WaveDiffusion: Exploring Full Waveform Inversion via Joint Diffusion in the Latent Space

Weakness: Unfortunately, the idea of using an AE with common feature spaces for data and model is not new. See https://paperswithcode.com/paper/paired-autoencoders-for-inverse-problems This is the main problem that the paper have. I understand that in this fast moving field some papers are missed but in this case, the work that was already done makes much of the paper not relevant. I would recommend the authors to withdraw the paper, concentrate of the diffusion aspect of the paper and resubmit to a different venue.

Thank you for bringing this reference to our attention. We will ensure it is cited appropriately in our revised manuscript. However, we respectfully disagree with the recommendation to withdraw the paper, as the primary contribution of our work is not the dual autoencoder architecture but rather the joint diffusion process in a shared latent space. The dual autoencoder serves only as a foundational step, enabling the creation of a shared latent representation for the two modalities, seismic data and velocity maps. The key novelty of our work lies in the use of this shared latent space for a joint diffusion process that refines the generated data pairs while maintaining physical consistency.

To further illustrate this, we conducted experiments with a modified autoencoder architecture that uses a single encoder and two decoders (one-in-two-out configuration) in the first stage. This setup demonstrates the flexibility of our approach, where the joint diffusion process continues to perform effectively, preserving the discoveries outlined in the paper. The generation results for this modified architecture are available at https://bit.ly/4hVLg1q.

Additionally, the one-in-two-out autoencoder configuration allows for direct application to standard FWI tasks, as shown in our response to Reviewer 1caw. In this setup, seismic data alone is used as input to the model, and the joint diffusion process generates both seismic and velocity outputs, making the approach suitable for practical applications. Example results for this experiment can be found at https://bit.ly/4eBIJX2.

In summary, while we acknowledge the existence of prior work on paired autoencoders, our contribution is orthogonal to this line of research. We focus on the integration of joint diffusion into the latent space, which we believe is a novel and significant advancement in the field. We hope this clarification and the new experimental results address your concerns regarding the relevance and novelty of our work.

Question 1: How do you actually do coarse to fine?

Thank you for your question. We achieve coarse-to-fine refinement through the latent space joint diffusion process. Starting with approximate representations, the process iteratively refines these by scoring the deviation from the governing PDE in the latent space. The forward diffusion process perturbs the latent vector progressively, while the backward diffusion process denoises and aligns it to physically valid solutions. This approach ensures a gradual improvement, moving from coarse approximations to accurate representations governed by the PDE, adhering to the physical constraints of the problem.

Question 2: Given some data d how to you use diffusion to find an appropriate model?

As noted in our response to Weakness 1 to Reviewer 1caw and to your raised Weakness above, we have conducted a new experiment to address this question. The experiment demonstrates how our framework finds an appropriate velocity model given specific seismic data d.

To address this task, we utilize a "one-in-two-out" configuration within the autoencoder part of our framework. In this setup: only seismic data (d) is input to the encoder, generating a latent representation solely contributed by seismic input. This latent vector is then refined through the joint diffusion process, iteratively denoising and aligning it with the physical constraints. Finally, the refined latent vector is passed through individual VQ layers and decoders, generating both seismic and velocity outputs. This setup enables the inversion of seismic data to produce velocity maps, making our model directly applicable to standard FWI tasks. Example results from this experiment are available at http://bit.ly/4eBIJX2. We will incorporate these results into the revised manuscript to demonstrate this capability in greater detail.

Dear Reviewer,

Thank you for your detailed and thoughtful feedback on our manuscript. We sincerely value this opportunity to contribute to the conference and are committed to improving our work to meet the high standards of innovation and clarity expected. Thus, we have uploaded a revised version of our manuscript. Your critique has been invaluable in guiding our revisions, and we are confident that the updates made reflect our commitment to addressing your concerns.

Below are the key updates specific to your suggestions:

Section 4.6 on page 9 (WaveDiffusion for conventional FWI):

We have added a new experiment demonstrating how our WaveDiffusion model can perform conventional FWI when only seismic data is provided. This addresses the concern about the inversion process (i.e., obtaining a velocity model given seismic data). The experiment uses a “one-in-two-out” configuration of the autoencoder, where only seismic data is input, and the diffusion process generates both seismic and velocity outputs. Results show that our joint diffusion approach effectively reconstructs velocity maps from seismic inputs, providing a clear demonstration of our inversion strategy.

Section 3.2 on page 4 (Clarification of the Role of Dual Autoencoder):

We have explicitly cited the suggested references and clarified that the dual autoencoder is not the primary contribution of our paper. Instead, it is a preliminary step to establish the shared latent space required for the diffusion process. The true novelty lies in the joint diffusion process, which refines latent representations to adhere to the governing PDE. This distinction is clearly stated in the revised manuscript, with Section 3.3 and the subsequent experiments highlighting the core contributions.

Highlighting the Novelty of Diffusion and Inversion:

Throughout the paper, we have emphasized that our main contribution lies in the diffusion process and its application to solving FWI problems in a novel generative framework. By restructuring and refocusing key sections, we have aligned the narrative to better reflect the innovative aspects of the joint diffusion process and its inversion capabilities.

We genuinely believe that these changes align our work more closely with your suggestions and enhance its contribution to the field. We are excited about the opportunity to present our research at this conference, as we feel it introduces a unique and valuable perspective on FWI.

If there are still areas that you find unsatisfactory or needing improvement, we would deeply appreciate further guidance. Your feedback is critical to helping us refine our work and make it as impactful as possible.

Thank you again for your time and effort in reviewing our submission. We look forward to your insights.

Best regards,

The Authors

While the authors have attempted to answer my concerns, the paper needs many changes to reflect

1. The novelty in the diffusion process (and not in the auto-encoder)
2. The inversion process (getting a model given d)

My rating stands, the authors need to rewrite the paper focusing on the diffusion process and the inversion strategy. The paper as is, lacks the needed novelty. I am sure that they can make it better for future conference

Reviewer STHJ

Weakness 1: Lack of comparison to conditional generative methods.

Thank you for your suggestion. Our method is a joint diffusion generation framework without conditioning, distinct from conditional generative approaches like [1], which integrate finite difference modeling. Our focus is to find solutions in a shared latent space by scoring deviations from the wave equation, avoiding iterative forward modeling steps. Thus, a direct comparison to [1] is not feasible as the objectives and methodologies differ fundamentally. Our work aims to provide a novel perspective on FWI, emphasizing joint diffusion for physical consistency, rather than competing as a conditional generative model. We will clarify these distinctions in the revised manuscript.

Weakness 2: Need for additional reconstruction methods.

Thank you for this valuable suggestion. We agree that evaluating with alternative reconstruction methods strengthens our analysis. We expanded the experiments in Section 4.2.3 to include UPFWI and VelocityGAN alongside InversionNet. The results, detailed at https://bit.ly/3YYfTua, show slight performance improvements (SSIM +~2%) with these solvers, confirming our primary conclusion that generated samples enhance end-to-end mapping networks. We will integrate these results into the revised manuscript.

Weakness 3: Realism of experiments.

Thank you for your observation. InversionNet is used to evaluate our generated data's quality rather than outperform existing FWI baselines. For practical applications, we propose denoising as a key use case for our approach, particularly when dealing with noisy or incomplete data. By leveraging the joint diffusion process, our model can refine and reconstruct physically consistent seismic-velocity pairs, even under challenging data conditions. This will be emphasized in the revised manuscript.

Question 1: Comparison with conditional generative models.

Thank you for this insightful question. As noted in Weakness 1, our joint diffusion framework differs fundamentally from conditional generative models like [1], focusing on discovering solution spaces in a shared latent space. Conditional approaches rely on specific conditions and finite difference modeling. Given these fundamental differences, a direct comparison would not be equitable. Moreover, our primary goal is to introduce a novel perspective for FWI, focusing on joint diffusion's potential for refining data pairs in a physically consistent manner, rather than proposing a SOTA FWI solution. We will clarify these distinctions further in the revised manuscript to address this concern comprehensively.

Question 2: Alternative reconstruction methods.

As addressed in Weakness 2, we evaluated additional algorithms (UPFWI and VelocityGAN), confirming the robustness of our generated samples. The results are detailed at https://bit.ly/3YYfTua and will be added to the manuscript.

Question 3: Dataset overlap in training and evaluation.

Thank you for this question. We added an experiment demonstrating the denoising capabilities of our joint diffusion model. By adding Gaussian noise to both seismic data and velocity maps, we observed that direct autoencoder reconstruction from noisy latent vectors produced suboptimal outputs, with visible noise and distortions. However, applying the joint diffusion process refined the latent vectors, yielding clean and consistent reconstructions, as shown at https://bit.ly/4eDJAH0. This highlights our model's practical application for noisy or incomplete data, addressing real-world challenges. As mentioned in Weakness 3, InversionNet primarily serves as a quality evaluation tool, not a direct benchmark, in our study.

We will clarify these points and include the denoising experiment in the revised manuscript.

Question 4: Realism of the Gen+1% setup.

Thank you for your insightful question. The Gen+1% setup evaluates the model’s ability to enhance a small dataset using supplementary in-distribution data, providing a controlled scenario for assessing performance. We agree that training on two subsets and fine-tuning with 1% of a third introduces a distribution shift, presenting an OOD challenge.

While our study primarily focuses on in-distribution scenarios, reflecting many real-world applications, addressing OOD challenges is an exciting direction for future work. Techniques such as fine-tuning, domain adaptation, or transfer learning could be explored to extend our framework for handling unseen data distributions. Our joint diffusion model inherently refines latent representations toward physically valid solutions, providing a strong foundation for tackling OOD scenarios. These limitations and future directions will be highlighted in the revised manuscript.

Weakness: My major problem with this manuscript is that the main approach to generating two joint autoencoders is not novel. A similar approach including similar experiments has been proposed and published the approach on dual autoencoder before this submission (https://arxiv.org/pdf/2305.13314) and another publication on dual autoencoder can be found at (https://arxiv.org/pdf/2405.13220). These contributions are neither acknowledged nor cited. The remaining novelty is the diffusion process within the latent spaces which is by itself an interesting idea and should have been stated as the contribution of this manuscript.

Thank you for highlighting this point and bringing these references to our attention. We will ensure both references are cited appropriately in our revised manuscript. However, we would like to clarify that our primary contribution lies in the application of a joint diffusion process within a shared latent space to refine data pairs in a physically consistent manner, rather than in the use of dual autoencoders.

The dual autoencoder architecture only serves as a preliminary step in constructing the shared latent space, enabling the subsequent diffusion process. To demonstrate the flexibility of our approach, we also tested a modified one-in-two-out autoencoder architecture, where a single encoder and two decoders were employed. This setup maintained the shared latent space, and the joint diffusion process continued to perform effectively, following the discoveries presented in our paper.

Furthermore, the one-in-two-out autoencoder architecture allows for direct application to standard FWI tasks, as demonstrated in our new experiment results (https://bit.ly/4eBIJX2). This experiment highlights the capability of the joint diffusion model to generate accurate velocity maps solely from seismic data input, addressing both novelty and practical application concerns.

To further support our claims, we have provided an example of the generation results using the modified one-in-two-out network architecture at https://bit.ly/4hVLg1q. We hope these additional experiments and clarifications address your concerns regarding the novelty and contributions of our work.

Reviewer 1caw

Weakness 1: What to do when we are given with some specific seismic data?

Thank you for highlighting this practical consideration. To address inversion with specific seismic data, we have implemented an adaptable "one-in-two-out" configuration within the autoencoder component of our framework. In this setup, only seismic data is input into the encoder, generating a latent representation solely derived from the seismic input. This latent vector is then refined through the joint diffusion process and passed through individual VQ layers and decoders to produce both seismic and velocity outputs.

This approach demonstrates the model's applicability to typical FWI tasks, effectively generating velocity maps from seismic data. Example results and statistical evaluations can be found at http://bit.ly/4eBIJX2 and https://bit.ly/4fyN6DB, respectively. Our joint diffusion model achieves an SSIM of 0.6290, comparable to benchmarks such as InversionNet (SSIM 0.6727) and UPFWI (SSIM 0.6614). While VelocityGAN achieves slightly lower RMSE and MAE, our model is still of the same level in structural accuracy, which is critical for seismic applications.

Weakness 2: Section 4.2.3 comparing InversionNet lacks detail and context.

Thank you for your feedback. Section 4.2.3 has been expanded to include detailed context and quantitative comparisons. Results show that WaveDiffusion achieves reasonably same level structural accuracy (SSIM 0.6290) compared to InversionNet and other benchmarks.

Weakness 3: Geophysical terminology is unclear, and "acoustic" and "seismic" are incorrectly used interchangeably.

We agree that the terms "acoustic" and "seismic" need clarification. In Section 2, we will explicitly note that "seismic data" refers to "acoustic seismic data" in this study, simplifying wave phenomena for modeling purposes.

Question 1: How were the seismic data and velocity models preprocessed before training?

Seismic data and velocity models were resized from [5,70,1000]/[1,70,70] to [3,64,1024]/[1,64,64] (channel, height, depth) for consistency with our architecture. Both were normalized to [-1,1] to ensure compatibility and stability. This will be clarified in the revised manuscript.

Question 2: How much training time was required in terms of CPU/GPU hours?

Thank you for your question. Training required approximately 1000 GPU hours for the autoencoder (per dataset) and 2000 GPU hours for the joint diffusion model. This information will be added to the manuscript.

Question 3: Can the algorithm be extended to 3D data, and what are the computational implications?

Thank you for this insightful question. Adapting our approach to 3D data is indeed feasible, with some modifications. For instance, the autoencoder module would require a more sophisticated 3D architecture to handle volumetric data, while the latent diffusion process should remain fundamentally similar. A simpler alternative would involve treating the 3D velocity cube as a collection of 2D slices along one horizontal dimension, enabling the use of our current network architecture, like the Appendix M experiment in the OpenFWI paper, though this may reduce volumetric coherence. Computational complexity would increase significantly due to the higher dimensionality. We will add this discussion to the manuscript and explore it as future work.

Question 4: Does the model preserve symmetry in velocity maps when the input seismic data is symmetric?

Thank you for this insightful question. To address this, we conducted experiments using the FlatVel_B (FVB) subset, which features symmetric data configurations with pure flat velocity layers. The results, provided at https://bit.ly/3YTjslu, confirm that our WaveDiffusion model successfully preserves symmetry in the generated velocity maps when provided with symmetric seismic data. This demonstrates that the model respects the geometric properties of the input data and maintains consistency in symmetric inversion scenarios. We will include these results and the corresponding discussion in the revised manuscript.

Question 5: How does the algorithm compare to baselines? Which is better?

Thank you for your comment. Our primary contribution is not to improve baseline FWI solutions but to introduce a novel generative framework for FWI through the use of joint diffusion models. To the best of our knowledge, there is no prior work that adopts such a generative perspective for FWI, making direct comparisons to baseline methods with similar generative models challenging. Our goal is to provide a new perspective on FWI by exploring its potential as a generative process. However, results from the one-in-two-out experiment (http://bit.ly/4eBIJX2) show that WaveDiffusion achieves an SSIM of 0.6290, reasonably comparable to InversionNet (0.6727) and others in structural accuracy. These findings will be emphasized in the revised manuscript.

Dear Reviewer rGTJ,

Thank you for your latest response. We appreciate your continued engagement and the time you’ve dedicated to reviewing our manuscript. We take your concerns about the framing and emphasis of our contributions seriously and have worked to address them thoroughly in the revised version.

Regarding your comment on the joint diffusion idea being under-discussed in the original manuscript, we would like to note that Section 3.3 is only the steps to perform a joint diffusion process, not the technical analysis part. On the other hand, Section 3.4 provides detailed analysis and technical insights into the joint diffusion process. Additionally, we’ve added extensive experimental results (Sections 4.3–4.6 and Appendix A.2–A.8) to highlight the joint diffusion process as the central contribution of our work.

To ensure that this key contribution is clearly conveyed, we would greatly value your suggestions:

Experiments: Is the current experiments sufficient enough to support our joint diffusion part contribution? Are there additional experiments or metrics you would recommend to better validate or expand upon the novelty of the joint diffusion approach?

Section 3.4 Analysis: Does the analysis provide sufficient depth to demonstrate the role and impact of joint diffusion? If not, we would be grateful if you could point out areas that need further elaboration or clarification.

We are committed to improving our manuscript and incorporating constructive feedback to ensure it meets the highest standards. Your insights as an expert are invaluable, and we would greatly appreciate your guidance on how to refine the discussion and experiments to fully address your concerns.

Thank you again for your thoughtful comments and for helping us make this a stronger contribution to the field.

Best regards,

The Authors

I here agree with reviewer rGTJ that while joint diffusion may offer novel applications, it seems this wasn't the primary focus of the author's original intent. I will stay with my score too.

Dear Reviewer rGTJ,

Thank you for your continued engagement with our manuscript. We appreciate your comments and have worked diligently to address your concerns regarding the emphasis and presentation of the joint diffusion model, which is indeed the central contribution of our work. Below, we provide a detailed table of contents comparing the joint diffusion content in both the original and revised versions of our manuscript. These demonstrate that the joint diffusion process forms a significant part of the paper.

We kindly request your reconsideration of the manuscript in light of the revisions, expansions, and additional experiments explicitly focusing on the joint diffusion model. If any concerns remain, we would greatly appreciate your specific feedback to further strengthen our work.

Table of Contents Comparison: Joint Diffusion-Related Content

Original Manuscript

1. Section 1 (Page 2, Lines 71-96):
   General introduction to the joint diffusion model and its significance.

2. Section 3.1 (Page 3, Lines 145-157):
   Motivation for joint diffusion as a novel approach to refine the shared latent space.

3. Section 3.3 (Page 4, Lines 191-210):
   Detailed explanation of how the joint diffusion process is implemented on the shared latent space.

4. Section 3.4 (Page 4-5, Lines 212-265):
   Discoveries and analyses of the joint diffusion model, including:

   • Refinement of the solution space.
   • Deviation from the PDE evaluation during forward and backward diffusion processes.
   • Transformation of the governing PDE process to an SDE.

5. Section 4.2.2 (Page 6-8, Lines 312-380):
   Joint diffusion results and analysis, including FID scores and noise level relationships.

6. Section 4.2.3 (Page 8, Lines 381-431):
   Using joint diffusion-generated samples to train InversionNet.

7. Section 4.2.4 (Page 9, Lines 432-469):
   Comparison between joint diffusion and separate diffusion for seismic and velocity data.

8. Appendix A.1.2 (Page 13, Lines 665-677):
   Hyperparameter details for the joint diffusion model.

9. Appendix A.2 (Page 13, Lines 686-697):
   Joint diffusion results on FVB and FFB datasets.

10. Appendix A.3 (Page 13, Lines 686-697):
    Joint diffusion results on combined datasets.

Figures and Tables in Original Manuscript:

• Figure 1: Overview of WaveDiffusion and the joint diffusion process.
• Figure 4: Joint diffusion architecture.
• Figures 5–7: Analysis of joint diffusion results and deviation from the PDE.
• Table 1–4: FID scores, comparisons with baselines, and separate vs. joint diffusion results.

Revised Manuscript

To address your concerns and further emphasize the joint diffusion process, we added the following new content:

1. Section 4.6 (Page 9, Lines 436-457):
   Application of the joint diffusion model to conventional FWI problems with seismic data only.

2. Appendix A.4 (Page 14-15, Lines 743-789):
   Joint diffusion-generated samples visualization and data symmetry checks for flat velocity layers.

3. Appendix A.5 (Page 15-16, Lines 791-844):
   Joint diffusion handling noisy seismic and velocity data, demonstrating robust reconstruction.

4. Appendix A.6 (Page 16-18, Lines 846-960):
   FWI performance comparisons between multiple baseline algorithms and joint diffusion.

5. Appendix A.8 (Page 19, Lines 991-1023):
   Joint diffusion model performance on noisy seismic data input, highlighting its robustness and applicability.

New Figures and Tables in Revised Manuscript:

• Figure 10: Visualization of FWI results with joint diffusion.
• Figure 14: Joint diffusion-generated examples across multiple subsets.
• Figure 16: Reconstructions of noisy seismic and velocity data using joint diffusion.
• Figure 18: FWI results visualization with noisy input seismic data.
• Tables 5, 6, and 8: Comparative metrics for joint diffusion vs. baselines.

Key Revisions and Justifications

1. Expanded Technical Description (Sections 3.3, 3.4):
   Detailed implementation of joint diffusion, including:

   • Refinement of latent space through joint diffusion.
   • Evaluation of deviations from the governing PDE during diffusion processes.
   • Transforming PDE processes into SDEs for physical consistency.

2. Comprehensive Experiments (Sections 4.2.2–4.6, Appendices A.4–A.8):
   Extensive results showcasing the effectiveness of joint diffusion:

   • Robust handling of noisy seismic inputs (Appendix A.5).
   • Symmetry checks and visualizations across multiple datasets (Appendix A.4).
   • Comparisons with multiple baseline methods, including InversionNet, VelocityGAN, and UPFWI (Appendices A.6, A.8).

Thank you for your time and effort in reviewing our work.

Best regards,
The authors

Dear Reviewer rGTJ,

Thank you for your feedback and engagement with our manuscript. We deeply value the time and effort you have spent reviewing our work. While we understand your decision to maintain your score, we would like to kindly ask for any further comments or suggestions you may have regarding the latest revised version of the manuscript.

We have worked diligently to address the concerns raised in your earlier comments and made significant revisions to clarify and emphasize the novelty of the joint diffusion process, its applications, and its analysis. Your expertise is invaluable to us, and we are committed to further improving the manuscript based on your insights.

If there are specific aspects of the revised manuscript that you believe could benefit from additional refinement or focus, we would greatly appreciate your guidance. Your feedback will help us ensure the manuscript is of the highest quality and effectively communicates its contributions to the field.

Thank you again for your thoughtful review and input. We look forward to hearing any additional suggestions you may have.

Best regards,
The authors