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
