1,622 research outputs found
Nonlinear regularization techniques for seismic tomography
The effects of several nonlinear regularization techniques are discussed in
the framework of 3D seismic tomography. Traditional, linear, penalties
are compared to so-called sparsity promoting and penalties,
and a total variation penalty. Which of these algorithms is judged optimal
depends on the specific requirements of the scientific experiment. If the
correct reproduction of model amplitudes is important, classical damping
towards a smooth model using an norm works almost as well as
minimizing the total variation but is much more efficient. If gradients (edges
of anomalies) should be resolved with a minimum of distortion, we prefer
damping of Daubechies-4 wavelet coefficients. It has the additional
advantage of yielding a noiseless reconstruction, contrary to simple
minimization (`Tikhonov regularization') which should be avoided. In some of
our examples, the method produced notable artifacts. In addition we
show how nonlinear methods for finding sparse models can be
competitive in speed with the widely used methods, certainly under
noisy conditions, so that there is no need to shun penalizations.Comment: 23 pages, 7 figures. Typographical error corrected in accelerated
algorithms (14) and (20
On the use of sensitivity tests in seismic tomography
ACKNOWLEDGEMENTS This work was partly supported by ARC Discovery Project DP120103673 and by the Research Council of Norway through its Centres of Excellence funding scheme, project number 223272. We thank Maximilliano Bezada and an anonymous referee for constructive comments which improved the original version of the manuscript. We also thank the Editor, A. Morelli, for providing additional helpful comments.Peer reviewedPublisher PD
Optimization Methods for Inverse Problems
Optimization plays an important role in solving many inverse problems.
Indeed, the task of inversion often either involves or is fully cast as a
solution of an optimization problem. In this light, the mere non-linear,
non-convex, and large-scale nature of many of these inversions gives rise to
some very challenging optimization problems. The inverse problem community has
long been developing various techniques for solving such optimization tasks.
However, other, seemingly disjoint communities, such as that of machine
learning, have developed, almost in parallel, interesting alternative methods
which might have stayed under the radar of the inverse problem community. In
this survey, we aim to change that. In doing so, we first discuss current
state-of-the-art optimization methods widely used in inverse problems. We then
survey recent related advances in addressing similar challenges in problems
faced by the machine learning community, and discuss their potential advantages
for solving inverse problems. By highlighting the similarities among the
optimization challenges faced by the inverse problem and the machine learning
communities, we hope that this survey can serve as a bridge in bringing
together these two communities and encourage cross fertilization of ideas.Comment: 13 page
Eigenvector Model Descriptors for Solving an Inverse Problem of Helmholtz Equation: Extended Materials
We study the seismic inverse problem for the recovery of subsurface
properties in acoustic media. In order to reduce the ill-posedness of the
problem, the heterogeneous wave speed parameter to be recovered is represented
using a limited number of coefficients associated with a basis of eigenvectors
of a diffusion equation, following the regularization by discretization
approach. We compare several choices for the diffusion coefficient in the
partial differential equations, which are extracted from the field of image
processing. We first investigate their efficiency for image decomposition
(accuracy of the representation with respect to the number of variables and
denoising). Next, we implement the method in the quantitative reconstruction
procedure for seismic imaging, following the Full Waveform Inversion method,
where the difficulty resides in that the basis is defined from an initial model
where none of the actual structures is known. In particular, we demonstrate
that the method is efficient for the challenging reconstruction of media with
salt-domes. We employ the method in two and three-dimensional experiments and
show that the eigenvector representation compensates for the lack of low
frequency information, it eventually serves us to extract guidelines for the
implementation of the method.Comment: 45 pages, 37 figure
Iterative algorithms for total variation-like reconstructions in seismic tomography
A qualitative comparison of total variation like penalties (total variation,
Huber variant of total variation, total generalized variation, ...) is made in
the context of global seismic tomography. Both penalized and constrained
formulations of seismic recovery problems are treated. A number of simple
iterative recovery algorithms applicable to these problems are described. The
convergence speed of these algorithms is compared numerically in this setting.
For the constrained formulation a new algorithm is proposed and its convergence
is proven.Comment: 28 pages, 8 figures. Corrected sign errors in formula (25
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