Physics-Informed Inference of Historical Stair Usage from Geometric Wear Profiles in Heritage Structures

Wear on historic staircases is often used as evidence for conservation assessment and historical interpretation, yet existing studies are largely descriptive and rarely provide a quantitative explanation of how observed wear relates to long-term pedestrian use. To address this limitation, this paper proposes a physics-constrained inversion framework for analyzing directional preference and wear-related usage regimes from geometric wear profiles of heritage staircases. An Archard-type wear model is extended to account for spatial footfall distribution, cumulative abrasion, material deterioration, and environmental loss, and the reconstruction problem is formulated as an inverse parameter estimation task. Bayesian uncertainty quantification is introduced to estimate posterior distributions, credible intervals, and parameter coupling. A unified workflow is developed for staircase geometry representation, reference surface reconstruction, profile extraction, regularized height field construction, forward simulation, and inverse solution. Nine synthetic scenarios with different usage levels and directional preferences are tested under 1%, 3%, and 5% noise, and the method is further applied to a publicly available three-dimensional heritage staircase model. Under 3% noise, profile correlation coefficients for three representative scenarios reach 0.9646, 0.9807, and 0.9868, indicating strong recoverability of geometric wear morphology under model-consistent conditions. The results indicate that directional preference, material hardness, and some degradation-related parameters are identifiable, whereas pedestrian volume and the wear coefficient show strong compensation. Overall, the proposed framework provides a quantitative basis for identifying directional asymmetry, analyzing parameter identifiability, and supporting geometry-based interpretation in heritage staircase studies.