An inverse random source problem for the one-dimensional Helmholtz equation with attenuation

TL;DR

This paper investigates an inverse problem for the 1D stochastic Helmholtz equation with attenuation, focusing on identifying the micro-correlation strength of a Gaussian random source from wave field measurements.

Contribution

It establishes well-posedness for the direct problem and proves unique determination of the source's micro-correlation strength from measurements, including numerical validation.

Findings

Unique determination of the micro-correlation strength from wave data.

Well-posedness of the direct scattering problem in distribution sense.

Numerical experiments confirm the method's effectiveness for white noise sources.

Abstract

This paper is concerned with an inverse random source problem for the one-dimensional stochastic Helmholtz equation with attenuation. The source is assumed to be a microlocally isotropic Gaussian random field with its covariance operator being a classical pseudo-differential operator. The random sources under consideration are equivalent to the generalized fractional Gaussian random fields which include rough fields and can be even rougher than the white noise, and hence should be interpreted as distributions. The well-posedness of the direct source scattering problem is established in the distribution sense. The micro-correlation strength of the random source, which appears to be the strength in the principal symbol of the covariance operator, is proved to be uniquely determined by the wave field in an open measurement set. Numerical experiments are presented for the white noise model to demonstrate the validity and effectiveness of the proposed method.