A multistep reciprocity gap functional method for the inverse problem in a multilayered medium

We introduce a multistep reciprocity gap functional method to reconstruct the support of a target embedded in a multi-layered medium from near field measurements at a fixed frequency. The approach is based on the use of the standard reciprocity gap functional method for homogeneous background to strip the layers and recover the data at each interface by means of the so-called interior transmission problem and then, at the last step, to reconstruct the support of the scatterer. The advantage of this approach is that it avoids computing the Green's function for multi-layered medium. Numerical examples are given, showing the performance of our reconstruction algorithm.