|
Mathematics
in Geosciences, September 2001 - June 2002
Winter
2002
IMA Short Course:
Wavelet
Methods in Seismology:
Application of Wavelets to Multiscale Analysis
and Non-linear Estimation
Felix Herrmann
Earth
Resources Laboratory, MIT
6
hours of lectures, February 18-20, 2002
All
lectures in Lind 409
In this 3-day mini-course the application of the wavelet and
related transforms to seismology will be discussed. Attention
will be paid to characterization as well as to non-linear solution
of linear inversion problems, such as denoising and deconvolution.
First a brief overview of (exploration) seismology will be given,
followed by a review of the continuous wavelet transform as
a tool to characterize the scaling of broadband upper sedimentary
records (well-log data). Mallat's Wavelet Transform Modulus
Maxima method will be introduced to calculate the Hölder
regularity as well as the multifractal singularity spectra.
Secondly, methods will be discussed that aim to estimate local
coarse-grained Hölder exponents from essentially bandwidth
limited seismic data. These methods consist of extensions of
the Modulus Maxima framework and of a Matching Pursuit Algorithm
with Fractional Spline Wavelet Packets as a dictionary. Finaly,
examples will be shown how to apply basis function expansions
to the non-linear solution of linear inverse problems.
Feb 18, 9:30 am
Brief overview seismology with the emphasis on exploration seismology
- Outline
of the course emphasizing characterization versus inversion
- Relation
linearized (inverse) scattering and wavelets
Feb
18, 11:00am
Multiscale analysis by the Wavelet transform
- Modulus
Maxima Method
- Local
singularity order estimation via wavelet coefficient decay
along Modulus Maxima
- Open
problems: sensitivity to noise, bandwidth limitation.
Feb
19, 9:30 am
Multifractal Analysis
- Hausdorff
dimensions
- Partition
functions
- Singularity
Spectra
- Modulus
maxima partitioning
- Partition
function by Modulus Maxima
- Some
remarks on link to Regularity Estimation and Besov spaces
Feb
19, 2:00 pm
Monoscale Analysis I
- Generalization of the Modulus
Maxima Method by Fractional Calculus.
- Application to seismic stratigraphy
Feb
20, 9:30 am
Monoscale Analysis II
- Fractional
Splines
- Fractional
Spline Wavelets
- Wavelet
Packets
- Fractional
Spline Matching Pursuit
- Atomic
decomposition and reconstruction
Feb
20, 11:00 am
Non-linear estimation I
- Non-adaptive
denoising by Thresholding
- Deconvolution
using by combining Fourier and Wavelet Methods
- Wavelet-Vaguelette
methods
- Application
to Seismic data
Feb
20, 3:30 pm Seminar on Seismic
singularity extraction and its relation to sedimentary transitions
LIST OF CONFIRMED PARTICIPANTS
| Name |
Department |
Affiliation |
| Santiago Betelu |
Mathematics |
University of North Texas |
| Jamylle Carter |
|
Institute for Mathematics & its Applications |
| Christine Cheng |
Institute for Mathematics & its Applications |
University of Minnesota |
| Dacian Daescu |
University of Minnesota |
Institute for Mathematics and its Applications |
| Gregory S. Duane |
University of Minnesota |
Institute for Mathematics and its Applications |
| Michael Efroimsky |
University of Minnesota |
Institute for Mathematics and its Applications |
| Selim Esedoglu |
|
Institute for Mathematics & its Applications |
| Felix Herrmann |
Earth Resources Laboratory |
Massachusetts Institute of Technology |
| Daniel Kern |
|
|
| Anna Mazzucato |
Mathematics |
Yale University |
| Aurelia Minut |
University of Minnesota |
Institute for Mathematics and its Applications |
| M. Yvonne Ou |
University of Minnesota |
Institute for Mathematics and its Applications |
| Emmanouil I. Papadakis |
Mathematics |
University of Houston |
| Jianliang Qian |
|
Institute for Mathematics & its Applications |
Mathematics
in Geosciences, September 2001 - June 2002
|
