Uncertainty propagation and stochastic modelling in correlated two-phase structural load testing

Short-term structural load tests are often performed from a single, unchanged total-station setup, where observations before and after load application may share instrumental, geometric, or environmental error components. If such inter-epoch dependence is ignored, the propagated uncertainty of displacement and the resulting detectability assessment may be distorted. This study develops an analytical and numerically verified framework for uncertainty propagation in two-phase structural load testing with inter-epoch covariance. A joint two-epoch three-dimensional least-squares model is formulated using a block-structured observation covariance matrix. A reduced closed-form expression for the variance of vertical displacement is then derived for unchanged station geometry and constant instrument and prism heights. The expression is verified using a synthetic 3D adjustment model and 5,000 Monte Carlo realisations. For assumed total-station precision parameters of 1 mm +1 ppm for slope distance and 1″ for angular observations, the closed-form and full 3D least-squares covariance results agree to the displayed numerical precision. Monte Carlo standard deviations differ from the analytical values by less than 0.8%. For inter-epoch correlation scenarios of −0.8, 0, and +0.8, the standard uncertainty of vertical displacement is 0.1702 mm, 0.1268 mm, and 0.0567 mm, respectively. Across the interval −0.9 to +0.9, uncertainty varies from 0.1748 mm to 0.0401 mm. For an assumed 0.20 mm displacement, the standardised detectability ratio changes from 1.18 to 3.53. The correlation coefficients are treated as sensitivity parameters, not empirically estimated instrument characteristics.