Viscoelastic inverse problems from seismic to medical scale

By Florian Faucher

This project is funded by the Austrian Science Fund FWF, under the Lise Meitner fellowship grant number M2791-N. It is a two-years fellowship that is carried out at the Faculty of Mathematics at the University of Vienna, and that has started in October 2019.

The project focuses on the quantitative reconstruction of the physical parameters of a medium, with an emphasis on heterogeneous viscoelastic materials, in the context of non-invasive testing for seismic and medical imaging. The objective is to develop efficient algorithms to recover properties which are not directly measurable, such as the Earth’s crust (seismic imaging) or the interior of patients (medical imaging). This research will be carried out in collaboration with the Medical University of Vienna and the energy company OMV.

Despite the differences of our applications (e.g. the scale), both the Earth’s sub-surfaces and biological tissues are usually modeled as elastic media and imaging relies on the propagation of mechanical waves. Namely, to get access to the undisclosed properties inside a body, penetrating waves are used, as they carry information
on the medium in which they propagate. For instance, in the case of an earthquake, mechanical waves
travel through the Earth and are recorded at the surface. These measured seismograms are then used to identify the sub-surface (Earth’s crust and even deeper), see the illustration on the Figure below. The similar procedure holds with X-Ray medical imaging, where waves outside of the patient are measured to recompute the absorption coefficient inside the patient.

Illustration of seismic data: the wave propagates through the sub-surface, generating multiple reflections towards the surface.
The observational devices, located at the surface, record the amplitude of the waves (here the pressure field).
The inverse problem aims at the characterization of the sub-surface from the seismograms measured at the surface.

Then, the reconstruction relies on numerical simulations which necessitate to realistically model the medium to extract its properties. For instance, conventional medical ultrasound imaging assumes that the body is essentially water, while conventional seismic imaging assumes a linear elastic material. In this project, we shall study the case of viscous materials, which have less elasticity and tend to wrinkle. They are harder to model and therefore, it is also harder to “see inside” using computational methods. This project aims at developing an appropriate numerical process for the reconstruction of such materials (optimization method, deployment on supercomputers for large scale applications) and will also investigate the mathematical properties of the problem.

Eventually, this project can also find further reaching towards applications of various scale such that, e.g., extra-terrestrial imaging and helioseismology.

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