Inverse random source scattering for the Helmholtz equation in inhomogeneous media

Abstract

In this paper, a new model is proposed for the inverse random source scattering problem of the Helmholtz equation with attenuation. The source is assumed to be driven by a fractional Gaussian field whose covariance is represented by a classical pseudo-differential operator. The work contains three contributions. First, the connection is established between fractional Gaussian fields and rough sources characterized by their principal symbols. Second, the direct source scattering problem is shown to be well-posed in the distribution sense. Third, we demonstrate that the microcorrelation strength of the random source can be uniquely determined by the passive measurements of the wave field in a set which is disjoint with the support of the strength function. The analysis relies on careful studies on the Green function and Fourier integrals for the Helmholtz equation.

Stability on the inverse random source scattering problem for the one-dimensional Helmholtz equation

“…We have studied an inverse random source scattering problem for the one-dimensional Helmholtz equation with attenuation, which is to reconstruct the strength of the random source. Compared with higher dimensional cases studied in [22], the fundamental solution in the one-dimensional case is smooth, which makes it possible to deal with rougher random sources including the white noise. The strength is shown to be uniquely determined by the variance of the wave field in an open measurement set.…”

“…Let f be a real-valued centered microlocally isotropic Gaussian random field of order −m compactly supported in D ⊂ R d , i.e., the covariance operator of f is a pseudo-differential operator whose principal symbol has the form µ(x)|ξ| −m with the micro-correlation strength µ ∈ C ∞ 0 (D) and µ ≥ 0. It is shown in [22,Proposition 2.5] that the generalized Gaussian random field…”

“…Consequently, the regularity of random fields satisfying Assumption 1 can be obtained by investigating the regularity of the generalized Gaussian random fields, which is stated in the following lemma (cf. [22]).…”

“…The covariance operators of the random fields are assumed to be pseudo-differential operators with principal symbol µ(x)|ξ| −m , where the nonnegative function µ ∈ C ∞ 0 (D) is called the micro-correlation strength of the random source and is the statistics to be determined. It is shown in [22] that the microlocally isotropic Gaussian random field is equivalent to the generalized fractional Gaussian random field in the form…”

“…Instead, the microlocal analysis for large frequencies is applied to reconstruct the micro-correlation strength µ involved in the principal symbol of the covariance operator of f . In [22], the d-dimensional Helmholtz equation with attenuation is studied with…”