标题： 带有测量引导扩散模型的地球物理反问题 显示英文标题

标题： Geophysical inverse problems with measurement-guided diffusion models

摘要： 通过扩散模型的反向过程解决逆问题是一种有吸引力的方法，可以从不完整且可能带有噪声的测量中生成高度真实且多样化的解决方案，最终实现大规模的不确定性量化。 然而，由于似然项的得分函数（即$\nabla_{\mathbf{x}_t} p(\mathbf{y} | \mathbf{x}_t)$）难以处理，文献中提出了各种采样器，这些采样器使用不同的（更准确或不太准确）的梯度近似来引导扩散过程，使其朝着与观测结果匹配的解决方案发展。 在本工作中，我考虑了两种最近以 Diffusion Posterior Sampling (DPS) 和 Pseudo-inverse Guided Diffusion Model (PGDM) 命名的采样算法。 在 DPS 中，反向扩散过程中每一步使用的引导项是通过对建模算子的伴随算子应用于从解的一个去噪估计中获得的残差得到的。 另一方面，PGDM 使用了一个伪逆算子，这是由于一个去噪后的解不被假设为确定性的，而是被建模为高斯分布。 通过在两个地球物理逆问题（即地震插值和地震反演）上的大量数值例子，我展示了任何测量引导的扩散过程成功的关键方面是：i) 我们能够重新参数化逆问题，使得所寻求的模型介于 -1 和 1 之间（这是任何扩散模型的前提条件）；ii) 用于学习引导反向扩散过程的隐式先验的训练数据集的选择。 在合成和现场数据集上的数值例子表明，PGDM 在两种情况下都优于 DPS，且附加成本有限。 显示英文摘要

摘要： Solving inverse problems with the reverse process of a diffusion model represents an appealing avenue to produce highly realistic, yet diverse solutions from incomplete and possibly noisy measurements, ultimately enabling uncertainty quantification at scale. However, because of the intractable nature of the score function of the likelihood term (i.e., $\nabla_{\mathbf{x}_t} p(\mathbf{y} | \mathbf{x}_t)$), various samplers have been proposed in the literature that use different (more or less accurate) approximations of such a gradient to guide the diffusion process towards solutions that match the observations. In this work, I consider two sampling algorithms recently proposed under the name of Diffusion Posterior Sampling (DPS) and Pseudo-inverse Guided Diffusion Model (PGDM), respectively. In DSP, the guidance term used at each step of the reverse diffusion process is obtained by applying the adjoint of the modeling operator to the residual obtained from a one-step denoising estimate of the solution. On the other hand, PGDM utilizes a pseudo-inverse operator that originates from the fact that the one-step denoised solution is not assumed to be deterministic, rather modeled as a Gaussian distribution. Through an extensive set of numerical examples on two geophysical inverse problems (namely, seismic interpolation and seismic inversion), I show that two key aspects for the success of any measurement-guided diffusion process are: i) our ability to re-parametrize the inverse problem such that the sought after model is bounded between -1 and 1 (a pre-requisite for any diffusion model); ii) the choice of the training dataset used to learn the implicit prior that guides the reverse diffusion process. Numerical examples on synthetic and field datasets reveal that PGDM outperforms DPS in both scenarios at limited additional cost.