Stokes Equations in Geophysics: Forward and Inverse Problems
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Abstract
Motion of rock inside the Earth is commonly modeled by incompressible Stokes equations using various rheological models.
Most models for the effective viscosity are characterized by severe nonlinearities and large
viscosity jumps over small length scales as well as unknown or uncertain material properties, which lead to many theoretical and practical questions about the governing
systems. This thesis addresses two aspects of geodynamic Stokes equations: In the first part, well-posedness of Stokes
systems with two highly nonlinear rheologies, which are important for realistic simulations of flow in the Earth's mantle, is discussed from a
theoretical point of view. After formulating the instantaneous incompressible Stokes equations as employed throughout this thesis,
well-posedness of the linear viscous model is reviewed while introducing concepts of convex optimization.
Subsequently, we investigate Stokes systems using a non-Newtonian power-law viscosity that is also used
for one of the computational inverse problems we address later. Using methods
from convex analysis, well-posedness of the system is shown in a Sobolev space that depends
on the degree of nonlinearity.
As a next step, we investigate the corresponding Stokes system for a specific type of
visco-plastic rheology.
If a positive lower bound is added to the effective viscosity,
well-posedness can be shown in a Sobolev space by means of a convex energy minimization problem. However, without the lower bound
for the effective viscosity, a solution must be sought in the larger space of functions of bounded deformation.
Existence of a corresponding energy minimizer
is shown under a smallness assumption on the body forces acting on the fluid.
In the second part of this thesis, we address the issue of inferring unknown subsurface material
parameters and geometric structures from surface or near-surface measurements of the solution of the Stokes equations.
First, we introduce principal stress directions as a data type for geodynamical inversions.
Formulating the inversion as a regularized infinite-dimensional optimization problem constrained by the Stokes equations, the adjoint equations
are derived employing a formal Lagrangian approach. In simplified numerical examples, we demonstrate
the potential use of this data type in real-world applications.
Finally, we address the inverse problem of simultaneously reconstructing subsurface structures and material parameters from surface observations.
Employing the Bayesian inversion framework enables quantification of uncertainties in
the inferred parameters and shapes. We formulate an infinite-dimensional inverse problem using a level-set approach to describe boundaries
between different materials, which thus allows for jumps in the material parameters.
By proposing a heuristic for constructing tailored level-set priors representing seismic knowledge,
we build a bridge between seismic and mechanical models. In two numerical examples, we demonstrate the
potential of our approach by studying the uncertainties of and trade-offs between subsurface geometric structures,
densities, and viscosities using a dimension-robust Markov chain Monte
Carlo sampling method to approximate the posterior distribution.