This chapter summarizes three important methods for seismic transmission tomography: the interpretation of delays in onset times of seismic phases using ray theory, of cross-correlation delays using finite-frequency methods, and of full waveforms using adjoint techniques. Delay-time techniques differ importantly in one key aspect from full waveform inversions in that they are more linear. The inverse problem for onset times is usually small enough that it can be solved by matrix inversion; for waveform inversions gradient searches are generally needed, and for cross-correlation delays the solver depends on the size of the problem.
Onset times can simply be interpreted using the approximations of geometrical optics (ray theory). For cross-correlation delays one can use ray theory to compute the linearized dependency on model perturbations in a volume around the ray, if the observed phase travels a well-identified raypath. However, for diffracted pulses or headwaves, numerical solvers for the wavefield are needed. This is also the case for waveform inversions. Whatever the technique that is used, the resulting linearized system is usually underdetermined and needs to be regularized.
Progress in the near future is to be expected from efforts to densify the network of seismometers and extending it to the oceanic domain, as well as from the continued growth in the power of supercomputing that will soon push waveform inversions to embrace the full frequency range of observed seismic signals.
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