An inverse random source scattering problem in inhomogeneous media

作者：Li Peijun^*

来源：Inverse Problems, 2011, 27(3): 035004.

DOI：10.1088/0266-5611/27/3/035004

Consider the scattering problem for the one-dimensional stochastic Helmholtz equation in a slab of an inhomogeneous medium, where the source function is driven by the Wiener process. To determine the random wave field, the direct problem is equivalently formulated as a two-point stochastic boundary value problem. This problem is shown to have pathwise existence and uniqueness of a solution. Furthermore, the solution is explicitly deduced with an integral representation by solving the two-point boundary value problem. Since the source and hence the radiated field are stochastic, the inverse problem is to reconstruct the statistical structure, such as the mean and the variance, of the source function from physically realizable measurements of the radiated field on the boundary point. Based on the constructed solution for the direct problem, integral equations are derived for the reconstruction formulas, which connect the mean and the variance of the random source to those of the measured field. Numerical examples are presented to demonstrate the validity and effectiveness of the proposed method.

• 出版日期2011-3