Abstract: The given dissertation is devoted to the numerical solution of a number of planar tomography problems and its application to the solution of inverse kinematic problems of seismology. The aim of this work was the derivation of new inversion formulae and its effective numerical implementation. Tomography problems arise in different areas of medicine and industry that require the layer-by-layer reconstruction of the images of 3D heterogeneous objects.

In the first chapter the author writes down the inversion formula for the fan-beam Radon transform, obtained by the method of A-analytic functions, and estimates the error of the projection method that uses only the finite number of Fourier coefficients of the initial data. In the second chapter new inversion formulae for the emission tomography problem are derived (for the scalar and vector cases) that don't formally use the theory of A-analytic functions. In the vector case it becomes possible to reconstruct the full vector field (and not merely its solenoidal part like in the unattenuated case), provided that the attenuation function doesn't vanish. In the third chapter a singular value decomposition of the Radon transform of tensor fields was obtained in the framework of the fan-beam scanning geometry, it allows to characterize the range of the tensorial Radon transform, invert it and estimate the level of incorrectness. In the fourth chapter the author considers several statements of the inverse kinematic problem of seismology, chooses the stable ones and derives inversion formulae for the case of reflected rays (in a 2D layer) and for the case of refracted rays (in a 3D volume). All the listed above algorithms were numerically implemented.