Mathematical theory and computational methods for seismic full waveform inversion problems

在过去几年原油价格低迷期间，能源行业受到重创，特别是石油服务公司。与此同时，各级政府出台了更严格的环保政策。因此，具有最小环境影响的具有成本效益的方法对于碳氢化合物勘探是重要的，这需要潜在油田的准确的高分辨率地下图像。地震全波形反演（FWI）是一种基于模型的数据拟合方法，已广泛应用于勘探地球物理学中，以获得高分辨率的地下属性。 FWI能够创建高达传播波长一半的地球高分辨率地下图像。此外，它可以同时估计多个参数，并且需要对记录的地震数据进行最少的预处理。然而，FWI算法存在对初始模型敏感、计算量大、收敛速度慢、易陷入局部极小、易出现跳周期等问题。这些挑战阻碍了该方法在勘探物理领域的进一步广泛应用和接受。 本研究计划旨在发展先进的数学理论和有效的计算方法来解决这些问题的FWI。为此，我们将制定FWI作为一个偏微分方程（PDE）约束的非线性优化问题，其中测量观测数据和合成数据之间的差异的失配函数迭代最小化基于梯度的优化算法。约束偏微分方程是一个地震波方程，可以是亥姆霍兹方程和声波/弹性波方程。我们将专注于几个重要的方面，如建立精确的初始模型FWI，高效和高阶数值算法和预处理器，以减少计算成本的发展和分析。此外，各种正则化策略将被研究和应用，以减轻循环跳跃和局部极小值问题。通过研究计划，我们将为学生和博士后研究员提供高质量的培训，使他们在完成培训后为竞争激烈的就业市场做好准备，并满足能源行业和信息技术对高素质人才日益增长的需求。 研究成果也将为地球科学家和石油工程师提供一种经济、快速、准确地推断地下地质性质的方法。

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.