This thesis develops novel wave-equation based seismic imaging and inversion methods that invert for the high- and low-wavenumber components of P- and Svelocity models. To invert for the P- and S-wave velocity perturbations (highwavenumber component), I first propose a linearized elastic waveform inversion method denoted as elastic least-squares reverse time migration (LSRTM). Elastic LSRTM solves the linearized elastic-wave equation for forward modeling and the adjoint equations for backpropagating the residual wavefield. Both synthetic- and field-data results prove that this method can accurately reconstruct the P- and S-wave velocity perturbations. Compared with the elastic reverse time migration (RTM) method, the elastic LSRTM images have fewer artifacts, higher resolution and better amplitude balancing. In addition, elastic LSRTM mitigates the coupling effect between elastic parameters, and so gives accurate relative information about the P- and S-wave velocity distributions. Elastic LSRTM method suffers from a slow convergence rate because of blurring effects and crosstalk artifacts. To mitigate these problems, I propose a multiparameter deblurring filter that approximates the multiparameter inverse Hessian. This method significantly improves the quality for multiparameter migration images. Numerical tests show that the multiparameter deblurring filter can compute elastic migration images similar in quality to the ones inverted by elastic LSRTM at a much lower cost. It can also be used as a preconditioner to accelerate the convergence rate in multiparameter inversion. In general, the proposed method can also be applied to elastic full waveform inversion (FWI) or any multiparameter migration/inversion operator. One of most crucial problems for elastic inversion is the accurate estimation of the background P- and S-wave velocity models (low-wavenumber component). To accurately estimate the velocity models, I propose a joint PP and PS wave-equation migration velocity analysis method using plane-wave common image gathers (CIGs) with depth consistency. Both the moveout residuals of CIGs and relative depth shifts between PP and PS images are transformed into weighted image perturbations for updating the velocity models. Numerical tests with synthetic and multicomponent field data demonstrate that this method can accurately invert for P- and S-wave velocity models.
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