Inverse Random Source Problems for Time-Harmonic Acoustic Waves in a Two-Layered Medium

Abstract. This work studies inverse problems for random sources in the Helmholtz equation in a two-layered medium. The source is represented by a centered generalized Gaussian field with microlocal isotropy, whose covariance and relation operators are taken as classical pseudodifferential operators. For the one-dimensional problem with a real-valued source, we prove that the principal symbol of the covariance operator can be uniquely identified from measurements of the wave field’s second moment on an open subset. For the two- and three-dimensional problems with complex-valued sources, we establish that the principal symbols of both the covariance and relation operators can be uniquely determined, with probability one, from a single realization of the far-field pattern averaged over a frequency band. To demonstrate the applicability of the theory, numerical experiments are carried out for the one-dimensional case with a white noise source model, confirming the effectiveness and stability of the proposed approach.