## Workshop 3: November 21-25, 2011

# Wave Propagation and Scattering, Inverse Problems and Applications in Energy and the Environment

## Organizers

Ivan G. Graham, University of Bath, UK

Ulrich Langer, Johann Radon Institute & University of Linz, Austria

Markus Melenk, Vienna University of Technology, Austria

Mourad Sini, Johann Radon Institute, Austria

## Synopsis and Main Topics

The efficient computation of wave propagation and scattering is a core problem in numerical mathematics, which is currently
of great research interest and is central to many applications in energy and the environment. Two generic applications
which resonate strongly with the central aims of this special semester are forward wave propagation in heterogeneous media
and seismic inversion for subsurface imaging.

As an example of the first application, modelling of absorption and scattering of radiation by clouds, aerosol and
precipitation is used as a tool for interpretation of (e.g.) solar, infrared and radar measurements, and as a component in
larger weather/climate prediction models in numerical weather forecasting. One key numerical component in this modelling
is the prediction of the total optical properties and the full scattering matrix from an ensemble of irregular particles.
The underlying mathematical problem is that of accurately computing high frequency wave propagation in a highly heterogeneous
medium.

As an example of the second application, inverse problems in wave propagation in heterogeneous media arise in the
problem of imaging the subsurface below land or marine deposits. Solutions to this problem have a number of environmental
uses, for example in the location of hydrocarbon-bearing rocks, in the monitoring of pollution in groundwater or in
earthquake modelling. A seismic source is directed into the ground and the material properties of the subsurface are inferred
by analysing the observed scattered field, recorded by sensors. The inversion process (a large scale optimisation problem) is
complicated by the presence of multiple reflections and the fact that the scales involved in the exploration of sub-marine,
sub-basalt or sub-salt oil reservoirs can be many kilometres in extent, leading to a challenging multi-scale problem. Current
iterative methods for solving the inverse problem involve repeated solution of the forward problem (the computational kernel),
which is typically a frequency-domain reduction of the elastic (or scalar) wave equation with high frequency and typically
highly spatially varying wave speed. If the inversion is to be competitive, the key underlying problem to be overcome is the
design of robust and scalable solvers for the large highly indefinite linear systems arising from these problems.

The workshop will bring together key numerical mathematicians whose interest is in the analysis and computation of wave propagation and scattering problems, and in inverse problems, together with practitioners from engineering and industry whose interest is in the applications of these core problems. Particular problems to be considered will be (i) The design of accurate methods for solving frequency domain problems; (ii) The use of wave enriched and other hybrid approximation strategies in the solution of high-frequency problems; (iii) Fast linear algebra solvers for frequency domain problems in heterogeneous media; (iv) advanced inverse problem approaches for wave problems such as reverse time migration which move away from traditional ray-based approaches in the high frequency case.