In this paper, we study an inverse transmission scattering problem of a time-harmonic acoustic wave from the viewpoint of Bayesian statistics. In Bayesian inversion, the solution of the inverse problem is the posterior distribution of the unknown parameters conditioned on the observational data. The shape of the scatterer will be reconstructed from full-aperture and limited-aperture far-field measurement data. We first prove a well-posedness result for the posterior distribution in the sense of the Hellinger metric. Then, we employ the Markov chain Monte Carlo method based on the preconditioned Crank-Nicolson algorithm to extract the posterior distribution information. Numerical results are given to demonstrate the effectiveness of the proposed method.

在本文中，我们从贝叶斯统计的角度研究了时谐声波的逆传输散射问题。在贝叶斯反演中，逆问题的解决方案是以观测数据为条件的未知参数的后验分布。散射体的形状将根据全孔径和有限孔径远场测量数据重建。我们首先证明了 Hellinger 度量意义上的后验分布的适定结果。然后，我们采用基于预处理Crank-Nicolson 算法的马尔可夫链蒙特卡罗方法来提取后验分布信息。数值结果证明了所提出方法的有效性。