3D impedance inversion based on a gradient-enhanced diffusion model

doi:10.3969/j.issn.2096-1693.2026.01.012

Abstract

Abstract:

Traditional model-driven inversion methods rely on prior information and regularization, which often lead to oversimplification of geological features. With the rapid development of deep learning, diffusion models have emerged as a powerful alternative for solving inverse problems, as they can learn the complex data distribution characteristics and provide better prior information. Inspired by this, this paper introduces diffusion models to enhance the reliability and stability of inversion results. The method learns the data distribution through the processes of noise addition and denoising applied to a synthetic impedance model. Subsequently, by utilizing posterior sampling conditioned on seismic data, it incorporates low-frequency model constraints, 3D lateral constraints, and momentum estimation to improve lateral continuity and the stability of gradient updates, thereby achieving a robust mapping between seismic data and the impedance model. Application results on both synthetic and real data demonstrate that the new method can recover impedance models that are both detailed and geologically plausible. Compared to traditional model-driven methods, the proposed method improves the accuracy of single-trace inversion by 5%. The new inversion framework reduces reliance on prior information and significantly enhances generalizability and reliability, while also providing new approaches for solving other complex geophysical inverse problems.

Key words: diffusion model, 3D inversion, acoustic impedance, gradient enhancement, lateral constraint

摘要：

传统的模型驱动反演方法依赖于先验信息和正则化，往往会出现过度简化地质特征的现象。随着深度学习的快速发展，扩散模型由于其能学习数据的复杂分布特征，为求解反问题提供了更好的先验信息。受此启发，本文引入扩散模型提高反演结果的可靠性和稳定性。该方法通过对合成阻抗模型的加噪和去噪过程学习其数据分布，之后利用以地震数据为条件的后验采样，引入低频模型约束、三维横向约束和动量估计，提高横向连续性和梯度更新的稳定性，实现地震数据和阻抗模型的稳健映射。合成数据和实际数据的应用结果表明，新方法能够生成服从观测地震数据且细节丰富的阻抗模型，与传统模型驱动方法相比，本文方法对单井反演准确率提升了5%。新的反演框架减弱了对先验信息的依赖并且大大提高了泛化性和可靠性，同时也为求解其他复杂的地球物理反问题提供了新思路。

关键词: 扩散模型, 3D反演, 声波阻抗, 梯度增强, 横向约束

CLC Number:

LIU Zeyang, LI Jingye, LIU Dawei, ZHANG Wei, LIU Guochang, CHEN Xiaohong. 3D impedance inversion based on a gradient-enhanced diffusion model[J]. Petroleum Science Bulletin, 2026, 11(2): 415-428.

刘泽洋, 李景叶, 刘达伟, 张伟, 刘国昌, 陈小宏. 基于梯度增强扩散模型的三维阻抗反演[J]. 石油科学通报, 2026, 11(2): 415-428.

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URL: https://sykxtb.cup.edu.cn/EN/10.3969/j.issn.2096-1693.2026.01.012

https://sykxtb.cup.edu.cn/EN/Y2026/V11/I2/415

Figures/Tables 11

References 37

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Algorithm 1 DPS	Algorithm 2 GDPS
Require：$T, \boldsymbol{y}, \Sigma_{t}, l_{r}$	Require：$T, \boldsymbol{y}, \Sigma_{t}, l_{r}, \lambda_{1}, \lambda_{2}, \hat{\boldsymbol{x}}_{\text {Low }}$
1: $\boldsymbol{x}_{T} \sim \mathcal{N}(0, \mathbf{I})$	1: $\boldsymbol{x}_{T} \sim \mathcal{N}(0, \mathbf{I}), m_{T+1}=0, v_{T+1}=0$
2: for $t=T$ tol do	2: for $t=T$ tol do
3: $\hat{\boldsymbol{s}} \leftarrow \boldsymbol{s}_{\theta^{*}}\left(\boldsymbol{x}_{t}, t\right)$	3: $\hat{\boldsymbol{s}} \leftarrow \boldsymbol{s}_{\theta^{*}}\left(\boldsymbol{x}_{t}, t\right)$
4:$\hat{\boldsymbol{x}}_{0} \leftarrow\left(\boldsymbol{x}_{t}+\left(1-\bar{\alpha}_{t}\right) \hat{\boldsymbol{s}}\right) / \sqrt{\bar{\alpha}_{t}}$	4:$\hat{\boldsymbol{x}}_{0} \leftarrow\left(\boldsymbol{x}_{t}+\left(1-\bar{\alpha}_{t}\right) \hat{\boldsymbol{s}}\right) / \sqrt{\bar{\alpha}_{t}}$
5:$z \sim \mathcal{N}(0, \mathbf{I})$	5:$\boldsymbol{z} \sim \mathcal{N}(0, \mathbf{I})$
6:$\boldsymbol{x}_{t-1}^{\prime} \leftarrow\left(\boldsymbol{x}_{t}+\left(1-\alpha_{t}\right) \hat{\boldsymbol{s}}\right) / \sqrt{\alpha_{t}}+\Sigma_{t} z$	6:$\boldsymbol{x}_{t-1}^{\prime} \leftarrow\left(\boldsymbol{x}_{t}+\left(1-\alpha_{t}\right) \hat{\boldsymbol{s}}\right) / \sqrt{\alpha_{t}}+\Sigma_{t} z$
7:$\boldsymbol{x}_{t-1} \leftarrow \boldsymbol{x}_{t-1}^{\prime}-l_{r} \nabla_{\boldsymbol{x}_{t}}\left\\|\boldsymbol{y}-\boldsymbol{G}\left(\hat{\boldsymbol{x}}_{0}\right)\right\\|_{2}^{2}$	7:$\mathrm{g}_{t} \leftarrow \nabla_{x_{t}}\left(\boldsymbol{y}-\boldsymbol{G}\left(\hat{\boldsymbol{x}}_{0}\right)_{2}^{2}+\lambda_{1}\left\\|\hat{\boldsymbol{x}}_{0}-\hat{\boldsymbol{x}}_{\text {Low }}\right\\|_{2}^{2}+\lambda_{2}\left\\|\mathrm{D} \hat{\boldsymbol{x}}_{0}\right\\|_{2}^{2}\right)$
8: return $\hat{\boldsymbol{x}}_{0}$	8:$\begin{array}{l}\hat{m}_{t}=\left(\beta_{1} m_{t+1}+\left(1-\beta_{1}\right) \mathrm{g}_{t}\right) /\left(1-\beta_{1}^{T-t+1}\right), \\\hat{v}_{t}=\sqrt{\left(\beta_{2} v_{t+1}+\left(1-\beta_{2}\right) \mathrm{g}_{t}^{2}\right) /\left(1-\beta_{2}^{T-t+1}\right)}\end{array}$
	9:$\boldsymbol{x}_{t-1} \leftarrow \boldsymbol{x}_{t-1}^{\prime}-l_{r} \hat{m}_{t} /\left(\sqrt{\hat{v}_{t}}+\epsilon\right)$
	10:return $\hat{\boldsymbol{x}}_{0}$

Algorithm 1 DPS	Algorithm 2 GDPS
Require：$T, \boldsymbol{y}, \Sigma_{t}, l_{r}$	Require：$T, \boldsymbol{y}, \Sigma_{t}, l_{r}, \lambda_{1}, \lambda_{2}, \hat{\boldsymbol{x}}_{\text {Low }}$
1: $\boldsymbol{x}_{T} \sim \mathcal{N}(0, \mathbf{I})$	1: $\boldsymbol{x}_{T} \sim \mathcal{N}(0, \mathbf{I}), m_{T+1}=0, v_{T+1}=0$
2: for $t=T$ tol do	2: for $t=T$ tol do
3: $\hat{\boldsymbol{s}} \leftarrow \boldsymbol{s}_{\theta^{*}}\left(\boldsymbol{x}_{t}, t\right)$	3: $\hat{\boldsymbol{s}} \leftarrow \boldsymbol{s}_{\theta^{*}}\left(\boldsymbol{x}_{t}, t\right)$
4:$\hat{\boldsymbol{x}}_{0} \leftarrow\left(\boldsymbol{x}_{t}+\left(1-\bar{\alpha}_{t}\right) \hat{\boldsymbol{s}}\right) / \sqrt{\bar{\alpha}_{t}}$	4:$\hat{\boldsymbol{x}}_{0} \leftarrow\left(\boldsymbol{x}_{t}+\left(1-\bar{\alpha}_{t}\right) \hat{\boldsymbol{s}}\right) / \sqrt{\bar{\alpha}_{t}}$
5:$z \sim \mathcal{N}(0, \mathbf{I})$	5:$\boldsymbol{z} \sim \mathcal{N}(0, \mathbf{I})$
6:$\boldsymbol{x}_{t-1}^{\prime} \leftarrow\left(\boldsymbol{x}_{t}+\left(1-\alpha_{t}\right) \hat{\boldsymbol{s}}\right) / \sqrt{\alpha_{t}}+\Sigma_{t} z$	6:$\boldsymbol{x}_{t-1}^{\prime} \leftarrow\left(\boldsymbol{x}_{t}+\left(1-\alpha_{t}\right) \hat{\boldsymbol{s}}\right) / \sqrt{\alpha_{t}}+\Sigma_{t} z$
7:$\boldsymbol{x}_{t-1} \leftarrow \boldsymbol{x}_{t-1}^{\prime}-l_{r} \nabla_{\boldsymbol{x}_{t}}\left\\|\boldsymbol{y}-\boldsymbol{G}\left(\hat{\boldsymbol{x}}_{0}\right)\right\\|_{2}^{2}$	7:$\mathrm{g}_{t} \leftarrow \nabla_{x_{t}}\left(\boldsymbol{y}-\boldsymbol{G}\left(\hat{\boldsymbol{x}}_{0}\right)_{2}^{2}+\lambda_{1}\left\\|\hat{\boldsymbol{x}}_{0}-\hat{\boldsymbol{x}}_{\text {Low }}\right\\|_{2}^{2}+\lambda_{2}\left\\|\mathrm{D} \hat{\boldsymbol{x}}_{0}\right\\|_{2}^{2}\right)$
8: return $\hat{\boldsymbol{x}}_{0}$	8:$\begin{array}{l}\hat{m}_{t}=\left(\beta_{1} m_{t+1}+\left(1-\beta_{1}\right) \mathrm{g}_{t}\right) /\left(1-\beta_{1}^{T-t+1}\right), \\\hat{v}_{t}=\sqrt{\left(\beta_{2} v_{t+1}+\left(1-\beta_{2}\right) \mathrm{g}_{t}^{2}\right) /\left(1-\beta_{2}^{T-t+1}\right)}\end{array}$
	9:$\boldsymbol{x}_{t-1} \leftarrow \boldsymbol{x}_{t-1}^{\prime}-l_{r} \hat{m}_{t} /\left(\sqrt{\hat{v}_{t}}+\epsilon\right)$
	10:return $\hat{\boldsymbol{x}}_{0}$

参数	含义
$T$	迭代次数
$\alpha_{t}, \Sigma_{t}$	次噪声水平和方差
$x_{0}$	原始阻抗
$x_{T}$	纯噪声
$x_{t}$	$x_{0}$添加噪声直至变为$x_{T}$的过程
$\boldsymbol{S}_{\theta^{*}}$	神经网络

参数	含义
$T$	迭代次数
$\alpha_{t}, \Sigma_{t}$	次噪声水平和方差
$x_{0}$	原始阻抗
$x_{T}$	纯噪声
$x_{t}$	$x_{0}$添加噪声直至变为$x_{T}$的过程
$\boldsymbol{S}_{\theta^{*}}$	神经网络

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