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.
Full waveform inversion (FWI) estimates subsurface velocity from seismic recordings, but its ill-posedness and nonlinearity make accurate reconstruction strongly dependent on initialization and prior information. Diffusion posterior sampling provides a learned geological prior, yet directly coupling its denoiser to the nonlinear wave solver can yield unreliable physical guidance. We propose Physical-State-Guided Diffusion Sampling (PSG), which couples a persistent physical velocity to the diffusion prior through a Gaussian bridge. The physical state is refined by waveform fitting regularized by the denoised velocity, and in turn guides the reverse diffusion process. This formulation separates the wave-equation and denoiser gradients while preserving conventional FWI initialization and accumulated optimization history. On four OpenFWI families, PSG's terminal denoised estimates outperform classical and diffusion-based baselines under clean and missing-trace acquisitions and maintain strong structural recovery under measurement noise. Repeated stochastic runs preserve the dominant geological structures, with ensemble variability concentrated near geological interfaces and positively associated with local inversion error. A frozen OpenFWI-trained prior further supports inversion of the larger Marmousi, Overthrust, and BP2004 Salt models, recovering complex geological structures without retraining.
Full waveform inversion (FWI) recovers subsurface velocity from seismic recordings by solving a severely ill-posed, nonconvex PDE-constrained optimization. Classical regularizers stabilize the inversion but fail to reproduce realistic geological structures; recent diffusion-prior methods improve realism at the cost of a fragile trade-off between data fidelity and prior consistency. We propose Decoupled Latent Optimization (DLO), which relaxes the standard latent-optimization formulation into a quadratic-penalty objective over an auxiliary physical variable and a latent variable. The data-fidelity gradient acts in physical space, the diffusion sampler contributes only through a decoded prior sample, and the standard smoothed-velocity initialization of classical FWI is preserved. On the OpenFWI benchmark, DLO outperforms classical regularizers and existing diffusion-based methods under clean, noisy, and missing-trace acquisitions. The prior, trained on 70*70 OpenFWI models, transfers directly to the Marmousi and Overthrust benchmarks, where DLO recovers intricate fault structures and remains robust to initialization smoothing and measurement noise.
We propose latent-space diffusion posterior sampling (L-DPS), an approximate Bayesian framework for high-dimensional inverse problems governed by partial differential equations (PDEs). The method addresses three challenges in PDE-constrained inversion: implicit sample-based priors without tractable densities, high-dimensional spatially distributed parameters, and the high cost of repeated forward-model evaluations during posterior sampling. L-DPS combines a variational autoencoder, an unconditional latent diffusion model, diffusion posterior sampling, and a differentiable neural surrogate. The VAE maps the parameter field to a lower-dimensional latent space, the diffusion model learns an implicit prior score in this latent space, and DPS combines this learned prior with likelihood-based guidance. The likelihood gradient is evaluated through the decoder-surrogate composition, avoiding repeated calls to the full numerical PDE solver. We evaluate the method on an inverse Darcy flow problem with an unknown spatially distributed permeability field inferred from sparse and noisy pressure observations. L-DPS produces accurate and robust inverse solutions, reduces inference cost relative to full-space DPS, and outperforms amortized inverse baselines such as conditional latent diffusion and inverse FNO in sparse and noisy regimes. We further compare L-DPS with a KLE-MAP baseline and study mixed-prior generalization and the sensitivity of inversion accuracy to surrogate forward-model error.