Global tomography using finite-frequency kernels in the wavelet domain

Finite-frequency tomographic methods find their origin in the recognition that seismic waves are sensitive to the earth's structure not only on but also in a neighborhood of the ray connecting source and receiver. The sensitivity kernels are therefore nonzero within some positive distance from this ray. Real-life tomographic applications often need to employ a coarse model parameterization to reduce the number of model parameters and make the inversion practical from a computational point of view. This coarse parameterization, however, substantially reduces the benefit in resolution of finite-frequency tomography when compared to classical tomographic methods; standard coarse parameterization effectively turns the finite-frequency sensitivity kernels into 'fat' rays. To overcome this we are developing global-scale finite-frequency tomography in the wavelet domain, where the sparseness of both the sensitivity kernel and the model can be exploited in carrying out the inversion. We work on the cubed sphere to allow us to use wavelet transforms in Cartesian coordinates. This cubed sphere is built through a one-to-one mapping of Cartesian coordinates on each face of the cube to the corresponding "faces of the sphere". At the edges of each of the faces of the cube, the mapping is singular; this induces artifical singularities in the model and kernel, which in the wavelet domain would show up as large coefficients. We avoid these artificially large wavelet coefficients by using domain-adapted wavelets based on the construction of wavelets on the interval. The inversion is based on an l1-norm minimization procedure. We will present some preliminary examples.