  Full Waveform Inversion (FWI) is a successful and well-established inverse
method for reconstructing material models from measured wave signals. In the
field of seismic exploration, FWI has proven particularly successful in the
reconstruction of smoothly varying material deviations. In contrast,
non-destructive testing (NDT) often requires the detection and specification of
sharp defects in a specimen. If the contrast between materials is low, FWI can
be successfully applied to these problems as well. However, so far the method
is not fully suitable to image defects such as voids, which are characterized
by a high contrast in the material parameters. In this paper, we introduce a
dimensionless scaling function $\gamma$ to model voids in the forward and
inverse scalar wave equation problem. Depending on which material parameters
this function $\gamma$ scales, different modeling approaches are presented,
leading to three formulations of mono-parameter FWI and one formulation of
two-parameter FWI. The resulting problems are solved by first-order
optimization, where the gradient is computed by an ajdoint state method. The
corresponding Fr\'echet kernels are derived for each approach and the
associated minimization is performed using an L-BFGS algorithm. A comparison
between the different approaches shows that scaling the density with $\gamma$
is most promising for parameterizing voids in the forward and inverse problem.
Finally, in order to consider arbitrary complex geometries known a priori, this
approach is combined with an immersed boundary method, the finite cell method
(FCM).
