Inversion of impact-echo data using a shape optimization approach

For the thickness measurement of refractory structures, the impact-echo measurement is one of the methods being used. This method is especially suited for the cases where high frequencies cannot be used due to the high damping and scattering in the materials. The data gathered with this method are often processed using a one-dimensional approach, which works well for simple layered structures. However, when the geometry is more complex, the thickness of the material needs to be estimated, as well as the shape of the interface. In this paper a method for inversion of impact-echo data in a two-dimensional setting is presented using multiple transducers. The algorithm was implemented using Fenics finite element software. The software is used to solve the elastic wave equation and the dolfin-adjoint toolbox is used for efficient computation of the shape gradient. In the paper, the algorithm is tested on two common erosion profiles. These examples show that, under reasonable initial assumptions, the solution converges close to the shape of the true interface. In practice, it is convenient to use a low number of sensors; for the two cases, a numerical experiment is carried out to determine the minimum number of sources required to arrive at a good result. The robustness of the algorithm is tested by adding roughness to the interface and by superimposing noise on the reference signals. The last experiment investigates how the solutions change as the number of samples increases.