Quantitative reconstruction of mantle structures and temperature field backwards in time requires a numerical tool for solving the inverse problem of thermal convection at infinite Prandtl number.
Data assimilation, i.e. incorporation of present (observations) and past (initial conditions) data in an explicit dynamical model, is a useful tool to resolve the problem. A variational approach to three-dimensional numerical assimilation of present temperature data into a thermoconvective mantle flow will be introduced. This approach is based on a search for the mantle temperature and flow in the geological past by minimizing differences between present mantle temperature derived from seismic velocities and that predicted by forward models of mantle flow for an initial temperature guess. I shall illustrate the applicability of the approach to assimilation of synthetic and real data and analyze how strong features of mantle plumes and lithospheric slabs in the geological past can be recovered after they have dissipated due to thermal diffusion. Also I shall discuss the numerical techniques developed (a combination of Eulerian FEM to solve the Stokes equations for the vector potential and FDM for the heat balance equation) and parallel algorithms used in this modeling.