^{2}$ is bounded for $s\in(0,1)$.

For $\bm{u}\in H^{\varepsilon,p}(D)^{2}$ and $M_{ij}\in C_{0}^{1}(D)\subset H_{0}^{-\varepsilon,p_{1}^{\prime}}(D)$, by Lemma 5, we obtain that $M_{ij}\bm{u}$ is a well-defined element of $H_{0}^{-\varepsilon,p^{\prime}}(D)^{2}$ and

[TABLE]

For some fixed $\varepsilon\in(0,\frac{2}{p})$, we define $\tilde{\delta}=\frac{1}{p}-\frac{\varepsilon}{2}\in(0,1)$ and $s=1-\tilde{\delta}\in(0,1)$. It is clear to note that $\frac{1}{2}-\frac{s}{2}<\frac{1}{p}-\frac{\varepsilon}{2}$. The Sobolev embedding theorem implies that $H^{s}(D)$ is embedded compactly into $H^{\varepsilon,p}(D)$ and $H_{0}^{-\varepsilon,p^{\prime}}(D)$ is embedded compactly into $H^{-s}_{0}(D)$. Noting that $K_{\omega}\bm{u}=H_{\omega}(\bm{M}\bm{u})$ and $\bm{M}\bm{u}\in H_{0}^{-\varepsilon,p^{\prime}}(D)^{2}$, which is embedded compactly into $H_{0}^{-s}(D)^{2}$, and that $H_{\omega}:H_{0}^{-s}(D)^{2}\rightarrow H^{s}(D)^{2}$ is bounded, we claim from (27) that $K_{\omega}:H^{\varepsilon,p}(D)^{2}\rightarrow H^{\varepsilon,p}(D)^{2}$ is bounded and compact. ∎

Now we present the existence of a unique solution of the direct scattering problem (1)–(2).

Theorem 12**.**

Let $\bm{f}\in H_{0}^{-\varepsilon,p^{\prime}}(D)^{2}$ with $0<\varepsilon<\frac{2}{p}$ and $\bm{M}$ satisfy Assumption 3. Then the Lippmann–Schwinger equation (22) has a unique solution $\bm{u}\in H_{\rm loc}^{\varepsilon,p}(\mathbb{R}^{2})^{2}$, which implies that the scattering problem (1)–(2) has a unique solution $\bm{u}\in H_{\rm loc}^{\varepsilon,p}(\mathbb{R}^{2})^{2}$ which satisfies the stability estimate

[TABLE]

Proof.

For the Lippmann–Schwinger equation $(I+K_{\omega})\bm{u}=-H_{\omega}\bm{f}$, by Lemma 11, we obtain that $H_{\omega}\bm{f}\in H^{\varepsilon,p}(D)^{2}$ for $\bm{f}\in H_{0}^{-\varepsilon,p^{\prime}}(D)^{2}$ and $I+K_{\omega}:H^{\varepsilon,p}(D)^{2}\rightarrow H^{\varepsilon,p}(D)^{2}$ is a Fredholm operator. Thus, by the Fredholm alternative, it suffices to show that $(I+K_{\omega})\bm{u}=0$ has only the trivial solution $\bm{u}=0$.

For $(I+K_{\omega})\bm{u}=0$, we have

[TABLE]

Thus we have $\bm{u}$ is smooth in $\mathbb{R}^{2}\setminus\overline{D}$ and

[TABLE]

which implies

[TABLE]

Taking the inverse Fourier transform of the above equation yields

[TABLE]

By the Helmholtz decomposition, there exists two scalar potential functions $\psi_{1}$ and $\psi_{2}$ such that

[TABLE]

Substituting (29) into (28) gives that

[TABLE]

which implies that

[TABLE]

Letting $\bm{u}_{\rm p}=\nabla\psi_{1}$ and $\bm{u}_{\rm s}={\bf curl}\psi_{2}$, we obtain that

[TABLE]

and

[TABLE]

Since ${\rm supp}M_{ij}\subset D$, it follows from (30)–(31) that $\bm{u}_{\rm p}$ and $\bm{u}_{\rm s}$ satisfy the homogeneous Helmholtz equation in $\mathbb{R}^{2}\setminus\overline{D}$ and the Sommerfeld radiation condition. Hence, $\bm{u}_{\rm p}$ and $\bm{u}_{\rm s}$ admit the following asymptotic expansions

[TABLE]

Noting that $\bm{u}_{\rm p}$ satisfies the Sommerfeld radiation condition, when $r\rightarrow\infty$, we have

[TABLE]

Combining the second Green theorem and (30)–(31), we get

[TABLE]

where $u_{{\rm p},1}$ and $u_{{\rm p},2}$ are the components of $\bm{u}_{\rm p}$. Since $M$ is real-valued and symmetric, taking the imaginary part of the above equation leads to ${\rm Im}\int_{\partial B_{r}}\bm{u}_{\rm p}\partial_{\nu}\overline{\bm{u}}_{\rm p}ds=0$ which yields $\lim\limits_{r\rightarrow\infty}\int_{\partial B_{r}}|\bm{u}_{\rm p}|^{2}dx=0$. Using (32), we obtain $\int_{\partial B_{1}}|\bm{u}_{{\rm p},\infty}|^{2}ds=0$, which implies $\bm{u}_{{\rm p},\infty}=0$, so $\bm{u}_{\rm p}(x)=0$ in $\mathbb{R}^{2}\setminus\overline{D}$. Similarly, we can obtain $\bm{u}_{\rm s}=0$ in $\mathbb{R}^{2}\setminus\overline{D}$. Thus, we have $\bm{u}=0$ in $\mathbb{R}^{2}\setminus\overline{D}$. Since $M_{ij}\in C_{0}^{1}(\overline{D})$, it follows from the unique continuation (e.g., [4]) that $\bm{u}=0$ in $\mathbb{R}^{2}$, which shows that $I+K_{\omega}$ is injective and completes the proof. ∎

4 Born approximation

As shown in the previous section, the direct scattering problem is equivalent to the Lippmann–Schwinger equation

[TABLE]

Consider the Born sequence of the Lippmann–Schwinger equation

[TABLE]

where the initial guess is given by

[TABLE]

which is called the Born approximation to the solution of the Lippmann–Schwinger equation. Here, $K_{\omega}$ and $H_{\omega}$ are operators given by (23) and (24), respectively.

We aim to show that for sufficient large $\omega$ and $x\in U$, the Born series $\sum_{n=0}^{\infty}\bm{u}_{n}(x)$ converges to the solution $\bm{u}(x)$ and the higher order terms decay in an appropriate way.

Lemma 13**.**

For any $1\leq p\leq 2\leq r\leq\infty$, $s\in(0,1)$ and $\omega\geq 1$, the following estimates hold

[TABLE]

where the constant $c=c(\hat{\omega})$ is finite almost surely.

The proof of Lemma 13 can be found in [32, Lemma 5]. By Lemma 13, we have for large enough $\omega$ that

[TABLE]

Since $(I+K_{\omega})^{-1}\bm{u}_{0}=\bm{u}$, taking the inverse of the operator $I+K_{\omega}$ in (34) leads to

[TABLE]

where $\bm{b}(x,\omega):=\sum_{n=2}^{\infty}\bm{u}_{n}(x,\omega)$. With the convergence of the Born approximation (35), we can analyze each item in the Born approximation. For the leading item $\bm{u}_{0}$, we have the following result [29, Theorem 4.6].

Theorem 14**.**

Let $\bm{f}$ satisfy Assumption 1. For all $x\in U$, it holds almost surely that

[TABLE]

where $a$ is a constant given in Theorem 4.

Now we analyze the item $\bm{b}(x,\omega)$. For $n\geq 2$, by Lemma 13, we get

[TABLE]

which gives

[TABLE]

Since $0<\varepsilon<\frac{2}{p}$ and $p>2$, we can choose suitable $\varepsilon,p$ such that $\varepsilon^{\prime}=6\varepsilon+5-\frac{10}{p}$ is small enough and

[TABLE]

Hence, when $Q\to\infty$,

[TABLE]

where $\alpha=m+2\varepsilon^{\prime}-3$. Note that $m\in[2,5/2)$, we have $\alpha\in(-1,0)$ which is used in (37).

5 The analysis of $\bm{u}_{1}(x,\omega)$

In this section, we consider the term $\bm{u}_{1}(x,\omega)$ in the Born series (33), which is given by.

[TABLE]

It turns out the term $\bm{u}_{1}(x,\omega)$ is very difficult to analyze. Fortunately, after tedious calculations, we find out that the contribution of $\bm{u}_{1}$ can be ignored. We present the main result of this section.

Theorem 15**.**

Let $\bm{f}$, $U$, and $\bm{M}$ satisfy Assumption 1, Assumption 2, and Assumption 3, respectively. Then for $x\in U$, it holds almost surely that

[TABLE]

Proof.

Recall the Green tensor in (18), a direct computation shows

[TABLE]

where $x-y=(x_{1}-y_{1},x_{2}-y_{2})^{\top}$ and $\Gamma_{1},\Gamma_{2}$ are given in (12), (13). Noting the definition of $H_{n,N}^{(1)}$ in (16), we define the notations $\Theta_{n}(z,\omega):=\kappa_{\rm s}^{n}H_{n,0}^{(1)}(\kappa_{\rm s}|z|)-\kappa_{\rm p}^{n}H_{n,0}^{(1)}(\kappa_{\rm p}|z|)$,

[TABLE]

and

[TABLE]

Now we estimate the order of the difference $\bm{u}_{1}-\bm{u}_{1,l}$ with respect to the angular frequency $\omega$. A simple calculation yields

[TABLE]

Since $x\in U$, $y\in D$, there exists $c_{1},c_{2}>0$ such that $c_{1}<|x-y|<c_{2}$. By (17), we have

[TABLE]

A direct computation shows that $\nabla\Gamma_{n,0}(\kappa|x-\cdot|)\lesssim\kappa^{-\frac{1}{2}}$. Hence

[TABLE]

By (41) and (42), we get

[TABLE]

Therefore,

[TABLE]

It follows from Lemma 13 that we obtain

[TABLE]

where we use the fact that $\|\bm{f}\|_{H_{0}^{-\varepsilon,p^{\prime}}(D)^{2}}$ is bounded almost surely. Denoting $\varepsilon_{l}=3\varepsilon+2(1-\frac{2}{p})$ which can be sufficient small for suitably chosen $\varepsilon$ and $p$ due to $p\geq 2$ and $0<\varepsilon<\frac{2}{p}$, we have from (43) and (44) that

[TABLE]

In order to analyze the term $\bm{u}_{1,l}$, we replace the Green tensor $\bm{G}(y,z,\omega)$ in $\bm{u}_{1,l}$ by $\bm{G}_{0}(y,z,\omega)$ and define

[TABLE]

Next is estimate the order of the difference $\bm{u}_{1,l}-\bm{u}_{1,r}$ which is given by

[TABLE]

where

[TABLE]

for $i,j,k,l=1,2$. Here, $G_{ij}$ and $G_{0,ij}$ represent the elements of the matrix $\bm{G}$ and $\bm{G}_{0}$, respectively.

Now we only focus on the analysis of the term $I_{111}^{(1)}$ and show the details, other terms can be analyzed in a similar way. In the dual sense, we have

[TABLE]

By (39) and (40), we can split $\bm{G}_{11}(y,z,\omega)-\bm{G}_{0,11}(y,z,\omega)$ into three terms

[TABLE]

with

[TABLE]

Note $y,z\in D$ and $D$ is a bounded domain. Next is to estimate the term $\|\bm{G}_{11}(y,z,\omega)-\bm{G}_{0,11}(y,z,\omega)\|_{H^{\varepsilon,\tilde{p}}(D\times D)}$, we only need to estimate $\|g_{j}(z,\omega)\|_{H^{\varepsilon,\tilde{p}}(B)},j=0,1,2$ for some bounded domain containing the origin.

We analyze the three terms one by one. For large $\kappa_{\rm s}|z|$, it is easy to note from (17) that

[TABLE]

For small $\kappa_{\rm s}|z|$, using (8) and (16) gives that

[TABLE]

By (47) and (48), we obtain

[TABLE]

holds for $\tilde{p}<\frac{4}{3}$, where $R:=\max\{|z|,z\in B\}$. Since

[TABLE]

Hence, we have for large $\kappa_{\rm s}|z|$ that

[TABLE]

For small $\kappa_{\rm s}|z|$, we get

[TABLE]

By (50) and (51), we conclude for $\tilde{p}<\frac{4}{3}$ that

[TABLE]

Using (49) and (52), we have for $\tilde{p}<\frac{4}{3}$ that

[TABLE]

Now we analyze the term $g_{1}(z,\omega)$ which is given by

[TABLE]

For large $\omega|z|$, it follows from (17) that

[TABLE]

For small $\omega|z|$, by (7) and (10), we have

[TABLE]

Combining (54) and (55) implies for $\tilde{p}<\frac{4}{3}$ that

[TABLE]

For convenience, we split $g_{1}$ into two parts by $g_{1}(z,\omega)=g_{11}(z,\omega)+g_{12}(z,\omega)$ with

[TABLE]

For large $\omega|z|$, by (9), we have

[TABLE]

For small $\omega|z|$, by (7), we have

[TABLE]

Combining (56) and (57) yields for $\tilde{p}<\frac{4}{3}$ that

[TABLE]

For the $\nabla g_{11}(z,\omega)$, we have

[TABLE]

For large $\omega|z|$, (9) implies

[TABLE]

For small $\omega|z|$, (8) implies

[TABLE]

Following (59) and (60), we get for $\tilde{p}<\frac{4}{3}$ that

[TABLE]

Using (58) and (61), we have that

[TABLE]

Since

[TABLE]

it suffices to prove that $\omega^{-\frac{3}{2}}|z|^{-\frac{3}{2}}\in H^{\varepsilon,\tilde{p}}(B)$. By the Slobodeckij semi-norm, we need to prove

[TABLE]

To prove (63), we need the following two lemmas, one is the integrability criterion and the other is Young’s inequality for convolutions [2, Theorem 2.24].

Lemma 16**.**

For the $n-$dimensional space, we have

[TABLE]

This lemma is fundamental and can be easily proved by using the polar coordinates.

Lemma 17**.**

Let $s_{1},s_{2},s_{3}\geq 1$ and suppose that $\frac{1}{s_{1}}+\frac{1}{s_{2}}+\frac{1}{s_{3}}=2$. It holds that

[TABLE]

for any $h_{1}\in L^{s_{1}}(\mathbb{R}^{n})$, $h_{2}\in L^{s_{2}}(\mathbb{R}^{n})$, $h_{3}\in L^{s_{3}}(\mathbb{R}^{n})$.

Since

[TABLE]

hence,

[TABLE]

We choose $\tilde{p}=\frac{10}{9}$, $\varepsilon=\frac{1}{5}$, $s_{1}=\frac{89}{50}$, $s_{2}=\frac{59}{50}$ and $s_{3}=\frac{5251}{3102}$, then we have $\frac{1}{s_{1}}+\frac{1}{s_{2}}+\frac{1}{s_{3}}=2$ and $\tilde{p}s_{1}<2$, $\frac{3}{2}\tilde{p}s_{2}<2$, $(2+\varepsilon\tilde{p}-\tilde{p})s_{3}<2$. A direct application of Lemmas 16 and 17 leads to

[TABLE]

where $B_{2R}$ is the ball with radius $2R$ and center at the origin, and $\chi_{B}$ is the characteristic function of the domain $B$ which equals to 1 in $B$ and vanishes outside of $B$. We can prove $I_{2}<\infty$ by a similar argument. Therefore,

[TABLE]

Next we analyze the term $g_{2}(z,\omega)$ which is given by

[TABLE]

For large $\omega|z|$, (17) shows that

[TABLE]

For small $\omega|z|$, from (13) we have

[TABLE]

Thus, (65) and (66) implies for $\tilde{p}<\frac{4}{3}$ that

[TABLE]

A direct computation shows that

[TABLE]

For large $\omega|z|$, from (9) we know

[TABLE]

For small $\omega|z|$, from (13) and (14), we obtain

[TABLE]

By (67) and (68), we conclude for $\tilde{p}<\frac{4}{3}$ that

[TABLE]

Using (67) and (69) and interpolation, we have for $\tilde{p}<\frac{4}{3}$ that

[TABLE]

Noting that $D$ is a bounded domain, and combining (53), (62), (64), and (69), we obtain for any $\varepsilon\in(0,\frac{1}{5}]$ and $\tilde{p}\in[1,\frac{10}{9}]$ that

[TABLE]

Since $\bm{G}_{0,11}(x,y,\omega)$ is smooth for $x\in U$ and $y\in D$, $M_{11}(y)\in C_{0}^{1}(\overline{D})$, and $f_{1}(z)\in H^{-\varepsilon,\tilde{p}}(D)$ for any $\varepsilon>0$ and $1<\tilde{p}<\infty$, we have $\bm{G}_{0,11}(x,y,\omega)M_{11}(y)f_{1}(z)\in H_{0}^{-\varepsilon,\tilde{p}}(D\times D)$. Moreover,

[TABLE]

Thus, we obtain for sufficient large $\omega$ that

[TABLE]

Substituting (69) and (70) into (46) yields $|I_{111}^{(1)}|\lesssim\omega^{-2+\varepsilon}$ holds for any $\varepsilon\in(0,\frac{1}{5}]$.

Using similar proofs, we can obtain estimates for $I_{112}^{(1)}$, $\cdots$, $I_{222}^{(2)}$ and get

[TABLE]

Noting (45), we have

[TABLE]

Since

[TABLE]

and

[TABLE]

for $m\in[2,5/2)$ and small enough $\varepsilon$. To prove (38), it is sufficient to prove

[TABLE]

It follows from a straightforward but tedious calculation that the vector $\bm{u}_{1,r}(x,\omega)$ can be decomposed into three parts according to the order of $\omega$ in the following form

[TABLE]

where

[TABLE]

[TABLE]

[TABLE]

Here $\bm{J}(x-y)=(x-y)(x-y)^{\top}$ and $\bm{J}(y-z)=(y-z)(y-z)^{\top}$.

By (72) and the Cauchy–Schwartz inequality, we have

[TABLE]

Noting the facts that $|x-y|$ has a positive lower bound for $x\in U$, $y\in D$, $\||y-z|^{-\frac{3}{2}}\|_{H^{\frac{1}{5},\frac{10}{9}}(D\times D)}$ is bounded from the above analysis about $g_{12}$, $M_{ij}(y)\in C_{0}^{1}(\overline{D})$, and $\|f_{j}(z)\|_{H^{-\frac{1}{5},10}(D)}$ is bounded from the assumption, we conclude that

[TABLE]

Hence, we have as $\omega\to\infty$ that

[TABLE]

To prove (71), it suffices to prove that

[TABLE]

We claim that in order to prove (73), it will be enough to show that

[TABLE]

To show this, we notice that

[TABLE]

From the dominated convergence theorem, the last integral in the above inequality converges almost surely to zero as $Q\to\infty$, so the claim follows. The remaining part of the proof will focus on (74). To this end, we define

[TABLE]

where $c_{1},c_{2}>0$, $p_{1},...,l_{2}\geq 0$, $\tilde{f}$ denotes a generalized Gaussian random field which equals to $f_{1}$ or $f_{2}$, and $q(y)\in C_{0}^{1}(\overline{D})$ stands for $M_{ij}(y)$. From the formulation of $\bm{v}_{1}(x,\omega)$, we know that it is a linear combination of $g(x,\omega)$ for different $(l_{1},l_{2},p_{1},p_{2},p_{3},p_{4})\in S$ which is given by

[TABLE]

To prove (74), it is enough to show that

[TABLE]

In the following, we consider two cases.

Case 1. $m=2$. In this case, Lemma 8 claims that $\tilde{f}\in H^{-\varepsilon,p}(D)$ almost surely for any $\varepsilon>0$ and $1<p<\infty$. In order to avoid the distribution dualities, we introduce the modification $\tilde{f}_{\delta}:=\tilde{f}*\rho_{\delta}$ where $\rho_{\delta}:=\delta^{-2}\rho\left(\frac{x}{\delta}\right)$,$\rho\in C_{0}^{\infty}(\mathbb{R}^{2})$ is a radially symmetric function satisfying $\int_{\mathbb{R}^{2}}\rho(x)dx=1$. We denote $g_{\delta}$ by replacing $\tilde{f}$ by the standard mollification $\tilde{f}_{\delta}$ in (75). Let $M_{\delta}\tilde{f}:=\tilde{f}_{\delta}$ be the modification operator, and $C_{\delta}$ be the covariance operator of $\tilde{f}_{\delta}$. Then it is easy to verify that $C_{\delta}=M_{\delta}C_{\tilde{f}}M_{\delta}$ and $g_{\delta}(x,\omega)\to g(x,\omega)$ as $\delta\to 0$. To prove (76), we claim that it is enough to show that

[TABLE]

If (77) holds, applying the Fubini theorem and Fatou lemma implies that

[TABLE]

which shows that (76) holds immediately. So, we focus on the prove of (77) for this case. To this end, we look at the phase function $A(y,z)=c_{1}|x-y|+c_{2}|y-z|$ for some fixed $x\in U$. It is easy to see that $A(y,z)$ is smooth on $D\times D$ apart from the subset where $y=z$. A direct computation shows

[TABLE]

Hence,

[TABLE]

Since

[TABLE]

where $\theta$ denotes the angle between $y$ and $y-x$, noting the facts that the origin belongs to $U$ and $U$ is convex, we have $(y,z)\cdot\nabla A(y,z)$ has a positive lower bound for $(y,z)\in D\times D$ and $y\neq z$. So

[TABLE]

Our aim is to express $g_{\delta}(x,\omega)$ as a one-dimensional Fourier transform and get rid of the variable $\omega$. To this end, we define the following surface

[TABLE]

It is easy to see that there exists smallest and largest values $T_{0}=T_{0}(x)$ and $T_{1}=T_{1}(x)$ such that $\Gamma^{\prime}_{t}$ is nonempty only for $t\in[T_{0},T_{1}]$. Now we fix a $\tilde{t}\in[T_{0},T_{1}]$, then there exists $\eta=\eta(\tilde{t})$ and an open cone $K=K(\tilde{t})\subset{\mathbb{R}^{4}}$ with center at the origin such that for $t_{0}=\tilde{t}-\eta$ and $t_{1}=\tilde{t}+\eta$, we have

[TABLE]

Moreover, since $D$ has a positive distance to the origin we may also choose $\eta$ and $K$ such that

[TABLE]

Denote $\Gamma_{t}=\Gamma\cap\{(y,z):A(y,z)=t\}$. We obtain $\Gamma=\cup_{t_{0}\leq t\leq t_{1}}\Gamma_{t}$. By (78) and (79), we deduce that there is a radial stretch $B_{t}$ yielding a bi-Lipschitz chart $B_{t}:F\to\Gamma_{t}$ over a subdomain $F$ of the unit ball. The bi-Lip constant of $B_{t}$ is uniform over $t_{0}<t<t_{1}$ and each $B_{t}$ is actually a local diffeomorphism apart from $y=z$. By (78) and (79), we may write $B_{t}$ in the following form

[TABLE]

where the dependence $(w_{1},w_{2})\to\sigma(t,w_{1},w_{2})$ is Lipschitz with respect to $t$ with a uniform Lipschitz constant with respect to $w_{1},w_{2}$.

Let $h$ be a integrable Borel-function on $\Gamma$, note that $\Gamma=\cup_{t_{0}\leq t\leq t_{1}}\Gamma_{t}$, we get

[TABLE]

where the inner integral is with respect to the three-dimensional Hausdorff measure on $\Gamma_{t}$. By a change of variables, we have

[TABLE]

By (78) and (79), the Jacobian $E_{t}$ in (82) satisfies

[TABLE]

Since $B_{t}(w_{1},w_{2})$ is Lipschitz with respect to $t$, for our later purpose, we claim that the dependence $t\to E_{t}(w_{1},w_{2})$ is uniformly Lipschitz with respect to $t$. Using (81), we have

[TABLE]

where $S_{\delta}$ is given by

[TABLE]

Since $\Gamma_{t}$ is only nonempty for $t\in[T_{0},T_{1}]$, $S_{\delta}(t)$ is compactly supported inside $[T_{0},T_{1}]$. For fixed $x\in U$, let $L(x,y)$ be a smooth cutoff of the function $\frac{(x_{1}-y_{1})^{p_{1}}(x_{2}-y_{2})^{p_{2}}}{|x-y|^{l_{1}}}$ that vanishes outside $D$, hence, $L(x,\cdot)\in C_{0}^{\infty}({\mathbb{R}^{2}})$. Thus, we can rewrite $S_{\delta}(t)$ as

[TABLE]

Recall that our aim is to prove $\sup\limits_{\delta\in(0,1)}\int_{1}^{\infty}{\mathbb{E}}|g_{\delta}(x,\omega)|^{2}d\omega<\infty.$ It is sufficient to show that for each $\tilde{t}\in[T_{0},T_{1}]$, there exists a finite constant $M=M(\tilde{t})<\infty$ such that

[TABLE]

This can be seen by the following facts: by compactness, we can choose finitely many $\tilde{t}\in[T_{0},T_{1}]$ such that the union set of $[t_{0}(\tilde{t}),t_{1}(\tilde{t})]$ for these $\tilde{t}$ can cover $[T_{0},T_{1}]$. Hence, for any $t\in[T_{0},T_{1}]$, we have ${\mathbb{E}}|S_{\delta}(t)|^{2}\leq M^{\prime}$. The Parseval formula yields

[TABLE]

It remains to show (84). By (83), we have

[TABLE]

Noting that ${\mathbb{E}}(\tilde{f}_{\delta}(z)\tilde{f}_{\delta}(z^{\prime}))=C_{\delta}(z,z^{\prime})$ and $C_{\delta}=M_{\delta}C_{\tilde{f}}M_{\delta}$, we obtain from Lemma 9 that for any given $\beta>0$, there is a finite constant $C^{\prime}_{\beta}$ such that $C_{\delta}(z,z^{\prime})\leq C^{\prime}_{\beta}|z-z^{\prime}|^{-\beta}$ for any $\delta\in(0,1)$ and $(z,z^{\prime})\in D\times D$. Since $q\in C_{0}^{1}(\overline{D})$, an application of Hölder’s inequality arrives

[TABLE]

where we use the fact $l_{2}-p_{3}-p_{4}=\frac{1}{2}$ for $(l_{1},l_{2},p_{1},p_{2},p_{3},p_{4})\in S$. To show the integral in the right hand side of the above inequality is bounded, we need the following result [32, Lemma 6]).

Lemma 18**.**

Given $\gamma\in(0,2)$ there is a finite constant $c$ such that for every $t\in[t_{0},t_{1}]$ we have

[TABLE]

for $(\tilde{y},\tilde{z})=(y,y^{\prime}),(y,z^{\prime}),(z,y^{\prime}),(z,z^{\prime})$.

Choosing $\beta=\frac{1}{2}$ and applying Lemma 18 give (84). So Theorem 15 holds for the case $m=2$.

Case 2. $m\in(2,5/2)$. By Lemma 8, we know that in this case the realizations of $\tilde{f}$ are Hölder continuous with probability one. So it is not necessary to introduce the mollification, we define

[TABLE]

In order to prove (76), i.e., $\int_{1}^{\infty}\omega^{m-2}|g(x,\omega)|^{2}d\omega<\infty$, note that $g(x,\omega)=[{\mathcal{F}}^{-1}S](-\omega)$, it suffices to prove that $S(t)\in H^{\frac{m-2}{2}}_{\rm homog}(\mathbb{R})$ which denotes the homogeneous Sobolev space. By compactness, it is enough to show that $S(t)\in H^{\frac{m-2}{2}}_{\rm homog}(t_{0}(\tilde{t}),t_{1}(\tilde{t}))$ for each $\tilde{t}\in[T_{0},T_{1}]$. According to the Besov characterization of the homogeneous Sobolev space, it is sufficient to show

[TABLE]

The Fubini theorem shows that (85) holds as long as for some positive constant $M$ that

[TABLE]

We can rewrite $S(t)$ by

[TABLE]

Recall that the bi-Lipschitz chart $B_{t}:F\to\Gamma_{t}$ is given by

[TABLE]

Denote

[TABLE]

By (87), we can rewrite $S(t)$ by

[TABLE]

where the function

[TABLE]

is uniformly bounded and Lipschitz continuous with respect to $t$. Since

[TABLE]

where

[TABLE]

we have

[TABLE]

Since $|t-t^{\prime}|=|t-t^{\prime}|^{\frac{m-1}{2}}|t-t^{\prime}|^{\frac{3-m}{2}}\lesssim|t-t^{\prime}|^{\frac{m-1}{2}}$, it suffices to estimate $\|S_{1}(t)-S_{1}(t^{\prime})\|_{L^{2}(\Omega)}$. Similarly, we have

[TABLE]

where

[TABLE]

Note that

[TABLE]

Hence

[TABLE]

Now we estimate $\|S_{2}(t)-S_{2}(t^{\prime})\|_{L^{2}(\Omega)}$ which can be rewritten in a double integral as

[TABLE]

where $(y_{t},z_{t})=\sigma_{t}(w_{1},w_{2})$, $(y_{t^{\prime}},z_{t^{\prime}})=\sigma_{t^{\prime}}(w_{1},w_{2})$, $(s_{t},u_{t})=\sigma_{t}(v_{1},v_{2})$, $(s_{t^{\prime}},u_{t^{\prime}})=\sigma_{t^{\prime}}(v_{1},v_{2})$,

[TABLE]

and

[TABLE]

Recall that the covariance function has the form

[TABLE]

where $c_{0}\in C_{0}^{\infty}(D\times D)$ and $r_{1}\in C_{0}^{\alpha}(D\times D)$ for any $\alpha<1$. Combining the fact $q\in C_{0}^{1}(\overline{D})$ yields immediately that

[TABLE]

Denote $d=|z_{t}-u_{t}|=|\sigma_{t}(w_{2}-v_{2})|$ and $\delta=|u_{t}-u_{t^{\prime}}|=|(\sigma_{t}-\sigma_{t^{\prime}})v_{2}|$, if $\frac{\delta}{d}<1$, we have

[TABLE]

Hence, if $|t-t^{\prime}|<c|w_{2}-v_{2}|$ for some small enough $c>0$, we have

[TABLE]

Similarly, we have that

[TABLE]

holds if $|t-t^{\prime}|<c|w_{2}-v_{2}|$ for some small enough $c>0$. Thus, if we define a set

[TABLE]

then we have

[TABLE]

Dividing integration on $F\times F$ over the sets $P\cap F\times F$ and $(F\times F)\setminus P$, we obtain

[TABLE]

Observe that $|R(w_{1},w_{2})|\lesssim|w_{1}-w_{2}|^{-\frac{1}{2}}$ and $|R(v_{1},v_{2})|\lesssim|v_{1}-v_{2}|^{-\frac{1}{2}}$, using (88), Hölder inequality along with Lemma 18, we have

[TABLE]

For $I_{2}$, we have from (89) that

[TABLE]

where we use the Hölder inequality along with Lemma 18. Hence, we arrive

[TABLE]

which shows that (86) holds true. By the previous argument we have that (76) holds for this case. The proof is completed. ∎

With the convergence of the Born approximation, using Theorem 14 along with Theorem 15, we are ready to show the proof of Theorem 4.

Proof.

Recall the convergence of the Born approximation

[TABLE]

where $\bm{b}(x,\omega)=\sum_{n=2}^{\infty}\bm{u}_{n}(x,\omega)$. It follows from (36) that

[TABLE]

for some small enough $\varepsilon^{\prime}>0$. So

[TABLE]

as $Q\to\infty$, where we use the fact $m\in(2,5/2)$. Recalling Theorem 14 and Theorem 15, we have

[TABLE]

hold almost surely, where $a$ is a constant given in Theorem 4. Since

[TABLE]

along with (90)–(92) and the Cauchy-Schwartz inequality, it is to easy to verify that

[TABLE]

By Lemma 3.8 in [29], we know that the integral $\int_{\mathbb{R}^{2}}\frac{1}{|x-y|}\phi(y)dy$ for all $x\in U$ can uniquely determines the function $\phi$. The proof is completed. ∎

6 Conclusion

We have studied the inverse random source scattering problem for the two-dimensional elastic wave equation with an inhomogeneous, anisotropic mass density. The source is modeled as a generalized Gaussian random function and its covariance operator is described as a classical pseudo-differential operator. Both the direct and the inverse problems are considered. The direct problem is equivalently formulated as a Lippmann–Schwinger integral equation which is shown to have a unique solution. Combining the Born approximation and microlocal analysis, we deduce a relationship between the principle symbol of the covariance operator for the random source and the amplitude of the displacement generated from a single realization of the random source. Based on this connection, we obtain the uniqueness for the reconstruction of the principle symbol of the random source. In this paper, the mass density or the linear load is considered to be a smooth deterministic matrix. An ongoing project is to study the direct and inverse scattering problems when both the source and the mass density or the linear load are random. Another challenging problem is to study the random source scattering problem for three-dimensional elastic wave equation. We hope to be able to report the progress elsewhere in the future.