Invariant imbedding and inverse problems with applications to radiative transfer

In this paper, we developed two methods to solve the inverse problem of a nonlinear integro-differential equation. Both methods are based on the principle of invariant imbedding. The first method involves two auxiliary integro-differential equations. The inverse problem is solved by a sequence of approximation solutions of linear equations. The second method involves algebraic equations of scattering matrices under the so-called star-product. An application to a radiative transfer problem such as correction of atmospheric effects on remote sensing is discussed.