On the solution of some nonlinear inverse problems in geophysics

We show that a canonical three-dimensional domain can be mapped onto unknown source of masses with the help of dispersionless two-dimensional (2D) Toda integrable hierarchies. The maps are determined by a particular approximate solution to a set of 2D Toda hierarchies singled out by the condition known in the 2D case as 'string equations'. The same hierarchy locally solves 2D inverse potential problem, i.e. reconstruction of the 2D domain out of a set of its harmonic moments. We solve the non-linear inverse problem of geophysics by setting a conditional variational problem to determine the coefficients of the Laurent expansion of the function realizing the conformal map.