Priors learned from legacy reconstructions inherit undetectable overconfidence

Where truths are scarce (e.g., seismic and medical imaging), a prior for an ill-posed inverse problem is trained on an archive of legacy reconstructions---an older method's outputs---and its uncertainty is treated as data-driven. In the population limit, an archive of posterior samples is the regularizer that produced it, advanced one expectation-maximization step toward the truth. On directions the operator resolves, it improves the assumption; on its blind subspace, the step is the identity, so the assumption survives unchanged however often it is rebuilt. An archive of single-best reconstructions, one per survey, keeps no spread there: the blind interval collapses whatever the penalty was, so error becomes overconfidence. The assumption enters as the archive and leaves as a reported spread, and nothing in deployment tests it. Two truths differing only there share the data law, and no procedure using survey and archive alone can both report a finite blind interval and guarantee coverage over indistinguishable truths. The question requires information the survey does not carry. We provide a resolvability statement that names affected directions from the operator, state how many reference truths are needed to test a prior on them, and use those references to build an interval that contains the truth as claimed, even if the prior is wrong. On synthetic experiments with seismic and groundwater operators, the archive-trained prior's intervals contain the truth less often on the blind subspace than on resolved directions, while the truth-trained control shows no gap of that sign or size. On seismic, a random subspace of the same size gives the same result, so the separation follows the operator. On groundwater, rebuilding the archive using the survey and a handful of boreholes brings the prior's reports toward the truth on directions those boreholes reach, while leaving the rest unchanged.

Latent Variable-Mediated Cross-Learning for Few-Shot Acoustic Impedance Imaging

Acoustic impedance imaging is a fundamental yet severely ill-posed problem in subsurface analysis: the seismic wavelet is unknown, observations are band-limited, and labeled well-log samples are extremely scarce (typically <1% of all traces). Existing semi-supervised deep learning methods mitigate few-shot problem by incorporating forward modeling, yet they either rely on inaccurate prior wavelet assumptions or introduce auxiliary networks, leading to unstable optimization and degraded performance. We propose RD-SCL, a novel framework that integrates regularized deconvolution with semi-supervised cross-learning. At its core lies a differentiable, closed-form first-order Tikhonov deconvolution operator that dynamically estimates the latent wavelet in the frequency domain during training, providing stable physics-guided feedback without explicit auxiliary networks and fixed wavelet priors. Building on this operator, we design a symmetric cross-learning that enforces consistency between predictions on labeled and unlabeled data, thereby effectively exploiting abundant unlabeled traces. Extensive experiments on the SEAM and Marmousi 2 benchmarks demonstrate that RD-SCL consistently outperforms state-of-the-art supervised and semi-supervised methods, achieving substantial gains with lower computational cost. With only 56.5k learnable parameters and competitive runtime, RD-SCL offers a practical, physically consistent, and efficient solution for acoustic impedance imaging.

Joint Velocity Slope Diffusion Prior for Structurally Constrained Velocity Model Building

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.

Probabilistic Inversion with Flow Matching

We demonstrate the application of Flow Matching, a technique originating from generative Artificial Intelligence, to probabilistic inversion in geophysical settings, such as seismic Full-Waveform inversion. We adapt the well-established mathematical theory of Flow Matching from generative Artificial Intelligence to the context of probabilistic inversion. We evaluate the approach with two case studies: a simple 2D velocity model to illustrate the general features of the method, and the OpenFWI dataset to show its capabilities for probabilistic inversion of more complex seismic velocity models.

Decoupled Latent Optimization of Diffusion Models for Full Waveform Inversion

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.