The connection of different modifications of the method of linear integral representation is studied. Solutions of the related inverse problems based upon a ‘hybrid version of two approximations’ of the topography and geopotential fields enable more refined tuning of the method in solving the inverse problems of geophysics and geomorphology and more complete allowance for the a priori infor- mation about the surface elevation data and elements of anomalous fields. The technique for finding a stable approximate solution for the inverse problem of determining the mass distributions equivalent by the external gravitational (or any other potential) field is presented. The method is applied to study anomalous densities beneath one of the highest elevation Elysium Mons and deep Hellas basin on Mars. The results of the mathematical experiment are discussed.