Reconstruction stability in some problems of X-ray and seismic tomography

We study problems of recovering a function given by weighted integrals over plane curves of a special shape. The curves and weight functions are piecewise smooth. Such problems of integral geometry are connected with the problems of reconstruction of internal structure of an object from the boundary data. We reduce these problems to the investigation of Fredholm equations of first kind. Stability estimates for a solution to the considered problems in spaces of finite smoothness were obtained thereby demonstrating weak ill-posedness of the problem. We present also an efficient algorithm for stable solving of initial problem.