An experimental method for studying the local atomic structure with a depth resolution in low-contrast multilayered thin films is proposed. We consider combination of the X-ray reflectivity and the extended X-ray absorption fine structure spectroscopy with angular resolution. To obtain the structural characteristics, the most advanced methods for solving inverse ill-posed problems are used, namely, the Tikhonov regularization method in the case of linear integral equations and the Levenberg-Marquardt algorithm for nonlinear problems. The proposed mathematical algorithms do not require a priori information about the studied system, such as the width and shape of the interface, the width and depth of the bands of the individual elements, or detailed information about the atomic structure. A software for the method has been developed. It allows us to obtain the most complete information about the local atomic structure of interface and surface. To evaluate the capabilities of the method, we used the system of cluster-layered nanostructures Al2O3/Cr/Fe-Cr/Cr/Fe-Cr/Cr as a model. A depth resolution of 1-2 angstrom was reached.