Mathematics in Geosciences, September 2001 - June 2002

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