Three-dimensional (3D) velocity model building (VMB) remains computationally challenging with conventional approaches and native 3D neural networks, largely due to the prohibitive memory and training cost of full 3D formulations. Therefore, we introduce a depth-progressive framework that achieves 3D VMB using 2D generative diffusion models through a depth-as-channel formulation, in which the depth dimension of each 3D velocity patch is mapped to the channel dimension of a 2D U-Net. This allows the model to synthesize overlapping depth patches sequentially from shallow to deep while conditioning the prediction of the deeper patch on the shallower one, as well as on any available geophysical constraints, including well logs and structural attributes. During training, paired shallow-deep patch pairs with randomized overlaps are used to teach the network how deeper velocity structures relate to shallower information, and a classifier-free guidance strategy enables the model to handle missing constraints gracefully. At inference, starting from a shallow velocity prior, the model recursively generates deeper patches in a top-down manner, and a depth-attenuated Gaussian blending scheme merges overlapping predictions into spatially coherent 3D volumes while accounting for the progressive accumulation of uncertainty at greater depths. Numerical experiments on both in-distribution benchmark models, including Overthrust, SEAM Arid, and SEG/EAGE, and the out-of-distribution Marmousi model demonstrate that the method effectively propagates shallow velocity priors downward and produces high-quality 3D velocity volumes across a range of geological settings and conditioning scenarios. Compared to a 3D benchmark of equivalent architecture, the proposed method significantly reduces parameter count, training cost, and inference cost, while maintaining superior VMB accuracy.
Figures & tables
Figure 1: An illustration of the proposed network architecture. (a) The overall network structure. (b) Time embedding layer (where t is the diffusion timestep and dmax is the depth coordinate of the deepest grid point in vdeep ). (c) Input embedding layer (where vshallow is the shallow velocity patch). (d) Structure embedding layer (where s is the structural attribute). (e) The attention block (where Omid is the block output in (g)). (f) Well embedding layer (where w is the well velocity and l is the lateral position of the well). (g) The residual block (where Imid is the block input and Omid is the block output).
Figure 2: Representative training samples from each source model.
Figure 3: Statistical properties of the full augmented training dataset of 22000 velocity models. (a) Mean velocity volume. (b) Standard deviation volume.
Figure 4: Per-model statistics of the training samples derived from the three 3D industrial models used in subsequent experiments. The first row shows the mean velocity and the second row shows the standard deviation, from left to right corresponding to Overthrust, SEAM Arid, and SEG/EAGE.
Figure 5: Test configuration for the Overthrust model. (a) Reference 3D velocity model, with the red dashed line indicating the location of the single well constraint. (b) Convolution image used as the structural attribute. (c) Shallow velocity prior formed from the top 16 grid points of the reference model.
Figure 6: 3D VMB results on the Overthrust model obtained by the proposed method under four conditioning scenarios. Rows (a)-(d) correspond to unconditional, well-only, image-only, and well-plus-image conditioning, respectively. Columns 1-4 show a representative sample, the ensemble mean, the ensemble standard deviation, and the pointwise absolute error between the ensemble mean and reference model (Figure 5 a).
Figure 7: 3D VMB results on the Overthrust model obtained by the 3D network benchmark method under the same four conditioning scenarios as in Figure 6 .
Figure 8: Test configuration for the SEAM Arid model. (a) Reference 3D velocity model, with the red dashed line indicating the well location. (b) Convolution image used as the structural constraint. (c) Shallow velocity prior.
Figure 9: 3D VMB results on the SEAM Arid model obtained by the proposed method under four conditioning scenarios. Rows (a)-(d) correspond to unconditional, well-only, image-only, and well-plus-image conditioning, respectively. Columns 1-4 show a representative sample, the ensemble mean, the ensemble standard deviation, and the pointwise absolute error between the ensemble mean and reference model (Figure 8 a).
Figure 10: 3D VMB results on the SEAM Arid model obtained by the 3D network benchmark method under the same four conditioning scenarios as in Figure 9 .
Figure 11: Test configuration for the SEG/EAGE model. (a) Reference 3D velocity model, with the red dashed line indicating the well location. (b) Convolution image used as the structural constraint. (c) Shallow velocity prior.
Figure 12: 3D VMB results on the SEG/EAGE model obtained by the proposed method under four conditioning scenarios. Rows (a)-(d) correspond to unconditional, well-only, image-only, and well-plus-image conditioning, respectively. Columns 1-4 show a representative sample, the ensemble mean, the ensemble standard deviation, and the pointwise absolute error between the ensemble mean and reference model (Figure 11 a).
Figure 13: 3D VMB results on the SEG/EAGE model obtained by the 3D network benchmark method under the same four conditioning scenarios as in Figure 12 .
Figure 14: Test configuration for the Marmousi model. (a) Reference 3D velocity model, with the red dashed line indicating the well location. (b) Convolution image used as the structural constraint. (c) Shallow velocity prior.
Figure 15: 3D VMB results on the Marmousi model obtained by the proposed method under four conditioning scenarios. Rows (a)-(d) correspond to unconditional, well-only, image-only, and well-plus-image conditioning, respectively. Columns 1-4 show a representative sample, the ensemble mean, the ensemble standard deviation, and the pointwise absolute error between the ensemble mean and reference model (Figure 14 a).
Figure 16: 3D VMB results on the Marmousi model obtained by the 3D network benchmark method under the same four conditioning scenarios as in Figure 15 .
Proposed Method
3D Benchmark
Model
Metric
Uncond.
Well
Image
Well+Image
Uncond.
Well
Image
Well+Image
Overthrust
MAE
129.83
127.80
6.89
6.88
1239.42
1261.15
24.90
20.53
SSIM
0.757
0.761
0.999
0.999
0.410
0.410
0.993
0.996
SEAM Arid
MAE
227.49
247.79
11.56
11.33
932.69
958.78
28.58
24.56
SSIM
0.637
0.621
0.993
0.994
0.615
0.605
0.987
0.989
SEG/EAGE
MAE
247.53
253.35
11.38
11.01
271.83
268.69
58.43
56.72
Table 1: Quantitative comparison of MAE (m/s) and SSIM for the proposed method and the 3D benchmark under four conditioning scenarios across all test models. The best result in each row is shown in bold .
Figure 17: Effect of shallow prior extent on unconditional 3D VMB results for the Overthrust model. Columns (a)-(d) correspond to shallow prior depths of 16, 32, 64, and 96 grid points, respectively. Rows 1-5 show the shallow prior, a representative single sample, the ensemble mean over 50 samples, the standard deviation, and the pointwise absolute error between the ensemble mean and reference model (Figure 5 a).
Proposed method
3D Benchmark
Model
Total parameters (M)
526.35
1473.20
Training
Time: batch=1 (s)
27.26
99.34
Memory: fixed (GB)
8.05
27.66
Memory: per batch (GB)
1.47
27.60
Inference
Time: 25 samples (s)
25.43
749.45
Memory: 25 samples (GB)
6.68
75.10
Table 2: Computational comparison between the proposed method and the 3D benchmark under equivalent network configurations. Training cost is measured per 100 iterations at batch size 1. Inference cost is measured for generating 25 samples. Fixed memory refers to the memory occupied by model parameters and optimizer states, and per-batch memory refers to the additional memory incurred by each training sample. All measurements are conducted on a single NVIDIA A100 GPU.
High-resolution velocity models are crucial for reservoir characterization and subsurface delineation. However, the band limited nature of our surface recorded data limits resolution. Utilizing well measurements to enhance the resolution of our subsurface models is an important objective. To this end, we present a diffusion-guided framework for structurally preconditioned velocity-model reconstruction from sparse well-log information. The proposed approach combines plane-wave PDE regularization, structurally preconditioned inversion, and measurement-guided diffusion posterior sampling within a unified formulation. Local structural slopes estimated through plane-wave destruction are used both to propagate well information along geological dip directions and to guide the diffusion sampling process through a joint velocity--slope generative prior. Numerical experiments on the Volve synthetic model and the Viking Graben field dataset demonstrate that the proposed framework improves structural continuity, lateral consistency, and geological realism compared with conventional structurally preconditioned inversion approaches while maintaining computationally practical inference through DDIM sampling.
Francesco Brandolin, Tariq Alkhalifah
King Abdullah University of Science and Technology (KAUST) Thuwal, Kingdom of Saudi Arabia
Recent advances in generative networks have enabled new approaches to subsurface velocity model synthesis, offering a compelling alternative to traditional methods such as Full Waveform Inversion. However, these approaches predominantly rely on the availability of large-scale datasets of high-quality, geologically realistic subsurface velocity models, which are often difficult to obtain in practice. We introduce SAGE, a novel framework for statistically consistent proxy velocity generation from incomplete observations, specifically sparse well logs and migrated seismic images. During training, SAGE learns a proxy posterior over velocity models conditioned on both modalities (wells and seismic); at inference, it produces full-resolution velocity fields conditioned solely on migrated images, with well information implicitly encoded in the learned distribution. This enables the generation of geologically plausible and statistically accurate velocity realizations. We validate SAGE on both synthetic and field datasets, demonstrating its ability to capture complex subsurface variability under limited observational constraints. Furthermore, samples drawn from the learned proxy distribution can be leveraged to train downstream networks, supporting inversion workflows. Overall, SAGE provides a scalable and data-efficient pathway toward learning geological proxy posterior for seismic imaging and inversion. Repo link: https://github.com/slimgroup/SAGE.
Huseyin Tuna Erdinc, Ipsita Bhar, Rafael Orozco +2
Georgia Institute of Technology · Oxy · Huber Engineered Materials
Full waveform inversion (FWI) is the gold standard for subsurface imaging, with applications from carbon sequestration to energy and mineral exploration to earthquake hazard assessment. Machine learning approaches to FWI need field-scale, geologically diverse, and physically realistic training data, but existing resources such as Marmousi, SEAM, and OpenFWI fall short on spatial extent, temporal extent, geological diversity, and physical realism. We address these limitations with SubsurfaceGen, a GPU-accelerated generator for 3D velocity models and seismic data. Along with SubsurfaceGen, we release a paired dataset of 4,276 2D velocity slices, 5 s wavefields, and 8 s shot gathers drawn from 42 realistic, field-scale 3D velocity models, each spanning 10 km x 10 km laterally and 6.19 km deep at 10 m resolution. The dataset spans six geological settings -- four built with SubsurfaceGen and two drawn from prior sources -- relevant for carbon sequestration and hydrocarbon exploration. We use this dataset to evaluate neural operators on wavefield prediction and encoder-decoders on end-to-end velocity inversion, holding out one geological setting for out-of-distribution testing. These experiments surface failure modes at field-scale and demonstrate how SubsurfaceGen and the associated dataset can impact ML-based FWI.