Stability for a multi-frequency inverse random source problem

Inverse source scattering problem is concerned with recovering the unknown source from near-field or far-field data away from its support. Such problem has generated tremendous interest due to its wide applications in scientific and engineering fields such as seismology, telecommunications, medical imaging, antenna synthesis, radar technology, and magnetoencephalography [1, 3, 8, 15]. However, the non-uniqueness of the inverse source problem at a single frequency, caused by the existence of non-radiating sources, poses a challenge [9, 12]. Consequently, additional information is required for a unique determination of the source. To resolve this issue, the use of multi-frequency data has been realized to be an effective approach to regain the uniqueness and achieve enhanced stability [7, 8, 11, 24].

In many applications, the source is often considered as a random field due to uncertainties in the surrounding environment or random measurement noise [2]. The presence of randomness introduces additional challenges compared to deterministic source scattering. Specifically, the regularities of wave fields tend to be lower, and the measurements become statistical data. Inverse source problems driven by Wiener process have been extensively investigated [4, 5, 6, 19, 23]. In recent studies, uniqueness of inverse source problems have been studied in [17, 18, 21] by assuming the source to be a generalized Gaussian random field. The covariance of such random field is a classical pseudo-differential operator with a principal symbol taking the form $h(x)|\xi|^{-m}$, where $h$ is called micro-correlation strength. This model encompasses various important stochastic processes, including white noise, fractional Brownian motion, and Markov fields [16]. Compared with the many uniqueness results of the inverse random source problems, the stability has been much less studied. To the best of our knowledge, the only existing stability results were obtained in [20, 22] driven by Wiener process. The corresponding stability for the generalized Gaussian random field remains unsolved.

In numerical experiments in [4, 19], it has been observed that the ill-posedness of the inverse random source problem can be overcome by using multi-frequency data which yields increasing stability, i.e., as the frequency increases the inverse problem becomes more stable. The goal of this work is to mathematical verify the increasing stability with a generalized Gaussian random field. Specifically, consider the stochastic Helmholtz equation

where $k>0$ is the wavenumber. The source function $f$ is a microlocally isotropic general Gaussian random field of order $m$ in a bounded Lipschitz domain $D\subset\mathbb{R}^{d}$ (see Section 2 for a detailed definition). The wave field $u$ is required to satisfy the Sommerfeld radiation condition

Let $B_{R}=\{x:x\in\mathbb{R}^{d},\,|x|\leq R\}$ with boundary $\partial B_{R}$ and assume that $D\subset\subset B_{R}$. The inverse problem is to determine the principle symbol of $f$ from either the near-field data on $\partial B_{R}$ or far-field data.

Compared with the deterministic case, new challenges arise due to the roughness and randomness of the source. The well-posedness of the direct problem (1.1)–(1.2) has been discussed in [18]. However, it is not clear if the solution can be represented by the convolution of the Green function and the source as in the classical setting. By studying the regularity of the Green function, we establish an explicit integral representation of the solution which holds pointwisely. As a consequence, we obtain an enhanced regularity result. The analysis of the inverse problem employs microlocal analysis to achieve an estimate for the Fourier transform of the micro-correlation strength by the near-field correlation data and a high-frequency tail. Next, we show that the correlation data is analytic and derive an upper bound with respect to complex wave number. The stability estimate follows by an application of a novel analytic continuation developed in [25]. For the case of far-field data, the stability estimate can be derived by investigating the correlation of the far-field data. The stability has a unified form which consists of the Lipschitz data discrepancy and a logarithmic stability, where the latter decreases as the frequency increases.

The rest of this paper is organized as follows. Section 2 is devoted to the definition and some properties of the microlocally isotropic Gaussian random function. In Section 3, we derive a regularity result for the Green function of the Helmholtz equation which leads to an explicit integral representation of the direct problem. The main increasing stability results are presented in Section 4 and Section 5 for near-field data and far-field data, respectively. A conclusion is given in Section 6. Throughout this paper, $a\lesssim b$ stands for $a\leq Cb$, where $C>0$ is a generic constant whose special value is not required but should be clear from the context.

In this section, we state the properties of microlocally isotropic generalized Gaussian random functions. Let $(\Omega,\mathcal{A},\mathbb{P})$ be a complete probability space. Denote the test function space by $\mathcal{D}$, which consists of smooth functions with compact supports in $\mathbb{R}^{d}(d=2,3)$. Then the dual space of $\mathcal{D}$ is denoted as $\mathcal{D}^{\prime}$. A scalar field $f$ is said to be a real-valued generalized Gaussian random function if $f\,:\,\Omega\to\mathcal{D}^{\prime}$ is a distribution such that for each $\omega\in\Omega$ the path $f[\cdot](\omega)\in\mathcal{D}^{\prime}$ is a linear functional on $\mathcal{D}$, and $\omega\mapsto\langle f(\omega),\phi\rangle$ is a real-valued Gaussian random variable for all $\phi\in\mathcal{D}$. The expectation of $f$ is a generalized function defined by

and the covariance is a bilinear form given by

Then define the covariance operator $C_{f}\,:\,\mathcal{D}\to\mathcal{D}^{\prime}$ by

which is associated with a Schwartz kernel denoted by $K_{f}(x,y)$ as follows

In this paper, the random source is assumed to be characterized by a special class of generalized Gaussian random functions as follows.

The smooth function $h$ is called the micro-correlation strength of the random function $f$. Let $c(x,\xi)$ be the symbol of $C_{f}$. The operator can be represented by

where $\hat{\phi}(\xi)$ stands for the Fourier transform of $\phi$ defined by $\hat{\phi}(\xi)=\mathcal{F}\phi(\xi):=\int_{\mathbb{R}^{d}}e^{-ix\cdot\xi}\phi(x)\,\mathrm{d}x.$ The kernel $K_{f}$ can be represented as an oscillatory integral of the form

We introduce the following regularity result in [21] for microlocally isotropic Gaussian random functions.

By Sobolev embedding theorem, Lemma 2.1 gives the following corollary.

In this section, we investigate the direct scattering problem. The Green function of the Helmholtz equation $\Phi(x,y)$ has the explicit form

where $H^{(1)}_{0}$ is the Hankel function of the first kind with order zero. According to Corollary 2.1, for $m>d$, the source $f\in C^{0,\gamma}{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}(\bar{D})}\subset L^{2}(D)$ with some $\gamma>0$ almost surely. Hence, the scattering problem (1.1)-(1.2) is classical which admits a unique solution $u\in H^{2}_{loc}(\mathbb{R}^{d})$ with the following explicit integral form

However, for $m\leq d$, Lemma 2.1 shows that such source is too rough to exist pointwisely which should be taken as distribution. In this case, the following theorem proved in [18] gives the well-posedness and regularity of the direct problem.

For $m>d$, by classical acoustic wave scattering theory we know that the solution of the direct problem admits a pointwise explicit integral representation (3.4). However, for $m\leq d$, we only know that (3.4) holds in sense of distributions. In the rest of this section, we show that (3.4) holds pointwisely, which implies $u\in L^{\infty}(\mathbb{R}^{d})$.

The following lemma is useful in the subsequent analysis.

Combining Lemma 2.1 and 3.2, we have the following regularity result.

It can be verified that using the regularity result in Theorem 4.1 and Sobolev embedding theorem one can only have $u\in L^{\infty}(\mathbb{R}^{d})$ when $m>2(d-2)$. Hence, the above theorem enhances the regularity result in Theorem 4.1.

In this section, we derive the stability estimate by near-field data. Denote $a(x,\xi)=c(x,\xi)-h(x)|\xi|^{-m}$. Assume that $f$ satisfies the following assumption.

Assumption (A). The random source $f$ is a real-valued microlocally isotropic Gaussian random function of order $m>d-1$. The covariance operator has the symbol $c(x,\xi)$ with the principal symbol $h(x)|\xi|^{-m}$ satisfying (i) $|h(x)|\leq M$ for $x\in D$; (ii) $|c(x,\xi)|\leq M(1+|\xi|)^{-m}$ for $\xi\in\mathbb{R}^{d}$ and $x\in D$; (iii) $|a(x,\xi)|\leq M|\xi|^{-(m+1)}$ for $|\xi|\geq 1$ and $x\in D$. Here $M>0$ stands for a constant.

We first give a bound with a high frequency tail for the Fourier transform of $h$ in terms of the source function. Recalling the definitions in Section 2, the covariance of $\hat{f}$ can be expressed by

Therefore, we obtain

Taking $\xi=k\theta_{1}$ and $\eta=k\theta_{2}$ with $\theta_{1},\,\theta_{2}\in\mathbb{S}^{d-1}$, one arrives at

Next, we bound the term $\mathbb{E}(\hat{f}(k\theta_{1})\hat{f}(k\theta_{2}))$ by the correlation data on $\partial B_{R}$. Multiplying the governing equation (1.1) by the plane wave $e^{-ik\theta_{j}\cdot x}$ and integrating by parts yield

Taking expectation gives that

where $\hat{x}=x/|x|$. By inserting (4.3) into (4.1) and noticing

we derive the inequality

which holds for all $|\xi|\leq 2k$. For convenience, denote

with $k>0$ and $\theta_{1},\theta_{2}\in\mathbb{S}^{d-1}$. From the equation (1.1) and the Sommerfeld radiation condition (1.2), it can be verified that

where the notation $u(x;k)$ is used to exhibit the dependence of the solution on the wavenumber $k$. Therefore, the definition of $\epsilon^{2}(k,\theta_{1},\theta_{2})$ can be extended to $\mathbb{C}$ as

Consider the multi-frequency data characterized as

where $K>0$ is the upper bound of the frequency. Denote a sectorial domain by $\mathcal{R}=\{z\in\mathbb{C}:|\arg z|<\pi/4\}$. In what follows, we show that $\epsilon^{2}(k,\theta_{1},\theta_{2})$ is analytic and has an upper bound for $k\in\mathcal{R}$. We only consider the term $I_{1}$ since the discussions for $I_{2},I_{3},I_{4}$ are similar. Recalling (3.15), we deduce

where we denote

Obviously, $J$ is analytic with respective to $k\in\mathcal{R}$. In order to show $I_{1}$ is also analytic, we shall verify the following estimates