Regularization strategies for increasing efficiency and robustness of least-squares RTM and FWI
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Seismic imaging in geologically complicated areas is playing an extremely important role for hydrocarbon exploration. To improve the resolution and fidelity of subsurface image or velocity, an inverse problem is usually solved for obtaining an optimal solution, with which the synthetic data can match the observations. One popular inverse problem is least-squares reverse-time migration (LSRTM), which is linear and trys to obtain an image with a better amplitude fidelity. Another popular problem is full waveform inversion (FWI), which is non-linear and attempts to invert for a velocity model with a higher resolution. Both LSRTM and FWI can be solved in the framework of local optimization, and their solutions could be iteratively updated with a gradient at each iteration, until the data mismatch reaches a desired low level. Each gradient calculation requires to solve the full wave equations twice, and many iterations are usually performed for obtaining a desired solution. The need for iterations has made LSRTM and FWI prohibitively computationally expensive. Another challenge with the two inverse problems is the low robustness caused by the poor quality of the input seismic data and a possibly poor initial model. In this dissertation, I aim at developing regularization strategies for increasing efficiency and robustness of LSRTM and FWI. First, I utilize time-shift gathers to improve the efficiency of computing gradient/RTM. Time-shift gathers can be computed on a coarse grid, and then the information along the time-shift axis can be extracted to generate a final image on a much denser grid. Second, I propose to use attenuation-compensated gradient to accelerate the convergence rate, when the energy loss has been caused by viscous media. Fractional Laplacian wave equations having separate controls over phase distortion and amplitude loss effects are solved for viscoacoustic and viscoelastic modeling. Two methods are tried to solve the spatially varying-order fractional Laplacians: 1) local pseudo-spectral method and domain decomposition; 2) low-rank wave extrapolation. Attenuation-compensated FWI is formulated and implemented based on the low-rank wave extrapolation method. Third, I apply structure-oriented smoothing (SOS) operator as a model constraint to LSRTM and FWI for improving their robustness. In LSRTM, SOS has been formulated as a shaping operator and the inverse problem is solved in the framework of shaping regularization. In FWI, SOS is implemented by imposing sparsity promotion in the seislet domain, where a velocity model can be represented by basis functions aligned along locally planar structures. Finally, I add a smoothing kernel into the FWI misfit function to improve its robustness especially when the initial model is far from the true model. The smoothing is applied in both time and space directions, to emphasize low frequencies and low wavenumbers, which could mitigate the cycle-skipping problem. Numerous synthetic examples are used to test the practical application and accuracy of the proposed approaches.