The aim of full waveform inversion is the reconstruction of subsurface material parameters (e.g., the velocity of compression and shear waves and the attenuation of these waves) from measurements of the full elastic wave field which has been excited locally by controlled sources or globally by earthquakes. From a mathematical point of view, we have to deal with a nonlinear parameter identification problem for the acoustic, elastic, viscoacoustic or viscoelastic wave equation. The least squares approach for the full nonlinear problem is called full waveform inversion (FWI). It needs to be regularized due to its illposedness. The resulting optimization problem with PDE constraints is mostly solved by gradient-type schemes which require repeated solutions of the direct and the adjoint time-dependent PDE. This approach became reasonable just recently due to the tremendously increased performance of modern computers. Nevertheless, the approach suffers twofold:
- Its practicality is still restricted by the numerical effort needed, for instance, by explicit time integrators, by the storage requirements for the full wave propagation in space and time, by non-linearities of the inverse problem, and by the slow convergence of gradient-type schemes.
- A comprehensive mathematical analysis of the inverse problem justifying the numerics is missing. However, such an analysis is needed to achieve the best possible reconstruction within the uncertainty and incompleteness of the measurement data.
We work on both issues under the following guiding principle: Develop a sound mathematical theory driven by realistic scenarios. And set up regularization schemes adapted to the underlying physics (for instance, preserving discontinuities of the material parameters and taking into account strongly anisotropic and heterogeneous porous media properties).
Some of our achievements in the abstract analysis of the problem so far are:
- A theory on inverse problems in the context of abstract evolution equations.
- Mathematical proofs for the well-posedness of the physical wave models mentioned above under certain initial and boundary conditions.
- Mathematical proofs for the existence of the Fréchet derivative of the parameter-to-solution map between appropriate spaces.
- A mathematical proof for the ill-posedness of the inverse problem related to the elastic wave equation where measurements are taken throughout the whole body.
- A new formulation of the viscoacoustic and viscoelastic wave equation which allows the adjoint equations to be evaluated with the same numerical solver as the direct ones.
In particular we came closer towards the goal to incorporate dispersion, attenuation and anisotropy in a 3D setting, thus reducing the gap between mathematical idealization and practical needs by extending the model step by step to a realistic physical scenario.
Open analytical questions we still need to address are:
- Between which spaces is the parameter-to-solution map injective? In other words: Which subsurface information is uniquely determined by the measurements? What is the sensitivity of different combinations of material parameters and which combination of parameters is best resolvable?
- Which analytical setting is appropriate to describe a noise level and to analyze a discrepancy principle?
- Which kind of regularization fits the physics of the problem?
Concerning the computational aspects of the inversion, we construct implicit hierarchical space-time discretizations leading to efficient parallel schemes for solving the direct problem and for evaluating the Fréchet derivative of the parameter-to-solution map and its adjoint.
So far we achieved the following in this field:
- Parallel implementation of upwind discontinuous Galerkin discretizations in space using the framework described in  for the linear viscoacoustic and viscoelastic equation as considered in . To treat the evolution in time, we employ time stepping schemes as well as space-time discretizations in cooperation with project A3. All the solvers are realized using the M++ library. Finite difference parallel implementation of viscoacoustic and viscoelastic equations described in .
- Parallel implementation of First Order System Least-Squares (FOSLS) type space-time discretizations for acoustic waves including a variant of a discontinuous Petrov-Galerkin method in cooperation with A3 , .
- Development of a mathematically motivated software interface which couples the inversion algorithm and the numerical scheme for the wave equation. As a result, the same implementation of the inversion algorithms within this framework can be used by any wave-solver. This framework has been used to produce the inversion results in .
- Implementation of CG-REGINN, a gradient descent and LBFGS within the framework mentioned above.
- Implementation of an exact numerical solver for the one-dimensional elastic wave equation on a layered medium.
- Realization of numerical experiments using real field data which we acquired during several geophysical expeditions.
- Development of a playground software for educational purposes to experiment with acoustic waves in inhomogeneous media. Within the program, the user can observe a simple FWI-algorithm at work.
- Organization of a joint summer school on full waveform inversion.
Challenges in the numerical parameter reconstruction to be addressed next are:
- Systematic comparison of different inversion schemes in combination with different discretizations for the wave equation.
- Avoidance of the inverse crime by using a different forward solver during the inversion than for producing benchmark data in synthetic experiments.
- Investigate whether space-time discretizations with high-order approximation properties in time are advantageous for the inversion.
Our vision is to realize highly reliable reconstruction algorithms complying with industrial standards. To this end, we combine up-to-date techniques for inverse problems with massively parallel explicit and implicit direct solvers for the coarse level of adaptive hierarchy space-time discretizations which allow for locally very exact resolutions and provide efficient coarse approximation for preconditioning.
References and Publications
 Andreas Kirsch and Andreas Rieder. Inverse problems for abstract evolution equations with applications in electrodynamics and elasticity. Inverse Problems, 32(8):085001, 24, 2016.
 Willy Dörfler, Stefan Findeisen, Christian Wieners and Daniel Ziegler, Parallel adaptive discontinuous Galerkin discretizations in space and time for linear elastic and acoustic waves, Space-Time Methods. Applications to Partial Differential Equations (U. Langer and O. Steinbach, eds.), Radon Series on Computational and Applied Mathematics 21, Walter de Gruyter, 2018.
 Johannes Ernesti. Space-time methods for acoustic waves with applications to full waveform inversion. Dissertation, Karlsruhe Institute of Technology, 2018.
 Johannes Ernesti, Christian Wieners, A space-time discontinuous Petrov-Galerkin method for acoustic waves, Space-Time Methods. Applications to Partial Differential Equations (U. Langer and O. Steinbach, eds.), Radon Series on Computational and Applied Mathematics 21, Walter de Gruyter, 2018.
 Uwe Zeltmann. The viscoelastic seismic model: Existence, uniqueness and differentiability with respect to parameters. Dissertation, Karlsruhe Institute of Technology, to be published 2019.