Abstract: Full-waveform inversion (FWI) in seismic exploration is an effective tool to retrieve high-resolution subsurface properties of the Earth. This process helps geophysicists identify potential locations of hydrocarbon reservoirs. In my dissertation, I present an efficient wavefield inversion (EWI) method to address the nonlinearity in FWI in an efficient matter. EWI introduces a modified source function to absorb parameter perturbations. As a result, the wavefield is an independent variable linear to the new formulation. I use efficient inner iterations between the wavefield reconstruction and a modified source update to include multi-scattering information in the wavefield and medium parameter perturbations are computed by a direct division process. In complex media, more parameters other than the velocity are needed to be considered to describe the seismic wave propagation. In this case, multi-parameter inversion is needed to provide better data fitting. I propose to use EWI to perform a multi-parameter inversion for complex media. Finally, I use a physics-informed neural network (PINN) to generate the wavefields. After this step, I build another independent network to predict the velocity that fits the predicted wavefield to achieve machine-learning (ML) based WRI.