Gibbs sampling-enhanced foundation model for in-context characterization of multivariate geotechnical data from limited measurements

Accurately characterizing the spatial variability of multivariate geotechnical properties from limited site investigation data remains a critical challenge in geotechnical engineering. Existing geostatistical and machine learning methods often rely on predefined model structures or large training datasets for modeling, failing to capture high-frequency spatial variations and preserve multivariate dependency structures under sparse observations. Recently, in-context learning (ICL) with foundation models has provided a new paradigm for geotechnical site characterization. However, geotechnical data are often sparse, multivariate, and incomplete owing to different measurement locations, depths, and testing conditions, resulting in misaligned missing data. Such incomplete multi-input multi-output (MIMO) settings pose challenges for existing foundation models, which rely on well-structured data for inference. To address this issue, this study proposes an ICL with Gibbs-type updating framework for the joint inference of MIMO problems. The proposed method integrates the high-frequency representation capacity of ICL with the conditional iterative updating mechanism of Gibbs sampling, enabling joint probabilistic modeling of spatial variability and inter-variable dependencies. A nested Gibbs- type updating strategy is then developed to enhance inference stability and reduce interference from weakly correlated variables, and the corresponding predictive uncertainty is quantified via a bootstrapping approach. Furthermore, a sequential random sample generator is implemented to preserve both marginal distributions and correlation structures in multivariate samples. The effectiveness and applicability of the proposed approach are validated through real-world geotechnical data from New Zealand and Quebec, Canada.