  In this paper, a new model is proposed for the inverse random source
scattering problem of the Helmholtz equation with attenuation. The source is
assumed to be driven by a fractional Gaussian field whose covariance is
represented by a classical pseudo-differential operator. The work contains
three contributions. First, the connection is established between fractional
Gaussian fields and rough sources characterized by their principal symbols.
Second, the direct source scattering problem is shown to be well-posed in the
distribution sense. Third, we demonstrate that the micro-correlation strength
of the random source can be uniquely determined by the passive measurements of
the wave field in a set which is disjoint with the support of the strength
function. The analysis relies on careful studies on the Green function and
Fourier integrals for the Helmholtz equation.
