The inversion of magnetotelluric measurements require the computation of the quantity of interest (solution at the receivers) at different positions. Using a multigoal-oriented adaptive strategy for an hp Finite Element algorithm, we ensure accurate solutions in all receivers. With them, and using a limited memory algorithm for bound constraint optimization (L-BFGS-U method) to solve the Inverse Problem, we show that we are able to invert properly the simulated measurements in order to recover the subsurface conductivity distribution. We emphasize that the choice of the unknown of the Inverse Problem affects significantly the convergence of the inversion. We also exhibit the limitations of the electric field as the quantity of interest. When the distribution of the subsurface contains big jumps, the solution of the problem is not unique. The fact that the quantity of interest is not sensible enough to the variation of the conductivity suggests the use of the impedance as quantity of interest as a candidate to bring better results in a future research.