Mathematical theory and computational methods for seismic full waveform inversion problems

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During the crude price downturn over the last few years, the energy industry has been hit hard, particularly the oil service companies. Meanwhile, governments at all levels have introduced stricter environmental protection policies. Therefore, cost-effective approaches with minimal environmental impact are important for hydrocarbon exploration, which requires an accurate high-resolution subsurface image of the potential oil field. Seismic full waveform inversion (FWI) is a powerful model-based data fitting procedure that has been widely used in exploration geophysics to obtain high-resolution subsurface properties of the earth.  FWI is able to create high-resolution subsurface images of the earth up to half of the propagated wavelength. Moreover, it can estimate multiple parameters simultaneously and requires minimal preprocessing of the recorded seismic data. However, FWI suffers from a series of challenges, such as sensitivity on initial model, high computational cost, slow convergence, local minima, cycle-skipping, to name a few. These challenges hinder the further widespread application and acceptance of this method in the field of exploration physics. This research program aims to develop advanced mathematical theory and efficient computational methods to resolve these issues of FWI. To this end, we will formulate FWI as a partial differential equation (PDE)-constrained nonlinear optimization problem, where the misfit function measuring the difference between observational data and synthetic data is iteratively minimized by gradient-based optimization algorithms. The constraint PDE is a seismic wave equation which can be Helmholtz equation and acoustic/elastic wave equation. We will focus on several important aspects, such as building accurate initial models for FWI, development and analysis of efficient and higher-order numerical algorithms and preconditioners to reduce computational cost. Moreover, various regularization strategies will be studied and applied to mitigate the cycle-skipping and local minima issues. Through the research program, we will provide students and postdoctoral fellows with high-quality training to prepare them for the competitive job market upon the completion of training, and to meet the increasing demand for highly qualified personnel from the energy industry and information technology.  Our research result will also provide geoscientists and petroleum engineers with an economical and fast option to infer subsurface geological properties accurately.