Least-squares reverse time migration (LSRTM) and full waveform inversion (FWI) can both provide an image of the subsurface. In this thesis, I investigate the connections and differences between LSRTM and FWI, both in time and frequency domains in acoustic media. Generally, LSRTM can be treated as the inner loop of the FWI algorithm and solved in the image domain. Different from FWI, LSRTM uses reflection data and requires an accurate operator since linearization implies the operator (Born modeling) is independent of the model(reflectivity). FWI uses diving waves and reflection data and is more robust to velocity errors in the initial model because the operator is itself and changed during the optimization. Comparing implementations domains, LSRTM in the frequency domain is relatively easier to formulate than in the time domain. It has the advantage that the wavefield is solved simultaneously for many shots. Unfortunately, the frequency domain formulation becomes too expensive in 3D surveys in terms of memory requirements. Modern implementations of time domain LSRTM require less computer memory and are more efficient than the frequency domain for the 3D case. A similar situation happens for the FWI problem. One advantage of the frequency domain FWI is the easy formulation for the multigrid method. For this approach, updates of the velocity model start from the low frequency components, which can correct the model in the first several iterations and make the result more accurate.
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