A time-dependent batch Bayesian framework for joint inversion of sudden stiffness loss and unknown input using limited sensor data

Accurately determining seismic excitation during earthquakes poses significant challenges in structural health monitoring (SHM), particularly when simultaneously addressing sudden stiffness changes, uncertainty quantification, and joint inversion problems. This study presents a novel time-dependent batch Bayesian framework for the probabilistic joint inversion of unknown seismic inputs and sudden stiffness loss using limited sensor data. The framework introduces several innovative components: (1) a sparsity-inducing Automatic Relevance Determination (ARD) prior that adaptively constrains parameter variations across differently-sensitive structural components, enhancing robustness against ill-conditioning; (2) a two-stage time-dependent model class selection strategy with Bayesian evidence maximization to explicitly identify the most probable moment of sudden stiffness reduction, a capability absent in traditional static parameterization approaches; (3) a tailored Bayesian learning algorithm employing bound-constrained nonlinear least-squares (NLLS) optimization that handles high-dimensional parameter spaces without requiring source code modifications of underlying optimization routines; and (4) computational advancements including Gaussian-Newton approximation and Cholesky factorization to overcome numerical difficulties in ultra-high-dimensional Hessian matrix evaluations. This integrated approach enables simultaneous determination of sudden stiffness loss timing, damage location, damage magnitude, unknown seismic input, and comprehensive uncertainty quantification of all inversion results. As a preliminary investigation, the methodology is validated using dynamic response measurements from a five-story laboratory shear-building model subjected to various shaking table test scenarios with strategically induced damage patterns. Results demonstrate the framework's capability to accurately identify and quantify abrupt stiffness degradation and reconstruct unknown seismic inputs across multiple damage scenarios and sensor configurations, even with limited measurement data. © 2025 Elsevier Ltd.