Inverse Elastic Scattering for a Random Source

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.

Inverse random source scattering for the Helmholtz equation in inhomogeneous media

“…, the random function is a distribution in f ∈ W s− d 2 −ǫ,p for any ǫ > 0 and p ∈ (1, ∞) (cf. Lemma 2.6), which is rougher than those considered in [9,15,20,21]; (2) if σ = 0 and s ∈ [ d 2 , d 2 + 1), the results obtained in this paper coincides with the ones given in [20]. The paper is organized as follows.…”

The Nyström method for elastic wave scattering by unbounded rough surfaces

“…To get the exact expressions of B (1) jk (s, t) and C (1) jk (s, t) while s = t for numerical computation, we need to use the following asymptotic behavior of Bessel functions (see [15, (5.16.1)-(5.16.3)] and [16]): as…”

Inverse source problems for the stochastic wave equations: far-field patterns

“…Motivated by [13], a new model is developed for the random source, which is assumed to be a real-valued generalized microlocally isotropic Gaussian (GMIG) random field with its covariance operator being a classical pseudo-differential operator. It is shown that the principal symbol of the covariance operator can be uniquely determined by the amplitude of the near-field scattering data averaged over the frequency band, generated by a single realization of the random source, see [14,20] for acoustic waves, [14,15] for elastic waves, and [23] for biharmonic waves. The inverse random source problem for electromagnetic waves is considered in [21], where the source is modeled by a complex-valued centered GMIG random field whose real and imaginary parts are assumed to be independent and identically distributed, leading to the relation operator being zero.…”