Adjoint state method and numerical algorithms for full waveform inversion of seismic data

全波形反演（FWI）是一种基于模型的数据拟合方法，广泛应用于勘探地球物理，以获得高分辨率的地球地下图像。一方面，FWI技术是一种很有发展潜力的技术。例如，它能够以高达传播波长一半的高分辨率对地下进行成像。此外，它对记录地震数据的预处理要求最低，并且在反演过程中同时考虑了直射波和反射波。另一方面，FWI方法也存在一定的局限性，如需要精确的初始地球模型；由于FWI是一个欠定逆问题，计算成本高，特别是三维反演和解的非唯一性。本文将通过发展高效、精确的地震正演模拟数值方法、利用伴随状态法进行有效的梯度计算和使用混合时间层析成像方法建立精确的初始模型来解决这些困难。结合基于梯度的优化算法的最新进展，将开发一个高效的计算框架。在这个程序中，我们主要集中在声学和弹性波动方程，但扩展到更复杂的模型是可能的。在数学上，我们将FWI描述为一个偏微分方程（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差异的失拟函数通过基于梯度的优化算法迭代最小化。由于地震数据中没有低频分量，FWI的一个关键限制是，启动模型必须准确，并且包含从0到地震数据可以重建的最小波数的空间波数。可靠的初始模型对于避免虚假的局部最小值和补偿数据中低频成分的缺失至关重要。我们将通过开发混合时间层析成像方法来解决这一困难，以获得FWI的精确初始模型。FWI方法的另一个限制是计算成本高。我们将从两个方面解决这一困难：(1)为正演问题开发有效的数值方法，从而降低单次FWI迭代的计算成本；(2)采用基于梯度的优化算法，减少迭代次数。利用基于梯度的算法需要对失配函数进行精确求导，而伴随状态法可以有效地计算失配函数。我们将利用微扰理论和拉格朗日乘数法推导伴随方程，然后发展有效的数值方法来求解伴随变量的伴随方程。FWI是一个有许多解的严重欠定逆问题。这一问题与模型参数过多和数据中缺少低频成分有关。我们将通过开发规律性策略来解决这一问题，例如将测井数据合并到misfit函数中。综上所述，本课题将在以下几个方面进行研究和贡献：高效、准确的地震方程数值方法；有效边界反射吸收；混合走时层析成像方法；反问题正则化的数学分析同时，提出了高效、准确的伴随状态法。本项目的研究成果将在油藏开发和远景评价中得到应用，并为应用型数学和地球物理的高素质人才提供充分的跨学科培训机会。

Full waveform inversion (FWI) is a model-based data fitting procedure that is widely used in exploration geophysics to obtain high-resolution subsurface images of the Earth. On one hand, the FWI method is a promising technique with great potentials. For example, it is able to image the subsurface with high resolution up to half of the propagated wavelength. Moreover, it requires minimal preprocessing of the recorded seismic data and takes into account both direct and reflected waves in the inversion procedure. On the other hand, the FWI method has some limitations such as the requirement of an accurate initial earth model; the high computational cost, especially for 3D inversion and non-uniqueness of solutions since the FWI is an underdetermined inverse problem. This proposal will address these difficulties through the development of efficient and accurate numerical methods for forward seismic modeling, the utilization of the adjoint state method for efficient gradient calculation and building an accurate initial model using a hybrid time tomography method. Together with the recent developments on gradient-based optimization algorithm, an efficient computational framework will be developed. Within this program we mainly focus on acoustic and elastic wave equations, however extension to more complicated models is possible. Mathematically, we formulate the FWI as a partial differential equation (PDE)-constrained nonlinear optimization problem, where the misfit function measuring the difference between observational and synthetic data is iteratively minimized by a gradient-based optimization algorithm. Due to the absence of low frequency components in the seismic data, a key limitation of FWI is that the starting model must be accurate and contain spatial wavenumbers from 0 up to the minimum wave number that can be reconstructed by the seismic data. A reliable initial model is critical in avoiding spurious local minima and in compensating for the absence of low frequency components in the data. We will address this difficulty through the development of a hybrid time tomography method to obtain an accurate initial model for FWI. Another limitation of the FWI method is its high computational cost. We will address this difficulty from two aspects: (1) develop efficient numerical methods for the forward problem so the computational cost in a single FWI iteration can be reduced; (2) apply gradient-based optimization algorithm to reduce the number of iterations. Utilization of gradient-based algorithm necessitates an accurate derivative of the misfit function, which is efficiently calculated by adjoint state method. We will derive the adjoint equation using both perturbation theory and Lagrange multiplier method, and then develop efficient numerical methods to solve the adjoint equation for the adjoint variable. FWI is a severely underdetermined inverse problem with many solutions. This problem is related to the large number of model parameters and the absence of low frequency components in data. We will address this issue through the development of regularity strategies, such as the incorporation of well-log data in the misfit function. In summary, this research program will study and make contributions in the following areas: efficient and accurate numerical methods for seismic equations; effective boundary reflection absorption; hybrid travel-time tomography methods; mathematical analysis of inverse problem regularization; and, efficient and accurate adjoint state method. The results of this research program will find applications in reservoir exploitation and prospect evaluation, and provide ample interdisciplinary training opportunities in applied mathematics and geophysics for highly qualified personnel.