Seismic migration started out with a humble goal: trying to position lateral subsurface reflectors accurately enough for exploration with a drill bit. Over time, stimulated by powerful computers, it became more ambitious. This is an overview talk to show how mathematics has shaped migration from a simple structural imaging tool to a computationally intensive process which is approaching to its ultimate goal: rendering quantitatively accurate models of the earth. I will introduce the theory of true amplitude reverse time migration, and its extension to image the structures in anisotropic and attenuative earth media. I will review how possible we can extract reflectivity, velocity, and other rock properties from seismic data, and how we build up the connections between a migration and an inversion. Seismic data is five dimensional, embedding gigantic geological information and requiring exhausting computational effort to process. I will explain what geophysicists have done to make accurate and cost-effective imaging software for commercial purpose. Also, I will describe the major technical challenges in this field.
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