Integrating deterministic and stochastic uncertainty in terrestrial laser scanner-based deformation monitoring

Terrestrial laser scanning (TLS) is widely used for deformation monitoring, where decisions about structural safety depend on a reliable description of measurement uncertainty. Classical congruency tests typically treat this uncertainty in a purely stochastic manner. However, remaining systematic effects from instrumental, atmospheric, and modelling sources may still influence the estimated deformation field. This becomes critical when the expected deformation is close to the magnitude of these unmodeled effects, as apparent displacements could be caused either by actual structural change or by systematic influences. In such situations of uncertainty, forcing a purely binary test decision may be difficult to justify. To address this, this contribution introduces an interval-extended uncertainty propagation framework for TLS-based deformation monitoring. The uncertainty budget is separated into two components: stochastic variability, represented by random variables, and remaining systematic effects, represented as unknown-but-bounded intervals. Using sensitivity analysis of the TLS correction and measurement models, these bounded effects are propagated from physical influence parameters to the control-point (CP) domain of a B-spline surface model, and subsequently to the derived surface points (SP). This establishes a transparent link between physically interpretable TLS influence parameters and their impact on the final deformation decision. The resulting congruency test extends the classical binary approach with a ternary decision structure: strict acceptance, ambiguity, and strict rejection. In the CP domain, the method utilizes a correlation-aware localization procedure based on the Schur complement. Conversely, in the SP domain, the propagated results are interpreted strictly as a pointwise diagnostic map. This distinction is essential because dense surface points are derived evaluations of the underlying CP deformation field; treating them as independent estimated parameters would be mathematically singular. The results demonstrate that bounded systematic effects can significantly alter classical deformation decisions, and the interval-extended formulation makes this influence explicit. Rather than forcing uncertain cases into overconfident binary classifications, the proposed framework identifies regions where the deformation signal cannot be reliably separated from systematic uncertainty, providing a more cautious, transparent, and structurally safe basis for TLS-based deformation interpretation.