Seismic exploration heavily relies on the accurate processing of seismic data, as high-quality reconstructed data is essential for reliable imaging and interpretation. In recent years, data-driven approaches have shown great promise in seismic data processing. However, supervised learning methods require large amounts of labeled data, while generative models, such as GANs, often encounter issues like mode collapse and instability. On the other hand, generative diffusion models, leveraging principles from nonequilibrium thermodynamics and Markov processes, have emerged as powerful tools for capturing complex data distributions.
Despite these advantages, Denoising Diffusion Probabilistic Models (DDPM) purely generate data distributions from the latent space with the reliance on random noise, making it inadequate for seismic data reconstruction where the goal is to accurately recover missing traces. Thus, DDPM is lacks interpretability in seismic data restoration, and may disrupt the structured patterns crucial for interpolating seismic signals. Furthermore, we view the reverse process that starts from noise as unnecessary and inefficient for reconstruction task.
To address these challenges, we propose a novel Conditional Residual Diffusion Model (CRDM) that enhances both certainty and interpretability by incorporating residual diffusion and conditional constraints derived from observed seismic data (Fig.1). This approach better aligns with the inherent structure of seismic signals, enabling more accurate and interpretable reconstruction. The model is grounded in DDPM, with mathematical derivations for loss functions, conditional probability distributions, and reverse inference steps, ensuring both theoretical rigor and practical applicability.
Additionally, Our CRDM utilizes a shallow U-Net architecture featuring one down-sampling and one up-sampling layer integrated with Multi-Head Self-Attention (MHSA), which significantly enhances the model's efficiency and effectiveness. Experimental results (Fig.2) demonstrate that CRDM outperforms DDPM, denoising convolutional neural network (DnCNN), and fast projection onto convex sets (FPOCS), achieving a 15.1% improvement in reconstruction SNR and reducing computation time by 139 times compared to DDPM. Notably, CRDM achieves optimal results in a few diffusion steps, whereas DDPM typically requires thousands of steps.
The innovative approach generates data through residuals for determinism, while guiding the processing with noise for diversity. This not only enhances the interpretability and efficiency of seismic data reconstruction, but also positions the model as a promising tool for advancing data-driven seismic processing through flexible coefficient adjustment. Therefore, we believe this model has great potential for broader applications in geophysical data analysis, offering significant value in accurately depicting complex geological structures and providing more effective guidance for petroleum exploration.

Fig.1 The framework of CRDM. The model consists of two stage: (a) the training stage with forward diffusion process; (b) the sampling stage for seismic data reconstruction.

Fig.2 Reconstruction results and residuals of the 1994 BP seismic data with 50% irregular missing traces. (a) Complete data, reconstruction using (b) FPOCS (SNR=10.5dB), (c) DnCNN (SNR=15.6dB), (d) DDPM(SNR=18.4dB), (e) CRDM(SNR=21.2dB), and (f) observed seismic data with 50% missing traces. (g-j) display the residuals corresponding to reconstructions (b-e), respectively. The red box highlights a zoomed-in region, which is shown in detail in (k-t).