CMG RESEARCH: Combining Adjoint Tomography and Sparse Imaging Methods in Seismology

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The focus of this project is on inverse problems in seismography. Inferring undergound structure from seismic measurements is a nonlinear problem that is also notoriously ill-posed. The PIs will tackle this problem in its full nonlinearity via the adjoint method, by iteratively minimizing a variational functional in which, at each iteration step, the nonlinear effects of the approximate solution are fully taken into account for the computations in the next iteration.To deal with the ill-posedness of the problem, they will use a regularization method that incorporates efficient modeling of spatial distributions that can exhibit discontinuities as well as smooth behavior between discontinuous transitions. More precisely, they will model the distribution as a sparse superposition of wavelets and curvelets, and add the sum of the absolute values of the corresponding expansion coefficients as a penalty term to the variational functional to be minimized. The inclusion of such a term enforces the sparseness of the expansion, thus expressing the smoothness of the model between discontinuous transitions, and has been proved to be regularizing. Computational resources have reached the speed and scale at which it is now feasible to use this approach for realistic problems.Seismology seeks to gain insight into underground geological structure by measurements done at the surface. When vibrational signals are sent into the ground (whether by earthquakes, carefully tailored explosions or specially constructed vibrators), they propagate at different speeds through layers of different constitution, and are reflected at the often abrupt transitions between different layers. The goal of seismology is to reconstruct the underground structure traversed by these seismic waves from the complex signals, registered by seismographs, that result from the multiple reflections and their interaction. The corresponding numerical problem is of such great complexity that it has been necessary, in the past, to simplify it so as to keep the problem feasible; this led, of necessity, to approximate solutions. The PIs will make use of recent mathematical advances that make it possible to model more effectively heterogeneous underground structures, and of the continuing progress in speed of computational resources to tackle the problem without having to introduce some of the restrictive simplifications used previously. This is expected to result in more accurate maps of underground structure.