Sensor Informativeness, Identifiability, and Uncertainty in Bayesian Inverse Problems for Structural Health Monitoring

TL;DR

This paper tackles the ill-posed problem of recovering distributed flexural rigidity $EI(x)$ from sparse rotation data in structural health monitoring by formulating a Bayesian inverse problem. It couples a linearized Euler–Bernoulli forward model with Gaussian likelihoods, positivity-enforcing Gaussian Markov random-field priors on the log-compliance, and a MAP–Tikhonov interpretation to quantify both point estimates and uncertainty, using Laplace approximations for nonlinearity. A key contribution is the Fisher information framework that quantifies sensor informativeness and identifiability, guiding sensor placement and load-path design. The methodology is demonstrated on the openLAB two-span bridge with controlled axle crossings, yielding spatially varying posterior uncertainty that reveals practical non-identifiability near zero-moment regions and supports informed maintenance decisions and digital-twin calibration.

Abstract

In Structural Health Monitoring (SHM), the recovery of distributed mechanical parameters from sparse data is often ill-posed, raising critical questions about identifiability and the reliability of inferred states. While deterministic regularization methods such as Tikhonov stabilise the inversion, they provide little insight into the spatial limits of resolution or the inherent uncertainty of the solution. This paper presents a Bayesian inverse framework that rigorously quantifies these limits, using the identification of distributed flexural rigidity from rotation (tilt) influence lines as a primary case study. Fisher information is employed as a diagnostic metric to quantify sensor informativeness, revealing how specific sensor layouts and load paths constrain the recoverable spatial features of the parameter field. The methodology is applied to the full-scale openLAB research bridge (TU Dresden) using data from controlled vehicle passages. Beyond estimating the flexural rigidity profile, the Bayesian formulation produces credible intervals that expose regions of practical non-identifiability, which deterministic methods may obscure. The results demonstrate that while the measurement data carry high information content for the target parameters, their utility is spatially heterogeneous and strictly bounded by the experiment design. The proposed framework unifies identification with uncertainty quantification, providing a rigorous basis for optimising sensor placement and interpreting the credibility of SHM diagnostics.

Figures (13)

• Figure 1: Simply supported Euler--Bernoulli beam with span $L$, point load $P$ at $z$, bending moment diagram, and variable flexural rigidity $EI(x)=E(x)I(x)$.
• Figure 2: Fisher information $\mathcal{I}^{(i)}_{jj}$ for flexural rigidity $EI$ along the span (computed via \ref{['eq:fi-diag']}). Solid blue: sensor at $x/L=1/4$; dashed grey: reference sensors at the supports ($x/L=0$ and $1$). A kink occurs at $x/L=1/4$ due to the piecewise kernels $m_r(x)$ and $M(x;\xi)$. Under the adopted load sweep, the blue curve attains its maximum to the right of the sensor; vertical lines mark the numerically attained maxima for the plotted curves
• Figure 3: Fisher information $\mathcal{I}^{(i)}_{jj}$ for flexural rigidity $EI$ along a two-span continuous beam. Each curve corresponds to a single rotation sensor placed on the first span at the indicated position ($x/L=0,0.1,\dots,1.0$). The vertical dashed line marks the interior support at $x/L=1$. Upstream sensors retain diminishing sensitivity beyond the support, while downstream sensors dominate the identifiability of the second span.
• Figure 4: Posterior flexural rigidity profile with spatially heterogeneous uncertainty along the span. Solid line: true $EI$; dashed line: posterior mean; blue band: $\pm 2\sigma$ (approx. $95\%$ credible interval) from the Bayesian inversion ($N=24$ elements, $K=24$ load positions, $\lambda=7.44\times 10^{-4}$). Uncertainty is largest near the supports and moderately elevated around mid-span, reflecting weaker identifiability where rotations provide less leverage; it narrows in regions where the data are most informative.
• Figure 5: Rotations at two stations under a traversing point load; tilt-noise s.d. $\sigma=0.02$ mm/m.