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IMA Short Course

Wavelet Methods in Seismology

Application of Wavelets to Multiscale Analysis and Non-linear Estimation

Wavelet Methods in Seismology

Application of Wavelets to Multiscale Analysis and Non-linear Estimation

February 18-20, 2002

Mathematics in Geosciences, September 2001 - June 2002

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

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 |