Power-scaled Bayesian Inference with Score-based Generative Models

TL;DR

The paper tackles the challenge of balancing prior knowledge and observed data in seismic velocity inversion by introducing power-scaled Bayesian inference with score-based generative models. A single conditional score network trained with classifier-free guidance estimates the posterior and prior scores, enabling sampling from the power-scaled posterior $p^{\lambda,\alpha}(\mathbf{x}|\mathbf{y})$ without retraining; the key gradient relation is $\nabla_{\mathbf{x}} \log p^{\lambda,\alpha}(\mathbf{x}|\mathbf{y}) = \lambda \nabla_{\mathbf{x}} \log p(\mathbf{x}|\mathbf{y}) + (\alpha - \lambda) \nabla_{\mathbf{x}} \log p(\mathbf{x})$. Using RTM images as conditioning statistics on a synthetic Compass model, the authors show that increasing $\lambda$ improves data fidelity up to a threshold (≈2.0), while adjusting $\alpha$ controls regularization and diversity of the samples. The approach yields a flexible, retraining-free framework with per-sample generation times around 4 seconds, enabling sensitivity analysis and robust uncertainty quantification for seismic inversion. This work provides a principled mechanism to explore prior-data tradeoffs and enhances interpretability in data-driven seismic velocity model generation.

Abstract

We propose a score-based generative algorithm for sampling from power-scaled priors and likelihoods within the Bayesian inference framework. Our algorithm enables flexible control over prior-likelihood influence without requiring retraining for different power-scaling configurations. Specifically, we focus on synthesizing seismic velocity models conditioned on imaged seismic. Our method enables sensitivity analysis by sampling from intermediate power posteriors, allowing us to assess the relative influence of the prior and likelihood on samples of the posterior distribution. Through a comprehensive set of experiments, we evaluate the effects of varying the power parameter in different settings: applying it solely to the prior, to the likelihood of a Bayesian formulation, and to both simultaneously. The results show that increasing the power of the likelihood up to a certain threshold improves the fidelity of posterior samples to the conditioning data (e.g., seismic images), while decreasing the prior power promotes greater structural diversity among samples. Moreover, we find that moderate scaling of the likelihood leads to a reduced shot data residual, confirming its utility in posterior refinement.

Figures (4)

• Figure 1: Prior samples generated with increasing power $\alpha$ from $0.25$ to $1.5$. As $\alpha$ increases, the generated velocity models become progressively sharper and more structurally consistent, especially in deeper layers. Lower powers result in higher variability and less coherent geologic features.
• Figure 2: Posterior samples generated with varying likelihood power $\lambda$ from $0.0$ to $16.0$, with fixed prior power $\alpha = 1$. As $\lambda$ increases, the samples incorporate more structure from the conditioning RTM image. Maximum alignment with the ground truth occurs around $\lambda = 2.0$, beyond which overfitting and degradation in performance become apparent.
• Figure 3: The $\ell_2$-norm of data residual as a function of likelihood power $\lambda$, with fixed prior power $\alpha = 1$. Residual decreases as $\lambda$ increases, reaching a minimum around $\lambda = 2.0$, indicating optimal alignment with the ground truth. Beyond this point, performance begins to degrade, suggesting that excessive amplification of the likelihood leads to overfitting or reduced generalization.
• Figure 4: Posterior samples generated by jointly varying likelihood power $\lambda$ and prior power $\alpha$. Columns correspond to increasing prior power (left to right), while rows correspond to increasing likelihood power (bottom to top). Samples with larger $\lambda$ better align with RTM conditioning data, while larger $\alpha$ values enforce structural consistency and sharper geological layering. Lower prior powers increase generative diversity and loosen geological constraints.