  We propose a deterministic-statistical method for an inverse source problem
using multiple frequency limited aperture far field data. The direct sampling
method is used to obtain a disc such that it contains the compact support of
the source. The Dirichlet eigenfunctions of the disc are used to expand the
source function. Then the inverse problem is recast as a statistical inference
problem for the expansion coefficients and the Bayesian inversion is employed
to reconstruct the coefficients. The stability of the statistical inverse
problem with respect to the measured data is justified in the sense of
Hellinger distance. A preconditioned Crank-Nicolson (pCN) Metropolis-Hastings
(MH) algorithm is implemented to explore the posterior density function of the
unknowns. Numerical examples show that the proposed method is effective for
both smooth and non-smooth sources given limited-aperture data.
