Full waveform inversion (FWI) creates high resolution models of the Earth's subsurface structures from seismic waveform data. Due to the non-linearity and non-uniqueness of FWI problems, finding globally best-fitting model solutions is not necessarily desirable since they fit noise as well as the desired signal in data. Bayesian FWI calculates a so-called posterior probability distribution function, which describes all possible model solutions and their uncertainties. In this paper, we solve Bayesian FWI using variational inference, and propose a new methodology called physically structured variational inference, in which a physics-based structure is imposed on the variational distribution. In a simple example motivated by prior information from imaging inverse problems, we include parameter correlations between pairs of spatial locations within a dominant wavelength of each other, and set other correlations to zero. This makes the method far more efficient compared to other variational methods in terms of both memory requirements and computation, at the cost of some loss of generality in the solution found. We demonstrate the proposed method with a 2D acoustic FWI scenario, and compare the results with those obtained using other methods. This verifies that the method can produce accurate statistical information about the posterior distribution with hugely improved efficiency (in our FWI example, 1 order of magnitude reduction in computation). We further demonstrate that despite the possible reduction in generality of the solution, the posterior uncertainties can be used to solve post-inversion interrogation problems connected to estimating volumes of subsurface reservoirs and of stored ${\text{CO}}_2$, with minimal bias, creating a highly efficient FWI-based decision-making workflow.

中文翻译：

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用于贝叶斯全波形反演的物理结构变分推理


全波形反演 （FWI） 根据地震波形数据创建地球地下结构的高分辨率模型。由于 FWI 问题的非线性和非唯一性，找到全局最佳拟合模型解不一定是可取的，因为它们拟合了噪声以及数据中所需的信号。贝叶斯 FWI 计算所谓的后验概率分布函数，该函数描述了所有可能的模型解及其不确定性。在本文中，我们使用变分推理解决了贝叶斯 FWI，并提出了一种称为物理结构变分推理的新方法，其中基于物理的结构被施加在变分分布上。在一个由成像逆问题中的先验信息驱动的简单示例中，我们包括了彼此主波长内的空间位置对之间的参数相关性，并将其他相关性设置为零。与其他变分方法相比，这使得该方法在内存需求和计算方面都更加高效，但代价是所找到的解决方案的通用性有所损失。我们用 2D 声学 FWI 场景演示了所提出的方法，并将结果与使用其他方法获得的结果进行比较。这验证了该方法可以生成有关后验分布的准确统计信息，并且效率大大提高（在我们的 FWI 示例中，计算量减少了 1 个数量级）。 我们进一步证明，尽管解决方案的通用性可能会降低，但后验不确定性可用于解决与估计地下储层和储存的 ${\text{CO}}_2$ 体积相关的后询问问题，偏差最小，创建了一个高效的基于 FWI 的决策工作流程。