Stability for a multi-frequency inverse random source problem

Abstract We present increasing stability estimates for the inverse random source problem of the Helmholtz equation in two and three dimensions by multi-frequency near-field data. The random source is assumed to be a microlocally isotropic generalized Gaussian random function such that its covariance is a classical pseudodifferential operator. We consider the use of either expectation of the correlation data or the data obtained from a single realization of the random source. For the direct problem, by exploring the regularity of the Green function, we demonstrate that the direct problem admits a unique bounded solution with an explicit integral representation. For the inverse problem, the analysis employs microlocal analysis to derive an estimate for the Fourier transform of the micro-correlation strength by the near-field correlation data and a high-frequency tail. The stability is achieved by showing the analyticity of the data and applying a novel analytic continuation principle. Furthermore, with the help of ergodicity, the stability estimate can be extended to the case of using only a single realization of the data. The stability estimates have a unified form which consists of the Lipschitz type data discrepancy and the logarithmic stability. The latter decreases as the upper bound of the frequency increases, which exhibits the phenomenon of increasing stability.