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与数值算法

• 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: 2018
• 资助国家: 加拿大
• 起止时间: 2018-01-01 至 2019-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=657588
• 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.

全波反演法是一种基于模型的数据拟合方法，广泛应用于地球物理勘探中，以获得高分辨率的地球次表层图像。一方面，FWI方法是一种很有潜力的技术。例如，它能够以高达传播波长一半的高分辨率成像亚表面。此外，它只需要对记录的地震数据进行最小程度的预处理，并在反演过程中同时考虑了直达波和反射波。另一方面，FWI方法也存在一些局限性，如对初始地球模型的精度要求较高，计算成本较高，特别是对于三维反演，而且由于FWI是一个欠定的反问题，所以解的不唯一性。**这项建议将通过开发有效和准确的地震正演模拟的数值方法，利用伴随状态方法进行有效的梯度计算，以及使用混合时间层析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，将开发一个高效的计算框架。在本程序中，我们主要关注声波和弹性波方程，但也可以扩展到更复杂的模型。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，其中衡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。**由于地震数据中没有低频分量，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。一个可靠的初始模型对于避免虚假的局部极小值和补偿数据中的低频分量是至关重要的。我们将通过开发一种混合时间层析成像方法来解决这一困难，以获得FWI的准确初始模型。**FWI方法的另一个局限性是其计算成本较高。我们将从两个方面来解决这一困难：(1)开发有效的数值方法来求解正问题，从而减少单次FWI迭代的计算量；(2)采用基于梯度的优化算法来减少迭代次数。利用基于梯度的算法需要失配函数的精确导数，而失配函数的导数由伴随状态法有效地计算出来。我们将利用微扰理论和拉格朗日乘子法来推导伴随方程，然后发展有效的数值方法来求解伴随变量的伴随方程。**FWI是一个具有许多解的严重不足的反问题。这一问题与模型参数较多、数据中不存在低频成分有关。我们将通过发展正则性策略来解决这个问题，例如将测井数据纳入失配函数。**综上所述，本研究计划将在以下方面进行研究和做出贡献：地震方程的高效和准确的数值方法；有效的边界反射吸收；混合走时层析方法；反问题正则化的数学分析；以及高效和准确的伴随状态方法。这项研究成果将在油藏开发和远景评价中得到应用，并为高素质人才提供充足的应用数学和地球物理学科交叉培训机会。