Grey-box Gaussian processes for mode shape normalisation: damage localisation under environmental and operational variability

Structural health monitoring (SHM) offers a promising path towards automated, long-term monitoring of critical infrastructure such as offshore structures and bridges. A crucial component of such monitoring schemes is the localisation of damage, with model-based SHM providing a possible framework for this task. Here, damage localisation can be achieved by minimising the discrepancy between the modal properties of a damaged state and those of a healthy reference. However, this process is severely hindered by changes in the modal properties induced by environmental and operational variations (EOVs). This difficulty is compounded by the fact that measurement data are often limited: data may be unavailable across the full operational span, may be missing because of sensor failure and dropout, or may be only sparsely sampled because of hardware constraints. In particular, mode shapes are often used without adequately accounting for the limited coverage of EOVs in continuous model-based damage localisation frameworks, resulting in inaccurate localisation. This paper proposes a regression-based data normalisation scheme that learns mode shapes as functions of spatial coordinate and EOVs using Gaussian process regression, allowing them to vary across a structure’s operating envelope and thereby enabling model-based damage localisation under varying environmental and operational conditions. To further alleviate the problem of limited training data, two grey-box modelling strategies based on Gaussian processes that incorporate accessible engineering knowledge are considered: (i) a Hilbert-space Gaussian process enforcing boundary conditions, and (ii) a Gaussian process with a physics-based prior mean given by finite-element mode shapes. The focus is on long-term monitoring scenarios in which training data are limited and purely data-driven regression techniques perform poorly. Using the Leibniz University Test Structure for Monitoring, a representative real-world structure, this study demonstrates that incorporating accessible engineering-domain knowledge into Gaussian process models alleviates data scarcity and improves damage localisation under partially observed EOVs. These results highlight the practical value of the proposed grey-box models for SHM. They enable mode-shape normalisation for model-based damage localisation under environmental and operational variability, even when training data are limited, thereby enhancing continuous, long-term monitoring of operational structures.