  We consider the iterative reconstruction of both the internal geometry and
the values of an inhomogeneous acoustic refraction index through a piecewise
constant approximation. In this context, we propose two enhancements intended
to reduce the number of parameters to reconstruct, while preserving accuracy.
This is achieved through the use of geometrical informations obtained from a
previously developed defect localization method. The first enhancement consists
in a preliminary selection of relevant parameters, while the second one is an
adaptive refinement to enhance precision with a low number of parameters. Each
of them is numerically illustrated.
