Institut für Numerische und Angewandte Mathematik - Arbeitsgruppe Inverse Probleme

In helioseismology, the measurements are images of the velocity or magnetic field on the solar surface. They are taken by satellites like the HMI (Helioseismic and Magnetic Imager) instrument on board of the SDO (Solar Dynamics Observatory) satellite. For example, the image on the left represents the magnetic field at the surface of the Sun. The white areas represent a polarity showing north (outward) while the black ones are in direction of the south (inward). Intense magnetic fields can lead to solar flares or coronal mass ejections that are visible on the right picture.
Green's function Green's function
Courtesy of NASA/SDO and the AIA, EVE, and HMI science teams

Using these measurements, our aim in a collaboration with Laurent Gizon is to learn about the solar interior. To achieve this goal, we need to know how the quantities we want to learn about (velocity, sound speed, magnetic field, ...) are linked to the observations. This step is called the forward modeling.

Forward modeling
Here, we model by an operator F the propagation of acoustic waves in the Sun. This operator links the observations (denoted g) described above to the quantity we want to infer inside the Sun (denoted u) via an equation F(u) = g. This step is fundamental to understand how a change in the medium will influence what we are seeing at the surface of the Sun.
The picture shows how a source located at the pole propagates inside the Sun. For example, due to the sound speed profile, the waves propagate faster close to the surface than at the center of the Sun. If the sound speed is larger than expected close to the surface, the waves will arrive faster at the surface which will change our observations.

Inverse problem
Once the observations g and the forward model F are known, we need to reconstruct our unknown q, this is the inverse problem. As for most inverse problems, the diffuculty is that small measurement errors can lead to huge errors in the reconstruction. In our group, we are looking for methods to obtain the best reconstruction depending on the level of noise. This is done by regularization which consists in adding information about the solution (smoothness) during the reconstruction process.
Green's function
Propagation of acoustic waves in the Sun.