Abstract: Inverse problems are notoriously unstable and to stabilise them one needs to impose some a priori conditions. In the case of anisotropic IP or IP on manifolds, these conditions should be coordinate-invariant involving curvature, volume growth, etc. These bring the question of stabilizing IP into the framework of the Gromov-Cheeger theory of geometric convergence. We discuss these issues for the case when the imposed geometric conditions allow for the 1D collapse to orbifolds. This is a common work with M. Lassas and T. Yamaguchi