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程序，广泛应用于勘探地球物理学中，以获得地球的高分辨率地下图像。一方面，FWI 方法是一种具有巨大潜力的有前途的技术。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，它需要对记录的地震数据进行最少的预处理，并在反演过程中考虑直达波和反射波。另一方面，FWI方法也有一些局限性，例如需要准确的初始地球模型；计算成本高，特别是对于 3D 反演和解的非唯一性，因为 FWI 是一个欠定反问题。该提案将通过开发高效且准确的正演地震建模数值方法、利用伴随状态方法进行高效梯度计算以及使用混合时间层析成像方法建立准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发展，将开发出高效的计算框架。在这个程序中，我们主要关注声学和弹性波方程，但是可以扩展到更复杂的模型。在数学上，我们将 FWI 表述为偏微分方程 (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配函数通过基于梯度的优化算法迭代最小化。由于地震数据中缺乏低频分量，FWI 的一个关键限制是起始模型必须准确，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局部最小值和补偿数据中低频分量的缺失至关重要。我们将通过开发混合时间断层扫描方法来解决这一难题，以获得准确的 FWI 初始模型。 FWI 方法的另一个限制是其高计算成本。我们将从两个方面解决这个难题：（1）为前向问题开发有效的数值方法，从而减少单次 FWI 迭代的计算成本； (2)应用基于梯度的优化算法来减少迭代次数。基于梯度的算法的使用需要失配函数的精确导数，这可以通过伴随状态方法有效地计算。我们将使用微扰理论和拉格朗日乘子法推导伴随方程，然后开发有效的数值方法来求解伴随变量的伴随方程。 FWI 是一个严重欠定的反问题，有很多解。这个问题与模型参数较多、数据中缺乏低频成分有关。我们将通过制定规律性策略来解决这个问题，例如将测井数据纳入失配函数中。综上所述，本研究计划将在以下领域进行研究并做出贡献：高效、准确的地震方程数值方法；有效的边界反射吸收；混合走时断层扫描方法；逆问题正则化的数学分析；并且，高效且准确的伴随状态方法。该研究项目的成果将在油藏开发和前景评价中得到应用，并为高素质人才提供充足的应用数学和地球物理学跨学科培训机会。

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.