Inverse random scattering for the one-dimensional Helmholtz equation

This paper concerns the inverse random potential scattering problem for the Helmholtz equation in one dimension. The random potential is assumed to be a generalized microlocally isotropic Gaussian random field, whose covariance is a classical pseudodifferential operator. To investigate the inverse problem, as an effective mathematical tool, the scattering theory is employed to obtain an analytic domain and estimates for the resolvent of the elliptic operator with rough potentials. For the inverse problem, based on the resolvent estimates, we show that the principal symbol of the covariance operator can be reconstructed by a single realization of the boundary data averaged over the high-frequency band almost surely. Consequently, by analyticity of the resolvent, the uniqueness of the inverse problem can be achieved by data at multiple frequencies only in a finite interval. The method developed here is unified and can be applied to other stochastic inverse scattering problems in higher dimensions for various wave equations by boundary measurements.