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Value of information from vibration-based structural health monitoring extracted via Bayesian model updating

Antonios Kamariotis, Eleni Chatzi, Daniel Straub

arXiv:2103.07382v1stat.APcs.SYeess.SY

TL;DR

The paper addresses how to quantify the decision value of SHM when prior studies often simplified the monitoring information. It models sequential vibration data, Bayesian model and reliability updating, and heuristic maintenance optimization; results on a deteriorating bridge show positive VoI, with outcomes varying by deterioration case.

  • Problem

    Prior SHM value-of-information studies often rely on simplified or hypothetical monitoring models, and the benefit of SHM depends on the decisions its data inform.

    Source: §1 · p. 1

  • Method

    The framework processes sequential accelerometer data with output-only OMA, updates deterioration and structural reliability Bayesianly, and applies heuristic life-cycle optimization.

    Source: §1 · p. 2, §5.1 · p. 8

  • Results

    For scour, VoI is positive across cost-ratio cases and close to the full VPPI value; for corrosion at a cost ratio of 10^-2, VoI is 51% of VPPI.

    Source: §6.6.1 · p. 18, §6.6.2 · p. 18

  • Takeaways & Limitations

    The framework provides quantitative measures of SHM optimality in a specific decision context and can compare sensor arrangements by their VoI.

    Source: §6.6.3 · p. 19, §7 · p. 20

    Reliability estimation assumes deterministic known capacity conditional on deterioration and models uncertain demand using the annual maximum-load distribution.

    Source: §4.1 · p. 6

Abstract

from arXiv · show

Quantifying the value of the information extracted from a structural health monitoring (SHM) system is an important step towards convincing decision makers to implement these systems. We quantify this value by adaptation of the Bayesian decision analysis framework. In contrast to previous works, we model in detail the entire process of data generation to processing, model updating and reliability calculation, and investigate it on a deteriorating bridge system. The framework assumes that dynamic response data are obtained in a sequential fashion from deployed accelerometers, subsequently processed by an output-only operational modal analysis scheme for identifying the system's modal characteristics. We employ a classical Bayesian model updating methodology to sequentially learn the deterioration and estimate the structural damage evolution over time. This leads to sequential updating of the structural reliability, which constitutes the basis for a preposterior Bayesian decision analysis. Alternative actions are defined and a heuristic-based approach is employed for the life-cycle optimization. By solving the preposterior Bayesian decision analysis, one is able to quantify the benefit of the availability of long-term SHM vibrational data. Numerical investigations show that this framework can provide quantitative measures on the optimality of an SHM system in a specific decision context.

1. Introduction

SHM has advanced, but its real-world use remains limited, and its value is difficult to assess apart from the decisions monitoring data inform. Prior VoI studies often simplify the monitoring information, while this work models the SHM data-to-decision process for a deteriorating bridge.

1. Introduction

  • SHM remains predominantly in research, with its potential benefit appraised through decisions triggered by monitoring data.The open questions are how SHM information can support optimal decisions and how its VoI can be quantified and maximized.Source: §1 · p. 1
  • Prior SHM VoI studies often use hypothetical likelihoods or observation models, limiting transfer to realistic applications.The authors formalize and extend their earlier attempt to model the full SHM process.Source: §1 · p. 2
  • Output-only OMA identifies eigenfrequencies and mode shapes from ambient vibration without requiring measured loads or obstructing structural operation.Continuous monitoring supplies dynamic response data such as accelerations.Source: §1 · p. 2
  • No prior study systematically quantifies how continuous SHM data used in Bayesian model updating support optimal maintenance decisions.Existing Bayesian model updating studies largely use fixed modal data to detect, localize, or quantify artificial damage.Source: §1 · p. 2
  • The proposed framework sequentially processes simulated bridge accelerations through OMA, Bayesian model and reliability updating, then compares life-cycle costs with and without SHM.A heuristic-based approach solves the life-cycle optimization problem.Source: §1 · p. 2

2. VoI from SHM analysis

The paper represents SHM value-of-information analysis as a lifetime decision process linking monitoring, structural condition, maintenance actions, and costs. It models continuous vibrational data and reliability updating in detail, while simplifying the action decision and using heuristics for life-cycle optimization.

2. VoI from SHM analysis

  • The influence diagram connects uncertain structural parameters and processes to monitoring, Bayesian learning, maintenance decisions, and lifetime costs.Its nodes distinguish uncertain quantities, SHM and repair decisions, monitoring data, costs, and computational methods.Source: §2 · p. 3, Figure 1 · p. 3
  • The framework uses continuous acceleration time series processed by OMA to identify modal data, then applies sequential Bayesian model updating to estimate structural condition.The authors describe this as avoiding overly simplified modeling of SHM data while retaining simplifications elsewhere.Source: §2 · p. 3
  • The framework models structural performance and reliability updating realistically and computationally efficiently, but simplifies the action decision and uses heuristics for optimization.The authors identify these simplified components as necessary to provide a computationally viable VoI analysis.Source: §2 · pp. 3–4
  • VoI is quantified by comparing optimal expected total life-cycle costs with monitoring-informed repair decisions against costs based on prior information alone.The influence diagram represents decisions across the structure's lifetime.Source: §2 · pp. 3–4

3. Bayesian model updating

Bayesian model updating uses noisy modal data to infer deterioration parameters and their uncertainty. The paper formulates a sequential update from continuous SHM data and presents Laplace approximation and adaptive MCMC as solution methods.

3. Bayesian model updating

  • The Bayesian updating framework uses OMA-identified modal data to estimate deterioration parameters and their uncertainty for a parameterized FE model.The formulation is presented as state of the art.Source: §3 · p. 4, §3.1 · p. 4

3.1. Bayesian formulation

  • Modal data can include eigenvalues and mode shapes or curvatures; eigenvalues support global damage detection, while spatial mode-shape information is needed for localization.Accurate mode-shape displacement identification requires relatively many sensors, and curvatures are described as more sensitive to local damage.Source: §3.1 · p. 4
  • Posterior deterioration parameters are proportional to the modal-data likelihood times their prior density, with model evidence as the normalizing constant.The model evidence requires a d-dimensional integral.Source: §3.1 · p. 4
  • The likelihood accounts for measurement and model uncertainty, while the chosen prediction-error coefficients affect updating and are often poorly justified in published work.The error model includes random measurement noise and variance or bias induced by SSI, as well as environmental variability.Source: §3.1 · pp. 4–5
  • The sequential likelihood formulation updates deterioration-parameter posteriors as new modal data arrive, although independence across time instances may not hold.The authors note hierarchical modeling of θ as one way to address that dependence.Source: §3.1 · p. 5

3.2.1. Laplace approximation

  • The Laplace method approximates the posterior as Gaussian around the MAP estimate, using the inverse log-posterior Hessian for covariance and the previous MAP as the next initialization.The approximation is described for globally identifiable cases with enough experimental data.Source: §3.2.1 · p. 6

3.2.2. MCMC sampling

  • Adaptive MCMC samples the posterior using a proposal covariance that adapts, and reruns when data arrive, seeded by the previous posterior mean.The authors present MCMC as a route to more accurate posterior estimates than the Laplace approximation.Source: §3.2.2 · p. 6

4. Structural reliability of a deteriorating structural system and its updating

The paper computes time-varying reliability from deterioration-dependent capacity and uncertain demand, then updates failure probabilities using posterior deterioration parameters learned from modal data. Its simplifying reliability assumptions support computationally demanding VoI analysis.

4. Structural reliability of a deteriorating structural system and its updating

  • Structural reliability and its updating with vibrational data provide the basis for the paper's decision framework.The section refers readers to reviews of the underlying reliability concepts.Source: §4 · p. 6

4.1. Structural reliability analysis for a deteriorating structural system

  • The reliability model assumes known deterministic capacity conditional on deterioration and uncertain demand represented by the annual maximum-load distribution.This choice simplifies reliability estimation for the computationally expensive VoI analysis.Source: §4.1 · p. 6
  • The time-varying problem is discretized into annual intervals, with accumulated failure probability computed from interval failure probabilities and Monte Carlo samples of deterioration parameters.The hazard function expresses failure rate conditional on survival through the preceding interval.Source: §4.1 · pp. 6–7

4.2. Structural reliability updating using SHM modal data

  • Modal-data-driven posterior distributions of deterioration parameters condition the accumulated failure probability on observations available before each time step.The posterior integral is approximated using MCMC or Laplace-based samples, and the conditional hazard can then be obtained.Source: §4.2 · p. 7

5. Life-cycle cost with SHM

The section compares life-cycle costs with and without monitoring by optimizing a hazard-threshold maintenance strategy. Its preposterior analysis samples monitoring histories and updates maintenance decisions; partial perfect information provides an upper bound on SHM’s value.

5.1. Life-cycle optimization based on heuristics

  • Repairs occur when the estimated hazard exceeds a threshold, and the heuristic search minimizes expected total life-cycle cost.The model treats repair as replacing damaged components and stopping further cost accumulation.Source: §5.1 · p. 8

5.2. Computation of the expected total life-cycle cost in the prior case

  • In the prior case, expected total cost averages repair and failure costs over sampled deterioration parameters, with repair timing determined by the hazard threshold.Monte Carlo samples from the prior parameter distribution are used to estimate failure probability and hazard.Source: §5.2 · p. 8

5.3. Computation of the expected total life-cycle cost in the preposterior case

  • In the preposterior case, cost expectations cover deterioration parameters and monitoring outcomes; monitoring changes the repair time, while matched prior and preposterior samples give identical risk estimates up to repair.Annual acceleration data are processed by SSI to generate monitoring histories.Source: §5.3 · p. 9

5.4. Summary of the proposed methodology to calculate the VoI

  • The workflow defines the deterioration model, decision costs, and hazard threshold, then samples deterioration and optimizes the prior-case expected life-cycle cost.The prior analysis estimates failure probability and hazard from the sampled deterioration parameters.Source: §5.4 · p. 10
  • For each sampled deterioration history, the workflow generates noisy acceleration data, identifies modal data, updates deterioration and hazard estimates, then optimizes and computes VoI.The preposterior optimization minimizes expected life-cycle cost over monitoring outcomes.Source: §5.4 · p. 10

5.5. Value of Partial Perfect Information

  • Partial perfect information assumes exact knowledge of deterioration parameters but leaves load uncertainty; its value upper-bounds VoI and can provide an initial estimate of maximum SHM investment.The upper bound is computed more easily than VoI.Source: §5.5 · p. 10

6. Numerical investigations

The numerical investigation uses a continuously monitored two-span bridge benchmark, with simulated vertical accelerations from sensors at predefined finite-element nodes.

6. Numerical investigations

  • The benchmark is a two-span bridge model with vertical-acceleration sensors at predefined finite-element nodes and simulated ambient excitation.The spans are 12 m and 13 m, and the model uses elastic supports.Source: §6.1 · p. 11, Figure 2 · p. 11

6.2. Deterioration modeling

The deterioration model represents time-varying stiffness loss through uncertain parameters. The bridge examples model scour at the middle support and corrosion at both midspans.

6.2. Deterioration modeling

  • The deterioration model uses a rate equation with uncertain A and B parameters to represent deterioration rate and time-power nonlinearity.D(t) is a unitless deterioration parameter representing stiffness loss.Source: §6.2 · p. 11

6.2.1. Bridge system subject to scour

  • 10% probability of D(t = 50) > 9 is assumed for the 50-year scour case, which models progressive vertical stiffness loss at the middle support.The uncertain deterioration parameters are summarized in Table 1.Source: §6.2.1 · pp. 11–12, Table 1 · p. 12

6.2.2. Bridge system subject to corrosion deterioration

  • 1% probability of D(t = 50) > 9 is assumed for corrosion modeled as progressive stiffness loss in the bottom two finite-element rows at both midspans.The mean values of B1 and B2 are set to 0.5, corresponding in this empirical model to diffusion-controlled damage.Source: §6.2.2 · p. 12, Table 2 · p. 12

6.3. Synthetic monitoring data creation

Synthetic monitoring data are generated by applying the deterioration-dependent stiffness to the finite-element model, simulating accelerations, adding measurement noise, and identifying modal data with SSI.

6.3. Synthetic monitoring data creation

  • 2% RMS noise-to-signal ratio is added to simulated acceleration data before SSI identifies lower eigenvalues and mode shapes.The process simulates measurement error and uses output-only operational modal analysis.Source: §6.3 · p. 12

6.4. Continuous Bayesian model updating

The section demonstrates sequential Bayesian learning of deterioration models from annual SHM modal data for scour and corrosion scenarios. The estimated deterioration parameters are sensitive to the assumed prediction-error factors.

6.4. Continuous Bayesian model updating

  • The monitoring histories contain one OMA-identified modal-data set per year over 50 years, without environmental or operational variability in this example.The computational burden of repeated likelihood and FE evaluations motivates surrogate models for Bayesian updating.Source: §6.4 · p. 13

6.4.1. Bridge system subject to scour deterioration - Global damage identification

  • The scour global-damage scenario uses eigenvalue data from six modes and 12 sensors, with a nearest-neighbor database replacing FE modal-analysis runs.The sensor configuration is selected by engineering judgment; the surrogate database is built over a grid of deterioration values.Source: §6.4.1 · p. 13, Figure 4 · p. 13
  • Posterior predictive deterioration models shift toward the underlying scour model by year 10, while their uncertainty decreases further with data through years 25 and 50.Figure 6 presents posterior predictive means and 90% credible intervals at those monitoring horizons.Source: Figure 6 · p. 14, §6.4.1 · p. 14
  • As SHM data are incorporated over time, uncertainty in the deterioration-model parameters gradually decreases; their PDFs become narrower and peak around the underlying true values for which the data were created.The comparison shows prior and posterior PDFs at years 25 and 50, using 5,000 MCMC samples at each time step.Source: §6.4.1 · pp. 13–14

6.4.2. Bridge system subject corrosion deterioration - Damage detection and localization

  • For two-location corrosion detection and localization, the analysis uses 24 accelerometers, six modes, and mode-shape curvatures in the likelihood function.The curvature representation is reported to enhance localization capabilities; eigenvalue and mode-shape displacement data are used.Source: §6.4.2 · p. 14, Figure 7 · p. 14
  • Changing cλm and cΦm from 0.02 to 0.05 significantly affects the year-50 posterior distribution of corrosion deterioration parameters.The comparison is between Figures 10 and 12.Source: Figure 10 · p. 15, §6.4.2 · p. 15

6.5. Time-dependent structural reliability and its updating using monitoring data

The section estimates time-dependent structural reliability from deterioration models and compares prior estimates with those informed by continuous corrosion monitoring. In the corrosion case, monitoring-informed estimates have reduced reliability uncertainty.

6.5.1. Bridge system subject to scour deterioration

  • Scour capacity is estimated by static analyses over a grid of deterioration values, evaluating load-bearing-capacity loss relative to the undamaged structure.The critical quantity identified for scour at the middle support is normal stress at the middle of the longer midspan.Source: §6.5.1 · pp. 15–16
  • For scour, the prior time-dependent accumulated failure probability and hazard function are shown with 95% credible intervals; skewness places the mean curves outside those intervals.The curves and intervals are estimated from 10^4 prior samples.Source: Figure 13 · p. 16, §6.5.1 · p. 16

6.5.2. Bridge system subject to corrosion deterioration

  • For two-location corrosion, static analyses over a two-dimensional deterioration grid produce capacity-loss estimates, which are represented by a polynomial response surface.The resulting surface describes structural capacity as a function of the two corrosion deteriorations.Source: §6.5.2 · pp. 16–17
  • Monitoring-informed corrosion reliability estimates have reduced uncertainty relative to prior estimates.The comparison uses accumulated failure probability and hazard rate, with 5,000 Monte Carlo prior samples and 5,000 posterior MCMC samples at each time step.Source: Figure 14 · p. 17, §6.5.2 · p. 17

6.6. VoI analysis

The VoI analysis compares prior and monitoring-informed life-cycle costs, then tests the approach on bridge deterioration and sensor-placement examples. The results vary by deterioration and repair-cost ratio, while the sensor study shows that sensor count affects the resulting optimization and VoI.

6.6. VoI analysis

  • The analysis estimates prior and preposterior expected life-cycle costs by Monte Carlo sampling, with sequential Bayesian updating for each monitoring history.The repair threshold is selected by exhaustive search over a discrete set; computational cost depends on the number of samples, synthetic data generation, and sequential updating.Source: §6.6 · p. 17
  • The analysis evaluates repair-cost-to-failure-cost ratios of 10^-1, 10^-2, and 10^-3, using a 2% discount rate.The assumed failure cost is 10^7 euros.Source: §6.6 · p. 17

6.6.1. VoI results for bridge system subject to scour deterioration

  • For scour, 1,000 state samples are used; for corrosion, 2,000 are used with a Laplace approximation that is faster than MCMC but introduces posterior approximation error, especially early on.The scour example uses adaptive MCMC, with 2,000 posterior samples for accumulated failure probabilities; the corrosion example uses 10,000 draws from the approximate posterior.Source: §6.6.1 · pp. 17–18, §6.6.2 · p. 18
  • For every scour cost-ratio case, VoI is positive and close to the VPPI, providing almost the full value of perfect condition information.The estimates have greater variability when the prior optimal action is no repair, because few preposterior samples select a different repair time.Source: §6.6.1 · p. 18, Table 3 · p. 18

6.6.2. VoI results for bridge system subject to corrosion deterioration

  • At repair-to-failure cost ratios of 10^-1, 10^-2, and 10^-3, corrosion-case VoI is respectively zero, 51% of VPPI, and 22% of VPPI.At 10^-1, neither the prior analysis nor any preposterior sample selects a repair; the repair cost exceeds the risk-of-failure cost at every time step.Source: §6.6.2 · p. 18, Table 4 · p. 19

6.6.3. VoI results - Sensor placement study

  • Changing sensor count affects mode-shape information, Bayesian model updating, heuristic life-cycle optimization, and the resulting VoI.The study compares 24 and 12 uniformly distributed accelerometers for corrosion deterioration at two locations.Source: §6.6.3 · p. 19, Table 5 · p. 19

7. Concluding remarks

The paper presents a preposterior Bayesian decision analysis for quantifying the value of long-term SHM data across a modeled chain from acquisition through reliability updating. Its examples use a single repair action and omit varying environmental effects, and the resulting VoI estimates are computationally demanding to obtain.

7. Concluding remarks

  • The framework models SHM data through operational modal analysis, sequential Bayesian model updating, and structural reliability updating before preposterior decision analysis.A heuristic solution finds the optimal timing of a single repair action, and value of partial perfect information provides an upper bound associated with hypothetical perfect knowledge of structural condition.Source: §7 · p. 20
  • The reported VoI estimates are limited to a single repair action and do not account for varying environmental effects such as temperature.Integrating over monitoring data adds a computationally expensive layer to the analysis.Source: §7 · p. 20
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