Efficient Inversion Methods for Parameter Identification in the Earth Sciences
Parameter identification methods play a crucial role in Earth sciences, especially in the context of determining information on Earth’s past or present heterogeneous internal state which is not directly observable. Two key applications that are to be dealt with in this project are seismic tomography (present state) and the generation of initial conditions for mantle convection models by data-assimilation (past state). Both questions are directly related to our overall target, since e.g. convective processes in the Earth's mantle are a key component of the large-scale forcing of the Earth's lithosphere.
Within this project we aim for three goals. The first is to improve inversion methods with respect to robustness and efficiency by developing new techniques based on adjoint formulations and co-variance operators. The second goal is to provide highly parallel implementations of the resulting algorithms fit for high-resolution Earth models. The third goal is to derive a standard for the visualisation of different possible solutions of the identification problem in combination with their respective likelihood. The outcomes of this project are expected to be beneficial also for parameter identification problems in other areas.
The research on improved inversion methods will be twofold.
- Building on the initial adjoint models for seismology and mantle convection problems built by Prof. Bunge’s group and the expertise in efficient adjoint-based algorithms for PDE-constrained optimisation by Prof. Ulbrich's group we will enhance these methods and compare/combine them with other approaches like automatic differentiation.
- Inversion in seismology can also be based on Bayesian approaches to a least squares formulation. Here, the choice of an appropriate a priori covariance operator for the data error is crucial and its sensitivity has to be studied. Modelling of this covariance operator adjusting for time and location proximities using geostatistical dynamic models will be investigated to improve the fit and the forecast abilities of the models. Here expertise of Prof. Czado's group especially in spatial statistics will be utilised.
Further research topics:
- For questions of run-time efficient parallel implementation we will closely co-operate with Project B7.
- In cooperation with the group of Prof. Westermann, research will be pursued on interactive visual analytics techniques for the exploration of multi-scale dynamic Earth models. To be able to effectively analyze uncertainty in the simulation data, research on dynamic visualisation tools to aid the underlying statistical modelling process is required, e.g. for model diagnostics, assessment of robustness, and the visualisation of spatio-temporal fluctuations.
In subproject Westermann the challenge is to provide adequate and intuitive 3D visualization methods for a context-aware illustration of uncertainties in multi-dimensional data sets. Here, we concentrate both on the visual analysis of parameterized uncertainty and uncertainty given by stochastic simulation results (e.g. Monte Carlo). Thereby a strong focus is put on integrating uncertainty effects in interactive 3D data representations by direct and indirect volume rendering techniques. So far we proposed methods for positional and geometric variability analysis of surfaces in 3D as well as local and global correlation visualization for structural uncertainty analysis. Future visualization challenges comprise visual representation of multimodal probability distributions, visual hypothesis tests, sensitivity analysis and combined prior and posterior covariance visualization.
Within the subproject of mathematical optimization we have implemented a trust-region Newton-CG method for full-waveform seismic tomography. It features the adjoint-based computation of the gradient and Hessian-vector products and the simulation of elastic waves with MPI parallelization. Moreover, we provide a modular framework to test different misfit criteria and regularization methods.
|Prof. Dr. Hans-Peter Bunge (coordination)||Geophysics|
|Prof. Dr. Heiner Igel||Geophysics|
|Prof. Claudia Czado, Ph.D.||Mathematical Statistics|
|Prof. Dr. Rüdiger Westermann||Computer Graphics & Visualization|
|Prof. Dr. Michael Ulbrich||Mathematical Optimisation|
|Dr. Ran Zhang|
|Dr. Simon Stähler|
|Dr. Tobias Pfaffelmoser|
|Dr. Christian Böhm|
|Karin Sigloch, Ph.D. (coord.)|
Poster (PDF) - Christian Boehm
Poster (PDF) - Christian Boehm
Poster (PDF) - Tobias Pfaffelmoser
- Christian Boehm, Efficient Inversion Methods for Constrained Parameter Identification in Full-Waveform Seismic Tomography, Dissertation, Department of Mathematics, Technical University of Munich, 2015
- Ran Zhang, Efficient Parameter Estimation in the High-Dimensional Inverse Problem of Seismic Tomography, Dissertation, Department of Mathematics, Technical University of Munich, 2014
- Tobias Pfaffelmoser, Matthias Reitinger and Rüdiger Westermann, Visualizing the Positional and Geometrical Variability of Isosurfaces in Uncertain Scalar Fields, Computer Graphics Forum (Proceedings of EuroVis 2011)
- Stähler, S. C., K. Sigloch, and T. Nissen-Meyer, Triplicated P-wave measurements for waveform tomography of the mantle transition zone, Solid Earth (2012) http://www.solid-earth-discuss.net/4/783/2012/.
- Tobias Pfaffelmoser, Visualization of Uncertain Scalar Data Sets , Dissertation, 2013
- T. Pfaffelmoser, M. Mihai, R. Westermann, Visualizing the Variability of Gradients in Uncertain 2D Scalar Fields , IEEE Transactions on Visualization and Computer Graphics, 19(11), 2013
- T. Pfaffelmoser, R. Westermann, Correlation Visualization for Structural Uncertainty Analysis , International Journal for Uncertainty Quantification, 3(2), 2013.
- C. Boehm, M. Ulbrich, A Newton-CG Method for Full-Waveform Inversion in a Coupled Solid-Fluid System , Advanced Computing, Lecture Notes in CSE, vol. 93, 2013
- S. C. Stähler, K. Sigloch, Fully probabilistic seismic source inversion -- Part 1: Efficient parameterisation , Solid Earth Discussions 5, 2013
- S. C. Stähler, Finite-frequency tomography with complex body waves: taking into account data uncertainty and correlation, Dissertation, 2014