In this presentation, we will explore two advanced seismic inversion techniques for estimating subsurface elastic properties: wave-equation migration analysis and full waveform inversion. The talk will begin with a review of the tomography-based inversion scheme for estimating seismic velocity and Q models, followed by their applications in deep water oil fields. Next, we will introduce full waveform inversion, including its principles, advantages, and limitations, as well as numerical examples on synthetic models. Finally, we will discuss its applications in medical brain imaging.