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

地震数据全波形反演伴随状态法和数值算法

• 批准号: RGPIN-2014-04913
• 负责人: Liao, Wenyuan
• 金额: $ 0.8万
• 依托单位: University of Calgary
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2015
• 资助国家: 加拿大
• 起止时间: 2015-01-01 至 2016-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=588096
• 关键词: Adjoint; state; method; numerical; algorithms

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.

完整波形反演（FWI）是一种基于模型的数据拟合程序，可在探索地球物理学中广泛使用，以获得地球的高分辨率地下图像。一方面，FWI方法是具有巨大潜力的有前途的技术。例如，它能够以高分辨率的高分辨率成像传播波长的一半。此外，它需要对记录的地震数据的最小预处理，并考虑到反转过程中的直接和反射波。另一方面，FWI方法具有一些局限性，例如准确的初始地球模型的要求；高计算成本，特别是对于3D反转和解决方案的非唯一性，因为FWI是一个不确定的反问题。 该建议将通过开发有效，准确的数值方法来解决这些困难，用于正向地震建模，伴随状态方法的有效梯度计算，并使用混合时间断层扫描方法构建准确的初始模型。加上有关基于梯度的优化算法的最新发展，将开发有效的计算框架。 在此程序中，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。 从数学上讲，我们将FWI作为部分微分方程（PDE）构成的非线性优化问题，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组件，FWI的关键限制是起始模型必须是准确的，并且包含从0到最小波数的空间波数字，该空间模型可以通过旋转数据重建。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没有低频组件至关重要。我们将通过开发混合时间断层扫描方法来解决这一困难，以获得FWI准确的初始模型。 FWI方法的另一个局限性是其高计算成本。我们将从两个方面解决这一困难：（1）为远期问题开发有效的数值方法，因此可以降低单个FWI迭代中的计算成本； （2）应用基于梯度的优化算法来减少迭代次数。基于梯度的算法的利用需要精确的派生函数的精确导数，这是通过伴随状态方法有效计算的。我们将使用扰动理论和Lagrange乘数方法得出伴随方程，然后开发有效的数值方法来求解伴随变量的伴随方程。 FWI是许多解决方案的严重不足的逆问题。此问题与大量模型参数以及数据中没有低频组件有关。我们将通过制定规律性策略来解决这个问题，例如将井log数据纳入失误功能。 总而言之，该研究计划将在以下领域进行研究并做出贡献：地震方程的高效，准确的数值方法；有效的边界反射吸收；混合旅行时间断层扫描方法；逆问题正则化数学分析；并且，有效，准确的伴随状态方法。该研究计划的结果将在水库剥削和潜在客户评估中找到应用，并为高素质人员提供充分的跨学科培训机会，并为应用数学和地球物理学提供大量的跨学科培训机会。