Research

Finite-frequency tomography in the wavelet domain

Ray-theoretical tomographic methods are based on the principle that the propagation of waves is sensitive to the structure in the earth along the ray only. We have, however, all experienced "wavefront healing" in our youth when fishing and we saw some waves from nearby kids in a pond propagating beyond our bobber without seeming to notice the bobber at all! Clearly the waves are not sensitive to the structure along the ray only. Finite-frequency tomographic methods overcome this limitation by using that waves are sensitive to the earth's structure a finite distance away from the ray. In real-life tomographic applications, however, the model parameterization often needs to be "coarse" to reduce the number of unknown model parameters and make the inversion practical from a computational point of view. This coarse parameterization effectively turns finite-frequency kernels almost into rays (albeit fat; see Figures 1a and b, courtesy of Sebastien Chevrot). Of course, the banana-doughnut operator and the model are both expected to be "sparse" in the wavelet domain because they are "smooth". Therefore, when doing the inversion in the wavelet domain, we expect to be able to use much finer model parameterization, thus improving resolution and allowing clear demonstration of the benefit of finite-frequency tomography when compared to ray-based methods. As shown recently by Loris, Nolet, Daubechies, and Dahlen (GJI, 2007), attacking the inverse problem then with an l1-norm minimization procedure, results in the solution that "contains no more fine-scale structure than is necessary to fit the data to within its assigned errors" and contains both smooth and sharp features without significant blurring due to the source-receiver distribution.

Figure 1a. Finely sampled banana-doughnut kernel.	Figure 1b. Coursely sampled banana-doughnut kernel.

As a postdoc at Princeton I am collaborating with seismologists (Guust Nolet and Tony Dahlen) and applied mathematicians (Ingrid Daubechies, Massimo Fornasier, and Ignace Loris) to develop finite-frequency tomography in the wavelet domain for both global scale (i.e., in spherical coordinates) and regional-scale (i.e., cartesian coordinates) tomographic inverse problems. We are aiming to have our first preliminary results end of summer 2007. Stay tuned!