What is a geophysical inverse problem? Basically, it means the inference of various properties of the Earth's interior from observations of physical fields at the surface or in boreholes. The physical fields may be, among others, gravity, temperature, electric and magnetic fields, deformation and seismic wavefields. Corresponding Earth properties are density, heat conductivity, electric conductivity and permittivity, magnetization, elastic, and more generally, rheological properties in the earth's interior. And earth's interior refers to everything below the surface.
All properties are functions of space and may sometimes even change in time with infinitely many possible variations. In contrast, there will always be a finite number of observations from which we want to estimate the (infinitely many) spatial variations of these properties. For this reason, any serious geophysical inverse problem is always underdetermined. Our observations are never sufficient to obtain a full and unique estimate of physical properties. On top of that, our observations have errors and may be contradictory. This fact further reduces our ability to make inferences of Earth's interior.
Nevertheless, we can learn something from our observations. If the observation sites are well chosen and sample the physical field densely, the permissible solutions of the inverse problem (i.e. functions of physical properties which explain our observations called earth models) often differ in small-scale variations while the large-scale patterns can be determined robustly. The basic strategy to solve a geophysical inverse problem is hence to search for the least-detailed solution that can explain our data according to Einstein's rule: ``Make things as simple as possible but not simpler''. We will see that this philosophy leads to an optimization (minimization) problem with constraints.
Solving optimization problems is a very important branch of the mathematical theories of linear algebra and functional analysis. For this reason, we will start this lecture with some very useful ideas and concepts of linear algebra and functional analysis that we will need later to obtain a mathematical formulation of an inverse problem. We will see that we can describe the earth models as elements of an abstract vector space. We will measure the spatial variations of an earth model by defining an appropriate size or length of an earth model and we will search for the smallest earth models that explains our data within their errors.
One may think that inverse theory gives us a recipe to create models from observations. In some way, this is true. But owing to the non-uniqueness, it cannot tell us which models make sense and which do not. In the end, it is the well-educated geophysicist who has to judge whether a model is reasonable or not and who has to steer the mathematical algorithm towards reasonable models. This fact is what guarantees your job and inhibits that you are replaced by a machine which works 24 hours a day without holidays.
- Kursleiter/in: Wolfgang Friederich