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Self-attention summary networks for subsurface velocity-model building from common-image gathers
Shiqin Zeng, Yunlin Zeng, Abhinav Prakash Gahlot, Zijun Deng, Felix J. Herrmann
TL;DR
Velocity inversion is high-dimensional and non-unique, while CIGs provide offset-dependent indicators of model mismatch. The paper summarizes CIGs with multiscale self-attention and uses the embeddings to condition flow-matching velocity generation; the reported results improve posterior inference over raw CIG conditioning and show robustness to background-model errors.
Problem
Velocity-model building is high-dimensional, nonlinear, and non-unique, while CIGs carry offset-dependent information about reflector focusing and residual moveout.
Source: §1 · p. 1Method
A multiscale self-attention encoder maps CIGs to compact embeddings that condition a flow-matching model for probabilistic velocity-model generation.
Source: §1 · p. 1, §2.2 · p. 2Results
Multiscale summary conditioning improves posterior accuracy over raw CIG conditioning and yields more accurate samples with more concentrated uncertainty under background-model errors.
Source: §3.4 · pp. 5, 7, §4 · p. 7Takeaways & Limitations
Multiscale self-attention with conditional flow matching provides an approach to uncertainty-aware velocity inference from CIGs.
Source: §4 · p. 7
Abstract
from arXiv · showhide
Common-image gathers (CIGs) contain physically meaningful information about velocity-model errors through reflector focusing and residual moveout, but in conventional imaging workflows they are typically used only as diagnostic tools. In this work, we propose a multiscale self-attention summary network that maps high-dimensional 3D CIG volumes into compact conditioning embeddings for probabilistic subsurface velocity inversion. These learned embeddings preserve offset-dependent kinematic structure and spatial coherence while reducing variability caused by background-velocity mismatch. Conditioned on these summary embeddings, a flow-matching model learns a transport from a Gaussian source distribution to the posterior distribution of plausible velocity fields. Numerical experiments show that, compared with direct conditioning on raw CIGs, the proposed summary network improves posterior velocity inference. In particular, the multiscale attention design provides greater robustness to background-model mismatch, yielding more accurate posterior reconstructions and lower predictive uncertainty.
1 Introduction
Velocity-model building is a high-dimensional, nonlinear, non-unique inverse problem, and conventional optimization can be sensitive to initialization and yield a single estimate. The paper uses CIG information in a probabilistic inversion framework, pairing a self-attention summary encoder with conditional flow matching.
1 Introduction
- CIGs encode offset-dependent reflector focusing and residual moveout, making them informative indicators of velocity-model mismatch.Conventional full-waveform inversion can be sensitive to initialization, prone to cycle skipping, and typically yields a single deterministic estimate.Source: §1 · p. 1
- The framework conditions probabilistic velocity inversion on compact CIG embeddings produced by a self-attention summary encoder and a conditional flow-matching model.The authors report improved reconstruction accuracy and uncertainty quantification.Source: §1 · p. 1
2 Method
The method represents CIGs as offset-indexed spatial tensors, summarizes them with multiscale local self-attention, and conditions a flow-matching model to generate velocity models.
2.1 3D CIG representation
- The extended imaging condition produces a spatial tensor of offset-dependent responses that encode local kinematic and structural information relative to the background model.With a kinematically consistent background, reflected energy focuses near zero offset; model error produces residual moveout and defocusing across offsets.Source: §2.1 · p. 2
2.2 Multiscale self-attention summary network
- A learned encoder maps CIGs to a low-dimensional embedding using local self-attention at multiple window sizes to aggregate spatially coherent offset responses and reduce redundancy.Multiple scales capture local kinematic variations and larger-scale continuity; a channel-mixing projection compresses the features for downstream velocity generation.Source: §2.2 · pp. 2–3
2.3 Velocity-model generation via conditional flow matching
- Conditional flow matching jointly learns the summary encoder and conditional vector field, then generates velocity models by integrating the learned ODE from Gaussian samples.The method uses a linear rectified-flow probability path; repeated sampling enables uncertainty quantification.Source: §2.3 · p. 3
3 Experiments and results
Multiscale summary conditioning improves posterior velocity inference over raw-CIG and single-scale conditioning, while experiments examine latent size, background-model mismatch, and attention-window scale. The selected channel size is C=8, and the window sizes capture complementary spatial information.
3.1 Compact latent representation from summary encoding
- The singular value ratio (σmin/σmax) falls from 7.2% at C=4 to 0.22% at C=10, while C=6–10 improves downstream performance over C=4.C=8 is the default; C=12 brings a larger singular value ratio and degraded downstream performance.Source: §3.1 · p. 4, Figure 2 · p. 4
3.2 Behavior under background-velocity model errors
- Under the 1D background, attention shifts emphasis away from noisy near-zero-offset responses and redistributes it across other offset components.The network adapts to mismatch-dependent CIG patterns rather than directly recovering correct focusing.Source: §3.2 · p. 4, Figure 3 · p. 5
3.3 Effect of multiscale attention windows
- Smaller attention windows preserve sharper local detail, while larger windows provide broader context; at ws=16, features become overly smooth and blocky.The complementary scales motivate multiscale attention to combine fine detail with broader structural continuity.Source: §3.3 · p. 5, Figure 4 · p. 6
3.4 Summary conditioning for posterior inference
- The multiscale summary network yields posterior means closer to ground truth, monotonically lower RMSE, higher SSIM, and more concentrated uncertainty than single-scale or raw-CIG conditioning.Its spatial errors are more localized and its uncertainty field is best structured overall.Source: §3.4 · pp. 5, 7, Figure 5 · p. 6
4 Conclusion and future work
The paper combines a multiscale self-attention summary network for CIGs with conditional flow matching to generate posterior velocity models. The reported results show improved robustness and inference over direct raw-CIG conditioning, while generalization across unseen background models remains future work.
4 Conclusion and future work
- The proposed framework combines multiscale self-attention summaries of subsurface-offset CIGs with conditional flow matching to generate posterior velocity models.The encoder converts high-dimensional CIGs into compact conditioning representations.Source: §4 · p. 7
- Compared with direct raw-CIG conditioning, the combination improves robustness to background-model errors and yields more accurate posterior samples with more concentrated uncertainty fields.The paper describes multiscale self-attention and conditional flow matching as an effective approach to uncertainty-aware velocity inference from CIGs.Source: §4 · p. 7
- Future work will test generalization by training on CIGs with different velocity errors and evaluating on unseen background models.This evaluation is intended to provide stronger evidence that the summary network learns adaptive mismatch-aware representations.Source: §4 · p. 7