The identification of distributed elastic moduli of damaged materials or crack like defects in elastic media are inverse problems known as generalized elastic tomography. It consists of recovering the damaged zone or the crack in a 3D body using mechanical overdetermined boundary data. For 21 distributed unknowns which are small perturbations of elastic isotropy, a linear system of rank 5 may be derived directly from the observation equations which involves both domains and boundary integrals, with actual mechanical fields and the proposed adjoint fields. It is found that the generalization of Calderon’s method in elasto-statics provided a linear system of rank 5, hence identification problems for small symmetry up to 5 elastic moduli fields could be solved. Finally, the problem of identification of a plane crack in 3D elastic body illustrates the ability of the observation equation method to provide closed form solution for the identification of the crack plane and the crack geometry.