Reducing non-uniqueness in seismic inverse problems

The scientific investigation of the solid Earth's complex
processes, including their interactions with the oceans and the
atmosphere, is an interdisciplinary field in which seismology has
one key role. Major contributions of modern seismology are (1) the
development of high-resolution tomographic images of the Earth's
structure and (2) the investigation of earthquake source processes.
In both disciplines the challenge lies in solving a seismic inverse
problem, i.e. in obtaining information about physical parameters
that are not directly observable. Seismic inverse studies usually
aim to find realistic models through the minimization of the misfit
between observed and theoretically computed (synthetic) ground
motions. In general, this approach depends on the numerical
simulation of seismic waves propagating in a specified Earth model
(forward problem) and the acquisition of illuminating data. While
the former is routinely solved using spectral-element methods, many
seismic inverse problems still suffer from the lack of information
typically leading to ill-posed inverse problems with multiple
solutions and trade-offs between the model parameters.
Non-linearity in forward modeling and the non-convexity of misfit
functions aggravate the inversion for structure and source. This
situation requires an efficient exploitation of the available data.
However, a careful analysis of whether individual models can be
considered a reasonable approximation of the true solution
(deterministic approach) or if single models should be replaced
with statistical distributions of model parameters (probabilistic
or Bayesian approach) is inevitable. Deterministic inversion
attempts to find the model that provides the best explanation of
the data, typically using iterative optimization techniques. To
prevent the inversion process from being trapped in a meaningless
local minimum an accurate initial low frequency model is
indispensable. Regularization, e.g. in terms of smoothing or
damping, is necessary to avoid artifacts from the mapping of high
frequency information. However, regularization increases parameter
trade-offs and is subjective to some degree, which means that
resolution estimates tend to be biased. Probabilistic (or Bayesian)
inversions overcome the drawbacks of the deterministic approach by
using a global model search that provides unbiased measures of
resolution and trade-offs. Critical aspects are computational
costs, the appropriate incorporation of prior knowledge and the
difficulties in interpreting and processing the results. This work
studies both the deterministic and the probabilistic approach.
Recent observations of rotational ground motions, that complement
translational ground motion measurements from conventional
seismometers, motivated the research. It is investigated if
alternative seismic observables, including rotations and dynamic
strain, have the potential to reduce non-uniqueness and parameter
trade-offs in seismic inverse problems. In the framework of
deterministic full waveform inversion a novel approach to seismic
tomography is applied for the first time to (synthetic) collocated
measurements of translations, rotations and strain. The concept is
based on the definition of new observables combining translation
and rotation, and translation and strain measurements,
respectively. Studying the corresponding sensitivity kernels
assesses the capability of the new observables to constrain various
aspects of a three-dimensional Earth structure. These observables
are generally sensitive only to small-scale near-receiver
structures. It follows, for example, that knowledge of deeper Earth
structure are not required in tomographic inversions for local
structure based on the new observables. Also in the context of
deterministic full waveform inversion a new method for the design
of seismic observables with focused sensitivity to a target model
parameter class, e.g. density structure, is developed. This is
achieved through the optimal linear combination of fundamental
observables that can be any scalar measurement extracted from
seismic recordings. A series of examples illustrate that the
resulting optimal observables are able to minimize inter-parameter
trade-offs that result from regularization in ill-posed
multi-parameter inverse problems. The inclusion of alternative and
the design of optimal observables in seismic tomography also affect
more general objectives in geoscience. The investigation of the
history and the dynamics of tectonic plate motion benefits, for
example, from the detailed knowledge of small-scale heterogeneities
in the crust and the upper mantle. Optimal observables focusing on
density help to independently constrain the Earth's temperature and
composition and provide information on convective flow. Moreover,
the presented work analyzes for the first time if the inclusion of
rotational ground motion measurements enables a more detailed
description of earthquake source processes. The complexities of
earthquake rupture suggest a probabilistic (or Bayesian) inversion
approach. The results of the synthetic study indicate that the
incorporation of rotational ground motion recordings can
significantly reduce the non-uniqueness in finite source
inversions, provided that measurement uncertainties are similar to
or below the uncertainties of translational velocity recordings. If
this condition is met, the joint processing of rotational and
translational ground motion provides more detailed information
about earthquake dynamics, including rheological fault properties
and friction law parameters. Both are critical e.g. for the
reliable assessment of seismic hazards.