A new three-staged technique based on various versions of the linear integral representation method (S-, F- and R-approximations) is applied to solve the problem of analytical modelling of the magnetic field on the surface of Mars. At the first step, we build an analytical approximation from the given data set (in local or regional cases). In the second step, we analyze the previous stage results and solve an ordinary differential equation system to find the integral curve of the gravity or magnetic field. The points of this curve are included in an extended data set, and we repeat the procedure described above for the first stage. The final approximation of the anomalous field elements or the topographic data allows us to find various linear transformations of the field under investigation, e.g. higher derivatives of the potential or Fourier-spectra of the field elements. Analytical downward continuations of the magnetic field of Mars at various distances, including the planet's surface is presented.