In my work I am interested in applying methods from Statistical Physics and Applied Harmonic Analysis to Geophysical Inversion problems. Specifically, I am working on problems in imaging where I not only aim to improve the image quality but where I also try to characterize and understand the physical mechanisms responsible for the main features in the image. Despite its focus on geophysical applications my work also applies to other imaging problems involving multiscale complexity and noise.I am interested in theoretical aspects of exploration and global seismology. For the upper crust, my research is directed towards creating a fundamental understanding of seismic imaging and inversion, as well as establishing a direct relationship between seismic reflectivity and the sedimentary record. For this purpose, I have developed a new methodology which not only models the onsets of sedimentary layers by scale exponents (10,24,19,9), but which also estimates these exponents from bandwidth-limited seismic data (21,11,20,18,13,15). Scale exponents characterize the abruptness of transitions. Besides the method's ability to detect subtle lithology changes, it has the additional advantage that singularity orders are preserved during imaging which makes them less prone to error compared to amplitudes (28,15). Also, the scaling aspect of the exponent-indexed transitions sheds new light on the interpretation of tuning effects within amplitude versus offset (AVO) analysis (12,14).
Within the upper mantle, I am working on the fine-structure of the 410 and 660 discontinuities. Here I apply similar techniques to establish sharpness constraints on these transitions from seismic data. I show that the estimated exponents are consistent with a mixture model yielding a critical behavior for the elastic moduli as a function of the concentration of the stronger, high pressure phases. Contrary to current approaches, where the seismic discontinuities are related to mineralogical phase transformations (22,27,31) via averaging (27), I explain the globally observed seismic discontinuities as critical points being predicted by a percolation phenomenon (29,4,26). As the concentration of one of the constituent composites reaches a critical value, site percolation predicts a cusp-like singularity for the elastic moduli of bi-compositional mixtures with differing elastic properties. At that critical point the elastic moduli undergo a change whose abruptness is characterized by a similar exponent. Since seismic data provide access to the value of the exponent (17,16), I am able to impose additional constraints on the mineralogy near the critical point. Obviously, these constraints will allow us to gain extra information on the composition, temperature and pressure at the transitions.
For sedimentary systems I am following a similar approach. I try to relate subtle changes in the sorting to apparent seismic reflections. At this point I am able to detect changes in the lithology along reflectors from variations in the sharpness (24,19). Even though my model provides geologically consistent results (19), I would like to better understand the mechanics behind the creation of singularities which are responsible for the reflections. Again, I think that the ideas behind percolation theory may provide new insights into this important issue.
Techniques I developed for the estimation of scale exponents draw heavily upon a wealth of new developments in the field of computational Harmonic Analysis (25,6). So far, most of these exciting developments have been applied outside our field. From the results I have obtained so far (23,8,11,13), I conclude that these techniques will significantly improve our ability to solve geophysical inverse problems. The crux of these developments is the recent capability to work with data containing jumps, i.e. non-stationary data. Traditional approaches assume stationarity, which leads to unnecessary additional smoothing of the inversion results (3,5). By applying these new techniques, which include atomic decompositions with respect dictionaries defined by fractional spline wavelets (8,11,30), to seismic imaging and inversion as well as to characterization, we will be able to significantly improve our inversion and characterization results.
One of the major challenges is to generalize existing ideas to include higher dimensional data such as seismic images. Wavelets and their associated basis functions (e.g. wavelet packets) have proven their superiority in solving a whole suite of inverse problems, both linear and non-linear (3,25,5). This superiority derives from their capability to isolate singularities confined to points. Geophysical data, however, typically contain contain curved reflectors, in which case wavelets cease to be optimal. Newly developed basis functions such as curvelets (3,2) remove this shortcoming, since they have directional selectivity build into them. With these truly multi-dimensional basis functions, we not only can significantly improve seismic imaging, but we will also be able to decompose geological structures into highly directional basis functions, whose parameterization will provide us with useful geological information.
Finally, I am starting a project in which I combine the results obtained from time-reversal (7,1) to recent results in time-lapse seismic imaging. In this way I hope to very accurately detect small temporal changes by using time-lapse wavefields including the coda. This promising technique has obvious applications in other fields such as non-destructive testing and non-evasive medical imaging.
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