Usually, inverse problems are ill-posed. The solution to the inverse problem is indeterminate, meaning that for given observational data, there may be multiple possible solutions. It is not sufficient to give a definite solution to common seismic inverse problems. In this study, we provide stochastic solutions for seismic inverse problems (denoising and reconstruction). We sample a range of possible and high-quality solutions for a given observation with various degradations from the posterior distribution through Langevin dynamics with conditional score function, all shown to be reasonable results; for example, the stochastic solutions we sampled may contain as many geological structures of interest to the expert as possible. Experimental results on synthetic and field data verify the superiority of posterior sampling. In particular, our method has obvious advantages over other methods, such as traditional and (supervised, self-supervised, and unsupervised) deep learning (DL) methods, especially in denoising under extremely low signal-to-noise ratio (SNR) and reconstruction for data with consecutively missing traces and noise. We also analyze the advantages of our approach and concluded that successful generative modeling of seismic data by the score-based generative models (SGMs) is the key to posterior sampling for the inverse problems, which all benefit from the seismic data prior implicit in the trained score network in the SGM.

中文翻译：

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通过基于分数的生成模型同步地震数据去噪和重建的随机解决方案


通常，逆问题是不适定的。反问题的解是不确定的，这意味着对于给定的观测数据，可能有多种可能的解。对常见的地震反问题给出确定的解是不够的。在这项研究中，我们为地震反演问题（去噪和重建）提供了随机解决方案。我们通过带有条件评分函数的朗之万动力学，对给定的观察结果采样了一系列可能的高质量解决方案，这些解决方案具有后验分布的各种退化，所有这些都被证明是合理的结果；例如，我们采样的随机解可能包含尽可能多的专家感兴趣的地质结构。合成数据和现场数据的实验结果验证了后验采样的优越性。特别是，我们的方法比其他方法，例如传统的和（监督、自监督和无监督）深度学习（DL）方法具有明显的优势，特别是在极低信噪比（SNR）下的去噪和重建方面对于连续丢失痕迹和噪声的数据。我们还分析了我们的方法的优点，并得出结论，通过基于分数的生成模型（SGM）成功地对地震数据进行生成建模是反问题后验采样的关键，这都受益于训练中隐含的地震数据先验。 SGM 中的评分网络。